FOMODEL0: Preliminaries to Classical First-order Model Theory

:: Preliminaries to Classical First-order Model Theory
:: by Marco B. Caminati
::
:: Received December 29, 2010
:: Copyright (c) 2010-2021 Association of Mizar Users

::################### Preliminary Definitions #########################
::#######################################################################

definition

let X be set ;
let f be Function;

attr f is X -one-to-one means :: FOMODEL0:def 1
f | X is one-to-one ;

end;

:: deftheorem defines -one-to-one FOMODEL0:def 1 :

definition

let D be non empty set ;
let f be BinOp of D;
let X be set ;

attr X is f -unambiguous means :: FOMODEL0:def 2
f is [:X,D:] -one-to-one ;

end;

:: deftheorem defines -unambiguous FOMODEL0:def 2 :

definition

let D be non empty set ;
let X be set ;

attr X is D -prefix means :: FOMODEL0:def 3
X is D -concatenation -unambiguous ;

end;

:: deftheorem defines -prefix FOMODEL0:def 3 :

definition

let f be Function-yielding Function;
let g be Function;

func ^^^f,g__ -> Function means :: FOMODEL0:def 4
( dom it = (dom f) /\ (dom g) & ( for x being set st x in dom it holds
it . x = (f . x) . (g . x) ) );

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem defines ^^^ FOMODEL0:def 4 :

definition

let D be set ;

func D -pr1 -> BinOp of D equals :: FOMODEL0:def 5
pr1 (D,D);

coherence ;

end;

:: deftheorem defines -pr1 FOMODEL0:def 5 :

Lm1: for Y being set
for g being Function st g is Y -valued & g is FinSequence-like holds
g is FinSequence of Y
by FINSEQ_1:def 4;

Lm2: for A, B, Y being set st Y c= Funcs (A,B) holds
union Y c= [:A,B:]

proof end;

Lm3: for X being set
for f being Function holds
( f is X -one-to-one iff for x, y being set st x in (dom f) /\ X & y in (dom f) /\ X & f . x = f . y holds
x = y )

proof end;

Lm4: for A, B being set
for D being non empty set
for f being BinOp of D st A c= B & f is B -one-to-one holds
f is A -one-to-one

proof end;

Lm5: for A, B being set
for m, n being Nat st m -tuples_on A meets n -tuples_on B holds
m = n

proof end;

Lm6: for D being set holds 0 -tuples_on D = {{}}

proof end;

Lm7: for i being Nat
for Y being set holds i -tuples_on Y = Funcs ((Seg i),Y)

proof end;

Lm8: for A, B being set
for m being Nat holds (m -tuples_on A) /\ (B *) c= m -tuples_on B

proof end;

theorem Th1: :: FOMODEL0:1

for A, B being set
for m being Nat holds (m -tuples_on A) /\ (B *) = (m -tuples_on A) /\ (m -tuples_on B)

proof end;

Lm9: for A, B, C being set holds (Funcs (A,B)) /\ (Funcs (A,C)) = Funcs (A,(B /\ C))

proof end;

theorem Th2: :: FOMODEL0:2

for A, B being set
for m being Nat holds (m -tuples_on A) /\ (B *) = m -tuples_on (A /\ B)

proof end;

theorem Th3: :: FOMODEL0:3

for A, B being set
for m being Nat holds m -tuples_on (A /\ B) = (m -tuples_on A) /\ (m -tuples_on B)

proof end;

Lm10: for D being non empty set
for x, y being FinSequence of D holds (D -concatenation) . (x,y) = x ^ y

proof end;

theorem Th4: :: FOMODEL0:4

for U being non empty set
for x, y being FinSequence st x is U -valued & y is U -valued holds
(U -concatenation) . (x,y) = x ^ y

proof end;

theorem Th5: :: FOMODEL0:5

for D being non empty set
for x being set holds
( x is non empty FinSequence of D iff x in (D *) \ {{}} )

proof end;

Lm11: for A, B being set
for D being non empty set
for f being BinOp of D st B is f -unambiguous & A c= B holds
A is f -unambiguous

proof end;

Lm12: for D being non empty set
for f being BinOp of D holds {} is f -unambiguous

proof end;

Lm13: for f, g being FinSequence holds dom <:f,g:> = Seg (min ((len f),(len g)))

proof end;

::##################### Preliminary lemmas ############################
::############################## END ##################################
::############################ Type tuning ############################

registration

let D be non empty set ;

cluster D -pr1 -> associative for BinOp of D;

coherence

proof end;

end;

registration

let D be set ;

cluster Relation-like [:D,D:] -defined D -valued Function-like total quasi_total associative for Element of bool [:[:D,D:],D:];

existence

proof end;

end;

definition

let X be set ;
let Y be Subset of X;
:: original: *
redefine func Y * -> non empty Subset of (X *);
coherence by FINSEQ_1:62;

end;

registration

let D be non empty set ;

cluster D -concatenation -> associative for BinOp of (D *);

coherence by MONOID_0:67;

end;

registration

let D be non empty set ;

cluster (D *) \ {{}} -> non empty ;

coherence

proof end;

end;

registration

let D be non empty set ;
let m be Nat;

cluster Relation-like omega -defined D -valued Function-like finite m -element FinSequence-like FinSubsequence-like countable finite-support for Element of D * ;

existence

proof end;

end;

definition

let X be set ;
let f be Function;

redefine attr f is X -one-to-one means :Def6: :: FOMODEL0:def 6
for x, y being set st x in X /\ (dom f) & y in X /\ (dom f) & f . x = f . y holds
x = y;

compatibility by Lm3;

end;

:: deftheorem Def6 defines -one-to-one FOMODEL0:def 6 :

registration

let D be non empty set ;
let f be BinOp of D;

cluster f -unambiguous for set ;

existence

proof end;

end;

registration

let f be Function;
let x be set ;

cluster f | {x} -> one-to-one for Function;

coherence

proof end;

end;

registration

let e be empty set ;

identify e * with {e};

compatibility by FUNCT_7:17;

identify {e} with e * ;

compatibility ;

end;

registration

cluster empty -> empty-membered for set ;

coherence ;
let e be empty set ;

cluster {e} -> empty-membered ;

coherence

proof end;

end;

registration

let U be non empty set ;
let m1 be non zero Nat;

cluster m1 -tuples_on U -> with_non-empty_elements ;

coherence

proof end;

end;

registration

let X be empty-membered set ;

cluster -> empty-membered for Element of bool X;

coherence

proof end;

end;

registration

let A be set ;
let m0 be zero number ;

cluster m0 -tuples_on A -> empty-membered for set ;

coherence by Lm6;

end;

registration

let e be empty set ;
let m1 be non zero Nat;

cluster m1 -tuples_on e -> empty for set ;

coherence by FINSEQ_3:119;

end;

registration

let D be non empty set ;
let f be BinOp of D;
let e be empty set ;

cluster e typed/\ f -> f -unambiguous for set ;

coherence by Lm12;

end;

registration

let U be non empty set ;
let e be empty set ;

cluster e typed/\ U -> U -prefix for set ;

coherence

proof end;

end;

registration

let U be non empty set ;

cluster U -prefix for set ;

existence

proof end;

end;

::############################ Type tuning ############################
::############################## END ##################################
::########################### MultPlace ###############################

definition

let D be non empty set ;
let f be BinOp of D;
let x be FinSequence of D;

func MultPlace (f,x) -> Function means :Def7: :: FOMODEL0:def 7
( dom it = NAT & it . 0 = x . 1 & ( for n being Nat holds it . (n + 1) = f . ((it . n),(x . (n + 2))) ) );

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def7 defines MultPlace FOMODEL0:def 7 :

Lm14: for D being non empty set
for f being BinOp of D
for x being FinSequence of D
for m being Nat holds
( ( m + 1 <= len x implies (MultPlace (f,x)) . m in D ) & ( for n being Nat st n >= m + 1 holds
(MultPlace (f,(x | n))) . m = (MultPlace (f,x)) . m ) )

proof end;

definition

let D be non empty set ;
let f be BinOp of D;
let x be Element of (D *) \ {{}};

func MultPlace (f,x) -> Function equals :: FOMODEL0:def 8
MultPlace (f,x);

coherence ;

end;

:: deftheorem defines MultPlace FOMODEL0:def 8 :

definition

let D be non empty set ;
let f be BinOp of D;

::# MultPlace is an operator which transforms a BinOp f into a function
::# operating # on an arbitrary (positive and natural) number of arguments by
::# recursively # associating f on the left # Too late I realized something
::# similar (yielding a functor rather than # a function, though) was
::# introduced in FINSOP_1

func MultPlace f -> Function of ((D *) \ {{}}),D means :Def9: :: FOMODEL0:def 9
for x being Element of (D *) \ {{}} holds it . x = (MultPlace (f,x)) . ((len x) - 1);

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def9 defines MultPlace FOMODEL0:def 9 :

Lm15: for D being non empty set
for f being BinOp of D
for x being non empty FinSequence of D
for y being Element of D holds
( (MultPlace f) . <*y*> = y & (MultPlace f) . (x ^ <*y*>) = f . (((MultPlace f) . x),y) )

proof end;

Lm16: for D being non empty set
for f being BinOp of D
for X being set holds
( f is [:X,D:] -one-to-one iff for x, y, d1, d2 being set st x in X /\ D & y in X /\ D & d1 in D & d2 in D & f . (x,d1) = f . (y,d2) holds
( x = y & d1 = d2 ) )

proof end;

definition

let D be non empty set ;
let f be BinOp of D;
let X be set ;

redefine attr X is f -unambiguous means :Def10: :: FOMODEL0:def 10
for x, y, d1, d2 being set st x in X /\ D & y in X /\ D & d1 in D & d2 in D & f . (x,d1) = f . (y,d2) holds
( x = y & d1 = d2 );

compatibility by Lm16;

end;

:: deftheorem Def10 defines -unambiguous FOMODEL0:def 10 :

Lm17: for D being non empty set
for f being BinOp of D st f is associative holds
for d being Element of D
for m being Nat
for x being Element of (m + 1) -tuples_on D holds (MultPlace f) . (<*d*> ^ x) = f . (d,((MultPlace f) . x))

proof end;

Lm18: for D being non empty set
for X being non empty Subset of D
for f being BinOp of D
for m being Nat st f is associative & X is f -unambiguous holds
(MultPlace f) .: ((m + 1) -tuples_on X) is f -unambiguous

proof end;

Lm19: for Y being set
for D being non empty set
for f being BinOp of D
for m being Nat st f is associative & Y is f -unambiguous holds
(MultPlace f) .: ((m + 1) -tuples_on Y) is f -unambiguous

proof end;

Lm20: for D being non empty set
for X being non empty Subset of D
for f being BinOp of D
for m being Nat st f is associative & X is f -unambiguous holds
MultPlace f is (m + 1) -tuples_on X -one-to-one

proof end;

Lm21: for Y being set
for D being non empty set
for f being BinOp of D
for m being Nat st f is associative & Y is f -unambiguous holds
MultPlace f is (m + 1) -tuples_on Y -one-to-one

proof end;

::########################### MultPlace ###############################
::############################## END ##################################
::########################### FirstChar ###############################

definition

let D be non empty set ;

::# A function-like extracting the first item of a non empty FinSequence of D

func D -firstChar -> Function of ((D *) \ {{}}),D equals :: FOMODEL0:def 11
MultPlace (D -pr1);

coherence ;

end;

:: deftheorem defines -firstChar FOMODEL0:def 11 :

Lm22: for D being non empty set
for w being non empty FinSequence of D holds (D -firstChar) . w = w . 1

proof end;

theorem Th6: :: FOMODEL0:6

for U being non empty set
for p being FinSequence st p is U -valued & not p is empty holds
(U -firstChar) . p = p . 1

proof end;

::########################### FirstChar ###############################
::############################## END ##################################
::########################### MultiCat #################################
::#When f is concatenation of strings, MultPlace(f) can be extended to the
::#empty finsequence of strings in the immediate way, obtaining the multiCat
::#function, which chains an arbitrary (natural) number of strings

definition

let D be non empty set ;

func D -multiCat -> Function equals :: FOMODEL0:def 12
({} .--> {}) +* (MultPlace (D -concatenation));

coherence ;

end;

:: deftheorem defines -multiCat FOMODEL0:def 12 :

definition

let D be non empty set ;
:: original: -multiCat
redefine func D -multiCat -> Function of ((D *) *),(D *);
coherence

proof end;

end;

Lm23: for D being non empty set holds (D -multiCat) | (((D *) *) \ {{}}) = MultPlace (D -concatenation)

proof end;

Lm24: for Y being set
for D being non empty set holds (D -multiCat) | (Y \ {{}}) = (MultPlace (D -concatenation)) | Y

proof end;

Lm25: for D being non empty set holds {} .--> {} tolerates MultPlace (D -concatenation)

proof end;

registration

let D be non empty set ;
let e be empty set ;

cluster (D -multiCat) . e -> empty for set ;

coherence

proof end;

end;

registration

let D be non empty set ;

cluster -> D -prefix for Element of bool (1 -tuples_on D);

coherence

proof end;

end;

theorem Th7: :: FOMODEL0:7

for A being set
for D being non empty set
for m being Nat st A is D -prefix holds
(D -multiCat) .: (m -tuples_on A) is D -prefix

proof end;

theorem :: FOMODEL0:8

for A being set
for D being non empty set
for m being Nat st A is D -prefix holds
D -multiCat is m -tuples_on A -one-to-one

proof end;

theorem :: FOMODEL0:9

for Y being set
for m being Nat holds (m + 1) -tuples_on Y c= (Y *) \ {{}}

proof end;

theorem :: FOMODEL0:10

for Y being set
for m being Nat st m is zero holds
m -tuples_on Y = {{}} by Lm6;

theorem :: FOMODEL0:11

for Y being set
for i being Nat holds i -tuples_on Y = Funcs ((Seg i),Y) by Lm7;

::#extending FUNCT_2:111

theorem :: FOMODEL0:12

for A, x being set
for m being Nat st x in m -tuples_on A holds
x is FinSequence of A

proof end;

definition

let A, X be set ;
:: original: chi
redefine func chi (A,X) -> Function of X,BOOLEAN;
coherence

proof end;

end;

theorem :: FOMODEL0:13

for D being non empty set
for d being Element of D
for f being BinOp of D holds
( (MultPlace f) . <*d*> = d & ( for x being non empty FinSequence of D holds (MultPlace f) . (x ^ <*d*>) = f . (((MultPlace f) . x),d) ) ) by Lm15;

theorem Th14: :: FOMODEL0:14

for D being non empty set
for d being non empty Element of (D *) * holds (D -multiCat) . d = (MultPlace (D -concatenation)) . d

proof end;

theorem :: FOMODEL0:15

for D being non empty set
for d1, d2 being Element of D * holds (D -multiCat) . <*d1,d2*> = d1 ^ d2

proof end;

registration

let f, g be FinSequence;

cluster <:f,g:> -> FinSequence-like ;

coherence

proof end;

end;

registration

let m be Nat;
let f, g be m -element FinSequence;

cluster <:f,g:> -> m -element ;

coherence

proof end;

end;

registration

let X, Y be set ;
let f be X -defined Function;
let g be Y -defined Function;

cluster <:f,g:> -> X /\ Y -defined for Function;

coherence

proof end;

end;

registration

let X be set ;
let f, g be X -defined Function;

cluster <:f,g:> -> X -defined for Function;

coherence ;

end;

registration

let X, Y be set ;
let f be X -defined total Function;
let g be Y -defined total Function;

cluster <:f,g:> -> X /\ Y -defined total for X /\ Y -defined Function;

coherence

proof end;

end;

registration

let X be set ;
let f, g be X -defined total Function;

cluster <:f,g:> -> X -defined total for X -defined Function;

coherence ;

end;

registration

let X, Y be set ;
let f be X -valued Function;
let g be Y -valued Function;

cluster <:f,g:> -> [:X,Y:] -valued for Function;

coherence

proof end;

end;

registration

let D be non empty set ;

cluster Relation-like omega -defined D -valued Function-like finite FinSequence-like FinSubsequence-like countable finite-support for set ;

existence

proof end;

end;

registration

let D be non empty set ;
let m be Nat;

cluster Relation-like omega -defined D -valued Function-like finite m -element FinSequence-like FinSubsequence-like countable finite-support for set ;

existence

proof end;

end;

registration

let X, Y be non empty set ;
let f be Function of X,Y;
let p be X -valued FinSequence;

cluster p >*> f -> FinSequence-like ;

coherence

proof end;

end;

registration

let X, Y be non empty set ;
let m be Nat;
let f be Function of X,Y;
let p be X -valued m -element FinSequence;

cluster p >*> f -> m -element for set ;

coherence

proof end;

end;

definition

let D be non empty set ;
let f be BinOp of D;
let p, q be Element of D * ;

func f AppliedPairwiseTo (p,q) -> FinSequence of D equals :: FOMODEL0:def 13
f * <:p,q:>;

coherence

proof end;

end;

:: deftheorem defines AppliedPairwiseTo FOMODEL0:def 13 :

registration

let D be non empty set ;
let f be BinOp of D;
let m be Nat;
let p, q be m -element Element of D * ;

cluster f AppliedPairwiseTo (p,q) -> m -element for set ;

coherence ;

end;

notation

let D be non empty set ;
let f be BinOp of D;
let p, q be Element of D * ;
synonym f _\ (p,q) for f AppliedPairwiseTo (p,q);

end;

definition

redefine func INT equals :: FOMODEL0:def 14
NAT \/ ([:{0},NAT:] \ {[0,0]});

compatibility

proof end;

end;

:: deftheorem defines INT FOMODEL0:def 14 :

theorem Th16: :: FOMODEL0:16

for Y being set
for m being Nat
for p being FinSequence st p is Y -valued & p is m -element holds
p in m -tuples_on Y

proof end;

notation

let A, B be set ;
synonym A typed/\ B for A /\ B;

end;

definition

let A, B be set ;
:: original: typed/\
redefine func A typed/\ B -> Subset of A;
coherence by XBOOLE_1:17;

end;

registration

let A, B be set ;

identify A typed/\ B with B typed/\ A;

compatibility ;

end;

registration

let A, B be set ;

identify A typed/\ B with A typed/\ B;

compatibility ;

end;

definition

let B, A be object ;

func A null B -> set equals :: FOMODEL0:def 15
A;

coherence by TARSKI:1;

end;

:: deftheorem defines null FOMODEL0:def 15 :

registration

let A, B be set ;

reduce A null B to A;

reducibility ;

end;

registration

let A be set ;
let B be Subset of A;

identify A /\ B with B null A;

compatibility by XBOOLE_1:28;

identify B null A with A /\ B;

compatibility ;

end;

registration

let A, B, C be set ;

cluster (B \ A) typed/\ (A /\ C) -> empty for set ;

coherence

proof end;

end;

definition

let A, B be set ;

func A typed\ B -> Subset of A equals :: FOMODEL0:def 16
A \ B;

coherence ;

end;

:: deftheorem defines typed\ FOMODEL0:def 16 :

registration

let A, B be set ;

identify A \ B with A typed\ B;

compatibility ;

end;

definition

let A, B be set ;

::# to automatize like: A null B c= A\/B; then A c= A\/B; this is mainly
::# not to have to remember each time what XBOOLE_1 theorem is to be invoked,
::# as long as this file is imported via definitions directive

func A \typed/ B -> Subset of (A \/ B) equals :: FOMODEL0:def 17
A;

coherence by XBOOLE_1:7;

end;

:: deftheorem defines \typed/ FOMODEL0:def 17 :

registration

let A, B be set ;

identify A \typed/ B with A null B;

compatibility ;

identify A null B with A \typed/ B;

compatibility ;

end;

registration

let A be set ;
let B be Subset of A;

identify A \/ B with A null B;

compatibility by XBOOLE_1:12;

identify A null B with A \/ B;

compatibility ;

end;

Lm26: for g being Function
for m being Nat
for f being Function st f c= g holds
iter (f,m) c= iter (g,m)

proof end;

Lm27: for R being Relation holds R [*] is_transitive_in field R

proof end;

Lm28: for R being Relation holds field (R [*]) c= field R

proof end;

Lm29: for R being Relation holds R [*] is_reflexive_in field R

proof end;

Lm30: for R being Relation holds field (R [*]) = field R
by LANG1:18, RELAT_1:16, Lm28;

registration

let R be Relation;

cluster R [*] -> transitive for Relation;

coherence

proof end;

cluster R [*] -> reflexive for Relation;

coherence

proof end;

end;

Lm31: for f being Function holds iter (f,0) c= f [*]

proof end;

Lm32: for m being Nat
for f being Function holds iter (f,(m + 1)) c= f [*]

proof end;

Lm33: for m being Nat
for f being Function holds iter (f,m) c= f [*]

proof end;

Lm34: for x being set
for g being Function
for m being Nat
for f being Function st rng f c= dom f & x in dom f & g . 0 = x & ( for i being Nat st i < m holds
g . (i + 1) = f . (g . i) ) holds
g . m = (iter (f,m)) . x

proof end;

definition

func plus -> Function of COMPLEX,COMPLEX means :Def18: :: FOMODEL0:def 18
for z being Complex holds it . z = z + 1;

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def18 defines plus FOMODEL0:def 18 :

Lm35: for x being set
for p being FinSequence
for m being Nat
for f being Function st rng f c= dom f & x in dom f & p . 1 = x & ( for i being Nat st i >= 1 & i < m + 1 holds
p . (i + 1) = f . (p . i) ) holds
p . (m + 1) = (iter (f,m)) . x

proof end;

Lm36: for z being set
for f being Function st z in f [*] & rng f c= dom f holds
ex n being Nat st z in iter (f,n)

proof end;

Lm37: for f being Function st rng f c= dom f holds
f [*] = union { (iter (f,mm)) where mm is Element of NAT : verum }

proof end;

theorem :: FOMODEL0:17

for f being Function st rng f c= dom f holds
f [*] = union { (iter (f,mm)) where mm is Element of NAT : verum } by Lm37;

theorem :: FOMODEL0:18

for g being Function
for m being Nat
for f being Function st f c= g holds
iter (f,m) c= iter (g,m) by Lm26;

registration

let X be functional set ;

cluster union X -> Relation-like for set ;

coherence

proof end;

end;

theorem :: FOMODEL0:19

for A, B, Y being set st Y c= Funcs (A,B) holds
union Y c= [:A,B:] by Lm2;

registration

let Y be set ;
let R be Y -valued Relation;

identify Y |` R with R null Y;

compatibility ;

end;

registration

let Y be set ;

cluster Y \ Y -> empty for set ;

coherence by XBOOLE_1:37;

end;

registration

let D be non empty set ;
let d be Element of D;

cluster {((id D) . d)} \ {d} -> empty for set ;

coherence ;

end;

registration

let p be FinSequence;
let q be empty FinSequence;

identify p ^ q with p null q;

compatibility by FINSEQ_1:34;

identify p null q with p ^ q;

compatibility ;

identify q ^ p with p null q;

compatibility by FINSEQ_1:34;

identify p null q with q ^ p;

compatibility ;

end;

registration

let Y be set ;
let R be Y -defined Relation;

identify R | Y with R null Y;

compatibility ;

identify R null Y with R | Y;

compatibility ;

end;

theorem Th20: :: FOMODEL0:20

for f being Function holds f = { [x,(f . x)] where x is Element of dom f : x in dom f }

proof end;

theorem Th21: :: FOMODEL0:21

for Y being set
for R being b₁ -defined total Relation holds id Y c= R * (R ~)

proof end;

theorem :: FOMODEL0:22

for D being non empty set
for m, n being Nat holds (m + n) -tuples_on D = (D -concatenation) .: [:(m -tuples_on D),(n -tuples_on D):]

proof end;

theorem Th23: :: FOMODEL0:23

for Y being set
for P, Q being Relation holds (P \/ Q) " Y = (P " Y) \/ (Q " Y)

proof end;

Lm38: for A, B being set holds chi (A,B) = (B --> 0) +* ((A /\ B) --> 1)

proof end;

Lm39: for A, B being set holds chi (A,B) = ((B \ A) --> 0) +* ((A /\ B) --> 1)

proof end;

Lm40: for A, B being set holds chi (A,B) = ((B \ A) --> 0) \/ ((A /\ B) --> 1)

proof end;

theorem :: FOMODEL0:24

for A, B being set holds
( (chi (A,B)) " {0} = B \ A & (chi (A,B)) " {1} = A /\ B )

proof end;

theorem :: FOMODEL0:25

for x being set
for f being Function
for y being non empty set holds
( y = f . x iff x in f " {y} )

proof end;

theorem :: FOMODEL0:26

for Y being set
for f being Function st f is Y -valued & f is FinSequence-like holds
f is FinSequence of Y by Lm1;

registration

let Y be set ;
let X be Subset of Y;

cluster Relation-like X -valued -> Y -valued for set ;

coherence by XBOOLE_1:1;

end;

Lm41: for Y being set
for R being b₁ -defined total Relation ex f being Function of Y,(rng R) st
( f c= R & dom f = Y )

proof end;

Lm42: for R being Relation
for f being Function st dom f c= dom R & R c= f holds
R = f

proof end;

Lm43: for Y being set
for P, R being Relation
for Q being b₁ -defined Relation st Q is total & R is Y -defined & ((P * Q) * (Q ~)) * R is Function-like & rng P c= dom R holds
((P * Q) * (Q ~)) * R = P * R

proof end;

registration

let A be set ;
let U be non empty set ;

cluster quasi_total -> total for Element of bool [:A,U:];

coherence ;

end;

theorem :: FOMODEL0:27

for A, B being set
for U1, U2 being non empty set
for Q being quasi_total Relation of B,U1
for R being quasi_total Relation of B,U2
for P being Relation of A,B st ((P * Q) * (Q ~)) * R is Function-like holds
((P * Q) * (Q ~)) * R = P * R

proof end;

theorem :: FOMODEL0:28

for p, q being FinSequence st not p is empty holds
(p ^ q) . 1 = p . 1

proof end;

registration

let U be non empty set ;
let p, q be U -valued FinSequence;

cluster p ^ q -> U -valued for FinSequence;

coherence

proof end;

end;

definition

let X be set ;
:: original: FinSequence
redefine mode FinSequence of X -> Element of X * ;
coherence by FINSEQ_1:def 11;

end;

definition

let U be non empty set ;
let X be set ;

redefine attr X is U -prefix means :: FOMODEL0:def 19
for p1, q1, p2, q2 being U -valued FinSequence st p1 in X & p2 in X & p1 ^ q1 = p2 ^ q2 holds
( p1 = p2 & q1 = q2 );

compatibility

proof end;

end;

:: deftheorem defines -prefix FOMODEL0:def 19 :

registration

let X be set ;

cluster -> X -valued for Element of X * ;

coherence ;

end;

registration

let U be non empty set ;
let m be Nat;
let X be U -prefix set ;

cluster (U -multiCat) .: (m -tuples_on X) -> U -prefix for set ;

coherence by Th7;

end;

theorem Th29: :: FOMODEL0:29

for Y, X being set holds
( ( X \+\ Y = {} implies X = Y ) & ( X = Y implies X \+\ Y = {} ) & ( X \ Y = {} implies X c= Y ) & ( X c= Y implies X \ Y = {} ) & ( for x being object holds
( {x} \ Y = {} iff x in Y ) ) )

proof end;

registration

let P be Relation;

cluster P \ [:(dom P),(rng P):] -> empty ;

coherence by RELAT_1:7, Th29;

end;

registration

let X, Y, Z be set ;

cluster ((X \/ Y) \/ Z) \+\ (X \/ (Y \/ Z)) -> empty ;

coherence by XBOOLE_1:4, Th29;

end;

registration

let x be set ;

cluster (id {x}) \+\ {[x,x]} -> empty for set ;

coherence

proof end;

end;

registration

let x, y be set ;

cluster (x .--> y) \+\ {[x,y]} -> empty for set ;

coherence

proof end;

end;

registration

let x be set ;

cluster (id {x}) \+\ (x .--> x) -> empty for set ;

coherence

proof end;

end;

theorem :: FOMODEL0:30

for x being set
for D being non empty set holds
( x in (D *) \ {{}} iff ( x is b₂ -valued FinSequence & not x is empty ) )

proof end;

theorem Th31: :: FOMODEL0:31

for D being non empty set
for d being Element of D
for f being BinOp of D holds
( (MultPlace f) . <*d*> = d & ( for x being b₁ -valued FinSequence st not x is empty holds
(MultPlace f) . (x ^ <*d*>) = f . (((MultPlace f) . x),d) ) )

proof end;

registration

let p be FinSequence;
let x, y be set ;

cluster p +~ (x,y) -> FinSequence-like ;

coherence

proof end;

end;

definition

let x, y be set ;
let p be FinSequence;

func (x,y) -SymbolSubstIn p -> FinSequence equals :: FOMODEL0:def 20
p +~ (x,y);

coherence ;

end;

:: deftheorem defines -SymbolSubstIn FOMODEL0:def 20 :

registration

let x, y be set ;
let m be Nat;
let p be m -element FinSequence;

cluster (x,y) -SymbolSubstIn p -> m -element for FinSequence;

coherence

proof end;

end;

registration

let x be set ;
let U be non empty set ;
let u be Element of U;
let p be U -valued FinSequence;

cluster (x,u) -SymbolSubstIn p -> U -valued for FinSequence;

coherence ;

end;

definition

let X, x, y be set ;
let p be X -valued FinSequence;

:: original: -SymbolSubstIn
redefine func (x,y) -SymbolSubstIn p -> FinSequence equals :: FOMODEL0:def 21
((id X) +* (x,y)) * p;

compatibility

proof end;

end;

:: deftheorem defines -SymbolSubstIn FOMODEL0:def 21 :

definition

let U be non empty set ;
let x be set ;
let u be Element of U;
let q be U -valued FinSequence;
:: original: -SymbolSubstIn
redefine func (x,u) -SymbolSubstIn q -> FinSequence of U;
coherence by Lm1;

end;

Lm44: for x2 being set
for U being non empty set
for u, u1 being Element of U holds
( ( u = u1 implies (u1,x2) -SymbolSubstIn <*u*> = <*x2*> ) & ( u <> u1 implies (u1,x2) -SymbolSubstIn <*u*> = <*u*> ) )

proof end;

Lm45: for x being set
for U being non empty set
for u being Element of U
for q1, q2 being b₂ -valued FinSequence holds (x,u) -SymbolSubstIn (q1 ^ q2) = ((x,u) -SymbolSubstIn q1) ^ ((x,u) -SymbolSubstIn q2)

proof end;

definition

let U be non empty set ;
let x be set ;
let u be Element of U;
set D = U * ;
deffunc H₁( Element of U * ) -> FinSequence of U = (x,u) -SymbolSubstIn $1;

func x SubstWith u -> Function of (U *),(U *) means :Def22: :: FOMODEL0:def 22
for q being U -valued FinSequence holds it . q = (x,u) -SymbolSubstIn q;

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def22 defines SubstWith FOMODEL0:def 22 :

Lm46: for U being non empty set
for u, u1, u2 being Element of U holds
( ( u = u1 implies (u1 SubstWith u2) . <*u*> = <*u2*> ) & ( u <> u1 implies (u1 SubstWith u2) . <*u*> = <*u*> ) )

proof end;

registration

let U be non empty set ;
let x be set ;
let u be Element of U;

cluster x SubstWith u -> FinSequence-yielding for Function;

coherence

proof end;

end;

Lm47: for x being set
for U being non empty set
for u being Element of U
for q1, q2 being b₂ -valued FinSequence holds (x SubstWith u) . (q1 ^ q2) = ((x SubstWith u) . q1) ^ ((x SubstWith u) . q2)

proof end;

registration

let U be non empty set ;
let x be set ;
let u be Element of U;
let m be Nat;
let p be U -valued m -element FinSequence;

cluster (x SubstWith u) . p -> m -element for FinSequence;

coherence

proof end;

end;

registration

let U be non empty set ;
let x be set ;
let u be Element of U;
let e be empty set ;

cluster (x SubstWith u) . e -> empty for set ;

coherence

proof end;

end;

registration

let U be non empty set ;

cluster U -multiCat -> FinSequence-yielding for Function;

coherence

proof end;

end;

registration

let U be non empty set ;
let m1 be non zero Nat;
let n be Nat;
let p be U -valued m1 + n -element FinSequence;

cluster {(p . m1)} \ U -> empty for set ;

coherence

proof end;

end;

registration

let U be non empty set ;
let m, n be Nat;
let p be (m + 1) + n -element Element of U * ;

cluster {(p . (m + 1))} \ U -> empty for set ;

coherence ;

end;

registration

let x be set ;

cluster <*x*> \+\ {[1,x]} -> empty for set ;

coherence ;

end;

registration

let m be Nat;
let p be m + 1 -element FinSequence;

cluster ((p | (Seg m)) ^ <*(p . (m + 1))*>) \+\ p -> empty for set ;

coherence

proof end;

end;

registration

let m, n be Nat;
let p be m + n -element FinSequence;

cluster p | (Seg m) -> m -element for FinSequence;

coherence

proof end;

end;

registration

cluster Relation-like {{}} -valued -> empty-yielding for set ;

coherence ;

cluster Relation-like empty-yielding -> {{}} -valued for set ;

coherence ;

end;

theorem Th32: :: FOMODEL0:32

for x being set
for U being non empty set holds (U -multiCat) . x = (MultPlace (U -concatenation)) . x

proof end;

theorem Th33: :: FOMODEL0:33

for U being non empty set
for p being FinSequence
for q being b₁ -valued FinSequence st p is U * -valued holds
(U -multiCat) . (p ^ <*q*>) = ((U -multiCat) . p) ^ q

proof end;

Lm48: for x being set
for U being non empty set
for u being Element of U
for p being FinSequence st p is U * -valued holds
(x SubstWith u) . ((U -multiCat) . p) = (U -multiCat) . ((x SubstWith u) * p)

proof end;

registration

let Y be set ;
let X be Subset of Y;
let R be Y -defined total Relation;

cluster R | X -> X -defined total for X -defined Relation;

coherence

proof end;

end;

theorem :: FOMODEL0:34

for x2 being set
for U being non empty set
for u, u1 being Element of U holds
( ( u = u1 implies (u1,x2) -SymbolSubstIn <*u*> = <*x2*> ) & ( u <> u1 implies (u1,x2) -SymbolSubstIn <*u*> = <*u*> ) ) by Lm44;

theorem :: FOMODEL0:35

for U being non empty set
for u, u1, u2 being Element of U holds
( ( u = u1 implies (u1 SubstWith u2) . <*u*> = <*u2*> ) & ( u <> u1 implies (u1 SubstWith u2) . <*u*> = <*u*> ) ) by Lm46;

theorem :: FOMODEL0:36

for x being set
for U being non empty set
for u being Element of U
for q1, q2 being b₂ -valued FinSequence holds (x SubstWith u) . (q1 ^ q2) = ((x SubstWith u) . q1) ^ ((x SubstWith u) . q2) by Lm47;

theorem :: FOMODEL0:37

for x being set
for U being non empty set
for u being Element of U
for p being FinSequence st p is U * -valued holds
(x SubstWith u) . ((U -multiCat) . p) = (U -multiCat) . ((x SubstWith u) * p) by Lm48;

theorem :: FOMODEL0:38

for U being non empty set holds (U -concatenation) .: (id (1 -tuples_on U)) = { <*u,u*> where u is Element of U : verum }

proof end;

registration

let f be Function;
let U be non empty set ;
let u be Element of U;

cluster ((f | U) . u) \+\ (f . u) -> empty for set ;

coherence

proof end;

end;

registration

let f be Function;
let U1, U2 be non empty set ;
let u be Element of U1;
let g be Function of U1,U2;

cluster ((f * g) . u) \+\ (f . (g . u)) -> empty for set ;

coherence

proof end;

end;

registration

cluster integer non negative -> natural for object ;

coherence

proof end;

end;

registration

let x, y be Real;

cluster (max (x,y)) - x -> non negative for ExtReal;

coherence

proof end;

end;

theorem :: FOMODEL0:39

for x being set st x is boolean holds
( x = 1 iff x <> 0 ) by XBOOLEAN:def 3;

registration

let Y be set ;
let X be Subset of Y;

cluster X \ Y -> empty for set ;

coherence by XBOOLE_1:37;

end;

registration

let x, y be object ;

cluster {x} \ {x,y} -> empty for set ;

coherence

proof end;

end;

registration

let x, y be set ;

cluster ([x,y] `1) \+\ x -> empty for set ;

coherence ;

end;

registration

let x, y be set ;

cluster ([x,y] `2) \+\ y -> empty for set ;

coherence ;

end;

registration

let n be positive Nat;
let X be non empty set ;

cluster Relation-like omega -defined Function-like finite n -element FinSequence-like FinSubsequence-like countable finite-support for Element of (X *) \ {{}};

existence

proof end;

end;

registration

let m1 be non zero Nat;

cluster Relation-like Function-like m1 + 0 -element FinSequence-like -> non empty for set ;

coherence ;

end;

registration

let R be Relation;
let x be set ;

cluster R null x -> Relation-like for set ;

coherence ;

end;

registration

let f be Function-like set ;
let x be set ;

cluster f null x -> Function-like for set ;

coherence ;

end;

registration

let p be FinSequence-like Relation;
let x be set ;

cluster p null x -> FinSequence-like for Relation;

coherence ;

end;

registration

let p be FinSequence;
let x be set ;

cluster p null x -> len p -element for FinSequence;

coherence by CARD_1:def 7;

end;

registration

let p be non empty FinSequence;

cluster len p -> non zero for number ;

coherence ;

end;

registration

let R be Relation;
let X be set ;

cluster R | X -> X -defined for Relation;

coherence by RELAT_1:58;

end;

registration

let x be set ;
let e be empty set ;

cluster e null x -> empty for set ;

coherence ;

end;

registration

let X be set ;
let e be empty set ;

cluster e null X -> X -valued for Relation;

coherence

proof end;

end;

registration

let Y be non empty FinSequence-membered set ;

cluster Relation-like Y -valued Function-like -> FinSequence-yielding for set ;

coherence

proof end;

end;

registration

let X, Y be set ;

cluster -> FinSequence-yielding for Element of Funcs (X,(Y *));

coherence ;

end;

theorem Th40: :: FOMODEL0:40

for X, x being set
for f being Function st f is X * -valued holds
f . x in X *

proof end;

registration

let m, n be Nat;
let p be m -element FinSequence;

cluster p null n -> Seg (m + n) -defined for Relation;

coherence

proof end;

end;

Lm49: for m being Nat
for p1, p2, q1, q2 being FinSequence st p1 is m -element & q1 is m -element & p1 ^ p2 = q1 ^ q2 holds
( p1 = q1 & p2 = q2 )

proof end;

registration

let m, n be Nat;
let p be m -element FinSequence;
let q be n -element FinSequence;

cluster p ^ q -> m + n -element for FinSequence;

coherence ;

end;

theorem :: FOMODEL0:41

for m being Nat
for p1, p2, q1, q2 being FinSequence st p1 is m -element & q1 is m -element & ( p1 ^ p2 = q1 ^ q2 or p2 ^ p1 = q2 ^ q1 ) holds
( p1 = q1 & p2 = q2 )

proof end;

theorem :: FOMODEL0:42

for x being set
for U, U1 being non empty set st (U -multiCat) . x is U1 -valued & x in (U *) * holds
x is FinSequence of U1 *

proof end;

registration

let U be non empty set ;

cluster Relation-like U -defined U -valued total reflexive for Element of bool [:U,U:];

existence

proof end;

end;

registration

let m be Nat;

cluster Relation-like Function-like m + 1 -element FinSequence-like -> non empty for set ;

coherence ;

end;

registration

let U be non empty set ;
let u be Element of U;

cluster ((id U) . u) \+\ u -> empty for set ;

coherence ;

end;

registration

let U be non empty set ;
let p be U -valued non empty FinSequence;

cluster {(p . 1)} \ U -> empty for set ;

coherence

proof end;

end;

theorem :: FOMODEL0:43

for x1, x2, y1, y2 being set
for f being Function holds
( ( x1 = x2 implies (f +* (x1 .--> y1)) +* (x2 .--> y2) = f +* (x2 .--> y2) ) & ( x1 <> x2 implies (f +* (x1 .--> y1)) +* (x2 .--> y2) = (f +* (x2 .--> y2)) +* (x1 .--> y1) ) )

proof end;

registration

let X be set ;
let U be non empty set ;

cluster Relation-like X -defined U -valued Function-like total for set ;

existence

proof end;

end;

registration

let X be set ;
let U be non empty set ;
let P be X -defined U -valued total Relation;
let Q be U -defined total Relation;

cluster P >*> Q -> X -defined total for X -defined Relation;

coherence

proof end;

end;

theorem :: FOMODEL0:44

for X being set
for p, p1, p2 being FinSequence st (p ^ p1) ^ p2 is X -valued holds
( p2 is X -valued & p1 is X -valued & p is X -valued )

proof end;

registration

let X be set ;
let R be Relation;

cluster R null X -> X \/ (rng R) -valued for Relation;

coherence

proof end;

end;

registration

let X, Y be functional set ;

cluster X \/ Y -> functional ;

coherence

proof end;

end;

registration

cluster FinSequence-membered -> finite-membered for set ;

coherence

proof end;

end;

definition

let X be functional set ;

func SymbolsOf X -> set equals :: FOMODEL0:def 23
union { (rng x) where x is Element of X \/ {{}} : x in X } ;

coherence ;

end;

:: deftheorem defines SymbolsOf FOMODEL0:def 23 :

Lm50: for X being functional set st X is finite & X is finite-membered holds
SymbolsOf X is finite

proof end;

registration

cluster functional non empty trivial finite-membered FinSequence-membered for set ;

existence

proof end;

end;

registration

let X be functional finite finite-membered set ;

cluster SymbolsOf X -> finite for set ;

coherence by Lm50;

end;

registration

let X be finite FinSequence-membered set ;

cluster SymbolsOf X -> finite for set ;

coherence ;

end;

theorem Th45: :: FOMODEL0:45

for f being Function holds SymbolsOf {f} = rng f

proof end;

registration

let P be Function;

cluster (SymbolsOf {P}) \+\ (rng P) -> empty ;

coherence by Th29, Th45;

end;

registration

let z be non zero Complex;

cluster |.z.| -> positive for ExtReal;

coherence by COMPLEX1:47;

end;

scheme :: FOMODEL0:sch 1
Sc1{ F₁() -> set , F₂() -> set , F₃( set ) -> set } :

{ F₃(x) where x is Element of F₁() : x in F₁() } = { F₃(x) where x is Element of F₂() : x in F₁() }

provided

A1: F₁() c= F₂()

proof end;

definition

let X be functional set ;

redefine func SymbolsOf X equals :: FOMODEL0:def 24
union { (rng x) where x is Element of X : x in X } ;

compatibility

proof end;

end;

:: deftheorem defines SymbolsOf FOMODEL0:def 24 :

Lm51: for B being functional set
for A being Subset of B holds { (rng x) where x is Element of A : x in A } c= { (rng x) where x is Element of B : x in B }

proof end;

theorem :: FOMODEL0:46

for B being functional set
for A being Subset of B holds SymbolsOf A c= SymbolsOf B by Lm51, ZFMISC_1:77;

theorem Th47: :: FOMODEL0:47

for A, B being functional set holds SymbolsOf (A \/ B) = (SymbolsOf A) \/ (SymbolsOf B)

proof end;

registration

let X be set ;
let F be Subset of (bool X);

cluster (union F) \ X -> empty for set ;

coherence ;

end;

theorem Th48: :: FOMODEL0:48

for X, Y being set holds X = (X \ Y) \/ (X /\ Y)

proof end;

theorem :: FOMODEL0:49

for A, B being set
for m, n being Nat st m -tuples_on A meets n -tuples_on B holds
m = n by Lm5;

theorem :: FOMODEL0:50

for D being non empty set
for A, B being set st B is D -prefix & A c= B holds
A is D -prefix ;

theorem Th51: :: FOMODEL0:51

for f, g being Function holds
( f c= g iff for x being set st x in dom f holds
( x in dom g & f . x = g . x ) )

proof end;

registration

let X, Y be set ;

cluster ((X \ Y) \/ (X /\ Y)) \+\ X -> empty ;

coherence by Th29, Th48;
let Z be set ;

cluster ((X /\ Y) \/ (X /\ Z)) \+\ (X /\ (Y \/ Z)) -> empty ;

coherence by XBOOLE_1:23, Th29;

cluster ((X \ Y) \ Z) \+\ (X \ (Y \/ Z)) -> empty ;

coherence by XBOOLE_1:41, Th29;

end;

registration

let X, Y be functional set ;

cluster ((SymbolsOf X) \/ (SymbolsOf Y)) \+\ (SymbolsOf (X \/ Y)) -> empty ;

coherence by Th47, Th29;

end;

registration

let U be non empty set ;

cluster non empty -> non empty-yielding for Element of ((U *) \ {{}}) * ;

coherence

proof end;

end;

registration

let e be empty set ;

cluster -> empty for Element of e * ;

coherence ;

end;

theorem Th52: :: FOMODEL0:52

for x being set
for U1, U2 being non empty set
for p being FinSequence holds
( ( (U1 -multiCat) . x <> {} & (U2 -multiCat) . x <> {} implies (U1 -multiCat) . x = (U2 -multiCat) . x ) & ( p is {} * -valued implies (U1 -multiCat) . p = {} ) & ( (U1 -multiCat) . p = {} & p is U1 * -valued implies p is {} * -valued ) )

proof end;

registration

let U be non empty set ;
let x be set ;

cluster (U -multiCat) . x -> U -valued ;

coherence

proof end;

end;

definition

let x be set ;

func x null -> object equals :: FOMODEL0:def 25
x;

coherence ;

end;

:: deftheorem defines null FOMODEL0:def 25 :

registration

let x be set ;

reduce x null to x;

reducibility ;

end;

registration

let Y be with_non-empty_elements set ;

cluster Relation-like Y -valued non empty -> non empty-yielding Y -valued for set ;

coherence

proof end;

end;

registration

let X be set ;

cluster X \ {{}} -> with_non-empty_elements ;

coherence

proof end;

end;

registration

let X be with_non-empty_elements set ;

cluster -> with_non-empty_elements for Element of bool X;

coherence

proof end;

end;

registration

let U be non empty set ;

cluster U * -> infinite for set ;

coherence

proof end;

end;

registration

let U be non empty set ;

cluster U * -> with_non-empty_element for set ;

coherence ;

end;

registration

let X be with_non-empty_element set ;

cluster non empty with_non-empty_elements for Element of bool X;

existence

proof end;

end;

Lm52: for Y being set
for U being non empty set
for p being FinSequence st p <> {} & p is Y -valued & Y c= U * & Y is with_non-empty_elements holds
(U -multiCat) . p <> {}

proof end;

theorem :: FOMODEL0:53

for Y being set
for U1, U2 being non empty set
for p being FinSequence st U1 c= U2 & Y c= U1 * & p is Y -valued & p <> {} & Y is with_non-empty_elements holds
(U1 -multiCat) . p = (U2 -multiCat) . p

proof end;

theorem :: FOMODEL0:54

for X, x being set
for U being non empty set st ex p being FinSequence st
( x = p & p is X * -valued ) holds
(U -multiCat) . x is X -valued

proof end;

registration

let X be set ;
let m be Nat;

cluster (m -tuples_on X) \ (X *) -> empty for set ;

coherence

proof end;

end;

theorem :: FOMODEL0:55

for A, B being set holds (A /\ B) * = (A *) /\ (B *)

proof end;

theorem Th56: :: FOMODEL0:56

for X being set
for P, Q being Relation holds (P \/ Q) | X = (P | X) \/ (Q | X)

proof end;

registration

let X be set ;

cluster (bool X) \ X -> non empty for set ;

coherence

proof end;

end;

registration

let X be set ;
let R be Relation;

cluster R null X -> X \/ (dom R) -defined for Relation;

coherence

proof end;

end;

theorem Th57: :: FOMODEL0:57

for X being set
for f, g being Function holds (f | X) +* g = (f | (X \ (dom g))) \/ g

proof end;

registration

let X be set ;
let f be X -defined Function;
let g be X -defined total Function;

identify f +* g with g null f;

compatibility

proof end;

identify g null f with f +* g;

compatibility ;

end;

theorem Th58: :: FOMODEL0:58

for A, X, y being set st not y in proj2 X holds
[:A,{y}:] misses X

proof end;

definition

let X be set ;

func X -freeCountableSet -> set equals :: FOMODEL0:def 26
[:NAT,{ the Element of (bool (proj2 X)) \ (proj2 X)}:];

coherence ;

end;

:: deftheorem defines -freeCountableSet FOMODEL0:def 26 :

theorem Th59: :: FOMODEL0:59

for X being set holds
( (X -freeCountableSet) /\ X = {} & X -freeCountableSet is infinite )

proof end;

registration

let X be set ;

cluster X -freeCountableSet -> infinite for set ;

coherence ;

end;

registration

let X be set ;

cluster (X -freeCountableSet) typed/\ X -> empty ;

coherence by Th59;

end;

registration

let X be set ;

cluster X -freeCountableSet -> countable for set ;

coherence by CARD_4:7;

end;

registration

cluster NAT \ INT -> empty ;

coherence

proof end;

end;

registration

let x be set ;
let p be FinSequence;

cluster ((<*x*> ^ p) . 1) \+\ x -> empty for set ;

coherence

proof end;

end;

registration

let m be Nat;
let m0 be zero number ;
let p be m -element FinSequence;

cluster p null m0 -> Seg (m + m0) -defined total for Seg (m + m0) -defined Relation;

coherence

proof end;

end;

registration

let U be non empty set ;
let q1, q2 be U -valued FinSequence;

cluster ((U -multiCat) . <*q1,q2*>) \+\ (q1 ^ q2) -> empty for set ;

coherence

proof end;

end;

registration

let f be Function;
let e be empty set ;

identify f +* e with f null e;

compatibility ;

identify f null e with f +* e;

compatibility ;

identify e +* f with f null e;

compatibility ;

identify f null e with e +* f;

compatibility ;

end;

registration

let X be infinite set ;

cluster denumerable for Element of bool X;

existence

proof end;

end;

registration

let X be finite-membered countable set ;

cluster union X -> countable for set ;

coherence

proof end;

end;

registration

let X be functional finite-membered countable set ;

cluster SymbolsOf X -> countable for set ;

coherence

proof end;

end;

registration

let A, B be countable set ;

cluster A \/ B -> countable for set ;

coherence by CARD_2:85;

end;

theorem Th60: :: FOMODEL0:60

for X, Y being set st union X c= Y holds
X c= bool Y

proof end;

theorem Th61: :: FOMODEL0:61

for A, B, Y, X being set st A is_finer_than B & X is_finer_than Y holds
A \/ X is_finer_than B \/ Y

proof end;

theorem Th62: :: FOMODEL0:62

for A, B being set st A is_finer_than B holds
A \/ B is_finer_than B

proof end;

theorem Th63: :: FOMODEL0:63

for A, B being set st B is c=directed & A is_finer_than B holds
A \/ B is c=directed

proof end;

theorem Th64: :: FOMODEL0:64

for X, Y being set holds INTERSECTION (X,Y) is_finer_than X

proof end;

theorem :: FOMODEL0:65

for Y being set st Y is c=directed holds
for X being finite Subset of (union Y) ex y being set st
( y in Y & X c= y )

proof end;

Lm53: for A, B, X being set st X is Subset of (Funcs (A,B)) holds
X is Subset-Family of [:A,B:]

proof end;

theorem :: FOMODEL0:66

for A, B, X being set st X is Subset of (Funcs (A,B)) holds
X is Subset-Family of [:A,B:] by Lm53;

Lm54: for Y being set
for U being non empty set
for x, y being Element of U holds
not ( ( x in Y implies y in Y ) & ( y in Y implies x in Y ) & not [((chi (Y,U)) . x),((chi (Y,U)) . y)] in id BOOLEAN )

proof end;

theorem :: FOMODEL0:67

for Y being set
for U being non empty set
for x, y being Element of U holds
not ( ( x in Y implies y in Y ) & ( y in Y implies x in Y ) & not [((chi (Y,U)) . x),((chi (Y,U)) . y)] in id BOOLEAN ) by Lm54;

definition

let A be set ;
let R be Relation;

func R typed| A -> Subset of R equals :: FOMODEL0:def 27
R | A;

coherence by RELAT_1:59;

end;

:: deftheorem defines typed| FOMODEL0:def 27 :

registration

let A be set ;
let R be Relation;

identify R typed| A with R | A;

compatibility ;

identify R | A with R typed| A;

compatibility ;

end;

Lm55: for A, B being set
for R being Relation holds R | (A \ B) = (R | A) \ [:B,(rng R):]

proof end;

Lm56: for f, g being Function holds f +* g = (f \ [:(dom g),(rng f):]) \/ g

proof end;

Lm57: for A, B, C being set holds A /\ C = A \ ((A \/ B) \ C)

proof end;

registration

let A, B, C be set ;

cluster (A \ ((A \/ B) \ C)) \+\ (A /\ C) -> empty ;

coherence by Lm57, Th29;

end;

definition

let X, P be set ;

func P |1 X -> set equals :: FOMODEL0:def 28
P /\ [:X,(proj2 P):];

coherence ;

end;

:: deftheorem defines |1 FOMODEL0:def 28 :

registration

let X be set ;
let P be Relation;

identify P |1 X with P | X;

compatibility by RELAT_1:67;

identify P | X with P |1 X;

compatibility ;

end;

Lm58: for X being set
for P being Relation holds P | ((dom P) \ X) = P \ [:X,(rng P):]

proof end;

definition

let P, Q be Relation;

func P +*1 Q -> object equals :: FOMODEL0:def 29
(P \ [:(dom Q),(rng P):]) \/ Q;

coherence ;

func P +*2 Q -> object equals :: FOMODEL0:def 30
(P | ((dom P) \ (dom Q))) \/ Q;

coherence ;

func P +*3 Q -> object equals :: FOMODEL0:def 31
((P | (dom P)) \ (P | (dom Q))) \/ Q;

coherence ;

end;

:: deftheorem defines +*1 FOMODEL0:def 29 :

:: deftheorem defines +*2 FOMODEL0:def 30 :

:: deftheorem defines +*3 FOMODEL0:def 31 :

registration

let P, Q be Relation;

identify P +*1 Q with P +*2 Q;

compatibility by Lm58;

identify P +*2 Q with P +*3 Q;

compatibility by RELAT_1:80;

end;

registration

let f, g be Function;

identify f +* g with f +*1 g;

compatibility by Lm56;

identify f +*1 g with f +* g;

compatibility ;

end;

registration

let U be non empty set ;
let u be Element of U;
let q be U -valued FinSequence;
set f = U -firstChar ;
set p = <*u*>;
set P = <*u*> ^ q;

cluster ((U -firstChar) . (<*u*> ^ q)) \+\ u -> empty ;

coherence

proof end;

end;

registration

cluster V109() integer ext-real negative complex for object ;

existence

proof end;

cluster integer positive -> natural for object ;

coherence ;

end;

registration

let X, Y be set ;
let x be Subset of X;
let y be Subset of Y;

cluster (x \ Y) \ (X \ y) -> empty for set ;

coherence

proof end;

end;

registration

let X, Y be set ;
let x be Subset of X;
let y be Subset of Y;

cluster (x \ Y) \ (X \ y) -> empty for set ;

coherence ;

end;

registration

let X, Y be set ;
let x be Subset of X;

cluster (x \ Y) \ (X \ Y) -> empty for set ;

coherence

proof end;

end;

registration

let X, Y be set ;
let x be Subset of X;
let y be Subset of Y;

cluster ((x \/ y) \ Y) \ X -> empty for set ;

coherence

proof end;

end;

theorem :: FOMODEL0:68

for R being Relation
for f being Function st dom f c= dom R & R c= f holds
R = f by Lm42;

registration

let T be trivial set ;
let x, y be Element of T;

cluster x \+\ y -> empty for set ;

coherence

proof end;

end;

scheme :: FOMODEL0:sch 2
FraenkelTrivial{ F₁() -> trivial set , F₂( object ) -> object , P₁[ set , set ] } :

{ F₂(x) where x is Element of F₁() : P₁[x,F₁()] } c= {F₂( the Element of F₁())}

proof end;

scheme :: FOMODEL0:sch 3
OnSingleTonsGen{ F₁() -> set , F₂( set ) -> set , P₁[ set ] } :

{ [o,F₂(o)] where o is Element of F₁() : P₁[o] } is Function

proof end;

scheme :: FOMODEL0:sch 4
FraenkelInclusion{ F₁() -> set , F₂( set , set ) -> set , P₁[ set , set ] } :

F₁() c= { F₂(x,F₁()) where x is Element of F₁() : P₁[x,F₁()] }

provided

A1: for y being set st y in F₁() holds
ex x being set st
( x in F₁() & P₁[x,F₁()] & y = F₂(x,F₁()) )

proof end;

scheme :: FOMODEL0:sch 5
FraenkelInclusion2{ F₁() -> set , P₁[ set , set ] } :

F₁() c= { x where x is Element of F₁() : P₁[x,F₁()] }

provided

A1: for x being set st x in F₁() holds
P₁[x,F₁()]

proof end;

scheme :: FOMODEL0:sch 6
FraenkelEq{ F₁() -> non empty set , P₁[ set ] } :

F₁() = { x where x is Element of F₁() : P₁[x] }

provided

A1: for x being set st x in F₁() holds
P₁[x]

proof end;

registration

let Y be set ;
let X be Subset of Y;

cluster (proj1 X) \ (proj1 Y) -> empty ;

coherence by XTUPLE_0:8, Th29;

cluster (proj2 X) \ (proj2 Y) -> empty ;

coherence by XTUPLE_0:9, Th29;

end;

registration

let X, Y be set ;

cluster (proj1 [:X,Y:]) \ X -> empty ;

coherence by FUNCT_5:10, Th29;

cluster ((proj1 [:X,Y:]) /\ X) \+\ (proj1 [:X,Y:]) -> empty ;

coherence

proof end;

cluster (proj2 [:X,Y:]) \ Y -> empty ;

coherence by FUNCT_5:10, Th29;

cluster ((proj2 [:X,Y:]) /\ Y) \+\ (proj2 [:X,Y:]) -> empty ;

coherence

proof end;

cluster ((proj2 X) \/ (proj2 Y)) \+\ (proj2 (X \/ Y)) -> empty ;

coherence by XTUPLE_0:27, Th29;

cluster ((proj1 X) \/ (proj1 Y)) \+\ (proj1 (X \/ Y)) -> empty ;

coherence by XTUPLE_0:23, Th29;
let y be Subset of Y;

cluster (X \ Y) typed/\ y -> empty ;

coherence

proof end;

end;

registration

let X be set ;
let U be non empty set ;

cluster (proj1 [:X,U:]) \+\ X -> empty ;

coherence

proof end;

cluster (proj2 [:U,X:]) \+\ X -> empty ;

coherence

proof end;

end;

registration

let X be set ;
let Y be non empty set ;

reduce proj1 [:X,Y:] to X;

reducibility

proof end;

reduce proj2 [:Y,X:] to X;

reducibility

proof end;

end;

registration

let R be Relation;
let X be set ;

cluster (R .: X) \ (rng R) -> empty ;

coherence by RELAT_1:111, Th29;

end;

registration

let A, B, X, Y be set ;

cluster [:A,B:] \ [:(A \/ X),(B \/ Y):] -> empty ;

coherence

proof end;

end;

registration

let X, Y, Z be set ;

cluster ((X /\ Y) /\ Z) \+\ (X /\ (Y /\ Z)) -> empty ;

coherence by Th29, XBOOLE_1:16;

end;

registration

let X, Y be set ;

reduce (X \ Y) \/ (X /\ Y) to X;

reducibility by Th48;

end;

registration

let P be Relation;
let X be Subset of (dom P);

reduce dom (P | X) to X;

reducibility by RELAT_1:62;

end;

registration

let X be set ;
let x, y be Subset of X;

cluster (x \/ y) \ X -> empty ;

coherence ;

end;

registration

let X be set ;

cluster id X -> onto ;

coherence ;

end;

registration

cluster Relation-like Function-like symmetric -> involutive for set ;

coherence

proof end;

end;

registration

cluster Relation-like Function-like non empty involutive for set ;

existence

proof end;

end;

registration

let f be non empty involutive Function;
let x be Element of dom f;

reduce f . (f . x) to x;

reducibility by PARTIT_2:def 2;

end;

registration

let X, Y be set ;

cluster [:X,Y:] -> Y -valued ;

coherence

proof end;

end;

registration

let x, y be set ;

cluster [:{x},{y}:] +* [:{y},{x}:] -> symmetric ;

coherence

proof end;

end;

registration

let X be set ;
let P be Relation;
let x be Subset of X;

cluster (P " x) \ (P " X) -> empty ;

coherence by RELAT_1:143, Th29;

end;

registration

let X be set ;
let P be Relation;

cluster (P " (X \/ (rng P))) \+\ (dom P) -> empty ;

coherence

proof end;

end;

registration

let X be set ;
let R be X -defined total Relation;

cluster X \+\ (dom R) -> empty ;

coherence by PARTFUN1:def 2, Th29;

end;

registration

let P be non empty Relation;

cluster -> pair for Element of P;

coherence

proof end;

end;

registration

let X be set ;
let Y be non empty set ;
let f be X -defined Y -valued total Function;

cluster {f} \ (Funcs (X,Y)) -> empty ;

coherence

proof end;

end;

registration

let X be non empty set ;
let f be X -defined X -valued total Function;

cluster (rng f) \ (dom f) -> empty ;

coherence

proof end;

end;

registration

let X be set ;
let Y be Subset of X;

reduce (id X) .: Y to Y;

reducibility by FUNCT_1:92;

end;

registration

let A, B be set ;
let a be Subset of A;

cluster (a \ (A \ B)) \+\ (a /\ B) -> empty ;

coherence

proof end;

end;

definition

let a, b be set ;

func a doubleton b -> object equals :: FOMODEL0:def 32
{a} \/ {b};

coherence ;

end;

:: deftheorem defines doubleton FOMODEL0:def 32 :

registration

let a, b be set ;

identify {a,b} with a doubleton b;

compatibility by ENUMSET1:1;

identify a doubleton b with {a,b};

compatibility ;

identify a doubleton b with b doubleton a;

compatibility ;

end;

definition

let X, Y be set ;

func X .:1 Y -> set equals :: FOMODEL0:def 33
proj2 (X |1 Y);

coherence ;

end;

:: deftheorem defines .:1 FOMODEL0:def 33 :

registration

let P be Relation;
let X be set ;

identify P .: X with P .:1 X;

compatibility by RELAT_1:115;

identify P .:1 X with P .: X;

compatibility ;

end;

notation

let P, Q be Relation;
synonym P >*> Q for P * Q;

end;

notation

let f, g be Function;
synonym g <*< f for g * f;

end;

definition

let P, Q be Relation;
:: original: +*1
redefine func P +*1 Q -> Subset of (P \/ Q);
coherence

proof end;

end;

Lm59: for f, g being Function
for x being Element of [:f,g:] st f <> {} & g <> {} holds
( x `1 is pair & x `2 is pair )

proof end;

registration

let f, g be non empty Function;
let x be Element of [:f,g:];

cluster x `1 -> pair ;

coherence by Lm59;

cluster x `2 -> pair ;

coherence by Lm59;

end;

Lm60: for B, C being set
for P being Relation holds P * [:B,C:] c= [:(P " B),C:]

proof end;

Lm61: for B, C being set
for P being Relation holds P * [:B,C:] = [:(P " B),C:]

proof end;

Lm62: for X, Y, Z, y being set st y <> {} & y c= Y holds
[:X,y:] * [:Y,Z:] = [:X,Z:]

proof end;

Lm63: for X being set
for P being Relation holds P * [:(rng P),X:] = [:(dom P),X:]

proof end;

Lm64: for f being Function st rng f = dom f holds
f is onto Function of (dom f),(dom f)

proof end;

Lm65: for f being Function st f is involutive holds
f * f c= id (dom f)

proof end;

Lm66: for X being set holds X /\ (id ((proj1 X) /\ (proj2 X))) = X /\ (id (proj1 X))

proof end;

Lm67: for X, Y being set holds (id X) | Y = id (X /\ Y)

proof end;

Lm68: for b1, b2 being set
for f1, f2 being Function st (rng f1) /\ (rng f2) = {} & b1 <> b2 & rng f1 c= dom f2 & rng f2 c= dom f1 holds
[:(rng f1),{b2}:] \/ [:(rng f2),{b1}:] c= ((f1 \/ f2) >*> ([:(rng f1),{b2}:] \/ [:(rng f2),{b1}:])) >*> ((b1,b2) --> (b2,b1))

proof end;

Lm69: for a, a1 being set
for f being Function st a in {a1,(f . a1)} & f is involutive & a1 in dom f holds
{a,(f . a)} = {a1,(f . a1)}

proof end;

Lm70: for a, a1 being set
for f being Function st ( a in {a1,(f . a1)} or f . a in {a1,(f . a1)} ) & f is involutive & a in dom f & a1 in dom f holds
{a,(f . a)} = {a1,(f . a1)}

proof end;

Lm71: for X being set
for f being Function st X c= f holds
dom (f \ X) = (dom f) \ (proj1 X)

proof end;

Lm72: for X being set st X <> {} holds
{ x where x is Element of X : verum } = X

proof end;

Lm73: for a1, a2, b1, b2 being set
for f, g being Function holds
( [[a1,a2],[b1,b2]] in [:f,g:] iff ( [a1,b1] in f & [a2,b2] in g ) )

proof end;

Lm74: for X, Y being set holds
( X = Y iff for x being set st x in X \/ Y holds
( x in X iff x in Y ) )

proof end;

Lm75: for x being set
for f, g being Function st x in [:f,g:] holds
x = [[((x `1) `1),((x `1) `2)],[((x `2) `1),((x `2) `2)]]

proof end;

Lm76: for f1, f2, g1, g2 being Function holds [:f1,f2:] /\ [:g1,g2:] = [:(f1 /\ g1),(f2 /\ g2):]

proof end;

definition

let P be Relation;

attr P is oneone means :Def34: :: FOMODEL0:def 34
P ~ is Function-like ;

end;

:: deftheorem Def34 defines oneone FOMODEL0:def 34 :

registration

cluster Relation-like Function-like one-to-one -> oneone for set ;

coherence ;

cluster Relation-like Function-like oneone -> one-to-one for set ;

coherence

proof end;

end;

registration

cluster Relation-like Function-like one-to-one oneone for set ;

existence

proof end;

end;

registration

let P be oneone Relation;

cluster P ~ -> Function-like ;

coherence by Def34;

end;

registration

cluster Relation-like Function-like one-to-one non empty for set ;

existence

proof end;

let P be oneone Relation;

cluster -> oneone for Element of bool P;

coherence

proof end;

end;

definition

let R be set ;

func fixpoints R -> set equals :: FOMODEL0:def 35
proj1 (R /\ (id (proj1 R)));

coherence ;

func fixpoints1 R -> set equals :: FOMODEL0:def 36
proj1 (R /\ (id ((proj1 R) /\ (proj2 R))));

coherence ;

end;

:: deftheorem defines fixpoints FOMODEL0:def 35 :

:: deftheorem defines fixpoints1 FOMODEL0:def 36 :

registration

let X be set ;
let R be Subset of (id X);

reduce R ~ to R;

reducibility

proof end;

end;

Lm77: for R being Relation holds fixpoints1 R = fixpoints1 (R ~)

proof end;

registration

let R be Relation;

cluster (fixpoints1 (R ~)) \+\ (fixpoints1 R) -> empty ;

coherence by Lm77, Th29;

end;

registration

let f be one-to-one Function;

cluster (fixpoints1 (f ")) \+\ (fixpoints1 f) -> empty ;

coherence

proof end;

end;

registration

let R be set ;

identify fixpoints R with fixpoints1 R;

compatibility by Lm66;

identify fixpoints1 R with fixpoints R;

compatibility ;

end;

Lm78: for f being Function
for x being object holds
( x in fixpoints f iff x is_a_fixpoint_of f )

proof end;

registration

let X be set ;

reduce fixpoints (id X) to X;

reducibility ;

end;

Lm79: for f, g being Function holds fixpoints [:f,g:] = [:(fixpoints f),(fixpoints g):]

proof end;

registration

let X be non empty set ;

cluster id X -> with_fixpoint ;

coherence

proof end;

end;

registration

let X be non empty set ;

cluster Relation-like X -defined X -valued Function-like one-to-one non empty total quasi_total onto bijective symmetric with_fixpoint oneone for Element of bool [:X,X:];

existence

proof end;

end;

registration

cluster Relation-like Function-like non empty symmetric with_fixpoint for set ;

existence

proof end;

end;

registration

let f be with_fixpoint Function;

cluster dom f -> non empty ;

coherence

proof end;

end;

registration

cluster Relation-like Function-like empty -> without_fixpoints for set ;

coherence

proof end;

end;

registration

cluster Relation-like Function-like empty without_fixpoints for set ;

existence

proof end;

end;

registration

let f be with_fixpoint Function;

cluster fixpoints f -> non empty ;

coherence

proof end;

end;

registration

let f be without_fixpoints Function;

cluster fixpoints f -> empty ;

coherence by Lm78, ABIAN:def 5;

end;

definition

let X be set ;

attr X is constanT means :Def37: :: FOMODEL0:def 37
proj2 X is trivial ;

end;

:: deftheorem Def37 defines constanT FOMODEL0:def 37 :

registration

cluster empty -> constanT for set ;

coherence ;

end;

registration

cluster Relation-like Function-like empty constanT for set ;

existence

proof end;

end;

registration

cluster Relation-like Function-like constant -> constanT for set ;

coherence ;

cluster Relation-like Function-like constanT -> constant for set ;

coherence ;

end;

Lm80: for X being set
for P being Relation st not X is empty & X is trivial & P is X -valued holds
P is constant Function

proof end;

registration

let x be trivial set ;

cluster Relation-like x -valued -> Function-like for set ;

coherence

proof end;

end;

registration

let x be trivial set ;

cluster Relation-like x -valued Function-like -> constant for set ;

coherence ;

end;

registration

let P be Relation;
let Q be constanT Relation;

cluster P >*> Q -> constanT ;

coherence

proof end;

end;

registration

cluster constanT -> Function-like for set ;

coherence

proof end;

end;

registration

let P be Relation;
let c be constanT Relation;

cluster P >*> c -> Function-like ;

coherence ;

end;

registration

let X, Y be set ;
let x be trivial set ;

cluster [:X,Y:] >*> [:Y,x:] -> Function-like ;

coherence ;

end;

theorem :: FOMODEL0:69

for x being set
for f being Function holds
( x in fixpoints f iff x is_a_fixpoint_of f ) by Lm78;

theorem :: FOMODEL0:70

for a, a1 being set
for f being Function st a in {a1,(f . a1)} & f is involutive & a1 in dom f holds
{a,(f . a)} = {a1,(f . a1)} by Lm69;

theorem :: FOMODEL0:71

for a, a1 being set
for f being Function st ( a in {a1,(f . a1)} or f . a in {a1,(f . a1)} ) & f is involutive & a in dom f & a1 in dom f holds
{a,(f . a)} = {a1,(f . a1)} by Lm70;

theorem :: FOMODEL0:72

for f being Function st f is involutive holds
f * f c= id (dom f) by Lm65;

theorem :: FOMODEL0:73

for X, Y, Z, y being set st y <> {} & y c= Y holds
[:X,y:] * [:Y,Z:] = [:X,Z:] by Lm62;

theorem :: FOMODEL0:74

for X being set st X <> {} holds
{ x where x is Element of X : verum } = X by Lm72;

theorem :: FOMODEL0:75

for f being Function st rng f = dom f holds
f is onto Function of (dom f),(dom f) by Lm64;

theorem :: FOMODEL0:76

for f, g being Function holds fixpoints [:f,g:] = [:(fixpoints f),(fixpoints g):] by Lm79;

theorem :: FOMODEL0:77

for b1, b2 being set
for f1, f2 being Function st (rng f1) /\ (rng f2) = {} & b1 <> b2 & rng f1 c= dom f2 & rng f2 c= dom f1 holds
[:(rng f1),{b2}:] \/ [:(rng f2),{b1}:] c= ((f1 \/ f2) >*> ([:(rng f1),{b2}:] \/ [:(rng f2),{b1}:])) >*> ((b1,b2) --> (b2,b1)) by Lm68;

theorem :: FOMODEL0:78

for X being set
for f being Function st X c= f holds
dom (f \ X) = (dom f) \ (proj1 X) by Lm71;