# Theory Poly_Mod_Finite_Field

```(*
Authors:      Jose Divasón
Sebastiaan Joosten
René Thiemann
*)
subsection ‹Polynomials in a Finite Field›
text ‹We connect polynomials in a prime field with integer polynomials modulo some prime.›

theory Poly_Mod_Finite_Field
imports
Finite_Field
Polynomial_Interpolation.Ring_Hom_Poly
"HOL-Types_To_Sets.Types_To_Sets"
More_Missing_Multiset
Poly_Mod
begin

(* TODO: Move -- General transfer rule *)
declare rel_mset_Zero[transfer_rule]

lemma mset_transfer[transfer_rule]: "(list_all2 rel ===> rel_mset rel) mset mset"
proof (intro rel_funI)
show "list_all2 rel xs ys ⟹ rel_mset rel (mset xs) (mset ys)" for xs ys
proof (induct xs arbitrary: ys)
case Nil
then show ?case by auto
next
case IH: (Cons x xs)
then show ?case by (auto dest!:msed_rel_invL simp: list_all2_Cons1 intro!:rel_mset_Plus)
qed
qed

abbreviation to_int_poly :: "'a :: finite mod_ring poly ⇒ int poly" where
"to_int_poly ≡ map_poly to_int_mod_ring"

interpretation to_int_poly_hom: map_poly_inj_zero_hom to_int_mod_ring ..

lemma irreducible⇩d_def_0:
fixes f :: "'a :: {comm_semiring_1,semiring_no_zero_divisors} poly"
shows "irreducible⇩d f = (degree f ≠ 0 ∧
(∀ g h. degree g ≠ 0 ⟶ degree h ≠ 0 ⟶ f ≠ g * h))"
proof-
have "degree g ≠ 0 ⟹ g ≠ 0" for g :: "'a poly" by auto
note 1 = degree_mult_eq[OF this this, simplified]
then show ?thesis by (force elim!: irreducible⇩dE)
qed

subsection ‹Transferring to class-based mod-ring›

locale poly_mod_type = poly_mod m
for m and ty :: "'a :: nontriv itself" +
assumes m: "m = CARD('a)"
begin

lemma m1: "m > 1" using nontriv[where 'a = 'a] by (auto simp:m)

sublocale poly_mod_2 using m1 by unfold_locales

definition MP_Rel :: "int poly ⇒ 'a mod_ring poly ⇒ bool"
where "MP_Rel f f' ≡ (Mp f = to_int_poly f')"

definition M_Rel :: "int ⇒ 'a mod_ring ⇒ bool"
where "M_Rel x x' ≡ (M x = to_int_mod_ring x')"

definition "MF_Rel ≡ rel_prod M_Rel (rel_mset MP_Rel)"

lemma to_int_mod_ring_plus: "to_int_mod_ring ((x :: 'a mod_ring) + y) = M (to_int_mod_ring x + to_int_mod_ring y)"
unfolding M_def using m by (transfer, auto)

lemma to_int_mod_ring_times: "to_int_mod_ring ((x :: 'a mod_ring) * y) = M (to_int_mod_ring x * to_int_mod_ring y)"
unfolding M_def using m by (transfer, auto)

lemma degree_MP_Rel [transfer_rule]: "(MP_Rel ===> (=)) degree_m degree"
unfolding MP_Rel_def rel_fun_def
by (auto intro!: degree_map_poly)

lemma eq_M_Rel[transfer_rule]: "(M_Rel ===> M_Rel ===> (=)) (λ x y. M x = M y) (=)"
unfolding M_Rel_def rel_fun_def by auto

interpretation to_int_mod_ring_hom: map_poly_inj_zero_hom to_int_mod_ring..

lemma eq_MP_Rel[transfer_rule]: "(MP_Rel ===> MP_Rel ===> (=)) (=m) (=)"
unfolding MP_Rel_def rel_fun_def by auto

lemma eq_Mf_Rel[transfer_rule]: "(MF_Rel ===> MF_Rel ===> (=)) (λ x y. Mf x = Mf y) (=)"
proof (intro rel_funI, goal_cases)
case (1 cfs Cfs dgs Dgs)
obtain c fs where cfs: "cfs = (c,fs)" by force
obtain C Fs where Cfs: "Cfs = (C,Fs)" by force
obtain d gs where dgs: "dgs = (d,gs)" by force
obtain D Gs where Dgs: "Dgs = (D,Gs)" by force
note pairs = cfs Cfs dgs Dgs
from 1[unfolded pairs MF_Rel_def rel_prod.simps]
have *[transfer_rule]: "M_Rel c C" "M_Rel d D" "rel_mset MP_Rel fs Fs" "rel_mset MP_Rel gs Gs"
by auto
have eq1: "(M c = M d) = (C = D)" by transfer_prover
from *(3)[unfolded rel_mset_def] obtain fs' Fs' where fs_eq: "mset fs' = fs" "mset Fs' = Fs"
and rel_f: "list_all2 MP_Rel fs' Fs'" by auto
from *(4)[unfolded rel_mset_def] obtain gs' Gs' where gs_eq: "mset gs' = gs" "mset Gs' = Gs"
and rel_g: "list_all2 MP_Rel gs' Gs'" by auto
have eq2: "(image_mset Mp fs = image_mset Mp gs) = (Fs = Gs)"
using *(3-4)
proof (induct fs arbitrary: Fs gs Gs)
case (empty Fs gs Gs)
from empty(1) have Fs: "Fs = {#}" unfolding rel_mset_def by auto
with empty show ?case by (cases gs; cases Gs; auto simp: rel_mset_def)
next
case (add f fs Fs' gs' Gs')
obtain Fs F where Fs': "Fs' = Fs + {#F#}" and rel[transfer_rule]:
"MP_Rel f F" "rel_mset MP_Rel fs Fs" by auto
{
from add(3)[unfolded rel_mset_def] obtain gs Gs where id: "mset gs = gs'" "mset Gs = Gs'"
and rel: "list_all2 MP_Rel gs Gs" by auto
have "Mp f ∈# image_mset Mp gs' ⟷ F ∈# Gs'"
proof -
have "?thesis = ((Mp f ∈ Mp ` set gs) = (F ∈ set Gs))"
unfolding id[symmetric] by simp
also have … using rel
proof (induct gs Gs rule: list_all2_induct)
case (Cons g gs G Gs)
note [transfer_rule] = Cons(1-2)
have id: "(Mp g = Mp f) = (F = G)" by (transfer, auto)
show ?case using id Cons(3) by auto
qed auto
finally show ?thesis by simp
qed
} note id = this
show ?case
proof (cases "Mp f ∈# image_mset Mp gs'")
case False
have "Mp f ∈# image_mset Mp (fs + {#f#})" by auto
with False have F: "image_mset Mp (fs + {#f#}) ≠ image_mset Mp gs'" by metis
with False[unfolded id] show ?thesis unfolding Fs' by auto
next
case True
then obtain g where fg: "Mp f = Mp g" and g: "g ∈# gs'" by auto
from g obtain gs where gs': "gs' = add_mset g gs" by (rule mset_add)
obtain Gs G where Gs': "Gs' = Gs + {# G #}" and gG[transfer_rule]: "MP_Rel g G" and
gsGs: "rel_mset MP_Rel gs Gs" by auto
have FG: "F = G" by (transfer, simp add: fg)
note IH = IH[OF gsGs]
show ?thesis unfolding gs' Fs' Gs' by (simp add: fg IH FG)
qed
qed
show "(Mf cfs = Mf dgs) = (Cfs = Dgs)" unfolding pairs Mf_def split
qed

lemmas coeff_map_poly_of_int = coeff_map_poly[of of_int, OF of_int_0]

lemma plus_MP_Rel[transfer_rule]: "(MP_Rel ===> MP_Rel ===> MP_Rel) (+) (+)"
unfolding MP_Rel_def
proof (intro rel_funI, goal_cases)
case (1 x f y g)
have "Mp (x + y) = Mp (Mp x + Mp y)" by simp
also have "… = Mp (map_poly to_int_mod_ring f + map_poly to_int_mod_ring g)" unfolding 1 ..
also have "… = map_poly to_int_mod_ring (f + g)" unfolding poly_eq_iff Mp_coeff
by (auto simp: to_int_mod_ring_plus)
finally show ?case .
qed

lemma times_MP_Rel[transfer_rule]: "(MP_Rel ===> MP_Rel ===> MP_Rel) ((*)) ((*))"
unfolding MP_Rel_def
proof (intro rel_funI, goal_cases)
case (1 x f y g)
have "Mp (x * y) = Mp (Mp x * Mp y)" by simp
also have "… = Mp (map_poly to_int_mod_ring f * map_poly to_int_mod_ring g)" unfolding 1 ..
also have "… = map_poly to_int_mod_ring (f * g)"
proof -
{ fix n :: nat
define A where "A = {.. n}"
have "finite A" unfolding A_def by auto
then have "M (∑i≤n. to_int_mod_ring (coeff f i) * to_int_mod_ring (coeff g (n - i))) =
to_int_mod_ring (∑i≤n. coeff f i * coeff g (n - i))"
unfolding A_def[symmetric]
proof (induct A)
case (insert a A)
have "?case = ?case" (is "(?l = ?r) = _") by simp
have "?r = to_int_mod_ring (coeff f a * coeff g (n - a) + (∑i∈ A. coeff f i * coeff g (n - i)))"
using insert(1-2) by auto
note r = this[unfolded to_int_mod_ring_plus to_int_mod_ring_times]
from insert(1-2) have "?l = M (to_int_mod_ring (coeff f a) * to_int_mod_ring (coeff g (n - a))
+ M (∑i∈A. to_int_mod_ring (coeff f i) * to_int_mod_ring (coeff g (n - i))))"
by simp
also have "M (∑i∈A. to_int_mod_ring (coeff f i) * to_int_mod_ring (coeff g (n - i))) = to_int_mod_ring (∑i∈A. coeff f i * coeff g (n - i))"
unfolding insert ..
finally
show ?case unfolding r by simp
qed auto
}
then show ?thesis by (auto intro!:poly_eqI simp: coeff_mult  Mp_coeff)
qed
finally show ?case .
qed

lemma smult_MP_Rel[transfer_rule]: "(M_Rel ===> MP_Rel ===> MP_Rel) smult smult"
unfolding MP_Rel_def M_Rel_def
proof (intro rel_funI, goal_cases)
case (1 x x' f f')
thus ?case unfolding poly_eq_iff coeff Mp_coeff
coeff_smult M_def
proof (intro allI, goal_cases)
case (1 n)
have "x * coeff f n mod m = (x mod m) * (coeff f n mod m) mod m"
also have "… = to_int_mod_ring x' * (to_int_mod_ring (coeff f' n)) mod m"
using 1 by auto
also have " … = to_int_mod_ring (x' * coeff f' n)"
unfolding to_int_mod_ring_times M_def by simp
finally show ?case by auto
qed
qed

lemma one_M_Rel[transfer_rule]: "M_Rel 1 1"
unfolding M_Rel_def M_def
unfolding m by auto

lemma one_MP_Rel[transfer_rule]: "MP_Rel 1 1"
unfolding MP_Rel_def poly_eq_iff Mp_coeff M_def
unfolding m by auto

lemma zero_M_Rel[transfer_rule]: "M_Rel 0 0"
unfolding M_Rel_def M_def
unfolding m by auto

lemma zero_MP_Rel[transfer_rule]: "MP_Rel 0 0"
unfolding MP_Rel_def poly_eq_iff Mp_coeff M_def
unfolding m by auto

lemma listprod_MP_Rel[transfer_rule]: "(list_all2 MP_Rel ===> MP_Rel) prod_list prod_list"
proof (intro rel_funI, goal_cases)
case (1 xs ys)
thus ?case
proof (induct xs ys rule: list_all2_induct)
case (Cons x xs y ys)
note [transfer_rule] = this
show ?case by simp transfer_prover
qed

lemma prod_mset_MP_Rel[transfer_rule]: "(rel_mset MP_Rel ===> MP_Rel) prod_mset prod_mset"
proof (intro rel_funI, goal_cases)
case (1 xs ys)
have "(MP_Rel ===> MP_Rel ===> MP_Rel) ((*)) ((*))" "MP_Rel 1 1" by transfer_prover+
from 1 this show ?case
proof (induct xs ys rule: rel_mset_induct)
case (add R x xs y ys)
note [transfer_rule] = this
show ?case by simp transfer_prover
qed

lemma right_unique_MP_Rel[transfer_rule]: "right_unique MP_Rel"
unfolding right_unique_def MP_Rel_def by auto

lemma M_to_int_mod_ring: "M (to_int_mod_ring (x :: 'a mod_ring)) = to_int_mod_ring x"
unfolding M_def unfolding m by (transfer, auto)

lemma Mp_to_int_poly: "Mp (to_int_poly (f :: 'a mod_ring poly)) = to_int_poly f"
by (auto simp: poly_eq_iff Mp_coeff M_to_int_mod_ring)

lemma right_total_M_Rel[transfer_rule]: "right_total M_Rel"
unfolding right_total_def M_Rel_def using M_to_int_mod_ring by blast

lemma left_total_M_Rel[transfer_rule]: "left_total M_Rel"
unfolding left_total_def M_Rel_def[abs_def]
proof
fix x
show "∃ x' :: 'a mod_ring. M x = to_int_mod_ring x'"  unfolding M_def unfolding m
by (rule exI[of _ "of_int x"], transfer, simp)
qed

lemma bi_total_M_Rel[transfer_rule]: "bi_total M_Rel"
using right_total_M_Rel left_total_M_Rel by (metis bi_totalI)

lemma right_total_MP_Rel[transfer_rule]: "right_total MP_Rel"
unfolding right_total_def MP_Rel_def
proof
fix f :: "'a mod_ring poly"
show "∃x. Mp x = to_int_poly f"
by (intro exI[of _ "to_int_poly f"], simp add: Mp_to_int_poly)
qed

lemma to_int_mod_ring_of_int_M: "to_int_mod_ring (of_int x :: 'a mod_ring) = M x" unfolding M_def
unfolding m by transfer auto

lemma Mp_f_representative: "Mp f = to_int_poly (map_poly of_int f :: 'a mod_ring poly)"
unfolding Mp_def by (auto intro: poly_eqI simp: coeff_map_poly to_int_mod_ring_of_int_M)

lemma left_total_MP_Rel[transfer_rule]: "left_total MP_Rel"
unfolding left_total_def MP_Rel_def[abs_def] using Mp_f_representative by blast

lemma bi_total_MP_Rel[transfer_rule]: "bi_total MP_Rel"
using right_total_MP_Rel left_total_MP_Rel by (metis bi_totalI)

lemma bi_total_MF_Rel[transfer_rule]: "bi_total MF_Rel"
unfolding MF_Rel_def[abs_def]
by (intro prod.bi_total_rel multiset.bi_total_rel bi_total_MP_Rel bi_total_M_Rel)

lemma right_total_MF_Rel[transfer_rule]: "right_total MF_Rel"
using bi_total_MF_Rel unfolding bi_total_alt_def by auto

lemma left_total_MF_Rel[transfer_rule]: "left_total MF_Rel"
using bi_total_MF_Rel unfolding bi_total_alt_def by auto

lemma domain_RT_rel[transfer_domain_rule]: "Domainp MP_Rel = (λ f. True)"
proof
fix f :: "int poly"
show "Domainp MP_Rel f = True" unfolding MP_Rel_def[abs_def] Domainp.simps
by (auto simp: Mp_f_representative)
qed

lemma mem_MP_Rel[transfer_rule]: "(MP_Rel ===> rel_set MP_Rel ===> (=)) (λ x Y. ∃y ∈ Y. eq_m x y) (∈)"
proof (intro rel_funI iffI)
fix x y X Y assume xy: "MP_Rel x y" and XY: "rel_set MP_Rel X Y"
{ assume "∃x' ∈ X. x =m x'"
then obtain x' where x'X: "x' ∈ X" and xx': "x =m x'" by auto
with xy have x'y: "MP_Rel x' y" by (auto simp: MP_Rel_def)
from rel_setD1[OF XY x'X] obtain y' where "MP_Rel x' y'" and "y' ∈ Y" by auto
with x'y
show "y ∈ Y" by (auto simp: MP_Rel_def)
}
assume "y ∈ Y"
from rel_setD2[OF XY this] obtain x' where x'X: "x' ∈ X" and x'y: "MP_Rel x' y" by auto
from xy x'y have "x =m x'" by (auto simp: MP_Rel_def)
with x'X show "∃x' ∈ X. x =m x'" by auto
qed

lemma conversep_MP_Rel_OO_MP_Rel [simp]: "MP_Rel¯¯ OO MP_Rel = (=)"
using Mp_to_int_poly by (intro ext, auto simp: OO_def MP_Rel_def)

lemma MP_Rel_OO_conversep_MP_Rel [simp]: "MP_Rel OO MP_Rel¯¯ = eq_m"
by (intro ext, auto simp: OO_def MP_Rel_def Mp_f_representative)

lemma conversep_MP_Rel_OO_eq_m [simp]: "MP_Rel¯¯ OO eq_m = MP_Rel¯¯"
by (intro ext, auto simp: OO_def MP_Rel_def)

lemma eq_m_OO_MP_Rel [simp]: "eq_m OO MP_Rel = MP_Rel"
by (intro ext, auto simp: OO_def MP_Rel_def)

lemma eq_mset_MP_Rel [transfer_rule]: "(rel_mset MP_Rel ===> rel_mset MP_Rel ===> (=)) (rel_mset eq_m) (=)"
proof (intro rel_funI iffI)
fix A B X Y
assume AX: "rel_mset MP_Rel A X" and BY: "rel_mset MP_Rel B Y"
{
assume AB: "rel_mset eq_m A B"
from AX have "rel_mset MP_Rel¯¯ X A" by (simp add: multiset.rel_flip)
note rel_mset_OO[OF this AB]
note rel_mset_OO[OF this BY]
then show "X = Y" by (simp add: multiset.rel_eq)
}
assume "X = Y"
with BY have "rel_mset MP_Rel¯¯ X B" by (simp add: multiset.rel_flip)
from rel_mset_OO[OF AX this]
show "rel_mset eq_m A B" by simp
qed

lemma dvd_MP_Rel[transfer_rule]: "(MP_Rel ===> MP_Rel ===> (=)) (dvdm) (dvd)"
unfolding dvdm_def[abs_def] dvd_def[abs_def]
by transfer_prover

lemma irreducible_MP_Rel [transfer_rule]: "(MP_Rel ===> (=)) irreducible_m irreducible"
unfolding irreducible_m_def irreducible_def
by transfer_prover

lemma irreducible⇩d_MP_Rel [transfer_rule]: "(MP_Rel ===> (=)) irreducible⇩d_m irreducible⇩d"
unfolding irreducible⇩d_m_def[abs_def] irreducible⇩d_def[abs_def]
by transfer_prover

lemma UNIV_M_Rel[transfer_rule]: "rel_set M_Rel {0..<m} UNIV"
unfolding rel_set_def M_Rel_def[abs_def] M_def
by (auto simp: M_def m, goal_cases, metis to_int_mod_ring_of_int_mod_ring, (transfer, auto)+)

lemma coeff_MP_Rel [transfer_rule]: "(MP_Rel ===> (=) ===> M_Rel) coeff coeff"
unfolding rel_fun_def M_Rel_def MP_Rel_def Mp_coeff[symmetric] by auto

lemma M_1_1: "M 1 = 1" unfolding M_def unfolding m by simp

lemma square_free_MP_Rel [transfer_rule]: "(MP_Rel ===> (=)) square_free_m square_free"
unfolding square_free_m_def[abs_def] square_free_def[abs_def]
by (transfer_prover_start, transfer_step+, auto)

lemma mset_factors_m_MP_Rel [transfer_rule]: "(rel_mset MP_Rel ===> MP_Rel ===> (=)) mset_factors_m mset_factors"
unfolding mset_factors_def mset_factors_m_def
by (transfer_prover_start, transfer_step+, auto dest:eq_m_irreducible_m)

lemma coprime_MP_Rel [transfer_rule]: "(MP_Rel ===> MP_Rel ===> (=)) coprime_m coprime"
unfolding coprime_m_def[abs_def] coprime_def' [abs_def]
by (transfer_prover_start, transfer_step+, auto)

lemma prime_elem_MP_Rel [transfer_rule]: "(MP_Rel ===> (=)) prime_elem_m prime_elem"
unfolding prime_elem_m_def prime_elem_def by transfer_prover

end

context poly_mod_2 begin

lemma non_empty: "{0..<m} ≠ {}" using m1 by auto

lemma type_to_set:
assumes type_def: "∃(Rep :: 'b ⇒ int) Abs. type_definition Rep Abs {0 ..< m :: int}"
shows "class.nontriv (TYPE('b))" (is ?a) and "m = int CARD('b)" (is ?b)
proof -
from type_def obtain rep :: "'b ⇒ int" and abs :: "int ⇒ 'b" where t: "type_definition rep abs {0 ..< m}" by auto
have "card (UNIV :: 'b set) = card {0 ..< m}" using t by (rule type_definition.card)
also have "… = m" using m1 by auto
finally show ?b ..
then show ?a unfolding class.nontriv_def using m1 by auto
qed

end

locale poly_mod_prime_type = poly_mod_type m ty for m :: int and
ty :: "'a :: prime_card itself"
begin

lemma factorization_MP_Rel [transfer_rule]:
"(MP_Rel ===> MF_Rel ===> (=)) factorization_m (factorization Irr_Mon)"
unfolding rel_fun_def
proof (intro allI impI, goal_cases)
case (1 f F cfs Cfs)
note [transfer_rule] = 1(1)
obtain c fs where cfs: "cfs = (c,fs)" by force
obtain C Fs where Cfs: "Cfs = (C,Fs)" by force
from 1(2)[unfolded rel_prod.simps cfs Cfs MF_Rel_def]
have tr[transfer_rule]: "M_Rel c C" "rel_mset MP_Rel fs Fs" by auto
have eq: "(f =m smult c (prod_mset fs) = (F = smult C (prod_mset Fs)))"
by transfer_prover
have "set_mset Fs ⊆ Irr_Mon = (∀ x ∈# Fs. irreducible⇩d x ∧ monic x)" unfolding Irr_Mon_def by auto
also have "… = (∀f∈#fs. irreducible⇩d_m f ∧ monic (Mp f))"
proof (rule sym, transfer_prover_start, transfer_step+)
{
fix f
assume "f ∈# fs"
have "monic (Mp f) ⟷ M (coeff f (degree_m f)) = M 1"
unfolding Mp_coeff[symmetric] by simp
}
thus "(∀f∈#fs. irreducible⇩d_m f ∧ monic (Mp f)) =
(∀x∈#fs. irreducible⇩d_m x ∧ M (coeff x (degree_m x)) = M 1)" by auto
qed
finally
show "factorization_m f cfs = factorization Irr_Mon F Cfs" unfolding cfs Cfs
factorization_m_def factorization_def split eq by simp
qed

lemma unique_factorization_MP_Rel [transfer_rule]: "(MP_Rel ===> MF_Rel ===> (=))
unique_factorization_m (unique_factorization Irr_Mon)"
unfolding rel_fun_def
proof (intro allI impI, goal_cases)
case (1 f F cfs Cfs)
note [transfer_rule] = 1(1,2)
let ?F = "factorization Irr_Mon F"
let ?f = "factorization_m f"
let ?R = "Collect ?F"
let ?L = "Mf ` Collect ?f"
note X_to_x = right_total_MF_Rel[unfolded right_total_def, rule_format]
{
fix X
assume "X ∈ ?R"
hence F: "?F X" by simp
from X_to_x[of X] obtain x where rel[transfer_rule]: "MF_Rel x X" by blast
from F[untransferred] have "Mf x ∈ ?L" by blast
with rel have "∃ x. Mf x ∈ ?L ∧ MF_Rel x X" by blast
} note R_to_L = this
show "unique_factorization_m f cfs = unique_factorization Irr_Mon F Cfs" unfolding
unique_factorization_m_def unique_factorization_def
proof -
have fF: "?F Cfs = ?f cfs" by transfer simp
have "(?L = {Mf cfs}) = (?L ⊆ {Mf cfs} ∧ Mf cfs ∈ ?L)" by blast
also have "?L ⊆ {Mf cfs} = (∀ dfs. ?f dfs ⟶ Mf dfs = Mf cfs)" by blast
also have "… = (∀ y. ?F y ⟶ y = Cfs)" (is "?left = ?right")
proof (rule; intro allI impI)
fix Dfs
assume *: ?left and F: "?F Dfs"
from X_to_x[of Dfs] obtain dfs where [transfer_rule]: "MF_Rel dfs Dfs" by auto
from F[untransferred] have f: "?f dfs" .
from *[rule_format, OF f] have eq: "Mf dfs = Mf cfs" by simp
have "(Mf dfs = Mf cfs) = (Dfs = Cfs)" by (transfer_prover_start, transfer_step+, simp)
thus "Dfs = Cfs" using eq by simp
next
fix dfs
assume *: ?right and f: "?f dfs"
from left_total_MF_Rel obtain Dfs where
rel[transfer_rule]: "MF_Rel dfs Dfs" unfolding left_total_def by blast
have "?F Dfs" by (transfer, rule f)
from *[rule_format, OF this] have eq: "Dfs = Cfs" .
have "(Mf dfs = Mf cfs) = (Dfs = Cfs)" by (transfer_prover_start, transfer_step+, simp)
thus "Mf dfs = Mf cfs" using eq by simp
qed
also have "Mf cfs ∈ ?L = (∃ dfs. ?f dfs ∧ Mf cfs = Mf dfs)" by auto
also have "… = ?F Cfs"  unfolding fF
proof
assume "∃ dfs. ?f dfs ∧ Mf cfs  = Mf dfs"
then obtain dfs where f: "?f dfs" and id: "Mf dfs = Mf cfs" by auto
from f have "?f (Mf dfs)" by simp
from this[unfolded id] show "?f cfs" by simp
qed blast
finally show "(?L = {Mf cfs}) = (?R = {Cfs})" by auto
qed
qed

end

context begin
private lemma 1: "poly_mod_type TYPE('a :: nontriv) m = (m = int CARD('a))"
and 2: "class.nontriv TYPE('a) = (CARD('a) ≥ 2)"
unfolding poly_mod_type_def class.prime_card_def class.nontriv_def poly_mod_prime_type_def by auto

private lemma 3: "poly_mod_prime_type TYPE('b) m = (m = int CARD('b))"
and 4: "class.prime_card TYPE('b :: prime_card) = prime CARD('b :: prime_card)"
unfolding poly_mod_type_def class.prime_card_def class.nontriv_def poly_mod_prime_type_def by auto

lemmas poly_mod_type_simps = 1 2 3 4
end

lemma remove_duplicate_premise: "(PROP P ⟹ PROP P ⟹ PROP Q) ≡ (PROP P ⟹ PROP Q)" (is "?l ≡ ?r")
proof (intro Pure.equal_intr_rule)
assume p: "PROP P" and ppq: "PROP ?l"
from ppq[OF p p] show "PROP Q".
next
assume p: "PROP P" and pq: "PROP ?r"
from pq[OF p] show "PROP Q".
qed

context poly_mod_prime begin

lemma type_to_set:
assumes type_def: "∃(Rep :: 'b ⇒ int) Abs. type_definition Rep Abs {0 ..< p :: int}"
shows "class.prime_card (TYPE('b))" (is ?a) and "p = int CARD('b)" (is ?b)
proof -
from prime have p2: "p ≥ 2" by (rule prime_ge_2_int)
from type_def obtain rep :: "'b ⇒ int" and abs :: "int ⇒ 'b" where t: "type_definition rep abs {0 ..< p}" by auto
have "card (UNIV :: 'b set) = card {0 ..< p}" using t by (rule type_definition.card)
also have "… = p" using p2 by auto
finally show ?b ..
then show ?a unfolding class.prime_card_def using prime p2 by auto
qed
end

(* it will be nice to be able to automate this *)

lemmas (in poly_mod_type) prime_elem_m_dvdm_multD = prime_elem_dvd_multD
[where 'a = "'a mod_ring poly",untransferred]
lemmas (in poly_mod_2) prime_elem_m_dvdm_multD = poly_mod_type.prime_elem_m_dvdm_multD
[unfolded poly_mod_type_simps, internalize_sort "'a :: nontriv", OF type_to_set, unfolded remove_duplicate_premise, cancel_type_definition, OF non_empty]

lemmas(in poly_mod_prime_type) degree_m_mult_eq = degree_mult_eq
[where 'a = "'a mod_ring", untransferred]
lemmas(in poly_mod_prime) degree_m_mult_eq = poly_mod_prime_type.degree_m_mult_eq
[unfolded poly_mod_type_simps, internalize_sort "'a :: prime_card", OF type_to_set, unfolded remove_duplicate_premise, cancel_type_definition, OF non_empty]

lemma(in poly_mod_prime) irreducible⇩d_lifting:
assumes n: "n ≠ 0"
and deg: "poly_mod.degree_m (p^n) f = degree_m f"
and irr: "irreducible⇩d_m f"
shows "poly_mod.irreducible⇩d_m (p^n) f"
proof -
interpret q: poly_mod_2 "p^n" unfolding poly_mod_2_def using n m1 by auto
show "q.irreducible⇩d_m f"
proof (rule q.irreducible⇩d_mI)
from deg irr show "q.degree_m f > 0" by (auto elim: irreducible⇩d_mE)
then have pdeg_f: "degree_m f ≠ 0" by (simp add: deg)
note pMp_Mp = Mp_Mp_pow_is_Mp[OF n m1]
fix g h
assume deg_g: "degree g < q.degree_m f" and deg_h: "degree h < q.degree_m f"
and eq: "q.eq_m f (g * h)"
from eq have p_f: "f =m (g * h)" using pMp_Mp by metis
have "¬g =m 0" and "¬h =m 0"
apply (metis degree_0 mult_zero_left Mp_0 p_f pdeg_f poly_mod.mult_Mp(1))
by (metis degree_0 mult_eq_0_iff Mp_0 mult_Mp(2) p_f pdeg_f)
note [simp] = degree_m_mult_eq[OF this]
from degree_m_le[of g] deg_g
have 2: "degree_m g < degree_m f" by (fold deg, auto)
from degree_m_le[of h] deg_h
have 3: "degree_m h < degree_m f" by (fold deg, auto)
from irreducible⇩d_mD(2)[OF irr 2 3] p_f
show False by auto
qed
qed

(* Lifting UFD properties *)
lemmas (in poly_mod_prime_type) mset_factors_exist =
mset_factors_exist[where 'a = "'a mod_ring poly",untransferred]
lemmas (in poly_mod_prime) mset_factors_exist = poly_mod_prime_type.mset_factors_exist
[unfolded poly_mod_type_simps, internalize_sort "'a :: prime_card", OF type_to_set, unfolded remove_duplicate_premise, cancel_type_definition, OF non_empty]

lemmas (in poly_mod_prime_type) mset_factors_unique =
mset_factors_unique[where 'a = "'a mod_ring poly",untransferred]
lemmas (in poly_mod_prime) mset_factors_unique = poly_mod_prime_type.mset_factors_unique
[unfolded poly_mod_type_simps, internalize_sort "'a :: prime_card", OF type_to_set, unfolded remove_duplicate_premise, cancel_type_definition, OF non_empty]

lemmas (in poly_mod_prime_type) prime_elem_iff_irreducible =
prime_elem_iff_irreducible[where 'a = "'a mod_ring poly",untransferred]
lemmas (in poly_mod_prime) prime_elem_iff_irreducible[simp] = poly_mod_prime_type.prime_elem_iff_irreducible
[unfolded poly_mod_type_simps, internalize_sort "'a :: prime_card", OF type_to_set, unfolded remove_duplicate_premise, cancel_type_definition, OF non_empty]

lemmas (in poly_mod_prime_type) irreducible_connect =
irreducible_connect_field[where 'a = "'a mod_ring", untransferred]
lemmas (in poly_mod_prime) irreducible_connect[simp] = poly_mod_prime_type.irreducible_connect
[unfolded poly_mod_type_simps, internalize_sort "'a :: prime_card", OF type_to_set, unfolded remove_duplicate_premise, cancel_type_definition, OF non_empty]

lemmas (in poly_mod_prime_type) irreducible_degree =
irreducible_degree_field[where 'a = "'a mod_ring", untransferred]
lemmas (in poly_mod_prime) irreducible_degree = poly_mod_prime_type.irreducible_degree
[unfolded poly_mod_type_simps, internalize_sort "'a :: prime_card", OF type_to_set, unfolded remove_duplicate_premise, cancel_type_definition, OF non_empty]

end
```