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theory Algebra4(** Algebra4
author Hidetsune Kobayashi
Lingjun Chen (part of Chap 4. section 2,
with revision by H. Kobayashi)
Group You Santo
Department of Mathematics
Nihon University
h_coba@math.cst.nihon-u.ac.jp
May 3, 2004.
April 6, 2007 (revised)
chapter 3. Group Theory. Focused on Jordan Hoelder theorem (continued)
section 20. abelian groups
subsection 20-1. Homomorphism of abelian groups
subsection 20-2 quotient abelian group
section 21 direct product and direct sum of abelian groups,
in general case
chapter 4. Ring theory
section 1. Definition of a ring and an ideal
section 2. Calculation of elements
section 3. ring homomorphisms
section 4. quotient rings
section 5. primary ideals, prime ideals
**)
theory Algebra4
imports Algebra3 Binomial Zorn
begin
(*<*)hide const ring(*>*)
section "20. Abelian groups"
record 'a aGroup = "'a carrier" +
pop :: "['a, 'a ] => 'a" (infixl "±\<index>" 62)
mop :: "'a => 'a" ("(-a\<index> _)" [64]63 )
zero :: "'a" ("\<zero>\<index>")
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 = \<zero>"
and ex_zero: "\<zero> ∈ carrier A"
and l_zero:"a ∈ carrier A ==> \<zero> ± a = a"
constdefs (structure A)
b_ag::"_ =>
(|carrier:: 'a set, top:: ['a, 'a] => 'a , iop:: 'a => 'a, one:: 'a |)),"
"b_ag A == (|carrier = carrier A, top = pop A, iop = mop A,
one = zero A |)),"
asubGroup :: "[_ , 'a set] => bool"
"asubGroup A H == (b_ag A) » H"
constdefs (structure A)
aqgrp :: "[_ , 'a set] =>
(| carrier::'a set set, pop::['a set, 'a set] => 'a set,
mop::'a set => 'a set, zero :: 'a set |)),"
"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 |)),"
ag_idmap::"_ => ('a => 'a)" ("(aI_)")
"aIA == λx∈carrier A. x"
syntax (xsymbols)
"@ASubG" :: "[('a, 'more) aGroup_scheme, 'a set] => bool"
(infixl "+>" 58)
translations
"A +> H" == "asubGroup A H"
constdefs (structure A)
Ag_ind ::"[_ , 'a => 'd] => 'd aGroup"
"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) ±
((invfun (carrier A) (f`(carrier A)) f) y)),
mop = λx∈(f`(carrier A)). f (-a ((invfun (carrier A) (f`(carrier A)) f) x)),
zero = f (\<zero> ) |)),"
Agii ::"[_ , 'a => 'd] => ('a => 'd)"
"Agii A f == λx∈carrier A. f x" (** Ag_induced_isomorphism **)
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 "op ± " "carrier A" "carrier A -> carrier A" "x"],
assumption+)
apply (rule funcset_mem[of "op ± 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" (*agpop_gtop*)
apply (simp add:b_ag_def)
done
lemma (in aGroup) agiop_giop:"iop (b_ag A) = mop A" (*agmop_giop*)
apply (simp add:b_ag_def)
done
lemma (in aGroup) agunit_gone:"one (b_ag A) = \<zero>"
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 ==> \<zero> ∈ 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:"\<zero> ∈ carrier A"
by (simp add:ex_zero)
lemma (in aGroup) ag_l_zero:"x ∈ carrier A ==> \<zero> ± x = x"
by (simp add:l_zero)
lemma (in aGroup) ag_r_zero:"x ∈ carrier A ==> x ± \<zero> = 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 = \<zero>"
by (simp add:l_m)
lemma (in aGroup) ag_r_inv1:"x ∈ carrier A ==> x ± (-a x) = \<zero>"
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 = \<zero>|] ==>
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 \<zero> = \<zero>"
apply (cut_tac ex_zero)
apply (frule l_zero[of "\<zero>"])
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) = \<zero>)"
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 = \<zero>)"
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) ≠ \<zero>)"
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 = \<zero>)"
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) \<triangleright> 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 +> {\<zero>}"
apply (rule asubg_test[of "{\<zero>}"])
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 bivar_func_test)
apply (rule ballI)+
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+)
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:funcset_mem[of "f" "carrier A" "D"])
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 univar_func_test)
apply (rule ballI)
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:funcset_mem[of "f" "carrier A" "D"])
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)
apply (simp add: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)
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 "20-1. Homomorphism of abelian groups"
constdefs
aHom::"[('a, 'm) aGroup_scheme, ('b, 'm1) aGroup_scheme] => ('a => 'b) set"
"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))}"
constdefs
compos::"[('a, 'm) aGroup_scheme, 'b => 'c, 'a => 'b] => 'a => 'c"
"compos A g f == compose (carrier A) g f"
constdefs
ker::"[('a, 'm) aGroup_scheme, ('b, 'm1) aGroup_scheme] => ('a => 'b)
=> 'a set" ("(3ker_,_ _)" [82,82,83]82)
"kerF,G f == {a. a ∈ carrier F ∧ f a = (\<zero>G)}"
constdefs
injec::"[('a, 'm) aGroup_scheme, ('b, 'm1) aGroup_scheme, 'a => 'b]
=> bool" ("(3injec_,_ _)" [82,82,83]82)
"injecF,G f == f ∈ aHom F G ∧ kerF,G f = {\<zero>F}"
constdefs
surjec::"[('a, 'm) aGroup_scheme, ('b, 'm1) aGroup_scheme, 'a => 'b]
=> bool" ("(3surjec_,_ _)" [82,82,83]82)
"surjecF,G f == f ∈ aHom F G ∧ surj_to f (carrier F) (carrier G)"
constdefs
bijec::"[('a, 'm) aGroup_scheme, ('b, 'm1) aGroup_scheme, 'a => 'b]
=> bool" ("(3bijec_,_ _)" [82,82,83]82)
"bijecF,G f == injecF,G f ∧surjecF,G f"
constdefs
ainvf::"[('a, 'm) aGroup_scheme, ('b, 'm1) aGroup_scheme, 'a => 'b]
=> ('b => 'a)" ("(3ainvf_,_ _)" [82,82,83]82)
"ainvfF,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:funcset_mem)
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 (\<zero>F) = \<zero>G"
apply (frule aGroup.ag_inc_zero [of "F"])
apply (subst aGroup.ag_l_zero [THEN sym, of "F" "\<zero>F"], assumption+)
apply (simp add:aHom_add)
apply (frule aGroup.ag_l_zero [THEN sym, of "F" "\<zero>F"], assumption+)
apply (subgoal_tac "f (\<zero>F) = f (\<zero>F ±F \<zero>F)") prefer 2 apply simp
apply (thin_tac "\<zero>F = \<zero>F ±F \<zero>F")
apply (simp add:aHom_add) apply (frule sym)
apply (thin_tac "f \<zero>F = f \<zero>F ±G f \<zero>F")
apply (frule aHom_mem[of "F" "G" "f" "\<zero>F"], assumption+)
apply (frule aGroup.ag_mOp_closed[of "G" "f \<zero>F"], assumption+)
apply (frule aGroup.aassoc[of "G" "-aG (f \<zero>F)" "f \<zero>F" "f \<zero>F"], assumption+)
apply (simp add:aGroup.l_m aGroup.l_zero)
done
lemma ker_inc_zero:"[|aGroup F; aGroup G; f ∈ aHom F G|] ==> \<zero>F ∈ kerF,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 (-aF a) = -aG (f a)"
apply (frule aGroup.ag_l_inv1 [of "F" "a"], assumption+,
frule sym, thin_tac "-aF a ±F a = \<zero>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 "\<zero>F = -aF a ±F a")
apply (frule aHom_mem[of "F" "G" "f" "-aF a"], assumption+,
frule aHom_mem[of "F" "G" "f" "a"], assumption+,
simp only:aGroup.ag_pOp_commute[of "G" "f (-aF 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 univar_func_test) apply (rule ballI)
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; injecF,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 -aF y ∈ {a ∈ carrier F. f a = \<zero>G}",
simp)
apply (subst aGroup.ag_eq_diffzero[of "F"], assumption+)
apply (frule_tac x = x and y = "-aF y" in aGroup.ag_pOp_closed[of "F"],
assumption+)
apply simp apply blast
done
lemma surjec_surj_to:"surjecR,S f ==> surj_to f (carrier R) (carrier S)"
by (simp add:surjec_def)
lemma compos_bijec:"[|aGroup E; aGroup F; aGroup G; bijecE,F f; bijecF,G g|] ==>
bijecE,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 = \<zero>G}", simp)
apply (subgoal_tac "x ∈ {a ∈ carrier E. f a = \<zero>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; bijecF,G f|] ==>
ainvfF,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 b = "(Ifn F G f) (a ±G b)" in forball_spec1, assumption,
thin_tac "∀x∈carrier F. ∀y∈carrier F. f x = f y --> x = y")
apply (frule_tac b = "(Ifn F G f) a ±F (Ifn F G f) b" in forball_spec1,
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; bijecF,G f|] ==> bijecG,F (ainvfF,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 "ainvfF,G f" "carrier G" "carrier F"],
assumption+)
apply (rule ballI,
thin_tac "∀a∈carrier G. ∀b∈carrier G.
(ainvfF,G f) (a ±G b) = (ainvfF,G f) a ±F (ainvfF,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; bijecE,F f; x ∈ carrier E|] ==>
(ainvfE,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:"aIA ∈ aHom A A"
apply (simp add:aHom_def)
apply (rule conjI,
rule univar_func_test, rule ballI, simp add:ag_idmap_def)
apply (simp add:ag_idmap_def ag_pOp_closed)
done
lemma compos_aI_l:"[|aGroup A; aGroup B; f ∈ aHom A B|] ==> compos A aIB 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 aIA = 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 = aIA|] ==> surjecB,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 = aIA b",
thin_tac "compos A g f = aIA")
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 = aIA|] ==> injecA,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 = aIA x",
thin_tac "compos A g f = aIA")
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 (rule univar_func_test)
apply (rule ballI, simp add:Agii_def funcset_mem)
apply (simp add:Agii_def)
apply (rule ballI)+
apply (simp add:Ag_ind_def)
apply (simp add:funcset_mem)+
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)|] ==> bijecA, (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 = \<zero>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" "\<zero>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
constdefs
aimg ::"[('b, 'm1) aGroup_scheme, _, 'b => 'a]
=> 'a aGroup" ("(3aimg_,_ _)" [82,82,83]82)
"aimgF,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 +> kerF,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 "20-2 quotient abelian group"
constdefs (structure A)
ar_coset :: "['a, _ , 'a set] => 'a set" (** a_rcs **)
("(3_ \<uplus>_ _)" [66,66,67]66)
"ar_coset a A H == H •(b_ag A) a"
set_ar_cos:: "[_ , 'a set] => 'a set set"
"set_ar_cos A I == {X. ∃a∈carrier A. X = ar_coset a A I}"
aset_sum :: "[_ , 'a set, 'a set] => 'a set "
"aset_sum A H K == s_top (b_ag A) H K"
syntax
"@ASBOP1" :: "['a set, _ , 'a set] => 'a set" (infix "\<minusplus>\<index>" 60)
translations
"H \<minusplus>A K" == "aset_sum A H K"
lemma (in aGroup) ag_a_in_ar_cos:"[|A +> H; a ∈ carrier A|] ==> a ∈ a \<uplus>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 \<minusplus> 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 \<minusplus> 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 \<minusplus> 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 \<minusplus> 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 \<uplus>A H) (b \<uplus>A H) = (a ± b) \<uplus>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 \<uplus>A H) = (-a a) \<uplus>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 \<uplus>A H = (h ± a) \<uplus>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
constdefs
rind_hom :: "[('a, 'more) aGroup_scheme, ('b, 'more1) aGroup_scheme,
('a => 'b)] => ('a set => 'b )"
"rind_hom A B f == λX∈(set_ar_cos A (kerA,B f)). f (SOME x. x ∈ X)"
syntax
"@RIND_HOM"::"['a => 'b, ('a, 'm) aGroup_scheme, ('b, 'm1) aGroup_scheme]
=> ('a set => 'b )" ("(3_°_,_)" [82,82,83]82)
translations
"f°F,G " == "rind_hom F G f"
section "21 direct product and direct sum of abelian groups, in general case"
constdefs
Un_carrier::"['i set, 'i => ('a, 'more) aGroup_scheme] => 'a set"
"Un_carrier I A == \<Union>{X. ∃i∈I. X = carrier (A i)}"
carr_prodag::"['i set, 'i => ('a, 'more) aGroup_scheme] =>
('i => 'a ) set"
"carr_prodag I A == {f. f ∈ extensional I ∧ f ∈ I -> (Un_carrier I A) ∧
(∀i∈I. f i ∈ carrier (A i))}"
prod_pOp::"['i set, 'i => ('a, 'more) aGroup_scheme] =>
('i => 'a) => ('i => 'a) => ('i => 'a)"
"prod_pOp I A == λf∈carr_prodag I A. λg∈carr_prodag I A.
λx∈I. (f x) ±(A x) (g x)"
prod_mOp::"['i set, 'i => ('a, 'more) aGroup_scheme] =>
('i => 'a) => ('i => 'a)"
"prod_mOp I A == λf∈carr_prodag I A. λx∈I. (-a(A x) (f x))"
prod_zero::"['i set, 'i => ('a, 'more) aGroup_scheme] => ('i => 'a)"
"prod_zero I A == λx∈I. \<zero>(A x)"
prodag::"['i set, 'i => ('a, 'more) aGroup_scheme] => ('i => 'a) aGroup"
"prodag I A == (| carrier = carr_prodag I A,
pop = prod_pOp I A, mop = prod_mOp I A,
zero = prod_zero I A|)),"
PRoject::"['i set, 'i => ('a, 'more) aGroup_scheme, 'i]
=> ('i => 'a) => 'a" ("(3π_,_,_)" [82,82,83]82)
"PRoject I A x == λf ∈ carr_prodag I A. f x"
syntax
"@PRODag" :: "['i set, 'i => ('a, 'more) aGroup_scheme] =>
('i => 'a ) set" ("(aΠ_ _)" [72,73]72)
translations
"aΠI 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 bivar_func_test)
apply (rule ballI)+
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 univar_func_test)
apply (rule ballI)
apply (simp add:prod_pOp_def)
apply (subst Un_carrier_def) apply (simp add:CollectI)
apply (frule_tac b = x in forball_spec1, 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 b = x in forball_spec1, assumption,
thin_tac "∀i∈I. a i ∈ carrier (A i)",
frule_tac b = x in forball_spec1, 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 univar_func_test)
apply (rule ballI)
apply (simp add:prod_mOp_def carr_prodag_def)
apply (erule conjE)+
apply (rule conjI)
apply (rule univar_func_test)
apply (rule ballI) 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 b = j in forball_spec1, assumption,
thin_tac "∀k∈I. aGroup (A k)",
frule_tac b = j in forball_spec1, 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:funcset_mem)
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 univar_func_test)
apply (rule ballI) 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 b = i in forball_spec1, 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 = \<zero>(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 b = l in forball_spec1, 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 b = l in forball_spec1, 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 b = l in forball_spec1, 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 b = l in forball_spec1, 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 univar_func_test, rule ballI)
apply (subst Un_carrier_def, simp, blast)
apply (rule subsetI)
apply (simp add:carr_prodag_def, (erule conjE)+)
apply (rule univar_func_test, rule ballI)
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 (rule univar_func_test)
apply (rule ballI)
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
constdefs
A_to_prodag :: "[('a, 'm) aGroup_scheme, 'i set, 'i =>('a => 'b),
'i => ('b, 'm1) aGroup_scheme] => ('a => ('i =>'b))"
"A_to_prodag A I S B == λa∈carrier A. λk∈I. S k a"
(* I is an index set, A is an abelian group, S: I -> carrier A ->
carrier (prodag I B), s i ∈ carrier A -> B i *)
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 b = xa in forball_spec1, 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 b = i in forball_spec1, assumption,
thin_tac "∀k∈I. S k ∈ aHom A (B k)")
apply (simp add:aHom_def) apply (erule conjE)+
apply (simp add:funcset_mem)
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 (aΠI B)"
apply (simp add:aHom_def [of "A" "aΠI B"])
apply (rule conjI)
apply (rule univar_func_test)
apply (rule ballI) apply (simp add:prodag_def)
apply (simp add: 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 "aΠI 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 b = l in forball_spec1, assumption,
thin_tac "∀k∈I. S k ∈ aHom A (B k)")
apply (simp add:prodag_def prod_pOp_def)
apply (frule_tac b = l in forball_spec1, assumption,
thin_tac "∀k∈I. aGroup (B k)")
apply (simp add: aHom_add)
done
constdefs
finiteHom::"['i set, 'i => ('a, 'more) aGroup_scheme, 'i => 'a] => bool"
"finiteHom I A f == f ∈ carr_prodag I A ∧ (∃H. H ⊆ I ∧ finite H ∧ (
∀j ∈ (I - H). (f j) = \<zero>(A j)))"
constdefs
carr_dsumag::"['i set, 'i => ('a, 'more) aGroup_scheme] =>
('i => 'a ) set"
"carr_dsumag I A == {f. finiteHom I A f}"
dsumag::"['i set, 'i => ('a, 'more) aGroup_scheme] => ('i => 'a) aGroup"
"dsumag I A == (| carrier = carr_dsumag I A,
pop = prod_pOp I A, mop = prod_mOp I A,
zero = prod_zero I A|)),"
dProj::"['i set, 'i => ('a, 'more) aGroup_scheme, 'i]
=> ('i => 'a) => 'a"
"dProj I A x == λf∈carr_dsumag I A. f x"
syntax
"@DSUMag" :: "['i set, 'i => ('a, 'more) aGroup_scheme] =>
('i => 'a ) set" ("(a\<Oplus>_ _)" [72,73]72)
translations
"a\<Oplus>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 bivar_func_test)
apply (rule ballI)+
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 a b j = \<zero>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 b = j in forball_spec1, assumption,
thin_tac "∀k∈I. aGroup (A k)",
frule_tac b = j in forball_spec1, simp,
thin_tac "∀j∈I - H. a j = \<zero>A j",
frule_tac b = j in forball_spec1, simp,
thin_tac "∀j∈I - Ha. b j = \<zero>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 univar_func_test)
apply (rule ballI)
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. -aA j (x j) = \<zero>A j")
apply blast
apply (rule ballI)
apply (frule_tac b = j in forball_spec1, simp,
thin_tac "∀k∈I. aGroup (A k)",
frule_tac b = j in forball_spec1, simp,
thin_tac "∀j∈I - H. x j = \<zero>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:funcset_mem)
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 = \<zero>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 b = l in forball_spec1, 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 b = l in forball_spec1, assumption,
thin_tac "∀i∈I. a i ∈ carrier (A i)",
frule_tac b = l in forball_spec1, assumption,
thin_tac "∀i∈I. b i ∈ carrier (A i)",
frule_tac b = l in forball_spec1, 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 b = l in forball_spec1, 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 b = l in forball_spec1, 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 b = l in forball_spec1, 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 b = l in forball_spec1, 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 (aΠI A);
g ∈ carrier (aΠI 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 (aΠI A)",
thin_tac "∀j∈I. (πI,A,j) f = (πI,A,j) g",
simp add:prodag_def carr_prodag_def)
apply (thin_tac "∀j∈I. aGroup (A j)",
thin_tac "f ∈ carrier (aΠI A)",
thin_tac "∀j∈I. (πI,A,j) f = (πI,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 (aΠI A)"
apply (simp add:prodag_def carr_prodag_def)
apply (simp add:aHom_mem)
apply (rule univar_func_test, rule ballI, simp add:Un_carrier_def)
apply (frule_tac b = xa in forball_spec1, assumption,
thin_tac "∀j∈I. aGroup (A j)",
frule_tac b = xa in forball_spec1, 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 (aΠI A)"
apply (subst aHom_def, simp)
apply (simp add:aGroup.ag_pOp_closed)
apply (rule conjI)
apply (rule univar_func_test, rule ballI)
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 "aΠI 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 (aΠI 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 (πI,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" "aΠI 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" "aΠI A"], assumption+,
frule_tac f = f1 and a = x in aHom_mem[of "S" "aΠI 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 b = j in forball_spec1, assumption,
thin_tac "∀j∈I. compos S (πI,A,j) f = g j",
frule_tac b = j in forball_spec1, assumption,
thin_tac "∀j∈I. compos S (πI,A,j) f1 = g j",
simp add:compos_def compose_def)
apply (subgoal_tac "(λx∈carrier S. (πI,A,j) (f x)) x = g j x",
subgoal_tac "(λx∈carrier S. (πI,A,j) (f1 x)) x = g j x",
thin_tac "(λx∈carrier S. (πI,A,j) (f x)) = g j",
thin_tac "(λx∈carrier S. (πI,A,j) (f1 x)) = g j",
simp+)
done
lemma Ag_ind_triangle:"[|∀j∈I. aGroup (A j); j ∈ I; f ∈ carrier (aΠI A) -> B;
bij_to f (carrier (aΠI A)) (B::'d set); j ∈ I|] ==>
compos (aΠI A) (compos (Ag_ind (aΠI A) f)(PRoject I A j) (ainvf(aΠI A),
(Ag_ind (aΠI A) f) (Agii (aΠI A) f))) (Agii (aΠI A) f) =
PRoject I A j"
apply (frule prodag_aGroup[of "I" "A"])
apply (frule aGroup.Ag_ind_aGroup[of "aΠI A" "f" "B"], assumption+)
apply (simp add:compos_def)
apply (rule funcset_eq[of _ "carrier (aΠI 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 "aΠI A" "f" "B"], assumption+)
apply (frule_tac x = x in ainvf_l[of "aΠI A" "Ag_ind (aΠI A) f"
"Agii (aΠI A) f"], assumption+)
apply simp
done
(** Note f'
aΠI A -> Ag_ind (aΠI A) f
\ |
\ |
PRoject I A j \ | (PRoject I A j) o (f'¯1)
\ |
A j , where f' = Agii (aΠI A) f **)
constdefs
ProjInd :: "['i set, 'i => ('a, 'm) aGroup_scheme, ('i => 'a) => 'd, 'i] =>
('d => 'a)"
"ProjInd I A f j == compos (Ag_ind (aΠI A) f)(PRoject I A j) (ainvf(aΠI A), (Ag_ind (aΠI A) f) (Agii (aΠI A) f))"
(** Note f'
aΠI A -> Ag_ind (aΠI A) f
\ |
\ |
PRoject I A j \ | PRojInd I A f j
\ |
A j **)
lemma ProjInd_aHom:"[|∀j∈ I. aGroup (A j); j ∈ I; f ∈ carrier (aΠI A) -> B;
bij_to f (carrier (aΠI A)) (B::'d set); j ∈ I|] ==>
(ProjInd I A f j) ∈ aHom (Ag_ind (aΠI A) f) (A j)"
apply (frule prodag_aGroup[of "I" "A"])
apply (frule aGroup.Ag_ind_aGroup[of "aΠI A" "f" "B"], assumption+)
apply (frule_tac b = j in forball_spec1, assumption)
apply (frule aGroup.Ag_ind_aHom[of "aΠI A" "f" "B"], assumption+)
apply (simp add:ProjInd_def)
apply (frule Ag_ind_bijec[of "aΠI A" "f" "B"], assumption+)
apply (frule ainvf_aHom[of "aΠI A" "Ag_ind (aΠI A) f" "Agii (aΠI 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 (aΠI A) -> B;
bij_to f (carrier (aΠI A)) (B::'d set)|] ==>
∀j∈I. (ProjInd I A f j) ∈ aHom (Ag_ind (aΠI A) f) (A j)"
apply (rule ballI)
apply (simp add:ProjInd_aHom)
done
lemma ProjInd_mem_eq:"[|∀j∈I. aGroup (A j); f ∈ carrier (aΠI A) -> B;
bij_to f (carrier (aΠI A)) B; aGroup S; x ∈ carrier (Ag_ind (aΠI A) f);
y ∈ carrier (Ag_ind (aΠI 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 "aΠI A" "f" "B"], assumption+)
apply (frule aGroup.Ag_ind_aHom[of "aΠI A" "f" "B"], assumption+)
apply (frule Ag_ind_bijec[of "aΠI A" "f" "B"], assumption+)
apply (frule ainvf_aHom[of "aΠI A" "Ag_ind (aΠI A) f" "Agii (aΠI A) f"],
assumption+)
apply (frule aHom_mem[of "Ag_ind (aΠI A) f" "aΠI A" "ainvf(aΠI A),Ag_ind (aΠI A) f Agii (aΠI A) f" "x"], assumption+,
frule aHom_mem[of "Ag_ind (aΠI A) f" "aΠI A" "ainvf(aΠI A),Ag_ind (aΠI A) f Agii (aΠI A) f" "y"], assumption+)
apply (frule direct_prod_mem_eq[of "I" "A" "(ainvf(aΠI A),Ag_ind (aΠI A) f Agii (aΠI A) f) x" "(ainvf(aΠI A),Ag_ind (aΠI A) f Agii (aΠI A) f) y"], assumption+)
apply (thin_tac "ainvf(aΠI A),Ag_ind (aΠI A) f Agii (aΠI A) f
∈ aHom (Ag_ind (aΠI A) f) (aΠI A)")
apply (frule ainvf_bijec[of "aΠI A" "Ag_ind (aΠI A) f" "Agii (aΠI A) f"],
assumption+)
apply (thin_tac "bijec(aΠI A),Ag_ind (aΠI A) f Agii (aΠI A) f")
apply (unfold bijec_def, frule conjunct1, fold bijec_def)
apply (frule injec_inj_on[of "Ag_ind (aΠI A) f" "aΠI A" "ainvf(aΠI A),Ag_ind (aΠI A) f Agii (aΠI A) f"], assumption+)
apply (simp add:injective_iff[THEN sym, of "ainvf(aΠI A),Ag_ind (aΠI A) f Agii (aΠI A) f" "carrier (Ag_ind (aΠI A) f)" "x" "y"])
done
lemma ProjInd_mem_eq1:"[|∀j∈I. aGroup (A j); f ∈ carrier (aΠI A) -> B;
bij_to f (carrier (aΠI A)) B; aGroup S;
h ∈ aHom (Ag_ind (aΠI A) f) (Ag_ind (aΠI A) f);
∀j∈I. compos (Ag_ind (aΠI A) f) (ProjInd I A f j) h = ProjInd I A f j|] ==> h = ag_idmap (Ag_ind (aΠI A) f)"
apply (rule funcset_eq[of _ "carrier (Ag_ind (aΠI 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 "aΠI A" "f" "B"], assumption+)
apply (frule_tac a = x in aHom_mem[of "Ag_ind (aΠI A) f" "Ag_ind (aΠI 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 b = j in forball_spec1, assumption,
thin_tac "∀j∈I. compos (Ag_ind (aΠI 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 (aΠI A) f). ProjInd I A f j (h x)) x
= ProjInd I A f j x",
thin_tac "(λx∈carrier (Ag_ind (aΠI 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 (aΠI A) -> B;
bij_to f (carrier (aΠI A)) (B::'d set); j ∈ I|] ==>
compos (aΠI A) (ProjInd I A f j) (Agii (aΠI 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 (aΠI A) -> B;
bij_to f (carrier (aΠI A)) (B::'d set); aGroup S;
∀j∈I. ((g j) ∈ aHom S (A j))|] ==> ∃!h. h ∈ aHom S (Ag_ind (aΠI 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 "aΠI A" "f" "B"], assumption+)
apply (frule Ag_ind_bijec[of "aΠI 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" "aΠI A" "Ag_ind (aΠI A) f" _
"Agii (aΠI A) f"], assumption+)
apply (subgoal_tac "∀j∈I. compos S (ProjInd I A f j)
(compos S (Agii (aΠI A) f) fa) = g j")
apply blast
apply (rule ballI)
apply (frule_tac N = "A j" and f = fa and g = "Agii (aΠI A) f" and
h = "ProjInd I A f j" in aHom_compos_assoc[of "S" "aΠI A" "Ag_ind (aΠI 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 (aΠI A) f) fa) =
compos S (compos (aΠI A) (ProjInd I A f j) (Agii (aΠI 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 (aΠI A) f"],
assumption+,
frule_tac f = h1 and a = x in aHom_mem[of "S" "Ag_ind (aΠI 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 b = j in forball_spec1, assumption,
thin_tac "∀j∈I. compose (carrier S) (ProjInd I A f j) h = g j",
frule_tac b = j in forball_spec1, 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 (aΠI A) -> B;
bij_to ff (carrier (aΠI A)) B;
h1 ∈ aHom (Ag_ind (aΠI A) ff) S;
∀j∈I. compos (Ag_ind (aΠI A) ff) (g j) h1 = ProjInd I A ff j;
h2 ∈ aHom S (Ag_ind (aΠI A) ff);
∀j∈I. compos S (ProjInd I A ff j) h2 = g j|]
==> ∀j∈I. compos (Ag_ind (aΠI A) ff) (ProjInd I A ff j)
(compos (Ag_ind (aΠI 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 "aΠI A" _ "B"], assumption+)
apply (frule_tac N = "A j" and h = "ProjInd I A ff j" in aHom_compos_assoc[of "Ag_ind (aΠI A) ff" "S" "Ag_ind (aΠI 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 b = j in forball_spec1, assumption,
thin_tac "∀j∈I. aGroup (A j)",
frule_tac b = j in forball_spec1, 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 (aΠI A) -> B;
bij_to ff (carrier (aΠI A)) B;
h1 ∈ aHom (Ag_ind (aΠI A) ff) S;
∀j∈I. compos (Ag_ind (aΠI A) ff) (g j) h1 = ProjInd I A ff j;
h2 ∈ aHom S (Ag_ind (aΠI A) ff);
∀j∈I. compos S (ProjInd I A ff j) h2 = g j|]
==> (compos (Ag_ind (aΠI A) ff) h2 h1) = ag_idmap (Ag_ind (aΠI 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 "aΠI A" "ff" "B"], assumption+)
apply (frule aHom_compos[of "Ag_ind (aΠI A) ff" "S" "Ag_ind (aΠI A) ff" "h1"
"h2"], assumption+)
apply (rule ProjInd_mem_eq1[of "I" "A" "ff" "B" "S"
"compos (Ag_ind (aΠI 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 (aΠI A) -> (B::'d set) ∧
bij_to ff (carrier (aΠI 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 "aΠI A" _ "B"], erule conjE,
assumption, simp, erule conjE)
apply (frule aGroup.Ag_ind_aGroup[of "aΠI A" _ "B"], assumption+,
frule_tac a = S in forall_spec, assumption+)
apply (rotate_tac -1,
frule_tac a = g in forall_spec1,
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 (aΠI 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 a = "ProjInd I A ff" in forall_spec1,
thin_tac "∀g'. ∀j∈I. g' j ∈ aHom (Ag_ind (aΠI A) ff) (A j) -->
(∃!f. f ∈ aHom (Ag_ind (aΠI A) ff) S ∧
(∀j∈I. compos (Ag_ind (aΠI 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 (aΠI A) ff) S ∧
(∀j∈I. compos (Ag_ind (aΠI 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 (aΠI 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 (aΠI A) ff" in map_family_triangle3[of "I"
"A" "S" _ "g"],
assumption+,
frule_tac f = h2 and g = h1 and M = "Ag_ind (aΠI A) ff" in
aHom_compos[of "S" _ "S" ], assumption+)
apply (erule ex1E)
apply (rotate_tac -1,
frule_tac a = "compos S h1 h2" in forall_spec1,
frule map_family_triangle4[of "I" "A" "S" "g"], assumption+,
frule aGroup.aI_aHom[of "S"])
apply (frule_tac a = "aIS" in forall_spec1,
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 (aΠI A) ff) (g j) h1 = ProjInd I A ff j")
apply (rotate_tac -1, frule sym, thin_tac "aIS = f", simp,
frule_tac A = "Ag_ind (aΠI A) ff" and f = h1 and g = h2 in
compos_aI_inj[of _ "S"], assumption+,
frule_tac B = "Ag_ind (aΠI 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 "aΠI A" _ "B"], assumption+,
frule_tac F = "Ag_ind (aΠI A) ff" and f = "Agii (aΠI A) ff" and g = h1
in compos_bijec[of "aΠI A" _ "S"], assumption+)
apply (subst bijec_def, simp)
apply (thin_tac "bijec(aΠI A),Ag_ind (aΠI A) ff Agii (aΠI A) ff",
thin_tac "injecAg_ind (aΠI A) ff,S h1",
thin_tac "surjecAg_ind (aΠI A) ff,S h1")
apply (rule exI, simp)
done
(*** Note.
f
S' -> S
\ |
g' j\ | g j
\ |
A j
***)
chapter "4. Ring theory"
section "1. Definition of a ring and an ideal"
record 'a Ring = "'a aGroup" +
tp :: "['a, 'a ] => 'a" (infixl "·r\<index>" 70)
un :: "'a" ("1r\<index>")
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 = \<zero>"
and ex_zero: "\<zero> ∈ carrier R"
and l_zero:"a ∈ carrier R ==> \<zero> ± 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: "(1r) ∈ 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 ==> (1r) ·r a = a"
constdefs
zeroring :: "('a, 'more) Ring_scheme => bool"
"zeroring R == Ring R ∧ carrier R = {\<zero>R}"
consts
nscal :: "('a, 'more) Ring_scheme => 'a => nat => 'a"
npow :: "('a, 'more) Ring_scheme => 'a => nat => 'a"
nsum :: "('a, 'more) aGroup_scheme => (nat => 'a) => nat => 'a"
nprod :: "('a, 'more) Ring_scheme => (nat => 'a) => nat => 'a"
primrec
nscal_0: "nscal R x 0 = \<zero>R"
nscal_suc: "nscal R x (Suc n) = (nscal R x n) ±R x"
primrec
npow_0: "npow R x 0 = 1rR"
npow_suc: "npow R x (Suc n) = (npow R x n) ·rR x"
primrec
nprod_0: "nprod R f 0 = f 0"
nprod_suc:"nprod R f (Suc n) = (nprod R f n) ·rR (f (Suc n))"
primrec
nsum_0: "nsum R f 0 = f 0"
nsum_suc: "nsum R f (Suc n) = (nsum R f n) ±R (f (Suc n))"
syntax
"@NSCAL" :: "[nat, ('a, 'more) Ring_scheme, 'a] => 'a"
("(3 _ ×_ _)" [75,75,76]75)
"@NPOW" :: "['a, ('a, 'more) Ring_scheme, nat] => 'a"
("(3_^_ _)" [77,77,78]77)
"@SUM" :: "('a, 'more) aGroup_scheme => (nat => 'a) => nat => 'a"
("(3Σe _ _ _)" [85,85,86]85)
"@NPROD"::"[('a, 'm) Ring_scheme, nat, nat => 'a] => 'a"
("(3eΠ_,_ _)" [98,98,99]98)
translations
"n ×R x" == "nscal R x n"
"a^R n" == "npow R a n"
"Σe G f n" == "nsum G f n"
"eΠR,n f" == "nprod R f n"
constdefs (structure A)
fSum::"[_, (nat => 'a), nat, nat] => 'a"
"fSum A f n m == if n ≤ m then nsum A (cmp f (slide n))(m - n)
else \<zero>"
syntax
"@FSUM" :: "[('a, 'more) aGroup_scheme, (nat => 'a), nat, nat] => 'a"
("(4Σf _ _ _ _)" [85,85,85,86]85)
translations
"Σf G f n m" == "fSum G f n m"
lemma (in aGroup) nsum_zeroGTr:"(∀j ≤ n. f j = \<zero>) --> nsum A f n = \<zero>"
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 \<zero>])
done
lemma (in aGroup) nsum_zeroA:"∀j ≤ n. f j = \<zero> ==> nsum A f n = \<zero>"
apply (simp add:nsum_zeroGTr)
done
constdefs (structure R)
sr::"[_ , 'a set] => bool"
"sr R S == S ⊆ carrier R ∧ 1r ∈ S ∧ (∀x∈S. ∀y ∈ S. x ± (-a y) ∈ S ∧
x ·r y ∈ S)"
Sr ::"[_ , 'a set] => _"
"Sr R S == R (|carrier := S, pop := λx∈S. λy∈S. x ±R y, mop := λx∈S. (-aR x),
zero := \<zero>R, tp := λx∈S. λy∈S. x ·rR y, un := 1rR |)),"
(** sr is a subring without ring structure, Sr is a subring with Ring structure
**)
lemma (in Ring) Ring:"Ring R"
by (unfold_locales)
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:"\<zero> ∈ carrier R"
by (simp add: ex_zero)
lemma (in Ring) ring_one:"1r ∈ 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 "op ·r" "carrier R" "carrier R -> carrier R"
"x"], assumption+,
thin_tac "op ·r ∈ carrier R -> carrier R -> carrier R")
apply (rule funcset_mem[of "op ·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 ==> 1r ·r x = x"
by (simp add:rg_l_unit)
lemma (in Ring) ring_r_one:"x ∈ carrier R ==> x ·r 1r = 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 ==> \<zero> ·r x = \<zero>"
apply (cut_tac ring_is_ag)
apply (cut_tac ring_zero)
apply (frule ring_distrib2 [of "x" "\<zero>" "\<zero>"], assumption+)
apply (simp add:aGroup.ag_l_zero [of "R" "\<zero>"])
apply (frule ring_tOp_closed [of "\<zero>" "x"], assumption+)
apply (frule sym, thin_tac "\<zero> ·r x = \<zero> ·r x ± \<zero> ·r x")
apply (frule aGroup.ag_eq_sol2 [of "R" "\<zero> ·r x" "\<zero> ·r x" "\<zero> ·r x"],
assumption+)
apply (thin_tac "\<zero> ·r x ± \<zero> ·r x = \<zero> ·r x")
apply (simp add:aGroup.ag_r_inv1)
done
lemma (in Ring) ring_times_x_0:"x ∈ carrier R ==> x ·r \<zero> = \<zero>"
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 = \<zero> ∨ y = \<zero> |] ==> x ·r y = \<zero>";
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" "\<zero>"],
assumption+)
apply (rule ring_zero, assumption+)
apply (thin_tac "a ·r b ± (-a a) ·r b = \<zero>")
apply (frule sym) apply (thin_tac "(-a a) ·r b = -a (a ·r b) ± \<zero>")
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)" "\<zero>"],
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 1r) ·r a"
apply (cut_tac ring_one)
apply (frule ring_inv1_1[of "1r" "a"], assumption+)
apply (simp add:ring_l_one)
done
lemma (in Ring) ring_times_minusr:"a ∈ carrier R ==> -a a = a ·r (-a 1r)"
apply (cut_tac ring_one)
apply (frule ring_inv1_2[of "a" "1r"], 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:"1r = \<zero> ==> 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) ==> 1r ≠ \<zero>"
apply (rule contrapos_pp, simp+,
cut_tac Zero_ring, simp+)
done
lemma (in Ring) Sr_one:"sr R S ==> 1r ∈ S"
apply (simp add:sr_def)
done
lemma (in Ring) Sr_zero:"sr R S ==> \<zero> ∈ S"
apply (cut_tac ring_is_ag, frule Sr_one[of "S"])
apply (simp add:sr_def) apply (erule conjE)+
apply (frule_tac b = "1r" in forball_spec1, assumption,
thin_tac "∀x∈S. ∀y∈S. x ± -a y ∈ S ∧ x ·r y ∈ S",
frule_tac b = "1r" in forball_spec1, assumption,
thin_tac "∀y∈S. 1r ± -a y ∈ S ∧ 1r ·r y ∈ S",
erule conjE)
apply (cut_tac ring_one,
simp add:aGroup.ag_r_inv1[of "R" "1r"])
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 b = "\<zero>" in forball_spec1, assumption,
thin_tac "∀x∈S. ∀y∈S. x ± -a y ∈ S ∧ x ·r y ∈ S",
frule_tac b = x in forball_spec1, assumption,
thin_tac "∀y∈S. \<zero> ± -a y ∈ S ∧ \<zero> ·r y ∈ S", erule conjE)
apply (frule subsetD[of "S" "carrier R" "\<zero>"], 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 b = x in forball_spec1, assumption,
thin_tac "∀x∈S. ∀y∈S. x ± -a y ∈ S ∧ x ·r y ∈ S",
frule_tac b = "-a y" in forball_spec1, 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)
apply (rule bivar_func_test, (rule ballI)+)
apply (frule_tac x = a and y = b in Sr_pOp_closed, assumption+,
simp)
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 (rule univar_func_test, rule ballI, 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)
apply (rule bivar_func_test, (rule ballI)+)
apply (simp add: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 "2. Calculation of elements"
(** The author of this part is L. Chen, revised by H. Murao and Y.
Santo **)
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 \<zero> n = \<zero>"
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 = \<zero> ==> nscal R x n = \<zero>";
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 = \<zero> |]
==>(m ≤ n) --> (npow R x n = \<zero> )";
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 = \<zero>")
apply (subst npMulDistr, assumption)
apply simp
done
lemma (in Ring) npGTPowZero:
"!! n. [| x ∈ carrier R; npow R x m = \<zero>; m ≤ n |]
==> npow R x n = \<zero>"
apply (cut_tac x = x and m = m and n = n in npGTPowZero_sub, assumption+)
apply simp
done
lemma (in Ring) npOne: " npow R (1r) n = 1r"
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 \<zero> n = \<zero>"
apply (induct_tac "n")
apply simp
apply simp
apply (cut_tac ring_zero,
frule_tac n = n in npClose[of "\<zero>"])
apply (simp add:ring_times_x_0)
done
lemma (in Ring) npZero: "0 < n ==> npow R \<zero> n = \<zero>"
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" (* Y. Santo*)
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 "Σf A g n (Suc (n + na)) =
Σf 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 b = "Suc (n + na)" in forball_spec1,
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 = \<zero>) -->
fSum A f n (n + m) = \<zero>"
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 b = "n + Suc na" in forball_spec1, 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 = \<zero> ==>
fSum A f n (n + m) = \<zero>"
by (simp add:fSum_zeroTr)
lemma (in aGroup) fSum_zero1:"[|n < m; ∀j ∈ nset (Suc n) m. f j = \<zero>|] ==>
fSum A f (Suc n) m = \<zero>"
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:funcset_mem,
cut_tac n_in_Nsetn[of "Suc 0"],
simp add:funcset_mem)
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 "Σe A (λi. f i ± g i) n = Σe A f n ± Σe 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 "Σe A (λi. if i ≤ n then f i else g (sliden (Suc n) i))
(n + na) ± g na = Σe A f n ± Σe 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) nsumEqFunc_sub:
"[| !! i. f i ∈ carrier R; !! i. g i ∈ carrier R |]
==> ( ∀ i. i ≤ n --> f i = g i) --> (nsum0 R f n = nsum0 R g n)";
apply (induct_tac "n")
apply simp+
done
lemma (in Ring) nsumEqFunc:
"[| !! i. f i ∈ carrier R; !! i. g i ∈ carrier R;
!! i. i ≤ n --> f i = g i |] ==> nsum0 R f n = nsum0 R g n"
apply (cut_tac nsumEqFunc_sub [of "f" "g" "n"])
apply simp+
done nsumEqFunc --> nsum_eq **)
(********)(********)
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) ±
(\<zero> ± ((x ·r (npow R x n)) ·r (1r)))"
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")
(*1*)
apply simp
apply (cut_tac ring_one, simp add:ring_r_one, simp add:aGroup.ag_l_zero)
(*1:done*)
apply (subst aGroup.nsumElmTail, assumption+)
apply (rule allI)
apply (simp add:nsClose ring_tOp_closed npClose)
(**
thm binomial_Suc_Suc
**)
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)
(* final part *)
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 = \<zero>) --> (nsum A f n = \<zero>)";
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 = \<zero>; npow R y n = \<zero> |]
==> npow R (x ± y) (m + n) = \<zero>"
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)
(* n=0 *)
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")
(* n=0 *)
apply simp
apply (rule ring_r_one [THEN sym]) apply (simp add:ring_one)
(* n>0 *)
apply (simp only:npow_suc)
apply (rule ring_tOp_rel[THEN sym])
apply (rule npClose, assumption+)+
done
section "3. ring homomorphisms"
constdefs
rHom :: "[('a, 'm) Ring_scheme, ('b, 'm1) Ring_scheme]
=> ('a => 'b) set"
"rHom A R == {f. f ∈ aHom A R ∧
(∀x∈carrier A. ∀y∈carrier A. f ( x ·rA y) = (f x) ·rR (f y))
∧ f (1rA) = (1rR)}"
constdefs
rInvim :: "[('a, 'm) Ring_scheme, ('b, 'm1) Ring_scheme, 'a => 'b, 'b set]
=> 'a set"
"rInvim A R f K == {a. a ∈ carrier A ∧ f a ∈ K}"
constdefs
rimg::"[('a, 'm) Ring_scheme, ('b, 'm1) Ring_scheme, 'a => 'b] =>
'b Ring"
"rimg A R f == (|carrier= f `(carrier A), pop = pop R, mop = mop R,
zero = zero R, tp = tp R, un = un R |)),"
constdefs
ridmap::"('a, 'm) Ring_scheme => ('a => 'a)"
"ridmap R == λx∈carrier R. x"
constdefs
r_isom::"[('a, 'm) Ring_scheme, ('b, 'm1) Ring_scheme] => bool"
(infixr "≅r" 100)
"r_isom R R' == ∃f∈rHom R R'. bijecR,R' f"
constdefs
Subring::"[('a, 'm) Ring_scheme, ('a, 'm1) Ring_scheme] => bool"
"Subring R S == Ring S ∧ (carrier S ⊆ carrier R) ∧ (ridmap S) ∈ rHom S R"
lemma ridmap_surjec:"Ring A ==> surjecA,A (ridmap A)"
apply (simp add:surjec_def)
apply (rule conjI,
simp add:aHom_def,
rule conjI, rule univar_func_test, rule ballI, simp add:ridmap_def)
apply (rule conjI,
simp add:ridmap_def)
apply ((rule ballI)+,
simp add:ridmap_def,
frule Ring.ring_is_ag[of "A"], simp add:aGroup.ag_pOp_closed)
apply (simp add:surj_to_def ridmap_def)
done
lemma rHom_aHom:"f ∈ rHom A R ==> f ∈ aHom A R"
apply (simp add:rHom_def)
done
lemma rimg_carrier:"f ∈ rHom A R ==> carrier (rimg A R f) = f ` (carrier A)"
apply (simp add:rimg_def)
done
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 ·rA y) = f x ·rR f y) ∧ f 1rA = 1rR")
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 (-aA x) = -aR (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 (\<zero>A) = \<zero>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 ·rA y) = (f x) ·rR (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 (1rA) = (1rR)"
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 bivar_func_test)
apply (rule ballI)+
apply (simp add:image_def)
apply (erule bexE)+
apply simp
apply (subst aHom_add [THEN sym, of "A" "R" "f"], assumption+)
apply (frule_tac x = x and y = xa in aGroup.ag_pOp_closed, assumption+,
blast)
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 univar_func_test, rule ballI)
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 bivar_func_test, (rule ballI)+)
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 (frule_tac x = x and y = xa in Ring.ring_tOp_closed, assumption+,
blast)
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
constdefs (structure R)
ideal::"[_ , 'a set] => bool"
"ideal R I == (R +> I) ∧ (∀r∈carrier R. ∀x∈I. (r ·r x ∈ I))"
translations
"f°F,G " == "rind_hom F G f"
(* tOp -> pOp *)
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 ==> \<zero> ∈ 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 --> \<zero> ∈ 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" "1r"], assumption+)
apply (frule_tac b = "-a 1r " in forball_spec1, assumption+,
thin_tac "∀r∈carrier R. ∀x∈I. r ·r x ∈ I",
rotate_tac -1,
frule_tac b = y in forball_spec1, assumption,
thin_tac "∀x∈I. (-a 1r) ·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 {\<zero>}"
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; 1r ∈ 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" "1r"], assumption+,
simp add:ring_r_one)
done
lemma (in Ring) ideal_inc_one1:"ideal R I ==>
(1r ∈ 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
constdefs (structure R)
Unit :: "_ => 'a => bool"
"Unit R a == a ∈ carrier R ∧ (∃b∈carrier R. a ·r b = 1r)"
lemma (in Ring) ideal_inc_unit:"[|ideal R I; a ∈ I; Unit R a|] ==> 1r ∈ 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; 1r ∉ 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
constdefs (structure R)
ideal_prod::"[_, 'a set, 'a set] => 'a set" (infix "♦r\<index>" 90 )
"ideal_prod R I J == \<Inter> {L. ideal R L ∧
{x.(∃i∈I. ∃j∈J. x = i ·r j)} ⊆ L}"
lemma (in Ring) set_sum_mem:"[|a ∈ I; b ∈ J; I ⊆ carrier R; J ⊆ carrier R|] ==>
a ± b ∈ I \<minusplus> 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 \<minusplus> 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 "\<zero>" "I1" "\<zero>" "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 \<minusplus> 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 ± \<zero> = x", blast)
done
lemma (in Ring) sum_ideals_la2:"[|ideal R I1; ideal R I2 |] ==> I2 ⊆ (I1 \<minusplus> 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 "\<zero> ± x = x", blast)
done
lemma (in Ring) sum_ideals_cont:"[|ideal R I; A ⊆ I; B ⊆ I |] ==> A \<minusplus> 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 \<minusplus> 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
constdefs (structure R)
Rxa :: "[_, 'a ] => 'a set" (infixl "♦p" 200)
"Rxa R a == {x. ∃r∈carrier R. x = (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 1r = 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 \<zero> = {\<zero>}"
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 "\<zero>"], blast)
done
lemma (in Ring) Rxa_nonzero:"[|a ∈ carrier R; a ≠ \<zero>|] ==> Rxa R a ≠ {\<zero>}"
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 \<minusplus> 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 \<minusplus> 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
consts
sum_pr_ideals::"[('a, 'm) Ring_scheme, nat => 'a, nat] => 'a set"
primrec
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))) \<minusplus>R (sum_pr_ideals R f n)"
lemma (in Ring) restrictfun_Nset:"f ∈ {i. i ≤ (Suc n)} -> carrier R
==> f ∈ {i. i ≤ n} -> carrier R"
apply (rule univar_func_test, rule ballI)
apply (rule_tac funcset_mem, assumption+)
apply (cut_tac Nsetn_sub_mem1[of n], simp)
done
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)
(** case n **)
apply (rule allI, rule impI)
apply (frule_tac a = f in forall_spec1,
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) \<minusplus> 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)) \<minusplus> 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 a = "Suc n" in forall_spec1,
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 (kerA,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"
constdefs
Zr::"int Ring"
"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 (rule bivar_func_test)
apply (rule ballI)+
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 (rule bivar_func_test)
apply ((rule ballI)+, 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:zadd_zmult_distrib2)
apply (rule allI, rule impI)
apply (simp add:Zr_def Zset_def)
done
lemma Zr_zero:"\<zero>Zr = 0"
by (simp add:Zr_def)
lemma Zr_one:"1rZr = 1"
by (simp add:Zr_def)
lemma Zr_minus:"-aZr 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 ·rZr m = n * m"
by (simp add:Zr_def Zset_def)
constdefs
lev :: "int set => int"
"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" "1rZr"], assumption)
apply (frule_tac x = a in Ring.ideal_ring_multiple[of "Zr" "I" _ "-aZr 1rZr"],
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 a = x and b = "lev I" in zmod_zdiv_equality)
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 zmult_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 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" (** principal ideal ring *)
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 "4. quotient rings"
lemma (in Ring) mem_set_ar_cos:"[|ideal R I; a ∈ carrier R|] ==>
a \<uplus>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 \<uplus>R I = b \<uplus>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 \<uplus>R I = b \<uplus>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 \<uplus>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 \<uplus>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 \<uplus>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 \<uplus>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 \<uplus>R I"
apply (frule ar_coset_same1 [of "I" "a" "x"], assumption+)
apply (subgoal_tac "x ∈ x \<uplus>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 \<uplus>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 \<uplus>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 \<uplus>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 \<uplus>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 \<uplus>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 \<uplus>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 \<uplus>R I) (b \<uplus>R I)) = (a ± b) \<uplus>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 \<uplus>R I = a2 \<uplus>R I; b1 \<uplus>R I = b2 \<uplus>R I|] ==>
(a1 ·r b1) \<uplus>R I = (a2 ·r b2) \<uplus>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
constdefs (structure R)
rcostOp :: "[_, 'a set] => (['a set, 'a set] => 'a set)"
"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 y) ± h = z}"
lemma (in Ring) rcostOp:"[|ideal R I; a ∈ carrier R; b ∈ carrier R|] ==>
rcostOp R I (a \<uplus>R I) (b \<uplus>R I) = (a ·r b) \<uplus>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
constdefs (structure R)
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 |)),"
"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 = 1r \<uplus>R I|)),"
syntax
"@QRING" :: "([('a, 'more) Ring_scheme, 'a set] => ('a set) Ring)"
(infixl "'/'r" 200)
translations
"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 bivar_func_test, (rule ballI)+)
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 (frule_tac a = aa and b = ab in Group.mult_closed[of "b_ag R"],
assumption+, blast)
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 (rule univar_func_test, rule ballI)
apply (simp add:qring_def)
apply (simp add: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 bivar_func_test, (rule ballI)+)
apply (simp add:qring_def aGroup.aqgrp_carrier)
apply (simp add:set_ar_cos_def, (erule bexE)+, simp add:rcostOp,
frule_tac x = aa and y = ab in ring_tOp_closed, assumption+,
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 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 \<uplus>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 \<uplus>R I",
blast)
apply (rule subsetI, simp, erule bexE, frule sym, thin_tac "a \<uplus>R I = x",
blast)
done
lemma (in Ring) qring_mem:"[|ideal R I; a ∈ carrier R|] ==>
a \<uplus>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 \<uplus>R I) (b \<uplus>R I) = (a ± b) \<uplus>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 \<uplus>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 \<uplus>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 \<uplus>R I|] ==>
x \<uplus>R I = a \<uplus>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 \<uplus>R I) (b \<uplus>R I) = (a ·r b) \<uplus>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 \<uplus>A (kerA,R f))"
apply (frule ker_ideal[of "A" "R" "f"], assumption+)
apply (frule Ring.mem_set_ar_cos[of "A" "kerA,R f" "a"], assumption+)
apply (simp add:rind_hom_def)
apply (rule someI2_ex)
apply (frule Ring.a_in_ar_coset [of "A" "kerA,R f" "a"], assumption+, blast)
apply (frule_tac x = x in Ring.ar_coset_mem2[of "A" "kerA,R f" "a"],
assumption+, erule bexE, simp,
frule_tac h = h in Ring.ideal_subset[of "A" "kerA,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) (kerA,R f) = set_ar_cos A (kerA,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 (kerA,R f)) R"
apply (simp add:rHom_def [of "qring A (kerA,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 univar_func_test)
apply (rule ballI)
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" "kerA,R f"],
assumption+)
apply (rule someI2_ex, blast)
apply (frule_tac I = "kerA,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 (kerA,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 (kerA,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 \<uplus>A kerA,R f")
apply (rule_tac a = a in Ring.Qring_fix1[of "A" _ "kerA,R f"], assumption+)
apply (simp add:ker_def)
apply (rule subsetI, simp)
apply (simp add:Ring.I_in_set_ar_cos[of "A" "kerA,R f"])
apply (frule Ring.ideal_zero[of "A" "kerA,R f"], assumption+,
frule Ring.ring_zero[of "A"])
apply (frule Ring.ar_coset_same4[of "A" "kerA,R f" "\<zero>A"], assumption+)
apply (frule rind_hom_well_def[THEN sym, of "A" "R" "f" "\<zero>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 (rule univar_func_test, rule ballI, 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 |] ==>
kerA,R f = kerA,(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+) (** extensional **)
apply (rule funcset_eq[of "f°A,rimg A R f " "carrier (A /r (kerA,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 (kerA,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" "kerA,R f"], assumption+)
apply blast
done
lemma surjec_ind_bijec:"[|Ring A; Ring R; f ∈ rHom A R; surjecA,R f|] ==>
bijec(qring A (kerA,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" "kerA,R f"], assumption+)
apply blast
done
lemma ridmap_ind_bijec:"Ring A ==>
bijec(qring A (kerA,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 ==> kerA,A (ridmap A) = {\<zero>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 {\<zero>A}),A ((ridmap A)°A,A)"
apply (frule ridmap_ind_bijec)
apply (simp add: ker_of_idmap)
done (** A /r {0A} ≅ A **)
constdefs
pj :: "[('a, 'm) Ring_scheme, 'a set] => ('a => 'a set)"
"pj R I == λx. Pj (b_ag R) I x"
(* pj is projection homomorphism *)
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 univar_func_test) apply (rule ballI)
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 \<uplus>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 = \<zero>(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 (rule subsetI) apply (simp add:CollectI)
apply (subgoal_tac "\<zero>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 = "-aR 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 = "-aR 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 ·rR xa" in Ring.ideal_subset [of "R" "J"],
assumption+)
apply (frule_tac x1 = "a ·rR 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 \<uplus>R I) n = (npow R a n) \<uplus>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 "5. Primary ideals, Prime ideals"
constdefs
maximal_set::"['a set set, 'a set] => bool"
"maximal_set S mx == mx ∈ S ∧ (∀s∈S. mx ⊆ s --> mx = s)"
constdefs (structure R)
nilpotent::"[_, 'a] => bool"
"nilpotent R a == ∃(n::nat). a^R n = \<zero>"
zero_divisor::"[_, 'a] => bool"
"zero_divisor R a == ∃x∈ carrier R. x ≠ \<zero> ∧ x ·r a = \<zero>"
primary_ideal::"[_, 'a set] => bool"
"primary_ideal R q == ideal R q ∧ (1r) ∉ q ∧
(∀x∈ carrier R. ∀y∈ carrier R.
x ·r y ∈ q --> (∃n. (npow R x n) ∈ q ∨ y ∈ q))"
prime_ideal::"[_, 'a set] => bool"
"prime_ideal R p == ideal R p ∧ (1r) ∉ p ∧ (∀x∈ carrier R. ∀y∈ carrier R.
(x ·r y ∈ p --> x ∈ p ∨ y ∈ p))"
maximal_ideal::"[_, 'a set] => bool"
"maximal_ideal R mx == ideal R mx ∧ 1r ∉ 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 ==> 1r ∉ 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 ==> 1r ∉ 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 ==> 1r ∉ 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 b = x in forball_spec1, assumption,
thin_tac "∀x∈carrier R. ∀y∈carrier R. x ·r y ∈ P --> x ∈ P ∨ y ∈ P",
frule_tac b = y in forball_spec1, 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 b = x in forball_spec1, assumption,
thin_tac "∀x∈carrier R. ∀y∈carrier R. x ·r y ∈ p --> x ∈ p ∨ y ∈ p",
frule_tac b = y in forball_spec1, 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 \<minusplus> (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 \<minusplus> (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 \<minusplus> (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_is_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 = "1r" and B = "{xa. ∃i∈mx. ∃j∈(Rxa R x). xa = i ± j}" in
eq_set_inc[of _ "carrier R"], assumption,
frule_tac a = "1r" 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 "1r = i ± j")
apply (drule CollectD, (erule bexE)+, rotate_tac -1,
frule sym, thin_tac "1r = 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 "1r ·r ia = i ·r ia ± j ·r ia",
thin_tac "1r ·r ja = i ·r ja ± j ·r ja",
simp add:ring_l_one[of "1r"])
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 "1r = 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 "1r = 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) chain_un:"[|c ∈ chain {I. ideal R I ∧ I ⊂ carrier R}; c ≠ {}|]
==> ideal R (\<Union>c)"
apply (rule ideal_condition1)
apply (rule Union_least[of "c" "carrier R"])
apply (simp add:chain_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:chain_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: chain_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 b = X in forball_spec1, assumption,
thin_tac "∀x∈c. ∀y∈c. x ⊆ y ∨ y ⊆ x",
frule_tac b = Xa in forball_spec1, 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:chain_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 = {\<zero>}",
frule sym, thin_tac "1r = \<zero>", simp, thin_tac "\<zero> = 1r")
apply (simp add:maximal_ideal_def)
done
lemma (in Ring) id_maximal_Exist:"¬(zeroring R) ==> ∃I. maximal_ideal R I"
apply (cut_tac S="{ 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 ≠ {\<zero>}")
apply (cut_tac ring_zero,
frule singleton_sub[of "\<zero>" "carrier R"],
thin_tac "\<zero> ∈ carrier R")
apply (subst psubset_eq)
apply blast
apply (subgoal_tac "\<Union>c ∈ {I. ideal R I ∧ I ⊂ carrier R}")
apply (subgoal_tac "∀x∈c. x ⊆ (\<Union>c)", blast)
apply (rule ballI, rule Union_upper, assumption)
apply (simp add:chain_un)
apply (cut_tac A = c in Union_least[of _ "carrier R"])
apply (simp add:chain_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 "\<Union>c = carrier R")
apply (frule_tac B = "\<Union>c" in eq_set_inc[of "1r" "carrier R"], assumption,
thin_tac "carrier R = \<Union>c")
apply (simp, erule bexE)
apply (simp add:chain_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 y", blast)
apply (simp add:maximal_ideal_def)
apply (rule conjI, rule contrapos_pp, simp+,
frule_tac I = y in ideal_inc_one, assumption+, simp)
apply (rule equalityI)
apply (rule subsetI, simp)
apply (erule conjE)
apply (frule_tac a = x in forall_spec1,
thin_tac "∀x. ideal R x ∧ x ⊂ carrier R --> y ⊆ x --> y = x", 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
constdefs (structure R)
ideal_Int::"[_, 'a set set] => 'a set"
"ideal_Int R S == \<Inter> S"
lemma (in Ring) ideal_Int_ideal:"[|S ⊆ {I. ideal R I}; S≠{}|] ==>
ideal R (\<Inter> 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 = "\<Inter>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 b = X in forball_spec1, assumption,
thin_tac "∀X∈S. x ∈ X",
frule_tac b = X in forball_spec1, 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 b = X in forball_spec1, 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) = \<Inter> 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) *}
consts
ideal_n_prod::"[('a, 'm) Ring_scheme, nat, nat => 'a set] => 'a set"
primrec
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) ♦rR (J (Suc n))"
syntax
"@IDNPROD"::"[('a, 'm) Ring_scheme, nat, nat => 'a set] => 'a set"
("(3iΠ_,_ _)" [98,98,99]98)
translations
"iΠR,n J" == "ideal_n_prod R n J"
consts
ideal_pow :: "['a set, ('a, 'more) Ring_scheme, nat] => 'a set"
("(3_/ ♦_ _)" [120,120,121]120)
primrec
ip0: "I ♦R 0 = carrier R"
ipSuc: "I ♦R (Suc n) = I ♦rR (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 a = "carrier R" in forall_spec1,
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 "\<zero> ∈ 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 a = xa in forall_spec1,
thin_tac "∀xa. ideal R xa ∧ {x. ∃i∈I. ∃j∈J. x = i ·r j} ⊆ xa
--> x ∈ xa",
frule_tac a = xa in forall_spec1,
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 a = xa in forall_spec1,
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{* iΠR,n J is the product of ideals *}
lemma (in Ring) ele_n_prodTr0:"[|∀k ≤ (Suc n). ideal R (J k);
a ∈ iΠR,(Suc n) J |] ==> a ∈ (iΠR,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 "iΠR,n J" "J (Suc n)"], assumption, simp+)
apply (rule ideal_prod_el2[of "iΠR,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)
(** n = 0 **)
apply simp
(** n **)
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) ♦rR 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 "1r ·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 "1r ·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 b = "x ·r xa" in forball_spec1, 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 b = x in forball_spec1, 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 b = xa in forball_spec1, 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 = "iΠR,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
constdefs (structure R)
ppa::"[_, nat => 'a set, 'a set, nat] => (nat => 'a)"
"ppa R P A i l == SOME x. x ∈ A ∧ x ∈ (P (skip i l)) ∧ x ∉ P i"
(** Note (ppa R P A) is used to prove prime_ideal_cont1,
some element x of A such that x ∈ P j for (i ≠ j) and 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 b = x in forball_spec1, assumption,
thin_tac "∀x∈carrier R. ∀y∈carrier R. x ·r y ∈ P --> x ∈ P ∨ y ∈ P",
frule_tac b = xa in forball_spec1, 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 a = i in forall_spec1,
thin_tac "∀l ≤ (Suc n). ¬ A ⊆ P l", simp)
apply (cut_tac l = l in skip_mem[of _ "n" "i"], simp,
frule_tac a = "skip i l" in forall_spec1,
thin_tac "∀k ≤ (Suc n). ∀l ≤ (Suc n). k = l ∨ ¬ P k ⊆ P l",
simp)
apply (rotate_tac -1,
frule_tac a = i in forall_spec1,
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 a = "skip i l" in forall_spec1,
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 a = l in forall_spec1,
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)
(** n = 0 **)
apply (rule impI, rule allI, rule impI)
apply simp
(** n **)
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
(** n **)
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)
(** n = 0 **)
apply simp
(** n **)
apply (rule impI)
apply (erule conjE)+
apply (erule exE)
apply (case_tac "j = Suc n", simp) apply (
thin_tac "(∃j≤n. f j ∉ A) --> Σe 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 = "Σe 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 b = "Suc n" in forball_spec1, simp)
apply (rule contrapos_pp, simp+)
apply (frule_tac a = "Σe 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)
--> Σe R f n ∉ A")
apply (subgoal_tac "∀l∈{i. i ≤ n} - {j}. f l ∈ A", blast,
thin_tac "Σe R f n ± f (Suc n) ∈ A",
thin_tac "Σe R f n ∈ A")
apply (rule ballI)
apply (frule_tac b = l in forball_spec1, 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_anti_sym, 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)
(** n = 0 **)
apply simp
(** n **)
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)
(** n = 0 **)
apply simp (* n = 0 done *)
(** n **)
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)
constdefs (structure R)
nilrad::"_ => 'a set"
"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 \<zero>")
apply blast
apply (simp add:nilpotent_def)
apply (frule np_1[of "\<zero>"], 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
constdefs (structure R)
rad_ideal :: "[_, 'a set ] => 'a set"
"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)"
(* thm invim_of_ideal *)
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, assumption+,
simp add:ideal_subset) (* thm npQring *)
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 b = "x^R n" in forball_spec1, assumption,
thin_tac "∀x∈carrier R. ∀y∈carrier R. x ·r y ∈ q -->
(∃n. x^R n ∈ q) ∨ y ∈ q",
frule_tac b = "y^R n" in forball_spec1, 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
constdefs (structure R)
mul_closed_set::"[_, 'a set ] => bool"
"mul_closed_set R S == S ⊆ carrier R ∧ (∀s∈S. ∀t∈S. s ·r t ∈ S)"
locale Idomain = Ring +
assumes idom:
"[|a ∈ carrier R; b ∈ carrier R; a ·r b = \<zero>|] ==> a = \<zero> ∨ b = \<zero>"
(* integral domain *)
locale Corps =
fixes K (structure)
assumes f_is_ring: "Ring K"
and f_inv: "∀x∈carrier K - {\<zero>}. ∃x' ∈ carrier K. x' ·r x = 1r"
(** integral domain **)
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 - {\<zero>}; x' ∈ carrier K;
x'' ∈ carrier K; x' ·r x = 1r; x'' ·r x = 1r |] ==> 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
constdefs (structure K)
invf:: "[_, 'a] => 'a"
"invf K x == THE y. y ∈ carrier K ∧ y ·r x = 1r"
lemma (in Corps) invf_inv:"x ∈ carrier K - {\<zero>} ==>
(invf K x) ∈ carrier K ∧ (invf K x) ·r x = 1r "
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
constdefs (structure K)
npowf :: "_ => 'a => int => 'a"
"npowf K x n ==
if 0 ≤ n then npow K x (nat n) else npow K (invf K x) (nat (- n))"
syntax
"@NPOWF" :: "['a, _, int] => 'a"
("(3___)" [77,77,78]77)
"@IOP" :: "['a, _] => 'a"
("(_ _)" [87,88]87)
translations
"aKn" == "npowf K a n "
"aK" == "invf K a"
lemma (in Idomain) idom_is_ring:"Ring R"
by unfold_locales
lemma (in Idomain) idom_tOp_nonzeros:"[|x ∈ carrier R;
y ∈ carrier R; x ≠ \<zero>; y ≠ \<zero>|] ==> x ·r y ≠ \<zero>"
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 ≠ \<zero>|] ==> npow R x n ≠ \<zero> "
apply (induct_tac n)
apply simp (* case 0 *)
apply (rule contrapos_pp, simp+)
apply (frule ring_l_one[of "x", THEN sym]) apply simp
apply (simp add:ring_times_0_x)
(* case (Suc n) *)
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 ≠ \<zero>; 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 ≠ \<zero>; 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 - {\<zero>} ==>
invf K x ∈ (carrier K) - {\<zero>}"
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 "\<zero> = 1r", simp, erule conjE)
apply (frule Ring.ring_l_one[of K x], assumption)
apply (rotate_tac -1, frule sym, thin_tac "1r ·r x = x",
simp add:Ring.ring_times_0_x)
done
lemma (in Corps) linvf:"x ∈ carrier K - {\<zero>} ==> (invf K x) ·r x = 1r"
by (simp add:invf_inv)
lemma (in Corps) field_is_ring:"Ring K"
by (simp add:f_is_ring)
lemma (in Corps) invf_one:"1r ≠ \<zero> ==> invf K (1r) = 1r"
apply (cut_tac field_is_ring)
apply (frule_tac Ring.ring_one)
apply (cut_tac invf_closed1 [of "1r"])
apply (cut_tac linvf[of "1r"])
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 ≠ \<zero>|] ==> (xK)K = x"
apply (cut_tac invf_closed1[of "x"])
apply (cut_tac invf_inv[of "xK"], 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 ≠ \<zero>|] ==>
x^K n ≠ \<zero>"
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 ≠ \<zero>|] ==> xKn ≠ \<zero>"
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 = \<zero>|] ==> x = \<zero>"
by (rule contrapos_pp, simp+, simp add:field_potent_nonzero)
lemma (in Corps) npowf_mem:"[|a ∈ carrier K; a ≠ \<zero>|] ==>
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 ≠ \<zero>|] ==>
npowf K a 0 = 1r"
by (cut_tac field_is_ring, simp add:npowf_def)
lemma (in Corps) npow_exp_minusTr1:"[|x ∈ carrier K; x ≠ \<zero>; 0 ≤ i|] ==>
0 ≤ i - (int j) --> xK(i - (int j)) = x^K (nat i) ·r (xK)^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:zadd_commute[of "1"])
apply (cut_tac z = i and w = "int n + 1" in zdiff,
simp only:zminus_zadd_distrib,
thin_tac "i - (int n + 1) = i + (- int n + - 1)")
apply (simp only:zadd_assoc[THEN sym])
apply (simp only:zdiff[THEN sym, of _ "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) - Suc 0)" 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 (rule Ring.ring_tOp_closed[of K], assumption+, simp)
apply arith
apply simp
done
lemma (in Corps) npow_exp_minusTr2:"[|x ∈ carrier K; x ≠ \<zero>; 0 ≤ i; 0 ≤ j;
0 ≤ i - j|] ==> xK(i - j) = x^K (nat i) ·r (xK)^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 ≠ \<zero>; 0 ≤ j|] ==> xKj = (xK)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 ≠ \<zero>; ¬ 0 ≤ j|] ==>
xKj = (xK)K(-j)"
apply (simp add:npowf_def)
done
lemma (in Corps) npowf_inverse:"[|x ∈ carrier K; x ≠ \<zero>|] ==> xKj = (xK)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 ≠ \<zero>; 0 ≤ i; 0 ≤ j;
0 ≤ i - j|] ==> xK(i - j) = xKi ·r xK(- j)"
apply (simp add:npow_exp_minusTr2)
apply (simp add:npowf_def)
done
lemma (in Corps) npowf_expTr2:"[|x ∈ carrier K; x ≠ \<zero>; 0 ≤ i + j|] ==>
xK(i + j) = xKi ·r xKj"
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:zadd_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 "xKi" "xKj"])
apply (simp add:npowf_def)
done
lemma (in Corps) npowf_exp_add:"[|x ∈ carrier K; x ≠ \<zero>|] ==>
xK(i + j) = xKi ·r xKj"
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 "xK" "-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 ≠ \<zero>|] ==>
xK(1 + j) = x ·r xKj"
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 ≠ \<zero>|] ==> (xKj)K = xK(- 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 "xKj"], simp, erule conjE,
frule Ring.ring_r_one[of "K" "(xKj)K"], assumption, simp,
thin_tac "1r = xKj ·r xK- 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 ≠ \<zero>(qring R mx)|] ==>∃y∈carrier (qring R mx). y ·r(qring R mx) x = 1r(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 \<uplus>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 \<minusplus> R ♦p a = carrier R")
apply (frule_tac B = "mx \<minusplus> R ♦p a" in eq_set_inc[of "1r" "carrier R"],
assumption+,
thin_tac "carrier R = mx \<minusplus> 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 = "1r" 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 = "1r ± -a (r ·r a)" and A = mx in
eq_elem_in, assumption+)
apply (frule_tac a = "r ·r a" and b = "1r" 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 \<uplus>R mx",
thin_tac "1r = h ± r ·r a",
thin_tac "h = 1r ± -a (r ·r a)", thin_tac "1r ± -a (r ·r a) ∈ mx")
apply (rename_tac b h k r) apply simp
apply blast
done
(*
constdefs (structure R)
field_cd::"_ => bool"
"field_cd R == ∀x∈(carrier R - {\<zero>}). ∃y∈carrier R.
y ·r x = 1r" *)
(* field condition *) (*
constdefs (structure R)
rIf :: "_ => 'a => 'a " *) (** rIf is ring_invf **) (*
"rIf R == λx. (SOME y. y ∈ carrier R ∧ y ·r x = 1r)"
*) (*
constdefs (structure R)
Rf::"_ => 'a field"
"Rf R == (|carrier = carrier R, pop = pop R, mop = mop R, zero = zero R,
tp = tp R, un = un R, invf = rIf R|))," *)
(*
constdefs (structure R)
Rf :: "_ => (| carrier :: 'a set,
pOp :: ['a, 'a] => 'a, mOp ::'a => 'a, zero :: 'a, tOp :: ['a, 'a] => 'a,
one ::'a, iOp ::'a => 'a|)),"
"Rf R == (| carrier = carrier R, pOp = pOp R, mOp = mOp R, zero = zero R,
tOp = tOp R, one = one R, iOp = ring_iOp R|))," *)
(*
lemma (in Ring) rIf_mem:"[|field_cd R; x ∈ carrier R - {\<zero>}|] ==>
rIf R x ∈ carrier R ∧ rIf R x ≠ \<zero>"
apply (simp add:rIf_def)
apply (rule someI2_ex)
apply (simp add:field_cd_def, blast)
apply (simp add:field_cd_def)
apply (thin_tac "∀x∈carrier R - {\<zero>}. ∃y∈carrier R. y ·r x = 1r")
apply (erule conjE)+
apply (rule contrapos_pp, simp+)
apply (frule sym, thin_tac "\<zero> ·r x = 1r", simp add:ring_times_0_x)
apply (frule ring_l_one[of "x"])
apply (simp add:ring_times_0_x)
done
lemma (in Ring) rIf:"[|field_cd R; x ∈ carrier R - {\<zero>}|] ==>
(rIf R x) ·r x = 1r"
apply (simp add:rIf_def)
apply (rule someI2_ex)
apply (simp add:field_cd_def, blast)
apply simp
done
lemma (in Ring) field_cd_integral:"field_cd R ==> Idomain R"
apply (rule Idomain.intro)
apply assumption
apply (rule Idomain_axioms.intro)
apply (rule contrapos_pp, simp+, erule conjE)
apply (cut_tac x = a in rIf_mem, assumption, simp, erule conjE)
apply (frule_tac x = "rIf R a" and y = a and z = b in ring_tOp_assoc,
assumption+, simp add:rIf)
apply (simp add:ring_l_one ring_times_x_0)
done
lemma (in Ring) Rf_field:"field_cd R ==> field (Rf R)"
apply (rule field.intro)
apply (simp add:Rf_def)
apply (rule Ring.intro)
apply (simp add:pop_closed)
apply ( cut_tac ring_is_ag, simp add:aGroup.ag_pOp_assoc)
apply (simp add:Rf_def,
cut_tac ring_is_ag, simp add:aGroup.ag_pOp_commute)
apply (simp add:mop_closed)
apply (simp add:
apply (rule conjI)
prefer 2
apply (rule conjI)
apply (rule univar_func_test, rule ballI)
apply (simp, erule conjE, simp add:Rf_def)
apply (rule rIf_mem, assumption+, simp)
apply (rule allI, rule impI)
apply (simp add:Rf_def)
apply (frule_tac x = x in rIf, simp, assumption)
apply (subst Rf_def, simp add:Ring_def)
apply (cut_tac ring_is_ag)
apply (rule conjI, simp add:aGroup_def)
apply (rule conjI, (rule allI, rule impI)+, simp add:aGroup.ag_pOp_assoc)
apply (rule conjI, (rule allI, rule impI)+, simp add:aGroup.ag_pOp_commute)
apply (rule conjI, rule univar_func_test, rule ballI,
simp add:aGroup.ag_mOp_closed)
apply (rule conjI, rule allI, rule impI, simp add:aGroup.ag_l_inv1)
apply (simp add:aGroup.ag_inc_zero)
apply (rule conjI, rule allI, rule impI, simp add:aGroup.ag_l_zero)
apply (rule conjI, rule bivar_func_test, (rule ballI)+,
simp add:ring_tOp_closed)
apply (rule conjI, (rule allI, rule impI)+, simp add:ring_tOp_assoc)
apply (rule conjI, (rule allI, rule impI)+, simp add:ring_tOp_commute)
apply (simp add:ring_one)
apply (rule conjI, (rule allI, rule impI)+, simp add:ring_distrib1)
apply (rule allI, rule impI, simp add:ring_l_one)
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) qRf_field:"maximal_ideal R mx ==> field (Rf (qring R mx))"
apply (frule maximal_ideal_ideal[of "mx"])
apply (frule qring_ring [of "mx"])
apply (frule residue_field_cd[of "mx"])
apply (rule Ring.Rf_field, assumption+)
done
lemma (in Ring) qRf_pj_rHom:"maximal_ideal R mx ==>
(pj R mx) ∈ rHom R (Rf (qring R mx))"
apply (frule maximal_ideal_ideal[of "mx"])
apply (frule pj_Hom[OF Ring, of "mx"])
apply (simp add:rHom_def aHom_def Rf_def)
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)
lemma (in Ring) mulDisj:"[|mul_closed_set R S; 1r ∈ S; \<zero> ∉ 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 \<minusplus> 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 \<minusplus> R ♦p x")
apply (frule_tac J = "mx \<minusplus> 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 \<minusplus> R ♦p y")
apply (frule_tac A = "S ∩ (mx \<minusplus> R ♦p x)" in nonempty_ex,
frule_tac A = "S ∩ (mx \<minusplus> R ♦p y)" in nonempty_ex,
(erule exE)+, simp, (erule conjE)+)
apply (rename_tac x y s1 s2,
thin_tac "S ∩ (mx \<minusplus> R ♦p x) ≠ {}",
thin_tac "S ∩ (mx \<minusplus> 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; \<zero> ∉ S; 1r ∈ S;
T = {I. ideal R I ∧ S ∩ I = {}}|] ==> ∃mx. maximal_set T mx"
apply (cut_tac S="{ 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 ∈ chain {I. ideal R I ∧ I ⊂ carrier R}")
apply (frule chain_un, assumption)
apply (subgoal_tac "S ∩ (\<Union> c) = {}")
apply (subgoal_tac "∀x∈c. x ⊆ \<Union> c", blast)
apply (rule ballI, rule subsetI, simp add:CollectI)
apply blast
apply (rule contrapos_pp, simp+)
apply (frule_tac A = S and B = "\<Union> c" in nonempty_int)
apply (erule exE)
apply (simp add:Inter_def, erule conjE, erule bexE)
apply (simp add:chain_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:chain_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; \<zero> ∉ S; 1r ∈ 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 = \<Inter> {p. prime_ideal R p}"
apply (rule equalityI)
(* nilrad R ⊆ \<Inter>Collect (prime_ideal R) *)
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 = \<zero>", 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_is_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 "\<zero> ∉ {s. ∃n. s = npow R x n} ∧
1r ∈ {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 "\<zero> ∉ {s. ∃n. s = x^R n} ∧ 1r ∈ {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 "\<zero> = 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 \<uplus>R (nilrad R))|] ==>
x \<uplus>R (nilrad R) = \<zero>(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 \<uplus>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 = {1r}; \<zero> ∉ S; ideal R I; I ∩ S = {};
T = {J. ideal R J ∧ S ∩ J = {} ∧ I ⊆ J}|] ==> ∃mx. maximal_set T mx"
apply (cut_tac S="{J. ideal R J ∧ S ∩ J = {} ∧ I ⊆ J}" in Zorn_Lemma2)
apply (rule ballI)
apply (case_tac "c = {}") (** case c = {} **)
apply blast (** case c = {} done **)
(** existence of sup in c **)
apply (subgoal_tac "\<Union>c∈{J. ideal R J ∧ S ∩ J = {} ∧ I ⊆ J} ∧
(∀x∈c. x ⊆ \<Union>c)")
apply blast
apply (rule conjI,
simp add:CollectI)
apply (subgoal_tac "c ∈ chain {I. ideal R I ∧ I ⊂ carrier R}")
apply (rule conjI,
simp add:chain_un)
apply (rule conjI)
apply (rule contrapos_pp, simp+, erule bexE)
apply (thin_tac " c ∈ chain {I. ideal R I ∧ I ⊂ carrier R}")
apply (simp add:chain_def, erule conjE)
apply (frule_tac c = x and A = c and B = "{J. ideal R J ∧ 1r ∉ J ∧ I ⊆ J}"
in subsetD, assumption+, simp,
thin_tac "c ∈ chain {I. ideal R I ∧ I ⊂ carrier R}")
apply (frule_tac A = c in nonempty_ex, erule exE, simp add:chain_def,
erule conjE,
frule_tac c = x and A = c and B = "{J. ideal R J ∧ 1r ∉ J ∧ I ⊆ J}" in
subsetD, assumption+, simp, (erule conjE)+)
apply (rule_tac A = I and B = x and C = "\<Union>c" in subset_trans, assumption,
rule_tac B = x and A = c in Union_upper, assumption+)
apply (simp add:chain_def, erule conjE)
apply (rule subsetI, simp)
apply (frule_tac c = x and A = c and B = "{J. ideal R J ∧ 1r ∉ 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 = {1r}; \<zero> ∉ 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)
(** {J. ideal R J ∧ mx ⊆ J} ⊆ {mx, carrier R} **)
apply (rule subsetI, simp add:CollectI, erule conjE)
apply (case_tac "x = mx", simp, simp)
apply (subgoal_tac "1r ∈ x")
apply (rule_tac I = x in ideal_inc_one, assumption+)
apply (rule contrapos_pp, simp+)
apply (subgoal_tac "mx = x",
rotate_tac -3, frule not_sym)
apply (simp, blast)
apply (rule subsetI,
simp add:CollectI)
apply (case_tac "x = mx", simp, simp)
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; 1r ∉ I|] ==>
∃mx. maximal_ideal R mx ∧ I ⊆ mx"
apply (cut_tac ex_contid_maximal[of "{1r}" "I"
"{J. ideal R J ∧ {1r} ∩ J = {} ∧ I ⊆ J}"])
apply (erule exE,
cut_tac mx = mx in contid_maximal[of "{1r}" "I"
"{J. ideal R J ∧ {1r} ∩ 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 "1r" "carrier R" "R ♦p a"], assumption,
thin_tac "carrier R = R ♦p a", thin_tac "1r ∈ carrier R")
apply (simp add:Rxa_def, erule bexE, simp add:ring_tOp_commute[of _ "a"],
frule sym, thin_tac "1r = 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
constdefs (structure R)
local_ring :: "_ => bool"
"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 b = a in forball_spec1, 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 b = xa in forball_spec1,
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 ± 1r) |] ==> 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 \<minusplus> R ♦p a = carrier R",
cut_tac ring_one,
frule_tac a = "1r" and A = "carrier R" and B = "mx \<minusplus> R ♦p a" in
eq_set_inc, assumption+,
thin_tac "carrier R = mx \<minusplus> 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 "1r = 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 = 1r")
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
constdefs (structure R)
J_rad ::"_ => 'a set"
"J_rad R == if (zeroring R) then (carrier R) else
\<Inter> {mx. maximal_ideal R mx}"
(** if zeroring R then \<Inter> {mx. maximal_ideal R mx} is UNIV, hence
we restrict UNIV to carrier R **)
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 (1r ± (-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 = "1r" and y = "-a (x ·r y)" in aGroup.ag_pOp_closed,
assumption+)
apply (frule_tac a = "1r ± -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 = "1r" and b = "-a (x ·r y)" in ideal_ele_sumTr2,
assumption+)
apply (simp add:maximal_ideal_def)
done
end
lemma ag_carrier_carrier:
carrier (b_ag A) = carrier A
lemma ag_pOp_closed:
[| x ∈ carrier A; y ∈ carrier A |] ==> x ± y ∈ carrier A
lemma ag_mOp_closed:
x ∈ carrier A ==> -a x ∈ carrier A
lemma asubg_subset:
@ASubG A H ==> H ⊆ carrier A
lemma ag_pOp_commute:
[| x ∈ carrier A; y ∈ carrier A |] ==> x ± y = y ± x
lemma b_ag_group:
Group (b_ag A)
lemma agop_gop:
Group.top (b_ag A) = op ±
lemma agiop_giop:
iop (b_ag A) = mop A
lemma agunit_gone:
\<one>b_ag A = \<zero>
lemma ag_pOp_add_r:
[| a ∈ carrier A; b ∈ carrier A; c ∈ carrier A; a = b |] ==> a ± c = b ± c
lemma ag_add_commute:
[| a ∈ carrier A; b ∈ carrier A |] ==> a ± b = b ± a
lemma ag_pOp_add_l:
[| a ∈ carrier A; b ∈ carrier A; c ∈ carrier A; a = b |] ==> c ± a = c ± b
lemma asubg_pOp_closed:
[| @ASubG A H; x ∈ H; y ∈ H |] ==> x ± y ∈ H
lemma asubg_mOp_closed:
[| @ASubG A H; x ∈ H |] ==> -a x ∈ H
lemma asubg_subset1:
[| @ASubG A H; x ∈ H |] ==> x ∈ carrier A
lemma asubg_inc_zero:
@ASubG A H ==> \<zero> ∈ H
lemma ag_inc_zero:
\<zero> ∈ carrier A
lemma ag_l_zero:
x ∈ carrier A ==> \<zero> ± x = x
lemma ag_r_zero:
x ∈ carrier A ==> x ± \<zero> = x
lemma ag_l_inv1:
x ∈ carrier A ==> -a x ± x = \<zero>
lemma ag_r_inv1:
x ∈ carrier A ==> x ± -a x = \<zero>
lemma ag_pOp_assoc:
[| x ∈ carrier A; y ∈ carrier A; z ∈ carrier A |] ==> x ± y ± z = x ± (y ± z)
lemma ag_inv_unique:
[| x ∈ carrier A; y ∈ carrier A; x ± y = \<zero> |] ==> y = -a x
lemma ag_inv_inj:
[| x ∈ carrier A; y ∈ carrier A; x ≠ y |] ==> -a x ≠ -a y
lemma pOp_assocTr41:
[| a ∈ carrier A; b ∈ carrier A; c ∈ carrier A; d ∈ carrier A |]
==> a ± b ± c ± d = a ± b ± (c ± d)
lemma pOp_assocTr42:
[| a ∈ carrier A; b ∈ carrier A; c ∈ carrier A; d ∈ carrier A |]
==> a ± b ± c ± d = a ± (b ± c) ± d
lemma pOp_assocTr43:
[| a ∈ carrier A; b ∈ carrier A; c ∈ carrier A; d ∈ carrier A |]
==> a ± b ± (c ± d) = a ± (b ± c) ± d
lemma pOp_assoc_cancel:
[| a ∈ carrier A; b ∈ carrier A; c ∈ carrier A |]
==> a ± -a b ± (b ± -a c) = a ± -a c
lemma ag_p_inv:
[| x ∈ carrier A; y ∈ carrier A |] ==> -a (x ± y) = -a x ± -a y
lemma gEQAddcross:
[| l1.0 ∈ carrier A; l2.0 ∈ carrier A; r1.0 ∈ carrier A; r1.0 ∈ carrier A;
l1.0 = r2.0; l2.0 = r1.0 |]
==> l1.0 ± l2.0 = r1.0 ± r2.0
lemma ag_eq_sol1:
[| a ∈ carrier A; x ∈ carrier A; b ∈ carrier A; a ± x = b |] ==> x = -a a ± b
lemma ag_eq_sol2:
[| a ∈ carrier A; x ∈ carrier A; b ∈ carrier A; x ± a = b |] ==> x = b ± -a a
lemma ag_add4_rel:
[| a ∈ carrier A; b ∈ carrier A; c ∈ carrier A; d ∈ carrier A |]
==> a ± b ± (c ± d) = a ± c ± (b ± d)
lemma ag_inv_inv:
x ∈ carrier A ==> -a (-a x) = x
lemma ag_inv_zero:
-a \<zero> = \<zero>
lemma ag_diff_minus:
[| a ∈ carrier A; b ∈ carrier A; c ∈ carrier A; a ± -a b = c |]
==> b ± -a a = -a c
lemma pOp_cancel_l:
[| a ∈ carrier A; b ∈ carrier A; c ∈ carrier A; c ± a = c ± b |] ==> a = b
lemma pOp_cancel_r:
[| a ∈ carrier A; b ∈ carrier A; c ∈ carrier A; a ± c = b ± c |] ==> a = b
lemma ag_eq_diffzero:
[| a ∈ carrier A; b ∈ carrier A |] ==> (a = b) = (a ± -a b = \<zero>)
lemma ag_eq_diffzero1:
[| a ∈ carrier A; b ∈ carrier A |] ==> (a = b) = (-a a ± b = \<zero>)
lemma ag_neq_diffnonzero:
[| a ∈ carrier A; b ∈ carrier A |] ==> (a ≠ b) = (a ± -a b ≠ \<zero>)
lemma ag_plus_zero:
[| x ∈ carrier A; y ∈ carrier A |] ==> (x = -a y) = (x ± y = \<zero>)
lemma asubg_nsubg:
@ASubG A H ==> b_ag A \<triangleright> H
lemma subg_asubg:
b_ag G » H ==> @ASubG G H
lemma asubg_test:
[| H ⊆ carrier A; H ≠ {}; ∀a∈H. ∀b∈H. a ± -a b ∈ H |] ==> @ASubG A H
lemma asubg_zero:
@ASubG A {\<zero>}
lemma asubg_whole:
@ASubG A (carrier A)
lemma Ag_ind_carrier:
bij_to f (carrier A) D ==> carrier (Ag_ind A f) = f ` carrier A
lemma Ag_ind_aGroup:
[| f ∈ carrier A -> D; bij_to f (carrier A) D |] ==> aGroup (Ag_ind A f)
lemma aHom_mem:
[| aGroup F; aGroup G; f ∈ aHom F G; a ∈ carrier F |] ==> f a ∈ carrier G
lemma aHom_func:
f ∈ aHom F G ==> f ∈ carrier F -> carrier G
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
lemma aHom_0_0:
[| aGroup F; aGroup G; f ∈ aHom F G |] ==> f \<zero>F = \<zero>G
lemma ker_inc_zero:
[| aGroup F; aGroup G; f ∈ aHom F G |] ==> \<zero>F ∈ kerF,G f
lemma aHom_inv_inv:
[| aGroup F; aGroup G; f ∈ aHom F G; a ∈ carrier F |] ==> f (-aF a) = -aG f a
lemma aHom_compos:
[| aGroup L; aGroup M; aGroup N; f ∈ aHom L M; g ∈ aHom M N |]
==> compos L g f ∈ aHom L N
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
lemma injec_inj_on:
[| aGroup F; aGroup G; injecF,G f |] ==> inj_on f (carrier F)
lemma surjec_surj_to:
surjecR,S f ==> surj_to f (carrier R) (carrier S)
lemma compos_bijec:
[| aGroup E; aGroup F; aGroup G; bijecE,F f; bijecF,G g |]
==> bijecE,G compos E g f
lemma ainvf_aHom:
[| aGroup F; aGroup G; bijecF,G f |] ==> ainvfF,G f ∈ aHom G F
lemma ainvf_bijec:
[| aGroup F; aGroup G; bijecF,G f |] ==> bijecG,F (ainvfF,G f)
lemma ainvf_l:
[| aGroup E; aGroup F; bijecE,F f; x ∈ carrier E |] ==> (ainvfE,F f) (f x) = x
lemma aI_aHom:
aIA ∈ aHom A A
lemma compos_aI_l:
[| aGroup A; aGroup B; f ∈ aHom A B |] ==> compos A aIB f = f
lemma compos_aI_r:
[| aGroup A; aGroup B; f ∈ aHom A B |] ==> compos A f aIA = f
lemma compos_aI_surj:
[| aGroup A; aGroup B; f ∈ aHom A B; g ∈ aHom B A; compos A g f = aIA |]
==> surjecB,A g
lemma compos_aI_inj:
[| aGroup A; aGroup B; f ∈ aHom A B; g ∈ aHom B A; compos A g f = aIA |]
==> injecA,B f
lemma Ag_ind_aHom:
[| f ∈ carrier A -> D; bij_to f (carrier A) D |]
==> Agii A f ∈ aHom A (Ag_ind A f)
lemma Agii_mem:
[| f ∈ carrier A -> D; x ∈ carrier A; bij_to f (carrier A) D |]
==> Agii A f x ∈ carrier (Ag_ind A f)
lemma Ag_ind_bijec:
[| aGroup A; f ∈ carrier A -> D; bij_to f (carrier A) D |]
==> bijecA,Ag_ind A f Agii A f
lemma ker_subg:
[| aGroup F; aGroup G; f ∈ aHom F G |] ==> @ASubG F (kerF,G f)
lemma ag_a_in_ar_cos:
[| @ASubG A H; a ∈ carrier A |] ==> a ∈ a \<uplus>A H
lemma r_cos_subset:
[| @ASubG A H; X ∈ set_rcs (b_ag A) H |] ==> X ⊆ carrier A
lemma asubg_costOp_commute:
[| @ASubG 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
lemma Subg_Qgroup:
@ASubG A H ==> aGroup (aqgrp A H)
lemma plus_subgs:
[| @ASubG A H1.0; @ASubG A H2.0 |] ==> @ASubG A (H1.0 \<minusplus> H2.0)
lemma set_sum:
[| H ⊆ carrier A; K ⊆ carrier A |]
==> H \<minusplus> K = {x. ∃h∈H. ∃k∈K. x = h ± k}
lemma mem_set_sum:
[| H ⊆ carrier A; K ⊆ carrier A; x ∈ H \<minusplus> K |]
==> ∃h∈H. ∃k∈K. x = h ± k
lemma mem_sum_subgs:
[| @ASubG A H; @ASubG A K; h ∈ H; k ∈ K |] ==> h ± k ∈ H \<minusplus> K
lemma aqgrp_carrier:
@ASubG A H ==> set_rcs (b_ag A) H = set_ar_cos A H
lemma unit_in_set_ar_cos:
@ASubG A H ==> H ∈ set_ar_cos A H
lemma aqgrp_pOp_maps:
[| @ASubG A H; a ∈ carrier A; b ∈ carrier A |]
==> a \<uplus>A H ±aqgrp A H b \<uplus>A H = (a ± b) \<uplus>A H
lemma aqgrp_mOp_maps:
[| @ASubG A H; a ∈ carrier A |]
==> -aaqgrp A H a \<uplus>A H = (-a a) \<uplus>A H
lemma aqgrp_zero:
@ASubG A H ==> \<zero>aqgrp A H = H
lemma arcos_fixed:
[| @ASubG A H; a ∈ carrier A; h ∈ H |] ==> a \<uplus>A H = (h ± a) \<uplus>A H
lemma prodag_comp_i:
[| a ∈ carr_prodag I A; i ∈ I |] ==> a i ∈ carrier (A i)
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
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
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
lemma prod_mOp_func:
∀k∈I. aGroup (A k) ==> prod_mOp I A ∈ carr_prodag I A -> carr_prodag I A
lemma prod_mOp_mem:
[| ∀j∈I. aGroup (A j); X ∈ carr_prodag I A |]
==> prod_mOp I A X ∈ carr_prodag I A
lemma prod_mOp_mem_i:
[| ∀j∈I. aGroup (A j); X ∈ carr_prodag I A; i ∈ I |]
==> prod_mOp I A X i = -aA i X i
lemma prod_zero_func:
∀k∈I. aGroup (A k) ==> prod_zero I A ∈ carr_prodag I A
lemma prod_zero_i:
[| ∀k∈I. aGroup (A k); i ∈ I |] ==> prod_zero I A i = \<zero>A i
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
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)
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
lemma prodag_aGroup:
∀k∈I. aGroup (A k) ==> aGroup (aΠI A)
lemma prodag_carrier:
∀k∈I. aGroup (A k) ==> carrier (aΠI A) = carr_prodag I A
lemma prodag_elemfun:
[| ∀k∈I. aGroup (A k); f ∈ carrier (aΠI A) |] ==> f ∈ extensional I
lemma prodag_component:
[| f ∈ carrier (aΠI A); i ∈ I |] ==> f i ∈ carrier (A i)
lemma prodag_pOp:
∀k∈I. aGroup (A k) ==> op ±aΠI A = prod_pOp I A
lemma prodag_iOp:
∀k∈I. aGroup (A k) ==> mop (aΠI A) = prod_mOp I A
lemma prodag_zero:
∀k∈I. aGroup (A k) ==> \<zero>aΠI A = prod_zero I A
lemma prodag_sameTr0:
[| ∀k∈I. aGroup (A k); ∀k∈I. A k = B k |] ==> Un_carrier I A = Un_carrier I B
lemma prodag_sameTr1:
[| ∀k∈I. aGroup (A k); ∀k∈I. A k = B k |] ==> carr_prodag I A = carr_prodag I B
lemma prodag_sameTr2:
[| ∀k∈I. aGroup (A k); ∀k∈I. A k = B k |] ==> prod_pOp I A = prod_pOp I B
lemma prodag_sameTr3:
[| ∀k∈I. aGroup (A k); ∀k∈I. A k = B k |] ==> prod_mOp I A = prod_mOp I B
lemma prodag_sameTr4:
[| ∀k∈I. aGroup (A k); ∀k∈I. A k = B k |] ==> prod_zero I A = prod_zero I B
lemma prodag_same:
[| ∀k∈I. aGroup (A k); ∀k∈I. A k = B k |] ==> aΠI A = aΠI B
lemma project_mem:
[| ∀k∈I. aGroup (A k); j ∈ I; x ∈ carrier (aΠI A) |]
==> (πI,A,j) x ∈ carrier (A j)
lemma project_aHom:
[| ∀k∈I. aGroup (A k); j ∈ I |] ==> πI,A,j ∈ aHom (aΠI A) (A j)
lemma project_aHom1:
∀k∈I. aGroup (A k) ==> ∀j∈I. πI,A,j ∈ aHom (aΠI A) (A j)
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
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 (aΠI B)
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
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
lemma dsum_iOp_func:
∀k∈I. aGroup (A k) ==> prod_mOp I A ∈ carr_dsumag I A -> carr_dsumag I A
lemma dsum_iOp_mem:
[| ∀j∈I. aGroup (A j); X ∈ carr_dsumag I A |]
==> prod_mOp I A X ∈ carr_dsumag I A
lemma dsum_zero_func:
∀k∈I. aGroup (A k) ==> prod_zero I A ∈ carr_dsumag I A
lemma dsumag_sub_prodag:
∀k∈I. aGroup (A k) ==> carr_dsumag I A ⊆ carr_prodag I A
lemma carrier_dsumag:
∀k∈I. aGroup (A k) ==> carrier (a\<Oplus>I A) = carr_dsumag I A
lemma dsumag_elemfun:
[| ∀k∈I. aGroup (A k); f ∈ carrier (a\<Oplus>I A) |] ==> f ∈ extensional I
lemma dsumag_aGroup:
∀k∈I. aGroup (A k) ==> aGroup (a\<Oplus>I A)
lemma dsumag_pOp:
∀k∈I. aGroup (A k) ==> op ±a\<Oplus>I A = prod_pOp I A
lemma dsumag_mOp:
∀k∈I. aGroup (A k) ==> mop (a\<Oplus>I A) = prod_mOp I A
lemma dsumag_zero:
∀k∈I. aGroup (A k) ==> \<zero>a\<Oplus>I A = prod_zero I A
lemma direct_prod_mem_eq:
[| ∀j∈I. aGroup (A j); f ∈ carrier (aΠI A); g ∈ carrier (aΠI A);
∀j∈I. (πI,A,j) f = (πI,A,j) g |]
==> f = g
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 (aΠI A)
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 (aΠI A)
lemma map_family_triangle:
[| ∀j∈I. aGroup (A j); aGroup S; ∀j∈I. g j ∈ aHom S (A j) |]
==> ∃!f. f ∈ aHom S (aΠI A) ∧ (∀j∈I. compos S (πI,A,j) f = g j)
lemma Ag_ind_triangle:
[| ∀j∈I. aGroup (A j); j ∈ I; f ∈ carrier (aΠI A) -> B;
bij_to f (carrier (aΠI A)) B; j ∈ I |]
==> compos (aΠI A)
(compos (Ag_ind (aΠI A) f) (πI,A,j)
(ainvf(aΠI A),Ag_ind (aΠI A) f Agii (aΠI A) f))
(Agii (aΠI A) f) =
πI,A,j
lemma ProjInd_aHom:
[| ∀j∈I. aGroup (A j); j ∈ I; f ∈ carrier (aΠI A) -> B;
bij_to f (carrier (aΠI A)) B; j ∈ I |]
==> ProjInd I A f j ∈ aHom (Ag_ind (aΠI A) f) (A j)
lemma ProjInd_aHom1:
[| ∀j∈I. aGroup (A j); f ∈ carrier (aΠI A) -> B; bij_to f (carrier (aΠI A)) B |]
==> ∀j∈I. ProjInd I A f j ∈ aHom (Ag_ind (aΠI A) f) (A j)
lemma ProjInd_mem_eq:
[| ∀j∈I. aGroup (A j); f ∈ carrier (aΠI A) -> B; bij_to f (carrier (aΠI A)) B;
aGroup S; x ∈ carrier (Ag_ind (aΠI A) f); y ∈ carrier (Ag_ind (aΠI A) f);
∀j∈I. ProjInd I A f j x = ProjInd I A f j y |]
==> x = y
lemma ProjInd_mem_eq1:
[| ∀j∈I. aGroup (A j); f ∈ carrier (aΠI A) -> B; bij_to f (carrier (aΠI A)) B;
aGroup S; h ∈ aHom (Ag_ind (aΠI A) f) (Ag_ind (aΠI A) f);
∀j∈I. compos (Ag_ind (aΠI A) f) (ProjInd I A f j) h = ProjInd I A f j |]
==> h = aIAg_ind (aΠI A) f
lemma Ag_ind_triangle1:
[| ∀j∈I. aGroup (A j); f ∈ carrier (aΠI A) -> B; bij_to f (carrier (aΠI A)) B;
j ∈ I |]
==> compos (aΠI A) (ProjInd I A f j) (Agii (aΠI A) f) = πI,A,j
lemma map_family_triangle1:
[| ∀j∈I. aGroup (A j); f ∈ carrier (aΠI A) -> B; bij_to f (carrier (aΠI A)) B;
aGroup S; ∀j∈I. g j ∈ aHom S (A j) |]
==> ∃!h. h ∈ aHom S (Ag_ind (aΠI A) f) ∧
(∀j∈I. compos S (ProjInd I A f j) h = g j)
lemma map_family_triangle2:
[| I ≠ {}; ∀j∈I. aGroup (A j); aGroup S; ∀j∈I. g j ∈ aHom S (A j);
ff ∈ carrier (aΠI A) -> B; bij_to ff (carrier (aΠI A)) B;
h1.0 ∈ aHom (Ag_ind (aΠI A) ff) S;
∀j∈I. compos (Ag_ind (aΠI A) ff) (g j) h1.0 = ProjInd I A ff j;
h2.0 ∈ aHom S (Ag_ind (aΠI A) ff);
∀j∈I. compos S (ProjInd I A ff j) h2.0 = g j |]
==> ∀j∈I. compos (Ag_ind (aΠI A) ff) (ProjInd I A ff j)
(compos (Ag_ind (aΠI A) ff) h2.0 h1.0) =
ProjInd I A ff j
lemma map_family_triangle3:
[| ∀j∈I. aGroup (A j); aGroup S; aGroup S1.0; ∀j∈I. f j ∈ aHom S (A j);
∀j∈I. g j ∈ aHom S1.0 (A j); h1.0 ∈ aHom S1.0 S; h2.0 ∈ aHom S S1.0;
∀j∈I. compos S (g j) h2.0 = f j; ∀j∈I. compos S1.0 (f j) h1.0 = g j |]
==> ∀j∈I. compos S (f j) (compos S h1.0 h2.0) = f j
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) aIS = f j
lemma prod_triangle:
[| I ≠ {}; ∀j∈I. aGroup (A j); aGroup S; ∀j∈I. g j ∈ aHom S (A j);
ff ∈ carrier (aΠI A) -> B; bij_to ff (carrier (aΠI A)) B;
h1.0 ∈ aHom (Ag_ind (aΠI A) ff) S;
∀j∈I. compos (Ag_ind (aΠI A) ff) (g j) h1.0 = ProjInd I A ff j;
h2.0 ∈ aHom S (Ag_ind (aΠI A) ff);
∀j∈I. compos S (ProjInd I A ff j) h2.0 = g j |]
==> compos (Ag_ind (aΠI A) ff) h2.0 h1.0 = aIAg_ind (aΠI A) ff
lemma characterization_prodag:
[| I ≠ {}; ∀j∈I. aGroup (A j); aGroup S; ∀j∈I. g j ∈ aHom S (A j);
∃ff. ff ∈ carrier (aΠI A) -> B ∧ bij_to ff (carrier (aΠI A)) B;
∀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))) |]
==> ∃h. bijec(aΠI A),S h
lemma nsum_zeroGTr:
(∀j≤n. f j = \<zero>) --> Σe A f n = \<zero>
lemma nsum_zeroA:
∀j≤n. f j = \<zero> ==> Σe A f n = \<zero>
lemma Ring:
Ring R
lemma ring_is_ag:
aGroup R
lemma ring_zero:
\<zero> ∈ carrier R
lemma ring_one:
1r ∈ carrier R
lemma ring_tOp_closed:
[| x ∈ carrier R; y ∈ carrier R |] ==> x ·r y ∈ carrier R
lemma ring_tOp_commute:
[| x ∈ carrier R; y ∈ carrier R |] ==> x ·r y = y ·r x
lemma ring_distrib1:
[| x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |]
==> x ·r (y ± z) = x ·r y ± x ·r z
lemma ring_distrib2:
[| x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |]
==> (y ± z) ·r x = y ·r x ± z ·r x
lemma 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
lemma rEQMulR:
[| x ∈ carrier R; y ∈ carrier R; z ∈ carrier R; x = y |] ==> x ·r z = y ·r z
lemma ring_tOp_assoc:
[| x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |]
==> x ·r y ·r z = x ·r (y ·r z)
lemma ring_l_one:
x ∈ carrier R ==> 1r ·r x = x
lemma ring_r_one:
x ∈ carrier R ==> x ·r 1r = x
lemma ring_times_0_x:
x ∈ carrier R ==> \<zero> ·r x = \<zero>
lemma ring_times_x_0:
x ∈ carrier R ==> x ·r \<zero> = \<zero>
lemma rMulZeroDiv:
[| x ∈ carrier R; y ∈ carrier R; x = \<zero> ∨ y = \<zero> |]
==> x ·r y = \<zero>
lemma ring_inv1:
[| a ∈ carrier R; b ∈ carrier R |]
==> -a a ·r b = (-a a) ·r b ∧ -a a ·r b = a ·r (-a b)
lemma ring_inv1_1:
[| a ∈ carrier R; b ∈ carrier R |] ==> -a a ·r b = (-a a) ·r b
lemma ring_inv1_2:
[| a ∈ carrier R; b ∈ carrier R |] ==> -a a ·r b = a ·r (-a b)
lemma ring_times_minusl:
a ∈ carrier R ==> -a a = (-a 1r) ·r a
lemma ring_times_minusr:
a ∈ carrier R ==> -a a = a ·r (-a 1r)
lemma ring_inv1_3:
[| a ∈ carrier R; b ∈ carrier R |] ==> a ·r b = (-a a) ·r (-a b)
lemma 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
lemma rMulLC:
[| x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |]
==> x ·r (y ·r z) = y ·r (x ·r z)
lemma Zero_ring:
1r = \<zero> ==> zeroring R
lemma Zero_ring1:
¬ zeroring R ==> 1r ≠ \<zero>
lemma Sr_one:
sr R S ==> 1r ∈ S
lemma Sr_zero:
sr R S ==> \<zero> ∈ S
lemma Sr_mOp_closed:
[| sr R S; x ∈ S |] ==> -a x ∈ S
lemma Sr_pOp_closed:
[| sr R S; x ∈ S; y ∈ S |] ==> x ± y ∈ S
lemma Sr_tOp_closed:
[| sr R S; x ∈ S; y ∈ S |] ==> x ·r y ∈ S
lemma Sr_ring:
sr R S ==> Ring (Sr R S)
lemma 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)
lemma nsClose:
x ∈ carrier R ==> n ×R x ∈ carrier R
lemma nsZero:
n ×R \<zero> = \<zero>
lemma nsZeroI:
x = \<zero> ==> n ×R x = \<zero>
lemma nsEqElm:
[| x ∈ carrier R; y ∈ carrier R; x = y |] ==> n ×R x = n ×R y
lemma nsDistr:
x ∈ carrier R ==> n ×R x ± m ×R x = (n + m) ×R x
lemma nsDistrL:
[| x ∈ carrier R; y ∈ carrier R |] ==> n ×R x ± n ×R y = n ×R (x ± y)
lemma nsMulDistrL:
[| x ∈ carrier R; y ∈ carrier R |] ==> x ·r n ×R y = n ×R (x ·r y)
lemma nsMulDistrR:
[| x ∈ carrier R; y ∈ carrier R |] ==> n ×R y ·r x = n ×R (y ·r x)
lemma npClose:
x ∈ carrier R ==> x^R n ∈ carrier R
lemma npMulDistr:
x ∈ carrier R ==> x^R n ·r x^R m = x^R (n + m)
lemma npMulExp:
x ∈ carrier R ==> x^R n^R m = x^R (n * m)
lemma npGTPowZero_sub:
[| x ∈ carrier R; x^R m = \<zero> |] ==> m ≤ n --> x^R n = \<zero>
lemma npGTPowZero:
[| x ∈ carrier R; x^R m = \<zero>; m ≤ n |] ==> x^R n = \<zero>
lemma npOne:
1r^R n = 1r
lemma npZero_sub:
0 < n --> \<zero>^R n = \<zero>
lemma npZero:
0 < n ==> \<zero>^R n = \<zero>
lemma npMulElmL:
[| x ∈ carrier R; 0 ≤ n |] ==> x ·r x^R n = x^R Suc n
lemma npMulEleL:
x ∈ carrier R ==> x^R n ·r x = x^R Suc n
lemma npMulElmR:
x ∈ carrier R ==> x^R n ·r x = x^R Suc n
lemma np_1:
a ∈ carrier R ==> a^R Suc 0 = a
lemma nsum_memTr:
(∀j≤n. f j ∈ carrier A) --> Σe A f n ∈ carrier A
lemma nsum_mem:
∀j≤n. f j ∈ carrier A ==> Σe A f n ∈ carrier A
lemma nsum_eqTr:
(∀j≤n. f j ∈ carrier A ∧ g j ∈ carrier A ∧ f j = g j) --> Σe A f n = Σe A g n
lemma nsum_eq:
[| ∀j≤n. f j ∈ carrier A; ∀j≤n. g j ∈ carrier A; ∀j≤n. f j = g j |]
==> Σe A f n = Σe A g n
lemma nsum_cmp_assoc:
[| ∀j≤n. f j ∈ carrier A; g ∈ {j. j ≤ n} -> {j. j ≤ n};
h ∈ {j. j ≤ n} -> {j. j ≤ n} |]
==> Σe A cmp (cmp f h) g n = Σe A cmp f (cmp h g) n
lemma fSum_Suc:
∀j∈nset n (n + Suc m). f j ∈ carrier A
==> Σf A f n (n + Suc m) = Σf A f n (n + m) ± f (n + Suc m)
lemma fSum_eqTr:
(∀j∈nset n (n + m). f j ∈ carrier A ∧ g j ∈ carrier A ∧ f j = g j) -->
Σf A f n (n + m) = Σf A g n (n + m)
lemma 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 |]
==> Σf A f n (n + m) = Σf A g n (n + m)
lemma 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 |]
==> Σf A f n m = Σf A g n m
lemma fSum_zeroTr:
(∀j∈nset n (n + m). f j = \<zero>) --> Σf A f n (n + m) = \<zero>
lemma fSum_zero:
∀j∈nset n (n + m). f j = \<zero> ==> Σf A f n (n + m) = \<zero>
lemma fSum_zero1:
[| n < m; ∀j∈nset (Suc n) m. f j = \<zero> |] ==> Σf A f Suc n m = \<zero>
lemma nsumMulEleL:
[| ∀i. f i ∈ carrier R; x ∈ carrier R |]
==> x ·r Σe R f n = Σe R (λi. x ·r f i) n
lemma nsumMulElmL:
[| ∀i. f i ∈ carrier R; x ∈ carrier R |]
==> x ·r Σe R f n = Σe R (λi. x ·r f i) n
lemma nsumTailTr:
(∀j≤Suc n. f j ∈ carrier A) --> Σe A f Suc n = Σe A (λi. f (Suc i)) n ± f 0
lemma nsumTail:
∀j≤Suc n. f j ∈ carrier A ==> Σe A f Suc n = Σe A (λi. f (Suc i)) n ± f 0
lemma nsumElmTail:
∀i. f i ∈ carrier A ==> Σe A f Suc n = Σe A (λi. f (Suc i)) n ± f 0
lemma nsum_addTr:
(∀j≤n. f j ∈ carrier A ∧ g j ∈ carrier A) -->
Σe A (λi. f i ± g i) n = Σe A f n ± Σe A g n
lemma nsum_add:
[| ∀j≤n. f j ∈ carrier A; ∀j≤n. g j ∈ carrier A |]
==> Σe A (λi. f i ± g i) n = Σe A f n ± Σe A g n
lemma nsumElmAdd:
[| ∀i. f i ∈ carrier A; ∀i. g i ∈ carrier A |]
==> Σe A (λi. f i ± g i) n = Σe A f n ± Σe A g n
lemma nsum_add_nmTr:
(∀j≤n. f j ∈ carrier A) ∧ (∀j≤m. g j ∈ carrier A) -->
Σe A jointfun n f m g Suc (n + m) = Σe A f n ± Σe A g m
lemma nsum_add_nm:
[| ∀j≤n. f j ∈ carrier A; ∀j≤m. g j ∈ carrier A |]
==> Σe A jointfun n f m g Suc (n + m) = Σe A f n ± Σe A g m
lemma npeSum2_sub_muly:
[| x ∈ carrier R; y ∈ carrier R |]
==> y ·r Σe R (λi. (n choose i) ×R (x^R (n - i) ·r y^R i)) n =
Σe R (λi. (n choose i) ×R (x^R (n - i) ·r y^R (i + 1))) n
lemma binomial_n0:
Suc n choose 0 = n choose 0
lemma binomial_ngt_diff:
n choose Suc n = Suc n choose Suc n - (n choose n)
lemma binomial_ngt_0:
n choose Suc n = 0
lemma diffLessSuc:
m ≤ n ==> Suc (n - m) = Suc n - m
lemma npow_suc_i:
[| x ∈ carrier R; i ≤ n |] ==> x^R (Suc n - i) = x ·r x^R (n - i)
lemma npeSum2_sub_mulx:
[| x ∈ carrier R; y ∈ carrier R |]
==> x ·r Σe R (λi. (n choose i) ×R (x^R (n - i) ·r y^R i)) n =
Σe R (λi. (n choose Suc i) ×R (x^R (Suc n - Suc i) ·r y^R Suc i)) n ±
(Suc n choose 0) ×R (x^R (Suc n - 0) ·r y^R 0)
lemma npeSum2_sub_mulx2:
[| x ∈ carrier R; y ∈ carrier R |]
==> x ·r Σe R (λi. (n choose i) ×R (x^R (n - i) ·r y^R i)) n =
Σe R (λi. (n choose Suc i) ×R (x^R (n - i) ·r (y^R i ·r y))) n ±
(\<zero> ± x ·r x^R n ·r 1r)
lemma npeSum2:
[| x ∈ carrier R; y ∈ carrier R |]
==> (x ± y)^R n = Σe R (λi. (n choose i) ×R (x^R (n - i) ·r y^R i)) n
lemma nsum_zeroTr:
(∀i≤n. f i = \<zero>) --> Σe A f n = \<zero>
lemma npAdd:
[| x ∈ carrier R; y ∈ carrier R; x^R m = \<zero>; y^R n = \<zero> |]
==> (x ± y)^R (m + n) = \<zero>
lemma npInverse:
x ∈ carrier R ==> (-a x)^R n = x^R n ∨ (-a x)^R n = -a x^R n
lemma npMul:
[| x ∈ carrier R; y ∈ carrier R |] ==> (x ·r y)^R n = x^R n ·r y^R n
lemma ridmap_surjec:
Ring A ==> surjecA,A ridmap A
lemma rHom_aHom:
f ∈ rHom A R ==> f ∈ aHom A R
lemma rimg_carrier:
f ∈ rHom A R ==> carrier (rimg A R f) = f ` carrier A
lemma rHom_mem:
[| f ∈ rHom A R; a ∈ carrier A |] ==> f a ∈ carrier R
lemma rHom_func:
f ∈ rHom A R ==> f ∈ carrier A -> carrier R
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
lemma rHom_inv_inv:
[| Ring A; Ring R; x ∈ carrier A; f ∈ rHom A R |] ==> f (-aA x) = -aR f x
lemma rHom_0_0:
[| Ring A; Ring R; f ∈ rHom A R |] ==> f \<zero>A = \<zero>R
lemma rHom_tOp:
[| Ring A; Ring R; x ∈ carrier A; y ∈ carrier A; f ∈ rHom A R |]
==> f (x ·rA y) = f x ·rR f y
lemma rHom_add:
[| f ∈ rHom A R; x ∈ carrier A; y ∈ carrier A |] ==> f (x ±A y) = f x ±R f y
lemma rHom_one:
[| Ring A; Ring R; f ∈ rHom A R |] ==> f 1rA = 1rR
lemma rHom_npow:
[| Ring A; Ring R; x ∈ carrier A; f ∈ rHom A R |] ==> f (x^A n) = f x^R n
lemma rHom_compos:
[| Ring A; Ring B; Ring C; f ∈ rHom A B; g ∈ rHom B C |]
==> compos A g f ∈ rHom A C
lemma rimg_ag:
[| Ring A; Ring R; f ∈ rHom A R |] ==> aGroup (rimg A R f)
lemma rimg_ring:
[| Ring A; Ring R; f ∈ rHom A R |] ==> Ring (rimg A R f)
lemma ideal_asubg:
ideal R I ==> @ASubG R I
lemma ideal_pOp_closed:
[| ideal R I; x ∈ I; y ∈ I |] ==> x ± y ∈ I
lemma ideal_nsum_closedTr:
ideal R I ==> (∀j≤n. f j ∈ I) --> Σe R f n ∈ I
lemma ideal_nsum_closed:
[| ideal R I; ∀j≤n. f j ∈ I |] ==> Σe R f n ∈ I
lemma ideal_subset1:
ideal R I ==> I ⊆ carrier R
lemma ideal_subset:
[| ideal R I; h ∈ I |] ==> h ∈ carrier R
lemma ideal_ring_multiple:
[| ideal R I; x ∈ I; r ∈ carrier R |] ==> r ·r x ∈ I
lemma ideal_ring_multiple1:
[| ideal R I; x ∈ I; r ∈ carrier R |] ==> x ·r r ∈ I
lemma ideal_npow_closedTr:
[| ideal R I; x ∈ I |] ==> 0 < n --> x^R n ∈ I
lemma ideal_npow_closed:
[| ideal R I; x ∈ I; 0 < n |] ==> x^R n ∈ I
lemma 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
lemma ideal_inv1_closed:
[| ideal R I; x ∈ I |] ==> -a x ∈ I
lemma ideal_zero:
ideal R I ==> \<zero> ∈ I
lemma ideal_zero_forall:
∀I. ideal R I --> \<zero> ∈ I
lemma ideal_ele_sumTr1:
[| ideal R I; a ∈ carrier R; b ∈ carrier R; a ± b ∈ I; a ∈ I |] ==> b ∈ I
lemma ideal_ele_sumTr2:
[| ideal R I; a ∈ carrier R; b ∈ carrier R; a ± b ∈ I; b ∈ I |] ==> a ∈ I
lemma 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
lemma 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
lemma zero_ideal:
ideal R {\<zero>}
lemma whole_ideal:
ideal R (carrier R)
lemma ideal_inc_one:
[| ideal R I; 1r ∈ I |] ==> I = carrier R
lemma ideal_inc_one1:
ideal R I ==> (1r ∈ I) = (I = carrier R)
lemma ideal_inc_unit:
[| ideal R I; a ∈ I; Unit R a |] ==> 1r ∈ I
lemma proper_ideal:
[| ideal R I; 1r ∉ I |] ==> I ≠ carrier R
lemma ideal_inc_unit1:
[| a ∈ carrier R; Unit R a; ideal R I; a ∈ I |] ==> I = carrier R
lemma int_ideal:
[| ideal R I; ideal R J |] ==> ideal R (I ∩ J)
lemma set_sum_mem:
[| a ∈ I; b ∈ J; I ⊆ carrier R; J ⊆ carrier R |] ==> a ± b ∈ I \<minusplus> J
lemma sum_ideals:
[| ideal R I1.0; ideal R I2.0 |] ==> ideal R (I1.0 \<minusplus> I2.0)
lemma sum_ideals_la1:
[| ideal R I1.0; ideal R I2.0 |] ==> I1.0 ⊆ I1.0 \<minusplus> I2.0
lemma sum_ideals_la2:
[| ideal R I1.0; ideal R I2.0 |] ==> I2.0 ⊆ I1.0 \<minusplus> I2.0
lemma sum_ideals_cont:
[| ideal R I; A ⊆ I; B ⊆ I |] ==> A \<minusplus> B ⊆ I
lemma ideals_set_sum:
[| ideal R A; ideal R B; x ∈ A \<minusplus> B |] ==> ∃h∈A. ∃k∈B. x = h ± k
lemma a_in_principal:
a ∈ carrier R ==> a ∈ R ♦p a
lemma principal_ideal:
a ∈ carrier R ==> ideal R (R ♦p a)
lemma rxa_in_Rxa:
[| a ∈ carrier R; r ∈ carrier R |] ==> r ·r a ∈ R ♦p a
lemma Rxa_one:
R ♦p 1r = carrier R
lemma Rxa_zero:
R ♦p \<zero> = {\<zero>}
lemma Rxa_nonzero:
[| a ∈ carrier R; a ≠ \<zero> |] ==> R ♦p a ≠ {\<zero>}
lemma ideal_cont_Rxa:
[| ideal R I; a ∈ I |] ==> R ♦p a ⊆ I
lemma Rxa_mult_smaller:
[| a ∈ carrier R; b ∈ carrier R |] ==> R ♦p (a ·r b) ⊆ R ♦p b
lemma id_ideal_psub_sum:
[| ideal R I; a ∈ carrier R; a ∉ I |] ==> I ⊂ I \<minusplus> R ♦p a
lemma mul_two_principal_idealsTr:
[| a ∈ carrier R; b ∈ carrier R; x ∈ R ♦p a; y ∈ R ♦p b |]
==> ∃r∈carrier R. x ·r y = r ·r (a ·r b)
lemma restrictfun_Nset:
f ∈ {i. i ≤ Suc n} -> carrier R ==> f ∈ {i. i ≤ n} -> carrier R
lemma sum_of_prideals0:
∀f. (∀l≤n. f l ∈ carrier R) --> ideal R (sum_pr_ideals R f n)
lemma sum_of_prideals:
∀l≤n. f l ∈ carrier R ==> ideal R (sum_pr_ideals R f n)
lemma sum_of_prideals1:
∀f. (∀l≤n. f l ∈ carrier R) --> f ` {i. i ≤ n} ⊆ sum_pr_ideals R f n
lemma sum_of_prideals2:
∀l≤n. f l ∈ carrier R ==> f ` {i. i ≤ n} ⊆ sum_pr_ideals R f n
lemma 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
lemma sum_of_prideals4:
[| ideal R I; ∀l≤n. f l ∈ carrier R; f ` {i. i ≤ n} ⊆ I |]
==> sum_pr_ideals R f n ⊆ I
lemma ker_ideal:
[| Ring A; Ring R; f ∈ rHom A R |] ==> ideal A (kerA,R f)
lemma ring_of_integers:
Ring Zr
lemma Zr_zero:
\<zero>Zr = 0
lemma Zr_one:
1rZr = 1
lemma Zr_minus:
-aZr n = - n
lemma Zr_add:
n ±Zr m = n + m
lemma Zr_times:
n ·rZr m = n * m
lemma Zr_gen_Zleast:
[| ideal Zr I; I ≠ {0} |] ==> Zr ♦p lev I = I
lemma Zr_pir:
ideal Zr I ==> ∃n. Zr ♦p n = I
lemma mem_set_ar_cos:
[| ideal R I; a ∈ carrier R |] ==> a \<uplus>R I ∈ set_ar_cos R I
lemma I_in_set_ar_cos:
ideal R I ==> I ∈ set_ar_cos R I
lemma ar_coset_same1:
[| ideal R I; a ∈ carrier R; b ∈ carrier R; b ± -a a ∈ I |]
==> a \<uplus>R I = b \<uplus>R I
lemma ar_coset_same2:
[| ideal R I; a ∈ carrier R; b ∈ carrier R; a \<uplus>R I = b \<uplus>R I |]
==> b ± -a a ∈ I
lemma ar_coset_same3:
[| ideal R I; a ∈ carrier R; a \<uplus>R I = I |] ==> a ∈ I
lemma ar_coset_same3_1:
[| ideal R I; a ∈ carrier R; a ∉ I |] ==> a \<uplus>R I ≠ I
lemma ar_coset_same4:
[| ideal R I; a ∈ I |] ==> a \<uplus>R I = I
lemma ar_coset_same4_1:
[| ideal R I; a \<uplus>R I ≠ I |] ==> a ∉ I
lemma belong_ar_coset1:
[| ideal R I; a ∈ carrier R; x ∈ carrier R; x ± -a a ∈ I |]
==> x ∈ a \<uplus>R I
lemma a_in_ar_coset:
[| ideal R I; a ∈ carrier R |] ==> a ∈ a \<uplus>R I
lemma ar_coset_subsetD:
[| ideal R I; a ∈ carrier R; x ∈ a \<uplus>R I |] ==> x ∈ carrier R
lemma ar_cos_mem:
[| ideal R I; a ∈ carrier R |] ==> a \<uplus>R I ∈ set_rcs (b_ag R) I
lemma mem_ar_coset1:
[| ideal R I; a ∈ carrier R; x ∈ a \<uplus>R I |] ==> ∃h∈I. h ± a = x
lemma ar_coset_mem2:
[| ideal R I; a ∈ carrier R; x ∈ a \<uplus>R I |] ==> ∃h∈I. x = a ± h
lemma belong_ar_coset2:
[| ideal R I; a ∈ carrier R; x ∈ a \<uplus>R I |] ==> x ± -a a ∈ I
lemma ar_c_top:
[| ideal R I; a ∈ carrier R; b ∈ carrier R |]
==> c_top (b_ag R) I (a \<uplus>R I) (b \<uplus>R I) = (a ± b) \<uplus>R I
lemma quotient_ring_tr1:
[| ideal R I; a1.0 ∈ carrier R; a2.0 ∈ carrier R; b1.0 ∈ carrier R;
b2.0 ∈ carrier R; a1.0 \<uplus>R I = a2.0 \<uplus>R I;
b1.0 \<uplus>R I = b2.0 \<uplus>R I |]
==> a1.0 ·r b1.0 \<uplus>R I = a2.0 ·r b2.0 \<uplus>R I
lemma rcostOp:
[| ideal R I; a ∈ carrier R; b ∈ carrier R |]
==> rcostOp R I (a \<uplus>R I) (b \<uplus>R I) = a ·r b \<uplus>R I
lemma carrier_qring:
ideal R I ==> carrier (R /r I) = set_rcs (b_ag R) I
lemma carrier_qring1:
ideal R I ==> carrier (R /r I) = set_ar_cos R I
lemma qring_ring:
ideal R I ==> Ring (R /r I)
lemma qring_carrier:
ideal R I ==> carrier (R /r I) = {X. ∃a∈carrier R. a \<uplus>R I = X}
lemma qring_mem:
[| ideal R I; a ∈ carrier R |] ==> a \<uplus>R I ∈ carrier (R /r I)
lemma qring_pOp:
[| ideal R I; a ∈ carrier R; b ∈ carrier R |]
==> a \<uplus>R I ±R /r I b \<uplus>R I = (a ± b) \<uplus>R I
lemma qring_zero:
ideal R I ==> \<zero>R /r I = I
lemma qring_zero_1:
[| a ∈ carrier R; ideal R I; a \<uplus>R I = I |] ==> a ∈ I
lemma Qring_fix1:
[| a ∈ carrier R; ideal R I; a ∈ I |] ==> a \<uplus>R I = I
lemma ar_cos_same:
[| a ∈ carrier R; ideal R I; x ∈ a \<uplus>R I |]
==> x \<uplus>R I = a \<uplus>R I
lemma qring_tOp:
[| ideal R I; a ∈ carrier R; b ∈ carrier R |]
==> (a \<uplus>R I) ·rR /r I (b \<uplus>R I) = a ·r b \<uplus>R I
lemma rind_hom_well_def:
[| Ring A; Ring R; f ∈ rHom A R; a ∈ carrier A |]
==> f a = (f°A,R) (a \<uplus>A kerA,R f)
lemma set_r_ar_cos:
ideal R I ==> set_rcs (b_ag R) I = set_ar_cos R I
lemma set_r_ar_cos_ker:
[| Ring A; Ring R; f ∈ rHom A R |]
==> set_rcs (b_ag A) (kerA,R f) = set_ar_cos A (kerA,R f)
lemma ind_hom_rhom:
[| Ring A; Ring R; f ∈ rHom A R |] ==> f°A,R ∈ rHom (A /r (kerA,R f)) R
lemma ind_hom_injec:
[| Ring A; Ring R; f ∈ rHom A R |] ==> injecA /r (kerA,R f),R (f°A,R)
lemma rhom_to_rimg:
[| Ring A; Ring R; f ∈ rHom A R |] ==> f ∈ rHom A (rimg A R f)
lemma ker_to_rimg:
[| Ring A; Ring R; f ∈ rHom A R |] ==> kerA,R f = kerA,rimg A R f f
lemma indhom_eq:
[| Ring A; Ring R; f ∈ rHom A R |] ==> f°A,rimg A R f = f°A,R
lemma indhom_bijec2_rimg:
[| Ring A; Ring R; f ∈ rHom A R |] ==> bijecA /r (kerA,R f),rimg A R f (f°A,R)
lemma surjec_ind_bijec:
[| Ring A; Ring R; f ∈ rHom A R; surjecA,R f |]
==> bijecA /r (kerA,R f),R (f°A,R)
lemma ridmap_ind_bijec:
Ring A ==> bijecA /r (kerA,A ridmap A),A (ridmap A°A,A)
lemma ker_of_idmap:
Ring A ==> kerA,A ridmap A = {\<zero>A}
lemma ring_natural_isom:
Ring A ==> bijecA /r {\<zero>A},A (ridmap A°A,A)
lemma pj_Hom:
[| Ring R; ideal R I |] ==> pj R I ∈ rHom R (R /r I)
lemma pj_mem:
[| Ring R; ideal R I; x ∈ carrier R |] ==> pj R I x = x \<uplus>R I
lemma pj_zero:
[| Ring R; ideal R I; x ∈ carrier R |] ==> (pj R I x = \<zero>R /r I) = (x ∈ I)
lemma pj_surj_to:
[| Ring R; ideal R J; X ∈ carrier (R /r J) |] ==> ∃r∈carrier R. pj R J r = X
lemma invim_of_ideal:
[| Ring R; ideal R I; ideal (R /r I) J |]
==> ideal R (rInvim R (R /r I) (pj R I) J)
lemma pj_invim_cont_I:
[| Ring R; ideal R I; ideal (R /r I) J |] ==> I ⊆ rInvim R (R /r I) (pj R I) J
lemma pj_invim_mono1:
[| Ring R; ideal R I; ideal (R /r I) J1.0; ideal (R /r I) J2.0; J1.0 ⊆ J2.0 |]
==> rInvim R (R /r I) (pj R I) J1.0 ⊆ rInvim R (R /r I) (pj R I) J2.0
lemma pj_img_ideal:
[| Ring R; ideal R I; ideal R J; I ⊆ J |] ==> ideal (R /r I) (pj R I ` J)
lemma npQring:
[| Ring R; ideal R I; a ∈ carrier R |]
==> (a \<uplus>R I)^R /r I n = a^R n \<uplus>R I
lemma maximal_ideal_ideal:
maximal_ideal R mx ==> ideal R mx
lemma maximal_ideal_proper:
maximal_ideal R mx ==> 1r ∉ mx
lemma prime_ideal_ideal:
prime_ideal R I ==> ideal R I
lemma prime_ideal_proper:
prime_ideal R I ==> I ≠ carrier R
lemma prime_ideal_proper1:
prime_ideal R p ==> 1r ∉ p
lemma primary_ideal_ideal:
primary_ideal R q ==> ideal R q
lemma primary_ideal_proper1:
primary_ideal R q ==> 1r ∉ q
lemma prime_elems_mult_not:
[| prime_ideal R P; x ∈ carrier R; y ∈ carrier R; x ∉ P; y ∉ P |] ==> x ·r y ∉ P
lemma prime_is_primary:
prime_ideal R p ==> primary_ideal R p
lemma maximal_prime_Tr0:
[| maximal_ideal R mx; x ∈ carrier R; x ∉ mx |]
==> mx \<minusplus> R ♦p x = carrier R
lemma maximal_is_prime:
maximal_ideal R mx ==> prime_ideal R mx
lemma chain_un:
[| c ∈ chain {I. ideal R I ∧ I ⊂ carrier R}; c ≠ {} |] ==> ideal R (Union c)
lemma zeroring_no_maximal:
zeroring R ==> ¬ (∃I. maximal_ideal R I)
lemma id_maximal_Exist:
¬ zeroring R ==> ∃I. maximal_ideal R I
lemma ideal_Int_ideal:
[| S ⊆ {I. ideal R I}; S ≠ {} |] ==> ideal R (Inter S)
lemma 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 = Inter S
lemma prod_mem_prod_ideals:
[| ideal R I; ideal R J; i ∈ I; j ∈ J |] ==> i ·r j ∈ I ♦r J
lemma ideal_prod_ideal:
[| ideal R I; ideal R J |] ==> ideal R (I ♦r J)
lemma ideal_prod_commute:
[| ideal R I; ideal R J |] ==> I ♦r J = J ♦r I
lemma ideal_prod_subTr:
[| ideal R I; ideal R J; ideal R C; ∀i∈I. ∀j∈J. i ·r j ∈ C |] ==> I ♦r J ⊆ C
lemma n_prod_idealTr:
(∀k≤n. ideal R (J k)) --> ideal R (iΠR,n J)
lemma n_prod_ideal:
∀k≤n. ideal R (J k) ==> ideal R (iΠR,n J)
lemma ideal_prod_la1:
[| ideal R I; ideal R J |] ==> I ♦r J ⊆ I
lemma ideal_prod_el1:
[| ideal R I; ideal R J; a ∈ I ♦r J |] ==> a ∈ I
lemma ideal_prod_la2:
[| ideal R I; ideal R J |] ==> I ♦r J ⊆ J
lemma ideal_prod_sub_Int:
[| ideal R I; ideal R J |] ==> I ♦r J ⊆ I ∩ J
lemma ideal_prod_el2:
[| ideal R I; ideal R J; a ∈ I ♦r J |] ==> a ∈ J
lemma ele_n_prodTr0:
[| ∀k≤Suc n. ideal R (J k); a ∈ iΠR,Suc n J |] ==> a ∈ iΠR,n J ∧ a ∈ J (Suc n)
lemma ele_n_prodTr1:
(∀k≤n. ideal R (J k)) ∧ a ∈ iΠR,n J --> (∀k≤n. a ∈ J k)
lemma ele_n_prod:
[| ∀k≤n. ideal R (J k); a ∈ iΠR,n J |] ==> ∀k≤n. a ∈ J k
lemma idealprod_whole_l:
ideal R I ==> carrier R ♦r I = I
lemma idealprod_whole_r:
ideal R I ==> I ♦r carrier R = I
lemma idealpow_1_self:
ideal R I ==> I ♦R Suc 0 = I
lemma ideal_pow_ideal:
ideal R I ==> ideal R (I ♦R n)
lemma ideal_prod_prime:
[| ideal R I; ideal R J; prime_ideal R P; I ♦r J ⊆ P |] ==> I ⊆ P ∨ J ⊆ P
lemma ideal_n_prod_primeTr:
prime_ideal R P ==> (∀k≤n. ideal R (J k)) --> iΠR,n J ⊆ P --> (∃i≤n. J i ⊆ P)
lemma ideal_n_prod_prime:
[| prime_ideal R P; ∀k≤n. ideal R (J k); iΠR,n J ⊆ P |] ==> ∃i≤n. J i ⊆ P
lemma prod_primeTr:
[| prime_ideal R P; ideal R A; ¬ A ⊆ P; ideal R B; ¬ B ⊆ P |]
==> ∃x. x ∈ A ∧ x ∈ B ∧ x ∉ P
lemma 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
lemma 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
lemma nsum_memrTr:
(∀i≤n. f i ∈ carrier R) --> (∀l≤n. Σe R f l ∈ carrier R)
lemma nsum_memr:
∀i≤n. f i ∈ carrier R ==> ∀l≤n. Σe R f l ∈ carrier R
lemma nsum_ideal_incTr:
ideal R A ==> (∀i≤n. f i ∈ A) --> Σe R f n ∈ A
lemma nsum_ideal_inc:
[| ideal R A; ∀i≤n. f i ∈ A |] ==> Σe R f n ∈ A
lemma 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) -->
Σe R f n ∉ A
lemma 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 |]
==> Σe R f n ∉ A
lemma nprod_memTr:
(∀i≤n. f i ∈ carrier R) --> (∀l≤n. eΠR,l f ∈ carrier R)
lemma nprod_mem:
[| ∀i≤n. f i ∈ carrier R; l ≤ n |] ==> eΠR,l f ∈ carrier R
lemma ideal_nprod_incTr:
ideal R A ==> (∀i≤n. f i ∈ carrier R) ∧ (∃l≤n. f l ∈ A) --> eΠR,n f ∈ A
lemma ideal_nprod_inc:
[| ideal R A; ∀i≤n. f i ∈ carrier R; ∃l≤n. f l ∈ A |] ==> eΠR,n f ∈ A
lemma nprod_excTr:
prime_ideal R P ==> (∀i≤n. f i ∈ carrier R) ∧ (∀l≤n. f l ∉ P) --> eΠR,n f ∉ P
lemma prime_nprod_exc:
[| prime_ideal R P; ∀i≤n. f i ∈ carrier R; ∀l≤n. f l ∉ P |] ==> eΠR,n f ∉ P
lemma id_nilrad_ideal:
ideal R (nilrad R)
lemma id_rad_invim:
ideal R I ==> rad_ideal R I = rInvim R (R /r I) (pj R I) (nilrad (R /r I))
lemma id_rad_ideal:
ideal R I ==> ideal R (rad_ideal R I)
lemma id_rad_cont_I:
ideal R I ==> I ⊆ rad_ideal R I
lemma id_rad_set:
ideal R I ==> rad_ideal R I = {x : carrier R. ∃n. x^R n ∈ I}
lemma rad_primary_prime:
primary_ideal R q ==> prime_ideal R (rad_ideal R q)
lemma npow_notin_prime:
[| prime_ideal R P; x ∈ carrier R; x ∉ P |] ==> ∀n. x^R n ∉ P
lemma npow_in_prime:
[| prime_ideal R P; x ∈ carrier R; ∃n. x^R n ∈ P |] ==> x ∈ P
lemma mul_closed_set_sub:
mul_closed_set R S ==> S ⊆ carrier R
lemma mul_closed_set_tOp_closed:
[| mul_closed_set R S; s ∈ S; t ∈ S |] ==> s ·r t ∈ S
lemma f_inv_unique:
[| x ∈ carrier K - {\<zero>}; x' ∈ carrier K; x'' ∈ carrier K; x' ·r x = 1r;
x'' ·r x = 1r |]
==> x' = x''
lemma invf_inv:
x ∈ carrier K - {\<zero>} ==> invf K x ∈ carrier K ∧ invf K x ·r x = 1r
lemma idom_is_ring:
Ring R
lemma idom_tOp_nonzeros:
[| x ∈ carrier R; y ∈ carrier R; x ≠ \<zero>; y ≠ \<zero> |]
==> x ·r y ≠ \<zero>
lemma idom_potent_nonzero:
[| x ∈ carrier R; x ≠ \<zero> |] ==> x^R n ≠ \<zero>
lemma idom_potent_unit:
[| a ∈ carrier R; 0 < n |] ==> Unit R a = Unit R (a^R n)
lemma idom_mult_cancel_r:
[| a ∈ carrier R; b ∈ carrier R; c ∈ carrier R; c ≠ \<zero>; a ·r c = b ·r c |]
==> a = b
lemma idom_mult_cancel_l:
[| a ∈ carrier R; b ∈ carrier R; c ∈ carrier R; c ≠ \<zero>; c ·r a = c ·r b |]
==> a = b
lemma invf_closed1:
x ∈ carrier K - {\<zero>} ==> x K ∈ carrier K - {\<zero>}
lemma linvf:
x ∈ carrier K - {\<zero>} ==> x K ·r x = 1r
lemma field_is_ring:
Ring K
lemma invf_one:
1r ≠ \<zero> ==> 1r K = 1r
lemma field_tOp_assoc:
[| x ∈ carrier K; y ∈ carrier K; z ∈ carrier K |]
==> x ·r y ·r z = x ·r (y ·r z)
lemma field_tOp_commute:
[| x ∈ carrier K; y ∈ carrier K |] ==> x ·r y = y ·r x
lemma field_inv_inv:
[| x ∈ carrier K; x ≠ \<zero> |] ==> x K K = x
lemma field_is_idom:
Idomain K
lemma field_potent_nonzero:
[| x ∈ carrier K; x ≠ \<zero> |] ==> x^K n ≠ \<zero>
lemma field_potent_nonzero1:
[| x ∈ carrier K; x ≠ \<zero> |] ==> xKn ≠ \<zero>
lemma field_nilp_zero:
[| x ∈ carrier K; x^K n = \<zero> |] ==> x = \<zero>
lemma npowf_mem:
[| a ∈ carrier K; a ≠ \<zero> |] ==> aKn ∈ carrier K
lemma field_npowf_exp_zero:
[| a ∈ carrier K; a ≠ \<zero> |] ==> aK0 = 1r
lemma npow_exp_minusTr1:
[| x ∈ carrier K; x ≠ \<zero>; 0 ≤ i |]
==> 0 ≤ i - int j --> xK(i - int j) = x^K nat i ·r x K^K j
lemma npow_exp_minusTr2:
[| x ∈ carrier K; x ≠ \<zero>; 0 ≤ i; 0 ≤ j; 0 ≤ i - j |]
==> xK(i - j) = x^K nat i ·r x K^K nat j
lemma npowf_inv:
[| x ∈ carrier K; x ≠ \<zero>; 0 ≤ j |] ==> xKj = x KK- j
lemma npowf_inv1:
[| x ∈ carrier K; x ≠ \<zero>; ¬ 0 ≤ j |] ==> xKj = x KK- j
lemma npowf_inverse:
[| x ∈ carrier K; x ≠ \<zero> |] ==> xKj = x KK- j
lemma npowf_expTr1:
[| x ∈ carrier K; x ≠ \<zero>; 0 ≤ i; 0 ≤ j; 0 ≤ i - j |]
==> xK(i - j) = xKi ·r xK- j
lemma npowf_expTr2:
[| x ∈ carrier K; x ≠ \<zero>; 0 ≤ i + j |] ==> xK(i + j) = xKi ·r xKj
lemma npowf_exp_add:
[| x ∈ carrier K; x ≠ \<zero> |] ==> xK(i + j) = xKi ·r xKj
lemma npowf_exp_1_add:
[| x ∈ carrier K; x ≠ \<zero> |] ==> xK(1 + j) = x ·r xKj
lemma npowf_minus:
[| x ∈ carrier K; x ≠ \<zero> |] ==> (xKj) K = xK- j
lemma residue_fieldTr:
[| maximal_ideal R mx; x ∈ carrier (R /r mx); x ≠ \<zero>R /r mx |]
==> ∃y∈carrier (R /r mx). y ·rR /r mx x = 1rR /r mx
lemma residue_field_cd:
maximal_ideal R mx ==> Corps (R /r mx)
lemma maximal_set_idealTr:
maximal_set {I. ideal R I ∧ S ∩ I = {}} mx ==> ideal R mx
lemma maximal_setTr:
[| maximal_set {I. ideal R I ∧ S ∩ I = {}} mx; ideal R J; mx ⊂ J |]
==> S ∩ J ≠ {}
lemma mulDisj:
[| mul_closed_set R S; 1r ∈ S; \<zero> ∉ S; T = {I. ideal R I ∧ S ∩ I = {}};
maximal_set T mx |]
==> prime_ideal R mx
lemma ex_mulDisj_maximal:
[| mul_closed_set R S; \<zero> ∉ S; 1r ∈ S; T = {I. ideal R I ∧ S ∩ I = {}} |]
==> ∃mx. maximal_set T mx
lemma ex_mulDisj_prime:
[| mul_closed_set R S; \<zero> ∉ S; 1r ∈ S |]
==> ∃mx. prime_ideal R mx ∧ S ∩ mx = {}
lemma nilradTr1:
¬ zeroring R ==> nilrad R = Inter {p. prime_ideal R p}
lemma nonilp_residue_nilrad:
[| ¬ zeroring R; x ∈ carrier R;
nilpotent (R /r nilrad R) (x \<uplus>R nilrad R) |]
==> x \<uplus>R nilrad R = \<zero>R /r nilrad R
lemma ex_contid_maximal:
[| S = {1r}; \<zero> ∉ S; ideal R I; I ∩ S = {};
T = {J. ideal R J ∧ S ∩ J = {} ∧ I ⊆ J} |]
==> ∃mx. maximal_set T mx
lemma contid_maximal:
[| S = {1r}; \<zero> ∉ S; ideal R I; I ∩ S = {};
T = {J. ideal R J ∧ S ∩ J = {} ∧ I ⊆ J}; maximal_set T mx |]
==> maximal_ideal R mx
lemma ideal_contained_maxid:
[| ¬ zeroring R; ideal R I; 1r ∉ I |] ==> ∃mx. maximal_ideal R mx ∧ I ⊆ mx
lemma nonunit_principal_id:
[| a ∈ carrier R; ¬ Unit R a |] ==> R ♦p a ≠ carrier R
lemma nonunit_contained_maxid:
[| ¬ zeroring R; a ∈ carrier R; ¬ Unit R a |]
==> ∃mx. maximal_ideal R mx ∧ a ∈ mx
lemma 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
lemma localring_unit:
[| ¬ zeroring R; maximal_ideal R mx; ∀x. x ∈ mx --> Unit R (x ± 1r) |]
==> local_ring R
lemma zeroring_J_rad_empty:
zeroring R ==> J_rad R = carrier R
lemma J_rad_mem:
x ∈ J_rad R ==> x ∈ carrier R
lemma J_rad_unit:
[| ¬ zeroring R; x ∈ J_rad R |]
==> ∀y. y ∈ carrier R --> Unit R (1r ± (-a x) ·r y)