author haftmann Thu, 03 Mar 2016 08:33:55 +0100 changeset 62499 4a5b81ff5992 parent 62498 5dfcc9697f29 child 62500 ff99681b3fd8
constructive formulation of factorization
 CONTRIBUTORS file | annotate | diff | comparison | revisions src/HOL/Number_Theory/Factorial_Ring.thy file | annotate | diff | comparison | revisions
```--- a/CONTRIBUTORS	Wed Mar 02 19:43:31 2016 +0100
+++ b/CONTRIBUTORS	Thu Mar 03 08:33:55 2016 +0100
@@ -10,6 +10,9 @@
Abolition of compound operators INFIMUM and SUPREMUM
for complete lattices.

+* March 2016: Florian Haftmann
+  Abstract factorial rings with unique factorization.
+

Contributions to Isabelle2016
-----------------------------```
```--- a/src/HOL/Number_Theory/Factorial_Ring.thy	Wed Mar 02 19:43:31 2016 +0100
+++ b/src/HOL/Number_Theory/Factorial_Ring.thy	Thu Mar 03 08:33:55 2016 +0100
@@ -5,15 +5,31 @@
section \<open>Factorial (semi)rings\<close>

theory Factorial_Ring
-imports Main Primes "~~/src/HOL/Library/Multiset" (*"~~/src/HOL/Library/Polynomial"*)
+imports Main Primes "~~/src/HOL/Library/Multiset"
+begin
+
+context algebraic_semidom
begin

+lemma dvd_mult_imp_div:
+  assumes "a * c dvd b"
+  shows "a dvd b div c"
+proof (cases "c = 0")
+  case True then show ?thesis by simp
+next
+  case False
+  from \<open>a * c dvd b\<close> obtain d where "b = a * c * d" ..
+  with False show ?thesis by (simp add: mult.commute [of a] mult.assoc)
+qed
+
+end
+
class factorial_semiring = normalization_semidom +
assumes finite_divisors: "a \<noteq> 0 \<Longrightarrow> finite {b. b dvd a \<and> normalize b = b}"
fixes is_prime :: "'a \<Rightarrow> bool"
assumes not_is_prime_zero [simp]: "\<not> is_prime 0"
and is_prime_not_unit: "is_prime p \<Longrightarrow> \<not> is_unit p"
-    and no_prime_divisorsI: "(\<And>b. b dvd a \<Longrightarrow> \<not> is_prime b) \<Longrightarrow> is_unit a"
+    and no_prime_divisorsI2: "(\<And>b. b dvd a \<Longrightarrow> \<not> is_prime b) \<Longrightarrow> is_unit a"
assumes is_primeI: "p \<noteq> 0 \<Longrightarrow> \<not> is_unit p \<Longrightarrow> (\<And>a. a dvd p \<Longrightarrow> \<not> is_unit a \<Longrightarrow> p dvd a) \<Longrightarrow> is_prime p"
and is_primeD: "is_prime p \<Longrightarrow> p dvd a * b \<Longrightarrow> p dvd a \<or> p dvd b"
begin
@@ -93,21 +109,6 @@
with that show thesis by blast
qed

-lemma prime_divisorE:
-  assumes "a \<noteq> 0" and "\<not> is_unit a"
-  obtains p where "is_prime p" and "p dvd a"
-  using assms no_prime_divisorsI [of a] by blast
-
-lemma prime_dvd_mult_iff:
-  assumes "is_prime p"
-  shows "p dvd a * b \<longleftrightarrow> p dvd a \<or> p dvd b"
-  using assms by (auto dest: is_primeD)
-
-lemma prime_dvd_power_iff:
-  assumes "is_prime p"
-  shows "p dvd a ^ n \<longleftrightarrow> p dvd a \<and> n > 0"
-  using assms by (induct n) (auto dest: is_prime_not_unit is_primeD)
-
lemma is_prime_normalize_iff [simp]:
"is_prime (normalize p) \<longleftrightarrow> is_prime p" (is "?P \<longleftrightarrow> ?Q")
proof
@@ -119,6 +120,157 @@
(insert is_prime_not_zeroI [of "normalize p"] is_prime_not_unit [of "normalize p"] is_primeD2 [of "normalize p"] \<open>?P\<close>, simp_all)
qed

+lemma no_prime_divisorsI:
+  assumes "\<And>b. b dvd a \<Longrightarrow> normalize b = b \<Longrightarrow> \<not> is_prime b"
+  shows "is_unit a"
+proof (rule no_prime_divisorsI2)
+  fix b
+  assume "b dvd a"
+  then have "normalize b dvd a"
+    by simp
+  moreover have "normalize (normalize b) = normalize b"
+    by simp
+  ultimately have "\<not> is_prime (normalize b)"
+    by (rule assms)
+  then show "\<not> is_prime b"
+    by simp
+qed
+
+lemma prime_divisorE:
+  assumes "a \<noteq> 0" and "\<not> is_unit a"
+  obtains p where "is_prime p" and "p dvd a"
+  using assms no_prime_divisorsI [of a] by blast
+
+lemma is_prime_associated:
+  assumes "is_prime p" and "is_prime q" and "q dvd p"
+  shows "normalize q = normalize p"
+using \<open>q dvd p\<close> proof (rule associatedI)
+  from \<open>is_prime q\<close> have "\<not> is_unit q"
+    by (simp add: is_prime_not_unit)
+  with \<open>is_prime p\<close> \<open>q dvd p\<close> show "p dvd q"
+    by (blast intro: is_primeD2)
+qed
+
+lemma prime_dvd_mult_iff:
+  assumes "is_prime p"
+  shows "p dvd a * b \<longleftrightarrow> p dvd a \<or> p dvd b"
+  using assms by (auto dest: is_primeD)
+
+lemma prime_dvd_msetprod:
+  assumes "is_prime p"
+  assumes dvd: "p dvd msetprod A"
+  obtains a where "a \<in># A" and "p dvd a"
+proof -
+  from dvd have "\<exists>a. a \<in># A \<and> p dvd a"
+  proof (induct A)
+    case empty then show ?case
+    using \<open>is_prime p\<close> by (simp add: is_prime_not_unit)
+  next
+    case (add A a)
+    then have "p dvd msetprod A * a" by simp
+    with \<open>is_prime p\<close> consider (A) "p dvd msetprod A" | (B) "p dvd a"
+      by (blast dest: is_primeD)
+    then show ?case proof cases
+      case B then show ?thesis by auto
+    next
+      case A
+      with add.hyps obtain b where "b \<in># A" "p dvd b"
+        by auto
+      then show ?thesis by auto
+    qed
+  qed
+  with that show thesis by blast
+qed
+
+lemma msetprod_eq_iff:
+  assumes "\<forall>p\<in>set_mset P. is_prime p \<and> normalize p = p" and "\<forall>p\<in>set_mset Q. is_prime p \<and> normalize p = p"
+  shows "msetprod P = msetprod Q \<longleftrightarrow> P = Q" (is "?R \<longleftrightarrow> ?S")
+proof
+  assume ?S then show ?R by simp
+next
+  assume ?R then show ?S using assms proof (induct P arbitrary: Q)
+    case empty then have Q: "msetprod Q = 1" by simp
+    have "Q = {#}"
+    proof (rule ccontr)
+      assume "Q \<noteq> {#}"
+      then obtain r R where "Q = R + {#r#}"
+        using multi_nonempty_split by blast
+      moreover with empty have "is_prime r" by simp
+      ultimately have "msetprod Q = msetprod R * r"
+        by simp
+      with Q have "msetprod R * r = 1"
+        by simp
+      then have "is_unit r"
+        by (metis local.dvd_triv_right)
+      with \<open>is_prime r\<close> show False by (simp add: is_prime_not_unit)
+    qed
+    then show ?case by simp
+  next
+    case (add P p)
+    then have "is_prime p" and "normalize p = p"
+      and "msetprod Q = msetprod P * p" and "p dvd msetprod Q"
+      by auto (metis local.dvd_triv_right)
+    with prime_dvd_msetprod
+      obtain q where "q \<in># Q" and "p dvd q"
+      by blast
+    with add.prems have "is_prime q" and "normalize q = q"
+      by simp_all
+    from \<open>is_prime p\<close> have "p \<noteq> 0"
+      by auto
+    from \<open>is_prime q\<close> \<open>is_prime p\<close> \<open>p dvd q\<close>
+      have "normalize p = normalize q"
+      by (rule is_prime_associated)
+    from \<open>normalize p = p\<close> \<open>normalize q = q\<close> have "p = q"
+      unfolding \<open>normalize p = normalize q\<close> by simp
+    with \<open>q \<in># Q\<close> have "p \<in># Q" by simp
+    have "msetprod P = msetprod (Q - {#p#})"
+      using \<open>p \<in># Q\<close> \<open>p \<noteq> 0\<close> \<open>msetprod Q = msetprod P * p\<close>
+      by (simp add: msetprod_minus)
+    then have "P = Q - {#p#}"
+      using add.prems(2-3)
+      by (auto intro: add.hyps dest: in_diffD)
+    with \<open>p \<in># Q\<close> show ?case by simp
+  qed
+qed
+
+lemma prime_dvd_power_iff:
+  assumes "is_prime p"
+  shows "p dvd a ^ n \<longleftrightarrow> p dvd a \<and> n > 0"
+  using assms by (induct n) (auto dest: is_prime_not_unit is_primeD)
+
+lemma prime_power_dvd_multD:
+  assumes "is_prime p"
+  assumes "p ^ n dvd a * b" and "n > 0" and "\<not> p dvd a"
+  shows "p ^ n dvd b"
+using \<open>p ^ n dvd a * b\<close> and \<open>n > 0\<close> proof (induct n arbitrary: b)
+  case 0 then show ?case by simp
+next
+  case (Suc n) show ?case
+  proof (cases "n = 0")
+    case True with Suc \<open>is_prime p\<close> \<open>\<not> p dvd a\<close> show ?thesis
+      by (simp add: prime_dvd_mult_iff)
+  next
+    case False then have "n > 0" by simp
+    from \<open>is_prime p\<close> have "p \<noteq> 0" by auto
+    from Suc.prems have *: "p * p ^ n dvd a * b"
+      by simp
+    then have "p dvd a * b"
+      by (rule dvd_mult_left)
+    with Suc \<open>is_prime p\<close> \<open>\<not> p dvd a\<close> have "p dvd b"
+      by (simp add: prime_dvd_mult_iff)
+    moreover def c \<equiv> "b div p"
+    ultimately have b: "b = p * c" by simp
+    with * have "p * p ^ n dvd p * (a * c)"
+      by (simp add: ac_simps)
+    with \<open>p \<noteq> 0\<close> have "p ^ n dvd a * c"
+      by simp
+    with Suc.hyps \<open>n > 0\<close> have "p ^ n dvd c"
+      by blast
+    with \<open>p \<noteq> 0\<close> show ?thesis
+      by (simp add: b)
+  qed
+qed
+
lemma is_prime_inj_power:
assumes "is_prime p"
shows "inj (op ^ p)"
@@ -150,47 +302,80 @@
ultimately show False using * [of m n] * [of n m] by auto
qed

-lemma prime_unique:
-  assumes "is_prime q" and "is_prime p" and "q dvd p"
-  shows "normalize q = normalize p"
-proof -
-  from assms have "p dvd q"
-    by (auto dest: is_primeD2 [of p] is_prime_not_unit [of q])
-  with assms show ?thesis
-    by (auto intro: associatedI)
-qed
+definition factorization :: "'a \<Rightarrow> 'a multiset option"
+  where "factorization a = (if a = 0 then None
+    else Some (setsum (\<lambda>p. replicate_mset (Max {n. p ^ n dvd a}) p)
+      {p. p dvd a \<and> is_prime p \<and> normalize p = p}))"
+
+lemma factorization_normalize [simp]:
+  "factorization (normalize a) = factorization a"
+  by (simp add: factorization_def)
+
+lemma factorization_0 [simp]:
+  "factorization 0 = None"
+  by (simp add: factorization_def)
+
+lemma factorization_eq_None_iff [simp]:
+  "factorization a = None \<longleftrightarrow> a = 0"
+  by (simp add: factorization_def)

-lemma exists_factorization:
-  assumes "a \<noteq> 0"
-  obtains P where "\<And>p. p \<in># P \<Longrightarrow> is_prime p" "msetprod P = normalize a"
-proof -
+lemma factorization_eq_Some_iff:
+  "factorization a = Some P \<longleftrightarrow>
+   normalize a = msetprod P \<and> 0 \<notin># P \<and> (\<forall>p \<in> set_mset P. is_prime p \<and> normalize p = p)"
+proof (cases "a = 0")
+  have [simp]: "0 = msetprod P \<longleftrightarrow> 0 \<in># P"
+    using msetprod_zero_iff [of P] by blast
+  case True
+  then show ?thesis by auto
+next
+  case False
let ?prime_factors = "\<lambda>a. {p. p dvd a \<and> is_prime p \<and> normalize p = p}"
-  have "?prime_factors a \<subseteq> {b. b dvd a \<and> normalize b = b}" by auto
-  moreover from assms have "finite {b. b dvd a \<and> normalize b = b}"
+  have "?prime_factors a \<subseteq> {b. b dvd a \<and> normalize b = b}"
+    by auto
+  moreover from \<open>a \<noteq> 0\<close> have "finite {b. b dvd a \<and> normalize b = b}"
by (rule finite_divisors)
-  ultimately have "finite (?prime_factors a)" by (rule finite_subset)
-  then show thesis using \<open>a \<noteq> 0\<close> that proof (induct "?prime_factors a" arbitrary: thesis a)
+  ultimately have "finite (?prime_factors a)"
+    by (rule finite_subset)
+  then show ?thesis using \<open>a \<noteq> 0\<close>
+  proof (induct "?prime_factors a" arbitrary: a P)
case empty then have
-      P: "\<And>b. is_prime b \<Longrightarrow> b dvd a \<Longrightarrow> normalize b \<noteq> b" and "a \<noteq> 0"
+      *: "{p. p dvd a \<and> is_prime p \<and> normalize p = p} = {}"
+        and "a \<noteq> 0"
by auto
-    have "is_unit a"
-    proof (rule no_prime_divisorsI)
-      fix b
-      assume "b dvd a"
-      then show "\<not> is_prime b"
-        using P [of "normalize b"] by auto
+    from \<open>a \<noteq> 0\<close> have "factorization a = Some {#}"
+      by (simp only: factorization_def *) simp
+    from * have "normalize a = 1"
+      by (auto intro: is_unit_normalize no_prime_divisorsI)
+    show ?case (is "?lhs \<longleftrightarrow> ?rhs") proof
+      assume ?lhs with \<open>factorization a = Some {#}\<close> \<open>normalize a = 1\<close>
+      show ?rhs by simp
+    next
+      assume ?rhs have "P = {#}"
+      proof (rule ccontr)
+        assume "P \<noteq> {#}"
+        then obtain q Q where "P = Q + {#q#}"
+          using multi_nonempty_split by blast
+        with \<open>?rhs\<close> \<open>normalize a = 1\<close>
+        have "1 = q * msetprod Q" and "is_prime q"
+          by (simp_all add: ac_simps)
+        then have "is_unit q" by (auto intro: dvdI)
+        with \<open>is_prime q\<close> show False
+          using is_prime_not_unit by blast
+      qed
+      with \<open>factorization a = Some {#}\<close> show ?lhs by simp
qed
-    then have "\<And>p. p \<in># {#} \<Longrightarrow> is_prime p" and "msetprod {#} = normalize a"
-      by (simp_all add: is_unit_normalize)
-    with empty show thesis by blast
next
-    case (insert p P)
-    from \<open>insert p P = ?prime_factors a\<close>
-    have "p dvd a" and "is_prime p" and "normalize p = p"
-      by auto
-    obtain n where "n > 0" and "p ^ n dvd a" and "\<not> p ^ Suc n dvd a"
-    proof (rule that)
+    case (insert p F)
+    from \<open>insert p F = ?prime_factors a\<close>
+    have "?prime_factors a = insert p F"
+      by simp
+    then have "p dvd a" and "is_prime p" and "normalize p = p" and "p \<noteq> 0"
+      by (auto intro!: is_prime_not_zeroI)
+    def n \<equiv> "Max {n. p ^ n dvd a}"
+    then have "n > 0" and "p ^ n dvd a" and "\<not> p ^ Suc n dvd a"
+    proof -
def N \<equiv> "{n. p ^ n dvd a}"
+      then have n_M: "n = Max N" by (simp add: n_def)
from is_prime_inj_power \<open>is_prime p\<close> have "inj (op ^ p)" .
then have "inj_on (op ^ p) U" for U
by (rule subset_inj_on) simp
@@ -202,43 +387,275 @@
then have "N \<noteq> {}" by blast
note * = \<open>finite N\<close> \<open>N \<noteq> {}\<close>
from N_def \<open>p dvd a\<close> have "1 \<in> N" by simp
-      with * show "Max N > 0"
+      with * have "Max N > 0"
by (auto simp add: Max_gr_iff)
+      then show "n > 0" by (simp add: n_M)
from * have "Max N \<in> N" by (rule Max_in)
-      then show "p ^ Max N dvd a" by (simp add: N_def)
+      then have "p ^ Max N dvd a" by (simp add: N_def)
+      then show "p ^ n dvd a" by (simp add: n_M)
from * have "\<forall>n\<in>N. n \<le> Max N"
by (simp add: Max_le_iff [symmetric])
then have "p ^ Suc (Max N) dvd a \<Longrightarrow> Suc (Max N) \<le> Max N"
by (rule bspec) (simp add: N_def)
-      then show "\<not> p ^ Suc (Max N) dvd a"
+      then have "\<not> p ^ Suc (Max N) dvd a"
by auto
+      then show "\<not> p ^ Suc n dvd a"
+        by (simp add: n_M)
qed
-    from \<open>p ^ n dvd a\<close> obtain c where "a = p ^ n * c" ..
-    with \<open>\<not> p ^ Suc n dvd a\<close> have "\<not> p dvd c"
+    def b \<equiv> "a div p ^ n"
+    with \<open>p ^ n dvd a\<close> have a: "a = p ^ n * b"
+      by simp
+    with \<open>\<not> p ^ Suc n dvd a\<close> have "\<not> p dvd b" and "b \<noteq> 0"
by (auto elim: dvdE simp add: ac_simps)
-    have "?prime_factors a - {p} = ?prime_factors c" (is "?A = ?B")
+    have "?prime_factors a = insert p (?prime_factors b)"
proof (rule set_eqI)
fix q
-      show "q \<in> ?A \<longleftrightarrow> q \<in> ?B"
-      using \<open>normalize p = p\<close> \<open>is_prime p\<close> \<open>a = p ^ n * c\<close> \<open>\<not> p dvd c\<close>
-        prime_unique [of q p]
-        by (auto simp add: prime_dvd_power_iff prime_dvd_mult_iff)
+      show "q \<in> ?prime_factors a \<longleftrightarrow> q \<in> insert p (?prime_factors b)"
+      using \<open>is_prime p\<close> \<open>normalize p = p\<close> \<open>n > 0\<close>
+        by (auto simp add: a prime_dvd_mult_iff prime_dvd_power_iff)
+          (auto dest: is_prime_associated)
qed
-    moreover from insert have "P = ?prime_factors a - {p}"
+    with \<open>\<not> p dvd b\<close> have "?prime_factors a - {p} = ?prime_factors b"
+      by auto
+    with insert.hyps have "F = ?prime_factors b"
by auto
-    ultimately have "P = ?prime_factors c"
+    then have "?prime_factors b = F"
+      by simp
+    with \<open>?prime_factors a = insert p (?prime_factors b)\<close> have "?prime_factors a = insert p F"
by simp
-    moreover from \<open>a \<noteq> 0\<close> \<open>a = p ^ n * c\<close> have "c \<noteq> 0"
-      by auto
-    ultimately obtain P where "\<And>p. p \<in># P \<Longrightarrow> is_prime p" "msetprod P = normalize c"
-      using insert(3) by blast
-    with \<open>is_prime p\<close> \<open>a = p ^ n * c\<close> \<open>normalize p = p\<close>
-    have "\<And>q. q \<in># P + (replicate_mset n p) \<longrightarrow> is_prime q" "msetprod (P + replicate_mset n p) = normalize a"
-      by (auto simp add: ac_simps normalize_mult normalize_power)
-    with insert(6) show thesis by blast
+    have equiv: "(\<Sum>p\<in>F. replicate_mset (Max {n. p ^ n dvd a}) p) =
+      (\<Sum>p\<in>F. replicate_mset (Max {n. p ^ n dvd b}) p)"
+    using refl proof (rule Groups_Big.setsum.cong)
+      fix q
+      assume "q \<in> F"
+      have "{n. q ^ n dvd a} = {n. q ^ n dvd b}"
+      proof -
+        have "q ^ m dvd a \<longleftrightarrow> q ^ m dvd b" (is "?R \<longleftrightarrow> ?S")
+          for m
+        proof (cases "m = 0")
+          case True then show ?thesis by simp
+        next
+          case False then have "m > 0" by simp
+          show ?thesis
+          proof
+            assume ?S then show ?R by (simp add: a)
+          next
+            assume ?R
+            then have *: "q ^ m dvd p ^ n * b" by (simp add: a)
+            from insert.hyps \<open>q \<in> F\<close>
+            have "is_prime q" "normalize q = q" "p \<noteq> q" "q dvd p ^ n * b"
+              by (auto simp add: a)
+            from \<open>is_prime q\<close> * \<open>m > 0\<close> show ?S
+            proof (rule prime_power_dvd_multD)
+              have "\<not> q dvd p"
+              proof
+                assume "q dvd p"
+                with \<open>is_prime q\<close> \<open>is_prime p\<close> have "normalize q = normalize p"
+                  by (blast intro: is_prime_associated)
+                with \<open>normalize p = p\<close> \<open>normalize q = q\<close> \<open>p \<noteq> q\<close> show False
+                  by simp
+              qed
+              with \<open>is_prime q\<close> show "\<not> q dvd p ^ n"
+                by (simp add: prime_dvd_power_iff)
+            qed
+          qed
+        qed
+        then show ?thesis by auto
+      qed
+      then show
+        "replicate_mset (Max {n. q ^ n dvd a}) q = replicate_mset (Max {n. q ^ n dvd b}) q"
+        by simp
+    qed
+    def Q \<equiv> "the (factorization b)"
+    with \<open>b \<noteq> 0\<close> have [simp]: "factorization b = Some Q"
+      by simp
+    from \<open>a \<noteq> 0\<close> have "factorization a =
+      Some (\<Sum>p\<in>?prime_factors a. replicate_mset (Max {n. p ^ n dvd a}) p)"
+      by (simp add: factorization_def)
+    also have "\<dots> =
+      Some (\<Sum>p\<in>insert p F. replicate_mset (Max {n. p ^ n dvd a}) p)"
+      by (simp add: \<open>?prime_factors a = insert p F\<close>)
+    also have "\<dots> =
+      Some (replicate_mset n p + (\<Sum>p\<in>F. replicate_mset (Max {n. p ^ n dvd a}) p))"
+      using \<open>finite F\<close> \<open>p \<notin> F\<close> n_def by simp
+    also have "\<dots> =
+      Some (replicate_mset n p + (\<Sum>p\<in>F. replicate_mset (Max {n. p ^ n dvd b}) p))"
+      using equiv by simp
+    also have "\<dots> = Some (replicate_mset n p + the (factorization b))"
+      using \<open>b \<noteq> 0\<close> by (simp add: factorization_def \<open>?prime_factors a = insert p F\<close> \<open>?prime_factors b = F\<close>)
+    finally have fact_a: "factorization a =
+      Some (replicate_mset n p + Q)"
+      by simp
+    moreover have "factorization b = Some Q \<longleftrightarrow>
+      normalize b = msetprod Q \<and>
+      0 \<notin># Q \<and>
+      (\<forall>p\<in>#Q. is_prime p \<and> normalize p = p)"
+      using \<open>F = ?prime_factors b\<close> \<open>b \<noteq> 0\<close> by (rule insert.hyps)
+    ultimately have
+      norm_a: "normalize a = msetprod (replicate_mset n p + Q)" and
+      prime_Q: "\<forall>p\<in>set_mset Q. is_prime p \<and> normalize p = p"
+      by (simp_all add: a normalize_mult normalize_power \<open>normalize p = p\<close>)
+    show ?case (is "?lhs \<longleftrightarrow> ?rhs") proof
+      assume ?lhs with fact_a
+      have "P = replicate_mset n p + Q" by simp
+      with \<open>n > 0\<close> \<open>is_prime p\<close> \<open>normalize p = p\<close> prime_Q
+        show ?rhs by (auto simp add: norm_a dest: is_prime_not_zeroI)
+    next
+      assume ?rhs
+      with \<open>n > 0\<close> \<open>is_prime p\<close> \<open>normalize p = p\<close> \<open>n > 0\<close> prime_Q
+      have "msetprod P = msetprod (replicate_mset n p + Q)"
+        and "\<forall>p\<in>set_mset P. is_prime p \<and> normalize p = p"
+        and "\<forall>p\<in>set_mset (replicate_mset n p + Q). is_prime p \<and> normalize p = p"
+        by (simp_all add: norm_a)
+      then have "P = replicate_mset n p + Q"
+        by (simp only: msetprod_eq_iff)
+      then show ?lhs
+        by (simp add: fact_a)
+    qed
qed
qed
-
+
+lemma factorization_cases [case_names 0 factorization]:
+  assumes "0": "a = 0 \<Longrightarrow> P"
+  assumes factorization: "\<And>A. a \<noteq> 0 \<Longrightarrow> factorization a = Some A \<Longrightarrow> msetprod A = normalize a
+    \<Longrightarrow> 0 \<notin># A \<Longrightarrow> (\<And>p. p \<in># A \<Longrightarrow> normalize p = p) \<Longrightarrow> (\<And>p. p \<in># A \<Longrightarrow> is_prime p) \<Longrightarrow> P"
+  shows P
+proof (cases "a = 0")
+  case True with 0 show P .
+next
+  case False
+  then have "factorization a \<noteq> None" by simp
+  then obtain A where "factorization a = Some A" by blast
+  moreover from this have "msetprod A = normalize a"
+    "0 \<notin># A" "\<And>p. p \<in># A \<Longrightarrow> normalize p = p" "\<And>p. p \<in># A \<Longrightarrow> is_prime p"
+    by (auto simp add: factorization_eq_Some_iff)
+  ultimately show P using \<open>a \<noteq> 0\<close> factorization by blast
+qed
+
+lemma factorizationE:
+  assumes "a \<noteq> 0"
+  obtains A u where "factorization a = Some A" "normalize a = msetprod A"
+    "0 \<notin># A" "\<And>p. p \<in># A \<Longrightarrow> is_prime p" "\<And>p. p \<in># A \<Longrightarrow> normalize p = p"
+  using assms by (cases a rule: factorization_cases) simp_all
+
+lemma prime_dvd_mset_prod_iff:
+  assumes "is_prime p" "normalize p = p" "\<And>p. p \<in># A \<Longrightarrow> is_prime p" "\<And>p. p \<in># A \<Longrightarrow> normalize p = p"
+  shows "p dvd msetprod A \<longleftrightarrow> p \<in># A"
+using assms proof (induct A)
+  case empty then show ?case by (auto dest: is_prime_not_unit)
+next
+  case (add A q) then show ?case
+    using is_prime_associated [of q p]
+    by (simp_all add: prime_dvd_mult_iff, safe, simp_all)
+qed
+
+end
+
+class factorial_semiring_gcd = factorial_semiring + gcd +
+  assumes gcd_unfold: "gcd a b =
+    (if a = 0 then normalize b
+     else if b = 0 then normalize a
+     else msetprod (the (factorization a) #\<inter> the (factorization b)))"
+  and lcm_unfold: "lcm a b =
+    (if a = 0 \<or> b = 0 then 0
+     else msetprod (the (factorization a) #\<union> the (factorization b)))"
+begin
+
+subclass semiring_gcd
+proof
+  fix a b
+  have comm: "gcd a b = gcd b a" for a b
+   by (simp add: gcd_unfold ac_simps)
+  have "gcd a b dvd a" for a b
+  proof (cases a rule: factorization_cases)
+    case 0 then show ?thesis by simp
+  next
+    case (factorization A) note fact_A = this
+    then have non_zero: "\<And>p. p \<in>#A \<Longrightarrow> p \<noteq> 0"
+      using normalize_0 not_is_prime_zero by blast
+    show ?thesis
+    proof (cases b rule: factorization_cases)
+      case 0 then show ?thesis by (simp add: gcd_unfold)
+    next
+      case (factorization B) note fact_B = this
+      have "msetprod (A #\<inter> B) dvd msetprod A"
+      using non_zero proof (induct B arbitrary: A)
+        case empty show ?case by simp
+      next
+        case (add B p) show ?case
+        proof (cases "p \<in># A")
+          case True then obtain C where "A = C + {#p#}"
+            by (metis insert_DiffM2)
+          moreover with True add have "p \<noteq> 0" and "\<And>p. p \<in># C \<Longrightarrow> p \<noteq> 0"
+            by auto
+          ultimately show ?thesis
+            using True add.hyps [of C]
+            by (simp add: inter_union_distrib_left [symmetric])
+        next
+          case False with add.prems add.hyps [of A] show ?thesis
+            by (simp add: inter_add_right1)
+        qed
+      qed
+      with fact_A fact_B show ?thesis by (simp add: gcd_unfold)
+    qed
+  qed
+  then have "gcd a b dvd a" and "gcd b a dvd b"
+    by simp_all
+  then show "gcd a b dvd a" and "gcd a b dvd b"
+    by (simp_all add: comm)
+  show "c dvd gcd a b" if "c dvd a" and "c dvd b" for c
+  proof (cases "a = 0 \<or> b = 0 \<or> c = 0")
+    case True with that show ?thesis by (auto simp add: gcd_unfold)
+  next
+    case False then have "a \<noteq> 0" and "b \<noteq> 0" and "c \<noteq> 0"
+      by simp_all
+    then obtain A B C where fact:
+      "factorization a = Some A" "factorization b = Some B" "factorization c = Some C"
+      and norm: "normalize a = msetprod A" "normalize b = msetprod B" "normalize c = msetprod C"
+      and A: "0 \<notin># A" "\<And>p. p \<in># A \<Longrightarrow> normalize p = p" "\<And>p. p \<in># A \<Longrightarrow> is_prime p"
+      and B: "0 \<notin># B" "\<And>p. p \<in># B \<Longrightarrow> normalize p = p" "\<And>p. p \<in># B \<Longrightarrow> is_prime p"
+      and C: "0 \<notin># C" "\<And>p. p \<in># C \<Longrightarrow> normalize p = p" "\<And>p. p \<in># C \<Longrightarrow> is_prime p"
+      by (blast elim!: factorizationE)
+    moreover from that have "normalize c dvd normalize a" and "normalize c dvd normalize b"
+      by simp_all
+    ultimately have "msetprod C dvd msetprod A" and "msetprod C dvd msetprod B"
+      by simp_all
+    with A B C have "msetprod C dvd msetprod (A #\<inter> B)"
+    proof (induct C arbitrary: A B)
+      case empty then show ?case by simp
+    next
+      case add: (add C p)
+      from add.prems
+        have p: "p \<noteq> 0" "is_prime p" "normalize p = p" by auto
+      from add.prems have prems: "msetprod C * p dvd msetprod A" "msetprod C * p dvd msetprod B"
+        by simp_all
+      then have "p dvd msetprod A" "p dvd msetprod B"
+        by (auto dest: dvd_mult_imp_div dvd_mult_right)
+      with p add.prems have "p \<in># A" "p \<in># B"
+        by (simp_all add: prime_dvd_mset_prod_iff)
+      then obtain A' B' where ABp: "A = {#p#} + A'" "B = {#p#} + B'"
+        by (auto dest!: multi_member_split simp add: ac_simps)
+      with add.prems prems p have "msetprod C dvd msetprod (A' #\<inter> B')"
+        by (auto intro: add.hyps simp add: ac_simps)
+      with p have "msetprod ({#p#} + C) dvd msetprod (({#p#} + A') #\<inter> ({#p#} + B'))"
+        by (simp add: inter_union_distrib_right [symmetric])
+      then show ?case by (simp add: ABp ac_simps)
+    qed
+    with \<open>a \<noteq> 0\<close> \<open>b \<noteq> 0\<close> that fact have "normalize c dvd gcd a b"
+      by (simp add: norm [symmetric] gcd_unfold fact)
+    then show ?thesis by simp
+  qed
+  show "normalize (gcd a b) = gcd a b"
+    apply (simp add: gcd_unfold)
+    apply safe
+    apply (rule normalized_msetprodI)
+    apply (auto elim: factorizationE)
+    done
+  show "lcm a b = normalize (a * b) div gcd a b"
+    by (auto elim!: factorizationE simp add: gcd_unfold lcm_unfold normalize_mult
+      union_diff_inter_eq_sup [symmetric] msetprod_diff inter_subset_eq_union)
+qed
+
end

instantiation nat :: factorial_semiring```