# Theory Factorization

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theory Factorization = Primes + Permutation:

```(*  Title:      HOL/NumberTheory/Factorization.thy
ID:         \$Id: Factorization.thy,v 1.5 2001/10/05 19:52:51 wenzelm Exp \$
Author:     Thomas Marthedal Rasmussen
Copyright   2000  University of Cambridge
*)

header {* Fundamental Theorem of Arithmetic (unique factorization into primes) *}

theory Factorization = Primes + Permutation:

subsection {* Definitions *}

consts
primel :: "nat list => bool "
nondec :: "nat list => bool "
prod :: "nat list => nat"
oinsert :: "nat => nat list => nat list"
sort :: "nat list => nat list"

defs
primel_def: "primel xs == set xs ⊆ prime"

primrec
"nondec [] = True"
"nondec (x # xs) = (case xs of [] => True | y # ys => x ≤ y ∧ nondec xs)"

primrec
"prod [] = Suc 0"
"prod (x # xs) = x * prod xs"

primrec
"oinsert x [] = [x]"
"oinsert x (y # ys) = (if x ≤ y then x # y # ys else y # oinsert x ys)"

primrec
"sort [] = []"
"sort (x # xs) = oinsert x (sort xs)"

subsection {* Arithmetic *}

lemma one_less_m: "(m::nat) ≠ m * k ==> m ≠ Suc 0 ==> Suc 0 < m"
apply (case_tac m)
apply auto
done

lemma one_less_k: "(m::nat) ≠ m * k ==> Suc 0 < m * k ==> Suc 0 < k"
apply (case_tac k)
apply auto
done

lemma mult_left_cancel: "(0::nat) < k ==> k * n = k * m ==> n = m"
apply auto
done

lemma mn_eq_m_one: "(0::nat) < m ==> m * n = m ==> n = Suc 0"
apply (case_tac n)
apply auto
done

lemma prod_mn_less_k:
"(0::nat) < n ==> 0 < k ==> Suc 0 < m ==> m * n = k ==> n < k"
apply (induct m)
apply auto
done

subsection {* Prime list and product *}

lemma prod_append: "prod (xs @ ys) = prod xs * prod ys"
apply (induct xs)
apply (simp_all add: mult_assoc)
done

lemma prod_xy_prod:
"prod (x # xs) = prod (y # ys) ==> x * prod xs = y * prod ys"
apply auto
done

lemma primel_append: "primel (xs @ ys) = (primel xs ∧ primel ys)"
apply (unfold primel_def)
apply auto
done

lemma prime_primel: "n ∈ prime ==> primel [n] ∧ prod [n] = n"
apply (unfold primel_def)
apply auto
done

lemma prime_nd_one: "p ∈ prime ==> ¬ p dvd Suc 0"
apply (unfold prime_def dvd_def)
apply auto
done

lemma hd_dvd_prod: "prod (x # xs) = prod ys ==> x dvd (prod ys)"
apply (unfold dvd_def)
apply (rule exI)
apply (rule sym)
apply simp
done

lemma primel_tl: "primel (x # xs) ==> primel xs"
apply (unfold primel_def)
apply auto
done

lemma primel_hd_tl: "(primel (x # xs)) = (x ∈ prime ∧ primel xs)"
apply (unfold primel_def)
apply auto
done

lemma primes_eq: "p ∈ prime ==> q ∈ prime ==> p dvd q ==> p = q"
apply (unfold prime_def)
apply auto
done

lemma primel_one_empty: "primel xs ==> prod xs = Suc 0 ==> xs = []"
apply (unfold primel_def prime_def)
apply (case_tac xs)
apply simp_all
done

lemma prime_g_one: "p ∈ prime ==> Suc 0 < p"
apply (unfold prime_def)
apply auto
done

lemma prime_g_zero: "p ∈ prime ==> 0 < p"
apply (unfold prime_def)
apply auto
done

lemma primel_nempty_g_one [rule_format]:
"primel xs --> xs ≠ [] --> Suc 0 < prod xs"
apply (unfold primel_def prime_def)
apply (induct xs)
apply (auto elim: one_less_mult)
done

lemma primel_prod_gz: "primel xs ==> 0 < prod xs"
apply (unfold primel_def prime_def)
apply (induct xs)
apply auto
done

subsection {* Sorting *}

lemma nondec_oinsert [rule_format]: "nondec xs --> nondec (oinsert x xs)"
apply (induct xs)
apply (case_tac  list)
apply (simp_all cong del: list.weak_case_cong)
done

lemma nondec_sort: "nondec (sort xs)"
apply (induct xs)
apply simp_all
apply (erule nondec_oinsert)
done

lemma x_less_y_oinsert: "x ≤ y ==> l = y # ys ==> x # l = oinsert x l"
apply simp_all
done

lemma nondec_sort_eq [rule_format]: "nondec xs --> xs = sort xs"
apply (induct xs)
apply safe
apply simp_all
apply (case_tac list)
apply simp_all
apply (case_tac list)
apply simp
apply (rule_tac y = aa and ys = lista in x_less_y_oinsert)
apply simp_all
done

lemma oinsert_x_y: "oinsert x (oinsert y l) = oinsert y (oinsert x l)"
apply (induct l)
apply auto
done

subsection {* Permutation *}

lemma perm_primel [rule_format]: "xs <~~> ys ==> primel xs --> primel ys"
apply (unfold primel_def)
apply (erule perm.induct)
apply simp_all
done

lemma perm_prod [rule_format]: "xs <~~> ys ==> prod xs = prod ys"
apply (erule perm.induct)
apply (simp_all add: mult_ac)
done

lemma perm_subst_oinsert: "xs <~~> ys ==> oinsert a xs <~~> oinsert a ys"
apply (erule perm.induct)
apply auto
done

lemma perm_oinsert: "x # xs <~~> oinsert x xs"
apply (induct xs)
apply auto
done

lemma perm_sort: "xs <~~> sort xs"
apply (induct xs)
apply (auto intro: perm_oinsert elim: perm_subst_oinsert)
done

lemma perm_sort_eq: "xs <~~> ys ==> sort xs = sort ys"
apply (erule perm.induct)
apply (simp_all add: oinsert_x_y)
done

subsection {* Existence *}

lemma ex_nondec_lemma:
"primel xs ==> ∃ys. primel ys ∧ nondec ys ∧ prod ys = prod xs"
apply (blast intro: nondec_sort perm_prod perm_primel perm_sort perm_sym)
done

lemma not_prime_ex_mk:
"Suc 0 < n ∧ n ∉ prime ==>
∃m k. Suc 0 < m ∧ Suc 0 < k ∧ m < n ∧ k < n ∧ n = m * k"
apply (unfold prime_def dvd_def)
apply (auto intro: n_less_m_mult_n n_less_n_mult_m one_less_m one_less_k)
done

lemma split_primel:
"primel xs ==> primel ys ==> ∃l. primel l ∧ prod l = prod xs * prod ys"
apply (rule exI)
apply safe
apply (rule_tac  prod_append)
apply (simp add: primel_append)
done

lemma factor_exists [rule_format]: "Suc 0 < n --> (∃l. primel l ∧ prod l = n)"
apply (induct n rule: nat_less_induct)
apply (rule impI)
apply (case_tac "n ∈ prime")
apply (rule exI)
apply (erule prime_primel)
apply (cut_tac n = n in not_prime_ex_mk)
apply (auto intro!: split_primel)
done

lemma nondec_factor_exists: "Suc 0 < n ==> ∃l. primel l ∧ nondec l ∧ prod l = n"
apply (erule factor_exists [THEN exE])
apply (blast intro!: ex_nondec_lemma)
done

subsection {* Uniqueness *}

lemma prime_dvd_mult_list [rule_format]:
"p ∈ prime ==> p dvd (prod xs) --> (∃m. m:set xs ∧ p dvd m)"
apply (induct xs)
apply (force simp add: prime_def)
apply (force dest: prime_dvd_mult)
done

lemma hd_xs_dvd_prod:
"primel (x # xs) ==> primel ys ==> prod (x # xs) = prod ys
==> ∃m. m ∈ set ys ∧ x dvd m"
apply (rule prime_dvd_mult_list)
apply (simp add: primel_hd_tl)
apply (erule hd_dvd_prod)
done

lemma prime_dvd_eq: "primel (x # xs) ==> primel ys ==> m ∈ set ys ==> x dvd m ==> x = m"
apply (rule primes_eq)
apply (auto simp add: primel_def primel_hd_tl)
done

lemma hd_xs_eq_prod:
"primel (x # xs) ==>
primel ys ==> prod (x # xs) = prod ys ==> x ∈ set ys"
apply (frule hd_xs_dvd_prod)
apply auto
apply (drule prime_dvd_eq)
apply auto
done

lemma perm_primel_ex:
"primel (x # xs) ==>
primel ys ==> prod (x # xs) = prod ys ==> ∃l. ys <~~> (x # l)"
apply (rule exI)
apply (rule perm_remove)
apply (erule hd_xs_eq_prod)
apply simp_all
done

lemma primel_prod_less:
"primel (x # xs) ==>
primel ys ==> prod (x # xs) = prod ys ==> prod xs < prod ys"
apply (auto intro: prod_mn_less_k prime_g_one primel_prod_gz simp add: primel_hd_tl)
done

lemma prod_one_empty:
"primel xs ==> p * prod xs = p ==> p ∈ prime ==> xs = []"
apply (auto intro: primel_one_empty simp add: prime_def)
done

lemma uniq_ex_aux:
"∀m. m < prod ys --> (∀xs ys. primel xs ∧ primel ys ∧
prod xs = prod ys ∧ prod xs = m --> xs <~~> ys) ==>
primel list ==> primel x ==> prod list = prod x ==> prod x < prod ys
==> x <~~> list"
apply simp
done

lemma factor_unique [rule_format]:
"∀xs ys. primel xs ∧ primel ys ∧ prod xs = prod ys ∧ prod xs = n
--> xs <~~> ys"
apply (induct n rule: nat_less_induct)
apply safe
apply (case_tac xs)
apply (force intro: primel_one_empty)
apply (rule perm_primel_ex [THEN exE])
apply simp_all
apply (rule perm.trans [THEN perm_sym])
apply assumption
apply (rule perm.Cons)
apply (case_tac "x = []")
apply (simp add: perm_sing_eq primel_hd_tl)
apply (rule_tac p = a in prod_one_empty)
apply simp_all
apply (erule uniq_ex_aux)
apply (auto intro: primel_tl perm_primel simp add: primel_hd_tl)
apply (rule_tac k = a and n = "prod list" and m = "prod x" in mult_left_cancel)
apply (rule_tac  x = a in primel_prod_less)
apply (rule_tac  prod_xy_prod)
apply (rule_tac  s = "prod ys" in HOL.trans)
apply (erule_tac  perm_prod)
apply (erule_tac  perm_prod [symmetric])
apply (auto intro: perm_primel prime_g_zero)
done

lemma perm_nondec_unique:
"xs <~~> ys ==> nondec xs ==> nondec ys ==> xs = ys"
apply (rule HOL.trans)
apply (rule HOL.trans)
apply (erule nondec_sort_eq)
apply (erule perm_sort_eq)
apply (erule nondec_sort_eq [symmetric])
done

lemma unique_prime_factorization [rule_format]:
"∀n. Suc 0 < n --> (∃!l. primel l ∧ nondec l ∧ prod l = n)"
apply safe
apply (erule nondec_factor_exists)
apply (rule perm_nondec_unique)
apply (rule factor_unique)
apply simp_all
done

end
```

### Arithmetic

lemma one_less_m:

`  [| m ≠ m * k; m ≠ Suc 0 |] ==> Suc 0 < m`

lemma one_less_k:

`  [| m ≠ m * k; Suc 0 < m * k |] ==> Suc 0 < k`

lemma mult_left_cancel:

`  [| 0 < k; k * n = k * m |] ==> n = m`

lemma mn_eq_m_one:

`  [| 0 < m; m * n = m |] ==> n = Suc 0`

lemma prod_mn_less_k:

`  [| 0 < n; 0 < k; Suc 0 < m; m * n = k |] ==> n < k`

### Prime list and product

lemma prod_append:

`  prod (xs @ ys) = prod xs * prod ys`

lemma prod_xy_prod:

`  prod (x # xs) = prod (y # ys) ==> x * prod xs = y * prod ys`

lemma primel_append:

`  primel (xs @ ys) = (primel xs ∧ primel ys)`

lemma prime_primel:

`  n ∈ prime ==> primel [n] ∧ prod [n] = n`

lemma prime_nd_one:

`  p ∈ prime ==> ¬ p dvd Suc 0`

lemma hd_dvd_prod:

`  prod (x # xs) = prod ys ==> x dvd prod ys`

lemma primel_tl:

`  primel (x # xs) ==> primel xs`

lemma primel_hd_tl:

`  primel (x # xs) = (x ∈ prime ∧ primel xs)`

lemma primes_eq:

`  [| p ∈ prime; q ∈ prime; p dvd q |] ==> p = q`

lemma primel_one_empty:

`  [| primel xs; prod xs = Suc 0 |] ==> xs = []`

lemma prime_g_one:

`  p ∈ prime ==> Suc 0 < p`

lemma prime_g_zero:

`  p ∈ prime ==> 0 < p`

lemma primel_nempty_g_one:

`  [| primel xs; xs ≠ [] |] ==> Suc 0 < prod xs`

lemma primel_prod_gz:

`  primel xs ==> 0 < prod xs`

### Sorting

lemma nondec_oinsert:

`  nondec xs ==> nondec (oinsert x xs)`

lemma nondec_sort:

`  nondec (sort xs)`

lemma x_less_y_oinsert:

`  [| x ≤ y; l = y # ys |] ==> x # l = oinsert x l`

lemma nondec_sort_eq:

`  nondec xs ==> xs = sort xs`

lemma oinsert_x_y:

`  oinsert x (oinsert y l) = oinsert y (oinsert x l)`

### Permutation

lemma perm_primel:

`  [| xs <~~> ys; primel xs |] ==> primel ys`

lemma perm_prod:

`  xs <~~> ys ==> prod xs = prod ys`

lemma perm_subst_oinsert:

`  xs <~~> ys ==> oinsert a xs <~~> oinsert a ys`

lemma perm_oinsert:

`  x # xs <~~> oinsert x xs`

lemma perm_sort:

`  xs <~~> sort xs`

lemma perm_sort_eq:

`  xs <~~> ys ==> sort xs = sort ys`

### Existence

lemma ex_nondec_lemma:

`  primel xs ==> ∃ys. primel ys ∧ nondec ys ∧ prod ys = prod xs`

lemma not_prime_ex_mk:

```  Suc 0 < n ∧ n ∉ prime
==> ∃m k. Suc 0 < m ∧ Suc 0 < k ∧ m < n ∧ k < n ∧ n = m * k```

lemma split_primel:

`  [| primel xs; primel ys |] ==> ∃l. primel l ∧ prod l = prod xs * prod ys`

lemma factor_exists:

`  Suc 0 < n ==> ∃l. primel l ∧ prod l = n`

lemma nondec_factor_exists:

`  Suc 0 < n ==> ∃l. primel l ∧ nondec l ∧ prod l = n`

### Uniqueness

lemma prime_dvd_mult_list:

`  [| p ∈ prime; p dvd prod xs |] ==> ∃m. m ∈ set xs ∧ p dvd m`

lemma hd_xs_dvd_prod:

```  [| primel (x # xs); primel ys; prod (x # xs) = prod ys |]
==> ∃m. m ∈ set ys ∧ x dvd m```

lemma prime_dvd_eq:

`  [| primel (x # xs); primel ys; m ∈ set ys; x dvd m |] ==> x = m`

lemma hd_xs_eq_prod:

`  [| primel (x # xs); primel ys; prod (x # xs) = prod ys |] ==> x ∈ set ys`

lemma perm_primel_ex:

`  [| primel (x # xs); primel ys; prod (x # xs) = prod ys |] ==> ∃l. ys <~~> x # l`

lemma primel_prod_less:

`  [| primel (x # xs); primel ys; prod (x # xs) = prod ys |] ==> prod xs < prod ys`

lemma prod_one_empty:

`  [| primel xs; p * prod xs = p; p ∈ prime |] ==> xs = []`

lemma uniq_ex_aux:

```  [| ∀m<prod ys.
∀xs ys.
primel xs ∧ primel ys ∧ prod xs = prod ys ∧ prod xs = m --> xs <~~> ys;
primel list; primel x; prod list = prod x; prod x < prod ys |]
==> x <~~> list```

lemma factor_unique:

`  primel xs ∧ primel ys ∧ prod xs = prod ys ∧ prod xs = n ==> xs <~~> ys`

lemma perm_nondec_unique:

`  [| xs <~~> ys; nondec xs; nondec ys |] ==> xs = ys`

lemma unique_prime_factorization:

`  Suc 0 < n ==> ∃!l. primel l ∧ nondec l ∧ prod l = n`