FOMODEL0: Preliminaries to Classical First-order Model Theory

end;

:: deftheorem defines -pr1 FOMODEL0:def 4 :

Lm45: for Y being set
for g being Function st g is Y -valued & g is FinSequence-like holds
g is FinSequence of Y

proof end;

Lm28: for Y, A, B being set st Y c= Funcs (A,B) holds
union Y c= [:A,B:]

proof end;

LmOneOne: 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;

Lm12: for D being non empty set
for A, B being 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;

Lm18: 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;

Lm21: for i being Nat
for Y being set holds i -tuples_on Y = Funcs ((Seg i),Y)

proof end;

Lm11: for A, B being set
for m being Nat holds (m -tuples_on A) /\ (B *) c= m -tuples_on B

proof end;

theorem Lm22: :: 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;

Lm20: for A, B, C being set holds (Funcs (A,B)) /\ (Funcs (A,C)) = Funcs (A,(B /\ C))

proof end;

theorem Lm23: :: 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 Lm24: :: 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;

ThConcatenation: 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 Lm2: :: 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;

Lm13: for D being non empty set
for B, A being set
for f being BinOp of D st B is f -unambiguous & A c= B holds
A is f -unambiguous

proof end;

Lm14: for D being non empty set
for f being BinOp of D holds {} is f -unambiguous

proof end;

Lm27: 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;

proof end;

end;

registration

let D be set ;

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

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 ;

proof end;

end;

registration

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

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

proof end;

end;

definition

let X be set ;
let f be Function;

redefine attr f is X -one-to-one means :DefInj2: :: FOMODEL0:def 5
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 LmOneOne;

end;

:: deftheorem DefInj2 defines -one-to-one FOMODEL0:def 5 :

registration

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

cluster f -unambiguous for set ;

proof end;

end;

registration

let f be Function;
let x be set ;

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

proof end;

end;

registration

let e be empty set ;

identify e * with {e};

compatibility by FUNCT_7:17;

identify {e} with e * ;

end;

registration

cluster empty -> empty-membered for set ;

coherence ;
let e be empty set ;

cluster {e} -> empty-membered ;

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 ;

proof end;

end;

registration

let X be empty-membered set ;

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

proof end;

end;

registration

let A be set ;
let m0 be zero number ;

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

coherence by COMPUT_1:5;

end;

registration

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

cluster m1 -tuples_on e -> empty ;

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 /\ f -> f -unambiguous ;

coherence by Lm14;

end;

registration

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

cluster e /\ U -> U -prefix ;

proof end;

end;

registration

let U be non empty set ;

cluster U -prefix for set ;

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 :DefMultPlace: :: FOMODEL0:def 6
( dom it = NAT & it . 0 = x . 1 & ( for n being Nat holds it . (n + 1) = f . ((it . n),(x . (n + 2))) ) );

proof end;

proof end;

end;

:: deftheorem DefMultPlace defines MultPlace FOMODEL0:def 6 :

Lm1: 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 7
MultPlace (f,x);

end;

:: deftheorem defines MultPlace FOMODEL0:def 7 :

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 :DefMultPlace2: :: FOMODEL0:def 8
for x being Element of (D *) \ {{}} holds it . x = (MultPlace (f,x)) . ((len x) - 1);

proof end;

proof end;

end;

:: deftheorem DefMultPlace2 defines MultPlace FOMODEL0:def 8 :

LmMultPlace: 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;

Lm3: 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 :DefUnambiguous2: :: FOMODEL0:def 9
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;

end;

:: deftheorem DefUnambiguous2 defines -unambiguous FOMODEL0:def 9 :

Lm19: 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;

Lm5: 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;

Lm15: for D being non empty set
for Y being 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;

Lm17: 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;

Lm26: for D being non empty set
for Y being 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 10
MultPlace (D -pr1);

end;

:: deftheorem defines -firstChar FOMODEL0:def 10 :

Th2: for D being non empty set
for w being non empty FinSequence of D holds (D -firstChar) . w = w . 1

proof end;

theorem :: 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 11
({} .--> {}) +* (MultPlace (D -concatenation));

end;

:: deftheorem defines -multiCat FOMODEL0:def 11 :

definition

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

proof end;

end;

Lm6: for D being non empty set holds (D -multiCat) | (((D *) *) \ {{}}) = MultPlace (D -concatenation)

proof end;

Lm16: for D being non empty set
for Y being set holds (D -multiCat) | (Y \ {{}}) = (MultPlace (D -concatenation)) | Y

proof end;

Lm9: 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 ;

proof end;

end;

Lm59: for D being non empty set
for B, A being set st B is D -prefix & A c= B holds
A is D -prefix

proof end;

registration

let D be non empty set ;

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

proof end;

end;

theorem Lm25: :: FOMODEL0:7

for D being non empty set
for A being 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 D being non empty set
for A being 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 COMPUT_1:5;

theorem :: FOMODEL0:11

for Y being set
for i being Nat holds i -tuples_on Y = Funcs ((Seg i),Y) by Lm21;

::#extending FUNCT_2:111

theorem :: FOMODEL0:12

for x, A 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 LmMultPlace;

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 ;

proof end;

end;

registration

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

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

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;

proof end;

end;

registration

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

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

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;

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;

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;

proof end;

end;

registration

let D be non empty set ;

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

proof end;

end;

registration

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

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

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 ;

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 ;

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 12
f * <:p,q:>;

proof end;

end;

:: deftheorem defines AppliedPairwiseTo FOMODEL0:def 12 :

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 ;

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 13
NAT \/ ([:{0},NAT:] \ {[0,0]});

proof end;

end;

:: deftheorem defines INT FOMODEL0:def 13 :

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;

::##############################Automatizations###############################
::############################################################################
::#To automatize things like A/\B c= A

definition

let A, B be set ;

::##############################Automatizations###############################
::############################################################################
::#To automatize things like A/\B c= A

func A typed/\ B -> Subset of A equals :: FOMODEL0:def 14
A /\ B;

coherence by XBOOLE_1:17;

func A /\typed B -> Subset of B equals :: FOMODEL0:def 15
A /\ B;

coherence by XBOOLE_1:17;

end;

:: deftheorem defines typed/\ FOMODEL0:def 14 :

:: deftheorem defines /\typed FOMODEL0:def 15 :

registration

let A, B be set ;

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

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

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

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

end;

definition

let B, A be set ;

func A null B -> set equals :: FOMODEL0:def 16
A;

end;

:: deftheorem defines null FOMODEL0:def 16 :

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;

end;

registration

let A, B, C be set ;

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

proof end;

end;

definition

let A, B be set ;

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

end;

:: deftheorem defines typed\ FOMODEL0:def 17 :

registration

let A, B be set ;

identify A \ B with A typed\ B;

identify A typed\ B with A \ B;

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 element as this file is imported via definitions directive

func A \typed/ B -> Subset of (A \/ B) equals :: FOMODEL0:def 18
A;

coherence by XBOOLE_1:7;

end;

:: deftheorem defines \typed/ FOMODEL0:def 18 :

registration

let A, B be set ;

identify A \typed/ B with A null B;

identify A null B with A \typed/ B;

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;

end;

Lm43: for m being Nat
for g, f being Function st f c= g holds
iter (f,m) c= iter (g,m)

proof end;

Lm31: for R being Relation holds R [*] is_transitive_in field R

proof end;

Lm32: for R being Relation holds field (R [*]) c= field R

proof end;

Lm37: for R being Relation holds R [*] is_reflexive_in field R

proof end;

Lm34: for R being Relation holds field (R [*]) = field R

proof end;

registration

let R be Relation;

cluster R [*] -> transitive for Relation;

proof end;

cluster R [*] -> reflexive for Relation;

proof end;

end;

Lm38: for f being Function holds iter (f,0) c= f [*]

proof end;

Lm39: for m being Nat
for f being Function holds iter (f,(m + 1)) c= f [*]

proof end;

Lm40: for m being Nat
for f being Function holds iter (f,m) c= f [*]

proof end;

Lm35: for x being set
for m being Nat
for g, 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 :DefPlus: :: FOMODEL0:def 19
for z being complex number holds it . z = z + 1;

proof end;

proof end;

end;

:: deftheorem DefPlus defines plus FOMODEL0:def 19 :

Lm36: for x being set
for m being Nat
for f being Function
for p being FinSequence 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;

Lm41: 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;

Lm42: 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 Lm42;

theorem :: FOMODEL0:18

for m being Nat
for g, f being Function st f c= g holds
iter (f,m) c= iter (g,m) by Lm43;

registration

let X be functional set ;

cluster union X -> Relation-like ;

proof end;

end;

theorem :: FOMODEL0:19

for Y, A, B being set st Y c= Funcs (A,B) holds
union Y c= [:A,B:] by Lm28;

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 ;

proof end;

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;

identify q ^ p with p null q;

compatibility by FINSEQ_1:34;

identify p null q with q ^ p;

end;

registration

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

identify R | Y with R null Y;

proof end;

identify R null Y with R | Y;

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;

Lm4: for A, B being set holds chi (A,B) = (B --> 0) +* ((A /\ B) --> 1)

proof end;

Lm10: for A, B being set holds chi (A,B) = ((B \ A) --> 0) +* ((A /\ B) --> 1)

proof end;

Lm29: 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 Lm45;

registration

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

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

proof end;

end;

Lm46: 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;

Lm47: for R being Relation
for f being Function st dom f c= dom R & R c= f holds
R = f

proof end;

Lm48: for Y being set
for R, P 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;

theorem :: FOMODEL0:27

for B, A 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;

proof end;

end;

definition

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

redefine attr X is U -prefix means :: FOMODEL0:def 20
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 );

proof end;

end;

:: deftheorem defines -prefix FOMODEL0:def 20 :

registration

let X be set ;

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

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 Lm25;

end;

theorem Th29: :: FOMODEL0:29

for Y, X being set holds
( X \+\ Y = {} iff X = Y )

proof end;

registration

let x be set ;

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

proof end;

end;

registration

let x, y be set ;

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

proof end;

end;

registration

let x be set ;

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

proof end;

end;

theorem :: FOMODEL0:30

for D being non empty set
for x being set holds
( x in (D *) \ {{}} iff ( x is b₁ -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 ;

proof end;

end;

definition

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

func (x,y) -SymbolSubstIn p -> FinSequence equals :: FOMODEL0:def 21
p +~ (x,y);

end;

:: deftheorem defines -SymbolSubstIn FOMODEL0:def 21 :

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;

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;

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 22
((id X) +* (x,y)) * p;

proof end;

end;

:: deftheorem defines -SymbolSubstIn FOMODEL0:def 22 :

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 Lm45;

end;

Lm53: 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;

Lm50: 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 :DefSubst: :: FOMODEL0:def 23
for q being U -valued FinSequence holds it . q = (x,u) -SymbolSubstIn q;

proof end;

proof end;

end;

:: deftheorem DefSubst defines SubstWith FOMODEL0:def 23 :

Lm54: 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;

proof end;

end;

registration

let F be FinSequence-yielding Function;
let x be set ;

cluster F . x -> FinSequence-like ;

proof end;

end;

Lm55: 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;

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 ;

proof end;

end;

registration

let U be non empty set ;

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

proof end;

end;

registration

let U be non empty set ;

cluster Relation-like NAT -defined U -valued Function-like non empty finite FinSequence-like FinSubsequence-like countable finite-support for set ;

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 ;

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 ;

end;

registration

let x be set ;

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

end;

registration

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

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

proof end;

end;

registration

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

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

proof end;

end;

Lm51: for R being Relation holds
( R is {{}} -valued iff R is empty-yielding )

proof end;

registration

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

coherence by Lm51;

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

coherence by Lm51;

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;

Lm52: 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;

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 Lm53;

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 Lm54;

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 Lm55;

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 Lm52;

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 ;

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 ;

proof end;

end;

registration

cluster integer non negative -> natural integer for set ;

proof end;

end;

registration

let x, y be real number ;

cluster (max (x,y)) - x -> ext-real non negative for ext-real number ;

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 set ;

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

proof end;

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

proof end;

end;

registration

let x, y be set ;

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

proof end;

end;

registration

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

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

proof end;

end;

registration

let m1 be non zero Nat;

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

proof end;

end;

registration

let R be Relation;
let x be set ;

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

end;

registration

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

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

end;

registration

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

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

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 ;

end;

registration

let R be Relation;
let X be set ;

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

proof end;

end;

registration

let x be set ;
let e be empty set ;

cluster e null x -> empty for set ;

end;

registration

let X be set ;
let e be empty set ;

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

proof end;

end;

registration

let Y be non empty FinSequence-membered set ;

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

proof end;

end;

registration

let X, Y be set ;

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

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;

proof end;

end;

Lm56: 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;

proof end;

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 m be Nat;

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

proof end;

end;

registration

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

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

proof end;

end;

registration

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

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

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 ;

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;

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;

proof end;

end;

registration

let X, Y be functional set ;

cluster X \/ Y -> functional ;

proof end;

end;

registration

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

proof end;

end;

definition

let X be functional set ;

func SymbolsOf X -> set equals :: FOMODEL0:def 24
union { (rng x) where x is Element of X \/ {{}} : x in X } ;

end;

:: deftheorem defines SymbolsOf FOMODEL0:def 24 :

Lm58: for X being functional set st X is finite & X is finite-membered holds
SymbolsOf X is finite

proof end;

registration

cluster non empty trivial FinSequence-membered for set ;

proof end;

end;

registration

let X be functional finite finite-membered set ;

cluster SymbolsOf X -> finite ;

coherence by Lm58;

end;

registration

let X be finite FinSequence-membered set ;

cluster SymbolsOf X -> finite ;

end;

theorem :: FOMODEL0:45

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

proof end;

registration

let z be non zero complex number ;

cluster |.z.| -> ext-real positive for ext-real number ;

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

B0: F₁() c= F₂()

proof end;

definition

let X be functional set ;

redefine func SymbolsOf X equals :: FOMODEL0:def 25
union { (rng x) where x is Element of X : x in X } ;

proof end;

end;

:: deftheorem defines SymbolsOf FOMODEL0:def 25 :

Lm57: 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 Lm57, ZFMISC_1:77;

theorem :: 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 ;

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 Lm18;

theorem :: FOMODEL0:50

for D being non empty set
for B, A being set st B is D -prefix & A c= B holds
A is D -prefix by Lm59;

theorem :: 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 U be non empty set ;

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

proof end;

end;

registration

let e be empty set ;

cluster -> empty for Element of e * ;

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 ;

proof end;

end;

definition

let x be set ;

func x null -> set equals :: FOMODEL0:def 26
x;

end;

:: deftheorem defines null FOMODEL0:def 26 :

registration

let Y be with_non-empty_elements set ;

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

proof end;

end;

registration

let X be set ;

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

proof end;

end;

registration

let X be with_non-empty_elements set ;

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

proof end;

end;

registration

let U be non empty set ;

cluster U * -> infinite for set ;

proof end;

end;

registration

let U be non empty set ;

cluster U * -> with_non-empty_element ;

end;

registration

let X be with_non-empty_element set ;

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