# Theory Field_ZF

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section ‹Fields - introduction›

theory Field_ZF imports Ring_ZF

begin

text‹This theory covers basic facts about fields.›

subsection‹Definition and basic properties›

text‹In this section we define what is a field and list the basic properties
of fields.›

text‹Field is a notrivial commutative ring such that all
non-zero elements have an inverse. We define the notion of being a field as
a statement about three sets. The first set, denoted ‹K› is the
carrier of the field. The second set, denoted ‹A› represents the
additive operation on ‹K› (recall that in ZF set theory functions
are sets). The third set ‹M› represents the multiplicative operation
on ‹K›.›

definition
"IsAfield(K,A,M) ≡
(IsAring(K,A,M) ∧ (M {is commutative on} K) ∧
TheNeutralElement(K,A) ≠ TheNeutralElement(K,M) ∧
(∀a∈K. a≠TheNeutralElement(K,A)⟶
(∃b∈K. M⟨a,b⟩ = TheNeutralElement(K,M))))"

text‹The ‹field0› context extends the ‹ring0›
context adding field-related assumptions and notation related to the
multiplicative inverse.›

locale field0 = ring0 K A M for K A M +
assumes mult_commute: "M {is commutative on} K"

assumes not_triv: "𝟬 ≠ 𝟭"

assumes inv_exists: "∀a∈K. a≠𝟬 ⟶ (∃b∈K. a⋅b = 𝟭)"

fixes non_zero ("K⇩0")
defines non_zero_def[simp]: "K⇩0 ≡ K-{𝟬}"

fixes inv ("_¯ "  97)
defines inv_def[simp]: "a¯ ≡ GroupInv(K⇩0,restrict(M,K⇩0×K⇩0))(a)"

text‹The next lemma assures us that we are talking fields
in the ‹field0› context.›

lemma (in field0) Field_ZF_1_L1: shows "IsAfield(K,A,M)"
using ringAssum mult_commute not_triv inv_exists IsAfield_def
by simp

text‹We can use theorems proven in the ‹field0› context whenever we

lemma field_field0: assumes "IsAfield(K,A,M)"
shows "field0(K,A,M)"
using assms IsAfield_def field0_axioms.intro ring0_def field0_def
by simp

text‹Let's have an explicit statement that the multiplication
in fields is commutative.›

lemma (in field0) field_mult_comm: assumes "a∈K"  "b∈K"
shows "a⋅b = b⋅a"
using mult_commute assms IsCommutative_def by simp

text‹Fields do not have zero divisors.›

lemma (in field0) field_has_no_zero_divs: shows "HasNoZeroDivs(K,A,M)"
proof -
{ fix a b assume A1: "a∈K"  "b∈K" and A2: "a⋅b = 𝟬"  and A3: "b≠𝟬"
from inv_exists A1 A3 obtain c where I: "c∈K" and II: "b⋅c = 𝟭"
by auto
from A2 have "a⋅b⋅c = 𝟬⋅c" by simp
with A1 I have "a⋅(b⋅c) = 𝟬"
using Ring_ZF_1_L11 Ring_ZF_1_L6 by simp
with A1 II have "a=𝟬 "using Ring_ZF_1_L3 by simp }
then have "∀a∈K.∀b∈K. a⋅b = 𝟬 ⟶ a=𝟬 ∨ b=𝟬" by auto
then show ?thesis using HasNoZeroDivs_def by auto
qed

text‹$K_0$ (the set of nonzero field elements is closed with respect
to multiplication.›

lemma (in field0) Field_ZF_1_L2:
shows "K⇩0 {is closed under} M"
using Ring_ZF_1_L4 field_has_no_zero_divs Ring_ZF_1_L12
IsOpClosed_def by auto

text‹Any nonzero element has a right inverse that is nonzero.›

lemma (in field0) Field_ZF_1_L3: assumes A1: "a∈K⇩0"
shows "∃b∈K⇩0. a⋅b = 𝟭"
proof -
from inv_exists A1 obtain b where "b∈K" and "a⋅b = 𝟭"
by auto
with not_triv A1 show "∃b∈K⇩0. a⋅b = 𝟭"
using Ring_ZF_1_L6 by auto
qed

text‹If we remove zero, the field with multiplication
becomes a group and we can use all theorems proven in
‹group0› context.›

theorem (in field0) Field_ZF_1_L4: shows
"IsAgroup(K⇩0,restrict(M,K⇩0×K⇩0))"
"group0(K⇩0,restrict(M,K⇩0×K⇩0))"
"𝟭 = TheNeutralElement(K⇩0,restrict(M,K⇩0×K⇩0))"
proof-
let ?f = "restrict(M,K⇩0×K⇩0)"
have
"M {is associative on} K"
"K⇩0 ⊆ K"  "K⇩0 {is closed under} M"
using Field_ZF_1_L1 IsAfield_def IsAring_def IsAgroup_def
IsAmonoid_def Field_ZF_1_L2 by auto
then have "?f {is associative on} K⇩0"
using func_ZF_4_L3 by simp
moreover
from not_triv have
I: "𝟭∈K⇩0 ∧ (∀a∈K⇩0. ?f⟨𝟭,a⟩ = a ∧  ?f⟨a,𝟭⟩ = a)"
using Ring_ZF_1_L2 Ring_ZF_1_L3 by auto
then have "∃n∈K⇩0. ∀a∈K⇩0. ?f⟨n,a⟩ = a ∧  ?f⟨a,n⟩ = a"
by blast
ultimately have II: "IsAmonoid(K⇩0,?f)" using IsAmonoid_def
by simp
then have "monoid0(K⇩0,?f)" using monoid0_def by simp
moreover note I
ultimately show "𝟭 = TheNeutralElement(K⇩0,?f)"
by (rule monoid0.group0_1_L4)
then have "∀a∈K⇩0.∃b∈K⇩0. ?f⟨a,b⟩ =  TheNeutralElement(K⇩0,?f)"
using Field_ZF_1_L3 by auto
with II show "IsAgroup(K⇩0,?f)" by (rule definition_of_group)
then show "group0(K⇩0,?f)" using group0_def by simp
qed

text‹The inverse of a nonzero field element is nonzero.›

lemma (in field0) Field_ZF_1_L5: assumes A1: "a∈K"  "a≠𝟬"
shows "a¯ ∈ K⇩0"  "(a¯)⇧2 ∈ K⇩0"   "a¯ ∈ K"  "a¯ ≠ 𝟬"
proof -
from A1 have "a ∈ K⇩0" by simp
then show "a¯ ∈ K⇩0" using Field_ZF_1_L4 group0.inverse_in_group
by auto
then show  "(a¯)⇧2 ∈ K⇩0"  "a¯ ∈ K"  "a¯ ≠ 𝟬"
using Field_ZF_1_L2 IsOpClosed_def by auto
qed

text‹The inverse is really the inverse.›

lemma (in field0) Field_ZF_1_L6: assumes A1: "a∈K"  "a≠𝟬"
shows "a⋅a¯ = 𝟭"  "a¯⋅a = 𝟭"
proof -
let ?f = "restrict(M,K⇩0×K⇩0)"
from A1 have
"group0(K⇩0,?f)"
"a ∈ K⇩0"
using Field_ZF_1_L4 by auto
then have
"?f⟨a,GroupInv(K⇩0, ?f)(a)⟩ = TheNeutralElement(K⇩0,?f) ∧
?f⟨GroupInv(K⇩0,?f)(a),a⟩ = TheNeutralElement(K⇩0, ?f)"
by (rule group0.group0_2_L6)
with A1 show "a⋅a¯ = 𝟭"  "a¯⋅a = 𝟭"
using Field_ZF_1_L5 Field_ZF_1_L4 by auto
qed

text‹A lemma with two field elements and cancelling.›

lemma (in field0) Field_ZF_1_L7: assumes "a∈K" "b∈K" "b≠𝟬"
shows
"a⋅b⋅b¯ = a"
"a⋅b¯⋅b = a"
using assms Field_ZF_1_L5 Ring_ZF_1_L11 Field_ZF_1_L6 Ring_ZF_1_L3
by auto

subsection‹Equations and identities›

text‹This section deals with more specialized identities that are true in
fields.›

text‹$a/(a^2) = 1/a$ .›

lemma (in field0) Field_ZF_2_L1: assumes A1: "a∈K"  "a≠𝟬"
shows "a⋅(a¯)⇧2 = a¯"
proof -
have "a⋅(a¯)⇧2 = a⋅(a¯⋅a¯)" by simp
also from A1 have "… =  (a⋅a¯)⋅a¯"
using Field_ZF_1_L5 Ring_ZF_1_L11
by simp
also from A1 have "… = a¯"
using Field_ZF_1_L6 Field_ZF_1_L5 Ring_ZF_1_L3
by simp
finally show "a⋅(a¯)⇧2 = a¯" by simp
qed

text‹If we multiply two different numbers by a nonzero number, the results
will be different.›

lemma (in field0) Field_ZF_2_L2:
assumes "a∈K"  "b∈K"  "c∈K"  "a≠b"  "c≠𝟬"
shows "a⋅c¯ ≠ b⋅c¯"
using assms field_has_no_zero_divs Field_ZF_1_L5 Ring_ZF_1_L12B
by simp

text‹We can put a nonzero factor on the other side of non-identity
(is this the best way to call it?) changing it to the inverse.›

lemma (in field0) Field_ZF_2_L3:
assumes A1: "a∈K"  "b∈K"  "b≠𝟬"  "c∈K"   and A2: "a⋅b ≠ c"
shows "a ≠ c⋅b¯"
proof -
from A1 A2 have "a⋅b⋅b¯ ≠ c⋅b¯"
using  Ring_ZF_1_L4 Field_ZF_2_L2 by simp
with A1 show "a ≠ c⋅b¯" using Field_ZF_1_L7
by simp
qed

text‹If if the inverse of $b$ is different than $a$, then the
inverse of $a$ is different than $b$.›

lemma (in field0) Field_ZF_2_L4:
assumes "a∈K"  "a≠𝟬" and "b¯ ≠ a"
shows "a¯ ≠ b"
using assms Field_ZF_1_L4 group0.group0_2_L11B
by simp

text‹An identity with two field elements, one and an inverse.›

lemma (in field0) Field_ZF_2_L5:
assumes "a∈K"  "b∈K" "b≠𝟬"
shows "(𝟭 \<ra> a⋅b)⋅b¯ = a \<ra> b¯"
using assms Ring_ZF_1_L4 Field_ZF_1_L5 Ring_ZF_1_L2 ring_oper_distr
Field_ZF_1_L7 Ring_ZF_1_L3 by simp

text‹An identity with three field elements, inverse and cancelling.›

lemma (in field0) Field_ZF_2_L6: assumes A1: "a∈K"  "b∈K"  "b≠𝟬"  "c∈K"
shows "a⋅b⋅(c⋅b¯) = a⋅c"
proof -
from A1 have T: "a⋅b ∈ K"  "b¯ ∈ K"
using Ring_ZF_1_L4 Field_ZF_1_L5 by auto
with mult_commute A1 have "a⋅b⋅(c⋅b¯) = a⋅b⋅(b¯⋅c)"
using IsCommutative_def by simp
moreover
from A1 T have "a⋅b ∈ K"  "b¯ ∈ K"  "c∈K"
by auto
then have "a⋅b⋅b¯⋅c = a⋅b⋅(b¯⋅c)"
by (rule Ring_ZF_1_L11)
ultimately have "a⋅b⋅(c⋅b¯) = a⋅b⋅b¯⋅c" by simp
with A1 show "a⋅b⋅(c⋅b¯) = a⋅c"
using Field_ZF_1_L7 by simp
qed

subsection‹1/0=0›

text‹In ZF if $f: X\rightarrow Y$ and $x\notin X$ we have $f(x)=\emptyset$.
Since $\emptyset$ (the empty set) in ZF is the same as zero of natural numbers we
can claim that $1/0=0$ in certain sense. In this section we prove a theorem that
makes makes it explicit.›

text‹The next locale extends the ‹field0› locale to introduce notation
for division operation.›

locale fieldd = field0 +
fixes division
defines division_def[simp]: "division ≡ {⟨p,fst(p)⋅snd(p)¯⟩. p∈K×K⇩0}"

fixes fdiv (infixl "\<fd>" 95)
defines fdiv_def[simp]: "x\<fd>y ≡ division⟨x,y⟩"

text‹Division is a function on $K\times K_0$ with values in $K$.›

lemma (in fieldd) div_fun: shows "division: K×K⇩0 → K"
proof -
have "∀p ∈ K×K⇩0. fst(p)⋅snd(p)¯ ∈ K"
proof
fix p assume "p ∈ K×K⇩0"
hence "fst(p) ∈ K" and "snd(p) ∈ K⇩0" by auto
then show  "fst(p)⋅snd(p)¯ ∈ K" using Ring_ZF_1_L4 Field_ZF_1_L5 by auto
qed
then have "{⟨p,fst(p)⋅snd(p)¯⟩. p∈K×K⇩0}: K×K⇩0 → K"
by (rule ZF_fun_from_total)
thus ?thesis by simp
qed

text‹So, really $1/0=0$. The essential lemma is ‹apply_0› from standard
Isabelle's ‹func.thy›.›

theorem (in fieldd) one_over_zero: shows "𝟭\<fd>𝟬 = 0"
proof-
have "domain(division) = K×K⇩0" using div_fun func1_1_L1
by simp
hence "⟨𝟭,𝟬⟩ ∉ domain(division)" by auto
then show ?thesis using apply_0 by simp
qed

end