Theory Algebra4
theory Algebra4
imports Algebra3
begin
section "Abelian groups"
record 'a aGroup = "'a carrier" +
pop :: "['a, 'a ] ⇒ 'a" (infixl "±ı" 62)
mop :: "'a ⇒ 'a" ("(-⇩aı _)" [64]63 )
zero :: "'a" ("𝟬ı")
locale aGroup =
fixes A (structure)
assumes
pop_closed: "pop A ∈ carrier A → carrier A → carrier A"
and aassoc : "⟦a ∈ carrier A; b ∈ carrier A; c ∈ carrier A⟧ ⟹
(a ± b) ± c = a ± (b ± c)"
and pop_commute:"⟦a ∈ carrier A; b ∈ carrier A⟧ ⟹ a ± b = b ± a"
and mop_closed:"mop A ∈ carrier A → carrier A"
and l_m :"a ∈ carrier A ⟹ (-⇩a a) ± a = 𝟬"
and ex_zero: "𝟬 ∈ carrier A"
and l_zero:"a ∈ carrier A ⟹ 𝟬 ± a = a"
definition
b_ag :: "_ ⇒
⦇carrier:: 'a set, top:: ['a, 'a] ⇒ 'a , iop:: 'a ⇒ 'a, one:: 'a ⦈" where
"b_ag A = ⦇carrier = carrier A, top = pop A, iop = mop A, one = zero A ⦈"
definition
asubGroup :: "[_ , 'a set] ⇒ bool" where
"asubGroup A H ⟷ (b_ag A) » H"
definition
aqgrp :: "[_ , 'a set] ⇒
⦇ carrier::'a set set, pop::['a set, 'a set] ⇒ 'a set,
mop::'a set ⇒ 'a set, zero :: 'a set ⦈" where
"aqgrp A H = ⦇carrier = set_rcs (b_ag A) H,
pop = λX. λY. (c_top (b_ag A) H X Y),
mop = λX. (c_iop (b_ag A) H X), zero = H ⦈"
definition
ag_idmap :: "_ ⇒ ('a ⇒ 'a)" ("(aI⇘_⇙)") where
"aI⇘A⇙ = (λx∈carrier A. x)"
abbreviation
ASubG :: "[('a, 'more) aGroup_scheme, 'a set] => bool" (infixl "+>" 58) where
"A +> H == asubGroup A H"
definition
Ag_ind :: "[_ , 'a ⇒ 'd] ⇒ 'd aGroup" where
"Ag_ind A f = ⦇carrier = f`(carrier A),
pop = λx ∈ f`(carrier A). λy ∈ f`(carrier A).
f(((invfun (carrier A) (f`(carrier A)) f) x) ±⇘A⇙
((invfun (carrier A) (f`(carrier A)) f) y)),
mop = λx∈(f`(carrier A)). f (-⇩a⇘A⇙ ((invfun (carrier A) (f`(carrier A)) f) x)),
zero = f (𝟬⇘A⇙)⦈"
definition
Agii :: "[_ , 'a ⇒ 'd] ⇒ ('a ⇒ 'd)" where
"Agii A f = (λx∈carrier A. f x)"
lemma (in aGroup) ag_carrier_carrier:"carrier (b_ag A) = carrier A"
by (simp add:b_ag_def)
lemma (in aGroup) ag_pOp_closed:"⟦x ∈ carrier A; y ∈ carrier A⟧ ⟹
pop A x y ∈ carrier A"
apply (cut_tac pop_closed)
apply (frule funcset_mem[of "(±) " "carrier A" "carrier A → carrier A" "x"],
assumption+)
apply (rule funcset_mem[of "(±) x" "carrier A" "carrier A" "y"], assumption+)
done
lemma (in aGroup) ag_mOp_closed:"x ∈ carrier A ⟹ (-⇩a x) ∈ carrier A"
apply (cut_tac mop_closed)
apply (rule funcset_mem[of "mop A" "carrier A" "carrier A" "x"], assumption+)
done
lemma (in aGroup) asubg_subset:"A +> H ⟹ H ⊆ carrier A"
apply (simp add:asubGroup_def)
apply (simp add:sg_def, (erule conjE)+)
apply (simp add:ag_carrier_carrier)
done
lemma (in aGroup) ag_pOp_commute:"⟦x ∈ carrier A; y ∈ carrier A⟧ ⟹
pop A x y = pop A y x"
by (simp add:pop_commute)
lemma (in aGroup) b_ag_group:"Group (b_ag A)"
apply (unfold Group_def)
apply (simp add:b_ag_def)
apply (simp add:pop_closed mop_closed ex_zero)
apply (rule conjI)
apply (rule allI, rule impI)+
apply (simp add:aassoc)
apply (rule conjI)
apply (rule allI, rule impI)
apply (simp add:l_m)
apply (rule allI, rule impI)
apply (simp add:l_zero)
done
lemma (in aGroup) agop_gop:"top (b_ag A) = pop A"
apply (simp add:b_ag_def)
done
lemma (in aGroup) agiop_giop:"iop (b_ag A) = mop A"
apply (simp add:b_ag_def)
done
lemma (in aGroup) agunit_gone:"one (b_ag A) = 𝟬"
apply (simp add:b_ag_def)
done
lemma (in aGroup) ag_pOp_add_r:"⟦a ∈ carrier A; b ∈ carrier A; c ∈ carrier A;
a = b⟧ ⟹ a ± c = b ± c"
apply simp
done
lemma (in aGroup) ag_add_commute:"⟦a ∈ carrier A; b ∈ carrier A⟧ ⟹
a ± b = b ± a"
by (simp add:pop_commute)
lemma (in aGroup) ag_pOp_add_l:"⟦a ∈ carrier A; b ∈ carrier A; c ∈ carrier A;
a = b⟧ ⟹ c ± a = c ± b"
apply simp
done
lemma (in aGroup) asubg_pOp_closed:"⟦asubGroup A H; x ∈ H; y ∈ H⟧
⟹ pop A x y ∈ H"
apply (simp add:asubGroup_def)
apply (cut_tac b_ag_group)
apply (frule Group.sg_mult_closed [of "b_ag A" "H" "x" "y"], assumption+)
apply (simp only:agop_gop)
done
lemma (in aGroup) asubg_mOp_closed:"⟦asubGroup A H; x ∈ H⟧ ⟹ -⇩a x ∈ H"
apply (simp add:asubGroup_def)
apply (cut_tac b_ag_group)
apply (frule Group.sg_i_closed[of "b_ag A" "H" "x"], assumption+)
apply (simp add:agiop_giop)
done
lemma (in aGroup) asubg_subset1:"⟦asubGroup A H; x ∈ H⟧ ⟹ x ∈ carrier A"
apply (simp add:asubGroup_def)
apply (cut_tac b_ag_group)
apply (frule Group.sg_subset_elem[of "b_ag A" "H" "x"], assumption+)
apply (simp add:ag_carrier_carrier)
done
lemma (in aGroup) asubg_inc_zero:"asubGroup A H ⟹ 𝟬 ∈ H"
apply (simp add:asubGroup_def)
apply (cut_tac b_ag_group)
apply (frule Group.sg_unit_closed[of "b_ag A" "H"], assumption)
apply (simp add:b_ag_def)
done
lemma (in aGroup) ag_inc_zero:"𝟬 ∈ carrier A"
by (simp add:ex_zero)
lemma (in aGroup) ag_l_zero:"x ∈ carrier A ⟹ 𝟬 ± x = x"
by (simp add:l_zero)
lemma (in aGroup) ag_r_zero:"x ∈ carrier A ⟹ x ± 𝟬 = x"
apply (cut_tac ex_zero)
apply (subst pop_commute, assumption+)
apply (rule ag_l_zero, assumption)
done
lemma (in aGroup) ag_l_inv1:"x ∈ carrier A ⟹ (-⇩a x) ± x = 𝟬"
by (simp add:l_m)
lemma (in aGroup) ag_r_inv1:"x ∈ carrier A ⟹ x ± (-⇩a x) = 𝟬"
by (frule ag_mOp_closed[of "x"],
subst ag_pOp_commute, assumption+,
simp add:ag_l_inv1)
lemma (in aGroup) ag_pOp_assoc:"⟦x ∈ carrier A; y ∈ carrier A; z ∈ carrier A⟧
⟹ (x ± y) ± z = x ± (y ± z)"
by (simp add:aassoc)
lemma (in aGroup) ag_inv_unique:"⟦x ∈ carrier A; y ∈ carrier A; x ± y = 𝟬⟧ ⟹
y = -⇩a x"
apply (frule ag_mOp_closed[of "x"],
frule aassoc[of "-⇩a x" "x" "y"], assumption+,
simp add:l_m l_zero ag_r_zero)
done
lemma (in aGroup) ag_inv_inj:"⟦x ∈ carrier A; y ∈ carrier A; x ≠ y⟧ ⟹
(-⇩a x) ≠ (-⇩a y)"
apply (rule contrapos_pp, simp+)
apply (frule ag_mOp_closed[of "y"],
frule aassoc[of "y" "-⇩a y" "x"], assumption+)
apply (simp only:ag_r_inv1,
frule sym, thin_tac "-⇩a x = -⇩a y", simp add:l_m)
apply (simp add:l_zero ag_r_zero)
done
lemma (in aGroup) pOp_assocTr41:"⟦a ∈ carrier A; b ∈ carrier A; c ∈ carrier A;
d ∈ carrier A⟧ ⟹ a ± b ± c ± d = a ± b ± (c ± d)"
by (frule ag_pOp_closed[of "a" "b"], assumption+,
rule aassoc[of "a ± b" "c" "d"], assumption+)
lemma (in aGroup) pOp_assocTr42:"⟦a ∈ carrier A; b ∈ carrier A;
c ∈ carrier A; d ∈ carrier A⟧ ⟹ a ± b ± c ± d = a ± (b ± c) ± d"
by (simp add:aassoc[THEN sym, of "a" "b" "c"])
lemma (in aGroup) pOp_assocTr43:"⟦a ∈ carrier A; b ∈ carrier A;
c ∈ carrier A; d ∈ carrier A⟧ ⟹ a ± b ± (c ± d) = a ± (b ± c) ± d"
by (subst pOp_assocTr41[THEN sym], assumption+,
rule pOp_assocTr42, assumption+)
lemma (in aGroup) pOp_assoc_cancel:"⟦a ∈ carrier A; b ∈ carrier A;
c ∈ carrier A⟧ ⟹ a ± -⇩a b ± (b ± -⇩a c) = a ± -⇩a c"
apply (subst pOp_assocTr43, assumption)
apply (simp add:ag_l_inv1 ag_mOp_closed)+
apply (simp add:ag_r_zero)
done
lemma (in aGroup) ag_p_inv:"⟦x ∈ carrier A; y ∈ carrier A⟧ ⟹
(-⇩a (x ± y)) = (-⇩a x) ± (-⇩a y)"
apply (frule ag_mOp_closed[of "x"], frule ag_mOp_closed[of "y"],
frule ag_pOp_closed[of "x" "y"], assumption+)
apply (frule aassoc[of "x ± y" "-⇩a x" "-⇩a y"], assumption+,
simp add:pOp_assocTr43, simp add:pop_commute[of "y" "-⇩a x"],
simp add:aassoc[THEN sym, of "x" "-⇩a x" "y"],
simp add:ag_r_inv1 l_zero)
apply (frule ag_pOp_closed[of "-⇩a x" "-⇩a y"], assumption+,
simp add:pOp_assocTr41,
rule ag_inv_unique[THEN sym, of "x ± y" "-⇩a x ± -⇩a y"], assumption+)
done
lemma (in aGroup) gEQAddcross: "⟦l1 ∈ carrier A; l2 ∈ carrier A;
r1 ∈ carrier A; r1 ∈ carrier A; l1 = r2; l2 = r1⟧ ⟹
l1 ± l2 = r1 ± r2"
apply (simp add:ag_pOp_commute)
done
lemma (in aGroup) ag_eq_sol1:"⟦a ∈ carrier A; x∈ carrier A; b∈ carrier A;
a ± x = b⟧ ⟹ x = (-⇩a a) ± b"
apply (frule ag_mOp_closed[of "a"])
apply (frule aassoc[of "-⇩a a" "a" "x"], assumption+)
apply (simp add:l_m l_zero)
done
lemma (in aGroup) ag_eq_sol2:"⟦a ∈ carrier A; x∈ carrier A; b∈ carrier A;
x ± a = b⟧ ⟹ x = b ± (-⇩a a)"
apply (frule ag_mOp_closed[of "a"],
frule aassoc[of "x" "a" "-⇩a a"], assumption+,
simp add:ag_r_inv1 ag_r_zero)
done
lemma (in aGroup) ag_add4_rel:"⟦a ∈ carrier A; b ∈ carrier A; c ∈ carrier A;
d ∈ carrier A ⟧ ⟹ a ± b ± (c ± d) = a ± c ± (b ± d)"
apply (simp add:pOp_assocTr43[of "a" "b" "c" "d"],
simp add:ag_pOp_commute[of "b" "c"],
simp add:pOp_assocTr43[THEN sym, of "a" "c" "b" "d"])
done
lemma (in aGroup) ag_inv_inv:"x ∈ carrier A ⟹ -⇩a (-⇩a x) = x"
by (frule ag_l_inv1[of "x"], frule ag_mOp_closed[of "x"],
rule ag_inv_unique[THEN sym, of "-⇩a x" "x"], assumption+)
lemma (in aGroup) ag_inv_zero:"-⇩a 𝟬 = 𝟬"
apply (cut_tac ex_zero)
apply (frule l_zero[of "𝟬"])
apply (rule ag_inv_unique[THEN sym], assumption+)
done
lemma (in aGroup) ag_diff_minus:"⟦a ∈ carrier A; b ∈ carrier A; c ∈ carrier A;
a ± (-⇩a b) = c⟧ ⟹ b ± (-⇩a a) = (-⇩a c)"
apply (frule sym, thin_tac "a ± -⇩a b = c", simp, thin_tac "c = a ± -⇩a b")
apply (frule ag_mOp_closed[of "b"], frule ag_mOp_closed[of "a"],
subst ag_p_inv, assumption+, subst ag_inv_inv, assumption)
apply (simp add:ag_pOp_commute)
done
lemma (in aGroup) pOp_cancel_l:"⟦a ∈ carrier A; b ∈ carrier A; c ∈ carrier A; c ± a = c ± b ⟧ ⟹ a = b"
apply (frule ag_mOp_closed[of "c"],
frule aassoc[of "-⇩a c" "c" "a"], assumption+,
simp only:l_m l_zero)
apply (simp only:aassoc[THEN sym, of "-⇩a c" "c" "b"],
simp only:l_m l_zero)
done
lemma (in aGroup) pOp_cancel_r:"⟦a ∈ carrier A; b ∈ carrier A; c ∈ carrier A; a ± c = b ± c ⟧ ⟹ a = b"
by (simp add:ag_pOp_commute pOp_cancel_l)
lemma (in aGroup) ag_eq_diffzero:"⟦a ∈ carrier A; b ∈ carrier A⟧ ⟹
(a = b) = (a ± (-⇩a b) = 𝟬)"
apply (rule iffI)
apply (simp add:ag_r_inv1)
apply (frule ag_mOp_closed[of "b"])
apply (simp add:ag_pOp_commute[of "a" "-⇩a b"])
apply (subst ag_inv_unique[of "-⇩a b" "a"], assumption+,
simp add:ag_inv_inv)
done
lemma (in aGroup) ag_eq_diffzero1:"⟦a ∈ carrier A; b ∈ carrier A⟧ ⟹
(a = b) = ((-⇩a a) ± b = 𝟬)"
apply (frule ag_mOp_closed[of a],
simp add:ag_pOp_commute)
apply (subst ag_eq_diffzero[THEN sym], assumption+)
apply (rule iffI, rule sym, assumption)
apply (rule sym, assumption)
done
lemma (in aGroup) ag_neq_diffnonzero:"⟦a ∈ carrier A; b ∈ carrier A⟧ ⟹
(a ≠ b) = (a ± (-⇩a b) ≠ 𝟬)"
apply (rule iffI)
apply (rule contrapos_pp, simp+)
apply (simp add:ag_eq_diffzero[THEN sym])
apply (rule contrapos_pp, simp+)
apply (simp add:ag_r_inv1)
done
lemma (in aGroup) ag_plus_zero:"⟦x ∈ carrier A; y ∈ carrier A⟧ ⟹
(x = -⇩a y) = (x ± y = 𝟬)"
apply (rule iffI)
apply (simp add:ag_l_inv1)
apply (simp add:ag_pOp_commute[of "x" "y"])
apply (rule ag_inv_unique[of "y" "x"], assumption+)
done
lemma (in aGroup) asubg_nsubg:"A +> H ⟹ (b_ag A) ▹ H"
apply (cut_tac b_ag_group)
apply (simp add:asubGroup_def)
apply (rule Group.cond_nsg[of "b_ag A" "H"], assumption+)
apply (rule ballI)+
apply(simp add:agop_gop agiop_giop)
apply (frule Group.sg_subset[of "b_ag A" "H"], assumption)
apply (simp add:ag_carrier_carrier)
apply (frule_tac c = h in subsetD[of "H" "carrier A"], assumption+)
apply (subst ag_pOp_commute, assumption+)
apply (frule_tac x = a in ag_mOp_closed)
apply (subst aassoc, assumption+, simp add:ag_r_inv1 ag_r_zero)
done
lemma (in aGroup) subg_asubg:"b_ag G » H ⟹ G +> H"
apply (simp add:asubGroup_def)
done
lemma (in aGroup) asubg_test:"⟦H ⊆ carrier A; H ≠ {};
∀a∈H. ∀b∈H. (a ± (-⇩a b) ∈ H)⟧ ⟹ A +> H"
apply (simp add:asubGroup_def) apply (cut_tac b_ag_group)
apply (rule Group.sg_condition [of "b_ag A" "H"], assumption+)
apply (simp add:ag_carrier_carrier) apply assumption
apply (rule allI)+ apply (rule impI)
apply (simp add:agop_gop agiop_giop)
done
lemma (in aGroup) asubg_zero:"A +> {𝟬}"
apply (rule asubg_test[of "{𝟬}"])
apply (simp add:ag_inc_zero)
apply simp
apply (simp, cut_tac ag_inc_zero, simp add:ag_r_inv1)
done
lemma (in aGroup) asubg_whole:"A +> carrier A"
apply (rule asubg_test[of "carrier A"])
apply (simp,
cut_tac ag_inc_zero, simp add:nonempty)
apply ((rule ballI)+,
rule ag_pOp_closed, assumption,
rule_tac x = b in ag_mOp_closed, assumption)
done
lemma (in aGroup) Ag_ind_carrier:"bij_to f (carrier A) (D::'d set) ⟹
carrier (Ag_ind A f) = f ` (carrier A)"
by (simp add:Ag_ind_def)
lemma (in aGroup) Ag_ind_aGroup:"⟦f ∈ carrier A → D;
bij_to f (carrier A) (D::'d set)⟧ ⟹ aGroup (Ag_ind A f)"
apply (simp add:bij_to_def, frule conjunct1, frule conjunct2, fold bij_to_def)
apply (simp add:aGroup_def)
apply (rule conjI)
apply (rule Pi_I)+
apply (simp add:Ag_ind_carrier surj_to_def)
apply (frule_tac b = x in invfun_mem1[of "f" "carrier A" "D"], assumption+,
frule_tac b = xa in invfun_mem1[of "f" "carrier A" "D"], assumption+)
apply (simp add:Ag_ind_def)
apply (rule funcset_mem[of "f" "carrier A" "D"], assumption)
apply (simp add:ag_pOp_closed)
apply (rule conjI)
apply (rule allI, rule impI)+
apply (simp add: Ag_ind_carrier surj_to_def)
apply (frule_tac b = a in invfun_mem1[of "f" "carrier A" "D"], assumption+,
frule_tac b = b in invfun_mem1[of "f" "carrier A" "D"], assumption+,
frule_tac b = c in invfun_mem1[of "f" "carrier A" "D"], assumption+)
apply (simp add:Ag_ind_def)
apply (frule_tac x = "invfun (carrier A) D f a" and
y = "invfun (carrier A) D f b" in ag_pOp_closed, assumption+,
frule_tac x = "invfun (carrier A) D f b" and
y = "invfun (carrier A) D f c" in ag_pOp_closed, assumption+)
apply (simp add:Pi_def)
apply (unfold bij_to_def, frule conjunct1, fold bij_to_def)
apply (simp add:invfun_l)
apply (subst injective_iff[of "f" "carrier A", THEN sym], assumption)
apply (simp add:ag_pOp_closed)+
apply (simp add:ag_pOp_assoc)
apply (rule conjI)
apply (rule allI, rule impI)+
apply (simp add:Ag_ind_def)
apply (subst injective_iff[of "f" "carrier A", THEN sym], assumption)
apply (frule_tac b = a in invfun_mem1[of "f" "carrier A" "D"], assumption+,
frule_tac b = b in invfun_mem1[of "f" "carrier A" "D"], assumption+)
apply (simp add:surj_to_def) apply (simp add:surj_to_def)
apply (simp add:surj_to_def)
apply (frule_tac b = b in invfun_mem1[of "f" "carrier A" "D"], assumption+)
apply (simp add:ag_pOp_closed)
apply (simp add:surj_to_def)
apply (frule_tac b = a in invfun_mem1[of "f" "carrier A" "D"], assumption+,
frule_tac b = b in invfun_mem1[of "f" "carrier A" "D"], assumption+)
apply (simp add:ag_pOp_closed)
apply (simp add:surj_to_def)
apply (frule_tac b = a in invfun_mem1[of "f" "carrier A" "D"], assumption+,
frule_tac b = b in invfun_mem1[of "f" "carrier A" "D"], assumption+)
apply (simp add:ag_pOp_commute)
apply (rule conjI)
apply (rule Pi_I)
apply (simp add:Ag_ind_def surj_to_def)
apply (rule funcset_mem[of "f" "carrier A" "D"], assumption)
apply (frule_tac b = x in invfun_mem1[of "f" "carrier A" "D"], assumption+)
apply (simp add:ag_mOp_closed)
apply (rule conjI)
apply (rule allI, rule impI)
apply (simp add:Ag_ind_def surj_to_def)
apply (frule_tac b = a in invfun_mem1[of "f" "carrier A" "D"], assumption+)
apply (frule_tac x = "invfun (carrier A) D f a" in ag_mOp_closed)
apply (simp add:Pi_def)
apply (subst injective_iff[of "f" "carrier A", THEN sym], assumption)
apply (unfold bij_to_def, frule conjunct1, fold bij_to_def)
apply (simp add:invfun_l)
apply (simp add:ag_pOp_closed)
apply (simp add:ag_inc_zero)
apply (unfold bij_to_def, frule conjunct1, fold bij_to_def)
apply (simp add:invfun_l l_m)
apply (rule conjI)
apply (simp add:Ag_ind_def surj_to_def)
apply (rule funcset_mem[of "f" "carrier A" "D"], assumption)
apply (simp add:ag_inc_zero)
apply (rule allI, rule impI)
apply (simp add:Ag_ind_def surj_to_def)
apply (cut_tac ag_inc_zero, simp add:funcset_mem del:Pi_I)
apply (unfold bij_to_def, frule conjunct1, fold bij_to_def)
apply (simp add:invfun_l)
apply (frule_tac b = a in invfun_mem1[of "f" "carrier A" "D"], assumption+)
apply (simp add:l_zero)
apply (simp add:invfun_r)
done
subsection "Homomorphism of abelian groups"
definition
aHom :: "[('a, 'm) aGroup_scheme, ('b, 'm1) aGroup_scheme] ⇒ ('a ⇒ 'b) set" where
"aHom A B = {f. f ∈ carrier A → carrier B ∧ f ∈ extensional (carrier A) ∧
(∀a∈carrier A. ∀b∈carrier A. f (a ±⇘A⇙ b) = (f a) ±⇘B⇙ (f b))}"
definition
compos :: "[('a, 'm) aGroup_scheme, 'b ⇒ 'c, 'a ⇒ 'b] ⇒ 'a ⇒ 'c" where
"compos A g f = compose (carrier A) g f"
definition
ker :: "[('a, 'm) aGroup_scheme, ('b, 'm1) aGroup_scheme] ⇒ ('a ⇒ 'b)
⇒ 'a set" ("(3ker⇘_,_⇙ _)" [82,82,83]82) where
"ker⇘F,G⇙ f = {a. a ∈ carrier F ∧ f a = (𝟬⇘G⇙)}"
definition
injec :: "[('a, 'm) aGroup_scheme, ('b, 'm1) aGroup_scheme, 'a ⇒ 'b]
⇒ bool" ("(3injec⇘_,_⇙ _)" [82,82,83]82) where
"injec⇘F,G⇙ f ⟷ f ∈ aHom F G ∧ ker⇘F,G⇙ f = {𝟬⇘F⇙}"
definition
surjec :: "[('a, 'm) aGroup_scheme, ('b, 'm1) aGroup_scheme, 'a ⇒ 'b]
⇒ bool" ("(3surjec⇘_,_⇙ _)" [82,82,83]82) where
"surjec⇘F,G⇙ f ⟷ f ∈ aHom F G ∧ surj_to f (carrier F) (carrier G)"
definition
bijec :: "[('a, 'm) aGroup_scheme, ('b, 'm1) aGroup_scheme, 'a ⇒ 'b]
⇒ bool" ("(3bijec⇘_,_⇙ _)" [82,82,83]82) where
"bijec⇘F,G⇙ f ⟷ injec⇘F,G⇙ f ∧ surjec⇘F,G⇙ f"
definition
ainvf :: "[('a, 'm) aGroup_scheme, ('b, 'm1) aGroup_scheme, 'a ⇒ 'b]
⇒ ('b ⇒ 'a)" ("(3ainvf⇘_,_⇙ _)" [82,82,83]82) where
"ainvf⇘F,G⇙ f = invfun (carrier F) (carrier G) f"
lemma aHom_mem:"⟦aGroup F; aGroup G; f ∈ aHom F G; a ∈ carrier F⟧ ⟹
f a ∈ carrier G"
apply (simp add:aHom_def) apply (erule conjE)+
apply (simp add:Pi_def)
done
lemma aHom_func:"f ∈ aHom F G ⟹ f ∈ carrier F → carrier G"
by (simp add:aHom_def)
lemma aHom_add:"⟦aGroup F; aGroup G; f ∈ aHom F G; a ∈ carrier F;
b ∈ carrier F⟧ ⟹ f (a ±⇘F⇙ b) = (f a) ±⇘G⇙ (f b)"
apply (simp add:aHom_def)
done
lemma aHom_0_0:"⟦aGroup F; aGroup G; f ∈ aHom F G⟧ ⟹ f (𝟬⇘F⇙) = 𝟬⇘G⇙"
apply (frule aGroup.ag_inc_zero [of "F"])
apply (subst aGroup.ag_l_zero [THEN sym, of "F" "𝟬⇘F⇙"], assumption+)
apply (simp add:aHom_add)
apply (frule aGroup.ag_l_zero [THEN sym, of "F" "𝟬⇘F⇙"], assumption+)
apply (subgoal_tac "f (𝟬⇘F⇙) = f (𝟬⇘F⇙ ±⇘F⇙ 𝟬⇘F⇙)") prefer 2 apply simp
apply (thin_tac "𝟬⇘F⇙ = 𝟬⇘F⇙ ±⇘F⇙ 𝟬⇘F⇙")
apply (simp add:aHom_add) apply (frule sym)
apply (thin_tac "f 𝟬⇘F⇙ = f 𝟬⇘F⇙ ±⇘G⇙ f 𝟬⇘F⇙")
apply (frule aHom_mem[of "F" "G" "f" "𝟬⇘F⇙"], assumption+)
apply (frule aGroup.ag_mOp_closed[of "G" "f 𝟬⇘F⇙"], assumption+)
apply (frule aGroup.aassoc[of "G" "-⇩a⇘G⇙ (f 𝟬⇘F⇙)" "f 𝟬⇘F⇙" "f 𝟬⇘F⇙"], assumption+)
apply (simp add:aGroup.l_m aGroup.l_zero)
done
lemma ker_inc_zero:"⟦aGroup F; aGroup G; f ∈ aHom F G⟧ ⟹ 𝟬⇘F⇙ ∈ ker⇘F,G⇙ f"
by (frule aHom_0_0[of "F" "G" "f"], assumption+,
simp add:ker_def, simp add:aGroup.ag_inc_zero [of "F"])
lemma aHom_inv_inv:"⟦aGroup F; aGroup G; f ∈ aHom F G; a ∈ carrier F⟧ ⟹
f (-⇩a⇘F⇙ a) = -⇩a⇘G⇙ (f a)"
apply (frule aGroup.ag_l_inv1 [of "F" "a"], assumption+,
frule sym, thin_tac "-⇩a⇘F⇙ a ±⇘F⇙ a = 𝟬⇘F⇙",
frule aHom_0_0[of "F" "G" "f"], assumption+,
frule aGroup.ag_mOp_closed[of "F" "a"], assumption+)
apply (simp add:aHom_add, thin_tac "𝟬⇘F⇙ = -⇩a⇘F⇙ a ±⇘F⇙ a")
apply (frule aHom_mem[of "F" "G" "f" "-⇩a⇘F⇙ a"], assumption+,
frule aHom_mem[of "F" "G" "f" "a"], assumption+,
simp only:aGroup.ag_pOp_commute[of "G" "f (-⇩a⇘F⇙ a)" "f a"])
apply (rule aGroup.ag_inv_unique[of "G"], assumption+)
done
lemma aHom_compos:"⟦aGroup L; aGroup M; aGroup N; f ∈ aHom L M; g ∈ aHom M N ⟧
⟹ compos L g f ∈ aHom L N"
apply (simp add:aHom_def [of "L" "N"])
apply (rule conjI)
apply (rule Pi_I)
apply (simp add:compos_def compose_def)
apply (rule aHom_mem [of "M" "N" "g"], assumption+)
apply (simp add:aHom_mem [of "L" "M" "f"])
apply (rule conjI)
apply (simp add:compos_def compose_def extensional_def)
apply (rule ballI)+
apply (simp add:compos_def compose_def)
apply (simp add:aGroup.ag_pOp_closed)
apply (simp add:aHom_add)
apply (rule aHom_add, assumption+)
apply (simp add:aHom_mem)+
done
lemma aHom_compos_assoc:"⟦aGroup K; aGroup L; aGroup M; aGroup N; f ∈ aHom K L;
g ∈ aHom L M; h ∈ aHom M N ⟧ ⟹
compos K h (compos K g f) = compos K (compos L h g) f"
apply (simp add:compos_def compose_def)
apply (rule funcset_eq[of _ "carrier K"])
apply (simp add:restrict_def extensional_def)
apply (simp add:restrict_def extensional_def)
apply (rule ballI, simp)
apply (simp add:aHom_mem)
done
lemma injec_inj_on:"⟦aGroup F; aGroup G; injec⇘F,G⇙ f⟧ ⟹ inj_on f (carrier F)"
apply (simp add:inj_on_def)
apply (rule ballI)+ apply (rule impI)
apply (simp add:injec_def, erule conjE)
apply (frule_tac a = x in aHom_mem[of "F" "G" "f"], assumption+,
frule_tac a = x in aHom_mem[of "F" "G" "f"], assumption+)
apply (frule_tac x = "f x" in aGroup.ag_r_inv1[of "G"], assumption+)
apply (simp only:aHom_inv_inv[THEN sym, of "F" "G" "f"])
apply (frule sym, thin_tac "f x = f y", simp)
apply (frule_tac x = y in aGroup.ag_mOp_closed[of "F"], assumption+)
apply (simp add:aHom_add[THEN sym], simp add:ker_def)
apply (subgoal_tac "x ±⇘F⇙ -⇩a⇘F⇙ y ∈ {a ∈ carrier F. f a = 𝟬⇘G⇙}",
simp)
apply (subst aGroup.ag_eq_diffzero[of "F"], assumption+)
apply (frule_tac x = x and y = "-⇩a⇘F⇙ y" in aGroup.ag_pOp_closed[of "F"],
assumption+)
apply simp apply blast
done
lemma surjec_surj_to:"surjec⇘R,S⇙ f ⟹ surj_to f (carrier R) (carrier S)"
by (simp add:surjec_def)
lemma compos_bijec:"⟦aGroup E; aGroup F; aGroup G; bijec⇘E,F⇙ f; bijec⇘F,G⇙ g⟧ ⟹
bijec⇘E,G⇙ (compos E g f)"
apply (simp add:bijec_def, (erule conjE)+)
apply (rule conjI)
apply (simp add:injec_def, (erule conjE)+)
apply (simp add:aHom_compos[of "E" "F" "G" "f" "g"])
apply (rule equalityI, rule subsetI, simp add:ker_def, erule conjE)
apply (simp add:compos_def compose_def)
apply (frule_tac a = x in aHom_mem[of "E" "F" "f"], assumption+)
apply (subgoal_tac "(f x) ∈ {a ∈ carrier F. g a = 𝟬⇘G⇙}", simp)
apply (subgoal_tac "x ∈ {a ∈ carrier E. f a = 𝟬⇘F⇙}", simp)
apply blast apply blast
apply (rule subsetI, simp)
apply (simp add:ker_def compos_def compose_def)
apply (simp add:aGroup.ag_inc_zero) apply (simp add:aHom_0_0)
apply (simp add:surjec_def, (erule conjE)+)
apply (simp add:aHom_compos)
apply (simp add:aHom_def, (erule conjE)+) apply (simp add:compos_def)
apply (rule compose_surj[of "f" "carrier E" "carrier F" "g" "carrier G"],
assumption+)
done
lemma ainvf_aHom:"⟦aGroup F; aGroup G; bijec⇘F,G⇙ f⟧ ⟹
ainvf⇘F,G⇙ f ∈ aHom G F"
apply (subst aHom_def, simp)
apply (simp add:ainvf_def)
apply (simp add:bijec_def, erule conjE)
apply (frule injec_inj_on[of "F" "G" "f"], assumption+)
apply (simp add:surjec_def, (erule conjE)+)
apply (simp add:aHom_def, (erule conjE)+)
apply (frule inv_func[of "f" "carrier F" "carrier G"], assumption+, simp)
apply (rule conjI)
apply (simp add:invfun_def)
apply (rule ballI)+
apply (frule_tac x = a in funcset_mem[of "Ifn F G f" "carrier G" "carrier F"],
assumption+,
frule_tac x = b in funcset_mem[of "Ifn F G f" "carrier G" "carrier F"],
assumption+,
frule_tac x = a and y = b in aGroup.ag_pOp_closed[of "G"], assumption+,
frule_tac x = "a ±⇘G⇙ b" in funcset_mem[of "Ifn F G f" "carrier G"
"carrier F"], assumption+)
apply (frule_tac a = "(Ifn F G f) a" and b = "(Ifn F G f) b" in
aHom_add[of "F" "G" "f"], assumption+, simp add:injec_def,
assumption+,
thin_tac "∀a∈carrier F. ∀b∈carrier F. f (a ±⇘F⇙ b) = f a ±⇘G⇙ f b")
apply (simp add:invfun_r[of "f" "carrier F" "carrier G"])
apply (frule_tac x = a and y = b in aGroup.ag_pOp_closed[of "G"], assumption+) apply (frule_tac b = "a ±⇘G⇙ b" in invfun_r[of "f" "carrier F" "carrier G"],
assumption+)
apply (simp add:inj_on_def)
apply (frule_tac x = "(Ifn F G f) a" and y = "(Ifn F G f) b" in
aGroup.ag_pOp_closed, assumption+)
apply (frule_tac x = "(Ifn F G f) (a ±⇘G⇙ b)" in bspec, assumption,
thin_tac "∀x∈carrier F. ∀y∈carrier F. f x = f y ⟶ x = y")
apply (frule_tac x = "(Ifn F G f) a ±⇘F⇙ (Ifn F G f) b" in bspec,
assumption,
thin_tac "∀y∈carrier F.
f ((Ifn F G f) (a ±⇘G⇙ b)) = f y ⟶ (Ifn F G f) (a ±⇘G⇙ b) = y")
apply simp
done
lemma ainvf_bijec:"⟦aGroup F; aGroup G; bijec⇘F,G⇙ f⟧ ⟹ bijec⇘G,F⇙ (ainvf⇘F,G⇙ f)"
apply (subst bijec_def)
apply (simp add:injec_def surjec_def)
apply (simp add:ainvf_aHom)
apply (rule conjI)
apply (rule equalityI)
apply (rule subsetI, simp add:ker_def, erule conjE)
apply (simp add:ainvf_def)
apply (simp add:bijec_def,(erule conjE)+, simp add:surjec_def,
(erule conjE)+, simp add:aHom_def, (erule conjE)+)
apply (frule injec_inj_on[of "F" "G" "f"], assumption+)
apply (subst invfun_r[THEN sym, of "f" "carrier F" "carrier G"], assumption+)
apply (simp add:injec_def, (erule conjE)+, simp add:aHom_0_0)
apply (rule subsetI, simp add:ker_def)
apply (simp add:aGroup.ex_zero)
apply (frule ainvf_aHom[of "F" "G" "f"], assumption+)
apply (simp add:aHom_0_0)
apply (frule ainvf_aHom[of "F" "G" "f"], assumption+,
simp add:aHom_def, (erule conjE)+,
rule surj_to_test[of "ainvf⇘F,G⇙ f" "carrier G" "carrier F"],
assumption+)
apply (rule ballI,
thin_tac "∀a∈carrier G. ∀b∈carrier G.
(ainvf⇘F,G⇙ f) (a ±⇘G⇙ b) = (ainvf⇘F,G⇙ f) a ±⇘F⇙ (ainvf⇘F,G⇙ f) b")
apply (simp add:bijec_def, erule conjE)
apply (frule injec_inj_on[of "F" "G" "f"], assumption+)
apply (simp add:surjec_def aHom_def, (erule conjE)+)
apply (subst ainvf_def)
apply (frule_tac a = b in invfun_l[of "f" "carrier F" "carrier G"],
assumption+,
frule_tac x = b in funcset_mem[of "f" "carrier F" "carrier G"],
assumption+, blast)
done
lemma ainvf_l:"⟦aGroup E; aGroup F; bijec⇘E,F⇙ f; x ∈ carrier E⟧ ⟹
(ainvf⇘E,F⇙ f) (f x) = x"
apply (simp add:bijec_def, erule conjE)
apply (frule injec_inj_on[of "E" "F" "f"], assumption+)
apply (simp add:surjec_def aHom_def, (erule conjE)+)
apply (frule invfun_l[of "f" "carrier E" "carrier F" "x"], assumption+)
apply (simp add:ainvf_def)
done
lemma (in aGroup) aI_aHom:"aI⇘A⇙ ∈ aHom A A"
by (simp add:aHom_def ag_idmap_def ag_idmap_def ag_pOp_closed)
lemma compos_aI_l:"⟦aGroup A; aGroup B; f ∈ aHom A B⟧ ⟹ compos A aI⇘B⇙ f = f"
apply (simp add:compos_def)
apply (rule funcset_eq[of _ "carrier A"])
apply (simp add:compose_def extensional_def)
apply (simp add:aHom_def)
apply (rule ballI)
apply (frule_tac a = x in aHom_mem[of "A" "B" "f"], assumption+)
apply (simp add:compose_def ag_idmap_def)
done
lemma compos_aI_r:"⟦aGroup A; aGroup B; f ∈ aHom A B⟧ ⟹ compos A f aI⇘A⇙ = f"
apply (simp add:compos_def)
apply (rule funcset_eq[of _ "carrier A"])
apply (simp add:compose_def extensional_def)
apply (simp add:aHom_def)
apply (rule ballI)
apply (simp add:compose_def ag_idmap_def)
done
lemma compos_aI_surj:"⟦aGroup A; aGroup B; f ∈ aHom A B; g ∈ aHom B A;
compos A g f = aI⇘A⇙⟧ ⟹ surjec⇘B,A⇙ g"
apply (simp add:surjec_def)
apply (rule surj_to_test[of "g" "carrier B" "carrier A"])
apply (simp add:aHom_def)
apply (rule ballI)
apply (subgoal_tac "compos A g f b = aI⇘A⇙ b",
thin_tac "compos A g f = aI⇘A⇙")
apply (simp add:compos_def compose_def ag_idmap_def)
apply (frule_tac a = b in aHom_mem[of "A" "B" "f"], assumption+, blast)
apply simp
done
lemma compos_aI_inj:"⟦aGroup A; aGroup B; f ∈ aHom A B; g ∈ aHom B A;
compos A g f = aI⇘A⇙⟧ ⟹ injec⇘A,B⇙ f"
apply (simp add:injec_def)
apply (simp add:ker_def)
apply (rule equalityI)
apply (rule subsetI, simp, erule conjE)
apply (subgoal_tac "compos A g f x = aI⇘A⇙ x",
thin_tac "compos A g f = aI⇘A⇙")
apply (simp add:compos_def compose_def)
apply (simp add:aHom_0_0 ag_idmap_def) apply simp
apply (rule subsetI, simp)
apply (simp add:aGroup.ag_inc_zero aHom_0_0)
done
lemma (in aGroup) Ag_ind_aHom:"⟦f ∈ carrier A → D;
bij_to f (carrier A) (D::'d set)⟧ ⟹ Agii A f ∈ aHom A (Ag_ind A f)"
apply (simp add:aHom_def)
apply (unfold bij_to_def, frule conjunct1, frule conjunct2, fold bij_to_def)
apply (simp add:Ag_ind_carrier surj_to_def)
apply (rule conjI)
apply (simp add:Agii_def Pi_def)
apply (simp add:Agii_def)
apply (simp add:Ag_ind_def Pi_def)
apply (unfold bij_to_def, frule conjunct1, fold bij_to_def)
apply (simp add:invfun_l)
apply (simp add:ag_pOp_closed)
done
lemma (in aGroup) Agii_mem:"⟦f ∈ carrier A → D; x ∈ carrier A;
bij_to f (carrier A) (D::'d set)⟧ ⟹ Agii A f x ∈ carrier (Ag_ind A f)"
apply (simp add:Agii_def Ag_ind_carrier)
done
lemma Ag_ind_bijec:"⟦aGroup A; f ∈ carrier A → D;
bij_to f (carrier A) (D::'d set)⟧ ⟹ bijec⇘A, (Ag_ind A f)⇙ (Agii A f)"
apply (frule aGroup.Ag_ind_aHom[of "A" "f" "D"], assumption+)
apply (frule aGroup.Ag_ind_aGroup[of "A" "f" "D"], assumption+)
apply (simp add:bijec_def)
apply (rule conjI)
apply (simp add:injec_def)
apply (rule equalityI)
apply (rule subsetI)
apply (simp add:ker_def, erule conjE)
apply (frule aHom_0_0[of "A" "Ag_ind A f" "Agii A f"], assumption+)
apply (rotate_tac -2, frule sym, thin_tac "Agii A f x = 𝟬⇘Ag_ind A f⇙", simp)
apply (frule aGroup.ag_inc_zero[of "A"], simp add:Agii_def)
apply (unfold bij_to_def, frule conjunct2, fold bij_to_def)
apply (frule aGroup.ag_inc_zero[of "A"])
apply (simp add:injective_iff[THEN sym, of "f" "carrier A" "𝟬⇘A⇙"])
apply (rule subsetI, simp)
apply (subst ker_def, simp)
apply (simp add:aGroup.ag_inc_zero, simp add:aHom_0_0)
apply (subst surjec_def)
apply (unfold bij_to_def, frule conjunct1, fold bij_to_def, simp)
apply (simp add:aGroup.Ag_ind_carrier surj_to_def Agii_def)
done
definition
aimg :: "[('b, 'm1) aGroup_scheme, _, 'b ⇒ 'a]
⇒ 'a aGroup" ("(3aimg⇘_,_⇙ _)" [82,82,83]82) where
"aimg⇘F,A⇙ f = A ⦇ carrier := f ` (carrier F), pop := pop A, mop := mop A,
zero := zero A⦈"
lemma ker_subg:"⟦aGroup F; aGroup G; f ∈ aHom F G ⟧ ⟹ F +> ker⇘F,G⇙ f"
apply (rule aGroup.asubg_test, assumption+)
apply (rule subsetI)
apply (simp add:ker_def)
apply (simp add:ker_def)
apply (frule aHom_0_0 [of "F" "G" "f"], assumption+)
apply (frule aGroup.ex_zero [of "F"]) apply blast
apply (rule ballI)+
apply (simp add:ker_def) apply (erule conjE)+
apply (frule_tac x = b in aGroup.ag_mOp_closed[of "F"], assumption+)
apply (rule conjI)
apply (rule aGroup.ag_pOp_closed, assumption+)
apply (simp add:aHom_add)
apply (simp add:aHom_inv_inv)
apply (simp add:aGroup.ag_inv_zero[of "G"])
apply (cut_tac aGroup.ex_zero[of "G"], simp add:aGroup.l_zero)
apply assumption
done
subsection "Quotient abelian group"
definition
ar_coset :: "['a, _ , 'a set] ⇒ 'a set"
("(3_ ⊎⇘_⇙ _)" [66,66,67]66) where
"ar_coset a A H = H ∙⇘(b_ag A)⇙ a"
definition
set_ar_cos :: "[_ , 'a set] ⇒ 'a set set" where
"set_ar_cos A I = {X. ∃a∈carrier A. X = ar_coset a A I}"
definition
aset_sum :: "[_ , 'a set, 'a set] ⇒ 'a set" where
"aset_sum A H K = s_top (b_ag A) H K"
abbreviation
ASBOP1 (infix "∓ı" 60) where
"H ∓⇘A⇙ K == aset_sum A H K"
lemma (in aGroup) ag_a_in_ar_cos:"⟦A +> H; a ∈ carrier A⟧ ⟹ a ∈ a ⊎⇘A⇙ H"
apply (simp add:ar_coset_def)
apply (simp add:asubGroup_def)
apply (cut_tac b_ag_group)
apply (rule Group.a_in_rcs[of "b_ag A" "H" "a"], assumption+)
apply (simp add:ag_carrier_carrier[THEN sym])
done
lemma (in aGroup) r_cos_subset:"⟦A +> H; X ∈ set_rcs (b_ag A) H⟧ ⟹
X ⊆ carrier A"
apply (simp add:asubGroup_def set_rcs_def)
apply (erule bexE)
apply (cut_tac b_ag_group)
apply (frule_tac a = a in Group.rcs_subset[of "b_ag A" "H"], assumption+)
apply (simp add:ag_carrier_carrier)
done
lemma (in aGroup) asubg_costOp_commute:"⟦A +> H; x ∈ set_rcs (b_ag A) H;
y ∈ set_rcs (b_ag A) H⟧ ⟹
c_top (b_ag A) H x y = c_top (b_ag A) H y x"
apply (simp add:set_rcs_def, (erule bexE)+, simp)
apply (cut_tac b_ag_group)
apply (subst Group.c_top_welldef[THEN sym], assumption+,
simp add:asubg_nsubg,
(simp add:ag_carrier_carrier)+)
apply (subst Group.c_top_welldef[THEN sym], assumption+,
simp add:asubg_nsubg,
(simp add:ag_carrier_carrier)+)
apply (simp add:agop_gop)
apply (simp add:ag_pOp_commute)
done
lemma (in aGroup) Subg_Qgroup:"A +> H ⟹ aGroup (aqgrp A H)"
apply (frule asubg_nsubg[of "H"])
apply (cut_tac b_ag_group)
apply (simp add:aGroup_def)
apply (simp add:aqgrp_def)
apply (simp add:Group.Qg_top [of "b_ag A" "H"])
apply (simp add:Group.Qg_iop [of "b_ag A" "H"])
apply (frule Group.nsg_sg[of "b_ag A" "H"], assumption+,
simp add:Group.unit_rcs_in_set_rcs[of "b_ag A" "H"])
apply (simp add:Group.Qg_tassoc)
apply (simp add:asubg_costOp_commute)
apply (simp add:Group.Qg_i[of "b_ag A" "H"])
apply (simp add:Group.Qg_unit[of "b_ag A" "H"])
done
lemma (in aGroup) plus_subgs:"⟦A +> H1; A +> H2⟧ ⟹ A +> H1 ∓ H2"
apply (simp add:aset_sum_def)
apply (frule asubg_nsubg[of "H2"])
apply (simp add:asubGroup_def[of _ "H1"])
apply (cut_tac "b_ag_group")
apply (frule Group.smult_sg_nsg[of "b_ag A" "H1" "H2"], assumption+)
apply (simp add:asubGroup_def)
done
lemma (in aGroup) set_sum:"⟦H ⊆ carrier A; K ⊆ carrier A⟧ ⟹
H ∓ K = {x. ∃h∈H. ∃k∈K. x = h ± k}"
apply (cut_tac b_ag_group)
apply (rule equalityI)
apply (rule subsetI)
apply (simp add:aset_sum_def)
apply (simp add:agop_gop[THEN sym] s_top_def, (erule bexE)+,
frule sym, thin_tac "xa ⋅⇘b_ag A⇙ y = x", simp, blast)
apply (rule subsetI, simp add:aset_sum_def, (erule bexE)+)
apply (frule_tac c = h in subsetD[of H "carrier A"], assumption+,
frule_tac c = k in subsetD[of K "carrier A"], assumption+)
apply (simp add:agop_gop[THEN sym], simp add:s_top_def, blast)
done
lemma (in aGroup) mem_set_sum:"⟦H ⊆ carrier A; K ⊆ carrier A;
x ∈ H ∓ K ⟧ ⟹ ∃h∈H. ∃k∈K. x = h ± k"
by (simp add:set_sum)
lemma (in aGroup) mem_sum_subgs:"⟦A +> H; A +> K; h ∈ H; k ∈ K⟧ ⟹
h ± k ∈ H ∓ K"
apply (frule asubg_subset[of H],
frule asubg_subset[of K],
simp add:set_sum, blast)
done
lemma (in aGroup) aqgrp_carrier:"A +> H ⟹
set_rcs (b_ag A ) H = set_ar_cos A H"
apply (simp add:set_ar_cos_def)
apply (simp add:ag_carrier_carrier [THEN sym])
apply (simp add:ar_coset_def set_rcs_def)
done
lemma (in aGroup) unit_in_set_ar_cos:"A +> H ⟹ H ∈ set_ar_cos A H"
apply (simp add:aqgrp_carrier[THEN sym])
apply (cut_tac b_ag_group) apply (simp add:asubGroup_def)
apply (simp add:Group.unit_rcs_in_set_rcs[of "b_ag A" "H"])
done
lemma (in aGroup) aqgrp_pOp_maps:"⟦A +> H; a ∈ carrier A; b ∈ carrier A⟧ ⟹
pop (aqgrp A H) (a ⊎⇘A⇙ H) (b ⊎⇘A⇙ H) = (a ± b) ⊎⇘A⇙ H"
apply (simp add:aqgrp_def ar_coset_def)
apply (cut_tac b_ag_group)
apply (frule asubg_nsubg)
apply (simp add:ag_carrier_carrier [THEN sym])
apply (subst Group.c_top_welldef [THEN sym], assumption+)
apply (simp add:agop_gop)
done
lemma (in aGroup) aqgrp_mOp_maps:"⟦A +> H; a ∈ carrier A⟧ ⟹
mop (aqgrp A H) (a ⊎⇘A⇙ H) = (-⇩a a) ⊎⇘A⇙ H"
apply (simp add:aqgrp_def ar_coset_def)
apply (cut_tac b_ag_group)
apply (frule asubg_nsubg)
apply (simp add:ag_carrier_carrier [THEN sym])
apply (subst Group.c_iop_welldef, assumption+)
apply (simp add:agiop_giop)
done
lemma (in aGroup) aqgrp_zero:"A +> H ⟹ zero (aqgrp A H) = H"
apply (simp add:aqgrp_def)
done
lemma (in aGroup) arcos_fixed:"⟦A +> H; a ∈ carrier A; h ∈ H ⟧ ⟹
a ⊎⇘A⇙ H = (h ± a) ⊎⇘A⇙ H"
apply (cut_tac b_ag_group)
apply (simp add:agop_gop[THEN sym])
apply (simp add:ag_carrier_carrier[THEN sym])
apply (simp add:ar_coset_def)
apply (simp add:asubGroup_def)
apply (simp add:Group.rcs_fixed1[of "b_ag A" "H"])
done
definition
rind_hom :: "[('a, 'more) aGroup_scheme, ('b, 'more1) aGroup_scheme,
('a ⇒ 'b)] ⇒ ('a set ⇒ 'b )" where
"rind_hom A B f = (λX∈(set_ar_cos A (ker⇘A,B⇙ f)). f (SOME x. x ∈ X))"
abbreviation
RIND_HOM ("(3_°⇘_,_⇙)" [82,82,83]82) where
"f°⇘F,G⇙ == rind_hom F G f"
section "Direct product and direct sum of abelian groups, in general case"
definition
Un_carrier :: "['i set, 'i ⇒ ('a, 'more) aGroup_scheme] ⇒ 'a set" where
"Un_carrier I A = ⋃{X. ∃i∈I. X = carrier (A i)}"
definition
carr_prodag :: "['i set, 'i ⇒ ('a, 'more) aGroup_scheme] ⇒ ('i ⇒ 'a ) set" where
"carr_prodag I A = {f. f ∈ extensional I ∧ f ∈ I → (Un_carrier I A) ∧
(∀i∈I. f i ∈ carrier (A i))}"
definition
prod_pOp :: "['i set, 'i ⇒ ('a, 'more) aGroup_scheme] ⇒
('i ⇒ 'a) ⇒ ('i ⇒ 'a) ⇒ ('i ⇒ 'a)" where
"prod_pOp I A = (λf∈carr_prodag I A. λg∈carr_prodag I A.
λx∈I. (f x) ±⇘(A x)⇙ (g x))"
definition
prod_mOp :: "['i set, 'i ⇒ ('a, 'more) aGroup_scheme] ⇒
('i ⇒ 'a) ⇒ ('i ⇒ 'a)" where
"prod_mOp I A = (λf∈carr_prodag I A. λx∈I. (-⇩a⇘(A x)⇙ (f x)))"
definition
prod_zero :: "['i set, 'i ⇒ ('a, 'more) aGroup_scheme] ⇒ ('i ⇒ 'a)" where
"prod_zero I A = (λx∈I. 𝟬⇘(A x)⇙)"
definition
prodag :: "['i set, 'i ⇒ ('a, 'more) aGroup_scheme] ⇒ ('i ⇒ 'a) aGroup" where
"prodag I A = ⦇ carrier = carr_prodag I A,
pop = prod_pOp I A, mop = prod_mOp I A,
zero = prod_zero I A⦈"
definition
PRoject :: "['i set, 'i ⇒ ('a, 'more) aGroup_scheme, 'i]
⇒ ('i ⇒ 'a) ⇒ 'a" ("(3π⇘_,_,_⇙)" [82,82,83]82) where
"PRoject I A x = (λf ∈ carr_prodag I A. f x)"
abbreviation
PRODag ("(aΠ⇘_⇙ _)" [72,73]72) where
"aΠ⇘I⇙ A == prodag I A"
lemma prodag_comp_i:"⟦a ∈ carr_prodag I A; i ∈ I⟧ ⟹ (a i) ∈ carrier (A i)"
by (simp add:carr_prodag_def)
lemma prod_pOp_func:"∀k∈I. aGroup (A k) ⟹
prod_pOp I A ∈ carr_prodag I A → carr_prodag I A → carr_prodag I A"
apply (rule Pi_I)+
apply(rename_tac a b)
apply (subst carr_prodag_def) apply (simp add:CollectI)
apply (rule conjI)
apply (simp add:prod_pOp_def restrict_def extensional_def)
apply (rule conjI)
apply (rule Pi_I)
apply (simp add:prod_pOp_def)
apply (subst Un_carrier_def) apply (simp add:CollectI)
apply (frule_tac x = x in bspec, assumption,
thin_tac "∀k∈I. aGroup (A k)")
apply (simp add:carr_prodag_def) apply (erule conjE)+
apply (thin_tac "a ∈ I → Un_carrier I A")
apply (thin_tac "b ∈ I → Un_carrier I A")
apply (frule_tac x = x in bspec, assumption,
thin_tac "∀i∈I. a i ∈ carrier (A i)",
frule_tac x = x in bspec, assumption,
thin_tac "∀i∈I. b i ∈ carrier (A i)")
apply (frule_tac x = "a x" and y = "b x" in aGroup.ag_pOp_closed, assumption+)
apply blast
apply (rule ballI)
apply (simp add:prod_pOp_def)
apply (rule_tac A = "A i" and x = "a i" and y = "b i" in aGroup.ag_pOp_closed)
apply simp
apply (simp add:carr_prodag_def)+
done
lemma prod_pOp_mem:"⟦∀k∈I. aGroup (A k); X ∈ carr_prodag I A;
Y ∈ carr_prodag I A⟧ ⟹ prod_pOp I A X Y ∈ carr_prodag I A"
apply (frule prod_pOp_func)
apply (frule funcset_mem[of "prod_pOp I A"
"carr_prodag I A" "carr_prodag I A → carr_prodag I A"
"X"], assumption+)
apply (rule funcset_mem[of "prod_pOp I A X" "carr_prodag I A"
"carr_prodag I A" "Y"], assumption+)
done
lemma prod_pOp_mem_i:"⟦∀k∈I. aGroup (A k); X ∈ carr_prodag I A;
Y ∈ carr_prodag I A; i ∈ I⟧ ⟹ prod_pOp I A X Y i = (X i) ±⇘(A i)⇙ (Y i)"
apply (simp add:prod_pOp_def)
done
lemma prod_mOp_func:"∀k∈I. aGroup (A k) ⟹
prod_mOp I A ∈ carr_prodag I A → carr_prodag I A"
apply (rule Pi_I)
apply (simp add:prod_mOp_def carr_prodag_def)
apply (erule conjE)+
apply (rule conjI)
apply (rule Pi_I) apply simp
apply (rename_tac f j)
apply (frule_tac f = f and x = j in funcset_mem [of _ "I" "Un_carrier I A"],
assumption+)
apply (thin_tac "f ∈ I → Un_carrier I A")
apply (frule_tac x = j in bspec, assumption,
thin_tac "∀k∈I. aGroup (A k)",
frule_tac x = j in bspec, assumption,
thin_tac "∀i∈I. f i ∈ carrier (A i)")
apply (thin_tac "f j ∈ Un_carrier I A")
apply (simp add:Un_carrier_def)
apply (frule aGroup.ag_mOp_closed, assumption+)
apply blast
apply (rule ballI)
apply (rule_tac A = "A i" and x = "x i" in aGroup.ag_mOp_closed)
apply simp+
done
lemma prod_mOp_mem:"⟦∀j∈I. aGroup (A j); X ∈ carr_prodag I A⟧ ⟹
prod_mOp I A X ∈ carr_prodag I A"
apply (frule prod_mOp_func)
apply (simp add:Pi_def)
done
lemma prod_mOp_mem_i:"⟦∀j∈I. aGroup (A j); X ∈ carr_prodag I A; i ∈ I⟧ ⟹
prod_mOp I A X i = -⇩a⇘(A i)⇙ (X i)"
apply (simp add:prod_mOp_def)
done
lemma prod_zero_func:"∀k∈I. aGroup (A k) ⟹
prod_zero I A ∈ carr_prodag I A"
apply (simp add:prod_zero_def prodag_def)
apply (simp add:carr_prodag_def)
apply (rule conjI)
apply (rule Pi_I) apply simp
apply (subgoal_tac "aGroup (A x)") prefer 2 apply simp
apply (thin_tac "∀k∈I. aGroup (A k)")
apply (simp add:Un_carrier_def)
apply (frule aGroup.ex_zero)
apply auto
apply (frule_tac x = i in bspec, assumption,
thin_tac "∀k∈I. aGroup (A k)")
apply (simp add:aGroup.ex_zero)
done
lemma prod_zero_i:"⟦∀k∈I. aGroup (A k); i ∈ I⟧ ⟹
prod_zero I A i = 𝟬⇘(A i)⇙ "
by (simp add:prod_zero_def)
lemma carr_prodag_mem_eq:"⟦∀k∈I. aGroup (A k); X ∈ carr_prodag I A;
Y ∈ carr_prodag I A; ∀l∈I. (X l) = (Y l) ⟧ ⟹ X = Y"
apply (simp add:carr_prodag_def)
apply (erule conjE)+
apply (simp add:funcset_eq)
done
lemma prod_pOp_assoc:"⟦∀k∈I. aGroup (A k); a ∈ carr_prodag I A;
b ∈ carr_prodag I A; c ∈ carr_prodag I A⟧ ⟹
prod_pOp I A (prod_pOp I A a b) c =
prod_pOp I A a (prod_pOp I A b c)"
apply (frule_tac X = a and Y = b in prod_pOp_mem[of "I" "A"], assumption+,
frule_tac X = b and Y = c in prod_pOp_mem[of "I" "A"], assumption+,
frule_tac X = "prod_pOp I A a b" and Y = c in prod_pOp_mem[of "I"
"A"], assumption+,
frule_tac X = a and Y = "prod_pOp I A b c" in prod_pOp_mem[of "I"
"A"], assumption+)
apply (rule carr_prodag_mem_eq[of "I" "A"], assumption+,
rule ballI)
apply (simp add:prod_pOp_mem_i)
apply (frule_tac x = l in bspec, assumption,
thin_tac "∀k∈I. aGroup (A k)")
apply (rule aGroup.ag_pOp_assoc, assumption)
apply (simp add:prodag_comp_i)+
done
lemma prod_pOp_commute:"⟦∀k∈I. aGroup (A k); a ∈ carr_prodag I A;
b ∈ carr_prodag I A⟧ ⟹
prod_pOp I A a b = prod_pOp I A b a"
apply (frule_tac X = a and Y = b in prod_pOp_mem[of "I" "A"], assumption+,
frule_tac X = b and Y = a in prod_pOp_mem[of "I" "A"], assumption+)
apply (rule carr_prodag_mem_eq[of "I" "A"], assumption+,
rule ballI)
apply (simp add:prod_pOp_mem_i)
apply (frule_tac x = l in bspec, assumption,
thin_tac "∀k∈I. aGroup (A k)",
rule aGroup.ag_pOp_commute, assumption)
apply (simp add:prodag_comp_i)+
done
lemma prodag_aGroup:"∀k∈I. aGroup (A k) ⟹ aGroup (prodag I A)"
apply (simp add:aGroup_def [of "(prodag I A)"])
apply (simp add:prodag_def)
apply (simp add:prod_pOp_func)
apply (simp add:prod_mOp_func)
apply (simp add:prod_zero_func)
apply (rule conjI)
apply (rule allI, rule impI)+
apply (simp add:prod_pOp_assoc)
apply (rule conjI)
apply (rule allI, rule impI)+
apply (simp add:prod_pOp_commute)
apply (rule conjI)
apply (rule allI, rule impI)
apply (frule_tac X = a in prod_mOp_mem [of "I" "A"], assumption+)
apply (frule_tac X = "prod_mOp I A a" and Y = a in prod_pOp_mem[of "I" "A"],
assumption+)
apply (rule carr_prodag_mem_eq[of "I" "A"], assumption+)
apply (simp add:prod_zero_func)
apply (rule ballI)
apply (simp add:prod_pOp_mem_i,
simp add:prod_zero_i) apply (
simp add:prod_mOp_mem_i)
apply (frule_tac x = l in bspec, assumption,
thin_tac "∀k∈I. aGroup (A k)",
rule aGroup.l_m, assumption+, simp add:prodag_comp_i)
apply (rule allI, rule impI)
apply (frule_tac prod_zero_func[of "I" "A"],
frule_tac Y = a in prod_pOp_mem[of "I" "A" "prod_zero I A"],
assumption+)
apply (rule carr_prodag_mem_eq[of "I" "A"], assumption+)
apply (rule ballI)
apply (subst prod_pOp_mem_i[of "I" "A"], assumption+,
subst prod_zero_i[of "I" "A"], assumption+)
apply (frule_tac x = l in bspec, assumption,
rule aGroup.l_zero, assumption+,
simp add:prodag_comp_i)
done
lemma prodag_carrier:"∀k∈I. aGroup (A k) ⟹
carrier (prodag I A) = carr_prodag I A"
by (simp add:prodag_def)
lemma prodag_elemfun:"⟦∀k∈I. aGroup (A k); f ∈ carrier (prodag I A)⟧ ⟹
f ∈ extensional I"
apply (simp add:prodag_carrier)
apply (simp add:carr_prodag_def)
done
lemma prodag_component:"⟦f ∈ carrier (prodag I A); i ∈ I ⟧ ⟹
f i ∈ carrier (A i)"
by (simp add:prodag_def carr_prodag_def)
lemma prodag_pOp:"∀k∈I. aGroup (A k) ⟹
pop (prodag I A) = prod_pOp I A"
apply (simp add:prodag_def)
done
lemma prodag_iOp:"∀k∈I. aGroup (A k) ⟹
mop (prodag I A) = prod_mOp I A"
apply (simp add:prodag_def)
done
lemma prodag_zero:"∀k∈I. aGroup (A k) ⟹
zero (prodag I A) = prod_zero I A"
apply (simp add:prodag_def)
done
lemma prodag_sameTr0:"⟦∀k∈I. aGroup (A k); ∀k∈I. A k = B k⟧
⟹ Un_carrier I A = Un_carrier I B"
apply (simp add:Un_carrier_def)
done
lemma prodag_sameTr1:"⟦∀k∈I. aGroup (A k); ∀k∈I. A k = B k⟧
⟹ carr_prodag I A = carr_prodag I B"
apply (rule equalityI)
apply (rule subsetI)
apply (simp add:carr_prodag_def, (erule conjE)+)
apply (rule Pi_I)
apply (subst Un_carrier_def, simp, blast)
apply (rule subsetI)
apply (simp add:carr_prodag_def, (erule conjE)+)
apply (rule Pi_I)
apply (subst Un_carrier_def, simp)
apply blast
done
lemma prodag_sameTr2:"⟦∀k∈I. aGroup (A k); ∀k∈I. A k = B k⟧
⟹ prod_pOp I A = prod_pOp I B"
apply (frule prodag_sameTr1 [of "I" "A" "B"], assumption+)
apply (simp add:prod_pOp_def)
apply (rule bivar_func_eq)
apply (rule ballI)+
apply (rule funcset_eq [of _ "I"])
apply (simp add:restrict_def extensional_def)+
done
lemma prodag_sameTr3:"⟦∀k∈I. aGroup (A k); ∀k∈I. A k = B k⟧
⟹ prod_mOp I A = prod_mOp I B"
apply (frule prodag_sameTr1 [of "I" "A" "B"], assumption+)
apply (simp add:prod_mOp_def)
apply (rule funcset_eq [of _ "carr_prodag I B"])
apply (simp add:restrict_def extensional_def)
apply (simp add:restrict_def extensional_def)
apply (rule ballI)
apply (rename_tac g) apply simp
apply (rule funcset_eq [of _ "I"])
apply (simp add:restrict_def extensional_def)+
done
lemma prodag_sameTr4:"⟦∀k∈I. aGroup (A k); ∀k∈I. A k = B k⟧
⟹ prod_zero I A = prod_zero I B"
apply (simp add:prod_zero_def)
apply (rule funcset_eq [of _ "I"])
apply (simp add:restrict_def extensional_def)+
done
lemma prodag_same:"⟦∀k∈I. aGroup (A k); ∀k∈I. A k = B k⟧
⟹ prodag I A = prodag I B"
apply (frule prodag_sameTr1, assumption+)
apply (frule prodag_sameTr2, assumption+)
apply (frule prodag_sameTr3, assumption+)
apply (frule prodag_sameTr4, assumption+)
apply (simp add:prodag_def)
done
lemma project_mem:"⟦∀k∈I. aGroup (A k); j ∈ I; x ∈ carrier (prodag I A)⟧ ⟹
(PRoject I A j) x ∈ carrier (A j)"
apply (simp add:PRoject_def)
apply (simp add:prodag_def)
apply (simp add:carr_prodag_def)
done
lemma project_aHom:"⟦∀k∈I. aGroup (A k); j ∈ I⟧ ⟹
PRoject I A j ∈ aHom (prodag I A) (A j)"
apply (simp add:aHom_def)
apply (rule conjI)
apply (simp add:project_mem)
apply (rule conjI)
apply (simp add:PRoject_def restrict_def extensional_def)
apply (rule allI, rule impI, simp add:prodag_def)
apply (rule ballI)+
apply (simp add:prodag_def)
apply (simp add:prod_pOp_def)
apply (frule_tac X = a and Y = b in prod_pOp_mem[of I A], assumption+)
apply (simp add:prod_pOp_def)
apply (simp add:PRoject_def)
done
lemma project_aHom1:"∀k∈I. aGroup (A k) ⟹
∀j ∈ I. PRoject I A j ∈ aHom (prodag I A) (A j)"
apply (rule ballI)
apply (rule project_aHom, assumption+)
done
definition
A_to_prodag :: "[('a, 'm) aGroup_scheme, 'i set, 'i ⇒('a ⇒ 'b),
'i ⇒ ('b, 'm1) aGroup_scheme] ⇒ ('a ⇒ ('i ⇒'b))" where
"A_to_prodag A I S B = (λa∈carrier A. λk∈I. S k a)"
lemma A_to_prodag_mem:"⟦aGroup A; ∀k∈I. aGroup (B k); ∀k∈I. (S k) ∈
aHom A (B k); x ∈ carrier A ⟧ ⟹ A_to_prodag A I S B x ∈ carr_prodag I B"
apply (simp add:carr_prodag_def)
apply (rule conjI)
apply (simp add:A_to_prodag_def extensional_def restrict_def)
apply (simp add:Pi_def restrict_def A_to_prodag_def)
apply (rule conjI)
apply (rule allI) apply (rule impI)
apply (simp add:Un_carrier_def)
apply (rotate_tac 2,
frule_tac x = xa in bspec, assumption,
thin_tac "∀k∈I. S k ∈ aHom A (B k)")
apply (simp add:aHom_def) apply (erule conjE)+
apply (frule_tac f = "S xa" and A = "carrier A" and B = "carrier (B xa)"
and x = x in funcset_mem, assumption+)
apply blast
apply (rule ballI)
apply (rotate_tac 2,
frule_tac x = i in bspec, assumption,
thin_tac "∀k∈I. S k ∈ aHom A (B k)")
apply (simp add:aHom_def) apply (erule conjE)+
apply (simp add:Pi_def)
done
lemma A_to_prodag_aHom:"⟦aGroup A; ∀k∈I. aGroup (B k); ∀k∈I. (S k) ∈
aHom A (B k) ⟧ ⟹ A_to_prodag A I S B ∈ aHom A (aΠ⇘I⇙ B)"
apply (simp add:aHom_def [of "A" "aΠ⇘I⇙ B"])
apply (rule conjI)
apply (simp add:prodag_def A_to_prodag_mem)
apply (rule conjI)
apply (simp add:A_to_prodag_def restrict_def extensional_def)
apply (rule ballI)+
apply (frule_tac x = a and y = b in aGroup.ag_pOp_closed, assumption+)
apply (frule_tac x = "a ±⇘A⇙ b" in A_to_prodag_mem [of "A" "I" "B" "S"],
assumption+)
apply (frule_tac x = a in A_to_prodag_mem [of "A" "I" "B" "S"],
assumption+)
apply (frule_tac x = b in A_to_prodag_mem [of "A" "I" "B" "S"],
assumption+)
apply (frule prodag_aGroup [of "I" "B"])
apply (frule_tac x = a in A_to_prodag_mem[of "A" "I" "B" "S"], assumption+,
frule_tac x = b in A_to_prodag_mem[of "A" "I" "B" "S"], assumption+,
frule_tac x = "a ±⇘A⇙ b" in A_to_prodag_mem[of "A" "I" "B" "S"],
assumption+)
apply (frule prodag_aGroup[of "I" "B"],
frule_tac x = "A_to_prodag A I S B a" and
y = "A_to_prodag A I S B b" in aGroup.ag_pOp_closed [of "aΠ⇘I⇙ B"])
apply (simp add:prodag_carrier)
apply (simp add:prodag_carrier)
apply (rule carr_prodag_mem_eq, assumption+)
apply (simp add:prodag_carrier)
apply (rule ballI)
apply (simp add:A_to_prodag_def prod_pOp_def)
apply (rotate_tac 2,
frule_tac x = l in bspec, assumption,
thin_tac "∀k∈I. S k ∈ aHom A (B k)")
apply (simp add:prodag_def prod_pOp_def)
apply (frule_tac x = l in bspec, assumption,
thin_tac "∀k∈I. aGroup (B k)")
apply (simp add: aHom_add)
done
definition
finiteHom :: "['i set, 'i ⇒ ('a, 'more) aGroup_scheme, 'i ⇒ 'a] ⇒ bool" where
"finiteHom I A f ⟷ f ∈ carr_prodag I A ∧ (∃H. H ⊆ I ∧ finite H ∧ (
∀j ∈ (I - H). (f j) = 𝟬⇘(A j)⇙))"
definition
carr_dsumag :: "['i set, 'i ⇒ ('a, 'more) aGroup_scheme] ⇒ ('i ⇒ 'a ) set" where
"carr_dsumag I A = {f. finiteHom I A f}"
definition
dsumag :: "['i set, 'i ⇒ ('a, 'more) aGroup_scheme] ⇒ ('i ⇒ 'a) aGroup" where
"dsumag I A = ⦇ carrier = carr_dsumag I A,
pop = prod_pOp I A, mop = prod_mOp I A,
zero = prod_zero I A⦈"
definition
dProj :: "['i set, 'i ⇒ ('a, 'more) aGroup_scheme, 'i]
⇒ ('i ⇒ 'a) ⇒ 'a" where
"dProj I A x = (λf∈carr_dsumag I A. f x)"
abbreviation
DSUMag ("(a⨁⇘_⇙ _)" [72,73]72) where
"a⨁⇘I⇙ A == dsumag I A"
lemma dsum_pOp_func:"∀k∈I. aGroup (A k) ⟹
prod_pOp I A ∈ carr_dsumag I A → carr_dsumag I A → carr_dsumag I A"
apply (rule Pi_I)+
apply (subst carr_dsumag_def) apply (simp add:CollectI)
apply (simp add:finiteHom_def)
apply (rule conjI)
apply (simp add:carr_dsumag_def) apply (simp add:finiteHom_def)
apply (erule conjE)+ apply (simp add:prod_pOp_mem)
apply (simp add:carr_dsumag_def finiteHom_def) apply (erule conjE)+
apply ((erule exE)+, (erule conjE)+)
apply (frule_tac F = H and G = Ha in finite_UnI, assumption+)
apply (subgoal_tac "∀j∈I - (H ∪ Ha). prod_pOp I A x xa j = 𝟬⇘A j⇙")
apply (frule_tac A = H and B = Ha in Un_least[of _ "I"], assumption+)
apply blast
apply (rule ballI)
apply (simp, (erule conjE)+)
apply (frule_tac x = j in bspec, assumption,
thin_tac "∀k∈I. aGroup (A k)",
frule_tac x = j in bspec, simp,
thin_tac "∀j∈I - H. x j = 𝟬⇘A j⇙",
frule_tac x = j in bspec, simp,
thin_tac "∀j∈I - Ha. xa j = 𝟬⇘A j⇙")
apply (simp add:prod_pOp_def)
apply (rule aGroup.ag_l_zero) apply simp
apply (rule aGroup.ex_zero) apply assumption
done
lemma dsum_pOp_mem:"⟦∀k∈I. aGroup (A k); X ∈ carr_dsumag I A;
Y ∈ carr_dsumag I A⟧ ⟹ prod_pOp I A X Y ∈ carr_dsumag I A"
apply (frule dsum_pOp_func[of "I" "A"])
apply (frule funcset_mem[of "prod_pOp I A" "carr_dsumag I A"
"carr_dsumag I A → carr_dsumag I A" "X"], assumption+)
apply (rule funcset_mem[of "prod_pOp I A X" "carr_dsumag I A"
"carr_dsumag I A" "Y"], assumption+)
done
lemma dsum_iOp_func:"∀k∈I. aGroup (A k) ⟹
prod_mOp I A ∈ carr_dsumag I A → carr_dsumag I A"
apply (rule Pi_I)
apply (simp add:carr_dsumag_def) apply (simp add:finiteHom_def)
apply (erule conjE)+ apply (simp add:prod_mOp_mem)
apply (erule exE, (erule conjE)+)
apply (simp add:prod_mOp_def)
apply (subgoal_tac "∀j∈I - H. -⇩a⇘A j⇙ (x j) = 𝟬⇘A j⇙")
apply blast
apply (rule ballI)
apply (frule_tac x = j in bspec, simp,
thin_tac "∀k∈I. aGroup (A k)",
frule_tac x = j in bspec, simp,
thin_tac "∀j∈I - H. x j = 𝟬⇘A j⇙", simp add:aGroup.ag_inv_zero)
done
lemma dsum_iOp_mem:"⟦∀j∈I. aGroup (A j); X ∈ carr_dsumag I A⟧ ⟹
prod_mOp I A X ∈ carr_dsumag I A"
apply (frule dsum_iOp_func)
apply (simp add:Pi_def)
done
lemma dsum_zero_func:"∀k∈I. aGroup (A k) ⟹
prod_zero I A ∈ carr_dsumag I A"
apply (simp add:carr_dsumag_def) apply (simp add:finiteHom_def)
apply (rule conjI) apply (simp add:prod_zero_func)
apply (subgoal_tac "{} ⊆ I") prefer 2 apply simp
apply (subgoal_tac "finite {}") prefer 2 apply simp
apply (subgoal_tac "∀j∈I - {}. prod_zero I A j = 𝟬⇘A j⇙")
apply blast
apply (rule ballI) apply simp
apply (simp add:prod_zero_def)
done
lemma dsumag_sub_prodag:"∀k∈I. aGroup (A k) ⟹
carr_dsumag I A ⊆ carr_prodag I A"
by (rule subsetI,
simp add:carr_dsumag_def finiteHom_def)
lemma carrier_dsumag:"∀k∈I. aGroup (A k) ⟹
carrier (dsumag I A) = carr_dsumag I A"
apply (simp add:dsumag_def)
done
lemma dsumag_elemfun:"⟦∀k∈I. aGroup (A k); f ∈ carrier (dsumag I A)⟧ ⟹
f ∈ extensional I"
apply (simp add:carrier_dsumag)
apply (simp add:carr_dsumag_def) apply (simp add:finiteHom_def)
apply (erule conjE) apply (simp add:carr_prodag_def)
done
lemma dsumag_aGroup:"∀k∈I. aGroup (A k) ⟹ aGroup (dsumag I A)"
apply (simp add:aGroup_def [of "dsumag I A"])
apply (simp add:dsumag_def)
apply (simp add:dsum_pOp_func)
apply (simp add:dsum_iOp_func)
apply (simp add:dsum_zero_func)
apply (frule dsumag_sub_prodag[of "I" "A"])
apply (rule conjI)
apply (rule allI, rule impI)+
apply (frule_tac X = a and Y = b in dsum_pOp_mem, assumption+)
apply (frule_tac X = b and Y = c in dsum_pOp_mem, assumption+)
apply (frule_tac X = "prod_pOp I A a b" and Y = c in dsum_pOp_mem,
assumption+)
apply (frule_tac Y = "prod_pOp I A b c" and X = a in dsum_pOp_mem,
assumption+)
apply (rule carr_prodag_mem_eq [of "I" "A"], assumption+)
apply (simp add:subsetD) apply (simp add:subsetD)
apply (rule ballI)
apply (subst prod_pOp_mem_i, assumption+, (simp add:subsetD)+)
apply (subst prod_pOp_mem_i, assumption+)
apply (simp add:subsetD)+
apply (subst prod_pOp_mem_i, assumption+, (simp add:subsetD)+)
apply (subst prod_pOp_mem_i, assumption+) apply (simp add:subsetD)+
apply (thin_tac "prod_pOp I A a b ∈ carr_dsumag I A",
thin_tac "prod_pOp I A b c ∈ carr_dsumag I A",
thin_tac "prod_pOp I A (prod_pOp I A a b) c ∈ carr_dsumag I A",
thin_tac "prod_pOp I A a (prod_pOp I A b c) ∈ carr_dsumag I A",
thin_tac "carr_dsumag I A ⊆ carr_prodag I A")
apply (frule_tac x = l in bspec, assumption,
thin_tac "∀k∈I. aGroup (A k)",
simp add:carr_dsumag_def finiteHom_def, (erule conjE)+,
simp add:carr_prodag_def, (erule conjE)+)
apply (frule_tac x = l in bspec, assumption,
thin_tac "∀i∈I. a i ∈ carrier (A i)",
frule_tac x = l in bspec, assumption,
thin_tac "∀i∈I. b i ∈ carrier (A i)",
frule_tac x = l in bspec, assumption,
thin_tac "∀i∈I. c i ∈ carrier (A i)")
apply (simp add:aGroup.aassoc)
apply (rule conjI)
apply (rule allI, rule impI)+
apply (rule carr_prodag_mem_eq [of "I" "A"], assumption+)
apply (frule_tac X = a and Y = b in prod_pOp_mem[of "I" "A"],
(simp add:subsetD)+)
apply (frule_tac X = b and Y = a in prod_pOp_mem[of "I" "A"],
(simp add:subsetD)+)
apply (rule ballI,
subst prod_pOp_mem_i, assumption+, (simp add:subsetD)+)
apply (subst prod_pOp_mem_i, assumption+, (simp add:subsetD)+)
apply (frule_tac x = l in bspec, assumption,
thin_tac "∀k∈I. aGroup (A k)")
apply (frule_tac c = a in subsetD[of "carr_dsumag I A" "carr_prodag I A"],
assumption+, thin_tac "a ∈ carr_dsumag I A",
frule_tac c = b in subsetD[of "carr_dsumag I A" "carr_prodag I A"],
assumption+, thin_tac "b ∈ carr_dsumag I A",
thin_tac "carr_dsumag I A ⊆ carr_prodag I A")
apply (simp add:carr_prodag_def, (erule conjE)+,
simp add:aGroup.ag_pOp_commute)
apply (rule conjI)
apply (rule allI, rule impI)
apply (frule_tac X = a in prod_mOp_mem[of "I" "A"],
simp add:subsetD)
apply (frule_tac X = "prod_mOp I A a" and Y = a in prod_pOp_mem[of "I" "A"],
simp add:subsetD, simp add:subsetD)
apply (rule carr_prodag_mem_eq [of "I" "A"], assumption+,
simp add:prod_zero_func)
apply (rule ballI)
apply (subst prod_pOp_mem_i, assumption+,
simp add:subsetD, assumption)
apply (subst prod_mOp_mem_i, assumption+, simp add:subsetD, assumption)
apply (simp add:prod_zero_i)
apply (frule_tac x = l in bspec, assumption,
thin_tac "∀k∈I. aGroup (A k)",
thin_tac "prod_mOp I A a ∈ carr_prodag I A",
thin_tac "prod_pOp I A (prod_mOp I A a) a ∈ carr_prodag I A",
frule_tac c = a in subsetD[of "carr_dsumag I A" "carr_prodag I A"],
assumption,
thin_tac "carr_dsumag I A ⊆ carr_prodag I A",
simp add:carr_prodag_def, (erule conjE)+)
apply (frule_tac x = l in bspec, assumption,
thin_tac "∀i∈I. a i ∈ carrier (A i)")
apply (rule aGroup.l_m, assumption+)
apply (rule allI, rule impI)
apply (frule prod_zero_func[of "I" "A"])
apply (frule_tac X = "prod_zero I A" and Y = a in prod_pOp_mem[of "I" "A"],
assumption+, simp add:subsetD)
apply (rule carr_prodag_mem_eq [of "I" "A"], assumption+,
simp add:subsetD)
apply (rule ballI)
apply (subst prod_pOp_mem_i, assumption+)
apply (simp add:subsetD, assumption)
apply (simp add:prod_zero_i,
frule_tac x = l in bspec, assumption,
thin_tac "∀k∈I. aGroup (A k)",
frule_tac c = a in subsetD[of "carr_dsumag I A" "carr_prodag I A"],
assumption+,
thin_tac "carr_dsumag I A ⊆ carr_prodag I A",
thin_tac "a ∈ carr_dsumag I A",
thin_tac "prod_pOp I A (prod_zero I A) a ∈ carr_prodag I A")
apply (simp add:carr_prodag_def, (erule conjE)+)
apply (rule aGroup.l_zero, assumption)
apply blast
done
lemma dsumag_pOp:"∀k∈I. aGroup (A k) ⟹
pop (dsumag I A) = prod_pOp I A"
apply (simp add:dsumag_def)
done
lemma dsumag_mOp:"∀k∈I. aGroup (A k) ⟹
mop (dsumag I A) = prod_mOp I A"
apply (simp add:dsumag_def)
done
lemma dsumag_zero:"∀k∈I. aGroup (A k) ⟹
zero (dsumag I A) = prod_zero I A"
apply (simp add:dsumag_def)
done
subsection "Characterization of a direct product"
lemma direct_prod_mem_eq:"⟦∀j∈I. aGroup (A j); f ∈ carrier (aΠ⇘I⇙ A);
g ∈ carrier (aΠ⇘I⇙ A); ∀j∈I. (PRoject I A j) f = (PRoject I A j) g⟧ ⟹
f = g"
apply (rule funcset_eq[of "f" "I" "g"])
apply (thin_tac "∀j∈I. aGroup (A j)",
thin_tac "g ∈ carrier (aΠ⇘I⇙ A)",
thin_tac "∀j∈I. (π⇘I,A,j⇙) f = (π⇘I,A,j⇙) g",
simp add:prodag_def carr_prodag_def)
apply (thin_tac "∀j∈I. aGroup (A j)",
thin_tac "f ∈ carrier (aΠ⇘I⇙ A)",
thin_tac "∀j∈I. (π⇘I,A,j⇙) f = (π⇘I,A,j⇙) g",
simp add:prodag_def carr_prodag_def)
apply (simp add:PRoject_def prodag_def)
done
lemma map_family_fun:"⟦∀j∈I. aGroup (A j); aGroup S;
∀j∈I. ((g j) ∈ aHom S (A j)); x ∈ carrier S⟧ ⟹
(λy ∈ carrier S. (λj∈I. (g j) y)) x ∈ carrier (aΠ⇘I⇙ A)"
apply (simp add:prodag_def carr_prodag_def)
apply (simp add:aHom_mem)
apply (rule Pi_I, simp add:Un_carrier_def)
apply (frule_tac x = xa in bspec, assumption,
thin_tac "∀j∈I. aGroup (A j)",
frule_tac x = xa in bspec, assumption,
thin_tac "∀j∈I. g j ∈ aHom S (A j)")
apply (frule_tac G = "A xa" and f = "g xa" and a = x in aHom_mem[of "S"],
assumption+, blast)
done
lemma map_family_aHom:"⟦∀j∈I. aGroup (A j); aGroup S;
∀j∈I. ((g j) ∈ aHom S (A j))⟧ ⟹
(λy ∈ carrier S. (λj∈I. (g j) y)) ∈ aHom S (aΠ⇘I⇙ A)"
apply (subst aHom_def, simp)
apply (simp add:aGroup.ag_pOp_closed)
apply (rule conjI)
apply (rule Pi_I)
apply (rule map_family_fun[of "I" "A" "S" "g"], assumption+)
apply (rule ballI)+
apply (frule_tac x = a and y = b in aGroup.ag_pOp_closed[of "S"],
assumption+)
apply (frule_tac x = "a ±⇘S⇙ b" in map_family_fun[of "I" "A" "S" "g"],
assumption+, simp)
apply (frule_tac x = a in map_family_fun[of "I" "A" "S" "g"],
assumption+, simp,
frule_tac x = b in map_family_fun[of "I" "A" "S" "g"],
assumption+, simp)
apply (frule prodag_aGroup[of "I" "A"])
apply (frule_tac x = "(λj∈I. g j a)" and y = "(λj∈I. g j b)" in
aGroup.ag_pOp_closed[of "aΠ⇘I⇙ A"], assumption+)
apply (simp only:prodag_carrier)
apply (rule carr_prodag_mem_eq, assumption+)
apply (rule ballI)
apply (subst prodag_def, simp add:prod_pOp_def)
apply (simp add:aHom_add)
done
lemma map_family_triangle:"⟦∀j∈I. aGroup (A j); aGroup S;
∀j∈I. ((g j) ∈ aHom S (A j))⟧ ⟹ ∃!f. f ∈ aHom S (aΠ⇘I⇙ A) ∧
(∀j∈I. compos S (PRoject I A j) f = (g j))"
apply (rule ex_ex1I)
apply (frule map_family_aHom[of "I" "A" "S" "g"], assumption+)
apply (subgoal_tac "∀j∈I. compos S (π⇘I,A,j⇙) (λy∈carrier S. λj∈I. g j y) = g j")
apply blast
apply (rule ballI)
apply (simp add:compos_def)
apply (rule funcset_eq[of _ "carrier S"])
apply (simp add:compose_def) apply (simp add:aHom_def)
apply (rule ballI)
apply (frule prodag_aGroup[of "I" "A"])
apply (frule prodag_carrier[of "I" "A"])
apply (frule_tac f = "λy∈carrier S. λj∈I. g j y" and a = x in
aHom_mem[of "S" "aΠ⇘I⇙ A"], assumption+)
apply (simp add:compose_def, simp add:PRoject_def)
apply (rename_tac f f1)
apply (erule conjE)+
apply (rule funcset_eq[of _ "carrier S"])
apply (simp add:aHom_def, simp add:aHom_def)
apply (rule ballI)
apply (frule prodag_aGroup[of "I" "A"])
apply (frule_tac f = f and a = x in aHom_mem[of "S" "aΠ⇘I⇙ A"], assumption+,
frule_tac f = f1 and a = x in aHom_mem[of "S" "aΠ⇘I⇙ A"], assumption+)
apply (rule_tac f = "f x" and g = "f1 x" in direct_prod_mem_eq[of "I" "A"],
assumption+)
apply (rule ballI)
apply (rotate_tac 4,
frule_tac x = j in bspec, assumption,
thin_tac "∀j∈I. compos S (π⇘I,A,j⇙) f = g j",
frule_tac x = j in bspec, assumption,
thin_tac "∀j∈I. compos S (π⇘I,A,j⇙) f1 = g j",
simp add:compos_def compose_def)
apply (subgoal_tac "(λx∈carrier S. (π⇘I,A,j⇙) (f x)) x = g j x",
subgoal_tac "(λx∈carrier S. (π⇘I,A,j⇙) (f1 x)) x = g j x",
thin_tac "(λx∈carrier S. (π⇘I,A,j⇙) (f x)) = g j",
thin_tac "(λx∈carrier S. (π⇘I,A,j⇙) (f1 x)) = g j",
simp+)
done
lemma Ag_ind_triangle:"⟦∀j∈I. aGroup (A j); j ∈ I; f ∈ carrier (aΠ⇘I⇙ A) → B;
bij_to f (carrier (aΠ⇘I⇙ A)) (B::'d set); j ∈ I⟧ ⟹
compos (aΠ⇘I⇙ A) (compos (Ag_ind (aΠ⇘I⇙ A) f)(PRoject I A j) (ainvf⇘(aΠ⇘I⇙ A),
(Ag_ind (aΠ⇘I⇙ A) f)⇙ (Agii (aΠ⇘I⇙ A) f))) (Agii (aΠ⇘I⇙ A) f) =
PRoject I A j"
apply (frule prodag_aGroup[of "I" "A"])
apply (frule aGroup.Ag_ind_aGroup[of "aΠ⇘I⇙ A" "f" "B"], assumption+)
apply (simp add:compos_def)
apply (rule funcset_eq[of _ "carrier (aΠ⇘I⇙ A)"])
apply simp
apply (simp add:PRoject_def prodag_carrier extensional_def)
apply (rule ballI)
apply (simp add:compose_def invfun_l)
apply (simp add:aGroup.Agii_mem)
apply (frule Ag_ind_bijec[of "aΠ⇘I⇙ A" "f" "B"], assumption+)
apply (frule_tac x = x in ainvf_l[of "aΠ⇘I⇙ A" "Ag_ind (aΠ⇘I⇙ A) f"
"Agii (aΠ⇘I⇙ A) f"], assumption+)
apply simp
done
definition
ProjInd :: "['i set, 'i ⇒ ('a, 'm) aGroup_scheme, ('i ⇒ 'a) ⇒ 'd, 'i] ⇒
('d ⇒ 'a)" where
"ProjInd I A f j = compos (Ag_ind (aΠ⇘I⇙ A) f)(PRoject I A j) (ainvf⇘(aΠ⇘I⇙ A), (Ag_ind (aΠ⇘I⇙ A) f)⇙ (Agii (aΠ⇘I⇙ A) f))"
lemma ProjInd_aHom:"⟦∀j∈ I. aGroup (A j); j ∈ I; f ∈ carrier (aΠ⇘I⇙ A) → B;
bij_to f (carrier (aΠ⇘I⇙ A)) (B::'d set); j ∈ I⟧ ⟹
(ProjInd I A f j) ∈ aHom (Ag_ind (aΠ⇘I⇙ A) f) (A j)"
apply (frule prodag_aGroup[of "I" "A"])
apply (frule aGroup.Ag_ind_aGroup[of "aΠ⇘I⇙ A" "f" "B"], assumption+)
apply (frule_tac x = j in bspec, assumption)
apply (frule aGroup.Ag_ind_aHom[of "aΠ⇘I⇙ A" "f" "B"], assumption+)
apply (simp add:ProjInd_def)
apply (frule Ag_ind_bijec[of "aΠ⇘I⇙ A" "f" "B"], assumption+)
apply (frule ainvf_aHom[of "aΠ⇘I⇙ A" "Ag_ind (aΠ⇘I⇙ A) f" "Agii (aΠ⇘I⇙ A) f"],
assumption+)
apply (frule project_aHom[of "I" "A" "j"], assumption)
apply (simp add:aHom_compos)
done
lemma ProjInd_aHom1:"⟦∀j∈ I. aGroup (A j); f ∈ carrier (aΠ⇘I⇙ A) → B;
bij_to f (carrier (aΠ⇘I⇙ A)) (B::'d set)⟧ ⟹
∀j∈I. (ProjInd I A f j) ∈ aHom (Ag_ind (aΠ⇘I⇙ A) f) (A j)"
apply (rule ballI)
apply (simp add:ProjInd_aHom)
done
lemma ProjInd_mem_eq:"⟦∀j∈I. aGroup (A j); f ∈ carrier (aΠ⇘I⇙ A) → B;
bij_to f (carrier (aΠ⇘I⇙ A)) B; aGroup S; x ∈ carrier (Ag_ind (aΠ⇘I⇙ A) f);
y ∈ carrier (Ag_ind (aΠ⇘I⇙ A) f);
∀j∈I. (ProjInd I A f j x = ProjInd I A f j y)⟧ ⟹ x = y"
apply (simp add:ProjInd_def)
apply (simp add:compos_def compose_def)
apply (frule prodag_aGroup[of "I" "A"])
apply (frule aGroup.Ag_ind_aGroup[of "aΠ⇘I⇙ A" "f" "B"], assumption+)
apply (frule aGroup.Ag_ind_aHom[of "aΠ⇘I⇙ A" "f" "B"], assumption+)
apply (frule Ag_ind_bijec[of "aΠ⇘I⇙ A" "f" "B"], assumption+)
apply (frule ainvf_aHom[of "aΠ⇘I⇙ A" "Ag_ind (aΠ⇘I⇙ A) f" "Agii (aΠ⇘I⇙ A) f"],
assumption+)
apply (frule aHom_mem[of "Ag_ind (aΠ⇘I⇙ A) f" "aΠ⇘I⇙ A" "ainvf⇘(aΠ⇘I⇙ A),Ag_ind (aΠ⇘I⇙ A) f⇙ Agii (aΠ⇘I⇙ A) f" "x"], assumption+,
frule aHom_mem[of "Ag_ind (aΠ⇘I⇙ A) f" "aΠ⇘I⇙ A" "ainvf⇘(aΠ⇘I⇙ A),Ag_ind (aΠ⇘I⇙ A) f⇙ Agii (aΠ⇘I⇙ A) f" "y"], assumption+)
apply (frule direct_prod_mem_eq[of "I" "A" "(ainvf⇘(aΠ⇘I⇙ A),Ag_ind (aΠ⇘I⇙ A) f⇙ Agii (aΠ⇘I⇙ A) f) x" "(ainvf⇘(aΠ⇘I⇙ A),Ag_ind (aΠ⇘I⇙ A) f⇙ Agii (aΠ⇘I⇙ A) f) y"], assumption+)
apply (thin_tac "ainvf⇘(aΠ⇘I⇙ A),Ag_ind (aΠ⇘I⇙ A) f⇙ Agii (aΠ⇘I⇙ A) f
∈ aHom (Ag_ind (aΠ⇘I⇙ A) f) (aΠ⇘I⇙ A)")
apply (frule ainvf_bijec[of "aΠ⇘I⇙ A" "Ag_ind (aΠ⇘I⇙ A) f" "Agii (aΠ⇘I⇙ A) f"],
assumption+)
apply (thin_tac "bijec⇘(aΠ⇘I⇙ A),Ag_ind (aΠ⇘I⇙ A) f⇙ Agii (aΠ⇘I⇙ A) f")
apply (unfold bijec_def, frule conjunct1, fold bijec_def)
apply (frule injec_inj_on[of "Ag_ind (aΠ⇘I⇙ A) f" "aΠ⇘I⇙ A" "ainvf⇘(aΠ⇘I⇙ A),Ag_ind (aΠ⇘I⇙ A) f⇙ Agii (aΠ⇘I⇙ A) f"], assumption+)
apply (simp add:injective_iff[THEN sym, of "ainvf⇘(aΠ⇘I⇙ A),Ag_ind (aΠ⇘I⇙ A) f⇙ Agii (aΠ⇘I⇙ A) f" "carrier (Ag_ind (aΠ⇘I⇙ A) f)" "x" "y"])
done
lemma ProjInd_mem_eq1:"⟦∀j∈I. aGroup (A j); f ∈ carrier (aΠ⇘I⇙ A) → B;
bij_to f (carrier (aΠ⇘I⇙ A)) B; aGroup S;
h ∈ aHom (Ag_ind (aΠ⇘I⇙ A) f) (Ag_ind (aΠ⇘I⇙ A) f);
∀j∈I. compos (Ag_ind (aΠ⇘I⇙ A) f) (ProjInd I A f j) h = ProjInd I A f j⟧ ⟹ h = ag_idmap (Ag_ind (aΠ⇘I⇙ A) f)"
apply (rule funcset_eq[of _ "carrier (Ag_ind (aΠ⇘I⇙ A) f)"])
apply (simp add:aHom_def)
apply (simp add:ag_idmap_def)
apply (rule ballI)
apply (simp add:ag_idmap_def)
apply (frule prodag_aGroup[of "I" "A"],
frule aGroup.Ag_ind_aGroup[of "aΠ⇘I⇙ A" "f" "B"], assumption+)
apply (frule_tac a = x in aHom_mem[of "Ag_ind (aΠ⇘I⇙ A) f" "Ag_ind (aΠ⇘I⇙ A) f"
"h"], assumption+)
apply (rule_tac x = "h x" and y = x in ProjInd_mem_eq[of "I" "A" "f" "B" "S"],
assumption+)
apply (rotate_tac 1,
rule ballI,
frule_tac x = j in bspec, assumption,
thin_tac "∀j∈I. compos (Ag_ind (aΠ⇘I⇙ A) f) (ProjInd I A f j) h =
ProjInd I A f j")
apply (simp add:compos_def compose_def)
apply (subgoal_tac "(λx∈carrier (Ag_ind (aΠ⇘I⇙ A) f). ProjInd I A f j (h x)) x
= ProjInd I A f j x",
thin_tac "(λx∈carrier (Ag_ind (aΠ⇘I⇙ A) f). ProjInd I A f j (h x)) =
ProjInd I A f j")
apply simp+
done
lemma Ag_ind_triangle1:"⟦∀j∈I. aGroup (A j); f ∈ carrier (aΠ⇘I⇙ A) → B;
bij_to f (carrier (aΠ⇘I⇙ A)) (B::'d set); j ∈ I⟧ ⟹
compos (aΠ⇘I⇙ A) (ProjInd I A f j) (Agii (aΠ⇘I⇙ A) f) = PRoject I A j"
apply (simp add:ProjInd_def)
apply (simp add:Ag_ind_triangle)
done
lemma map_family_triangle1:"⟦∀j∈I. aGroup (A j); f ∈ carrier (aΠ⇘I⇙ A) → B;
bij_to f (carrier (aΠ⇘I⇙ A)) (B::'d set); aGroup S;
∀j∈I. ((g j) ∈ aHom S (A j))⟧ ⟹ ∃!h. h ∈ aHom S (Ag_ind (aΠ⇘I⇙ A) f) ∧
(∀j∈I. compos S (ProjInd I A f j) h = (g j))"
apply (frule prodag_aGroup[of "I" "A"])
apply (frule aGroup.Ag_ind_aGroup[of "aΠ⇘I⇙ A" "f" "B"], assumption+)
apply (frule Ag_ind_bijec[of "aΠ⇘I⇙ A" "f" "B"], assumption+)
apply (rule ex_ex1I)
apply (frule map_family_triangle[of "I" "A" "S" "g"], assumption+)
apply (frule ex1_implies_ex)
apply (erule exE)
apply (erule conjE)
apply (unfold bijec_def, frule conjunct2, fold bijec_def)
apply (unfold surjec_def, frule conjunct1, fold surjec_def)
apply (rename_tac fa,
frule_tac f = fa in aHom_compos[of "S" "aΠ⇘I⇙ A" "Ag_ind (aΠ⇘I⇙ A) f" _
"Agii (aΠ⇘I⇙ A) f"], assumption+)
apply (subgoal_tac "∀j∈I. compos S (ProjInd I A f j)
(compos S (Agii (aΠ⇘I⇙ A) f) fa) = g j")
apply blast
apply (rule ballI)
apply (frule_tac N = "A j" and f = fa and g = "Agii (aΠ⇘I⇙ A) f" and
h = "ProjInd I A f j" in aHom_compos_assoc[of "S" "aΠ⇘I⇙ A" "Ag_ind (aΠ⇘I⇙ A) f"],
assumption+) apply simp apply assumption+
apply (simp add:ProjInd_aHom)
apply simp
apply (thin_tac "compos S (ProjInd I A f j) (compos S (Agii (aΠ⇘I⇙ A) f) fa) =
compos S (compos (aΠ⇘I⇙ A) (ProjInd I A f j) (Agii (aΠ⇘I⇙ A) f)) fa")
apply (simp add:Ag_ind_triangle1)
apply (rename_tac h h1)
apply (erule conjE)+
apply (rule funcset_eq[of _ "carrier S"])
apply (simp add:aHom_def, simp add:aHom_def)
apply (rule ballI)
apply (simp add:compos_def)
apply (frule_tac f = h and a = x in aHom_mem[of "S" "Ag_ind (aΠ⇘I⇙ A) f"],
assumption+,
frule_tac f = h1 and a = x in aHom_mem[of "S" "Ag_ind (aΠ⇘I⇙ A) f"],
assumption+)
apply (rule_tac x = "h x" and y = "h1 x" in ProjInd_mem_eq[of "I" "A" "f"
"B" "S"], assumption+)
apply (rule ballI)
apply (rotate_tac 5,
frule_tac x = j in bspec, assumption,
thin_tac "∀j∈I. compose (carrier S) (ProjInd I A f j) h = g j",
frule_tac x = j in bspec, assumption,
thin_tac "∀j∈I. compose (carrier S) (ProjInd I A f j) h1 = g j")
apply (simp add:compose_def,
subgoal_tac "(λx∈carrier S. ProjInd I A f j (h x)) x = g j x",
thin_tac "(λx∈carrier S. ProjInd I A f j (h x)) = g j",
subgoal_tac "(λx∈carrier S. ProjInd I A f j (h1 x)) x = g j x",
thin_tac "(λx∈carrier S. ProjInd I A f j (h1 x)) = g j", simp+)
done
lemma map_family_triangle2:"⟦I ≠ {}; ∀j∈I. aGroup (A j); aGroup S;
∀j∈I. g j ∈ aHom S (A j); ff ∈ carrier (aΠ⇘I⇙ A) → B;
bij_to ff (carrier (aΠ⇘I⇙ A)) B;
h1 ∈ aHom (Ag_ind (aΠ⇘I⇙ A) ff) S;
∀j∈I. compos (Ag_ind (aΠ⇘I⇙ A) ff) (g j) h1 = ProjInd I A ff j;
h2 ∈ aHom S (Ag_ind (aΠ⇘I⇙ A) ff);
∀j∈I. compos S (ProjInd I A ff j) h2 = g j⟧
⟹ ∀j∈I. compos (Ag_ind (aΠ⇘I⇙ A) ff) (ProjInd I A ff j)
(compos (Ag_ind (aΠ⇘I⇙ A) ff) h2 h1) =
ProjInd I A ff j"
apply (rule ballI)
apply (frule prodag_aGroup[of "I" "A"])
apply (frule_tac f = ff in aGroup.Ag_ind_aGroup[of "aΠ⇘I⇙ A" _ "B"], assumption+)
apply (frule_tac N = "A j" and h = "ProjInd I A ff j" in aHom_compos_assoc[of "Ag_ind (aΠ⇘I⇙ A) ff" "S" "Ag_ind (aΠ⇘I⇙ A) ff" _ "h1" "h2"], assumption+)
apply simp apply assumption+ apply (simp add:ProjInd_aHom)
apply simp
done
lemma map_family_triangle3:"⟦∀j∈I. aGroup (A j); aGroup S; aGroup S1;
∀j∈I. f j ∈ aHom S (A j); ∀j∈I. g j ∈ aHom S1 (A j);
h1 ∈ aHom S1 S; h2 ∈ aHom S S1;
∀j∈I. compos S (g j) h2 = f j;
∀j∈I. compos S1 (f j) h1 = g j⟧
⟹ ∀j∈I. compos S (f j) (compos S h1 h2) = f j"
apply (rule ballI)
apply (frule_tac h = "f j" and N = "A j" in aHom_compos_assoc[of "S" "S1"
"S" _ "h2" "h1"], assumption+)
apply simp apply assumption+ apply simp
apply simp
done
lemma map_family_triangle4:"⟦∀j∈I. aGroup (A j); aGroup S;
∀j∈I. f j ∈ aHom S (A j)⟧ ⟹
∀j∈I. compos S (f j) (ag_idmap S) = f j"
apply (rule ballI)
apply (frule_tac x = j in bspec, assumption,
thin_tac "∀j∈I. aGroup (A j)",
frule_tac x = j in bspec, assumption,
thin_tac "∀j∈I. f j ∈ aHom S (A j)")
apply (simp add:compos_aI_r)
done
lemma prod_triangle:"⟦I ≠ {}; ∀j∈I. aGroup (A j); aGroup S;
∀j∈I. g j ∈ aHom S (A j); ff ∈ carrier (aΠ⇘I⇙ A) → B;
bij_to ff (carrier (aΠ⇘I⇙ A)) B;
h1 ∈ aHom (Ag_ind (aΠ⇘I⇙ A) ff) S;
∀j∈I. compos (Ag_ind (aΠ⇘I⇙ A) ff) (g j) h1 = ProjInd I A ff j;
h2 ∈ aHom S (Ag_ind (aΠ⇘I⇙ A) ff);
∀j∈I. compos S (ProjInd I A ff j) h2 = g j⟧
⟹ (compos (Ag_ind (aΠ⇘I⇙ A) ff) h2 h1) = ag_idmap (Ag_ind (aΠ⇘I⇙ A) ff)"
apply (frule map_family_triangle2[of "I" "A" "S" "g" "ff" "B" "h1" "h2"], assumption+)
apply (frule prodag_aGroup[of "I" "A"],
frule aGroup.Ag_ind_aGroup[of "aΠ⇘I⇙ A" "ff" "B"], assumption+)
apply (frule aHom_compos[of "Ag_ind (aΠ⇘I⇙ A) ff" "S" "Ag_ind (aΠ⇘I⇙ A) ff" "h1"
"h2"], assumption+)
apply (rule ProjInd_mem_eq1[of "I" "A" "ff" "B" "S"
"compos (Ag_ind (aΠ⇘I⇙ A) ff) h2 h1"], assumption+)
done
lemma characterization_prodag:"⟦I ≠ {}; ∀j∈(I::'i set). aGroup ((A j)::
('a, 'm) aGroup_scheme); aGroup (S::'d aGroup);
∀j∈I. ((g j) ∈ aHom S (A j)); ∃ff. ff ∈ carrier (aΠ⇘I⇙ A) → (B::'d set) ∧
bij_to ff (carrier (aΠ⇘I⇙ A)) B;
∀(S':: 'd aGroup). aGroup S' ⟶
(∀g'. (∀j∈I. (g' j) ∈ aHom S' (A j) ⟶
(∃! f. f ∈ aHom S' S ∧ (∀j∈I. compos S' (g j) f = (g' j)))))⟧ ⟹
∃h. bijec⇘(prodag I A),S⇙ h"
apply (frule prodag_aGroup[of "I" "A"])
apply (erule exE)
apply (frule_tac f = ff in aGroup.Ag_ind_aGroup[of "aΠ⇘I⇙ A" _ "B"], erule conjE,
assumption, simp, erule conjE)
apply (frule aGroup.Ag_ind_aGroup[of "aΠ⇘I⇙ A" _ "B"], assumption+,
frule_tac a = S in forall_spec, assumption+)
apply (rotate_tac -1,
frule_tac x = g in spec,
thin_tac "∀g'. ∀j∈I. g' j ∈ aHom S (A j) ⟶
(∃!f. f ∈ aHom S S ∧ (∀j∈I. compos S (g j) f = g' j))")
apply (frule_tac a = "Ag_ind (aΠ⇘I⇙ A) ff" in forall_spec, assumption+,
thin_tac "∀S'. aGroup S' ⟶ (∀g'. ∀j∈I. g' j ∈ aHom S' (A j) ⟶
(∃!f. f ∈ aHom S' S ∧ (∀j∈I. compos S' (g j) f = g' j)))")
apply (frule_tac x = "ProjInd I A ff" in spec,
thin_tac "∀g'. ∀j∈I. g' j ∈ aHom (Ag_ind (aΠ⇘I⇙ A) ff) (A j) ⟶
(∃!f. f ∈ aHom (Ag_ind (aΠ⇘I⇙ A) ff) S ∧
(∀j∈I. compos (Ag_ind (aΠ⇘I⇙ A) ff) (g j) f =
g' j))")
apply (frule_tac f = ff in ProjInd_aHom1[of "I" "A" _ "B"], assumption+)
apply (simp add:nonempty_ex[of "I"],
rotate_tac -2,
frule ex1_implies_ex,
thin_tac "∃!f. f ∈ aHom (Ag_ind (aΠ⇘I⇙ A) ff) S ∧
(∀j∈I. compos (Ag_ind (aΠ⇘I⇙ A) ff) (g j) f = ProjInd I A ff j)",
rotate_tac -1, erule exE, erule conjE)
apply (rename_tac ff h1,
frule_tac f = ff in map_family_triangle1[of "I" "A" _ "B" "S" "g"],
assumption+,
rotate_tac -1,
frule ex1_implies_ex,
thin_tac "∃!h. h ∈ aHom S (Ag_ind (aΠ⇘I⇙ A) ff) ∧
(∀j∈I. compos S (ProjInd I A ff j) h = g j)",
rotate_tac -1,
erule exE, erule conjE)
apply (rename_tac ff h1 h2)
apply (frule_tac ff = ff and ?h1.0 = h1 and ?h2.0 = h2 in prod_triangle[of "I"
"A" "S" "g" _ "B"], assumption+,
frule_tac ?S1.0 = "Ag_ind (aΠ⇘I⇙ A) ff" in map_family_triangle3[of "I"
"A" "S" _ "g"],
assumption+,
frule_tac f = h2 and g = h1 and M = "Ag_ind (aΠ⇘I⇙ A) ff" in
aHom_compos[of "S" _ "S" ], assumption+)
apply (erule ex1E)
apply (rotate_tac -1,
frule_tac x = "compos S h1 h2" in spec,
frule map_family_triangle4[of "I" "A" "S" "g"], assumption+,
frule aGroup.aI_aHom[of "S"])
apply (frule_tac x = "aI⇘S⇙" in spec,
thin_tac "∀y. y ∈ aHom S S ∧ (∀j∈I. compos S (g j) y = g j) ⟶ y = f",
simp,
thin_tac "∀j∈I. compos S (ProjInd I A ff j) h2 = g j",
thin_tac "∀j∈I. compos S (g j) f = g j",
thin_tac "∀j∈I. compos (Ag_ind (aΠ⇘I⇙ A) ff) (g j) h1 = ProjInd I A ff j")
apply (rotate_tac -1, frule sym, thin_tac "aI⇘S⇙ = f", simp,
frule_tac A = "Ag_ind (aΠ⇘I⇙ A) ff" and f = h1 and g = h2 in
compos_aI_inj[of _ "S"], assumption+,
frule_tac B = "Ag_ind (aΠ⇘I⇙ A) ff" and f = h2 and g = h1 in
compos_aI_surj[of "S"], assumption+)
apply (frule_tac f = ff in Ag_ind_bijec[of "aΠ⇘I⇙ A" _ "B"], assumption+,
frule_tac F = "Ag_ind (aΠ⇘I⇙ A) ff" and f = "Agii (aΠ⇘I⇙ A) ff" and g = h1
in compos_bijec[of "aΠ⇘I⇙ A" _ "S"], assumption+)
apply (subst bijec_def, simp)
apply (thin_tac "bijec⇘(aΠ⇘I⇙ A),Ag_ind (aΠ⇘I⇙ A) ff⇙ Agii (aΠ⇘I⇙ A) ff",
thin_tac "injec⇘Ag_ind (aΠ⇘I⇙ A) ff,S⇙ h1",
thin_tac "surjec⇘Ag_ind (aΠ⇘I⇙ A) ff,S⇙ h1")
apply (rule exI, simp)
done
chapter "Ring theory"
section "Definition of a ring and an ideal"
record 'a Ring = "'a aGroup" +
tp :: "['a, 'a ] ⇒ 'a" (infixl "⋅⇩rı" 70)
un :: "'a" ("1⇩rı")
locale Ring =
fixes R (structure)
assumes
pop_closed: "pop R ∈ carrier R → carrier R → carrier R"
and pop_aassoc : "⟦a ∈ carrier R; b ∈ carrier R; c ∈ carrier R⟧ ⟹
(a ± b) ± c = a ± (b ± c)"
and pop_commute:"⟦a ∈ carrier R; b ∈ carrier R⟧ ⟹ a ± b = b ± a"
and mop_closed:"mop R ∈ carrier R → carrier R"
and l_m :"a ∈ carrier R ⟹ (-⇩a a) ± a = 𝟬"
and ex_zero: "𝟬 ∈ carrier R"
and l_zero:"a ∈ carrier R ⟹ 𝟬 ± a = a"
and tp_closed: "tp R ∈ carrier R → carrier R → carrier R"
and tp_assoc : "⟦a ∈ carrier R; b ∈ carrier R; c ∈ carrier R⟧ ⟹
(a ⋅⇩r b) ⋅⇩r c = a ⋅⇩r (b ⋅⇩r c)"
and tp_commute: "⟦a ∈ carrier R; b ∈ carrier R⟧ ⟹ a ⋅⇩r b = b ⋅⇩r a"
and un_closed: "(1⇩r) ∈ carrier R"
and rg_distrib: "⟦a ∈ carrier R; b ∈ carrier R; c ∈ carrier R⟧ ⟹
a ⋅⇩r (b ± c) = a ⋅⇩r b ± a ⋅⇩r c"
and rg_l_unit: "a ∈ carrier R ⟹ (1⇩r) ⋅⇩r a = a"
definition
zeroring :: "('a, 'more) Ring_scheme ⇒ bool" where
"zeroring R ⟷ Ring R ∧ carrier R = {𝟬⇘R⇙}"
primrec nscal :: "('a, 'more) Ring_scheme => 'a => nat => 'a"
where
nscal_0: "nscal R x 0 = 𝟬⇘R⇙"
| nscal_suc: "nscal R x (Suc n) = (nscal R x n) ±⇘R⇙ x"
primrec npow :: "('a, 'more) Ring_scheme => 'a => nat => 'a"
where
npow_0: "npow R x 0 = 1⇩r⇘R⇙"
| npow_suc: "npow R x (Suc n) = (npow R x n) ⋅⇩r⇘R⇙ x"
primrec nprod :: "('a, 'more) Ring_scheme => (nat => 'a) => nat => 'a"
where
nprod_0: "nprod R f 0 = f 0"
| nprod_suc: "nprod R f (Suc n) = (nprod R f n) ⋅⇩r⇘R⇙ (f (Suc n))"
primrec nsum :: "('a, 'more) aGroup_scheme => (nat => 'a) => nat => 'a"
where
nsum_0: "nsum R f 0 = f 0"
| nsum_suc: "nsum R f (Suc n) = (nsum R f n) ±⇘R⇙ (f (Suc n))"
abbreviation
NSCAL :: "[nat, ('a, 'more) Ring_scheme, 'a] ⇒ 'a"
("(3 _ ×⇘_⇙ _)" [75,75,76]75) where
"n ×⇘R⇙ x == nscal R x n"
abbreviation
NPOW :: "['a, ('a, 'more) Ring_scheme, nat] ⇒ 'a"
("(3_^⇗_ _⇖)" [77,77,78]77) where
"a^⇗R n⇖ == npow R a n"
abbreviation
SUM :: "('a, 'more) aGroup_scheme => (nat => 'a) => nat => 'a"
("(3Σ⇩e _ _ _)" [85,85,86]85) where
"Σ⇩e G f n == nsum G f n"
abbreviation
NPROD :: "[('a, 'm) Ring_scheme, nat, nat ⇒ 'a] ⇒ 'a"
("(3eΠ⇘_,_⇙ _)" [98,98,99]98) where
"eΠ⇘R,n⇙ f == nprod R f n"
definition
fSum :: "[_, (nat => 'a), nat, nat] ⇒ 'a" where
"fSum A f n m = (if n ≤ m then nsum A (cmp f (slide n))(m - n)
else 𝟬⇘A⇙)"
abbreviation
FSUM :: "[('a, 'more) aGroup_scheme, (nat ⇒ 'a), nat, nat] ⇒ 'a"
("(4Σ⇩f _ _ _ _)" [85,85,85,86]85) where
"Σ⇩f G f n m == fSum G f n m"
lemma (in aGroup) nsum_zeroGTr:"(∀j ≤ n. f j = 𝟬) ⟶ nsum A f n = 𝟬"
apply (induct_tac n)
apply (rule impI, simp)
apply (rule impI)
apply (cut_tac n = n in Nsetn_sub_mem1, simp)
apply (cut_tac ex_zero)
apply (simp add:l_zero[of 𝟬])
done
lemma (in aGroup) nsum_zeroA:"∀j ≤ n. f j = 𝟬 ⟹ nsum A f n = 𝟬"
apply (simp add:nsum_zeroGTr)
done
definition
sr :: "[_ , 'a set] ⇒ bool" where
"sr R S == S ⊆ carrier R ∧ 1⇩r⇘R⇙ ∈ S ∧ (∀x∈S. ∀y ∈ S. x ±⇘R⇙ (-⇩a⇘R⇙ y) ∈ S ∧
x ⋅⇩r⇘R⇙ y ∈ S)"
definition
Sr :: "[_ , 'a set] ⇒ _" where
"Sr R S = R ⦇carrier := S, pop := λx∈S. λy∈S. x ±⇘R⇙ y, mop := λx∈S. (-⇩a⇘R⇙ x),
zero := 𝟬⇘R⇙, tp := λx∈S. λy∈S. x ⋅⇩r⇘R⇙ y, un := 1⇩r⇘R⇙ ⦈"
lemma (in Ring) Ring: "Ring R" ..
lemma (in Ring) ring_is_ag:"aGroup R"
apply (rule aGroup.intro,
rule pop_closed,
rule pop_aassoc, assumption+,
rule pop_commute, assumption+,
rule mop_closed,
rule l_m, assumption,
rule ex_zero,
rule l_zero, assumption)
done
lemma (in Ring) ring_zero:"𝟬 ∈ carrier R"
by (simp add: ex_zero)
lemma (in Ring) ring_one:"1⇩r ∈ carrier R"
by (simp add:un_closed)
lemma (in Ring) ring_tOp_closed:"⟦ x ∈ carrier R; y ∈ carrier R⟧ ⟹
x ⋅⇩r y ∈ carrier R"
apply (cut_tac tp_closed)
apply (frule funcset_mem[of "(⋅⇩r)" "carrier R" "carrier R → carrier R"
"x"], assumption+,
thin_tac "(⋅⇩r) ∈ carrier R → carrier R → carrier R")
apply (rule funcset_mem[of "(⋅⇩r) x" "carrier R" "carrier R" "y"],
assumption+)
done
lemma (in Ring) ring_tOp_commute:"⟦x ∈ carrier R; y ∈ carrier R⟧ ⟹
x ⋅⇩r y = y ⋅⇩r x"
by (simp add:tp_commute)
lemma (in Ring) ring_distrib1:"⟦x ∈ carrier R; y ∈ carrier R; z ∈ carrier R ⟧
⟹ x ⋅⇩r (y ± z) = x ⋅⇩r y ± x ⋅⇩r z"
by (simp add:rg_distrib)
lemma (in Ring) ring_distrib2:"⟦x ∈ carrier R; y ∈ carrier R; z ∈ carrier R ⟧
⟹ (y ± z) ⋅⇩r x = y ⋅⇩r x ± z ⋅⇩r x"
apply (subst tp_commute[of "y ± z" "x"])
apply (cut_tac ring_is_ag, simp add:aGroup.ag_pOp_closed)
apply assumption
apply (subst ring_distrib1, assumption+)
apply (simp add:tp_commute)
done
lemma (in Ring) ring_distrib3:"⟦a ∈ carrier R; b ∈ carrier R; x ∈ carrier R;
y ∈ carrier R ⟧ ⟹ (a ± b) ⋅⇩r (x ± y) =
a ⋅⇩r x ± a ⋅⇩r y ± b ⋅⇩r x ± b ⋅⇩r y"
apply (subst ring_distrib2)+
apply (cut_tac ring_is_ag)
apply (rule aGroup.ag_pOp_closed, assumption+)
apply ((subst ring_distrib1)+, assumption+)
apply (subst ring_distrib1, assumption+)
apply (rule pop_aassoc [THEN sym, of "a ⋅⇩r x ± a ⋅⇩r y" "b ⋅⇩r x" "b ⋅⇩r y"])
apply (cut_tac ring_is_ag, rule aGroup.ag_pOp_closed, assumption)
apply (simp add:ring_tOp_closed)+
done
lemma (in Ring) rEQMulR:
"⟦x ∈ carrier R; y ∈ carrier R; z ∈ carrier R; x = y ⟧
⟹ x ⋅⇩r z = y ⋅⇩r z"
by simp
lemma (in Ring) ring_tOp_assoc:"⟦x ∈ carrier R; y ∈ carrier R; z ∈ carrier R ⟧
⟹ (x ⋅⇩r y) ⋅⇩r z = x ⋅⇩r (y ⋅⇩r z)"
by (simp add:tp_assoc)
lemma (in Ring) ring_l_one:"x ∈ carrier R ⟹ 1⇩r ⋅⇩r x = x"
by (simp add:rg_l_unit)
lemma (in Ring) ring_r_one:"x ∈ carrier R ⟹ x ⋅⇩r 1⇩r = x"
apply (subst ring_tOp_commute, assumption+)
apply (simp add:un_closed)
apply (simp add:ring_l_one)
done
lemma (in Ring) ring_times_0_x:"x ∈ carrier R ⟹ 𝟬 ⋅⇩r x = 𝟬"
apply (cut_tac ring_is_ag)
apply (cut_tac ring_zero)
apply (frule ring_distrib2 [of "x" "𝟬" "𝟬"], assumption+)
apply (simp add:aGroup.ag_l_zero [of "R" "𝟬"])
apply (frule ring_tOp_closed [of "𝟬" "x"], assumption+)
apply (frule sym, thin_tac "𝟬 ⋅⇩r x = 𝟬 ⋅⇩r x ± 𝟬 ⋅⇩r x")
apply (frule aGroup.ag_eq_sol2 [of "R" "𝟬 ⋅⇩r x" "𝟬 ⋅⇩r x" "𝟬 ⋅⇩r x"],
assumption+)
apply (thin_tac "𝟬 ⋅⇩r x ± 𝟬 ⋅⇩r x = 𝟬 ⋅⇩r x")
apply (simp add:aGroup.ag_r_inv1)
done
lemma (in Ring) ring_times_x_0:"x ∈ carrier R ⟹ x ⋅⇩r 𝟬 = 𝟬"
apply (cut_tac ring_zero)
apply (subst ring_tOp_commute, assumption+, simp add:ring_zero)
apply (simp add:ring_times_0_x)
done
lemma (in Ring) rMulZeroDiv:
"⟦ x ∈ carrier R; y ∈ carrier R; x = 𝟬 ∨ y = 𝟬 ⟧ ⟹ x ⋅⇩r y = 𝟬"
apply (erule disjE, simp)
apply (rule ring_times_0_x, assumption+)
apply (simp, rule ring_times_x_0, assumption+)
done
lemma (in Ring) ring_inv1:"⟦ a ∈ carrier R; b ∈ carrier R ⟧ ⟹
-⇩a (a ⋅⇩r b) = (-⇩a a) ⋅⇩r b ∧ -⇩a (a ⋅⇩r b) = a ⋅⇩r (-⇩a b)"
apply (cut_tac ring_is_ag)
apply (rule conjI)
apply (frule ring_distrib2 [THEN sym, of "b" "a" "-⇩a a"], assumption+)
apply (frule aGroup.ag_mOp_closed [of "R" "a"], assumption+)
apply (simp add:aGroup.ag_r_inv1 [of "R" "a"])
apply (simp add:ring_times_0_x)
apply (frule aGroup.ag_mOp_closed [of "R" "a"], assumption+)
apply (frule ring_tOp_closed [of "a" "b"], assumption+)
apply (frule ring_tOp_closed [of "-⇩a a" "b"], assumption+)
apply (frule aGroup.ag_eq_sol1 [of "R" "a ⋅⇩r b" "(-⇩a a) ⋅⇩r b" "𝟬"],
assumption+)
apply (rule ring_zero, assumption+)
apply (thin_tac "a ⋅⇩r b ± (-⇩a a) ⋅⇩r b = 𝟬")
apply (frule sym) apply (thin_tac "(-⇩a a) ⋅⇩r b = -⇩a (a ⋅⇩r b) ± 𝟬")
apply (frule aGroup.ag_mOp_closed [of "R" " a ⋅⇩r b"], assumption+)
apply (simp add:aGroup.ag_r_zero)
apply (frule ring_distrib1 [THEN sym, of "a" "b" "-⇩a b"], assumption+)
apply (simp add:aGroup.ag_mOp_closed)
apply (simp add:aGroup.ag_r_inv1 [of "R" "b"])
apply (simp add:ring_times_x_0)
apply (frule aGroup.ag_mOp_closed [of "R" "b"], assumption+)
apply (frule ring_tOp_closed [of "a" "b"], assumption+)
apply (frule ring_tOp_closed [of "a" "-⇩a b"], assumption+)
apply (frule aGroup.ag_eq_sol1 [THEN sym, of "R" "a ⋅⇩r b" "a ⋅⇩r (-⇩a b)" "𝟬"],
assumption+)
apply (simp add:ring_zero) apply assumption
apply (frule aGroup.ag_mOp_closed [of "R" " a ⋅⇩r b"], assumption+)
apply (simp add:aGroup.ag_r_zero)
done
lemma (in Ring) ring_inv1_1:"⟦a ∈ carrier R; b ∈ carrier R ⟧ ⟹
-⇩a (a ⋅⇩r b) = (-⇩a a) ⋅⇩r b"
apply (simp add:ring_inv1)
done
lemma (in Ring) ring_inv1_2:"⟦ a ∈ carrier R; b ∈ carrier R ⟧ ⟹
-⇩a (a ⋅⇩r b) = a ⋅⇩r (-⇩a b)"
apply (frule ring_inv1 [of "a" "b"], assumption+)
apply (frule conjunct2)
apply (thin_tac "-⇩a a ⋅⇩r b = (-⇩a a) ⋅⇩r b ∧ -⇩a (a ⋅⇩r b) = a ⋅⇩r (-⇩a b)")
apply simp
done
lemma (in Ring) ring_times_minusl:"a ∈ carrier R ⟹ -⇩a a = (-⇩a 1⇩r) ⋅⇩r a"
apply (cut_tac ring_one)
apply (frule ring_inv1_1[of "1⇩r" "a"], assumption+)
apply (simp add:ring_l_one)
done
lemma (in Ring) ring_times_minusr:"a ∈ carrier R ⟹ -⇩a a = a ⋅⇩r (-⇩a 1⇩r)"
apply (cut_tac ring_one)
apply (frule ring_inv1_2[of "a" "1⇩r"], assumption+)
apply (simp add:ring_r_one)
done
lemma (in Ring) ring_inv1_3:"⟦a ∈ carrier R; b ∈ carrier R⟧ ⟹
a ⋅⇩r b = (-⇩a a) ⋅⇩r (-⇩a b)"
apply (cut_tac ring_is_ag)
apply (subst aGroup.ag_inv_inv[THEN sym], assumption+)
apply (frule aGroup.ag_mOp_closed[of "R" "a"], assumption+)
apply (subst ring_inv1_1[THEN sym, of "-⇩a a" "b"], assumption+)
apply (subst ring_inv1_2[of "-⇩a a" "b"], assumption+, simp)
done
lemma (in Ring) ring_distrib4:"⟦a ∈ carrier R; b ∈ carrier R;
x ∈ carrier R; y ∈ carrier R ⟧ ⟹
a ⋅⇩r b ± (-⇩a x ⋅⇩r y) = a ⋅⇩r (b ± (-⇩a y)) ± (a ± (-⇩a x)) ⋅⇩r y"
apply (cut_tac ring_is_ag)
apply (subst ring_distrib1, assumption+)
apply (rule aGroup.ag_mOp_closed, assumption+)
apply (subst ring_distrib2, assumption+)
apply (rule aGroup.ag_mOp_closed, assumption+)
apply (subst aGroup.pOp_assocTr43, assumption+)
apply (rule ring_tOp_closed, assumption+)+
apply (rule aGroup.ag_mOp_closed, assumption+)
apply (rule ring_tOp_closed, assumption+)
apply (rule ring_tOp_closed)
apply (simp add:aGroup.ag_mOp_closed)+
apply (subst ring_distrib1 [THEN sym, of "a" _], assumption+)
apply (rule aGroup.ag_mOp_closed, assumption+)
apply (simp add:aGroup.ag_l_inv1)
apply (simp add:ring_times_x_0)
apply (subst aGroup.ag_r_zero, assumption+)
apply (simp add:ring_tOp_closed)
apply (simp add: ring_inv1_1)
done
lemma (in Ring) rMulLC:
"⟦x ∈ carrier R; y ∈ carrier R; z ∈ carrier R⟧
⟹ x ⋅⇩r (y ⋅⇩r z) = y ⋅⇩r (x ⋅⇩r z)"
apply (subst ring_tOp_assoc [THEN sym], assumption+)
apply (subst ring_tOp_commute [of "x" "y"], assumption+)
apply (subst ring_tOp_assoc, assumption+)
apply simp
done
lemma (in Ring) Zero_ring:"1⇩r = 𝟬 ⟹ zeroring R"
apply (simp add:zeroring_def)
apply (rule conjI)
apply (rule Ring_axioms)
apply (rule equalityI)
apply (rule subsetI)
apply (frule_tac x = x in ring_r_one, simp add:ring_times_x_0)
apply (simp add:ring_zero)
done
lemma (in Ring) Zero_ring1:"¬ (zeroring R) ⟹ 1⇩r ≠ 𝟬"
apply (rule contrapos_pp, simp+,
cut_tac Zero_ring, simp+)
done
lemma (in Ring) Sr_one:"sr R S ⟹ 1⇩r ∈ S"
apply (simp add:sr_def)
done
lemma (in Ring) Sr_zero:"sr R S ⟹ 𝟬 ∈ S"
apply (cut_tac ring_is_ag, frule Sr_one[of "S"])
apply (simp add:sr_def) apply (erule conjE)+
apply (frule_tac x = "1⇩r" in bspec, assumption,
thin_tac "∀x∈S. ∀y∈S. x ± -⇩a y ∈ S ∧ x ⋅⇩r y ∈ S",
frule_tac x = "1⇩r" in bspec, assumption,
thin_tac "∀y∈S. 1⇩r ± -⇩a y ∈ S ∧ 1⇩r ⋅⇩r y ∈ S",
erule conjE)
apply (cut_tac ring_one,
simp add:aGroup.ag_r_inv1[of "R" "1⇩r"])
done
lemma (in Ring) Sr_mOp_closed:"⟦sr R S; x ∈ S⟧ ⟹ -⇩a x ∈ S"
apply (frule Sr_zero[of "S"])
apply (simp add:sr_def, (erule conjE)+)
apply (cut_tac ring_is_ag)
apply (frule_tac x = "𝟬" in bspec, assumption,
thin_tac "∀x∈S. ∀y∈S. x ± -⇩a y ∈ S ∧ x ⋅⇩r y ∈ S",
frule_tac x = x in bspec, assumption,
thin_tac "∀y∈S. 𝟬 ± -⇩a y ∈ S ∧ 𝟬 ⋅⇩r y ∈ S", erule conjE)
apply (frule subsetD[of "S" "carrier R" "𝟬"], assumption+,
frule subsetD[of "S" "carrier R" "x"], assumption+)
apply (frule aGroup.ag_mOp_closed [of "R" "x"], assumption)
apply (simp add:aGroup.ag_l_zero)
done
lemma (in Ring) Sr_pOp_closed:"⟦sr R S; x ∈ S; y ∈ S⟧ ⟹ x ± y ∈ S"
apply (frule Sr_mOp_closed[of "S" "y"], assumption+)
apply (unfold sr_def, (erule conjE)+)
apply (frule_tac x = x in bspec, assumption,
thin_tac "∀x∈S. ∀y∈S. x ± -⇩a y ∈ S ∧ x ⋅⇩r y ∈ S",
frule_tac x = "-⇩a y" in bspec, assumption,
thin_tac "∀y∈S. x ± -⇩a y ∈ S ∧ x ⋅⇩r y ∈ S", erule conjE)
apply (cut_tac ring_is_ag )
apply (frule subsetD[of "S" "carrier R" "y"], assumption+)
apply (simp add:aGroup.ag_inv_inv)
done
lemma (in Ring) Sr_tOp_closed:"⟦sr R S; x ∈ S; y ∈ S⟧ ⟹ x ⋅⇩r y ∈ S"
by (simp add:sr_def)
lemma (in Ring) Sr_ring:"sr R S ⟹ Ring (Sr R S)"
apply (simp add:Ring_def [of "Sr R S"],
cut_tac ring_is_ag)
apply (rule conjI)
apply (simp add:Sr_def Sr_pOp_closed)
apply (rule conjI)
apply (rule allI, rule impI)+
apply (simp add:Sr_def,
frule_tac x = a and y = b in Sr_pOp_closed, assumption+,
frule_tac x = b and y = c in Sr_pOp_closed, assumption+,
simp add:Sr_def sr_def, (erule conjE)+)
apply (frule_tac c = a in subsetD[of "S" "carrier R"], assumption+,
frule_tac c = b in subsetD[of "S" "carrier R"], assumption+,
frule_tac c = c in subsetD[of "S" "carrier R"], assumption+)
apply (simp add:aGroup.ag_pOp_assoc)
apply (rule conjI)
apply (rule allI, rule impI)+
apply (simp add:Sr_def sr_def, (erule conjE)+,
frule_tac c = a in subsetD[of "S" "carrier R"], assumption+,
frule_tac c = b in subsetD[of "S" "carrier R"], assumption+)
apply (simp add:aGroup.ag_pOp_commute)
apply (rule conjI)
apply ((subst Sr_def)+, simp)
apply (simp add:Sr_mOp_closed)
apply (rule conjI)
apply (rule allI)
apply ((subst Sr_def)+, simp add:Sr_mOp_closed, rule impI)
apply (unfold sr_def, frule conjunct1, fold sr_def,
frule_tac c = a in subsetD[of "S" "carrier R"], assumption+,
simp add:aGroup.ag_l_inv1)
apply (rule conjI)
apply (simp add:Sr_def Sr_zero)
apply (rule conjI)
apply (rule allI, simp add:Sr_def Sr_zero)
apply (rule impI)
apply (unfold sr_def, frule conjunct1, fold sr_def,
frule_tac c = a in subsetD[of "S" "carrier R"], assumption+,
simp add:aGroup.ag_l_zero)
apply (rule conjI)
apply (simp add:Sr_def Sr_tOp_closed)
apply (rule conjI)
apply (rule allI, rule impI)+
apply (simp add:Sr_def Sr_tOp_closed)
apply (unfold sr_def, frule conjunct1, fold sr_def,
frule_tac c = a in subsetD[of "S" "carrier R"], assumption+,
frule_tac c = b in subsetD[of "S" "carrier R"], assumption+,
frule_tac c = c in subsetD[of "S" "carrier R"], assumption+)
apply (simp add:ring_tOp_assoc)
apply (rule conjI)
apply ((rule allI, rule impI)+, simp add:Sr_def)
apply (unfold sr_def, frule conjunct1, fold sr_def,
frule_tac c = a in subsetD[of "S" "carrier R"], assumption+,
frule_tac c = b in subsetD[of "S" "carrier R"], assumption+,
simp add:ring_tOp_commute)
apply (rule conjI)
apply (simp add:Sr_def Sr_one)
apply (rule conjI)
apply (simp add:Sr_def Sr_pOp_closed Sr_tOp_closed)
apply (rule allI, rule impI)+
apply (unfold sr_def, frule conjunct1, fold sr_def,
frule_tac c = a in subsetD[of "S" "carrier R"], assumption+,
frule_tac c = b in subsetD[of "S" "carrier R"], assumption+,
frule_tac c = c in subsetD[of "S" "carrier R"], assumption+)
apply (simp add:ring_distrib1)
apply (simp add:Sr_def Sr_one)
apply (rule allI, rule impI)
apply (unfold sr_def, frule conjunct1, fold sr_def,
frule_tac c = a in subsetD[of "S" "carrier R"], assumption+)
apply (simp add:ring_l_one)
done
section "Calculation of elements"
subsection "nscale"
lemma (in Ring) ring_tOp_rel:"⟦x∈carrier R; xa∈carrier R; y∈carrier R;
ya ∈ carrier R ⟧ ⟹ (x ⋅⇩r xa) ⋅⇩r (y ⋅⇩r ya) = (x ⋅⇩r y) ⋅⇩r (xa ⋅⇩r ya)"
apply (frule ring_tOp_closed[of "y" "ya"], assumption+,
simp add:ring_tOp_assoc[of "x" "xa"])
apply (simp add:ring_tOp_assoc[THEN sym, of "xa" "y" "ya"],
simp add:ring_tOp_commute[of "xa" "y"],
simp add:ring_tOp_assoc[of "y" "xa" "ya"])
apply (frule ring_tOp_closed[of "xa" "ya"], assumption+,
simp add:ring_tOp_assoc[THEN sym, of "x" "y"])
done
lemma (in Ring) nsClose:
"⋀ n. ⟦ x ∈ carrier R ⟧ ⟹ nscal R x n ∈ carrier R"
apply (induct_tac n)
apply (simp add:ring_zero)
apply (cut_tac ring_is_ag, simp add:aGroup.ag_pOp_closed)
done
lemma (in Ring) nsZero:
"nscal R 𝟬 n = 𝟬"
apply (cut_tac ring_is_ag)
apply (induct_tac n)
apply simp
apply simp
apply (cut_tac ring_zero, simp add:aGroup.ag_l_zero)
done
lemma (in Ring) nsZeroI: "⋀ n. x = 𝟬 ⟹ nscal R x n = 𝟬"
by (simp only:nsZero)
lemma (in Ring) nsEqElm: "⟦ x ∈ carrier R; y ∈ carrier R; x = y ⟧
⟹ (nscal R x n) = (nscal R y n)"
by simp
lemma (in Ring) nsDistr: "x ∈ carrier R
⟹ (nscal R x n) ± (nscal R x m) = nscal R x (n + m)"
apply (cut_tac ring_is_ag)
apply (induct_tac m)
apply simp
apply (frule nsClose[of "x" "n"])
apply ( simp add:aGroup.ag_r_zero)
apply simp
apply (frule_tac x = x and n = n in nsClose,
frule_tac x = x and n = na in nsClose)
apply (subst aGroup.ag_pOp_assoc[THEN sym], assumption+, simp)
done
lemma (in Ring) nsDistrL: "⟦x ∈ carrier R; y ∈ carrier R ⟧
⟹ (nscal R x n) ± (nscal R y n) = nscal R (x ± y) n"
apply (cut_tac ring_is_ag)
apply (induct_tac n)
apply simp
apply (cut_tac ring_zero,
simp add:aGroup.ag_l_zero)
apply simp
apply (frule_tac x = x and n = n in nsClose,
frule_tac x = y and n = n in nsClose)
apply (subst aGroup.pOp_assocTr43[of R _ x _ y], assumption+)
apply (frule_tac x = x and y = "n ×⇘R⇙ y" in aGroup.ag_pOp_commute[of "R"],
assumption+)
apply simp
apply (subst aGroup.pOp_assocTr43[THEN sym, of R _ _ x y], assumption+)
apply simp
done
lemma (in Ring) nsMulDistrL:"⟦ x ∈ carrier R; y ∈ carrier R ⟧
⟹ x ⋅⇩r (nscal R y n) = nscal R (x ⋅⇩r y) n"
apply (induct_tac n)
apply simp
apply (simp add:ring_times_x_0)
apply simp apply (subst ring_distrib1, assumption+)
apply (rule nsClose, assumption+)
apply simp
done
lemma (in Ring) nsMulDistrR:"⟦ x ∈ carrier R; y ∈ carrier R⟧
⟹ (nscal R y n) ⋅⇩r x = nscal R (y ⋅⇩r x) n"
apply (frule_tac x = y and n = n in nsClose,
simp add:ring_tOp_commute[of "n ×⇘R⇙ y" "x"],
simp add:nsMulDistrL,
simp add:ring_tOp_commute[of "y" "x"])
done
subsection "npow"
lemma (in Ring) npClose:"x ∈ carrier R ⟹ npow R x n ∈ carrier R"
apply (induct_tac n)
apply simp apply (simp add:ring_one)
apply simp
apply (rule ring_tOp_closed, assumption+)
done
lemma (in Ring) npMulDistr:"⋀ n m. x ∈ carrier R ⟹
(npow R x n) ⋅⇩r (npow R x m) = npow R x (n + m)"
apply (induct_tac m)
apply simp apply (rule ring_r_one, simp add:npClose)
apply simp
apply (frule_tac x = x and n = n in npClose,
frule_tac x = x and n = na in npClose)
apply (simp add:ring_tOp_assoc[THEN sym])
done
lemma (in Ring) npMulExp:"⋀n m. x ∈ carrier R
⟹ npow R (npow R x n) m = npow R x (n * m)"
apply (induct_tac m)
apply simp
apply simp
apply (simp add:npMulDistr)
apply (simp add:add.commute)
done
lemma (in Ring) npGTPowZero_sub:
" ⋀ n. ⟦ x ∈ carrier R; npow R x m = 𝟬 ⟧
⟹(m ≤ n) ⟶ (npow R x n = 𝟬 )"
apply (rule impI)
apply (subgoal_tac "npow R x n = (npow R x (n-m)) ⋅⇩r (npow R x m)")
apply simp
apply (rule ring_times_x_0) apply (simp add:npClose)
apply (thin_tac "x^⇗R m⇖ = 𝟬")
apply (subst npMulDistr, assumption)
apply simp
done
lemma (in Ring) npGTPowZero:
"⋀ n. ⟦ x ∈ carrier R; npow R x m = 𝟬; m ≤ n ⟧
⟹ npow R x n = 𝟬"
apply (cut_tac x = x and m = m and n = n in npGTPowZero_sub, assumption+)
apply simp
done
lemma (in Ring) npOne: " npow R (1⇩r) n = 1⇩r"
apply (induct_tac n) apply simp
apply simp
apply (rule ring_r_one, simp add:ring_one)
done
lemma (in Ring) npZero_sub: "0 < n ⟶ npow R 𝟬 n = 𝟬"
apply (induct_tac "n")
apply simp
apply simp
apply (cut_tac ring_zero,
frule_tac n = n in npClose[of "𝟬"])
apply (simp add:ring_times_x_0)
done
lemma (in Ring) npZero: "0 < n ⟹ npow R 𝟬 n = 𝟬"
apply (simp add:npZero_sub)
done
lemma (in Ring) npMulElmL: "⋀ n. ⟦ x ∈ carrier R; 0 ≤ n⟧
⟹ x ⋅⇩r (npow R x n) = npow R x (Suc n)"
apply (simp only:npow_suc,
frule_tac n = n and x = x in npClose,
simp add:ring_tOp_commute)
done
lemma (in Ring) npMulEleL: "⋀ n. x ∈ carrier R
⟹ (npow R x n) ⋅⇩r x = npow R x (Suc n)"
by (simp add:npMulElmL[THEN sym])
lemma (in Ring) npMulElmR: "⋀ n. x ∈ carrier R
⟹ (npow R x n) ⋅⇩r x = npow R x (Suc n)"
apply ( frule_tac n = n in npClose[of "x"])
apply (simp only:ring_tOp_commute,
subst npMulElmL, assumption, simp, simp)
done
lemma (in Ring) np_1:"a ∈ carrier R ⟹ npow R a (Suc 0) = a"
apply simp
apply (simp add:ring_l_one)
done
subsection "nsum and fSum"
lemma (in aGroup) nsum_memTr: "(∀j ≤ n. f j ∈ carrier A) ⟶
nsum A f n ∈ carrier A"
apply (induct_tac "n")
apply simp
apply (rule impI)
apply (cut_tac n = n in Nsetn_sub_mem1, simp)
apply (frule_tac a = "Suc n" in forall_spec, simp,
thin_tac "∀j≤Suc n. f j ∈ carrier A")
apply (rule ag_pOp_closed, assumption+)
done
lemma (in aGroup) nsum_mem:"∀j ≤ n. f j ∈ carrier A ⟹
nsum A f n ∈ carrier A"
apply (simp add:nsum_memTr)
done
lemma (in aGroup) nsum_eqTr:"(∀j ≤ n. f j ∈ carrier A ∧
g j ∈ carrier A ∧
f j = g j)
⟶ nsum A f n = nsum A g n"
apply (induct_tac n)
apply simp
apply (rule impI)
apply (cut_tac n = n in Nsetn_sub_mem1, simp)
done
lemma (in aGroup) nsum_eq:"⟦∀j ≤ n. f j ∈ carrier A; ∀j ≤ n. g j ∈ carrier A;
∀j ≤ n. f j = g j⟧ ⟹ nsum A f n = nsum A g n"
by (simp add:nsum_eqTr)
lemma (in aGroup) nsum_cmp_assoc:"⟦∀j ≤ n. f j ∈ carrier A;
g ∈ {j. j ≤ n} → {j. j ≤ n}; h ∈ {j. j ≤ n} → {j. j ≤ n}⟧ ⟹
nsum A (cmp (cmp f h) g) n = nsum A (cmp f (cmp h g)) n"
apply (rule nsum_eq)
apply (rule allI, rule impI, simp add:cmp_def)
apply (frule_tac x = j in funcset_mem[of g "{j. j ≤ n}" "{j. j ≤ n}"], simp,
frule_tac x = "g j" in funcset_mem[of h "{j. j ≤ n}" "{j. j ≤ n}"],
assumption, simp)
apply (rule allI, rule impI, simp add:cmp_def,
frule_tac x = j in funcset_mem[of g "{j. j ≤ n}" "{j. j ≤ n}"], simp,
frule_tac x = "g j" in funcset_mem[of h "{j. j ≤ n}" "{j. j ≤ n}"],
assumption, simp)
apply (rule allI, simp add:cmp_def)
done
lemma (in aGroup) fSum_Suc:"∀j ∈ nset n (n + Suc m). f j ∈ carrier A ⟹
fSum A f n (n + Suc m) = fSum A f n (n + m) ± f (n + Suc m)"
by (simp add:fSum_def, simp add:cmp_def slide_def)
lemma (in aGroup) fSum_eqTr:"(∀j ∈ nset n (n + m). f j ∈ carrier A ∧
g j ∈ carrier A ∧ f j = g j) ⟶
fSum A f n (n + m) = fSum A g n (n + m)"
apply (induct_tac m)
apply (simp add:fSum_def,
simp add:cmp_def slide_def,
simp add:nset_def)
apply (rule impI)
apply (subst fSum_Suc,
rule ballI, simp, simp)
apply (cut_tac n = n and m = na and f = g in fSum_Suc,
rule ballI, simp, simp,
thin_tac "Σ⇩f A g n (Suc (n + na)) =
Σ⇩f A g n (n + na) ± g (Suc (n + na))")
apply (cut_tac n = n and m = na in nsetnm_sub_mem, simp,
thin_tac "∀j. j ∈ nset n (n + na) ⟶ j ∈ nset n (Suc (n + na))")
apply (frule_tac x = "Suc (n + na)" in bspec,
simp add:nset_def, simp)
done
lemma (in aGroup) fSum_eq:"⟦ ∀j ∈ nset n (n + m). f j ∈ carrier A;
∀j ∈ nset n (n + m). g j ∈ carrier A; (∀j∈ nset n (n + m). f j = g j)⟧
⟹
fSum A f n (n + m) = fSum A g n (n + m)"
by (simp add:fSum_eqTr)
lemma (in aGroup) fSum_eq1:"⟦n ≤ m; ∀j∈nset n m. f j ∈ carrier A;
∀j∈nset n m. g j ∈ carrier A; ∀j∈nset n m. f j = g j⟧ ⟹
fSum A f n m = fSum A g n m"
apply (cut_tac fSum_eq[of n "m - n" f g])
apply simp+
done
lemma (in aGroup) fSum_zeroTr:"(∀j ∈ nset n (n + m). f j = 𝟬) ⟶
fSum A f n (n + m) = 𝟬"
apply (induct_tac m)
apply (simp add:fSum_def cmp_def slide_def nset_def)
apply (rule impI)
apply (subst fSum_Suc)
apply (rule ballI, simp add:ag_inc_zero)
apply (frule_tac x = "n + Suc na" in bspec, simp add:nset_def,
simp)
apply (simp add:nset_def)
apply (cut_tac ag_inc_zero, simp add:ag_l_zero)
done
lemma (in aGroup) fSum_zero:"∀j ∈ nset n (n + m). f j = 𝟬 ⟹
fSum A f n (n + m) = 𝟬"
by (simp add:fSum_zeroTr)
lemma (in aGroup) fSum_zero1:"⟦n < m; ∀j ∈ nset (Suc n) m. f j = 𝟬⟧ ⟹
fSum A f (Suc n) m = 𝟬"
apply (cut_tac fSum_zero[of "Suc n" "m - Suc n" f])
apply simp+
done
lemma (in Ring) nsumMulEleL: "⋀ n. ⟦ ∀ i. f i ∈ carrier R; x ∈ carrier R ⟧
⟹ x ⋅⇩r (nsum R f n) = nsum R (λ i. x ⋅⇩r (f i)) n"
apply (cut_tac ring_is_ag)
apply (induct_tac "n")
apply simp
apply simp
apply (subst ring_distrib1, assumption)
apply (rule aGroup.nsum_mem, assumption)
apply (rule allI, simp+)
done
lemma (in Ring) nsumMulElmL:
"⋀ n. ⟦ ∀ i. f i ∈ carrier R; x ∈ carrier R ⟧
⟹ x ⋅⇩r (nsum R f n) = nsum R (λ i. x ⋅⇩r (f i)) n"
apply (cut_tac ring_is_ag)
apply (induct_tac "n")
apply simp
apply simp
apply (subst ring_distrib1, assumption+)
apply (simp add:aGroup.nsum_mem)+
done
lemma (in aGroup) nsumTailTr:
"(∀j≤(Suc n). f j ∈ carrier A) ⟶
nsum A f (Suc n) = (nsum A (λ i. (f (Suc i))) n) ± (f 0)"
apply (induct_tac "n")
apply simp
apply (rule impI,
rule ag_pOp_commute)
apply (cut_tac Nset_inc_0[of "Suc 0"],
simp add:Pi_def,
cut_tac n_in_Nsetn[of "Suc 0"],
simp add:Pi_def)
apply (rule impI)
apply (cut_tac n = "Suc n" in Nsetn_sub_mem1, simp)
apply (frule_tac a = 0 in forall_spec, simp,
frule_tac a = "Suc (Suc n)" in forall_spec, simp)
apply (cut_tac n = n in nsum_mem[of _ "λi. f (Suc i)"],
rule allI, rule impI,
frule_tac a = "Suc j" in forall_spec, simp, simp,
thin_tac "∀j≤Suc (Suc n). f j ∈ carrier A")
apply (subst ag_pOp_assoc, assumption+)
apply (simp add:ag_pOp_commute[of "f 0"])
apply (subst ag_pOp_assoc[THEN sym], assumption+)
apply simp
done
lemma (in aGroup) nsumTail:
"∀j ≤ (Suc n). f j ∈ carrier A ⟹
nsum A f (Suc n) = (nsum A (λ i. (f (Suc i))) n) ± (f 0)"
by (cut_tac nsumTailTr[of n f], simp)
lemma (in aGroup) nsumElmTail:
"∀i. f i ∈ carrier A
⟹ nsum A f (Suc n) = (nsum A (λ i. (f (Suc i))) n) ± (f 0)"
apply (cut_tac n = n and f = f in nsumTail,
rule allI, simp, simp)
done
lemma (in aGroup) nsum_addTr:
"(∀j ≤ n. f j ∈ carrier A ∧ g j ∈ carrier A) ⟶
nsum A (λ i. (f i) ± (g i)) n = (nsum A f n) ± (nsum A g n)"
apply (induct_tac "n")
apply simp
apply (simp, rule impI)
apply (cut_tac n = n in Nsetn_sub_mem1, simp)
apply (thin_tac "Σ⇩e A (λi. f i ± g i) n = Σ⇩e A f n ± Σ⇩e A g n")
apply (rule aGroup.ag_add4_rel, rule aGroup_axioms)
apply (rule aGroup.nsum_mem, rule aGroup_axioms, rule allI, simp)
apply (rule aGroup.nsum_mem, rule aGroup_axioms, rule allI, simp)
apply simp+
done
lemma (in aGroup) nsum_add:
"⟦ ∀j ≤ n. f j ∈ carrier A; ∀j ≤ n. g j ∈ carrier A⟧ ⟹
nsum A (λ i. (f i) ± (g i)) n = (nsum A f n) ± (nsum A g n)"
by (cut_tac nsum_addTr[of n f g], simp)
lemma (in aGroup) nsumElmAdd:
"⟦ ∀ i. f i ∈ carrier A; ∀ i. g i ∈ carrier A⟧
⟹ nsum A (λ i. (f i) ± (g i)) n = (nsum A f n) ± (nsum A g n)"
apply (cut_tac nsum_add[of n f g])
apply simp
apply (rule allI, simp)+
done
lemma (in aGroup) nsum_add_nmTr:
"(∀j ≤ n. f j ∈ carrier A) ∧ (∀j ≤ m. g j ∈ carrier A) ⟶
nsum A (jointfun n f m g) (Suc (n + m)) = (nsum A f n) ± (nsum A g m)"
apply (induct_tac m)
apply (simp add:jointfun_def sliden_def)
apply (rule impI)
apply (rule ag_pOp_add_r)
apply (rule nsum_mem, rule allI, erule conjE, rule impI, simp)
apply (erule conjE, simp add:nsum_mem, simp)
apply (rule nsum_eq[of n], simp+)
apply (simp add:jointfun_def)
apply (rule impI, simp)
apply (erule conjE, simp add:sliden_def)
apply (thin_tac "Σ⇩e A (λi. if i ≤ n then f i else g (sliden (Suc n) i))
(n + na) ± g na = Σ⇩e A f n ± Σ⇩e A g na")
apply (subst ag_pOp_assoc)
apply (simp add:nsum_mem)
apply (simp add:nsum_mem, simp)
apply simp
done
lemma (in aGroup) nsum_add_nm:
"⟦∀j ≤ n. f j ∈ carrier A; ∀j ≤ m. g j ∈ carrier A⟧ ⟹
nsum A (jointfun n f m g) (Suc (n + m)) = (nsum A f n) ± (nsum A g m)"
apply (cut_tac nsum_add_nmTr[of n f m g])
apply simp
done
lemma (in Ring) npeSum2_sub_muly:
"⟦ x ∈ carrier R; y ∈ carrier R ⟧ ⟹
y ⋅⇩r(nsum R (λi. nscal R ((npow R x (n-i)) ⋅⇩r (npow R y i))
(n choose i)) n)
= nsum R (λi. nscal R ((npow R x (n-i)) ⋅⇩r (npow R y (i+1)))
(n choose i)) n"
apply (cut_tac ring_is_ag)
apply (subst nsumMulElmL)
apply (rule allI)
apply (simp only:nsClose add:ring_tOp_closed
add:npClose)
apply assumption
apply (simp only:nsMulDistrL add:nsClose add:ring_tOp_closed
add:npClose)
apply (simp only: rMulLC [of "y"] add:npClose)
apply (simp del:npow_suc add:ring_tOp_commute[of y])
apply (rule aGroup.nsum_eq, assumption)
apply (rule allI, rule impI, rule nsClose,
rule ring_tOp_closed, simp add:npClose,
rule ring_tOp_closed, assumption, simp add:npClose)
apply (rule allI, rule impI, rule nsClose,
rule ring_tOp_closed, simp add:npClose,
rule npClose, assumption)
apply (rule allI, rule impI)
apply (frule_tac n = j in npClose[of y])
apply (simp add:ring_tOp_commute[of y])
done
lemma binomial_n0: "(Suc n choose 0) = (n choose 0)"
by simp
lemma binomial_ngt_diff:
"(n choose Suc n) = (Suc n choose Suc n) - (n choose n)"
by (subst binomial_Suc_Suc, arith)
lemma binomial_ngt_0: "(n choose Suc n) = 0"
apply (subst binomial_ngt_diff,
(subst binomial_n_n)+)
apply simp
done
lemma diffLessSuc: "m ≤ n ⟹ Suc (n-m) = Suc n - m"
by arith
lemma (in Ring) npow_suc_i:
"⟦ x ∈ carrier R; i ≤ n ⟧
⟹ npow R x (Suc n - i) = x ⋅⇩r (npow R x (n-i))"
apply (subst diffLessSuc [THEN sym, of "i" "n"], assumption)
apply (frule_tac n = "n - i" in npClose,
simp add:ring_tOp_commute[of x])
done
lemma (in Ring) npeSum2_sub_mulx: "⟦ x ∈ carrier R; y ∈ carrier R ⟧ ⟹
x ⋅⇩r (nsum R (λ i. nscal R ((npow R x (n-i)) ⋅⇩r (npow R y i))
(n choose i)) n)
= (nsum R (λi. nscal R
((npow R x (Suc n - Suc i)) ⋅⇩r (npow R y (Suc i)))
(n choose Suc i)) n) ±
(nscal R ((npow R x (Suc n - 0)) ⋅⇩r (npow R y 0))
(Suc n choose 0))"
apply (cut_tac ring_is_ag)
apply (simp only: binomial_n0)
apply (subst aGroup.nsumElmTail [THEN sym, of R "λ i. nscal R ((npow R x (Suc n - i)) ⋅⇩r (npow R y i)) (n choose i)"], assumption+)
apply (rule allI)
apply (simp only:nsClose add:ring_tOp_closed add:npClose)
apply (simp only:nsum_suc)
apply (subst binomial_ngt_0)
apply (simp only:nscal_0)
apply (subst aGroup.ag_r_zero, assumption)
apply (simp add:aGroup.nsum_mem nsClose ring_tOp_closed npClose)
apply (subst nsumMulElmL [of _ "x"])
apply (rule allI, rule nsClose, rule ring_tOp_closed, simp add:npClose,
simp add:npClose, assumption)
apply (simp add: nsMulDistrL [of "x"] ring_tOp_closed npClose)
apply (simp add:ring_tOp_assoc [THEN sym, of "x"] npClose)
apply (rule aGroup.nsum_eq, assumption)
apply (rule allI, rule impI,
rule nsClose, (rule ring_tOp_closed)+, assumption,
simp add:npClose, simp add:npClose)
apply (rule allI, rule impI,
rule nsClose, rule ring_tOp_closed,
simp add:npClose, simp add:npClose)
apply (rule allI, rule impI)
apply (frule_tac n = "n - j" in npClose[of x],
simp add:ring_tOp_commute[of x],
subst npow_suc[THEN sym])
apply (simp add:Suc_diff_le)
done
lemma (in Ring) npeSum2_sub_mulx2:
"⟦ x ∈ carrier R; y ∈ carrier R ⟧ ⟹
x ⋅⇩r (nsum R (λ i. nscal R ((npow R x (n-i)) ⋅⇩r (npow R y i))
(n choose i)) n)
= (nsum R (λi. nscal R
((npow R x (n - i)) ⋅⇩r ((npow R y i) ⋅⇩r y ))
(n choose Suc i)) n) ±
(𝟬 ± ((x ⋅⇩r (npow R x n)) ⋅⇩r (1⇩r)))"
apply (subst npeSum2_sub_mulx, assumption+, simp)
apply (frule npClose[of x n])
apply (subst ring_tOp_commute[of x], assumption+)
apply (cut_tac ring_is_ag)
apply (cut_tac aGroup.nsum_eq[of R n
"λi. (n choose Suc i) ×⇘R⇙ (x^⇗R (n - i)⇖ ⋅⇩r y^⇗R (Suc i)⇖)"
"λi. (n choose Suc i) ×⇘R⇙ (x^⇗R (n - i)⇖ ⋅⇩r (y^⇗R i⇖ ⋅⇩r y))"])
apply (simp del:npow_suc)+
apply (rule allI, rule impI,
rule nsClose, rule ring_tOp_closed, simp add:npClose,
simp only:npClose)
apply (rule allI, rule impI,
rule nsClose, rule ring_tOp_closed, simp add:npClose,
rule ring_tOp_closed, simp add:npClose, assumption)
apply (rule allI, rule impI)
apply (frule_tac n = j in npClose[of y])
apply simp
done
lemma (in Ring) npeSum2:
"⋀ n. ⟦ x ∈ carrier R; y ∈ carrier R ⟧
⟹ npow R (x ± y) n =
nsum R (λ i. nscal R ((npow R x (n-i)) ⋅⇩r (npow R y i))
( n choose i) ) n"
apply (cut_tac ring_is_ag)
apply (induct_tac "n")
apply simp
apply (cut_tac ring_one, simp add:ring_r_one, simp add:aGroup.ag_l_zero)
apply (subst aGroup.nsumElmTail, assumption+)
apply (rule allI)
apply (simp add:nsClose ring_tOp_closed npClose)
apply (simp only:binomial_Suc_Suc)
apply (simp only: nsDistr [THEN sym] add:npClose ring_tOp_closed)
apply (subst aGroup.nsumElmAdd, assumption+)
apply (rule allI,
simp add:nsClose ring_tOp_closed npClose)
apply (rule allI,
simp add:nsClose add:ring_tOp_closed npClose)
apply (subst aGroup.ag_pOp_assoc, assumption)
apply (rule aGroup.nsum_mem, assumption,
rule allI, rule impI, simp add:nsClose ring_tOp_closed npClose)
apply (rule aGroup.nsum_mem, assumption,
rule allI, rule impI, simp add:nsClose ring_tOp_closed npClose)
apply (simp add:nsClose ring_tOp_closed npClose)
apply (rule aGroup.ag_pOp_closed, assumption)
apply (simp add:aGroup.ag_inc_zero)
apply (rule ring_tOp_closed)+
apply (simp add:npClose, assumption, simp add:ring_one)
apply (subst npMulElmL [THEN sym, of "x ± y"],
simp add:aGroup.ag_pOp_closed, simp)
apply simp
apply (subst ring_distrib2 [of _ "x" "y"])
apply (rule aGroup.nsum_mem,assumption,
rule allI, rule impI, rule nsClose, rule ring_tOp_closed,
simp add:npClose, simp add:npClose, assumption+)
apply (rule aGroup.gEQAddcross [THEN sym], assumption+,
rule aGroup.nsum_mem, assumption, rule allI, rule impI, rule nsClose,
(rule ring_tOp_closed)+, simp add:npClose,
rule ring_tOp_closed, simp add:npClose, assumption)
apply (rule aGroup.ag_pOp_closed, assumption)
apply (rule aGroup.nsum_mem, assumption,
rule allI, rule impI, rule nsClose, rule ring_tOp_closed,
simp add:npClose, rule ring_tOp_closed, simp add:npClose, assumption)
apply (rule aGroup.ag_pOp_closed, assumption, simp add:ring_zero)
apply ((rule ring_tOp_closed)+,
simp add:npClose,assumption, simp add:ring_one)
apply (rule ring_tOp_closed, assumption,
rule aGroup.nsum_mem, assumption, rule allI, rule impI,
rule nsClose, rule ring_tOp_closed,
(simp add:npClose)+)
apply (rule ring_tOp_closed, assumption+,
rule aGroup.nsum_mem, assumption, rule allI, rule impI,
rule nsClose,
rule ring_tOp_closed,
simp add:npClose, simp add:npClose)
apply (subst npeSum2_sub_muly [of "x" "y"], assumption+, simp)
apply (subst npeSum2_sub_mulx2 [of x y], assumption+)
apply (frule_tac n = na in npClose[of x],
simp add:ring_tOp_commute[of _ x])
done
lemma (in aGroup) nsum_zeroTr:
"⋀ n. (∀ i. i ≤ n ⟶ f i = 𝟬) ⟶ (nsum A f n = 𝟬)"
apply (induct_tac "n")
apply simp
apply (rule impI)
apply (cut_tac n = na in Nsetn_sub_mem1, simp)
apply (subst aGroup.ag_l_zero, rule aGroup_axioms)
apply (simp add:ag_inc_zero)
apply simp
done
lemma (in Ring) npAdd:
"⟦ x ∈ carrier R; y ∈ carrier R;
npow R x m = 𝟬; npow R y n = 𝟬 ⟧
⟹ npow R (x ± y) (m + n) = 𝟬"
apply (subst npeSum2, assumption+)
apply (rule aGroup.nsum_zeroTr [THEN mp])
apply (simp add:ring_is_ag)
apply (rule allI, rule impI)
apply (rule nsZeroI)
apply (rule rMulZeroDiv, simp add:npClose, simp add:npClose)
apply (case_tac "i ≤ n")
apply (rule disjI1)
apply (rule npGTPowZero [of "x" "m"], assumption+)
apply arith
apply (rule disjI2)
apply (rule npGTPowZero [of "y" "n"], assumption+)
apply (arith)
done
lemma (in Ring) npInverse:
"⋀n. x ∈ carrier R
⟹ npow R (-⇩a x) n = npow R x n
∨ npow R (-⇩a x) n = -⇩a (npow R x n)"
apply (induct_tac n)
apply simp
apply (erule disjE)
apply simp
apply (subst ring_inv1_2,
simp add:npClose, assumption, simp)
apply (cut_tac ring_is_ag)
apply simp
apply (subst ring_inv1_2[THEN sym, of _ x])
apply (rule aGroup.ag_mOp_closed, assumption+,
simp add:npClose, assumption)
apply (thin_tac "(-⇩a x)^⇗R na⇖ = -⇩a (x^⇗R na⇖)",
frule_tac n = na in npClose[of x],
frule_tac x = "x^⇗R na⇖" in aGroup.ag_mOp_closed[of R], simp add:npClose)
apply (simp add: ring_inv1_1[of _ x])
apply (simp add:aGroup.ag_inv_inv[of R])
done
lemma (in Ring) npMul:
"⋀ n. ⟦ x ∈ carrier R; y ∈ carrier R ⟧
⟹ npow R (x ⋅⇩r y) n = (npow R x n) ⋅⇩r (npow R y n)"
apply (induct_tac "n")
apply simp
apply (rule ring_r_one [THEN sym]) apply (simp add:ring_one)
apply (simp only:npow_suc)
apply (rule ring_tOp_rel[THEN sym])
apply (rule npClose, assumption+)+
done
section "Ring homomorphisms"
definition
rHom :: "[('a, 'm) Ring_scheme, ('b, 'm1) Ring_scheme]
⇒ ('a ⇒ 'b) set" where
"rHom A R = {f. f ∈ aHom A R ∧
(∀x∈carrier A. ∀y∈carrier A. f ( x ⋅⇩r⇘A⇙ y) = (f x) ⋅⇩r⇘R⇙ (f y))
∧ f (1⇩r⇘A⇙) = (1⇩r⇘R⇙)}"
definition
rInvim :: "[('a, 'm) Ring_scheme, ('b, 'm1) Ring_scheme, 'a ⇒ 'b, 'b set]
⇒ 'a set" where
"rInvim A R f K = {a. a ∈ carrier A ∧ f a ∈ K}"
definition
rimg :: "[('a, 'm) Ring_scheme, ('b, 'm1) Ring_scheme, 'a ⇒ 'b] ⇒
'b Ring" where
"rimg A R f = ⦇carrier= f `(carrier A), pop = pop R, mop = mop R,
zero = zero R, tp = tp R, un = un R ⦈"
definition
ridmap :: "('a, 'm) Ring_scheme ⇒ ('a ⇒ 'a)" where
"ridmap R = (λx∈carrier R. x)"
definition
r_isom :: "[('a, 'm) Ring_scheme, ('b, 'm1) Ring_scheme] ⇒ bool"
(infixr "≅⇩r" 100) where
"r_isom R R' ⟷ (∃f∈rHom R R'. bijec⇘R,R'⇙ f)"
definition
Subring :: "[('a, 'm) Ring_scheme, ('a, 'm1) Ring_scheme] ⇒ bool" where
"Subring R S == Ring S ∧ (carrier S ⊆ carrier R) ∧ (ridmap S) ∈ rHom S R"
lemma ridmap_surjec:"Ring A ⟹ surjec⇘A,A⇙ (ridmap A)"
by(simp add:surjec_def aHom_def ridmap_def Ring.ring_is_ag aGroup.ag_pOp_closed surj_to_def)
lemma rHom_aHom:"f ∈ rHom A R ⟹ f ∈ aHom A R"
by (simp add:rHom_def)
lemma rimg_carrier:"f ∈ rHom A R ⟹ carrier (rimg A R f) = f ` (carrier A)"
by (simp add:rimg_def)
lemma rHom_mem:"⟦ f ∈ rHom A R; a ∈ carrier A ⟧ ⟹ f a ∈ carrier R"
apply (simp add:rHom_def, frule conjunct1)
apply (thin_tac "f ∈ aHom A R ∧
(∀x∈carrier A. ∀y∈carrier A. f (x ⋅⇩r⇘A⇙ y) = f x ⋅⇩r⇘R⇙ f y) ∧ f 1⇩r⇘A⇙ = 1⇩r⇘R⇙")
apply (simp add:aHom_def, frule conjunct1)
apply (thin_tac "f ∈ carrier A → carrier R ∧
f ∈ extensional (carrier A) ∧
(∀a∈carrier A. ∀b∈carrier A. f (a ±⇘A⇙ b) = f a ±⇘R⇙ f b)")
apply (simp add:funcset_mem)
done
lemma rHom_func:"f ∈ rHom A R ⟹ f ∈ carrier A → carrier R"
by (simp add:rHom_def aHom_def)
lemma ringhom1:"⟦ Ring A; Ring R; x ∈ carrier A; y ∈ carrier A;
f ∈ rHom A R ⟧ ⟹ f (x ±⇘A⇙ y) = (f x) ±⇘R⇙ (f y)"
apply (simp add:rHom_def) apply (erule conjE)
apply (frule Ring.ring_is_ag [of "A"])
apply (frule Ring.ring_is_ag [of "R"])
apply (rule aHom_add, assumption+)
done
lemma rHom_inv_inv:"⟦ Ring A; Ring R; x ∈ carrier A; f ∈ rHom A R ⟧
⟹ f (-⇩a⇘A⇙ x) = -⇩a⇘R⇙ (f x)"
apply (frule Ring.ring_is_ag [of "A"],
frule Ring.ring_is_ag [of "R"])
apply (simp add:rHom_def, erule conjE)
apply (simp add:aHom_inv_inv)
done
lemma rHom_0_0:"⟦ Ring A; Ring R; f ∈ rHom A R ⟧ ⟹ f (𝟬⇘A⇙) = 𝟬⇘R⇙"
apply (frule Ring.ring_is_ag [of "A"], frule Ring.ring_is_ag [of "R"])
apply (simp add:rHom_def, (erule conjE)+, simp add:aHom_0_0)
done
lemma rHom_tOp:"⟦ Ring A; Ring R; x ∈ carrier A; y ∈ carrier A;
f ∈ rHom A R ⟧ ⟹ f (x ⋅⇩r⇘A⇙ y) = (f x) ⋅⇩r⇘R⇙ (f y)"
by (simp add:rHom_def)
lemma rHom_add:"⟦f ∈ rHom A R; x ∈ carrier A; y ∈ carrier A⟧ ⟹
f (x ±⇘A⇙ y) = (f x) ±⇘R⇙ (f y)"
by (simp add:rHom_def aHom_def)
lemma rHom_one:"⟦ Ring A; Ring R;f ∈ rHom A R ⟧ ⟹ f (1⇩r⇘A⇙) = (1⇩r⇘R⇙)"
by (simp add:rHom_def)
lemma rHom_npow:"⟦ Ring A; Ring R; x ∈ carrier A; f ∈ rHom A R ⟧ ⟹
f (x^⇗A n⇖) = (f x)^⇗R n⇖"
apply (induct_tac n)
apply (simp add:rHom_one)
apply (simp,
frule_tac n = n in Ring.npClose[of "A" "x"], assumption+,
subst rHom_tOp[of "A" "R" _ "x" "f"], assumption+, simp)
done
lemma rHom_compos:"⟦Ring A; Ring B; Ring C; f ∈ rHom A B; g ∈ rHom B C⟧ ⟹
compos A g f ∈ rHom A C"
apply (subst rHom_def, simp)
apply (frule Ring.ring_is_ag[of "A"], frule Ring.ring_is_ag[of "B"],
frule Ring.ring_is_ag[of "C"],
frule rHom_aHom[of "f" "A" "B"], frule rHom_aHom[of "g" "B" "C"],
simp add:aHom_compos)
apply (rule conjI)
apply ((rule ballI)+, simp add:compos_def compose_def,
frule_tac x = x and y = y in Ring.ring_tOp_closed[of "A"], assumption+,
simp)
apply (simp add:rHom_tOp)
apply (frule_tac a = x in rHom_mem[of "f" "A" "B"], assumption+,
frule_tac a = y in rHom_mem[of "f" "A" "B"], assumption+,
simp add:rHom_tOp)
apply (frule Ring.ring_one[of "A"], frule Ring.ring_one[of "B"],
simp add:compos_def compose_def, simp add:rHom_one)
done
lemma rimg_ag:"⟦Ring A; Ring R; f ∈ rHom A R⟧ ⟹ aGroup (rimg A R f)"
apply (frule Ring.ring_is_ag [of "A"],
frule Ring.ring_is_ag [of "R"])
apply (simp add:rHom_def, (erule conjE)+)
apply (subst aGroup_def)
apply (simp add:rimg_def)
apply (rule conjI)
apply (rule Pi_I)+
apply (simp add:image_def)
apply (erule bexE)+
apply simp
apply (subst aHom_add [THEN sym, of "A" "R" "f"], assumption+)
apply (blast dest: aGroup.ag_pOp_closed)
apply (rule conjI)
apply ((rule allI, rule impI)+, simp add:image_def, (erule bexE)+, simp)
apply (frule_tac x = x and y = xa in aGroup.ag_pOp_closed, assumption+,
frule_tac x = xa and y = xb in aGroup.ag_pOp_closed, assumption+)
apply (simp add:aHom_add[of "A" "R" "f", THEN sym] aGroup.ag_pOp_assoc)
apply (rule conjI)
apply ((rule allI, rule impI)+, simp add:image_def, (erule bexE)+, simp)
apply (simp add:aHom_add[of "A" "R" "f", THEN sym] aGroup.ag_pOp_commute)
apply (rule conjI)
apply (rule Pi_I)
apply (simp add:image_def, erule bexE, simp)
apply (simp add:aHom_inv_inv[THEN sym],
frule_tac x = xa in aGroup.ag_mOp_closed[of "A"], assumption+, blast)
apply (rule conjI)
apply (rule allI, rule impI, simp add:image_def, (erule bexE)+, simp)
apply (simp add:aHom_inv_inv[THEN sym],
frule_tac x = x in aGroup.ag_mOp_closed[of "A"], assumption+,
simp add:aHom_add[of "A" "R" "f", THEN sym])
apply (simp add:aGroup.ag_l_inv1 aHom_0_0)
apply (rule conjI)
apply (simp add:image_def)
apply (frule aHom_0_0[THEN sym, of "A" "R" "f"], assumption+,
frule Ring.ring_zero[of "A"], blast)
apply (rule allI, rule impI,
simp add:image_def, erule bexE,
frule_tac a = x in aHom_mem[of "A" "R" "f"], assumption+, simp)
apply (simp add:aGroup.ag_l_zero)
done
lemma rimg_ring:"⟦Ring A; Ring R; f ∈ rHom A R ⟧ ⟹ Ring (rimg A R f)"
apply (unfold Ring_def [of "rimg A R f"])
apply (frule rimg_ag[of "A" "R" "f"], assumption+)
apply (rule conjI, simp add:aGroup_def[of "rimg A R f"])
apply(rule conjI)
apply (rule conjI, rule allI, rule impI)
apply (frule aGroup.ag_inc_zero[of "rimg A R f"],
subst aGroup.ag_pOp_commute, assumption+,
simp add:aGroup.ag_r_zero[of "rimg A R f"])
apply (rule conjI)
apply (rule Pi_I)+
apply (thin_tac "aGroup (rimg A R f)",
simp add:rimg_def, simp add:image_def, (erule bexE)+,
simp add:rHom_tOp[THEN sym])
apply (blast dest:Ring.ring_tOp_closed)
apply ((rule allI)+, (rule impI)+)
apply (thin_tac "aGroup (rimg A R f)", simp add:rimg_def,
simp add:image_def, (erule bexE)+, simp)
apply (frule_tac x = x and y = xa in Ring.ring_tOp_closed, assumption+,
frule_tac x = xa and y = xb in Ring.ring_tOp_closed, assumption+,
simp add:rHom_tOp[THEN sym],
simp add:Ring.ring_tOp_assoc)
apply (rule conjI, rule conjI, (rule allI)+, (rule impI)+)
apply (thin_tac "aGroup (rimg A R f)", simp add:rimg_def,
simp add:image_def, (erule bexE)+, simp,
simp add:rHom_tOp[THEN sym],
simp add:Ring.ring_tOp_commute)
apply (thin_tac "aGroup (rimg A R f)", simp add:rimg_def,
simp add:image_def)
apply (subst rHom_one [THEN sym, of "A" "R" "f"], assumption+,
frule Ring.ring_one[of "A"], blast)
apply (rule conjI, (rule allI)+, (rule impI)+)
apply (simp add:rimg_def, fold rimg_def,
simp add:image_def, (erule bexE)+, simp)
apply (frule rHom_aHom[of "f" "A" "R"],
frule Ring.ring_is_ag [of "A"],
frule Ring.ring_is_ag [of "R"],
simp add:aHom_add[THEN sym],
simp add:rHom_tOp[THEN sym])
apply (frule_tac x = xa and y = xb in aGroup.ag_pOp_closed[of "A"],
assumption+,
frule_tac x = x and y = xa in Ring.ring_tOp_closed[of "A"],
assumption+,
frule_tac x = x and y = xb in Ring.ring_tOp_closed[of "A"],
assumption+,
simp add:aHom_add[THEN sym],
simp add:rHom_tOp[THEN sym],
simp add:Ring.ring_distrib1)
apply (rule allI, rule impI,
thin_tac "aGroup (rimg A R f)")
apply (simp add:rimg_def,
simp add:image_def, erule bexE, simp add:rHom_tOp[THEN sym],
frule_tac a = x in rHom_mem[of "f" "A" "R"], assumption+,
simp add:Ring.ring_l_one)
done
definition
ideal :: "[_ , 'a set] ⇒ bool" where
"ideal R I ⟷ (R +> I) ∧ (∀r∈carrier R. ∀x∈I. (r ⋅⇩r⇘R⇙ x ∈ I))"
lemma (in Ring) ideal_asubg:"ideal R I ⟹ R +> I"
by (simp add:ideal_def)
lemma (in Ring) ideal_pOp_closed:"⟦ideal R I; x ∈ I; y ∈ I ⟧
⟹ x ± y ∈ I"
apply (unfold ideal_def, frule conjunct1, fold ideal_def)
apply (cut_tac ring_is_ag,
simp add:aGroup.asubg_pOp_closed)
done
lemma (in Ring) ideal_nsum_closedTr:"ideal R I ⟹
(∀j ≤ n. f j ∈ I) ⟶ nsum R f n ∈ I"
apply (induct_tac n)
apply (rule impI)
apply simp
apply (rule impI)
apply (cut_tac n = n in Nsetn_sub_mem1, simp)
apply (rule ideal_pOp_closed, assumption+)
apply simp
done
lemma (in Ring) ideal_nsum_closed:"⟦ideal R I; ∀j ≤ n. f j ∈ I⟧ ⟹
nsum R f n ∈ I"
by (simp add:ideal_nsum_closedTr)
lemma (in Ring) ideal_subset1:"ideal R I ⟹ I ⊆ carrier R"
apply (unfold ideal_def, frule conjunct1, fold ideal_def)
apply (simp add:asubGroup_def sg_def, (erule conjE)+)
apply (cut_tac ring_is_ag,
simp add:aGroup.ag_carrier_carrier)
done
lemma (in Ring) ideal_subset:"⟦ideal R I; h ∈ I⟧ ⟹ h ∈ carrier R"
by (frule ideal_subset1[of "I"],
simp add:subsetD)
lemma (in Ring) ideal_ring_multiple:"⟦ideal R I; x ∈ I; r ∈ carrier R⟧ ⟹
r ⋅⇩r x ∈ I"
by (simp add:ideal_def)
lemma (in Ring) ideal_ring_multiple1:"⟦ideal R I; x ∈ I; r ∈ carrier R ⟧ ⟹
x ⋅⇩r r ∈ I"
apply (frule ideal_subset[of "I" "x"], assumption+)
apply (simp add:ring_tOp_commute ideal_ring_multiple)
done
lemma (in Ring) ideal_npow_closedTr:"⟦ideal R I; x ∈ I⟧ ⟹
0 < n ⟶ x^⇗R n⇖ ∈ I"
apply (induct_tac n,
simp)
apply (rule impI)
apply simp
apply (case_tac "n = 0", simp)
apply (frule ideal_subset[of "I" "x"], assumption+,
simp add:ring_l_one)
apply simp
apply (frule ideal_subset[of "I" "x"], assumption+,
rule ideal_ring_multiple, assumption+,
simp add:ideal_subset)
done
lemma (in Ring) ideal_npow_closed:"⟦ideal R I; x ∈ I; 0 < n⟧ ⟹ x^⇗R n⇖ ∈ I"
by (simp add:ideal_npow_closedTr)
lemma (in Ring) times_modTr:"⟦a ∈ carrier R; a' ∈ carrier R; b ∈ carrier R;
b' ∈ carrier R; ideal R I; a ± (-⇩a b) ∈ I; a' ± (-⇩a b') ∈ I⟧ ⟹
a ⋅⇩r a' ± (-⇩a (b ⋅⇩r b')) ∈ I"
apply (cut_tac ring_is_ag)
apply (subgoal_tac "a ⋅⇩r a' ± (-⇩a (b ⋅⇩r b')) = a ⋅⇩r a' ± (-⇩a (a ⋅⇩r b'))
± (a ⋅⇩r b' ± (-⇩a (b ⋅⇩r b')))")
apply simp
apply (simp add:ring_inv1_2[of "a" "b'"], simp add:ring_inv1_1[of "b" "b'"])
apply (frule aGroup.ag_mOp_closed[of "R" "b'"], assumption+)
apply (simp add:ring_distrib1[THEN sym, of "a" "a'" "-⇩a b'"])
apply (frule aGroup.ag_mOp_closed[of "R" "b"], assumption+)
apply (frule ring_distrib2[THEN sym, of "b'" "a" "-⇩a b" ], assumption+)
apply simp
apply (thin_tac "a ⋅⇩r a' ± (-⇩a b) ⋅⇩r b' = a ⋅⇩r (a' ± -⇩a b') ± (a ± -⇩a b) ⋅⇩r b'",
thin_tac "a ⋅⇩r b' ± (-⇩a b) ⋅⇩r b' = (a ± -⇩a b) ⋅⇩r b'")
apply (frule ideal_ring_multiple[of "I" "a' ± (-⇩a b')" "a"], assumption+,
frule ideal_ring_multiple1[of "I" "a ± (-⇩a b)" "b'"], assumption+)
apply (simp add:ideal_pOp_closed)
apply (frule ring_tOp_closed[of "a" "a'"], assumption+,
frule ring_tOp_closed[of "a" "b'"], assumption+,
frule ring_tOp_closed[of "b" "b'"], assumption+,
frule aGroup.ag_mOp_closed[of "R" "b ⋅⇩r b'"], assumption+,
frule aGroup.ag_mOp_closed[of "R" "a ⋅⇩r b'"], assumption+)
apply (subst aGroup.ag_pOp_assoc[of "R"], assumption+)
apply (rule aGroup.ag_pOp_closed, assumption+)
apply (simp add:aGroup.ag_pOp_assoc[THEN sym, of "R" "-⇩a (a ⋅⇩r b')" "a ⋅⇩r b'"
"-⇩a (b ⋅⇩r b')"],
simp add:aGroup.ag_l_inv1 aGroup.ag_l_zero)
done
lemma (in Ring) ideal_inv1_closed:"⟦ ideal R I; x ∈ I ⟧ ⟹ -⇩a x ∈ I"
apply (cut_tac ring_is_ag)
apply (unfold ideal_def, frule conjunct1, fold ideal_def)
apply (simp add:aGroup.asubg_mOp_closed[of "R" "I"])
done
lemma (in Ring) ideal_zero:"ideal R I ⟹ 𝟬 ∈ I"
apply (cut_tac ring_is_ag)
apply (unfold ideal_def, frule conjunct1, fold ideal_def)
apply (simp add:aGroup.asubg_inc_zero)
done
lemma (in Ring) ideal_zero_forall:"∀I. ideal R I ⟶ 𝟬 ∈ I"
by (simp add:ideal_zero)
lemma (in Ring) ideal_ele_sumTr1:"⟦ ideal R I; a ∈ carrier R; b ∈ carrier R;
a ± b ∈ I; a ∈ I ⟧ ⟹ b ∈ I"
apply (frule ideal_inv1_closed[of "I" "a"], assumption+)
apply (frule ideal_pOp_closed[of "I" "-⇩a a" "a ± b"], assumption+)
apply (frule ideal_subset[of "I" "-⇩a a"], assumption+)
apply (cut_tac ring_is_ag,
simp add:aGroup.ag_pOp_assoc[THEN sym],
simp add:aGroup.ag_l_inv1,
simp add:aGroup.ag_l_zero)
done
lemma (in Ring) ideal_ele_sumTr2:"⟦ideal R I; a ∈ carrier R; b ∈ carrier R;
a ± b ∈ I; b ∈ I⟧ ⟹ a ∈ I"
apply (cut_tac ring_is_ag,
simp add:aGroup.ag_pOp_commute[of "R" "a" "b"])
apply (simp add:ideal_ele_sumTr1[of "I" "b" "a"])
done
lemma (in Ring) ideal_condition:"⟦I ⊆ carrier R; I ≠ {};
∀x∈I. ∀y∈I. x ± (-⇩a y) ∈ I; ∀r∈carrier R. ∀x∈I. r ⋅⇩r x ∈ I ⟧ ⟹
ideal R I"
apply (simp add:ideal_def)
apply (cut_tac ring_is_ag)
apply (rule aGroup.asubg_test[of "R" "I"], assumption+)
done
lemma (in Ring) ideal_condition1:"⟦I ⊆ carrier R; I ≠ {};
∀x∈I. ∀y∈I. x ± y ∈ I; ∀r∈carrier R. ∀x∈I. r ⋅⇩r x ∈ I ⟧ ⟹ ideal R I"
apply (rule ideal_condition[of "I"], assumption+)
apply (rule ballI)+
apply (cut_tac ring_is_ag,
cut_tac ring_one,
frule aGroup.ag_mOp_closed[of "R" "1⇩r"], assumption+)
apply (frule_tac x = "-⇩a 1⇩r " in bspec, assumption+,
thin_tac "∀r∈carrier R. ∀x∈I. r ⋅⇩r x ∈ I",
rotate_tac -1,
frule_tac x = y in bspec, assumption,
thin_tac "∀x∈I. (-⇩a 1⇩r) ⋅⇩r x ∈ I")
apply (frule_tac c = y in subsetD[of "I" "carrier R"], assumption+,
simp add:ring_times_minusl[THEN sym], simp add:ideal_pOp_closed)
done
lemma (in Ring) zero_ideal:"ideal R {𝟬}"
apply (cut_tac ring_is_ag)
apply (rule ideal_condition1)
apply (simp add:ring_zero)
apply simp
apply simp
apply (cut_tac ring_zero, simp add:aGroup.ag_l_zero)
apply simp
apply (rule ballI, simp add:ring_times_x_0)
done
lemma (in Ring) whole_ideal:"ideal R (carrier R)"
apply (rule ideal_condition1)
apply simp
apply (cut_tac ring_zero, blast)
apply (cut_tac ring_is_ag,
simp add:aGroup.ag_pOp_closed,
simp add:ring_tOp_closed)
done
lemma (in Ring) ideal_inc_one:"⟦ideal R I; 1⇩r ∈ I ⟧ ⟹ I = carrier R"
apply (rule equalityI)
apply (simp add:ideal_subset1)
apply (rule subsetI,
frule_tac r = x in ideal_ring_multiple[of "I" "1⇩r"], assumption+,
simp add:ring_r_one)
done
lemma (in Ring) ideal_inc_one1:"ideal R I ⟹
(1⇩r ∈ I) = (I = carrier R)"
apply (rule iffI)
apply (simp add:ideal_inc_one)
apply (frule sym, thin_tac "I = carrier R",
cut_tac ring_one, simp)
done
definition
Unit :: "_ ⇒ 'a ⇒ bool" where
"Unit R a ⟷ a ∈ carrier R ∧ (∃b∈carrier R. a ⋅⇩r⇘R⇙ b = 1⇩r⇘R⇙)"
lemma (in Ring) ideal_inc_unit:"⟦ideal R I; a ∈ I; Unit R a⟧ ⟹ 1⇩r ∈ I"
by (simp add:Unit_def, erule conjE, erule bexE,
frule_tac r = b in ideal_ring_multiple1[of "I" "a"], assumption+,
simp)
lemma (in Ring) proper_ideal:"⟦ideal R I; 1⇩r ∉ I⟧ ⟹ I ≠ carrier R"
apply (rule contrapos_pp, simp+)
apply (simp add: ring_one)
done
lemma (in Ring) ideal_inc_unit1:"⟦a ∈ carrier R; Unit R a; ideal R I; a ∈ I⟧
⟹ I = carrier R"
apply (frule ideal_inc_unit[of "I" "a"], assumption+)
apply (rule ideal_inc_one[of "I"], assumption+)
done
lemma (in Ring) int_ideal:"⟦ideal R I; ideal R J⟧ ⟹ ideal R (I ∩ J)"
apply (rule ideal_condition1)
apply (frule ideal_subset1[of "I"], frule ideal_subset1[of "J"])
apply blast
apply (frule ideal_zero[of "I"], frule ideal_zero[of "J"], blast)
apply ((rule ballI)+, simp, (erule conjE)+,
simp add:ideal_pOp_closed)
apply ((rule ballI)+, simp, (erule conjE)+)
apply (simp add:ideal_ring_multiple)
done
definition
ideal_prod::"[_, 'a set, 'a set] ⇒ 'a set" (infix "♢⇩rı" 90 ) where
"ideal_prod R I J == ⋂ {L. ideal R L ∧
{x.(∃i∈I. ∃j∈J. x = i ⋅⇩r⇘R⇙ j)} ⊆ L}"
lemma (in Ring) set_sum_mem:"⟦a ∈ I; b ∈ J; I ⊆ carrier R; J ⊆ carrier R⟧ ⟹
a ± b ∈ I ∓ J"
apply (cut_tac ring_is_ag)
apply (simp add:aGroup.set_sum, blast)
done
lemma (in Ring) sum_ideals:"⟦ideal R I1; ideal R I2⟧ ⟹ ideal R (I1 ∓ I2)"
apply (cut_tac ring_is_ag)
apply (frule ideal_subset1[of "I1"], frule ideal_subset1[of "I2"])
apply (rule ideal_condition1)
apply (rule subsetI, simp add:aGroup.set_sum, (erule bexE)+)
apply (frule_tac h = h in ideal_subset[of "I1"], assumption+,
frule_tac h = k in ideal_subset[of "I2"], assumption+,
cut_tac ring_is_ag,
simp add:aGroup.ag_pOp_closed)
apply (frule ideal_zero[of "I1"], frule ideal_zero[of "I2"],
frule set_sum_mem[of "𝟬" "I1" "𝟬" "I2"], assumption+, blast)
apply (rule ballI)+
apply (simp add:aGroup.set_sum, (erule bexE)+, simp)
apply (rename_tac x y i ia j ja)
apply (frule_tac h = i in ideal_subset[of "I1"], assumption+,
frule_tac h = ia in ideal_subset[of "I1"], assumption+,
frule_tac h = j in ideal_subset[of "I2"], assumption+,
frule_tac h = ja in ideal_subset[of "I2"], assumption+)
apply (subst aGroup.pOp_assocTr43, assumption+)
apply (frule_tac x = j and y = ia in aGroup.ag_pOp_commute[of "R"],
assumption+, simp)
apply (subst aGroup.pOp_assocTr43[THEN sym], assumption+)
apply (frule_tac x = i and y = ia in ideal_pOp_closed[of "I1"], assumption+,
frule_tac x = j and y = ja in ideal_pOp_closed[of "I2"], assumption+,
blast)
apply (rule ballI)+
apply (simp add:aGroup.set_sum, (erule bexE)+, simp)
apply (rename_tac r x i j)
apply (frule_tac h = i in ideal_subset[of "I1"], assumption+,
frule_tac h = j in ideal_subset[of "I2"], assumption+)
apply (simp add:ring_distrib1)
apply (frule_tac x = i and r = r in ideal_ring_multiple[of "I1"], assumption+,
frule_tac x = j and r = r in ideal_ring_multiple[of "I2"], assumption+,
blast)
done
lemma (in Ring) sum_ideals_la1:"⟦ideal R I1; ideal R I2⟧ ⟹ I1 ⊆ (I1 ∓ I2)"
apply (cut_tac ring_is_ag)
apply (rule subsetI)
apply (frule ideal_zero[of "I2"],
frule_tac h = x in ideal_subset[of "I1"], assumption+,
frule_tac x = x in aGroup.ag_r_zero[of "R"], assumption+)
apply (subst aGroup.set_sum, assumption,
simp add:ideal_subset1, simp add:ideal_subset1, simp,
frule sym, thin_tac "x ± 𝟬 = x", blast)
done
lemma (in Ring) sum_ideals_la2:"⟦ideal R I1; ideal R I2 ⟧ ⟹ I2 ⊆ (I1 ∓ I2)"
apply (cut_tac ring_is_ag)
apply (rule subsetI)
apply (frule ideal_zero[of "I1"],
frule_tac h = x in ideal_subset[of "I2"], assumption+,
frule_tac x = x in aGroup.ag_l_zero[of "R"], assumption+)
apply (subst aGroup.set_sum, assumption,
simp add:ideal_subset1, simp add:ideal_subset1, simp,
frule sym, thin_tac "𝟬 ± x = x", blast)
done
lemma (in Ring) sum_ideals_cont:"⟦ideal R I; A ⊆ I; B ⊆ I ⟧ ⟹ A ∓ B ⊆ I"
apply (cut_tac ring_is_ag)
apply (rule subsetI)
apply (frule ideal_subset1[of I],
frule subset_trans[of A I "carrier R"], assumption+,
frule subset_trans[of B I "carrier R"], assumption+)
apply (simp add:aGroup.set_sum[of R], (erule bexE)+, simp)
apply (frule_tac c = h in subsetD[of "A" "I"], assumption+,
frule_tac c = k in subsetD[of "B" "I"], assumption+)
apply (simp add:ideal_pOp_closed)
done
lemma (in Ring) ideals_set_sum:"⟦ideal R A; ideal R B; x ∈ A ∓ B⟧ ⟹
∃h∈A. ∃k∈B. x = h ± k"
apply (frule ideal_subset1[of A],
frule ideal_subset1[of B])
apply (cut_tac ring_is_ag,
simp add:aGroup.set_sum)
done
definition
Rxa :: "[_, 'a ] ⇒ 'a set" (infixl "♢⇩p" 200) where
"Rxa R a = {x. ∃r∈carrier R. x = (r ⋅⇩r⇘R⇙ a)}"
lemma (in Ring) a_in_principal:"a ∈ carrier R ⟹ a ∈ Rxa R a"
apply (cut_tac ring_one,
frule ring_l_one[THEN sym, of "a"])
apply (simp add:Rxa_def, blast)
done
lemma (in Ring) principal_ideal:"a ∈ carrier R ⟹ ideal R (Rxa R a)"
apply (rule ideal_condition1)
apply (rule subsetI,
simp add:Rxa_def, erule bexE, simp add:ring_tOp_closed)
apply (frule a_in_principal[of "a"], blast)
apply ((rule ballI)+,
simp add:Rxa_def, (erule bexE)+, simp,
subst ring_distrib2[THEN sym], assumption+,
cut_tac ring_is_ag,
frule_tac x = r and y = ra in aGroup.ag_pOp_closed, assumption+,
blast)
apply ((rule ballI)+,
simp add:Rxa_def, (erule bexE)+, simp,
simp add:ring_tOp_assoc[THEN sym])
apply (frule_tac x = r and y = ra in ring_tOp_closed, assumption, blast)
done
lemma (in Ring) rxa_in_Rxa:"⟦a ∈ carrier R; r ∈ carrier R⟧ ⟹
r ⋅⇩r a ∈ Rxa R a"
by (simp add:Rxa_def, blast)
lemma (in Ring) Rxa_one:"Rxa R 1⇩r = carrier R"
apply (rule equalityI)
apply (rule subsetI, simp add:Rxa_def, erule bexE)
apply (simp add:ring_r_one)
apply (rule subsetI, simp add:Rxa_def)
apply (frule_tac t = x in ring_r_one[THEN sym], blast)
done
lemma (in Ring) Rxa_zero:"Rxa R 𝟬 = {𝟬}"
apply (rule equalityI)
apply (rule subsetI)
apply (simp add:Rxa_def, erule bexE, simp add:ring_times_x_0)
apply (rule subsetI)
apply (simp add:Rxa_def)
apply (cut_tac ring_zero,
frule ring_times_x_0[THEN sym, of "𝟬"], blast)
done
lemma (in Ring) Rxa_nonzero:"⟦a ∈ carrier R; a ≠ 𝟬⟧ ⟹ Rxa R a ≠ {𝟬}"
apply (rule contrapos_pp, simp+)
apply (frule a_in_principal[of "a"])
apply simp
done
lemma (in Ring) ideal_cont_Rxa:"⟦ideal R I; a ∈ I⟧ ⟹ Rxa R a ⊆ I"
apply (rule subsetI)
apply (simp add:Rxa_def, erule bexE, simp)
apply (simp add:ideal_ring_multiple)
done
lemma (in Ring) Rxa_mult_smaller:"⟦ a ∈ carrier R; b ∈ carrier R⟧ ⟹
Rxa R (a ⋅⇩r b) ⊆ Rxa R b"
apply (frule rxa_in_Rxa[of b a], assumption,
frule principal_ideal[of b])
apply (rule ideal_cont_Rxa[of "R ♢⇩p b" "a ⋅⇩r b"], assumption+)
done
lemma (in Ring) id_ideal_psub_sum:"⟦ideal R I; a ∈ carrier R; a ∉ I⟧ ⟹
I ⊂ I ∓ Rxa R a"
apply (cut_tac ring_is_ag)
apply (simp add:psubset_eq)
apply (frule principal_ideal)
apply (rule conjI)
apply (rule sum_ideals_la1, assumption+)
apply (rule contrapos_pp) apply simp+
apply (frule sum_ideals_la2[of "I" "Rxa R a"], assumption+)
apply (frule a_in_principal[of "a"],
frule subsetD[of "Rxa R a" "I ∓ Rxa R a" "a"], assumption+)
apply simp
done
lemma (in Ring) mul_two_principal_idealsTr:"⟦a ∈ carrier R; b ∈ carrier R;
x ∈ Rxa R a; y ∈ Rxa R b⟧ ⟹ ∃r∈carrier R. x ⋅⇩r y = r ⋅⇩r (a ⋅⇩r b)"
apply (simp add:Rxa_def, (erule bexE)+)
apply simp
apply (frule_tac x = ra and y = b in ring_tOp_closed, assumption+)
apply (simp add:ring_tOp_assoc)
apply (simp add:ring_tOp_assoc[THEN sym, of a _ b])
apply (simp add:ring_tOp_commute[of a], simp add:ring_tOp_assoc)
apply (frule_tac x = a and y = b in ring_tOp_closed, assumption+,
thin_tac "ra ⋅⇩r b ∈ carrier R",
simp add:ring_tOp_assoc[THEN sym, of _ _ "a ⋅⇩r b"],
frule_tac x = r and y = ra in ring_tOp_closed, assumption+)
apply (simp add:ring_tOp_commute[of b a])
apply blast
done
primrec sum_pr_ideals::"[('a, 'm) Ring_scheme, nat ⇒ 'a, nat] ⇒ 'a set"
where
sum_pr0: "sum_pr_ideals R f 0 = Rxa R (f 0)"
| sum_prn: "sum_pr_ideals R f (Suc n) =
(Rxa R (f (Suc n))) ∓⇘R⇙ (sum_pr_ideals R f n)"
lemma (in Ring) sum_of_prideals0:
"∀f. (∀l ≤ n. f l ∈ carrier R) ⟶ ideal R (sum_pr_ideals R f n)"
apply (induct_tac n)
apply (rule allI) apply (rule impI)
apply simp
apply (rule Ring.principal_ideal, rule Ring_axioms, assumption)
apply (rule allI, rule impI)
apply (frule_tac x = f in spec,
thin_tac "∀f. (∀l ≤ n. f l ∈ carrier R) ⟶
ideal R (sum_pr_ideals R f n)")
apply (cut_tac n = n in Nsetn_sub_mem1, simp)
apply (cut_tac a = "f (Suc n)" in principal_ideal,
simp)
apply (rule_tac ?I1.0 = "Rxa R (f (Suc n))" and
?I2.0 = "sum_pr_ideals R f n" in Ring.sum_ideals, rule Ring_axioms, assumption+)
done
lemma (in Ring) sum_of_prideals:"⟦∀l ≤ n. f l ∈ carrier R⟧ ⟹
ideal R (sum_pr_ideals R f n)"
apply (simp add:sum_of_prideals0)
done
text ‹later, we show ‹sum_pr_ideals› is the least ideal containing
‹{f 0, f 1,…, f n}››
lemma (in Ring) sum_of_prideals1:"∀f. (∀l ≤ n. f l ∈ carrier R) ⟶
f ` {i. i ≤ n} ⊆ (sum_pr_ideals R f n)"
apply (induct_tac n)
apply (rule allI, rule impI)
apply (simp, simp add:a_in_principal)
apply (rule allI, rule impI)
apply (frule_tac a = f in forall_spec,
thin_tac "∀f. (∀l ≤ n. f l ∈ carrier R) ⟶
f ` {i. i ≤ n} ⊆ sum_pr_ideals R f n")
apply (rule allI, cut_tac n = n in Nset_un, simp)
apply (subst Nset_un)
apply (cut_tac A = "{i. i ≤ (Suc n)}" and f = f and B = "carrier R" and
?A1.0 = "{i. i ≤ n}" and ?A2.0 = "{Suc n}" in im_set_un1,
simp, rule Nset_un)
apply (thin_tac "∀f. (∀l≤n. f l ∈ carrier R) ⟶
f ` {i. i ≤ n} ⊆ sum_pr_ideals R f n",
simp)
apply (cut_tac n = n and f = f in sum_of_prideals,
cut_tac n = n in Nsetn_sub_mem1, simp)
apply (cut_tac a = "f (Suc n)" in principal_ideal, simp)
apply (frule_tac ?I1.0 = "Rxa R (f (Suc n))" and ?I2.0 = "sum_pr_ideals R f n"
in sum_ideals_la1, assumption+,
cut_tac a = "f (Suc n)" in a_in_principal, simp,
frule_tac A = "R ♢⇩p f (Suc n)" and
B = "R ♢⇩p f (Suc n) ∓ sum_pr_ideals R f n" and c = "f (Suc n)" in
subsetD, simp+)
apply (frule_tac ?I1.0 = "Rxa R (f (Suc n))" and
?I2.0 = "sum_pr_ideals R f n" in sum_ideals_la2, assumption+)
apply (rule_tac A = "f ` {j. j ≤ n}" and B = "sum_pr_ideals R f n" and
C = "Rxa R (f (Suc n)) ∓ sum_pr_ideals R f n" in subset_trans,
assumption+)
done
lemma (in Ring) sum_of_prideals2:"∀l ≤ n. f l ∈ carrier R
⟹ f ` {i. i ≤ n} ⊆ (sum_pr_ideals R f n)"
apply (simp add:sum_of_prideals1)
done
lemma (in Ring) sum_of_prideals3:"ideal R I ⟹
∀f. (∀l ≤ n. f l ∈ carrier R) ∧ (f ` {i. i ≤ n} ⊆ I) ⟶
(sum_pr_ideals R f n ⊆ I)"
apply (induct_tac n)
apply (rule allI, rule impI, erule conjE)
apply simp
apply (rule ideal_cont_Rxa[of I], assumption+)
apply (rule allI, rule impI, erule conjE)
apply (frule_tac a = f in forall_spec,
thin_tac "∀f. (∀l ≤ n. f l ∈ carrier R) ∧ f `{i. i ≤ n} ⊆ I ⟶
sum_pr_ideals R f n ⊆ I")
apply (simp add:Nset_un)
apply (thin_tac "∀f. (∀l ≤ n. f l ∈ carrier R) ∧ f ` {i. i ≤ n} ⊆ I ⟶
sum_pr_ideals R f n ⊆ I")
apply (frule_tac x = "Suc n" in spec,
thin_tac "∀l ≤ (Suc n). f l ∈ carrier R", simp)
apply (cut_tac a = "Suc n" and A = "{i. i ≤ Suc n}" and
f = f in mem_in_image2, simp)
apply (frule_tac A = "f ` {i. i ≤ Suc n}" and B = I and c = "f (Suc n)" in
subsetD, assumption+)
apply (rule_tac A = "Rxa R (f (Suc n))" and B = "sum_pr_ideals R f n" in
sum_ideals_cont[of I], assumption)
apply (rule ideal_cont_Rxa[of I], assumption+)
done
lemma (in Ring) sum_of_prideals4:"⟦ideal R I; ∀l ≤ n. f l ∈ carrier R;
(f ` {i. i ≤ n} ⊆ I)⟧ ⟹ sum_pr_ideals R f n ⊆ I"
apply (simp add:sum_of_prideals3)
done
lemma ker_ideal:"⟦Ring A; Ring R; f ∈ rHom A R⟧ ⟹ ideal A (ker⇘A,R⇙ f)"
apply (frule Ring.ring_is_ag[of "A"], frule Ring.ring_is_ag[of "R"])
apply (rule Ring.ideal_condition1, assumption+)
apply (rule subsetI,
simp add:ker_def)
apply (simp add:rHom_def, frule conjunct1)
apply (frule ker_inc_zero[of "A" "R" "f"], assumption+, blast)
apply (rule ballI)+
apply (simp add:ker_def, (erule conjE)+)
apply (simp add:aGroup.ag_pOp_closed)
apply (simp add:rHom_def, frule conjunct1,
simp add:aHom_add,
frule Ring.ring_zero[of "R"],
simp add:aGroup.ag_l_zero)
apply (rule ballI)+
apply (simp add:ker_def, (erule conjE)+)
apply (simp add:Ring.ring_tOp_closed)
apply (simp add:rHom_tOp)
apply (frule_tac a = r in rHom_mem[of "f" "A" "R"], assumption+,
simp add:Ring.ring_times_x_0)
done
subsection "Ring of integers"
definition
Zr :: "int Ring" where
"Zr = ⦇ carrier = Zset, pop = λn∈Zset. λm∈Zset. (m + n),
mop = λl∈Zset. -l, zero = 0, tp = λm∈Zset. λn∈Zset. m * n, un = 1⦈"
lemma ring_of_integers:"Ring Zr"
apply (simp add:Ring_def)
apply (rule conjI)
apply (simp add:Zr_def Zset_def)
apply (rule conjI)
apply (simp add:Zr_def Zset_def)
apply (rule conjI)
apply (rule allI, rule impI)+
apply (simp add:Zr_def Zset_def)
apply (rule conjI)
apply (simp add:Zr_def Zset_def)
apply (rule conjI,
rule allI, rule impI, simp add:Zr_def Zset_def)
apply (rule conjI, simp add:Zr_def Zset_def)
apply (rule conjI,
rule allI, rule impI, simp add:Zr_def Zset_def)
apply (rule conjI)
apply (simp add:Zr_def Zset_def)
apply (rule conjI,
(rule allI, rule impI)+, simp add:Zr_def Zset_def)
apply (rule conjI,
(rule allI, rule impI)+, simp add:Zr_def Zset_def)
apply (rule conjI)
apply (simp add:Zr_def Zset_def)
apply (rule conjI,
(rule allI, rule impI)+, simp add:Zr_def Zset_def)
apply (simp add: distrib_left)
apply (rule allI, rule impI)
apply (simp add:Zr_def Zset_def)
done
lemma Zr_zero:"𝟬⇘Zr⇙ = 0"
by (simp add:Zr_def)
lemma Zr_one:"1⇩r⇘Zr⇙ = 1"
by (simp add:Zr_def)
lemma Zr_minus:"-⇩a⇘Zr⇙ n = - n"
by (simp add:Zr_def Zset_def)
lemma Zr_add:"n ±⇘Zr⇙ m = n + m"
by (simp add:Zr_def Zset_def)
lemma Zr_times:"n ⋅⇩r⇘Zr⇙ m = n * m"
by (simp add:Zr_def Zset_def)
definition
lev :: "int set ⇒ int" where
"lev I = Zleast {n. n ∈ I ∧ 0 < n}"
lemma Zr_gen_Zleast:"⟦ideal Zr I; I ≠ {0::int}⟧ ⟹
Rxa Zr (lev I) = I"
apply (cut_tac ring_of_integers)
apply (simp add:lev_def)
apply (subgoal_tac "{n. n ∈ I ∧ 0 < n} ≠ {}")
apply (subgoal_tac "{n. n ∈ I ∧ 0 < n} ⊆ Zset")
apply (subgoal_tac "LB {n. n ∈ I ∧ 0 < n} 0")
apply (frule_tac A = "{n. n ∈ I ∧ 0 < n}" and n = 0 in Zleast, assumption+)
apply (erule conjE)+
apply (fold lev_def)
defer
apply (simp add:LB_def)
apply (simp add:Zset_def)
apply (frule Ring.ideal_zero[of "Zr" "I"], assumption+, simp add:Zr_zero)
apply (frule singleton_sub[of "0" "I"])
apply (frule sets_not_eq[of "I" "{0}"], assumption+, erule bexE, simp)
apply (case_tac "0 < a", blast)
apply (frule Ring.ring_one[of "Zr"])
apply (frule Ring.ring_is_ag[of "Zr"],
frule aGroup.ag_mOp_closed[of "Zr" "1⇩r⇘Zr⇙"], assumption)
apply (frule_tac x = a in Ring.ideal_ring_multiple[of "Zr" "I" _ "-⇩a⇘Zr⇙ 1⇩r⇘Zr⇙"],
assumption+)
apply (simp add:Zr_one Zr_minus,
thin_tac "ideal Zr I", thin_tac "Ring Zr", thin_tac "1 ∈ carrier Zr",
thin_tac "-1 ∈ carrier Zr", thin_tac "aGroup Zr")
apply (simp add:Zr_def Zset_def)
apply (subgoal_tac "0 < - a", blast)
apply arith
apply (thin_tac "{n ∈ I. 0 < n} ≠ {}", thin_tac "{n ∈ I. 0 < n} ⊆ Zset",
thin_tac "LB {n ∈ I. 0 < n} 0")
apply simp
apply (erule conjE)
apply (frule Ring.ideal_cont_Rxa[of "Zr" "I" "lev I"], assumption+)
apply (rule equalityI, assumption,
thin_tac "Rxa Zr (lev I) ⊆ I")
apply (rule subsetI)
apply (simp add:Rxa_def, simp add:Zr_times)
apply (cut_tac t = x and b = "lev I" in mult_div_mod_eq [symmetric])
apply (subgoal_tac "x = (x div lev I) * (lev I)",
subgoal_tac "x div lev I ∈ carrier Zr", blast)
apply (simp add:Zr_def Zset_def)
apply (subgoal_tac "x mod lev I = 0", simp)
apply (subst mult.commute, assumption)
apply (subgoal_tac "x mod lev I ∈ I")
apply (thin_tac "x = lev I * (x div lev I) + x mod lev I")
apply (frule_tac a = x in Divides.pos_mod_conj[of "lev I"])
apply (rule contrapos_pp, simp+)
apply (erule conjE)
apply (frule_tac a = "x mod (lev I)" in forall_spec)
apply simp apply arith
apply (frule_tac r = "x div (lev I)" in
Ring.ideal_ring_multiple1[of "Zr" "I" "lev I"], assumption+,
simp add:Zr_def Zset_def)
apply (frule sym, thin_tac "x = lev I * (x div lev I) + x mod lev I")
apply (rule_tac a = "lev I * (x div lev I)" and b = "x mod lev I " in
Ring.ideal_ele_sumTr1[of "Zr" "I"], assumption+)
apply (simp add:Zr_def Zset_def)
apply (simp add:Zr_def Zset_def)
apply (subst Zr_add)
apply simp
apply (simp add:Zr_times)
done
lemma Zr_pir:"ideal Zr I ⟹ ∃n. Rxa Zr n = I"
apply (case_tac "I = {(0::int)}")
apply (subgoal_tac "Rxa Zr 0 = I") apply blast
apply (rule equalityI)
apply (rule subsetI) apply (simp add:Rxa_def)
apply (simp add:Zr_def Zset_def)
apply (rule subsetI)
apply (simp add:Rxa_def Zr_def Zset_def)
apply (frule Zr_gen_Zleast [of "I"], assumption+)
apply blast
done
section "Quotient rings"
lemma (in Ring) mem_set_ar_cos:"⟦ideal R I; a ∈ carrier R⟧ ⟹
a ⊎⇘R⇙ I ∈ set_ar_cos R I"
by (simp add:set_ar_cos_def, blast)
lemma (in Ring) I_in_set_ar_cos:"ideal R I ⟹ I ∈ set_ar_cos R I"
apply (cut_tac ring_is_ag,
frule ideal_asubg[of "I"],
rule aGroup.unit_in_set_ar_cos, assumption+)
done
lemma (in Ring) ar_coset_same1:"⟦ideal R I; a ∈ carrier R; b ∈ carrier R;
b ± (-⇩a a) ∈ I ⟧ ⟹ a ⊎⇘R⇙ I = b ⊎⇘R⇙ I"
apply (cut_tac ring_is_ag)
apply (frule aGroup.b_ag_group[of "R"])
apply (simp add:ideal_def asubGroup_def) apply (erule conjE)
apply (frule aGroup.ag_carrier_carrier[THEN sym, of "R"])
apply simp
apply (frule Group.rcs_eq[of "b_ag R" "I" "a" "b"], assumption+)
apply (frule aGroup.agop_gop [of "R"])
apply (frule aGroup.agiop_giop[of "R"]) apply simp
apply (simp add:ar_coset_def rcs_def)
done
lemma (in Ring) ar_coset_same2:"⟦ideal R I; a ∈ carrier R; b ∈ carrier R;
a ⊎⇘R⇙ I = b ⊎⇘R⇙ I⟧ ⟹ b ± (-⇩a a) ∈ I"
apply (cut_tac ring_is_ag)
apply (simp add:ar_coset_def)
apply (frule aGroup.b_ag_group[of "R"])
apply (simp add:ideal_def asubGroup_def, frule conjunct1, fold asubGroup_def,
fold ideal_def, simp add:asubGroup_def)
apply (subgoal_tac "a ∈ carrier (b_ag R)",
subgoal_tac "b ∈ carrier (b_ag R)")
apply (simp add:Group.rcs_eq[THEN sym, of "b_ag R" "I" "a" "b"])
apply (frule aGroup.agop_gop [of "R"])
apply (frule aGroup.agiop_giop[of "R"]) apply simp
apply (simp add:b_ag_def)+
done
lemma (in Ring) ar_coset_same3:"⟦ideal R I; a ∈ carrier R; a ⊎⇘R⇙ I = I⟧ ⟹
a∈I"
apply (cut_tac ring_is_ag)
apply (simp add:ar_coset_def)
apply (rule Group.rcs_fixed [of "b_ag R" "I" "a" ])
apply (rule aGroup.b_ag_group, assumption)
apply (simp add:ideal_def asubGroup_def)
apply (simp add:b_ag_def)
apply assumption
done
lemma (in Ring) ar_coset_same3_1:"⟦ideal R I; a ∈ carrier R; a ∉ I⟧ ⟹
a ⊎⇘R⇙ I ≠ I"
apply (rule contrapos_pp, simp+)
apply (simp add:ar_coset_same3)
done
lemma (in Ring) ar_coset_same4:"⟦ideal R I; a ∈ I⟧ ⟹
a ⊎⇘R⇙ I = I"
apply (cut_tac ring_is_ag)
apply (frule ideal_subset[of "I" "a"], assumption+)
apply (simp add:ar_coset_def)
apply (rule Group.rcs_Unit2 [of "b_ag R" "I""a"])
apply (rule aGroup.b_ag_group, assumption)
apply (simp add:ideal_def asubGroup_def)
apply assumption
done
lemma (in Ring) ar_coset_same4_1:"⟦ideal R I; a ⊎⇘R⇙ I ≠ I⟧ ⟹ a ∉ I"
apply (rule contrapos_pp, simp+)
apply (simp add:ar_coset_same4)
done
lemma (in Ring) belong_ar_coset1:"⟦ideal R I; a ∈ carrier R; x ∈ carrier R;
x ± (-⇩a a) ∈ I⟧ ⟹ x ∈ a ⊎⇘R⇙ I"
apply (frule ar_coset_same1 [of "I" "a" "x"], assumption+)
apply (subgoal_tac "x ∈ x ⊎⇘R⇙ I")
apply simp
apply (cut_tac ring_is_ag)
apply (subgoal_tac "carrier R = carrier (b_ag R)")
apply (frule aGroup.agop_gop[THEN sym, of "R"])
apply (frule aGroup.agiop_giop [THEN sym, of "R"])
apply (simp add:ar_coset_def)
apply (simp add:ideal_def asubGroup_def)
apply (rule Group.a_in_rcs [of "b_ag R" "I" "x"])
apply (simp add: aGroup.b_ag_group)
apply simp
apply simp
apply (simp add:b_ag_def)
done
lemma (in Ring) a_in_ar_coset:"⟦ideal R I; a ∈ carrier R⟧ ⟹ a ∈ a ⊎⇘R⇙ I"
apply (rule belong_ar_coset1, assumption+)
apply (cut_tac ring_is_ag)
apply (simp add:aGroup.ag_r_inv1)
apply (simp add:ideal_zero)
done
lemma (in Ring) ar_coset_subsetD:"⟦ideal R I; a ∈ carrier R; x ∈ a ⊎⇘R⇙ I ⟧ ⟹
x ∈ carrier R"
apply (subgoal_tac "carrier R = carrier (b_ag R)")
apply (cut_tac ring_is_ag)
apply (frule aGroup.agop_gop [THEN sym, of "R"])
apply (frule aGroup.agiop_giop [THEN sym, of "R"])
apply (simp add:ar_coset_def)
apply (simp add:ideal_def asubGroup_def)
apply (rule Group.rcs_subset_elem[of "b_ag R" "I" "a" "x"])
apply (simp add:aGroup.b_ag_group)
apply simp
apply assumption+
apply (simp add:b_ag_def)
done
lemma (in Ring) ar_cos_mem:"⟦ideal R I; a ∈ carrier R⟧ ⟹
a ⊎⇘R⇙ I ∈ set_rcs (b_ag R) I"
apply (cut_tac ring_is_ag)
apply (simp add:set_rcs_def ar_coset_def)
apply (frule aGroup.ag_carrier_carrier[THEN sym, of "R"]) apply simp
apply blast
done
lemma (in Ring) mem_ar_coset1:"⟦ideal R I; a ∈ carrier R; x ∈ a ⊎⇘R⇙ I⟧ ⟹
∃h∈I. h ± a = x"
apply (cut_tac ring_is_ag)
apply (frule aGroup.ag_carrier_carrier[THEN sym, of "R"])
apply (frule aGroup.agop_gop [THEN sym, of "R"])
apply (frule aGroup.agiop_giop [THEN sym, of "R"])
apply (simp add:ar_coset_def)
apply (simp add:ideal_def asubGroup_def)
apply (simp add:rcs_def)
done
lemma (in Ring) ar_coset_mem2:"⟦ideal R I; a ∈ carrier R; x ∈ a ⊎⇘R⇙ I⟧ ⟹
∃h∈I. x = a ± h"
apply (cut_tac ring_is_ag)
apply (frule mem_ar_coset1 [of "I" "a" "x"], assumption+)
apply (erule bexE,
frule_tac h = h in ideal_subset[of "I"], assumption+)
apply (simp add:aGroup.ag_pOp_commute[of "R" _ "a"],
frule sym, thin_tac "a ± h = x", blast)
done
lemma (in Ring) belong_ar_coset2:"⟦ideal R I; a ∈ carrier R; x ∈ a ⊎⇘R⇙ I ⟧
⟹ x ± (-⇩a a) ∈ I"
apply (cut_tac ring_is_ag)
apply (frule mem_ar_coset1, assumption+, erule bexE)
apply (frule sym, thin_tac "h ± a = x", simp)
apply (frule_tac h = h in ideal_subset[of "I"], assumption)
apply (frule aGroup.ag_mOp_closed[of "R" "a"], assumption)
apply (subst aGroup.ag_pOp_assoc, assumption+,
simp add:aGroup.ag_r_inv1,
simp add:aGroup.ag_r_zero)
done
lemma (in Ring) ar_c_top: "⟦ideal R I; a ∈ carrier R; b ∈ carrier R⟧
⟹ (c_top (b_ag R) I (a ⊎⇘R⇙ I) (b ⊎⇘R⇙ I)) = (a ± b) ⊎⇘R⇙ I"
apply (cut_tac ring_is_ag, frule ideal_asubg,
frule aGroup.asubg_nsubg[of "R" "I"], assumption,
frule aGroup.b_ag_group[of "R"])
apply (simp add:ar_coset_def)
apply (subst Group.c_top_welldef[THEN sym], assumption+)
apply (simp add:aGroup.ag_carrier_carrier)+
apply (simp add:aGroup.agop_gop)
done
text‹Following lemma is not necessary to define a quotient ring. But
it makes clear that the binary operation2 of the quotient ring is well
defined.›
lemma (in Ring) quotient_ring_tr1:"⟦ideal R I; a1 ∈ carrier R; a2 ∈ carrier R;
b1 ∈ carrier R; b2 ∈ carrier R;
a1 ⊎⇘R⇙ I = a2 ⊎⇘R⇙ I; b1 ⊎⇘R⇙ I = b2 ⊎⇘R⇙ I⟧ ⟹
(a1 ⋅⇩r b1) ⊎⇘R⇙ I = (a2 ⋅⇩r b2) ⊎⇘R⇙ I"
apply (rule ar_coset_same1, assumption+)
apply (simp add: ring_tOp_closed)+
apply (frule ar_coset_same2 [of "I" "a1" "a2"], assumption+)
apply (frule ar_coset_same2 [of "I" "b1" "b2"], assumption+)
apply (frule ring_distrib4[of "a2" "b2" "a1" "b1"], assumption+)
apply simp
apply (rule ideal_pOp_closed[of "I"], assumption)
apply (simp add:ideal_ring_multiple, simp add:ideal_ring_multiple1)
done
definition
rcostOp :: "[_, 'a set] ⇒ (['a set, 'a set] ⇒ 'a set)" where
"rcostOp R I = (λX∈(set_rcs (b_ag R) I). λY∈(set_rcs (b_ag R) I).
{z. ∃ x ∈ X. ∃ y ∈ Y. ∃h∈I. (x ⋅⇩r⇘R⇙ y) ±⇘R⇙ h = z})"
lemma (in Ring) rcostOp:"⟦ideal R I; a ∈ carrier R; b ∈ carrier R⟧ ⟹
rcostOp R I (a ⊎⇘R⇙ I) (b ⊎⇘R⇙ I) = (a ⋅⇩r b) ⊎⇘R⇙ I"
apply (cut_tac ring_is_ag)
apply (frule ar_cos_mem[of "I" "a"], assumption+)
apply (frule ar_cos_mem[of "I" "b"], assumption+)
apply (simp add:rcostOp_def)
apply (rule equalityI)
apply (rule subsetI, simp) apply (erule bexE)+
apply (rule belong_ar_coset1, assumption+)
apply (simp add:ring_tOp_closed)
apply (frule sym, thin_tac "xa ⋅⇩r y ± h = x", simp)
apply (rule aGroup.ag_pOp_closed, assumption)
apply (frule_tac x = xa in ar_coset_mem2[of "I" "a"], assumption+,
frule_tac x = y in ar_coset_mem2[of "I" "b"], assumption+,
(erule bexE)+, simp)
apply (rule ring_tOp_closed, rule aGroup.ag_pOp_closed, assumption+,
simp add:ideal_subset)
apply (rule aGroup.ag_pOp_closed, assumption+, simp add:ideal_subset,
simp add:ideal_subset)
apply (frule sym, thin_tac "xa ⋅⇩r y ± h = x", simp)
apply (frule_tac x = xa in belong_ar_coset2[of "I" "a"], assumption+,
frule_tac x = y in belong_ar_coset2[of "I" "b"], assumption+)
apply (frule_tac x = xa in ar_coset_subsetD[of "I" "a"], assumption+,
frule_tac x = y in ar_coset_subsetD[of "I" "b"], assumption+)
apply (subst aGroup.ag_pOp_commute, assumption,
simp add:ring_tOp_closed, simp add:ideal_subset)
apply (subst aGroup.ag_pOp_assoc, assumption,
simp add:ideal_subset, simp add:ring_tOp_closed,
rule aGroup.ag_mOp_closed, simp add:ring_tOp_closed,
simp add:ring_tOp_closed)
apply (rule ideal_pOp_closed, assumption+)
apply (rule_tac a = xa and a' = y and b = a and b' = b in times_modTr,
assumption+)
apply (rule subsetI, simp)
apply (frule_tac x = x in ar_coset_mem2[of "I" "a ⋅⇩r b"],
simp add:ring_tOp_closed, assumption)
apply (erule bexE) apply simp
apply (frule a_in_ar_coset[of "I" "a"], assumption+,
frule a_in_ar_coset[of "I" "b"], assumption+)
apply blast
done
definition
qring :: "[('a, 'm) Ring_scheme, 'a set] ⇒ ⦇ carrier :: 'a set set,
pop :: ['a set, 'a set] ⇒ 'a set, mop :: 'a set ⇒ 'a set,
zero :: 'a set, tp :: ['a set, 'a set] ⇒ 'a set, un :: 'a set ⦈" where
"qring R I = ⦇ carrier = set_rcs (b_ag R) I,
pop = c_top (b_ag R) I,
mop = c_iop (b_ag R) I,
zero = I,
tp = rcostOp R I,
un = 1⇩r⇘R⇙ ⊎⇘R⇙ I⦈"
abbreviation
QRING (infixl "'/⇩r" 200) where
"R /⇩r I == qring R I"
lemma (in Ring) carrier_qring:"ideal R I ⟹
carrier (qring R I) = set_rcs (b_ag R) I"
by (simp add:qring_def)
lemma (in Ring) carrier_qring1:"ideal R I ⟹
carrier (qring R I) = set_ar_cos R I"
apply (cut_tac ring_is_ag)
apply (simp add:carrier_qring set_rcs_def set_ar_cos_def)
apply (simp add:ar_coset_def aGroup.ag_carrier_carrier)
done
lemma (in Ring) qring_ring:"ideal R I ⟹ Ring (qring R I)"
apply (cut_tac ring_is_ag)
apply (frule ideal_asubg[of "I"],
frule aGroup.asubg_nsubg[of "R" "I"], assumption,
frule aGroup.b_ag_group[of "R"])
apply (subst Ring_def, simp)
apply (rule conjI)
apply (rule Pi_I)+
apply (simp add:carrier_qring, simp add:set_rcs_def, (erule bexE)+)
apply (subst qring_def, simp)
apply (subst Group.c_top_welldef[THEN sym, of "b_ag R" "I"], assumption+)
apply (blast dest: Group.mult_closed[of "b_ag R"])
apply (rule conjI)
apply (rule allI, rule impI)+
apply (simp add:qring_def)
apply (simp add:Group.Qg_tassoc[of "b_ag R" "I"])
apply (rule conjI)
apply (rule allI, rule impI)+
apply (simp add:qring_def)
apply (simp add:set_rcs_def, (erule bexE)+, simp)
apply (subst Group.c_top_welldef[THEN sym, of "b_ag R" "I"], assumption+)+
apply (simp add:aGroup.agop_gop)
apply (simp add:aGroup.ag_carrier_carrier)
apply (simp add:aGroup.ag_pOp_commute)
apply (rule conjI)
apply (simp add:qring_def Group.Qg_iop_closed)
apply (rule conjI)
apply (rule allI, rule impI)
apply (simp add:qring_def)
apply (simp add:Group.Qg_i[of "b_ag R" "I"])
apply (rule conjI)
apply (simp add:qring_def)
apply (frule Group.nsg_sg[of "b_ag R" "I"], assumption)
apply (simp add:Group.unit_rcs_in_set_rcs)
apply (rule conjI)
apply (rule allI, rule impI)
apply (simp add:qring_def)
apply (simp add:Group.Qg_unit[of "b_ag R" "I"])
apply (rule conjI)
apply(rule Pi_I)+
apply (simp add:qring_def aGroup.aqgrp_carrier)
apply (simp add:set_ar_cos_def, (erule bexE)+, simp add:rcostOp,
blast dest: ring_tOp_closed)
apply (rule conjI)
apply (rule allI, rule impI)+
apply (simp add:qring_def aGroup.aqgrp_carrier)
apply (simp add:set_ar_cos_def, (erule bexE)+, simp add:rcostOp)
apply (frule_tac x = aa and y = ab in ring_tOp_closed, assumption+,
frule_tac x = ab and y = ac in ring_tOp_closed, assumption+,
simp add:rcostOp, simp add:ring_tOp_assoc)
apply (rule conjI)
apply (rule allI, rule impI)+
apply (simp add:qring_def aGroup.aqgrp_carrier)
apply (simp add:set_ar_cos_def, (erule bexE)+, simp add:rcostOp,
simp add:ring_tOp_commute)
apply (rule conjI)
apply (simp add:qring_def aGroup.aqgrp_carrier)
apply (cut_tac ring_one, simp add:set_ar_cos_def, blast)
apply (rule conjI)
apply (rule allI, rule impI)+
apply (simp add:qring_def aGroup.aqgrp_carrier)
apply (simp add:set_ar_cos_def, (erule bexE)+, simp)
apply (simp add:ar_c_top rcostOp)
apply (frule_tac x = ab and y = ac in aGroup.ag_pOp_closed,
assumption+,
frule_tac x = aa and y = ab in ring_tOp_closed, assumption+ ,
frule_tac x = aa and y = ac in ring_tOp_closed, assumption+)
apply (simp add:ar_c_top rcostOp, simp add:ring_distrib1)
apply (rule allI, rule impI)
apply (simp add:qring_def aGroup.aqgrp_carrier)
apply (simp add:set_ar_cos_def, erule bexE, simp)
apply (cut_tac ring_one)
apply (simp add:rcostOp, simp add:ring_l_one)
done
lemma (in Ring) qring_carrier:"ideal R I ⟹
carrier (qring R I) = {X. ∃a∈ carrier R. a ⊎⇘R⇙ I = X}"
apply (simp add:carrier_qring1 set_ar_cos_def)
apply (rule equalityI)
apply (rule subsetI, simp, erule bexE, frule sym, thin_tac "x = a ⊎⇘R⇙ I",
blast)
apply (rule subsetI, simp, erule bexE, frule sym, thin_tac "a ⊎⇘R⇙ I = x",
blast)
done
lemma (in Ring) qring_mem:"⟦ideal R I; a ∈ carrier R⟧ ⟹
a ⊎⇘R⇙ I ∈ carrier (qring R I)"
apply (simp add:qring_carrier)
apply blast
done
lemma (in Ring) qring_pOp:"⟦ideal R I; a ∈ carrier R; b ∈ carrier R ⟧
⟹ pop (qring R I) (a ⊎⇘R⇙ I) (b ⊎⇘R⇙ I) = (a ± b) ⊎⇘R⇙ I"
by (simp add:qring_def, simp add:ar_c_top)
lemma (in Ring) qring_zero:"ideal R I ⟹ zero (qring R I) = I"
apply (simp add:qring_def)
done
lemma (in Ring) qring_zero_1:"⟦a ∈ carrier R; ideal R I; a ⊎⇘R⇙ I = I⟧ ⟹
a ∈ I"
by (frule a_in_ar_coset [of "I" "a"], assumption+, simp)
lemma (in Ring) Qring_fix1:"⟦a ∈ carrier R; ideal R I; a ∈ I⟧ ⟹ a ⊎⇘R⇙ I = I"
apply (cut_tac ring_is_ag, frule aGroup.b_ag_group)
apply (simp add:ar_coset_def)
apply (frule ideal_asubg[of "I"], simp add:asubGroup_def)
apply (simp add:Group.rcs_fixed2[of "b_ag R" "I"])
done
lemma (in Ring) ar_cos_same:"⟦a ∈ carrier R; ideal R I; x ∈ a ⊎⇘R⇙ I⟧ ⟹
x ⊎⇘R⇙ I = a ⊎⇘R⇙ I"
apply (cut_tac ring_is_ag)
apply (rule ar_coset_same1[of "I" "x" "a"], assumption+)
apply (rule ar_coset_subsetD[of "I"], assumption+)
apply (frule ar_coset_mem2[of "I" "a" "x"], assumption+,
erule bexE)
apply (frule_tac h = h in ideal_subset[of "I"], assumption,
simp add:aGroup.ag_p_inv)
apply (frule_tac x = a in aGroup.ag_mOp_closed[of "R"], assumption+,
frule_tac x = h in aGroup.ag_mOp_closed[of "R"], assumption+)
apply (simp add:aGroup.ag_pOp_assoc[THEN sym],
simp add:aGroup.ag_r_inv1 aGroup.ag_l_zero)
apply (simp add:ideal_inv1_closed)
done
lemma (in Ring) qring_tOp:"⟦ideal R I; a ∈ carrier R; b ∈ carrier R⟧ ⟹
tp (qring R I) (a ⊎⇘R⇙ I) (b ⊎⇘R⇙ I) = (a ⋅⇩r b) ⊎⇘R⇙ I"
by (simp add:qring_def, simp add:rcostOp)
lemma rind_hom_well_def:"⟦Ring A; Ring R; f ∈ rHom A R; a ∈ carrier A ⟧ ⟹
f a = (f°⇘A,R⇙) (a ⊎⇘A⇙ (ker⇘A,R⇙ f))"
apply (frule ker_ideal[of "A" "R" "f"], assumption+)
apply (frule Ring.mem_set_ar_cos[of "A" "ker⇘A,R⇙ f" "a"], assumption+)
apply (simp add:rind_hom_def)
apply (rule someI2_ex)
apply (frule Ring.a_in_ar_coset [of "A" "ker⇘A,R⇙ f" "a"], assumption+, blast)
apply (frule_tac x = x in Ring.ar_coset_mem2[of "A" "ker⇘A,R⇙ f" "a"],
assumption+, erule bexE, simp,
frule_tac h = h in Ring.ideal_subset[of "A" "ker⇘A,R⇙ f"], assumption+)
apply (frule_tac Ring.ring_is_ag[of "A"],
frule_tac Ring.ring_is_ag[of "R"],
simp add:rHom_def, frule conjunct1, simp add:aHom_add)
apply (simp add:ker_def)
apply (frule aHom_mem[of "A" "R" "f" "a"], assumption+,
simp add:aGroup.ag_r_zero)
done
lemma (in Ring) set_r_ar_cos:"ideal R I ⟹
set_rcs (b_ag R) I = set_ar_cos R I"
apply (simp add:set_ar_cos_def set_rcs_def ar_coset_def)
apply (cut_tac ring_is_ag)
apply (simp add:aGroup.ag_carrier_carrier)
done
lemma set_r_ar_cos_ker:"⟦Ring A; Ring R; f ∈ rHom A R ⟧ ⟹
set_rcs (b_ag A) (ker⇘A,R⇙ f) = set_ar_cos A (ker⇘A,R⇙ f)"
apply (frule ker_ideal[of "A" "R" "f"], assumption+)
apply (simp add:Ring.carrier_qring[THEN sym],
simp add:Ring.carrier_qring1[THEN sym])
done
lemma ind_hom_rhom:"⟦Ring A; Ring R; f ∈ rHom A R⟧ ⟹
(f°⇘A,R⇙) ∈ rHom (qring A (ker⇘A,R⇙ f)) R"
apply (simp add:rHom_def [of "qring A (ker⇘A,R⇙ f)" "R"])
apply (rule conjI)
apply (simp add:aHom_def)
apply (rule conjI)
apply (simp add:qring_def)
apply (simp add:rind_hom_def extensional_def)
apply (rule Pi_I)
apply (frule Ring.ring_is_ag [of "A"], frule Ring.ring_is_ag [of "R"],
frule aGroup.b_ag_group [of "R"])
apply (simp add:aGroup.ag_carrier_carrier [THEN sym])
apply (simp add:set_ar_cos_def)
apply (rule conjI)
apply (rule impI)
apply (erule bexE, simp)
apply (frule ker_ideal [of "A" "R" "f"], assumption+)
apply (frule_tac a = a in Ring.a_in_ar_coset [of "A" "ker⇘A,R⇙ f"],
assumption+)
apply (rule someI2_ex, blast)
apply (frule_tac I = "ker⇘A,R⇙ f" and a = a and x = xa in
Ring.ar_coset_subsetD[of "A"], assumption+)
apply (simp add:aGroup.ag_carrier_carrier, simp add:rHom_mem)
apply (simp add:set_r_ar_cos_ker, simp add:set_ar_cos_def, rule impI, blast)
apply (rule conjI)
apply (simp add:qring_def)
apply (simp add:set_r_ar_cos_ker)
apply (simp add:rind_hom_def extensional_def)
apply (rule ballI)+
apply (simp add:qring_def)
apply (simp add:set_r_ar_cos_ker)
apply (simp add:set_ar_cos_def)
apply ((erule bexE)+, simp)
apply (frule ker_ideal[of "A" "R" "f"], assumption+)
apply (simp add:Ring.ar_c_top)
apply (frule Ring.ring_is_ag[of "A"],
frule Ring.ring_is_ag[of "R"],
frule_tac x = aa and y = ab in aGroup.ag_pOp_closed[of "A"],
assumption+)
apply (simp add:rind_hom_well_def[THEN sym])
apply (simp add:rHom_def, frule conjunct1, simp add:aHom_add)
apply (rule conjI)
apply (rule ballI)+
apply (frule ker_ideal[of "A" "R" "f"], assumption+,
simp add:Ring.carrier_qring1, simp add:set_ar_cos_def,
(erule bexE)+, simp add:qring_def Ring.rcostOp)
apply (frule Ring.ring_is_ag[of "A"],
frule_tac x = a and y = aa in Ring.ring_tOp_closed[of "A"],
assumption+)
apply (simp add:rind_hom_well_def[THEN sym], simp add:rHom_tOp)
apply (simp add:qring_def)
apply (frule Ring.ring_one[of "A"],
simp add:rind_hom_well_def[THEN sym],
simp add:rHom_one)
done
lemma ind_hom_injec:"⟦Ring A; Ring R; f ∈ rHom A R⟧ ⟹
injec⇘(qring A (ker⇘A,R⇙ f)),R⇙ (f°⇘A,R⇙)"
apply (simp add:injec_def)
apply (frule ind_hom_rhom [of "A" "R" "f"], assumption+)
apply (frule rHom_aHom[of "f°⇘A,R⇙" "A /⇩r (ker⇘A,R⇙ f)" "R"], simp)
apply (simp add:ker_def[of _ _ "f°⇘A,R⇙"])
apply ((subst qring_def)+, simp)
apply (simp add:set_r_ar_cos_ker)
apply (frule Ring.ring_is_ag[of "A"],
frule Ring.ring_is_ag[of "R"],
frule ker_ideal[of "A" "R" "f"], assumption+)
apply (rule equalityI)
apply (rule subsetI)
apply (simp, erule conjE)
apply (simp add:set_ar_cos_def, erule bexE, simp)
apply (simp add:rind_hom_well_def[THEN sym, of "A" "R" "f"],
thin_tac "x = a ⊎⇘A⇙ ker⇘A,R⇙ f")
apply (rule_tac a = a in Ring.Qring_fix1[of "A" _ "ker⇘A,R⇙ f"], assumption+)
apply (simp add:ker_def)
apply (rule subsetI, simp)
apply (simp add:Ring.I_in_set_ar_cos[of "A" "ker⇘A,R⇙ f"])
apply (frule Ring.ideal_zero[of "A" "ker⇘A,R⇙ f"], assumption+,
frule Ring.ring_zero[of "A"])
apply (frule Ring.ar_coset_same4[of "A" "ker⇘A,R⇙ f" "𝟬⇘A⇙"], assumption+)
apply (frule rind_hom_well_def[THEN sym, of "A" "R" "f" "𝟬⇘A⇙"], assumption+)
apply simp
apply (rule rHom_0_0, assumption+)
done
lemma rhom_to_rimg:"⟦Ring A; Ring R; f ∈ rHom A R⟧ ⟹
f ∈ rHom A (rimg A R f)"
apply (frule Ring.ring_is_ag[of "A"], frule Ring.ring_is_ag[of "R"])
apply (subst rHom_def, simp)
apply (rule conjI)
apply (subst aHom_def, simp)
apply (rule conjI)
apply (simp add:rimg_def)
apply (rule conjI)
apply (simp add:rHom_def aHom_def)
apply ((rule ballI)+, simp add:rimg_def)
apply (rule aHom_add, assumption+)
apply (simp add:rHom_aHom, assumption+)
apply (rule conjI)
apply ((rule ballI)+, simp add:rimg_def, simp add:rHom_tOp)
apply (simp add:rimg_def, simp add:rHom_one)
done
lemma ker_to_rimg:"⟦Ring A; Ring R; f ∈ rHom A R ⟧ ⟹
ker⇘A,R⇙ f = ker⇘A,(rimg A R f)⇙ f"
apply (frule rhom_to_rimg [of "A" "R" "f"], assumption+)
apply (simp add:ker_def)
apply (simp add:rimg_def)
done
lemma indhom_eq:"⟦Ring A; Ring R; f ∈ rHom A R⟧ ⟹ f°⇘A,(rimg A R f)⇙ = f°⇘A,R⇙"
apply (frule rimg_ring[of "A" "R" "f"], assumption+)
apply (frule rhom_to_rimg[of "A" "R" "f"], assumption+,
frule ind_hom_rhom[of "A" "rimg A R f"], assumption+,
frule ind_hom_rhom[of "A" "R" "f"], assumption+)
apply (rule funcset_eq[of "f°⇘A,rimg A R f⇙ " "carrier (A /⇩r (ker⇘A,R⇙ f))" "f°⇘A,R⇙"])
apply (simp add:ker_to_rimg[THEN sym],
simp add:rHom_def[of _ "rimg A R f"] aHom_def)
apply (simp add:rHom_def[of _ "R"] aHom_def)
apply (simp add:ker_to_rimg[THEN sym])
apply (rule ballI)
apply (frule ker_ideal[of "A" "R" "f"], assumption+,
simp add:Ring.carrier_qring1)
apply (simp add:set_ar_cos_def, erule bexE, simp)
apply (simp add:rind_hom_well_def[THEN sym])
apply (frule rind_hom_well_def[THEN sym, of "A" "rimg A R f" "f"],
assumption+, simp add:ker_to_rimg[THEN sym])
done
lemma indhom_bijec2_rimg:"⟦Ring A; Ring R; f ∈ rHom A R⟧ ⟹
bijec⇘(qring A (ker⇘A,R⇙ f)),(rimg A R f)⇙ (f°⇘A,R⇙)"
apply (frule rimg_ring [of "A" "R" "f"], assumption+)
apply (frule rhom_to_rimg[of "A" "R" "f"], assumption+)
apply (frule ind_hom_rhom[of "A" "rimg A R f" "f"], assumption+)
apply (frule ker_to_rimg[THEN sym, of "A" "R" "f"], assumption+)
apply (frule indhom_eq[of "A" "R" "f"], assumption+)
apply simp
apply (simp add:bijec_def)
apply (rule conjI)
apply (simp add:injec_def)
apply (rule conjI)
apply (simp add:rHom_def)
apply (frule ind_hom_injec [of "A" "R" "f"], assumption+)
apply (simp add:injec_def)
apply (simp add:ker_def [of _ _ "f°⇘A,R⇙"])
apply (simp add:rimg_def)
apply (simp add:surjec_def)
apply (rule conjI)
apply (simp add:rHom_def)
apply (rule surj_to_test)
apply (simp add:rHom_def aHom_def)
apply (rule ballI)
apply (simp add:rimg_carrier)
apply (simp add:image_def)
apply (erule bexE, simp)
apply (frule_tac a1 = x in rind_hom_well_def[THEN sym, of "A" "R" "f"],
assumption+)
apply (frule ker_ideal[of "A" "R" "f"], assumption+,
simp add:Ring.carrier_qring1,
frule_tac a = x in Ring.mem_set_ar_cos[of "A" "ker⇘A,R⇙ f"], assumption+)
apply blast
done
lemma surjec_ind_bijec:"⟦Ring A; Ring R; f ∈ rHom A R; surjec⇘A,R⇙ f⟧ ⟹
bijec⇘(qring A (ker⇘A,R⇙ f)),R⇙ (f°⇘A,R⇙)"
apply (frule ind_hom_rhom[of "A" "R" "f"], assumption+)
apply (simp add:surjec_def)
apply (simp add:bijec_def)
apply (simp add:ind_hom_injec)
apply (simp add:surjec_def)
apply (simp add:rHom_aHom)
apply (rule surj_to_test)
apply (simp add:rHom_def aHom_def)
apply (rule ballI)
apply (simp add:surj_to_def, frule sym,
thin_tac "f ` carrier A = carrier R", simp,
thin_tac "carrier R = f ` carrier A")
apply (simp add:image_def, erule bexE)
apply (frule_tac a1 = x in rind_hom_well_def[THEN sym, of "A" "R" "f"],
assumption+)
apply (frule ker_ideal[of "A" "R" "f"], assumption+,
simp add:Ring.carrier_qring1,
frule_tac a = x in Ring.mem_set_ar_cos[of "A" "ker⇘A,R⇙ f"], assumption+)
apply blast
done
lemma ridmap_ind_bijec:"Ring A ⟹
bijec⇘(qring A (ker⇘A,A⇙ (ridmap A))),A⇙ ((ridmap A)°⇘A,A⇙)"
apply (frule ridmap_surjec[of "A"])
apply (rule surjec_ind_bijec [of "A" "A" "ridmap A"], assumption+)
apply (simp add:rHom_def, simp add:surjec_def)
apply (rule conjI)
apply (rule ballI)+
apply (frule_tac x = x and y = y in Ring.ring_tOp_closed[of "A"],
assumption+, simp add:ridmap_def)
apply (simp add:ridmap_def Ring.ring_one)
apply assumption
done
lemma ker_of_idmap:"Ring A ⟹ ker⇘A,A⇙ (ridmap A) = {𝟬⇘A⇙}"
apply (simp add:ker_def)
apply (simp add:ridmap_def)
apply (rule equalityI)
apply (rule subsetI) apply (simp add:CollectI)
apply (rule subsetI) apply (simp add:CollectI)
apply (simp add:Ring.ring_zero)
done
lemma ring_natural_isom:"Ring A ⟹
bijec⇘(qring A {𝟬⇘A⇙}),A⇙ ((ridmap A)°⇘A,A⇙)"
apply (frule ridmap_ind_bijec)
apply (simp add: ker_of_idmap)
done
definition
pj :: "[('a, 'm) Ring_scheme, 'a set] ⇒ ('a => 'a set)" where
"pj R I = (λx. Pj (b_ag R) I x)"
lemma pj_Hom:"⟦Ring R; ideal R I⟧ ⟹ (pj R I) ∈ rHom R (qring R I)"
apply (simp add:rHom_def)
apply (rule conjI)
apply (simp add:aHom_def)
apply (rule conjI)
apply (rule Pi_I)
apply (simp add:qring_def)
apply (frule Ring.ring_is_ag)
apply (simp add:aGroup.ag_carrier_carrier [THEN sym])
apply (simp add:pj_def Pj_def)
apply (simp add:set_rcs_def) apply blast
apply (rule conjI)
apply (simp add:pj_def Pj_def extensional_def)
apply (frule Ring.ring_is_ag) apply (simp add:aGroup.ag_carrier_carrier)
apply (rule ballI)+
apply (frule Ring.ring_is_ag)
apply (frule_tac x = a and y = b in aGroup.ag_pOp_closed, assumption+)
apply (simp add:aGroup.ag_carrier_carrier [THEN sym])
apply (simp add:pj_def Pj_def)
apply (simp add:qring_def) apply (frule aGroup.b_ag_group)
apply (simp add:aGroup.agop_gop [THEN sym])
apply (subst Group.c_top_welldef[of "b_ag R" "I"], assumption+)
apply (frule Ring.ideal_asubg[of "R" "I"], assumption+)
apply (simp add:aGroup.asubg_nsubg)
apply assumption+
apply simp
apply (rule conjI)
apply (rule ballI)+
apply (simp add: qring_def)
apply (frule_tac x = x and y = y in Ring.ring_tOp_closed, assumption+)
apply (frule Ring.ring_is_ag)
apply (simp add:aGroup.ag_carrier_carrier [THEN sym])
apply (simp add:pj_def Pj_def)
apply (simp add:aGroup.ag_carrier_carrier)
apply (frule_tac a1 = x and b1 = y in Ring.rcostOp [THEN sym, of "R" "I"],
assumption+)
apply (simp add:ar_coset_def)
apply (simp add:qring_def)
apply (frule Ring.ring_one)
apply (frule Ring.ring_is_ag)
apply (simp add:aGroup.ag_carrier_carrier [THEN sym])
apply (simp add:pj_def Pj_def)
apply (simp add:ar_coset_def)
done
lemma pj_mem:"⟦Ring R; ideal R I; x ∈ carrier R⟧ ⟹ pj R I x = x ⊎⇘R⇙ I"
apply (frule Ring.ring_is_ag)
apply (simp add:aGroup.ag_carrier_carrier [THEN sym])
apply (simp add:pj_def Pj_def)
apply (simp add:ar_coset_def)
done
lemma pj_zero:"⟦Ring R; ideal R I; x ∈ carrier R⟧ ⟹
(pj R I x = 𝟬⇘(R /⇩r I)⇙) = (x ∈ I)"
apply (rule iffI)
apply (simp add:pj_mem Ring.qring_zero,
simp add:Ring.qring_zero_1[of "R" "x" "I"])
apply (simp add:pj_mem Ring.qring_zero,
rule Ring.Qring_fix1, assumption+)
done
lemma pj_surj_to:"⟦Ring R; ideal R J; X ∈ carrier (R /⇩r J)⟧ ⟹
∃r∈ carrier R. pj R J r = X"
apply (simp add:qring_def set_rcs_def,
fold ar_coset_def, simp add:b_ag_def, erule bexE,
frule_tac x = a in pj_mem[of R J], assumption+, simp)
apply blast
done
lemma invim_of_ideal:"⟦Ring R; ideal R I; ideal (qring R I) J ⟧ ⟹
ideal R (rInvim R (qring R I) (pj R I) J)"
apply (rule Ring.ideal_condition, assumption)
apply (simp add:rInvim_def)
apply (subgoal_tac "𝟬⇘R⇙ ∈ rInvim R (qring R I) (pj R I) J")
apply (simp add:nonempty)
apply (simp add:rInvim_def)
apply (simp add: Ring.ring_zero)
apply (frule Ring.ring_is_ag)
apply (frule pj_Hom [of "R" "I"], assumption+)
apply (frule Ring.qring_ring [of "R" "I"], assumption+)
apply (frule rHom_0_0 [of "R" "R /⇩r I" "pj R I"], assumption+)
apply (simp add:Ring.ideal_zero)
apply (rule ballI)+
apply (simp add:rInvim_def) apply (erule conjE)+
apply (rule conjI)
apply (frule Ring.ring_is_ag)
apply (rule aGroup.ag_pOp_closed, assumption+)
apply (rule aGroup.ag_mOp_closed, assumption+)
apply (frule pj_Hom [of "R" "I"], assumption+)
apply (frule Ring.ring_is_ag)
apply (frule_tac x = y in aGroup.ag_mOp_closed [of "R"], assumption+)
apply (simp add:rHom_def) apply (erule conjE)+
apply (subst aHom_add [of "R" "R /⇩r I" "pj R I"], assumption+)
apply (simp add:Ring.qring_ring Ring.ring_is_ag)
apply assumption+
apply (frule Ring.qring_ring [of "R" "I"], assumption+)
apply (rule Ring.ideal_pOp_closed, assumption+)
apply (subst aHom_inv_inv[of "R" "R /⇩r I" "pj R I"], assumption+)
apply (simp add:Ring.ring_is_ag) apply assumption+
apply (frule_tac x = "pj R I y" in Ring.ideal_inv1_closed [of "R /⇩r I" "J"],
assumption+)
apply (rule ballI)+
apply (simp add:rInvim_def) apply (erule conjE)
apply (simp add:Ring.ring_tOp_closed)
apply (frule pj_Hom [of "R" "I"], assumption+)
apply (subst rHom_tOp [of "R" "R /⇩r I" _ _ "pj R I"], assumption+)
apply (frule Ring.qring_ring[of "R" "I"], assumption+)
apply (rule Ring.ideal_ring_multiple [of "R /⇩r I" "J"])
apply (simp add:Ring.qring_ring) apply assumption+
apply (simp add:rHom_mem)
done
lemma pj_invim_cont_I:"⟦Ring R; ideal R I; ideal (qring R I) J⟧ ⟹
I ⊆ (rInvim R (qring R I) (pj R I) J)"
apply (rule subsetI)
apply (simp add:rInvim_def)
apply (frule Ring.ideal_subset [of "R" "I"], assumption+)
apply simp
apply (frule pj_mem [of "R" "I" _], assumption+)
apply (simp add:Ring.ar_coset_same4)
apply (frule Ring.qring_ring[of "R" "I"], assumption+)
apply (frule Ring.ideal_zero [of "qring R I" "J"], assumption+)
apply (frule Ring.qring_zero[of "R" "I"], assumption)
apply simp
done
lemma pj_invim_mono1:"⟦Ring R; ideal R I; ideal (qring R I) J1;
ideal (qring R I) J2; J1 ⊆ J2 ⟧ ⟹
(rInvim R (qring R I) (pj R I) J1) ⊆ (rInvim R (qring R I) (pj R I) J2)"
apply (rule subsetI)
apply (simp add:rInvim_def)
apply (simp add:subsetD)
done
lemma pj_img_ideal:"⟦Ring R; ideal R I; ideal R J; I ⊆ J⟧ ⟹
ideal (qring R I) ((pj R I)`J)"
apply (rule Ring.ideal_condition [of "qring R I" "(pj R I) `J"])
apply (simp add:Ring.qring_ring)
apply (rule subsetI, simp add:image_def)
apply (erule bexE)
apply (frule_tac h = xa in Ring.ideal_subset [of "R" "J"], assumption+)
apply (frule pj_Hom [of "R" "I"], assumption+)
apply (simp add:rHom_mem)
apply (frule Ring.ideal_zero [of "R" "J"], assumption+)
apply (simp add:image_def) apply blast
apply (rule ballI)+
apply (simp add:image_def)
apply (erule bexE)+
apply (frule pj_Hom [of "R" "I"], assumption+)
apply (rename_tac x y s t)
apply (frule_tac h = s in Ring.ideal_subset [of "R" "J"], assumption+)
apply (frule_tac h = t in Ring.ideal_subset [of "R" "J"], assumption+)
apply (simp add:rHom_def) apply (erule conjE)+
apply (frule Ring.ring_is_ag)
apply (frule Ring.qring_ring [of "R" "I"], assumption+)
apply (frule Ring.ring_is_ag [of "R /⇩r I"])
apply (frule_tac x = t in aGroup.ag_mOp_closed [of "R"], assumption+)
apply (frule_tac a1 = s and b1 = "-⇩a⇘R⇙ t" in aHom_add [of "R" "R /⇩r I"
"pj R I", THEN sym], assumption+) apply (simp add:aHom_inv_inv)
apply (frule_tac x = t in Ring.ideal_inv1_closed [of "R" "J"], assumption+)
apply (frule_tac x = s and y = "-⇩a⇘R⇙ t" in Ring.ideal_pOp_closed [of "R" "J"],
assumption+)
apply blast
apply (rule ballI)+
apply (simp add:qring_def)
apply (simp add:Ring.set_r_ar_cos)
apply (simp add:set_ar_cos_def, erule bexE)
apply simp
apply (simp add:image_def)
apply (erule bexE)
apply (frule_tac x = xa in pj_mem [of "R" "I"], assumption+)
apply (simp add:Ring.ideal_subset) apply simp
apply (subst Ring.rcostOp, assumption+)
apply (simp add:Ring.ideal_subset)
apply (frule_tac x = xa and r = a in Ring.ideal_ring_multiple [of "R" "J"],
assumption+)
apply (frule_tac h = "a ⋅⇩r⇘R⇙ xa" in Ring.ideal_subset [of "R" "J"],
assumption+)
apply (frule_tac x1 = "a ⋅⇩r⇘R⇙ xa" in pj_mem [THEN sym, of "R" "I"],
assumption+)
apply simp
apply blast
done
lemma npQring:"⟦Ring R; ideal R I; a ∈ carrier R⟧ ⟹
npow (qring R I) (a ⊎⇘R⇙ I) n = (npow R a n) ⊎⇘R⇙ I"
apply (induct_tac n)
apply (simp add:qring_def)
apply (simp add:qring_def)
apply (rule Ring.rcostOp, assumption+)
apply (rule Ring.npClose, assumption+)
done
section "Primary ideals, Prime ideals"
definition
maximal_set :: "['a set set, 'a set] ⇒ bool" where
"maximal_set S mx ⟷ mx ∈ S ∧ (∀s∈S. mx ⊆ s ⟶ s = mx)"
definition
nilpotent :: "[_, 'a] ⇒ bool" where
"nilpotent R a ⟷ (∃(n::nat). a^⇗R n⇖ = 𝟬⇘R⇙)"
definition
zero_divisor :: "[_, 'a] ⇒ bool" where
"zero_divisor R a ⟷ (∃x∈ carrier R. x ≠ 𝟬⇘R⇙ ∧ x ⋅⇩r⇘R⇙ a = 𝟬⇘R⇙)"
definition
primary_ideal :: "[_, 'a set] ⇒ bool" where
"primary_ideal R q ⟷ ideal R q ∧ (1⇩r⇘R⇙) ∉ q ∧
(∀x∈ carrier R. ∀y∈ carrier R.
x ⋅⇩r⇘R⇙ y ∈ q ⟶ (∃n. (npow R x n) ∈ q ∨ y ∈ q))"
definition
prime_ideal :: "[_, 'a set] ⇒ bool" where
"prime_ideal R p ⟷ ideal R p ∧ (1⇩r⇘R⇙) ∉ p ∧ (∀x∈ carrier R. ∀y∈ carrier R.
(x ⋅⇩r⇘R⇙ y ∈ p ⟶ x ∈ p ∨ y ∈ p))"
definition
maximal_ideal :: "[_, 'a set] ⇒ bool" where
"maximal_ideal R mx ⟷ ideal R mx ∧ 1⇩r⇘R⇙ ∉ mx ∧
{J. (ideal R J ∧ mx ⊆ J)} = {mx, carrier R}"
lemma (in Ring) maximal_ideal_ideal:"⟦maximal_ideal R mx⟧ ⟹ ideal R mx"
by (simp add:maximal_ideal_def)
lemma (in Ring) maximal_ideal_proper:"maximal_ideal R mx ⟹ 1⇩r ∉ mx"
by (simp add:maximal_ideal_def)
lemma (in Ring) prime_ideal_ideal:"prime_ideal R I ⟹ ideal R I"
by (simp add:prime_ideal_def)
lemma (in Ring) prime_ideal_proper:"prime_ideal R I ⟹ I ≠ carrier R"
apply (simp add:prime_ideal_def, (erule conjE)+)
apply (simp add:proper_ideal)
done
lemma (in Ring) prime_ideal_proper1:"prime_ideal R p ⟹ 1⇩r ∉ p"
by (simp add:prime_ideal_def)
lemma (in Ring) primary_ideal_ideal:"primary_ideal R q ⟹ ideal R q"
by (simp add:primary_ideal_def)
lemma (in Ring) primary_ideal_proper1:"primary_ideal R q ⟹ 1⇩r ∉ q"
by (simp add:primary_ideal_def)
lemma (in Ring) prime_elems_mult_not:"⟦prime_ideal R P; x ∈ carrier R;
y ∈ carrier R; x ∉ P; y ∉ P ⟧ ⟹ x ⋅⇩r y ∉ P"
apply (simp add:prime_ideal_def, (erule conjE)+)
apply (rule contrapos_pp, simp+)
apply (frule_tac x = x in bspec, assumption,
thin_tac "∀x∈carrier R. ∀y∈carrier R. x ⋅⇩r y ∈ P ⟶ x ∈ P ∨ y ∈ P",
frule_tac x = y in bspec, assumption,
thin_tac "∀y∈carrier R. x ⋅⇩r y ∈ P ⟶ x ∈ P ∨ y ∈ P", simp)
done
lemma (in Ring) prime_is_primary:"prime_ideal R p ⟹ primary_ideal R p"
apply (unfold primary_ideal_def)
apply (rule conjI, simp add:prime_ideal_def)
apply (rule conjI, simp add:prime_ideal_def)
apply ((rule ballI)+, rule impI)
apply (simp add:prime_ideal_def, (erule conjE)+)
apply (frule_tac x = x in bspec, assumption,
thin_tac "∀x∈carrier R. ∀y∈carrier R. x ⋅⇩r y ∈ p ⟶ x ∈ p ∨ y ∈ p",
frule_tac x = y in bspec, assumption,
thin_tac "∀y∈carrier R. x ⋅⇩r y ∈ p ⟶ x ∈ p ∨ y ∈ p", simp)
apply (erule disjE)
apply (frule_tac t = x in np_1[THEN sym])
apply (frule_tac a = x and A = p and b = "x^⇗R (Suc 0)⇖" in eq_elem_in,
assumption)
apply blast
apply simp
done
lemma (in Ring) maximal_prime_Tr0:"⟦maximal_ideal R mx; x ∈ carrier R; x ∉ mx⟧
⟹ mx ∓ (Rxa R x) = carrier R"
apply (frule principal_ideal [of "x"])
apply (frule maximal_ideal_ideal[of "mx"])
apply (frule sum_ideals [of "mx" "Rxa R x"], assumption)
apply (frule sum_ideals_la1 [of "mx" "Rxa R x"], assumption)
apply (simp add:maximal_ideal_def)
apply (erule conjE)+
apply (subgoal_tac "mx ∓ (Rxa R x) ∈ {J. ideal R J ∧ mx ⊆ J}")
apply simp
apply (frule sum_ideals_la2 [of "mx" "Rxa R x"], assumption+)
apply (frule a_in_principal [of "x"])
apply (frule subsetD [of "Rxa R x" "mx ∓ (Rxa R x)" "x"], assumption+)
apply (thin_tac "{J. ideal R J ∧ mx ⊆ J} = {mx, carrier R}")
apply (erule disjE)
apply simp apply simp
apply (thin_tac "{J. ideal R J ∧ mx ⊆ J} = {mx, carrier R}")
apply simp
done
lemma (in Ring) maximal_prime:"maximal_ideal R mx ⟹ prime_ideal R mx"
apply (cut_tac ring_is_ag)
apply (simp add:prime_ideal_def)
apply (simp add:maximal_ideal_ideal)
apply (simp add:maximal_ideal_proper)
apply ((rule ballI)+, rule impI)
apply (rule contrapos_pp, simp+, erule conjE)
apply (frule_tac x = x in maximal_prime_Tr0[of "mx"], assumption+,
frule_tac x = y in maximal_prime_Tr0[of "mx"], assumption+,
frule maximal_ideal_ideal[of mx],
frule ideal_subset1[of mx],
frule_tac a = x in principal_ideal,
frule_tac a = y in principal_ideal,
frule_tac I = "R ♢⇩p x" in ideal_subset1,
frule_tac I = "R ♢⇩p y" in ideal_subset1)
apply (simp add:aGroup.set_sum)
apply (cut_tac ring_one)
apply (frule sym,
thin_tac "{xa. ∃h∈mx. ∃k∈R ♢⇩p x. xa = h ± k} = carrier R",
frule sym,
thin_tac "{x. ∃h∈mx. ∃k∈R ♢⇩p y. x = h ± k} = carrier R")
apply (frule_tac a = "1⇩r" and B = "{xa. ∃i∈mx. ∃j∈(Rxa R x). xa = i ± j}" in
eq_set_inc[of _ "carrier R"], assumption,
frule_tac a = "1⇩r" and B = "{xa. ∃i∈mx. ∃j∈(Rxa R y). xa = i ± j}" in
eq_set_inc[of _ "carrier R"], assumption,
thin_tac "carrier R = {xa. ∃i∈mx. ∃j∈(Rxa R x). xa = i ± j}",
thin_tac "carrier R = {x. ∃i∈mx. ∃j∈(Rxa R y). x = i ± j}")
apply (drule CollectD, (erule bexE)+,
frule sym, thin_tac "1⇩r = i ± j")
apply (drule CollectD, (erule bexE)+, rotate_tac -1,
frule sym, thin_tac "1⇩r = ia ± ja")
apply (frule_tac h = i in ideal_subset[of mx], assumption,
frule_tac h = ia in ideal_subset[of mx], assumption,
frule_tac h = j in ideal_subset, assumption+,
frule_tac h = ja in ideal_subset, assumption+)
apply (cut_tac ring_one)
apply (frule_tac x = i and y = j in aGroup.ag_pOp_closed, assumption+)
apply (frule_tac x = "i ± j" and y = ia and z = ja in ring_distrib1,
assumption+)
apply (frule_tac x = ia and y = i and z = j in ring_distrib2, assumption+,
frule_tac x = ja and y = i and z = j in ring_distrib2, assumption+,
simp)
apply (thin_tac "1⇩r ⋅⇩r ia = i ⋅⇩r ia ± j ⋅⇩r ia",
thin_tac "1⇩r ⋅⇩r ja = i ⋅⇩r ja ± j ⋅⇩r ja",
simp add:ring_l_one[of "1⇩r"])
apply (frule_tac x = ia and r = i in ideal_ring_multiple[of mx], assumption+,
frule_tac x = i and r = j in ideal_ring_multiple1[of mx], assumption+,
frule_tac x = i and r = ja in ideal_ring_multiple1[of mx], assumption+,
frule_tac r = j and x = ia in ideal_ring_multiple[of mx], assumption+)
apply (subgoal_tac "j ⋅⇩r ja ∈ mx")
apply (frule_tac x = "i ⋅⇩r ia" and y = "j ⋅⇩r ia" in ideal_pOp_closed[of mx],
assumption+) apply (
frule_tac x = "i ⋅⇩r ja" and y = "j ⋅⇩r ja" in ideal_pOp_closed[of mx],
assumption+)
apply (frule_tac x = "i ⋅⇩r ia ± j ⋅⇩r ia" and y = "i ⋅⇩r ja ± j ⋅⇩r ja" in
ideal_pOp_closed[of mx], assumption+,
thin_tac "i ± j = i ⋅⇩r ia ± j ⋅⇩r ia ± (i ⋅⇩r ja ± j ⋅⇩r ja)",
thin_tac "ia ± ja = i ⋅⇩r ia ± j ⋅⇩r ia ± (i ⋅⇩r ja ± j ⋅⇩r ja)")
apply (frule sym, thin_tac "1⇩r = i ⋅⇩r ia ± j ⋅⇩r ia ± (i ⋅⇩r ja ± j ⋅⇩r ja)",
simp)
apply (simp add:maximal_ideal_def)
apply (thin_tac "i ± j = i ⋅⇩r ia ± j ⋅⇩r ia ± (i ⋅⇩r ja ± j ⋅⇩r ja)",
thin_tac "ia ± ja = i ⋅⇩r ia ± j ⋅⇩r ia ± (i ⋅⇩r ja ± j ⋅⇩r ja)",
thin_tac "i ⋅⇩r ia ± j ⋅⇩r ia ± (i ⋅⇩r ja ± j ⋅⇩r ja) ∈ carrier R",
thin_tac "1⇩r = i ⋅⇩r ia ± j ⋅⇩r ia ± (i ⋅⇩r ja ± j ⋅⇩r ja)",
thin_tac "i ⋅⇩r j ∈ mx", thin_tac "i ⋅⇩r ja ∈ mx",
thin_tac "R ♢⇩p y ⊆ carrier R", thin_tac "R ♢⇩p x ⊆ carrier R",
thin_tac "ideal R (R ♢⇩p y)", thin_tac "ideal R (R ♢⇩p x)")
apply (simp add:Rxa_def, (erule bexE)+, simp)
apply (simp add:ring_tOp_assoc)
apply (simp add:ring_tOp_assoc[THEN sym])
apply (frule_tac x = x and y = ra in ring_tOp_commute, assumption+, simp)
apply (simp add:ring_tOp_assoc,
frule_tac x = x and y = y in ring_tOp_closed, assumption+)
apply (frule_tac x1 = r and y1 = ra and z1 = "x ⋅⇩r y" in
ring_tOp_assoc[THEN sym], assumption+, simp)
apply (frule_tac x = r and y = ra in ring_tOp_closed, assumption+,
rule ideal_ring_multiple[of mx], assumption+)
done
lemma (in Ring) chains_un:"⟦c ∈ chains {I. ideal R I ∧ I ⊂ carrier R}; c ≠ {}⟧
⟹ ideal R (⋃c)"
apply (rule ideal_condition1)
apply (rule Union_least[of "c" "carrier R"])
apply (simp add:chains_def,
erule conjE,
frule_tac c = X in subsetD[of "c" "{I. ideal R I ∧ I ⊂ carrier R}"],
assumption+, simp add:psubset_imp_subset)
apply (simp add:chains_def,
erule conjE)
apply (frule nonempty_ex[of "c"], erule exE)
apply (frule_tac c = x in subsetD[of "c" "{I. ideal R I ∧ I ⊂ carrier R}"],
assumption+, simp, erule conjE)
apply (frule_tac I = x in ideal_zero, blast)
apply (rule ballI)+
apply simp
apply (erule bexE)+
apply (simp add: chains_def chain_subset_def)
apply (frule conjunct1) apply (frule conjunct2)
apply (thin_tac "c ⊆ {I. ideal R I ∧ I ⊂ carrier R} ∧ (∀x∈c. ∀y∈c. x ⊆ y ∨ y ⊆ x)")
apply (frule_tac x = X in bspec, assumption,
thin_tac "∀x∈c. ∀y∈c. x ⊆ y ∨ y ⊆ x",
frule_tac x = Xa in bspec, assumption,
thin_tac "∀y∈c. X ⊆ y ∨ y ⊆ X")
apply (frule_tac c = Xa in subsetD[of "c" "{I. ideal R I ∧ I ⊂ carrier R}"],
assumption+,
frule_tac c = X in subsetD[of "c" "{I. ideal R I ∧ I ⊂ carrier R}"],
assumption+, simp)
apply (erule conjE)+
apply (erule disjE,
frule_tac c = x and A = X and B = Xa in subsetD, assumption+,
frule_tac x = x and y = y and I = Xa in ideal_pOp_closed, assumption+,
blast)
apply (frule_tac c = y and A = Xa and B = X in subsetD, assumption+,
frule_tac x = x and y = y and I = X in ideal_pOp_closed, assumption+,
blast)
apply (rule ballI)+
apply (simp, erule bexE)
apply (simp add:chains_def, erule conjE)
apply (frule_tac c = X in subsetD[of "c" "{I. ideal R I ∧ I ⊂ carrier R}"],
assumption+, simp, erule conjE)
apply (frule_tac I = X and x = x and r = r in ideal_ring_multiple,
assumption+, blast)
done
lemma (in Ring) zeroring_no_maximal:"zeroring R ⟹ ¬ (∃I. maximal_ideal R I)"
apply (rule contrapos_pp, simp+, erule exE,
frule_tac mx = x in maximal_ideal_ideal)
apply (frule_tac I = x in ideal_zero)
apply (simp add:zeroring_def, erule conjE,
cut_tac ring_one, simp, thin_tac "carrier R = {𝟬}",
frule sym, thin_tac "1⇩r = 𝟬", simp, thin_tac "𝟬 = 1⇩r")
apply (simp add:maximal_ideal_def)
done
lemma (in Ring) id_maximal_Exist:"¬(zeroring R) ⟹ ∃I. maximal_ideal R I"
apply (cut_tac A="{ I. ideal R I ∧ I ⊂ carrier R }" in Zorn_Lemma2)
apply (rule ballI)
apply (case_tac "C={}", simp)
apply (cut_tac zero_ideal)
apply (simp add:zeroring_def)
apply (cut_tac Ring, simp,
frule not_sym, thin_tac "carrier R ≠ {𝟬}")
apply (cut_tac ring_zero,
frule singleton_sub[of "𝟬" "carrier R"],
thin_tac "𝟬 ∈ carrier R")
apply (subst psubset_eq)
apply blast
apply (subgoal_tac "⋃C ∈ {I. ideal R I ∧ I ⊂ carrier R}")
apply (subgoal_tac "∀x∈C. x ⊆ (⋃C)", blast)
apply (rule ballI, rule Union_upper, assumption)
apply (simp add:chains_un)
apply (cut_tac A = C in Union_least[of _ "carrier R"])
apply (simp add:chains_def, erule conjE,
frule_tac c = X and A = C in
subsetD[of _ "{I. ideal R I ∧ I ⊂ carrier R}"], assumption+,
simp add:ideal_subset1, simp add:psubset_eq)
apply (rule contrapos_pp, simp+,
cut_tac ring_one, frule sym, thin_tac "⋃C = carrier R")
apply (frule_tac B = "⋃C" in eq_set_inc[of "1⇩r" "carrier R"], assumption,
thin_tac "carrier R = ⋃C")
apply (simp, erule bexE)
apply (simp add:chains_def, erule conjE)
apply (frule_tac c = X and A = C in
subsetD[of _ "{I. ideal R I ∧ I ⊆ carrier R ∧ I ≠ carrier R}"],
assumption+, simp, (erule conjE)+)
apply (frule_tac I = X in ideal_inc_one, assumption+, simp)
apply (erule bexE, simp, erule conjE)
apply (subgoal_tac "maximal_ideal R M", blast)
apply (simp add:maximal_ideal_def)
apply (rule conjI, rule contrapos_pp, simp+,
frule_tac I = M in ideal_inc_one, assumption+, simp)
apply (rule equalityI)
apply (rule subsetI, simp)
apply (erule conjE)
apply (frule_tac x = x in spec,
thin_tac "∀x. ideal R x ∧ x ⊂ carrier R ⟶ M ⊆ x ⟶ x = M", simp)
apply (frule_tac I = x in ideal_subset1, simp add:psubset_eq)
apply (case_tac "x = carrier R", simp)
apply simp
apply (rule subsetI, simp)
apply (erule disjE)
apply simp
apply (simp add:whole_ideal)
done
definition
ideal_Int :: "[_, 'a set set] ⇒ 'a set" where
"ideal_Int R S == ⋂ S"
lemma (in Ring) ideal_Int_ideal:"⟦S ⊆ {I. ideal R I}; S≠{}⟧ ⟹
ideal R (⋂ S)"
apply (rule ideal_condition1)
apply (frule nonempty_ex[of "S"], erule exE)
apply (frule_tac c = x in subsetD[of "S" "{I. ideal R I}"], assumption+)
apply (simp, frule_tac I = x in ideal_subset1)
apply (frule_tac B = x and A = S in Inter_lower)
apply (rule_tac A = "⋂S" and B = x and C = "carrier R" in subset_trans,
assumption+)
apply (cut_tac ideal_zero_forall, blast)
apply (simp, rule ballI)
apply (rule ballI)+
apply simp
apply (frule_tac x = X in bspec, assumption,
thin_tac "∀X∈S. x ∈ X",
frule_tac x = X in bspec, assumption,
thin_tac "∀X∈S. y ∈ X")
apply (frule_tac c = X in subsetD[of "S" "{I. ideal R I}"], assumption+,
simp, rule_tac x = x and y = y in ideal_pOp_closed, assumption+)
apply (rule ballI)+
apply (simp, rule ballI)
apply (frule_tac x = X in bspec, assumption,
thin_tac "∀X∈S. x ∈ X",
frule_tac c = X in subsetD[of "S" "{I. ideal R I}"], assumption+,
simp add:ideal_ring_multiple)
done
lemma (in Ring) sum_prideals_Int:"⟦∀l ≤ n. f l ∈ carrier R;
S = {I. ideal R I ∧ f ` {i. i ≤ n} ⊆ I}⟧ ⟹
(sum_pr_ideals R f n) = ⋂ S"
apply (rule equalityI)
apply (subgoal_tac "∀X∈S. sum_pr_ideals R f n ⊆ X")
apply blast
apply (rule ballI)
apply (simp, erule conjE)
apply (rule_tac I = X and n = n and f = f in sum_of_prideals4, assumption+)
apply (subgoal_tac "(sum_pr_ideals R f n) ∈ S")
apply blast
apply (simp add:CollectI)
apply (simp add: sum_of_prideals2)
apply (simp add: sum_of_prideals)
done
text‹This proves that ‹(sum_pr_ideals R f n)› is the smallest ideal containing
‹f ` (Nset n)››
primrec ideal_n_prod::"[('a, 'm) Ring_scheme, nat, nat ⇒ 'a set] ⇒ 'a set"
where
ideal_n_prod0: "ideal_n_prod R 0 J = J 0"
| ideal_n_prodSn: "ideal_n_prod R (Suc n) J =
(ideal_n_prod R n J) ♢⇩r⇘R⇙ (J (Suc n))"
abbreviation
IDNPROD ("(3iΠ⇘_,_⇙ _)" [98,98,99]98) where
"iΠ⇘R,n⇙ J == ideal_n_prod R n J"
primrec
ideal_pow :: "['a set, ('a, 'more) Ring_scheme, nat] ⇒ 'a set"
("(3_/ ⇗♢_ _⇖)" [120,120,121]120)
where
ip0: "I ⇗♢R 0⇖ = carrier R"
| ipSuc: "I ⇗♢R (Suc n)⇖ = I ♢⇩r⇘R⇙ (I ⇗♢R n⇖)"
lemma (in Ring) prod_mem_prod_ideals:"⟦ideal R I; ideal R J; i ∈ I; j ∈ J⟧ ⟹
i ⋅⇩r j ∈ (I ♢⇩r J)"
apply (simp add:ideal_prod_def)
apply (rule allI, rule impI, erule conjE, rename_tac X)
apply (rule_tac A = "{x. ∃i∈I. ∃j∈J. x = Ring.tp R i j}" and B = X and c = "i ⋅⇩r j" in subsetD, assumption)
apply simp apply blast
done
lemma (in Ring) ideal_prod_ideal:"⟦ideal R I; ideal R J ⟧ ⟹
ideal R (I ♢⇩r J)"
apply (rule ideal_condition1)
apply (simp add:ideal_prod_def)
apply (rule subsetI, simp)
apply (cut_tac whole_ideal)
apply (frule_tac x = "carrier R" in spec,
thin_tac "∀xa. ideal R xa ∧ {x. ∃i∈I. ∃j∈J. x = i ⋅⇩r j} ⊆ xa ⟶
x ∈ xa")
apply (subgoal_tac "{x. ∃i∈I. ∃j∈J. x = i ⋅⇩r j} ⊆ carrier R", simp)
apply (thin_tac "ideal R (carrier R) ∧
{x. ∃i∈I. ∃j∈J. x = i ⋅⇩r j} ⊆ carrier R ⟶ x ∈ carrier R")
apply (rule subsetI, simp, (erule bexE)+, simp)
apply (frule_tac h = i in ideal_subset[of "I"], assumption+,
frule_tac h = j in ideal_subset[of "J"], assumption+)
apply (rule_tac x = i and y = j in ring_tOp_closed, assumption+)
apply (frule ideal_zero[of "I"],
frule ideal_zero[of "J"],
subgoal_tac "𝟬 ∈ I ♢⇩r⇘ R⇙ J", blast)
apply (simp add:ideal_prod_def)
apply (rule allI, rule impI, erule conjE)
apply (rule ideal_zero, assumption)
apply (rule ballI)+
apply (simp add:ideal_prod_def)
apply (rule allI, rule impI)
apply (frule_tac x = xa in spec,
thin_tac "∀xa. ideal R xa ∧ {x. ∃i∈I. ∃j∈J. x = i ⋅⇩r j} ⊆ xa
⟶ x ∈ xa",
frule_tac x = xa in spec,
thin_tac "∀x. ideal R x ∧ {x. ∃i∈I. ∃j∈J. x = i ⋅⇩r j} ⊆ x ⟶ y ∈ x",
erule conjE, simp,
rule_tac x = x and y = y in ideal_pOp_closed, assumption+)
apply (rule ballI)+
apply (simp add:ideal_prod_def)
apply (rule allI, rule impI, erule conjE)
apply (frule_tac x = xa in spec,
thin_tac "∀xa. ideal R xa ∧ {x. ∃i∈I. ∃j∈J. x = i ⋅⇩r j}
⊆ xa ⟶ x ∈ xa", simp)
apply (simp add:ideal_ring_multiple)
done
lemma (in Ring) ideal_prod_commute:"⟦ideal R I; ideal R J⟧ ⟹
I ♢⇩r J = J ♢⇩r I"
apply (simp add:ideal_prod_def)
apply (subgoal_tac "{K. ideal R K ∧ {x. ∃i∈I. ∃j∈J. x = i ⋅⇩r j}
⊆ K} = {K. ideal R K ∧ {x. ∃i∈J. ∃j∈I. x = i ⋅⇩r j} ⊆ K}")
apply simp
apply (rule equalityI)
apply (rule subsetI, rename_tac X, simp, erule conjE)
apply (rule subsetI, simp)
apply ((erule bexE)+)
apply (subgoal_tac "x ∈ {x. ∃i∈I. ∃j∈J. x = i ⋅⇩r j}",
rule_tac c = x and A = "{x. ∃i∈I. ∃j∈J. x = i ⋅⇩r j}" and B = X in
subsetD, assumption+,
frule_tac h = i in ideal_subset[of "J"], assumption,
frule_tac h = j in ideal_subset[of "I"], assumption,
frule_tac x = i and y = j in ring_tOp_commute, assumption+, simp,
blast)
apply (rule subsetI, simp, erule conjE,
rule subsetI, simp,
(erule bexE)+,
subgoal_tac "xa ∈ {x. ∃i∈J. ∃j∈I. x = i ⋅⇩r j}",
rule_tac c = xa and A = "{x. ∃i∈J. ∃j∈I. x = i ⋅⇩r j}" and B = x in
subsetD, assumption+,
frule_tac h = i in ideal_subset[of "I"], assumption,
frule_tac h = j in ideal_subset[of "J"], assumption,
frule_tac x = i and y = j in ring_tOp_commute, assumption+, simp,
blast)
done
lemma (in Ring) ideal_prod_subTr:"⟦ideal R I; ideal R J; ideal R C;
∀i∈I. ∀j∈J. i ⋅⇩r j ∈ C⟧ ⟹ I ♢⇩r J ⊆ C"
apply (simp add:ideal_prod_def)
apply (rule_tac B = C and
A = "{L. ideal R L ∧ {x. ∃i∈I. ∃j∈J. x = i ⋅⇩r j} ⊆ L}" in
Inter_lower)
apply simp
apply (rule subsetI, simp, (erule bexE)+, simp)
done
lemma (in Ring) n_prod_idealTr:
"(∀k ≤ n. ideal R (J k)) ⟶ ideal R (ideal_n_prod R n J)"
apply (induct_tac n)
apply (rule impI)
apply simp
apply (rule impI)
apply (simp only:ideal_n_prodSn)
apply (cut_tac n = n in Nsetn_sub_mem1, simp)
apply (rule ideal_prod_ideal, assumption)
apply simp
done
lemma (in Ring) n_prod_ideal:"⟦∀k ≤ n. ideal R (J k)⟧
⟹ ideal R (ideal_n_prod R n J)"
apply (simp add:n_prod_idealTr)
done
lemma (in Ring) ideal_prod_la1:"⟦ideal R I; ideal R J⟧ ⟹ (I ♢⇩r J) ⊆ I"
apply (simp add:ideal_prod_def)
apply (rule subsetI)
apply (simp add:CollectI)
apply (subgoal_tac "{x. ∃i∈I. ∃j∈J. x = i ⋅⇩r j} ⊆ I")
apply blast
apply (thin_tac "∀xa. ideal R xa ∧ {x. ∃i∈I. ∃j∈J. x = i ⋅⇩r j} ⊆ xa
⟶ x ∈ xa")
apply (rule subsetI, simp add:CollectI,
(erule bexE)+, frule_tac h = j in ideal_subset[of "J"], assumption+)
apply (simp add:ideal_ring_multiple1)
done
lemma (in Ring) ideal_prod_el1:"⟦ideal R I; ideal R J; a ∈ (I ♢⇩r J)⟧ ⟹
a ∈ I"
apply (frule ideal_prod_la1 [of "I" "J"], assumption+)
apply (rule subsetD, assumption+)
done
lemma (in Ring) ideal_prod_la2:"⟦ideal R I; ideal R J ⟧ ⟹ (I ♢⇩r J) ⊆ J"
apply (subst ideal_prod_commute, assumption+,
rule ideal_prod_la1[of "J" "I"], assumption+)
done
lemma (in Ring) ideal_prod_sub_Int:"⟦ideal R I; ideal R J ⟧ ⟹
(I ♢⇩r J) ⊆ I ∩ J"
by (simp add:ideal_prod_la1 ideal_prod_la2)
lemma (in Ring) ideal_prod_el2:"⟦ideal R I; ideal R J; a ∈ (I ♢⇩r J)⟧ ⟹
a ∈ J"
by (frule ideal_prod_la2 [of "I" "J"], assumption+,
rule subsetD, assumption+)
text‹‹iΠ⇘R,n⇙ J› is the product of ideals›
lemma (in Ring) ele_n_prodTr0:"⟦∀k ≤ (Suc n). ideal R (J k);
a ∈ iΠ⇘R,(Suc n)⇙ J ⟧ ⟹ a ∈ (iΠ⇘R,n⇙ J) ∧ a ∈ (J (Suc n))"
apply (simp add:Nset_Suc[of n])
apply (cut_tac n_prod_ideal[of n J])
apply (rule conjI)
apply (rule ideal_prod_el1 [of "iΠ⇘R,n⇙ J" "J (Suc n)"], assumption, simp+)
apply (rule ideal_prod_el2[of "iΠ⇘R,n⇙ J" "J (Suc n)"], assumption+, simp+)
done
lemma (in Ring) ele_n_prodTr1:
"(∀k ≤ n. ideal R (J k)) ∧ a ∈ ideal_n_prod R n J ⟶
(∀k ≤ n. a ∈ (J k))"
apply (induct_tac n)
apply simp
apply (rule impI)
apply (rule allI, rule impI)
apply (cut_tac n = n in Nsetn_sub_mem1, simp)
apply (erule conjE)
apply (frule_tac n = n in ele_n_prodTr0[of _ J a])
apply simp
apply (erule conjE,
thin_tac "∀k≤Suc n. ideal R (J k)")
apply simp
apply (case_tac "k = Suc n", simp)
apply (frule_tac m = k and n = "Suc n" in noteq_le_less, assumption+,
thin_tac "k ≤ Suc n")
apply (frule_tac x = k and n = "Suc n" in less_le_diff, simp)
done
lemma (in Ring) ele_n_prod:"⟦∀k ≤ n. ideal R (J k);
a ∈ ideal_n_prod R n J ⟧ ⟹ ∀k ≤ n. a ∈ (J k)"
by (simp add: ele_n_prodTr1 [of "n" "J" "a"])
lemma (in Ring) idealprod_whole_l:"ideal R I ⟹ (carrier R) ♢⇩r⇘R⇙ I = I"
apply (rule equalityI)
apply (rule subsetI)
apply (simp add:ideal_prod_def)
apply (subgoal_tac "{x. ∃i∈carrier R. ∃j∈I. x = i ⋅⇩r j} ⊆ I")
apply blast
apply (thin_tac "∀xa. ideal R xa ∧ {x. ∃i∈carrier R. ∃j∈I.
x = i ⋅⇩r j} ⊆ xa ⟶ x ∈ xa")
apply (rule subsetI)
apply simp
apply ((erule bexE)+, simp)
apply (thin_tac "xa = i ⋅⇩r j", simp add:ideal_ring_multiple)
apply (rule subsetI)
apply (simp add:ideal_prod_def)
apply (rule allI, rule impI) apply (erule conjE)
apply (rename_tac xa X)
apply (cut_tac ring_one)
apply (frule_tac h = xa in ideal_subset[of "I"], assumption,
frule_tac x = xa in ring_l_one)
apply (subgoal_tac "1⇩r ⋅⇩r xa ∈ {x. ∃i∈carrier R. ∃j∈I. x = i ⋅⇩r j}")
apply (rule_tac c = xa and A = "{x. ∃i∈carrier R. ∃j∈I. x = i ⋅⇩r j}" and
B = X in subsetD, assumption+)
apply simp
apply simp
apply (frule sym, thin_tac "1⇩r ⋅⇩r xa = xa", blast)
done
lemma (in Ring) idealprod_whole_r:"ideal R I ⟹ I ♢⇩r (carrier R) = I"
by (cut_tac whole_ideal,
simp add:ideal_prod_commute[of "I" "carrier R"],
simp add:idealprod_whole_l)
lemma (in Ring) idealpow_1_self:"ideal R I ⟹ I ⇗♢R (Suc 0)⇖ = I"
apply simp
apply (simp add:idealprod_whole_r)
done
lemma (in Ring) ideal_pow_ideal:"ideal R I ⟹ ideal R (I ⇗♢R n⇖)"
apply (induct_tac n)
apply (simp add:whole_ideal)
apply simp
apply (simp add:ideal_prod_ideal)
done
lemma (in Ring) ideal_prod_prime:"⟦ideal R I; ideal R J; prime_ideal R P;
I ♢⇩r J ⊆ P ⟧ ⟹ I ⊆ P ∨ J ⊆ P"
apply (rule contrapos_pp, simp+)
apply (erule conjE, simp add:subset_eq, (erule bexE)+)
apply (frule_tac i = x and j = xa in prod_mem_prod_ideals[of "I" "J"],
assumption+)
apply (frule_tac x = "x ⋅⇩r xa" in bspec, assumption,
thin_tac "∀x∈I ♢⇩r⇘ R⇙ J. x ∈ P")
apply (simp add: prime_ideal_def, (erule conjE)+)
apply (frule_tac h = x in ideal_subset, assumption,
frule_tac x = x in bspec, assumption,
thin_tac "∀x∈carrier R. ∀y∈carrier R. x ⋅⇩r y ∈ P ⟶ x ∈ P ∨ y ∈ P",
frule_tac h = xa in ideal_subset, assumption,
frule_tac x = xa in bspec, assumption,
thin_tac "∀y∈carrier R. x ⋅⇩r y ∈ P ⟶ x ∈ P ∨ y ∈ P",
simp)
done
lemma (in Ring) ideal_n_prod_primeTr:"prime_ideal R P ⟹
(∀k ≤ n. ideal R (J k)) ⟶ (ideal_n_prod R n J ⊆ P) ⟶
(∃i ≤ n. (J i) ⊆ P)"
apply (induct_tac n)
apply simp
apply (rule impI)
apply (rule impI, simp)
apply (cut_tac I = "iΠ⇘R,n⇙ J" and J = "J (Suc n)" in
ideal_prod_prime[of _ _ "P"],
rule_tac n = n and J = J in n_prod_ideal,
rule allI, simp+)
apply (erule disjE, simp)
apply (cut_tac n = n in Nsetn_sub_mem1,
blast)
apply blast
done
lemma (in Ring) ideal_n_prod_prime:"⟦prime_ideal R P;
∀k ≤ n. ideal R (J k); ideal_n_prod R n J ⊆ P⟧ ⟹
∃i ≤ n. (J i) ⊆ P"
apply (simp add:ideal_n_prod_primeTr)
done
definition
ppa::"[_, nat ⇒ 'a set, 'a set, nat] ⇒ (nat ⇒ 'a)" where
"ppa R P A i l = (SOME x. x ∈ A ∧ x ∈ (P (skip i l)) ∧ x ∉ P i)"
lemma (in Ring) prod_primeTr:"⟦prime_ideal R P; ideal R A; ¬ A ⊆ P;
ideal R B; ¬ B ⊆ P ⟧ ⟹ ∃x. x ∈ A ∧ x ∈ B ∧ x ∉ P"
apply (simp add:subset_eq)
apply (erule bexE)+
apply (subgoal_tac "x ⋅⇩r xa ∈ A ∧ x ⋅⇩r xa ∈ B ∧ x ⋅⇩r xa ∉ P")
apply blast
apply (rule conjI)
apply (rule ideal_ring_multiple1, assumption+)
apply (simp add:ideal_subset)
apply (rule conjI)
apply (rule ideal_ring_multiple, assumption+)
apply (simp add:ideal_subset)
apply (rule contrapos_pp, simp+)
apply (simp add:prime_ideal_def, (erule conjE)+)
apply (frule_tac h = x in ideal_subset[of "A"], assumption+,
frule_tac h = xa in ideal_subset[of "B"], assumption+,
frule_tac x = x in bspec, assumption,
thin_tac "∀x∈carrier R. ∀y∈carrier R. x ⋅⇩r y ∈ P ⟶ x ∈ P ∨ y ∈ P",
frule_tac x = xa in bspec, assumption,
thin_tac "∀y∈carrier R. x ⋅⇩r y ∈ P ⟶ x ∈ P ∨ y ∈ P")
apply simp
done
lemma (in Ring) prod_primeTr1:"⟦∀k ≤ (Suc n). prime_ideal R (P k);
ideal R A; ∀l ≤ (Suc n). ¬ (A ⊆ P l);
∀k ≤ (Suc n). ∀l ≤ (Suc n). k = l ∨ ¬ (P k) ⊆ (P l); i ≤ (Suc n)⟧ ⟹
∀l ≤ n. ppa R P A i l ∈ A ∧
ppa R P A i l ∈ (P (skip i l)) ∧ ppa R P A i l ∉ (P i)"
apply (rule allI, rule impI)
apply (cut_tac i = i and l = l in skip_il_neq_i)
apply (rotate_tac 2)
apply (frule_tac x = i in spec,
thin_tac "∀l ≤ (Suc n). ¬ A ⊆ P l", simp)
apply (cut_tac l = l in skip_mem[of _ "n" "i"], simp,
frule_tac x = "skip i l" in spec,
thin_tac "∀k ≤ (Suc n). ∀l ≤ (Suc n). k = l ∨ ¬ P k ⊆ P l",
simp)
apply (rotate_tac -1,
frule_tac x = i in spec,
thin_tac "∀la ≤ (Suc n). skip i l = la ∨ ¬ P (skip i l) ⊆ P la",
simp)
apply (cut_tac P = "P i" and A = A and B = "P (skip i l)" in prod_primeTr,
simp, assumption+)
apply (frule_tac x = "skip i l" in spec,
thin_tac "∀k≤Suc n. prime_ideal R (P k)", simp,
rule prime_ideal_ideal, assumption+)
apply (simp add:ppa_def)
apply (rule someI2_ex, assumption+)
done
lemma (in Ring) ppa_mem:"⟦∀k ≤ (Suc n). prime_ideal R (P k); ideal R A;
∀l ≤ (Suc n). ¬ (A ⊆ P l);
∀k ≤ (Suc n). ∀l ≤ (Suc n). k = l ∨ ¬ (P k) ⊆ (P l);
i ≤ (Suc n); l ≤ n⟧ ⟹ ppa R P A i l ∈ carrier R"
apply (frule_tac prod_primeTr1[of n P A], assumption+)
apply (rotate_tac -1, frule_tac x = l in spec,
thin_tac "∀l≤n. ppa R P A i l ∈ A ∧
ppa R P A i l ∈ P (skip i l) ∧ ppa R P A i l ∉ P i", simp)
apply (simp add:ideal_subset)
done
lemma (in Ring) nsum_memrTr:"(∀i ≤ n. f i ∈ carrier R) ⟶
(∀l ≤ n. nsum R f l ∈ carrier R)"
apply (cut_tac ring_is_ag)
apply (induct_tac n)
apply (rule impI, rule allI, rule impI)
apply simp
apply (rule impI)
apply (rule allI, rule impI)
apply (rule aGroup.nsum_mem, assumption)
apply (rule allI, simp)
done
lemma (in Ring) nsum_memr:"∀i ≤ n. f i ∈ carrier R ⟹
∀l ≤ n. nsum R f l ∈ carrier R"
by (simp add:nsum_memrTr)
lemma (in Ring) nsum_ideal_incTr:"ideal R A ⟹
(∀i ≤ n. f i ∈ A) ⟶ nsum R f n ∈ A"
apply (induct_tac n)
apply (rule impI)
apply simp
apply (rule impI)
apply simp
apply (rule ideal_pOp_closed, assumption+)
apply simp
done
lemma (in Ring) nsum_ideal_inc:"⟦ideal R A; ∀i ≤ n. f i ∈ A⟧ ⟹
nsum R f n ∈ A"
by (simp add:nsum_ideal_incTr)
lemma (in Ring) nsum_ideal_excTr:"ideal R A ⟹
(∀i ≤ n. f i ∈ carrier R) ∧ (∃j ≤ n. (∀l ∈ {i. i ≤ n} -{j}. f l ∈ A)
∧ (f j ∉ A)) ⟶ nsum R f n ∉ A"
apply (induct_tac n)
apply simp
apply (rule impI)
apply (erule conjE)+
apply (erule exE)
apply (case_tac "j = Suc n", simp) apply (
thin_tac "(∃j≤n. f j ∉ A) ⟶ Σ⇩e R f n ∉ A")
apply (erule conjE)
apply (cut_tac n = n and f = f in nsum_ideal_inc[of A], assumption,
rule allI, simp)
apply (rule contrapos_pp, simp+)
apply (frule_tac a = "Σ⇩e R f n" and b = "f (Suc n)" in
ideal_ele_sumTr1[of A],
simp add:ideal_subset, simp, assumption+, simp)
apply (erule conjE,
frule_tac m = j and n = "Suc n" in noteq_le_less, assumption,
frule_tac x = j and n = "Suc n" in less_le_diff,
thin_tac "j ≤ Suc n", thin_tac "j < Suc n", simp,
cut_tac n = n in Nsetn_sub_mem1, simp)
apply (erule conjE,
frule_tac x = "Suc n" in bspec, simp)
apply (rule contrapos_pp, simp+)
apply (frule_tac a = "Σ⇩e R f n" and b = "f (Suc n)" in
ideal_ele_sumTr2[of A])
apply (cut_tac ring_is_ag,
rule_tac n = n in aGroup.nsum_mem[of R _ f], assumption+,
rule allI, simp, simp, assumption+, simp)
apply (subgoal_tac "∃j≤n. (∀l∈{i. i ≤ n} - {j}. f l ∈ A) ∧ f j ∉ A",
simp,
thin_tac "(∃j≤n. (∀l∈{i. i ≤ n} - {j}. f l ∈ A) ∧ f j ∉ A)
⟶ Σ⇩e R f n ∉ A")
apply (subgoal_tac "∀l∈{i. i ≤ n} - {j}. f l ∈ A", blast,
thin_tac "Σ⇩e R f n ± f (Suc n) ∈ A",
thin_tac "Σ⇩e R f n ∈ A")
apply (rule ballI)
apply (frule_tac x = l in bspec, simp, assumption)
done
lemma (in Ring) nsum_ideal_exc:"⟦ideal R A; ∀i ≤ n. f i ∈ carrier R;
∃j ≤ n. (∀l∈{i. i ≤ n} -{j}. f l ∈ A) ∧ (f j ∉ A) ⟧ ⟹ nsum R f n ∉ A"
by (simp add:nsum_ideal_excTr)
lemma (in Ring) nprod_memTr:"(∀i ≤ n. f i ∈ carrier R) ⟶
(∀l. l ≤ n ⟶ nprod R f l ∈ carrier R)"
apply (induct_tac n)
apply (rule impI, rule allI, rule impI, simp)
apply (rule impI, rule allI, rule impI)
apply (case_tac "l ≤ n")
apply (cut_tac n = n in Nset_Suc, blast)
apply (cut_tac m = l and n = "Suc n" in Nat.le_antisym, assumption)
apply (simp add: not_less)
apply simp
apply (rule ring_tOp_closed, simp)
apply (cut_tac n = n in Nset_Suc, blast)
done
lemma (in Ring) nprod_mem:"⟦∀i ≤ n. f i ∈ carrier R; l ≤ n⟧ ⟹
nprod R f l ∈ carrier R"
by (simp add:nprod_memTr)
lemma (in Ring) ideal_nprod_incTr:"ideal R A ⟹
(∀i ≤ n. f i ∈ carrier R) ∧
(∃l ≤ n. f l ∈ A) ⟶ nprod R f n ∈ A"
apply (induct_tac n)
apply simp
apply (rule impI)
apply (erule conjE)+
apply simp
apply (erule exE)
apply (case_tac "l = Suc n", simp)
apply (rule_tac x = "f (Suc n)" and r = "nprod R f n" in
ideal_ring_multiple[of "A"], assumption+)
apply (rule_tac n = "Suc n" and f = f and l = n in nprod_mem,
assumption+, simp)
apply (erule conjE)
apply (frule_tac m = l and n = "Suc n" in noteq_le_less, assumption,
frule_tac x = l and n = "Suc n" in less_le_diff,
thin_tac "l ≤ Suc n", thin_tac "l < Suc n", simp)
apply (rule_tac x = "nprod R f n" and r = "f (Suc n)" in
ideal_ring_multiple1[of "A"], assumption+)
apply blast
apply simp
done
lemma (in Ring) ideal_nprod_inc:"⟦ideal R A; ∀i ≤ n. f i ∈ carrier R;
∃l ≤ n. f l ∈ A⟧ ⟹ nprod R f n ∈ A"
by (simp add:ideal_nprod_incTr)
lemma (in Ring) nprod_excTr:"prime_ideal R P ⟹
(∀i ≤ n. f i ∈ carrier R) ∧ (∀l ≤ n. f l ∉ P) ⟶
nprod R f n ∉ P"
apply (induct_tac n)
apply simp
apply (rule impI)
apply (erule conjE)+
apply simp
apply (rule_tac y = "f (Suc n)" and x = "nprod R f n" in
prime_elems_mult_not[of "P"], assumption,
rule_tac n = n in nprod_mem, rule allI, simp+)
done
lemma (in Ring) prime_nprod_exc:"⟦prime_ideal R P; ∀i ≤ n. f i ∈ carrier R;
∀l ≤ n. f l ∉ P⟧ ⟹ nprod R f n ∉ P"
by (simp add:nprod_excTr)
definition
nilrad :: "_ ⇒ 'a set" where
"nilrad R = {x. x ∈ carrier R ∧ nilpotent R x}"
lemma (in Ring) id_nilrad_ideal:"ideal R (nilrad R)"
apply (cut_tac ring_is_ag)
apply (rule ideal_condition1[of "nilrad R"])
apply (rule subsetI) apply (simp add:nilrad_def CollectI)
apply (simp add:nilrad_def)
apply (cut_tac ring_zero)
apply (subgoal_tac "nilpotent R 𝟬")
apply blast
apply (simp add:nilpotent_def)
apply (frule np_1[of "𝟬"], blast)
apply (rule ballI)+
apply (simp add:nilrad_def nilpotent_def, (erule conjE)+)
apply (erule exE)+
apply (simp add:aGroup.ag_pOp_closed[of "R"])
apply (frule_tac x = x and y = y and m = n and n = na in npAdd,
assumption+, blast)
apply (rule ballI)+
apply (simp add:nilrad_def nilpotent_def, erule conjE, erule exE)
apply (simp add:ring_tOp_closed,
frule_tac x = r and y = x and n = n in npMul, assumption+,
simp,
frule_tac x = r and n = n in npClose)
apply (simp add:ring_times_x_0, blast)
done
definition
rad_ideal :: "[_, 'a set ] ⇒ 'a set" where
"rad_ideal R I = {a. a ∈ carrier R ∧ nilpotent (qring R I) ((pj R I) a)}"
lemma (in Ring) id_rad_invim:"ideal R I ⟹
rad_ideal R I = (rInvim R (qring R I) (pj R I ) (nilrad (qring R I)))"
apply (cut_tac ring_is_ag)
apply (rule equalityI)
apply (rule subsetI)
apply (simp add:rad_ideal_def)
apply (erule conjE)+
apply (simp add:rInvim_def)
apply (simp add:nilrad_def)
apply (subst pj_mem, rule Ring_axioms)
apply assumption+
apply (simp add:qring_def ar_coset_def set_rcs_def)
apply (simp add:aGroup.ag_carrier_carrier)
apply blast
apply (rule subsetI)
apply (simp add:rInvim_def nilrad_def)
apply (simp add: rad_ideal_def)
done
lemma (in Ring) id_rad_ideal:"ideal R I ⟹ ideal R (rad_ideal R I)"
apply (subst id_rad_invim [of "I"], assumption)
apply (rule invim_of_ideal, rule Ring_axioms, assumption)
apply (rule Ring.id_nilrad_ideal)
apply (simp add:qring_ring)
done
lemma (in Ring) id_rad_cont_I:"ideal R I ⟹ I ⊆ (rad_ideal R I)"
apply (simp add:rad_ideal_def)
apply (rule subsetI, simp,
simp add:ideal_subset)
apply (simp add:nilpotent_def)
apply (subst pj_mem, rule Ring_axioms, assumption+,
simp add:ideal_subset)
apply (frule_tac h = x in ideal_subset[of "I"], assumption,
frule_tac a = x in npQring[OF Ring, of "I" _ "Suc 0"], assumption,
simp only:np_1, simp only:Qring_fix1,
subst qring_zero[of "I"], assumption)
apply blast
done
lemma (in Ring) id_rad_set:"ideal R I ⟹
rad_ideal R I = {x. x ∈ carrier R ∧ (∃n. npow R x n ∈ I)}"
apply (simp add:rad_ideal_def)
apply (rule equalityI)
apply (rule subsetI)
apply (simp add:nilpotent_def, erule conjE, erule exE)
apply (simp add: pj_mem[OF Ring], simp add:npQring[OF Ring])
apply ( simp add:qring_zero)
apply (frule_tac x = x and n = n in npClose)
apply (frule_tac a = "x^⇗R n⇖" in ar_coset_same3[of "I"], assumption+,
blast)
apply (rule subsetI, simp, erule conjE, erule exE)
apply (simp add:nilpotent_def)
apply (simp add: pj_mem[OF Ring], simp add:npQring[OF Ring],
simp add:qring_zero)
apply (frule_tac a = "x^⇗R n⇖" in ar_coset_same4[of "I"], assumption+)
apply blast
done
lemma (in Ring) rad_primary_prime:"primary_ideal R q ⟹
prime_ideal R (rad_ideal R q)"
apply (simp add:prime_ideal_def)
apply (frule primary_ideal_ideal[of "q"])
apply (simp add:id_rad_ideal)
apply (rule conjI)
apply (rule contrapos_pp, simp+)
apply (simp add:id_rad_set, erule conjE, erule exE)
apply (simp add:npOne)
apply (simp add:primary_ideal_proper1[of "q"])
apply ((rule ballI)+, rule impI)
apply (rule contrapos_pp, simp+, erule conjE)
apply (simp add:id_rad_set, erule conjE, erule exE)
apply (simp add:npMul)
apply (simp add:primary_ideal_def, (erule conjE)+)
apply (frule_tac x = x and n = n in npClose,
frule_tac x = y and n = n in npClose)
apply (frule_tac x = "x^⇗R n⇖" in bspec, assumption,
thin_tac "∀x∈carrier R. ∀y∈carrier R. x ⋅⇩r y ∈ q ⟶
(∃n. x^⇗R n⇖ ∈ q) ∨ y ∈ q",
frule_tac x = "y^⇗R n⇖" in bspec, assumption,
thin_tac "∀y∈carrier R. x^⇗R n⇖ ⋅⇩r y ∈ q ⟶
(∃na. x^⇗R n⇖^⇗R na⇖ ∈ q) ∨ y ∈ q", simp)
apply (simp add:npMulExp)
done
lemma (in Ring) npow_notin_prime:"⟦prime_ideal R P; x ∈ carrier R; x ∉ P⟧
⟹ ∀n. npow R x n ∉ P"
apply (rule allI)
apply (induct_tac n)
apply simp
apply (simp add:prime_ideal_proper1)
apply simp
apply (frule_tac x = x and n = na in npClose)
apply (simp add:prime_elems_mult_not)
done
lemma (in Ring) npow_in_prime:"⟦prime_ideal R P; x ∈ carrier R;
∃n. npow R x n ∈ P ⟧ ⟹ x ∈ P"
apply (rule contrapos_pp, simp+)
apply (frule npow_notin_prime, assumption+)
apply blast
done
definition
mul_closed_set::"[_, 'a set ] ⇒ bool" where
"mul_closed_set R S ⟷ S ⊆ carrier R ∧ (∀s∈S. ∀t∈S. s ⋅⇩r⇘R⇙ t ∈ S)"
locale Idomain = Ring +
assumes idom:
"⟦a ∈ carrier R; b ∈ carrier R; a ⋅⇩r b = 𝟬⟧ ⟹ a = 𝟬 ∨ b = 𝟬"
locale Corps =
fixes K (structure)
assumes f_is_ring: "Ring K"
and f_inv: "∀x∈carrier K - {𝟬}. ∃x' ∈ carrier K. x' ⋅⇩r x = 1⇩r"
lemma (in Ring) mul_closed_set_sub:"mul_closed_set R S ⟹ S ⊆ carrier R"
by (simp add:mul_closed_set_def)
lemma (in Ring) mul_closed_set_tOp_closed:"⟦mul_closed_set R S; s ∈ S;
t ∈ S⟧ ⟹ s ⋅⇩r t ∈ S"
by (simp add:mul_closed_set_def)
lemma (in Corps) f_inv_unique:"⟦ x ∈ carrier K - {𝟬}; x' ∈ carrier K;
x'' ∈ carrier K; x' ⋅⇩r x = 1⇩r; x'' ⋅⇩r x = 1⇩r ⟧ ⟹ x' = x''"
apply (cut_tac f_is_ring)
apply (cut_tac x = x' and y = x and z = x'' in Ring.ring_tOp_assoc[of K],
assumption+, simp, assumption, simp)
apply (simp add:Ring.ring_l_one[of K],
simp add:Ring.ring_tOp_commute[of K x x''] Ring.ring_r_one[of K])
done
definition
invf :: "[_, 'a] ⇒ 'a" where
"invf K x = (THE y. y ∈ carrier K ∧ y ⋅⇩r⇘K⇙ x = 1⇩r⇘K⇙)"
lemma (in Corps) invf_inv:"x ∈ carrier K - {𝟬} ⟹
(invf K x) ∈ carrier K ∧ (invf K x) ⋅⇩r x = 1⇩r "
apply (simp add:invf_def)
apply (rule theI')
apply (rule ex_ex1I)
apply (cut_tac f_inv, blast)
apply (rule_tac x' = xa and x'' = y in f_inv_unique[of x])
apply simp+
done
definition
npowf :: "_ ⇒ 'a ⇒ int ⇒ 'a" where
"npowf K x n =
(if 0 ≤ n then npow K x (nat n) else npow K (invf K x) (nat (- n)))"
abbreviation
NPOWF :: "['a, _, int] ⇒ 'a" ("(3_⇘_⇙⇗_⇖)" [77,77,78]77) where
"a⇘K⇙⇗n⇖ == npowf K a n"
abbreviation
IOP :: "['a, _] ⇒ 'a" ("(_⇗‐ _⇖)" [87,88]87) where
"a⇗‐K⇖ == invf K a"
lemma (in Idomain) idom_is_ring: "Ring R" ..
lemma (in Idomain) idom_tOp_nonzeros:"⟦x ∈ carrier R;
y ∈ carrier R; x ≠ 𝟬; y ≠ 𝟬⟧ ⟹ x ⋅⇩r y ≠ 𝟬"
apply (rule contrapos_pp, simp+)
apply (cut_tac idom[of x y]) apply (erule disjE, simp+)
done
lemma (in Idomain) idom_potent_nonzero:
"⟦x ∈ carrier R; x ≠ 𝟬⟧ ⟹ npow R x n ≠ 𝟬 "
apply (induct_tac n)
apply simp
apply (rule contrapos_pp, simp+)
apply (frule ring_l_one[of "x", THEN sym]) apply simp
apply (simp add:ring_times_0_x)
apply (rule contrapos_pp, simp+)
apply (frule_tac n = n in npClose[of x],
cut_tac a = "x^⇗R n⇖" and b = x in idom, assumption+)
apply (erule disjE, simp+)
done
lemma (in Idomain) idom_potent_unit:"⟦a ∈ carrier R; 0 < n⟧
⟹ (Unit R a) = (Unit R (npow R a n))"
apply (rule iffI)
apply (simp add:Unit_def, erule bexE)
apply (simp add:npClose)
apply (frule_tac x1 = a and y1 = b and n1 = n in npMul[THEN sym], assumption,
simp add:npOne)
apply (frule_tac x = b and n = n in npClose, blast)
apply (case_tac "n = Suc 0", simp only: np_1)
apply (simp add:Unit_def, erule conjE, erule bexE)
apply (cut_tac x = a and n = "n - Suc 0" in npow_suc[of R], simp del:npow_suc,
thin_tac "a^⇗R n⇖ = a^⇗R (n - Suc 0)⇖ ⋅⇩r a",
frule_tac x = a and n = "n - Suc 0" in npClose,
frule_tac x = "a^⇗R (n - Suc 0)⇖" and y = a in ring_tOp_commute, assumption+,
simp add:ring_tOp_assoc,
frule_tac x = "a^⇗R (n - Suc 0)⇖" and y = b in ring_tOp_closed, assumption+)
apply blast
done
lemma (in Idomain) idom_mult_cancel_r:"⟦a ∈ carrier R;
b ∈ carrier R; c ∈ carrier R; c ≠ 𝟬; a ⋅⇩r c = b ⋅⇩r c⟧ ⟹ a = b"
apply (cut_tac ring_is_ag)
apply (frule ring_tOp_closed[of "a" "c"], assumption+,
frule ring_tOp_closed[of "b" "c"], assumption+)
apply (simp add:aGroup.ag_eq_diffzero[of "R" "a ⋅⇩r c" "b ⋅⇩r c"],
simp add:ring_inv1_1,
frule aGroup.ag_mOp_closed[of "R" "b"], assumption,
simp add:ring_distrib2[THEN sym, of "c" "a" "-⇩a b"])
apply (frule aGroup.ag_pOp_closed[of "R" "a" "-⇩a b"], assumption+)
apply (subst aGroup.ag_eq_diffzero[of R a b], assumption+)
apply (rule contrapos_pp, simp+)
apply (frule idom_tOp_nonzeros[of "a ± -⇩a b" c], assumption+, simp)
done
lemma (in Idomain) idom_mult_cancel_l:"⟦a ∈ carrier R;
b ∈ carrier R; c ∈ carrier R; c ≠ 𝟬; c ⋅⇩r a = c ⋅⇩r b⟧ ⟹ a = b"
apply (simp add:ring_tOp_commute)
apply (simp add:idom_mult_cancel_r)
done
lemma (in Corps) invf_closed1:"x ∈ carrier K - {𝟬} ⟹
invf K x ∈ (carrier K) - {𝟬}"
apply (frule invf_inv[of x], erule conjE)
apply (rule contrapos_pp, simp+)
apply (cut_tac f_is_ring) apply (
simp add:Ring.ring_times_0_x[of K])
apply (frule sym, thin_tac "𝟬 = 1⇩r", simp, erule conjE)
apply (frule Ring.ring_l_one[of K x], assumption)
apply (rotate_tac -1, frule sym, thin_tac "1⇩r ⋅⇩r x = x",
simp add:Ring.ring_times_0_x)
done
lemma (in Corps) linvf:"x ∈ carrier K - {𝟬} ⟹ (invf K x) ⋅⇩r x = 1⇩r"
by (simp add:invf_inv)
lemma (in Corps) field_is_ring:"Ring K"
by (simp add:f_is_ring)
lemma (in Corps) invf_one:"1⇩r ≠ 𝟬 ⟹ invf K (1⇩r) = 1⇩r"
apply (cut_tac field_is_ring)
apply (frule_tac Ring.ring_one)
apply (cut_tac invf_closed1 [of "1⇩r"])
apply (cut_tac linvf[of "1⇩r"])
apply (simp add:Ring.ring_r_one[of "K"])
apply simp+
done
lemma (in Corps) field_tOp_assoc:"⟦x ∈ carrier K; y ∈ carrier K; z ∈ carrier K⟧
⟹ x ⋅⇩r y ⋅⇩r z = x ⋅⇩r (y ⋅⇩r z)"
apply (cut_tac field_is_ring)
apply (simp add:Ring.ring_tOp_assoc)
done
lemma (in Corps) field_tOp_commute:"⟦x ∈ carrier K; y ∈ carrier K⟧
⟹ x ⋅⇩r y = y ⋅⇩r x"
apply (cut_tac field_is_ring)
apply (simp add:Ring.ring_tOp_commute)
done
lemma (in Corps) field_inv_inv:"⟦x ∈ carrier K; x ≠ 𝟬⟧ ⟹ (x⇗‐K⇖)⇗‐K⇖ = x"
apply (cut_tac invf_closed1[of "x"])
apply (cut_tac invf_inv[of "x⇗‐K⇖"], erule conjE)
apply (frule field_tOp_assoc[THEN sym, of "x⇗‐ K⇖⇗‐ K⇖" "x⇗‐ K⇖" "x"],
simp, assumption, simp)
apply (cut_tac field_is_ring,
simp add:Ring.ring_l_one Ring.ring_r_one, erule conjE,
cut_tac invf_inv[of x], erule conjE, simp add:Ring.ring_r_one)
apply simp+
done
lemma (in Corps) field_is_idom:"Idomain K"
apply (rule Idomain.intro)
apply (simp add:field_is_ring)
apply (cut_tac field_is_ring)
apply (rule Idomain_axioms.intro)
apply (rule contrapos_pp, simp+, erule conjE)
apply (cut_tac x = a in invf_closed1, simp, simp, erule conjE)
apply (frule_tac x = "a⇗‐ K⇖" and y = a and z = b in field_tOp_assoc,
assumption+)
apply (simp add:linvf Ring.ring_times_x_0 Ring.ring_l_one)
done
lemma (in Corps) field_potent_nonzero:"⟦x ∈ carrier K; x ≠ 𝟬⟧ ⟹
x^⇗K n⇖ ≠ 𝟬"
apply (cut_tac field_is_idom)
apply (cut_tac field_is_ring,
simp add:Idomain.idom_potent_nonzero)
done
lemma (in Corps) field_potent_nonzero1:"⟦x ∈ carrier K; x ≠ 𝟬⟧ ⟹ x⇘K⇙⇗n⇖ ≠ 𝟬"
apply (simp add:npowf_def)
apply (case_tac "0 ≤ n")
apply (simp add:field_potent_nonzero)
apply simp
apply (cut_tac invf_closed1[of "x"], simp+, (erule conjE)+)
apply (simp add:field_potent_nonzero)
apply simp
done
lemma (in Corps) field_nilp_zero:"⟦x ∈ carrier K; x^⇗K n⇖ = 𝟬⟧ ⟹ x = 𝟬"
by (rule contrapos_pp, simp+, simp add:field_potent_nonzero)
lemma (in Corps) npowf_mem:"⟦a ∈ carrier K; a ≠ 𝟬⟧ ⟹
npowf K a n ∈ carrier K"
apply (simp add:npowf_def)
apply (cut_tac field_is_ring)
apply (case_tac "0 ≤ n", simp,
simp add:Ring.npClose, simp)
apply (cut_tac invf_closed1[of "a"], simp, erule conjE,
simp add:Ring.npClose, simp)
done
lemma (in Corps) field_npowf_exp_zero:"⟦a ∈ carrier K; a ≠ 𝟬⟧ ⟹
npowf K a 0 = 1⇩r"
by (cut_tac field_is_ring, simp add:npowf_def)
lemma (in Corps) npow_exp_minusTr1:"⟦x ∈ carrier K; x ≠ 𝟬; 0 ≤ i⟧ ⟹
0 ≤ i - (int j) ⟶ x⇘K⇙⇗(i - (int j))⇖ = x^⇗K (nat i)⇖ ⋅⇩r (x⇗‐K⇖)^⇗K j⇖"
apply (cut_tac field_is_ring,
cut_tac invf_closed1[of "x"], simp,
simp add:npowf_def, erule conjE)
apply (induct_tac "j", simp)
apply (frule Ring.npClose[of "K" "x" "nat i"], assumption+,
simp add:Ring.ring_r_one)
apply (rule impI, simp)
apply (subst zdiff)
apply (simp add:add.commute[of "1"])
apply (cut_tac z = i and w = "int n + 1" in zdiff,
simp only:minus_add_distrib,
thin_tac "i - (int n + 1) = i + (- int n + - 1)")
apply (cut_tac z = "i + - int n" in nat_diff_distrib[of "1"],
simp, simp)
apply (simp only:zdiff[of _ "1"], simp)
apply (cut_tac field_is_idom)
apply (frule_tac n = "nat i" in Ring.npClose[of "K" "x"], assumption+,
frule_tac n = "nat i" in Ring.npClose[of "K" "x⇗‐ K⇖"], assumption+,
frule_tac n = n in Ring.npClose[of "K" "x⇗‐ K⇖"], assumption+ )
apply (rule_tac a = "x^⇗K (nat (i + (- int n - 1)))⇖" and
b = "x^⇗K (nat i)⇖ ⋅⇩r (x⇗‐ K⇖^⇗K n⇖ ⋅⇩r x⇗‐ K⇖)" and c = x in
Idomain.idom_mult_cancel_r[of "K"], assumption+)
apply (simp add:Ring.npClose, rule Ring.ring_tOp_closed, assumption+,
rule Ring.ring_tOp_closed, assumption+)
apply (subgoal_tac "0 < nat (i - int n)")
apply (subst Ring.npMulElmR, assumption+, simp,
simp add:field_tOp_assoc[THEN sym, of "x^⇗K (nat i)⇖" _ "x⇗‐ K⇖"])
apply (subst field_tOp_assoc[of _ _ x])
apply (rule Ring.ring_tOp_closed[of K], assumption+)
apply (simp add: linvf)
apply (subst Ring.ring_r_one[of K], assumption)
apply auto
apply (metis Ring.npClose)
apply (simp only: uminus_add_conv_diff [symmetric] add.assoc [symmetric])
apply (simp add: algebra_simps nat_diff_distrib Suc_diff_Suc)
apply (smt (verit) Ring.npMulElmR Suc_nat_eq_nat_zadd1 nat_diff_distrib' nat_int of_nat_0_le_iff)
done
lemma (in Corps) npow_exp_minusTr2:"⟦x ∈ carrier K; x ≠ 𝟬; 0 ≤ i; 0 ≤ j;
0 ≤ i - j⟧ ⟹ x⇘K⇙⇗(i - j)⇖ = x^⇗K (nat i)⇖ ⋅⇩r (x⇗‐K⇖)^⇗K (nat j)⇖"
apply (frule npow_exp_minusTr1[of "x" "i" "nat j"], assumption+)
apply simp
done
lemma (in Corps) npowf_inv:"⟦x ∈ carrier K; x ≠ 𝟬; 0 ≤ j⟧ ⟹ x⇘K⇙⇗j⇖ = (x⇗‐K⇖)⇘K⇙⇗(-j)⇖"
apply (simp add:npowf_def)
apply (rule impI, simp add:zle)
apply (simp add:field_inv_inv)
done
lemma (in Corps) npowf_inv1:"⟦x ∈ carrier K; x ≠ 𝟬; ¬ 0 ≤ j⟧ ⟹
x⇘K⇙⇗j⇖ = (x⇗‐K⇖)⇘K⇙⇗(-j)⇖"
apply (simp add:npowf_def)
done
lemma (in Corps) npowf_inverse:"⟦x ∈ carrier K; x ≠ 𝟬⟧ ⟹ x⇘K⇙⇗j⇖ = (x⇗‐K⇖)⇘K⇙⇗(-j)⇖"
apply (case_tac "0 ≤ j")
apply (simp add:npowf_inv, simp add:npowf_inv1)
done
lemma (in Corps) npowf_expTr1:"⟦x ∈ carrier K; x ≠ 𝟬; 0 ≤ i; 0 ≤ j;
0 ≤ i - j⟧ ⟹ x⇘K⇙⇗(i - j)⇖ = x⇘K⇙⇗i⇖ ⋅⇩r x⇘K⇙⇗(- j)⇖"
apply (simp add:npow_exp_minusTr2)
apply (simp add:npowf_def)
done
lemma (in Corps) npowf_expTr2:"⟦x ∈ carrier K; x ≠ 𝟬; 0 ≤ i + j⟧ ⟹
x⇘K⇙⇗(i + j)⇖ = x⇘K⇙⇗i⇖ ⋅⇩r x⇘K⇙⇗j⇖"
apply (cut_tac field_is_ring)
apply (case_tac "0 ≤ i")
apply (case_tac "0 ≤ j")
apply (simp add:npowf_def, simp add:nat_add_distrib,
rule Ring.npMulDistr[THEN sym], assumption+)
apply (subst zminus_minus[THEN sym, of "i" "j"],
subst npow_exp_minusTr2[of "x" "i" "-j"], assumption+)
apply (simp add:zle, simp add:zless_imp_zle, simp add:npowf_def)
apply (simp add:add.commute[of "i" "j"],
subst zminus_minus[THEN sym, of "j" "i"],
subst npow_exp_minusTr2[of "x" "j" "-i"], assumption+)
apply (simp add:zle, simp add:zless_imp_zle, simp)
apply (frule npowf_mem[of "x" "i"], assumption+,
frule npowf_mem[of "x" "j"], assumption+,
simp add:field_tOp_commute[of "x⇘K⇙⇗i⇖" "x⇘K⇙⇗j⇖"])
apply (simp add:npowf_def)
done
lemma (in Corps) npowf_exp_add:"⟦x ∈ carrier K; x ≠ 𝟬⟧ ⟹
x⇘K⇙⇗(i + j)⇖ = x⇘K⇙⇗i⇖ ⋅⇩r x⇘K⇙⇗j⇖"
apply (case_tac "0 ≤ i + j")
apply (simp add:npowf_expTr2)
apply (simp add:npowf_inv1[of "x" "i + j"])
apply (simp add:zle)
apply (subgoal_tac "0 < -i + -j") prefer 2 apply simp
apply (thin_tac "i + j < 0")
apply (frule zless_imp_zle[of "0" "-i + -j"])
apply (thin_tac "0 < -i + -j")
apply (cut_tac invf_closed1[of "x"])
apply (simp, erule conjE,
frule npowf_expTr2[of "x⇗‐K⇖" "-i" "-j"], assumption+)
apply (simp add:zdiff[THEN sym])
apply (simp add:npowf_inverse, simp)
done
lemma (in Corps) npowf_exp_1_add:"⟦x ∈ carrier K; x ≠ 𝟬⟧ ⟹
x⇘K⇙⇗(1 + j)⇖ = x ⋅⇩r x⇘K⇙⇗j⇖"
apply (simp add:npowf_exp_add[of "x" "1" "j"])
apply (cut_tac field_is_ring)
apply (simp add:npowf_def, simp add:Ring.ring_l_one)
done
lemma (in Corps) npowf_minus:"⟦x ∈ carrier K; x ≠ 𝟬⟧ ⟹ (x⇘K⇙⇗j⇖)⇗‐K⇖ = x⇘K⇙⇗(- j)⇖"
apply (frule npowf_exp_add[of "x" "j" "-j"], assumption+)
apply (simp add:field_npowf_exp_zero)
apply (cut_tac field_is_ring)
apply (frule npowf_mem[of "x" "j"], assumption+)
apply (frule field_potent_nonzero1[of "x" "j"], assumption+)
apply (cut_tac invf_closed1[of "x⇘K⇙⇗j⇖"], simp, erule conjE,
frule Ring.ring_r_one[of "K" "(x⇘K⇙⇗j⇖)⇗‐K⇖"], assumption, simp,
thin_tac "1⇩r = x⇘K⇙⇗j⇖ ⋅⇩r x⇘K⇙⇗- j⇖",
frule npowf_mem[of "x" "-j"], assumption+)
apply (simp add:field_tOp_assoc[THEN sym], simp add:linvf,
simp add:Ring.ring_l_one, simp)
done
lemma (in Ring) residue_fieldTr:"⟦maximal_ideal R mx; x ∈ carrier(qring R mx);
x ≠ 𝟬⇘(qring R mx)⇙⟧ ⟹∃y∈carrier (qring R mx). y ⋅⇩r⇘(qring R mx)⇙ x = 1⇩r⇘(qring R mx)⇙"
apply (frule maximal_ideal_ideal[of "mx"])
apply (simp add:qring_carrier)
apply (simp add:qring_zero)
apply (simp add:qring_def)
apply (erule bexE)
apply (frule sym, thin_tac "a ⊎⇘R⇙ mx = x", simp)
apply (frule_tac a = a in ar_coset_same4_1[of "mx"], assumption+)
apply (frule_tac x = a in maximal_prime_Tr0[of "mx"], assumption+)
apply (cut_tac ring_one)
apply (rotate_tac -2, frule sym, thin_tac "mx ∓ R ♢⇩p a = carrier R")
apply (frule_tac B = "mx ∓ R ♢⇩p a" in eq_set_inc[of "1⇩r" "carrier R"],
assumption+,
thin_tac "carrier R = mx ∓ R ♢⇩p a")
apply (frule ideal_subset1[of mx])
apply (frule_tac a = a in principal_ideal,
frule_tac I = "R ♢⇩p a" in ideal_subset1)
apply (cut_tac ring_is_ag,
simp add:aGroup.set_sum, (erule bexE)+)
apply (thin_tac "ideal R (R ♢⇩p a)", thin_tac "R ♢⇩p a ⊆ carrier R",
simp add:Rxa_def, (erule bexE)+, simp, thin_tac "k = r ⋅⇩r a")
apply (frule_tac a = r and b = a in rcostOp[of "mx"], assumption+)
apply (frule_tac x = r and y = a in ring_tOp_closed, assumption+)
apply (frule_tac a = "r ⋅⇩r a" and x = h and b = "1⇩r" in
aGroup.ag_eq_sol2[of "R"], assumption+)
apply (simp add:ideal_subset) apply (simp add:ring_one, simp)
apply (frule_tac a = h and b = "1⇩r ± -⇩a (r ⋅⇩r a)" and A = mx in
eq_elem_in, assumption+)
apply (frule_tac a = "r ⋅⇩r a" and b = "1⇩r" in ar_coset_same1[of "mx"],
rule ring_tOp_closed, assumption+, rule ring_one, assumption)
apply (frule_tac a1 = "r ⋅⇩r a" and h1 = h in aGroup.arcos_fixed[THEN sym,
of R mx], unfold ideal_def, erule conjE, assumption+,
thin_tac "R +> mx ∧ (∀r∈carrier R. ∀x∈mx. r ⋅⇩r x ∈ mx)",
thin_tac "x = a ⊎⇘R⇙ mx",
thin_tac "1⇩r = h ± r ⋅⇩r a",
thin_tac "h = 1⇩r ± -⇩a (r ⋅⇩r a)", thin_tac "1⇩r ± -⇩a (r ⋅⇩r a) ∈ mx")
apply (rename_tac b h k r) apply simp
apply blast
done
lemma (in Ring) residue_field_cd:"maximal_ideal R mx ⟹
Corps (qring R mx)"
apply (rule Corps.intro)
apply (rule Ring.qring_ring, rule Ring_axioms)
apply (simp add:maximal_ideal_ideal)
apply (simp add:residue_fieldTr[of "mx"])
done
lemma (in Ring) maximal_set_idealTr:
"maximal_set {I. ideal R I ∧ S ∩ I = {}} mx ⟹ ideal R mx"
by (simp add:maximal_set_def)
lemma (in Ring) maximal_setTr:"⟦maximal_set {I. ideal R I ∧ S ∩ I = {}} mx;
ideal R J; mx ⊂ J ⟧ ⟹ S ∩ J ≠ {}"
by (rule contrapos_pp, simp+, simp add:psubset_eq, erule conjE,
simp add:maximal_set_def, blast)
lemma (in Ring) mulDisj:"⟦mul_closed_set R S; 1⇩r ∈ S; 𝟬 ∉ S;
T = {I. ideal R I ∧ S ∩ I = {}}; maximal_set T mx ⟧ ⟹ prime_ideal R mx"
apply (simp add:prime_ideal_def)
apply (rule conjI, simp add:maximal_set_def,
rule conjI, simp add:maximal_set_def)
apply (rule contrapos_pp, simp+)
apply ((erule conjE)+, blast)
apply ((rule ballI)+, rule impI)
apply (rule contrapos_pp, simp+, (erule conjE)+)
apply (cut_tac a = x in id_ideal_psub_sum[of "mx"],
simp add:maximal_set_def, assumption+,
cut_tac a = y in id_ideal_psub_sum[of "mx"],
simp add:maximal_set_def, assumption+)
apply (frule_tac J = "mx ∓ R ♢⇩p x" in maximal_setTr[of "S" "mx"],
rule sum_ideals, simp add:maximal_set_def,
simp add:principal_ideal, assumption,
thin_tac "mx ⊂ mx ∓ R ♢⇩p x")
apply (frule_tac J = "mx ∓ R ♢⇩p y" in maximal_setTr[of "S" "mx"],
rule sum_ideals, simp add:maximal_set_def,
simp add:principal_ideal, assumption,
thin_tac "mx ⊂ mx ∓ R ♢⇩p y")
apply (frule_tac A = "S ∩ (mx ∓ R ♢⇩p x)" in nonempty_ex,
frule_tac A = "S ∩ (mx ∓ R ♢⇩p y)" in nonempty_ex,
(erule exE)+, simp, (erule conjE)+)
apply (rename_tac x y s1 s2,
thin_tac "S ∩ (mx ∓ R ♢⇩p x) ≠ {}",
thin_tac "S ∩ (mx ∓ R ♢⇩p y) ≠ {}")
apply (frule maximal_set_idealTr,
frule_tac a = x in principal_ideal,
frule_tac a = y in principal_ideal,
frule ideal_subset1[of mx],
frule_tac I = "R ♢⇩p x" in ideal_subset1,
frule_tac I = "R ♢⇩p y" in ideal_subset1)
apply (cut_tac ring_is_ag,
simp add:aGroup.set_sum[of R mx],
erule bexE, erule bexE, simp)
apply (frule_tac s = s1 and t = s2 in mul_closed_set_tOp_closed, simp,
assumption, simp,
frule_tac c = h in subsetD[of mx "carrier R"], assumption+,
frule_tac c = k and A = "R ♢⇩p x" in subsetD[of _ "carrier R"],
assumption+)
apply (
cut_tac mul_closed_set_sub,
frule_tac c = s2 in subsetD[of S "carrier R"], assumption+,
simp add:ring_distrib2)
apply ((erule bexE)+, simp,
frule_tac c = ha in subsetD[of mx "carrier R"], assumption+,
frule_tac c = ka and A = "R ♢⇩p y" in subsetD[of _ "carrier R"],
assumption+,
simp add:ring_distrib1)
apply (frule_tac x = h and r = ha in ideal_ring_multiple1[of mx], assumption+)
apply (frule_tac x = h and r = ka in ideal_ring_multiple1[of mx], assumption+,
frule_tac x = ha and r = k in ideal_ring_multiple[of mx], assumption+)
apply (frule_tac a = x and b = y and x = k and y = ka in
mul_two_principal_idealsTr, assumption+,
erule bexE,
frule_tac x = "x ⋅⇩r y" and r = r in ideal_ring_multiple[of mx],
assumption+,
rotate_tac -2, frule sym, thin_tac "k ⋅⇩r ka = r ⋅⇩r (x ⋅⇩r y)", simp)
apply (frule_tac x = "h ⋅⇩r ha ± h ⋅⇩r ka" and y = "k ⋅⇩r ha ± k ⋅⇩r ka" in
ideal_pOp_closed[of mx])
apply (rule ideal_pOp_closed, assumption+)+
apply (simp add:maximal_set_def)
apply blast
apply assumption
done
lemma (in Ring) ex_mulDisj_maximal:"⟦mul_closed_set R S; 𝟬 ∉ S; 1⇩r ∈ S;
T = {I. ideal R I ∧ S ∩ I = {}}⟧ ⟹ ∃mx. maximal_set T mx"
apply (cut_tac A="{ I. ideal R I ∧ S ∩ I = {}}" in Zorn_Lemma2)
prefer 2
apply (simp add:maximal_set_def)
apply (rule ballI)
apply (case_tac "C = {}")
apply (cut_tac zero_ideal, blast)
apply (subgoal_tac "C ∈ chains {I. ideal R I ∧ I ⊂ carrier R}")
apply (frule chains_un, assumption)
apply (subgoal_tac "S ∩ (⋃ C) = {}")
apply (subgoal_tac "∀x∈C. x ⊆ ⋃ C", blast)
apply (rule ballI, rule subsetI, simp add:CollectI)
apply blast
apply (rule contrapos_pp, simp+)
apply (frule_tac A = S and B = "⋃ C" in nonempty_int)
apply (erule exE)
apply (simp, erule conjE, erule bexE)
apply (simp add:chains_def, erule conjE)
apply (frule_tac c = X and A = C and B = "{I. ideal R I ∧ S ∩ I = {}}" in
subsetD, assumption+,
thin_tac "C ⊆ {I. ideal R I ∧ I ⊂ carrier R}",
thin_tac "C ⊆ {I. ideal R I ∧ S ∩ I = {}}")
apply (simp, blast)
apply (simp add:chains_def chain_subset_def, erule conjE)
apply (rule subsetI)
apply (frule_tac c = x and A = C and B = "{I. ideal R I ∧ S ∩ I = {}}" in
subsetD, assumption+,
thin_tac "C ⊆ {I. ideal R I ∧ S ∩ I = {}}",
thin_tac "T = {I. ideal R I ∧ S ∩ I = {}}")
apply (simp, thin_tac "∀x∈C. ∀y∈C. x ⊆ y ∨ y ⊆ x", erule conjE)
apply (simp add:psubset_eq ideal_subset1)
apply (rule contrapos_pp, simp+)
apply (rotate_tac -1, frule sym, thin_tac "x = carrier R",
thin_tac "carrier R = x")
apply (cut_tac ring_one, blast)
done
lemma (in Ring) ex_mulDisj_prime:"⟦mul_closed_set R S; 𝟬 ∉ S; 1⇩r ∈ S⟧ ⟹
∃mx. prime_ideal R mx ∧ S ∩ mx = {}"
apply (frule ex_mulDisj_maximal[of "S" "{I. ideal R I ∧ S ∩ I = {}}"],
assumption+, simp, erule exE)
apply (frule_tac mx = mx in mulDisj [of "S" "{I. ideal R I ∧ S ∩ I = {}}"],
assumption+, simp, assumption)
apply (simp add:maximal_set_def, (erule conjE)+, blast)
done
lemma (in Ring) nilradTr1:"¬ zeroring R ⟹ nilrad R = ⋂ {p. prime_ideal R p}"
apply (rule equalityI)
apply (rule subsetI)
apply (simp add:nilrad_def CollectI nilpotent_def)
apply (erule conjE, erule exE)
apply (rule allI, rule impI)
apply (frule_tac prime_ideal_ideal)
apply (frule sym, thin_tac "x^⇗R n⇖ = 𝟬", frule ideal_zero, simp)
apply (case_tac "n = 0", simp)
apply (frule Zero_ring1[THEN not_sym], simp)
apply (rule_tac P = xa and x = x in npow_in_prime,assumption+, blast)
apply (rule subsetI)
apply (rule contrapos_pp, simp+)
apply (frule id_maximal_Exist, erule exE,
frule maximal_prime)
apply (frule_tac a = I in forall_spec, assumption,
frule_tac I = I in prime_ideal_ideal,
frule_tac h = x and I = I in ideal_subset, assumption)
apply (subgoal_tac "𝟬 ∉ {s. ∃n. s = npow R x n} ∧
1⇩r ∈ {s. ∃n. s = npow R x n}")
apply (subgoal_tac "mul_closed_set R {s. ∃n. s = npow R x n}")
apply (erule conjE)
apply (frule_tac S = "{s. ∃n. s = npow R x n}" in ex_mulDisj_prime,
assumption+, erule exE, erule conjE)
apply (subgoal_tac "x ∈ {s. ∃n. s = x^⇗R n⇖}", blast)
apply simp
apply (cut_tac t = x in np_1[THEN sym], assumption, blast)
apply (thin_tac "𝟬 ∉ {s. ∃n. s = x^⇗R n⇖} ∧ 1⇩r ∈ {s. ∃n. s = x^⇗R n⇖}",
thin_tac "∀xa. prime_ideal R xa ⟶ x ∈ xa")
apply (subst mul_closed_set_def)
apply (rule conjI)
apply (rule subsetI, simp, erule exE)
apply (simp add:npClose)
apply ((rule ballI)+, simp, (erule exE)+, simp)
apply (simp add:npMulDistr, blast)
apply (rule conjI)
apply simp
apply (rule contrapos_pp, simp+, erule exE)
apply (frule sym, thin_tac "𝟬 = x^⇗R n⇖")
apply (simp add:nilrad_def nilpotent_def)
apply simp
apply (cut_tac x1 = x in npow_0[THEN sym, of "R"], blast)
done
lemma (in Ring) nonilp_residue_nilrad:"⟦¬ zeroring R; x ∈ carrier R;
nilpotent (qring R (nilrad R)) (x ⊎⇘R⇙ (nilrad R))⟧ ⟹
x ⊎⇘R⇙ (nilrad R) = 𝟬⇘(qring R (nilrad R))⇙"
apply (simp add:nilpotent_def)
apply (erule exE)
apply (cut_tac id_nilrad_ideal)
apply (simp add:qring_zero)
apply (cut_tac "Ring")
apply (simp add:npQring)
apply (frule_tac x = x and n = n in npClose)
apply (frule_tac I = "nilrad R" and a = "x^⇗R n⇖" in ar_coset_same3,
assumption+)
apply (rule_tac I = "nilrad R" and a = x in ar_coset_same4, assumption)
apply (thin_tac "x^⇗R n⇖ ⊎⇘R⇙ nilrad R = nilrad R",
simp add:nilrad_def nilpotent_def, erule exE)
apply (simp add:npMulExp, blast)
done
lemma (in Ring) ex_contid_maximal:"⟦ S = {1⇩r}; 𝟬 ∉ S; ideal R I; I ∩ S = {};
T = {J. ideal R J ∧ S ∩ J = {} ∧ I ⊆ J}⟧ ⟹ ∃mx. maximal_set T mx"
apply (cut_tac A="{J. ideal R J ∧ S ∩ J = {} ∧ I ⊆ J}" in Zorn_Lemma2)
apply (rule ballI)
apply (case_tac "C = {}")
apply blast
apply (subgoal_tac "⋃C∈{J. ideal R J ∧ S ∩ J = {} ∧ I ⊆ J} ∧
(∀x∈C. x ⊆ ⋃C)")
apply blast
apply (rule conjI,
simp add:CollectI)
apply (subgoal_tac "C ∈ chains {I. ideal R I ∧ I ⊂ carrier R}")
apply (rule conjI,
simp add:chains_un)
apply (rule conjI)
apply (rule contrapos_pp, simp+, erule bexE)
apply (thin_tac " C ∈ chains {I. ideal R I ∧ I ⊂ carrier R}")
apply (simp add:chains_def, erule conjE)
apply (frule_tac c = x and A = C and B = "{J. ideal R J ∧ 1⇩r ∉ J ∧ I ⊆ J}"
in subsetD, assumption+, simp,
thin_tac "C ∈ chains {I. ideal R I ∧ I ⊂ carrier R}")
apply (frule_tac A = C in nonempty_ex, erule exE, simp add:chains_def,
erule conjE,
frule_tac c = x and A = C and B = "{J. ideal R J ∧ 1⇩r ∉ J ∧ I ⊆ J}" in
subsetD, assumption+, simp, (erule conjE)+)
apply (rule_tac A = I and B = x and C = "⋃C" in subset_trans, assumption,
rule_tac B = x and A = C in Union_upper, assumption+)
apply (simp add:chains_def, erule conjE)
apply (rule subsetI, simp)
apply (frule_tac c = x and A = C and B = "{J. ideal R J ∧ 1⇩r ∉ J ∧ I ⊆ J}"
in subsetD, assumption+, simp, (erule conjE)+)
apply (subst psubset_eq, simp add:ideal_subset1)
apply (rule contrapos_pp, simp+, simp add:ring_one)
apply (rule ballI)
apply (rule Union_upper, assumption)
apply (erule bexE)
apply (simp add:maximal_set_def)
apply blast
done
lemma (in Ring) contid_maximal:"⟦S = {1⇩r}; 𝟬 ∉ S; ideal R I; I ∩ S = {};
T = {J. ideal R J ∧ S ∩ J = {} ∧ I ⊆ J}; maximal_set T mx⟧ ⟹
maximal_ideal R mx"
apply (simp add:maximal_set_def maximal_ideal_def)
apply (erule conjE)+
apply (rule equalityI)
apply (rule subsetI, simp add:CollectI, erule conjE)
apply (case_tac "x = mx", simp, simp)
apply (subgoal_tac "1⇩r ∈ x")
apply (rule_tac I = x in ideal_inc_one, assumption+)
apply (rule contrapos_pp, simp+)
apply (drule spec[of _ mx])
apply (simp add:whole_ideal,
rule subsetI, rule ideal_subset[of "mx"], assumption+)
done
lemma (in Ring) ideal_contained_maxid:"⟦¬(zeroring R); ideal R I; 1⇩r ∉ I⟧ ⟹
∃mx. maximal_ideal R mx ∧ I ⊆ mx"
apply (cut_tac ex_contid_maximal[of "{1⇩r}" "I"
"{J. ideal R J ∧ {1⇩r} ∩ J = {} ∧ I ⊆ J}"])
apply (erule exE,
cut_tac mx = mx in contid_maximal[of "{1⇩r}" "I"
"{J. ideal R J ∧ {1⇩r} ∩ J = {} ∧ I ⊆ J}"])
apply simp
apply (frule Zero_ring1, simp,
assumption, simp, simp, simp,
simp add:maximal_set_def, (erule conjE)+, blast,
simp, frule Zero_ring1, simp)
apply (assumption, simp, simp)
done
lemma (in Ring) nonunit_principal_id:"⟦a ∈ carrier R; ¬ (Unit R a)⟧ ⟹
(R ♢⇩p a) ≠ (carrier R)"
apply (rule contrapos_pp, simp+)
apply (frule sym, thin_tac "R ♢⇩p a = carrier R")
apply (cut_tac ring_one)
apply (frule eq_set_inc[of "1⇩r" "carrier R" "R ♢⇩p a"], assumption,
thin_tac "carrier R = R ♢⇩p a", thin_tac "1⇩r ∈ carrier R")
apply (simp add:Rxa_def, erule bexE, simp add:ring_tOp_commute[of _ "a"],
frule sym, thin_tac "1⇩r = a ⋅⇩r r")
apply (simp add:Unit_def)
done
lemma (in Ring) nonunit_contained_maxid:"⟦¬(zeroring R); a ∈ carrier R;
¬ Unit R a ⟧ ⟹ ∃mx. maximal_ideal R mx ∧ a ∈ mx"
apply (frule principal_ideal[of "a"],
frule ideal_contained_maxid[of "R ♢⇩p a"], assumption)
apply (rule contrapos_pp, simp+,
frule ideal_inc_one[of "R ♢⇩p a"], assumption,
simp add:nonunit_principal_id)
apply (erule exE, erule conjE)
apply (frule a_in_principal[of "a"])
apply (frule_tac B = mx in subsetD[of "R ♢⇩p a" _ "a"], assumption, blast)
done
definition
local_ring :: "_ ⇒ bool" where
"local_ring R == Ring R ∧ ¬ zeroring R ∧ card {mx. maximal_ideal R mx} = 1"
lemma (in Ring) local_ring_diff:"⟦¬ zeroring R; ideal R mx; mx ≠ carrier R;
∀a∈ (carrier R - mx). Unit R a ⟧ ⟹ local_ring R ∧ maximal_ideal R mx"
apply (subgoal_tac "{mx} = {m. maximal_ideal R m}")
apply (cut_tac singletonI[of "mx"], simp)
apply (frule sym, thin_tac "{mx} = {m. maximal_ideal R m}")
apply (simp add:local_ring_def, simp add:Ring)
apply (rule equalityI)
apply (rule subsetI, simp)
apply (simp add:maximal_ideal_def)
apply (simp add:ideal_inc_one1[of "mx", THEN sym])
apply (thin_tac "x = mx", simp)
apply (rule equalityI)
apply (rule subsetI, simp, erule conjE)
apply (case_tac "x ≠ mx")
apply (frule_tac A = x and B = mx in sets_not_eq, assumption)
apply (erule bexE)
apply (frule_tac h = a and I = x in ideal_subset, assumption+)
apply (frule_tac x = a in bspec, simp)
apply (frule_tac I = x and a = a in ideal_inc_unit1, assumption+,
simp)
apply simp
apply (rule subsetI, simp)
apply (erule disjE)
apply simp
apply (simp add:whole_ideal ideal_subset1)
apply (rule subsetI)
apply simp
apply (subgoal_tac "x ⊆ mx",
thin_tac "∀a∈carrier R - mx. Unit R a",
simp add:maximal_ideal_def, (erule conjE)+)
apply (subgoal_tac "mx ∈ {J. ideal R J ∧ x ⊆ J}", simp)
apply (thin_tac "{J. ideal R J ∧ x ⊆ J} = {x, carrier R}")
apply simp
apply (rule contrapos_pp, simp+)
apply (simp add:subset_eq, erule bexE)
apply (frule_tac mx = x in maximal_ideal_ideal,
frule_tac x = xa in bspec,
thin_tac "∀a∈carrier R - mx. Unit R a", simp,
simp add:ideal_subset)
apply (frule_tac I = x and a = xa in ideal_inc_unit, assumption+,
simp add:maximal_ideal_def)
done
lemma (in Ring) localring_unit:"⟦¬ zeroring R; maximal_ideal R mx;
∀x. x ∈ mx ⟶ Unit R (x ± 1⇩r) ⟧ ⟹ local_ring R"
apply (frule maximal_ideal_ideal[of "mx"])
apply (frule local_ring_diff[of "mx"], assumption)
apply (simp add:maximal_ideal_def, erule conjE)
apply (simp add:ideal_inc_one1[THEN sym, of "mx"])
apply (rule ballI, simp, erule conjE)
apply (frule_tac x = a in maximal_prime_Tr0[of "mx"], assumption+)
apply (frule sym, thin_tac "mx ∓ R ♢⇩p a = carrier R",
cut_tac ring_one,
frule_tac a = "1⇩r" and A = "carrier R" and B = "mx ∓ R ♢⇩p a" in
eq_set_inc, assumption+,
thin_tac "carrier R = mx ∓ R ♢⇩p a")
apply (frule_tac a = a in principal_ideal,
frule ideal_subset1[of mx],
frule_tac I = "R ♢⇩p a" in ideal_subset1)
apply (cut_tac ring_is_ag,
simp add:aGroup.set_sum, (erule bexE)+)
apply (simp add:Rxa_def, erule bexE, simp)
apply (frule sym, thin_tac "1⇩r = h ± r ⋅⇩r a",
frule_tac x = r and y = a in ring_tOp_closed, assumption+,
frule_tac h = h in ideal_subset[of "mx"], assumption+)
apply (frule_tac I = mx and x = h in ideal_inv1_closed, assumption)
apply (frule_tac a = "-⇩a h" in forall_spec, assumption,
thin_tac "∀x. x ∈ mx ⟶ Unit R (x ± (h ± r ⋅⇩r a))",
thin_tac "h ± r ⋅⇩r a = 1⇩r")
apply (frule_tac h = "-⇩a h" in ideal_subset[of "mx"], assumption,
frule_tac x1 = "-⇩a h" and y1 = h and z1 = "r ⋅⇩r a" in
aGroup.ag_pOp_assoc[THEN sym], assumption+,
simp add:aGroup.ag_l_inv1 aGroup.ag_l_zero,
thin_tac "k = r ⋅⇩r a", thin_tac "h ± r ⋅⇩r a ∈ carrier R",
thin_tac "h ∈ carrier R", thin_tac "-⇩a h ∈ mx",
thin_tac "-⇩a h ± (h ± r ⋅⇩r a) = r ⋅⇩r a")
apply (simp add:ring_tOp_commute, simp add:Unit_def, erule bexE,
simp add:ring_tOp_assoc,
frule_tac x = r and y = b in ring_tOp_closed, assumption+, blast)
apply simp
done
definition
J_rad ::"_ ⇒ 'a set" where
"J_rad R = (if (zeroring R) then (carrier R) else
⋂ {mx. maximal_ideal R mx})"
lemma (in Ring) zeroring_J_rad_empty:"zeroring R ⟹ J_rad R = carrier R"
by (simp add:J_rad_def)
lemma (in Ring) J_rad_mem:"x ∈ J_rad R ⟹ x ∈ carrier R"
apply (simp add:J_rad_def)
apply (case_tac "zeroring R", simp)
apply simp
apply (frule id_maximal_Exist, erule exE)
apply (frule_tac a = I in forall_spec, assumption,
thin_tac "∀xa. maximal_ideal R xa ⟶ x ∈ xa")
apply (frule maximal_ideal_ideal,
simp add:ideal_subset)
done
lemma (in Ring) J_rad_unit:"⟦¬ zeroring R; x ∈ J_rad R⟧ ⟹
∀y. (y∈ carrier R ⟶ Unit R (1⇩r ± (-⇩a x) ⋅⇩r y))"
apply (cut_tac ring_is_ag,
rule allI, rule impI,
rule contrapos_pp, simp+)
apply (frule J_rad_mem[of "x"],
frule_tac x = x and y = y in ring_tOp_closed, assumption,
frule_tac x = "x ⋅⇩r y" in aGroup.ag_mOp_closed, assumption+)
apply (cut_tac ring_one,
frule_tac x = "1⇩r" and y = "-⇩a (x ⋅⇩r y)" in aGroup.ag_pOp_closed,
assumption+)
apply (frule_tac a = "1⇩r ± -⇩a (x ⋅⇩r y)" in nonunit_contained_maxid,
assumption+, simp add:ring_inv1_1)
apply (erule exE, erule conjE)
apply (simp add:J_rad_def,
frule_tac a = mx in forall_spec, assumption,
thin_tac "∀xa. maximal_ideal R xa ⟶ x ∈ xa",
frule_tac mx = mx in maximal_ideal_ideal,
frule_tac I = mx and x = x and r = y in ideal_ring_multiple1,
assumption+)
apply (frule_tac I = mx and x = "x ⋅⇩r y" in ideal_inv1_closed,
assumption+)
apply (frule_tac I = mx and a = "1⇩r" and b = "-⇩a (x ⋅⇩r y)" in ideal_ele_sumTr2,
assumption+)
apply (simp add:maximal_ideal_def)
done
end