:: COMPUT_1 semantic presentation

begin


theorem Th1: :: COMPUT_1:1
for x, y, z being set holds
( <*x,y*> +* (1,z) = <*z,y*> & <*x,y*> +* (2,z) = <*x,z*> )
proof
let x, y, z be set ; ::_thesis: ( <*x,y*> +* (1,z) = <*z,y*> & <*x,y*> +* (2,z) = <*x,z*> )
set a = <*x,y*> +* (1,z);
set b = <*x,y*> +* (2,z);
 <*x,y*> . 1 = x by FINSEQ_1:44;
then A1: (<*x,y*> +* (2,z)) . 1 = x by FUNCT_7:32;
 <*x,y*> . 2 = y by FINSEQ_1:44;
then A2: (<*x,y*> +* (1,z)) . 2 = y by FUNCT_7:32;
A3: len <*x,y*> = 2 by FINSEQ_1:44;
then  1 in dom <*x,y*> by FINSEQ_3:25;
then A4: (<*x,y*> +* (1,z)) . 1 = z by FUNCT_7:31;
 len (<*x,y*> +* (1,z)) = 2 by A3, FUNCT_7:97;
hence  <*x,y*> +* (1,z) = <*z,y*> by A4, A2, FINSEQ_1:44; ::_thesis: <*x,y*> +* (2,z) = <*x,z*>
 2 in dom <*x,y*> by A3, FINSEQ_3:25;
then A5: (<*x,y*> +* (2,z)) . 2 = z by FUNCT_7:31;
 len (<*x,y*> +* (2,z)) = 2 by A3, FUNCT_7:97;
hence  <*x,y*> +* (2,z) = <*x,z*> by A1, A5, FINSEQ_1:44; ::_thesis: verum
end;

theorem Th2: :: COMPUT_1:2
for a, x, y, z being set
for f, g being Function st f +* (a,x) = g +* (a,y) holds
f +* (a,z) = g +* (a,z)
proof
let a, x, y, z be set ; ::_thesis: for f, g being Function st f +* (a,x) = g +* (a,y) holds
f +* (a,z) = g +* (a,z)
let f, g be Function; ::_thesis: ( f +* (a,x) = g +* (a,y) implies f +* (a,z) = g +* (a,z) )
set i = a;
assume A1: f +* (a,x) = g +* (a,y) ; ::_thesis: f +* (a,z) = g +* (a,z)
A2: dom (g +* (a,z)) = dom g by FUNCT_7:30;
A3: dom (g +* (a,y)) = dom g by FUNCT_7:30;
A4: dom (f +* (a,x)) = dom f by FUNCT_7:30;
now__::_thesis:_(_dom_(f_+*_(a,z))_=_dom_f_&_dom_(g_+*_(a,z))_=_dom_f_&_(_for_a_being_set_st_a_in_dom_f_holds_
(f_+*_(a,z))_._a_=_(g_+*_(a,z))_._a_)_)
thus  dom (f +* (a,z)) = dom f by FUNCT_7:30; ::_thesis: ( dom (g +* (a,z)) = dom f & ( for a being set st a in dom f holds
(f +* (a,z)) . b2 = (g +* (a,z)) . b2 ) )
thus  dom (g +* (a,z)) = dom f by A1, A3, A2, FUNCT_7:30; ::_thesis: for a being set st a in dom f holds
(f +* (a,z)) . b2 = (g +* (a,z)) . b2
let a be set ; ::_thesis: ( a in dom f implies (f +* (a,z)) . b1 = (g +* (a,z)) . b1 )
assume A5: a in dom f ; ::_thesis: (f +* (a,z)) . b1 = (g +* (a,z)) . b1
percases ( a = a or a <> a ) ;
supposeA6: a = a ; ::_thesis: (f +* (a,z)) . b1 = (g +* (a,z)) . b1
hence (f +* (a,z)) . a = z by A5, FUNCT_7:31
.= (g +* (a,z)) . a by A1, A4, A3, A5, A6, FUNCT_7:31 ;
::_thesis: verum
end;
supposeA7: a <> a ; ::_thesis: (f +* (a,z)) . b1 = (g +* (a,z)) . b1
hence (f +* (a,z)) . a = f . a by FUNCT_7:32
.= (g +* (a,y)) . a by A1, A7, FUNCT_7:32
.= g . a by A7, FUNCT_7:32
.= (g +* (a,z)) . a by A7, FUNCT_7:32 ;
::_thesis: verum
end;
end;
end;
hence  f +* (a,z) = g +* (a,z) by FUNCT_1:2; ::_thesis: verum
end;

theorem Th3: :: COMPUT_1:3
for i being Element of NAT
for x being set
for p being FinSequence holds Del ((p +* (i,x)),i) = Del (p,i)
proof
let i be Element of NAT ; ::_thesis: for x being set
for p being FinSequence holds Del ((p +* (i,x)),i) = Del (p,i)
let x be set ; ::_thesis: for p being FinSequence holds Del ((p +* (i,x)),i) = Del (p,i)
let p be FinSequence; ::_thesis: Del ((p +* (i,x)),i) = Del (p,i)
set f = p;
percases ( i in dom p or not i in dom p ) ;
supposeA1: i in dom p ; ::_thesis: Del ((p +* (i,x)),i) = Del (p,i)
then A2: i <= len p by FINSEQ_3:25;
 1 <= i by A1, FINSEQ_3:25;
then consider j being Nat such that
A3: len p = j + 1 by A2, NAT_1:6;
reconsider j = j as Element of NAT by ORDINAL1:def_12;
A4: dom (p +* (i,x)) = dom p by FUNCT_7:30;
then A5: len (p +* (i,x)) = len p by FINSEQ_3:29;
then  len (Del ((p +* (i,x)),i)) = j by A1, A3, A4, FINSEQ_3:109;
then A6: dom (Del ((p +* (i,x)),i)) = Seg j by FINSEQ_1:def_3;
now__::_thesis:_(_len_(Del_((p_+*_(i,x)),i))_=_j_&_len_(Del_(p,i))_=_j_&_(_for_a_being_Nat_st_a_in_dom_(Del_((p_+*_(i,x)),i))_holds_
(Del_((p_+*_(i,x)),i))_._a_=_(Del_(p,i))_._a_)_)
thus  len (Del ((p +* (i,x)),i)) = j by A1, A3, A4, A5, FINSEQ_3:109; ::_thesis: ( len (Del (p,i)) = j & ( for a being Nat st a in dom (Del ((p +* (i,x)),i)) holds
(Del ((p +* (i,x)),i)) . b2 = (Del (p,i)) . b2 ) )
thus  len (Del (p,i)) = j by A1, A3, FINSEQ_3:109; ::_thesis: for a being Nat st a in dom (Del ((p +* (i,x)),i)) holds
(Del ((p +* (i,x)),i)) . b2 = (Del (p,i)) . b2
let a be Nat; ::_thesis: ( a in dom (Del ((p +* (i,x)),i)) implies (Del ((p +* (i,x)),i)) . b1 = (Del (p,i)) . b1 )
assume  a in dom (Del ((p +* (i,x)),i)) ; ::_thesis: (Del ((p +* (i,x)),i)) . b1 = (Del (p,i)) . b1
then A7: a <= j by A6, FINSEQ_1:1;
percases ( a < i or i <= a ) ;
supposeA8: a < i ; ::_thesis: (Del ((p +* (i,x)),i)) . b1 = (Del (p,i)) . b1
hence (Del ((p +* (i,x)),i)) . a = (p +* (i,x)) . a by FINSEQ_3:110
.= p . a by A8, FUNCT_7:32
.= (Del (p,i)) . a by A8, FINSEQ_3:110 ;
::_thesis: verum
end;
supposeA9: i <= a ; ::_thesis: (Del ((p +* (i,x)),i)) . b1 = (Del (p,i)) . b1
then A10: i < a + 1 by NAT_1:13;
thus (Del ((p +* (i,x)),i)) . a = (p +* (i,x)) . (a + 1) by A1, A3, A4, A5, A7, A9, FINSEQ_3:111
.= p . (a + 1) by A10, FUNCT_7:32
.= (Del (p,i)) . a by A1, A3, A7, A9, FINSEQ_3:111 ; ::_thesis: verum
end;
end;
end;
hence  Del ((p +* (i,x)),i) = Del (p,i) by FINSEQ_2:9; ::_thesis: verum
end;
suppose not i in dom p ; ::_thesis: Del ((p +* (i,x)),i) = Del (p,i)
hence  Del ((p +* (i,x)),i) = Del (p,i) by FUNCT_7:def_3; ::_thesis: verum
end;
end;
end;

theorem Th4: :: COMPUT_1:4
for i being Element of NAT
for a being set
for p, q being FinSequence st p +* (i,a) = q +* (i,a) holds
Del (p,i) = Del (q,i)
proof
let i be Element of NAT ; ::_thesis: for a being set
for p, q being FinSequence st p +* (i,a) = q +* (i,a) holds
Del (p,i) = Del (q,i)
let a be set ; ::_thesis: for p, q being FinSequence st p +* (i,a) = q +* (i,a) holds
Del (p,i) = Del (q,i)
let p, q be FinSequence; ::_thesis: ( p +* (i,a) = q +* (i,a) implies Del (p,i) = Del (q,i) )
set x = p;
set y = q;
assume A1: p +* (i,a) = q +* (i,a) ; ::_thesis: Del (p,i) = Del (q,i)
set xi = p +* (i,a);
set yi = q +* (i,a);
set dx = Del (p,i);
set dy = Del (q,i);
A2: dom (p +* (i,a)) = dom p by FUNCT_7:30;
A3: dom (q +* (i,a)) = dom q by FUNCT_7:30;
A4: Seg (len q) = dom q by FINSEQ_1:def_3;
A5: Seg (len p) = dom p by FINSEQ_1:def_3;
percases ( i in dom p or not i in dom p ) ;
supposeA6: i in dom p ; ::_thesis: Del (p,i) = Del (q,i)
A7: dom (Del (p,i)) = Seg (len (Del (p,i))) by FINSEQ_1:def_3;
now__::_thesis:_(_len_(Del_(p,i))_=_len_(Del_(p,i))_&_len_(Del_(q,i))_=_len_(Del_(p,i))_&_(_for_j_being_Nat_st_j_in_dom_(Del_(p,i))_holds_
(Del_(p,i))_._j_=_(Del_(q,i))_._j_)_)
thus  len (Del (p,i)) = len (Del (p,i)) ; ::_thesis: ( len (Del (q,i)) = len (Del (p,i)) & ( for j being Nat st j in dom (Del (p,i)) holds
(Del (p,i)) . b2 = (Del (q,i)) . b2 ) )
 p <> {} by A6;
then consider m being Nat such that
A8: len p = m + 1 by NAT_1:6;
reconsider m = m as Element of NAT by ORDINAL1:def_12;
A9: len (Del (p,i)) = m by A6, A8, FINSEQ_3:109;
hence  len (Del (q,i)) = len (Del (p,i)) by A1, A2, A3, A5, A4, A6, A8, FINSEQ_1:6, FINSEQ_3:109; ::_thesis: for j being Nat st j in dom (Del (p,i)) holds
(Del (p,i)) . b2 = (Del (q,i)) . b2
let j be Nat; ::_thesis: ( j in dom (Del (p,i)) implies (Del (p,i)) . b1 = (Del (q,i)) . b1 )
assume  j in dom (Del (p,i)) ; ::_thesis: (Del (p,i)) . b1 = (Del (q,i)) . b1
then A10: j <= m by A7, A9, FINSEQ_1:1;
percases ( j < i or i <= j ) ;
supposeA11: j < i ; ::_thesis: (Del (p,i)) . b1 = (Del (q,i)) . b1
hence (Del (p,i)) . j = p . j by FINSEQ_3:110
.= (q +* (i,a)) . j by A1, A11, FUNCT_7:32
.= q . j by A11, FUNCT_7:32
.= (Del (q,i)) . j by A11, FINSEQ_3:110 ;
::_thesis: verum
end;
supposeA12: i <= j ; ::_thesis: (Del (p,i)) . b1 = (Del (q,i)) . b1
then A13: j + 1 > i by NAT_1:13;
thus (Del (p,i)) . j = p . (j + 1) by A6, A8, A10, A12, FINSEQ_3:111
.= (q +* (i,a)) . (j + 1) by A1, A13, FUNCT_7:32
.= q . (j + 1) by A13, FUNCT_7:32
.= (Del (q,i)) . j by A1, A2, A3, A5, A4, A6, A8, A10, A12, FINSEQ_1:6, FINSEQ_3:111 ; ::_thesis: verum
end;
end;
end;
hence  Del (p,i) = Del (q,i) by FINSEQ_2:9; ::_thesis: verum
end;
supposeA14: not i in dom p ; ::_thesis: Del (p,i) = Del (q,i)
then  p +* (i,a) = p by FUNCT_7:def_3;
hence  Del (p,i) = Del (q,i) by A1, A3, A14, FUNCT_7:def_3; ::_thesis: verum
end;
end;
end;

theorem Th5: :: COMPUT_1:5
for X being set holds 0 -tuples_on X = {{}}
proof
let X be set ; ::_thesis: 0 -tuples_on X = {{}}
set S =  {  s where s is Element of X * : len s = 0  }  ;
now__::_thesis:_for_x_being_set_holds_
(_(_x_in__{__s_where_s_is_Element_of_X_*_:_len_s_=_0__}__implies_x_in_{{}}_)_&_(_x_in_{{}}_implies_x_in__{__s_where_s_is_Element_of_X_*_:_len_s_=_0__}__)_)
let x be set ; ::_thesis: ( ( x in  {  s where s is Element of X * : len s = 0  }  implies x in {{}} ) & ( x in {{}} implies x in  {  s where s is Element of X * : len s = 0  }  ) )
hereby  ::_thesis: ( x in {{}} implies x in  {  s where s is Element of X * : len s = 0  }  )
assume  x in  {  s where s is Element of X * : len s = 0  }  ; ::_thesis: x in {{}}
then consider s being Element of X *  such that
A1: x = s and
A2: len s = 0 ;
 s = {} by A2;
hence  x in {{}} by A1, TARSKI:def_1; ::_thesis: verum
end;
assume  x in {{}} ; ::_thesis: x in  {  s where s is Element of X * : len s = 0  }
then A3: x = {} by TARSKI:def_1;
 <*> (X *) is Element of X * by FINSEQ_1:49;
hence  x in  {  s where s is Element of X * : len s = 0  }  by A3, CARD_1:27; ::_thesis: verum
end;
hence  0 -tuples_on X = {{}} by TARSKI:1; ::_thesis: verum
end;

theorem :: COMPUT_1:6
for n being Element of NAT st n <> 0 holds
n -tuples_on {} = {}
proof
let n be Element of NAT ; ::_thesis: ( n <> 0 implies n -tuples_on {} = {} )
assume that
A1: n <> 0 and
A2: n -tuples_on {} <> {} ; ::_thesis: contradiction
consider x being set  such that
A3: x in n -tuples_on {} by A2, XBOOLE_0:def_1;
 ex s being Element of {} * st
( s = x & len s = n ) by A3;
hence  contradiction by A1; ::_thesis: verum
end;

theorem Th7: :: COMPUT_1:7
for f being Function st {} in rng f holds
<:f:> = {}
proof
let f be Function; ::_thesis: ( {} in rng f implies <:f:> = {} )
A1: dom <:f:> = meet (doms f) by FUNCT_6:29
.= meet (rng (doms f)) by FUNCT_6:def_4 ;
assume  {} in rng f ; ::_thesis: <:f:> = {}
then consider x being set  such that
A2: x in dom f and
A3: f . x = {} by FUNCT_1:def_3;
A4: dom (doms f) = f " (SubFuncs (rng f)) by FUNCT_6:def_2;
then A5: x in dom (doms f) by A2, A3, FUNCT_6:19;
then  (doms f) . x = {} by A3, A4, FUNCT_5:8, FUNCT_6:def_2;
hence  <:f:> = {} by A1, A5, FUNCT_1:3, SETFAM_1:4; ::_thesis: verum
end;

theorem Th8: :: COMPUT_1:8
for D being non empty set
for f being Function st rng f = D holds
rng <:<*f*>:> = 1 -tuples_on D
proof
let D be non empty set ; ::_thesis: for f being Function st rng f = D holds
rng <:<*f*>:> = 1 -tuples_on D
let f be Function; ::_thesis: ( rng f = D implies rng <:<*f*>:> = 1 -tuples_on D )
set X = D;
A1: dom <:<*f*>:> = dom f by FINSEQ_3:141;
assume A2: rng f = D ; ::_thesis: rng <:<*f*>:> = 1 -tuples_on D
now__::_thesis:_for_x_being_set_holds_
(_(_x_in_rng_<:<*f*>:>_implies_x_in_1_-tuples_on_D_)_&_(_x_in_1_-tuples_on_D_implies_x_in_rng_<:<*f*>:>_)_)
let x be set ; ::_thesis: ( ( x in rng <:<*f*>:> implies x in 1 -tuples_on D ) & ( x in 1 -tuples_on D implies x in rng <:<*f*>:> ) )
hereby  ::_thesis: ( x in 1 -tuples_on D implies x in rng <:<*f*>:> )
assume  x in rng <:<*f*>:> ; ::_thesis: x in 1 -tuples_on D
then consider y being set  such that
A3: y in dom <:<*f*>:> and
A4: <:<*f*>:> . y = x by FUNCT_1:def_3;
reconsider fy = f . y as Element of D by A2, A1, A3, FUNCT_1:3;
A5: <*fy*> is Element of 1 -tuples_on D by FINSEQ_2:131;
 <:<*f*>:> . y = <*(f . y)*> by A1, A3, FINSEQ_3:141;
hence  x in 1 -tuples_on D by A4, A5; ::_thesis: verum
end;
assume  x in 1 -tuples_on D ; ::_thesis: x in rng <:<*f*>:>
then  x is Tuple of 1,D by FINSEQ_2:131;
then consider d being Element of D such that
A6: x = <*d*> by FINSEQ_2:97;
consider y being set  such that
A7: y in dom f and
A8: f . y = d by A2, FUNCT_1:def_3;
 <:<*f*>:> . y = <*d*> by A7, A8, FINSEQ_3:141;
hence  x in rng <:<*f*>:> by A1, A6, A7, FUNCT_1:3; ::_thesis: verum
end;
hence  rng <:<*f*>:> = 1 -tuples_on D by TARSKI:1; ::_thesis: verum
end;

theorem Th9: :: COMPUT_1:9
for i, n being Element of NAT
for D being non empty set st 1 <= i & i <= n + 1 holds
for p being Element of (n + 1) -tuples_on D holds Del (p,i) in n -tuples_on D
proof
let i, n be Element of NAT ; ::_thesis: for D being non empty set st 1 <= i & i <= n + 1 holds
for p being Element of (n + 1) -tuples_on D holds Del (p,i) in n -tuples_on D
let D be non empty set ; ::_thesis: ( 1 <= i & i <= n + 1 implies for p being Element of (n + 1) -tuples_on D holds Del (p,i) in n -tuples_on D )
set X = D;
assume that
A1: 1 <= i and
A2: i <= n + 1 ; ::_thesis: for p being Element of (n + 1) -tuples_on D holds Del (p,i) in n -tuples_on D
let p be Element of (n + 1) -tuples_on D; ::_thesis: Del (p,i) in n -tuples_on D
A3: len p = n + 1 by CARD_1:def_7;
then  i in dom p by A1, A2, FINSEQ_3:25;
then A4: len (Del (p,i)) = n by A3, FINSEQ_3:109;
 Del (p,i) is FinSequence of D by FINSEQ_3:105;
then  Del (p,i) is Element of n -tuples_on D by A4, FINSEQ_2:92;
hence  Del (p,i) in n -tuples_on D ; ::_thesis: verum
end;

theorem Th10: :: COMPUT_1:10
for X being set
for Y being FinSequenceSet of X holds Y c= X *
proof
let X be set ; ::_thesis: for Y being FinSequenceSet of X holds Y c= X *
let Y be FinSequenceSet of X; ::_thesis: Y c= X *
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Y or x in X * )
assume  x in Y ; ::_thesis: x in X *
then  x is FinSequence of X by FINSEQ_2:def_3;
hence  x in X * by FINSEQ_1:def_11; ::_thesis: verum
end;

begin

definition
let X be set ;
attrX is compatible  means :Def1: :: COMPUT_1:def 1
 for f, g being Function st f in X & g in X holds
f tolerates g;
end;

:: deftheorem Def1 defines compatible COMPUT_1:def_1_:_
 for X being set holds
( X is compatible iff for f, g being Function st f in X & g in X holds
f tolerates g );

registration
cluster non empty functional compatible for set ;
existence
 ex b1 being set st
( not b1 is empty & b1 is functional & b1 is compatible )
proof
set A = {{}};
take  {{}} ; ::_thesis: ( not {{}} is empty & {{}} is functional & {{}} is compatible )
 {{}} is compatible
proof
let f, g be Function; :: according to COMPUT_1:def_1 ::_thesis: ( f in {{}} & g in {{}} implies f tolerates g )
assume that
A1: f in {{}} and
 g in {{}} ; ::_thesis: f tolerates g
 f = {} by A1, TARSKI:def_1;
hence  f tolerates g by PARTFUN1:54, XBOOLE_1:2; ::_thesis: verum
end;
hence  ( not {{}} is empty & {{}} is functional & {{}} is compatible ) ; ::_thesis: verum
end;
end;

registration
let X be functional compatible set ;
cluster union X ->  Relation-like Function-like ;
coherence
 ( union X is Function-like & union X is Relation-like )
proof
thus  union X is Function-like ::_thesis: union X is Relation-like
proof
let x, y1, y2 be set ; :: according to FUNCT_1:def_1 ::_thesis: ( not [x,y1] in union X or not [x,y2] in union X or y1 = y2 )
assume that
A1: [x,y1] in union X and
A2: [x,y2] in union X ; ::_thesis: y1 = y2
consider f being set  such that
A3: [x,y1] in f and
A4: f in X by A1, TARSKI:def_4;
consider g being set  such that
A5: [x,y2] in g and
A6: g in X by A2, TARSKI:def_4;
reconsider f = f, g = g as Function by A4, A6;
A7: x in dom f by A3, XTUPLE_0:def_12;
then A8: f . x = y1 by A3, FUNCT_1:def_2;
A9: x in dom g by A5, XTUPLE_0:def_12;
then A10: g . x = y2 by A5, FUNCT_1:def_2;
A11: x in (dom f) /\ (dom g) by A7, A9, XBOOLE_0:def_4;
 f tolerates g by A4, A6, Def1;
hence  y1 = y2 by A8, A10, A11, PARTFUN1:def_4; ::_thesis: verum
end;
thus  union X is Relation-like ::_thesis: verum
proof
let x be set ; :: according to RELAT_1:def_1 ::_thesis: ( not x in union X or ex b1, b2 being set st x = [b1,b2] )
assume  x in union X ; ::_thesis: ex b1, b2 being set st x = [b1,b2]
then  ex f being set st
( x in f & f in X ) by TARSKI:def_4;
hence  ex b1, b2 being set st x = [b1,b2] by RELAT_1:def_1; ::_thesis: verum
end;
end;
end;

theorem Th11: :: COMPUT_1:11
for X being set holds
( ( X is functional & X is compatible ) iff union X is Function )
proof
let X be set ; ::_thesis: ( ( X is functional & X is compatible ) iff union X is Function )
now__::_thesis:_(_union_X_is_Function_implies_(_X_is_functional_&_X_is_compatible_)_)
assume A1: union X is Function ; ::_thesis: ( X is functional & X is compatible )
thus  X is functional ::_thesis: X is compatible
proof
let f be set ; :: according to FUNCT_1:def_13 ::_thesis: ( not f in X or f is set )
assume A2: f in X ; ::_thesis: f is set
A3: f is Function-like
proof
let x, y1, y2 be set ; :: according to FUNCT_1:def_1 ::_thesis: ( not [x,y1] in f or not [x,y2] in f or y1 = y2 )
assume that
A4: [x,y1] in f and
A5: [x,y2] in f ; ::_thesis: y1 = y2
A6: [x,y2] in union X by A2, A5, TARSKI:def_4;
 [x,y1] in union X by A2, A4, TARSKI:def_4;
hence  y1 = y2 by A1, A6, FUNCT_1:def_1; ::_thesis: verum
end;
 f is Relation-like
proof
let x be set ; :: according to RELAT_1:def_1 ::_thesis: ( not x in f or ex b1, b2 being set st x = [b1,b2] )
assume  x in f ; ::_thesis: ex b1, b2 being set st x = [b1,b2]
then  x in union X by A2, TARSKI:def_4;
hence  ex b1, b2 being set st x = [b1,b2] by A1, RELAT_1:def_1; ::_thesis: verum
end;
hence  f is set by A3; ::_thesis: verum
end;
thus  X is compatible ::_thesis: verum
proof
let f, g be Function; :: according to COMPUT_1:def_1 ::_thesis: ( f in X & g in X implies f tolerates g )
assume that
A7: f in X and
A8: g in X ; ::_thesis: f tolerates g
let x be set ; :: according to PARTFUN1:def_4 ::_thesis: ( not x in (dom f) /\ (dom g) or f . x = g . x )
assume A9: x in (dom f) /\ (dom g) ; ::_thesis: f . x = g . x
then A10: x in dom g by XBOOLE_0:def_4;
then consider y2 being set  such that
A11: [x,y2] in g by XTUPLE_0:def_12;
A12: [x,y2] in union X by A8, A11, TARSKI:def_4;
A13: x in dom f by A9, XBOOLE_0:def_4;
then consider y1 being set  such that
A14: [x,y1] in f by XTUPLE_0:def_12;
 [x,y1] in union X by A7, A14, TARSKI:def_4;
then A15: y1 = y2 by A1, A12, FUNCT_1:def_1;
thus f . x = y1 by A13, A14, FUNCT_1:def_2
.= g . x by A10, A11, A15, FUNCT_1:def_2 ; ::_thesis: verum
end;
end;
hence  ( ( X is functional & X is compatible ) iff union X is Function ) ; ::_thesis: verum
end;

registration
let X, Y be set ;
cluster non empty compatible for PartFunc-set of X,Y;
existence
 ex b1 being PFUNC_DOMAIN of X,Y st
( not b1 is empty & b1 is compatible )
proof
set A = {{}};
A1: {{}} is compatible
proof
let f, g be Function; :: according to COMPUT_1:def_1 ::_thesis: ( f in {{}} & g in {{}} implies f tolerates g )
assume that
A2: f in {{}} and
 g in {{}} ; ::_thesis: f tolerates g
 f = {} by A2, TARSKI:def_1;
hence  f tolerates g by PARTFUN1:54, XBOOLE_1:2; ::_thesis: verum
end;
now__::_thesis:_for_x_being_Element_of_{{}}_holds_x_is_PartFunc_of_X,Y
let x be Element of {{}}; ::_thesis: x is PartFunc of X,Y
 x = {} by TARSKI:def_1;
hence  x is PartFunc of X,Y by XBOOLE_1:2; ::_thesis: verum
end;
then  {{}} is PFUNC_DOMAIN of X,Y by RFUNCT_3:def_3;
hence  ex b1 being PFUNC_DOMAIN of X,Y st
( not b1 is empty & b1 is compatible ) by A1; ::_thesis: verum
end;
end;

theorem Th12: :: COMPUT_1:12
for X being non empty functional compatible set holds dom (union X) = union  {  (dom f) where f is Element of X : verum  }
proof
let X be non empty functional compatible set ; ::_thesis: dom (union X) = union  {  (dom f) where f is Element of X : verum  }
set F =  {  (dom f) where f is Element of X : verum  }  ;
now__::_thesis:_for_x_being_set_holds_
(_(_x_in_dom_(union_X)_implies_x_in_union__{__(dom_f)_where_f_is_Element_of_X_:_verum__}__)_&_(_x_in_union__{__(dom_f)_where_f_is_Element_of_X_:_verum__}__implies_x_in_dom_(union_X)_)_)
let x be set ; ::_thesis: ( ( x in dom (union X) implies x in union  {  (dom f) where f is Element of X : verum  }  ) & ( x in union  {  (dom f) where f is Element of X : verum  }  implies x in dom (union X) ) )
hereby  ::_thesis: ( x in union  {  (dom f) where f is Element of X : verum  }  implies x in dom (union X) )
assume  x in dom (union X) ; ::_thesis: x in union  {  (dom f) where f is Element of X : verum  }
then consider y being set  such that
A1: [x,y] in union X by XTUPLE_0:def_12;
consider Z being set  such that
A2: [x,y] in Z and
A3: Z in X by A1, TARSKI:def_4;
reconsider Z = Z as Element of X by A3;
A4: dom Z in  {  (dom f) where f is Element of X : verum  }  ;
 x in dom Z by A2, XTUPLE_0:def_12;
hence  x in union  {  (dom f) where f is Element of X : verum  }  by A4, TARSKI:def_4; ::_thesis: verum
end;
assume  x in union  {  (dom f) where f is Element of X : verum  }  ; ::_thesis: x in dom (union X)
then consider Z being set  such that
A5: x in Z and
A6: Z in  {  (dom f) where f is Element of X : verum  }  by TARSKI:def_4;
consider f being Element of X such that
A7: Z = dom f and
 verum by A6;
consider y being set  such that
A8: [x,y] in f by A5, A7, XTUPLE_0:def_12;
 [x,y] in union X by A8, TARSKI:def_4;
hence  x in dom (union X) by XTUPLE_0:def_12; ::_thesis: verum
end;
hence  dom (union X) = union  {  (dom f) where f is Element of X : verum  }  by TARSKI:1; ::_thesis: verum
end;

theorem Th13: :: COMPUT_1:13
for X being functional compatible set
for f being Function st f in X holds
( dom f c= dom (union X) & ( for x being set st x in dom f holds
(union X) . x = f . x ) )
proof
let X be functional compatible set ; ::_thesis: for f being Function st f in X holds
( dom f c= dom (union X) & ( for x being set st x in dom f holds
(union X) . x = f . x ) )
let f be Function; ::_thesis: ( f in X implies ( dom f c= dom (union X) & ( for x being set st x in dom f holds
(union X) . x = f . x ) ) )
assume A1: f in X ; ::_thesis: ( dom f c= dom (union X) & ( for x being set st x in dom f holds
(union X) . x = f . x ) )
thus  dom f c= dom (union X) ::_thesis: for x being set st x in dom f holds
(union X) . x = f . x
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom f or x in dom (union X) )
assume  x in dom f ; ::_thesis: x in dom (union X)
then consider y being set  such that
A2: [x,y] in f by XTUPLE_0:def_12;
 [x,y] in union X by A1, A2, TARSKI:def_4;
hence  x in dom (union X) by XTUPLE_0:def_12; ::_thesis: verum
end;
let x be set ; ::_thesis: ( x in dom f implies (union X) . x = f . x )
assume  x in dom f ; ::_thesis: (union X) . x = f . x
then consider y being set  such that
A3: [x,y] in f by XTUPLE_0:def_12;
 [x,y] in union X by A1, A3, TARSKI:def_4;
hence (union X) . x = y by FUNCT_1:1
.= f . x by A3, FUNCT_1:1 ;
::_thesis: verum
end;

theorem Th14: :: COMPUT_1:14
for X being non empty functional compatible set holds rng (union X) = union  {  (rng f) where f is Element of X : verum  }
proof
let X be non empty functional compatible set ; ::_thesis: rng (union X) = union  {  (rng f) where f is Element of X : verum  }
set F =  {  (rng f) where f is Element of X : verum  }  ;
now__::_thesis:_for_y_being_set_holds_
(_(_y_in_rng_(union_X)_implies_y_in_union__{__(rng_f)_where_f_is_Element_of_X_:_verum__}__)_&_(_y_in_union__{__(rng_f)_where_f_is_Element_of_X_:_verum__}__implies_y_in_rng_(union_X)_)_)
let y be set ; ::_thesis: ( ( y in rng (union X) implies y in union  {  (rng f) where f is Element of X : verum  }  ) & ( y in union  {  (rng f) where f is Element of X : verum  }  implies y in rng (union X) ) )
hereby  ::_thesis: ( y in union  {  (rng f) where f is Element of X : verum  }  implies y in rng (union X) )
assume  y in rng (union X) ; ::_thesis: y in union  {  (rng f) where f is Element of X : verum  }
then consider x being set  such that
A1: [x,y] in union X by XTUPLE_0:def_13;
consider Z being set  such that
A2: [x,y] in Z and
A3: Z in X by A1, TARSKI:def_4;
reconsider Z = Z as Element of X by A3;
A4: rng Z in  {  (rng f) where f is Element of X : verum  }  ;
 y in rng Z by A2, XTUPLE_0:def_13;
hence  y in union  {  (rng f) where f is Element of X : verum  }  by A4, TARSKI:def_4; ::_thesis: verum
end;
assume  y in union  {  (rng f) where f is Element of X : verum  }  ; ::_thesis: y in rng (union X)
then consider Z being set  such that
A5: y in Z and
A6: Z in  {  (rng f) where f is Element of X : verum  }  by TARSKI:def_4;
consider f being Element of X such that
A7: Z = rng f and
 verum by A6;
consider x being set  such that
A8: [x,y] in f by A5, A7, XTUPLE_0:def_13;
 [x,y] in union X by A8, TARSKI:def_4;
hence  y in rng (union X) by XTUPLE_0:def_13; ::_thesis: verum
end;
hence  rng (union X) = union  {  (rng f) where f is Element of X : verum  }  by TARSKI:1; ::_thesis: verum
end;

registration
let X, Y be set ;
cluster non empty  ->  functional for PartFunc-set of X,Y;
coherence
 for b1 being PFUNC_DOMAIN of X,Y holds b1 is functional
proof
let P be PFUNC_DOMAIN of X,Y; ::_thesis: P is functional
let x be set ; :: according to FUNCT_1:def_13 ::_thesis: ( not x in P or x is set )
assume  x in P ; ::_thesis: x is set
hence  x is set by RFUNCT_3:def_3; ::_thesis: verum
end;
end;

theorem Th15: :: COMPUT_1:15
for X, Y being set
for P being compatible PFUNC_DOMAIN of X,Y holds union P is PartFunc of X,Y
proof
let X, Y be set ; ::_thesis: for P being compatible PFUNC_DOMAIN of X,Y holds union P is PartFunc of X,Y
let D be compatible PFUNC_DOMAIN of X,Y; ::_thesis: union D is PartFunc of X,Y
set E =  {  (dom f) where f is Element of D : verum  }  ;
set F =  {  (rng f) where f is Element of D : verum  }  ;
reconsider u = union D as Function ;
A1: rng u c= Y
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng u or y in Y )
assume  y in rng u ; ::_thesis: y in Y
then  y in union  {  (rng f) where f is Element of D : verum  }  by Th14;
then consider Z being set  such that
A2: y in Z and
A3: Z in  {  (rng f) where f is Element of D : verum  }  by TARSKI:def_4;
consider f being Element of D such that
A4: Z = rng f and
 verum by A3;
 rng f c= Y by RELAT_1:def_19;
hence  y in Y by A2, A4; ::_thesis: verum
end;
 dom u c= X
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom u or x in X )
assume  x in dom u ; ::_thesis: x in X
then  x in union  {  (dom f) where f is Element of D : verum  }  by Th12;
then consider Z being set  such that
A5: x in Z and
A6: Z in  {  (dom f) where f is Element of D : verum  }  by TARSKI:def_4;
 ex f being Element of D st Z = dom f by A6;
hence  x in X by A5; ::_thesis: verum
end;
hence  union D is PartFunc of X,Y by A1, RELSET_1:4; ::_thesis: verum
end;

begin

notation
let f be Relation;
synonym  to-naturals f for  natural-valued ;
end;

registration
cluster Relation-like NAT * -defined Function-like to-naturals for set ;
existence
 ex b1 being Function st
( b1 is NAT * -defined & b1 is to-naturals )
proof
take f = {} ; ::_thesis: ( f is NAT * -defined & f is to-naturals )
thus  dom f c= NAT * by XBOOLE_1:2; :: according to RELAT_1:def_18 ::_thesis: f is to-naturals
thus  f is to-naturals ; ::_thesis: verum
end;
end;

definition
let f be NAT * -defined Relation;
attrf is len-total  means :Def2: :: COMPUT_1:def 2
 for x, y being FinSequence of NAT st len x = len y & x in dom f holds
y in dom f;
end;

:: deftheorem Def2 defines len-total COMPUT_1:def_2_:_
 for f being NAT * -defined Relation holds
( f is len-total iff for x, y being FinSequence of NAT st len x = len y & x in dom f holds
y in dom f );

theorem Th16: :: COMPUT_1:16
for n being Element of NAT
for D being non empty set
for R being Relation st dom R c= n -tuples_on D holds
R is homogeneous
proof
let n be Element of NAT ; ::_thesis: for D being non empty set
for R being Relation st dom R c= n -tuples_on D holds
R is homogeneous
let D be non empty set ; ::_thesis: for R being Relation st dom R c= n -tuples_on D holds
R is homogeneous
let R be Relation; ::_thesis: ( dom R c= n -tuples_on D implies R is homogeneous )
assume  dom R c= n -tuples_on D ; ::_thesis: R is homogeneous
hence  dom R is with_common_domain ; :: according to MARGREL1:def_21 ::_thesis: verum
end;

registration
cluster empty Relation-like  ->  homogeneous for set ;
coherence
 for b1 being Relation st b1 is empty holds
b1 is homogeneous
proof
let R be Relation; ::_thesis: ( R is empty implies R is homogeneous )
assume A1: R is empty ; ::_thesis: R is homogeneous
thus  dom R is with_common_domain by A1; :: according to MARGREL1:def_21 ::_thesis: verum
end;
end;

registration
let p be FinSequence;
let x be set ;
clusterp .--> x ->  non empty homogeneous ;
coherence
 ( not p .--> x is empty & p .--> x is homogeneous )
proof
set f = p .--> x;
thus  not p .--> x is empty ; ::_thesis: p .--> x is homogeneous
thus  dom (p .--> x) is with_common_domain ; :: according to MARGREL1:def_21 ::_thesis: verum
end;
end;

registration
cluster non empty Relation-like Function-like homogeneous for set ;
existence
 ex b1 being Function st
( not b1 is empty & b1 is homogeneous )
proof
set p = the FinSequence;
take  the FinSequence .--> 0 ; ::_thesis: ( not the FinSequence .--> 0 is empty & the FinSequence .--> 0 is homogeneous )
thus  ( not the FinSequence .--> 0 is empty & the FinSequence .--> 0 is homogeneous ) ; ::_thesis: verum
end;
end;

registration
let f be homogeneous Function;
let g be Function;
clusterf * g ->  homogeneous ;
coherence
 g * f is homogeneous
proof
 dom (g * f) c= dom f by RELAT_1:25;
hence  dom (g * f) is with_common_domain ; :: according to MARGREL1:def_21 ::_thesis: verum
end;
end;

registration
let X, Y be set ;
cluster Relation-like X * -defined Y -valued Function-like homogeneous for Element of K32(K33((X *),Y));
existence
 ex b1 being PartFunc of (X *),Y st b1 is homogeneous
proof
set f = {} ;
reconsider f = {} as PartFunc of (X *),Y by XBOOLE_1:2;
take  f ; ::_thesis: f is homogeneous
thus  dom f is with_common_domain ; :: according to MARGREL1:def_21 ::_thesis: verum
end;
end;

registration
let X, Y be non empty set ;
cluster non empty Relation-like X * -defined Y -valued Function-like homogeneous for Element of K32(K33((X *),Y));
existence
 ex b1 being PartFunc of (X *),Y st
( not b1 is empty & b1 is homogeneous )
proof
set y = the Element of Y;
reconsider Z = 0 -tuples_on X as non empty Subset of (X *) by Th10;
reconsider f = Z --> the Element of Y as PartFunc of (X *),Y ;
take  f ; ::_thesis: ( not f is empty & f is homogeneous )
thus  not f is empty ; ::_thesis: f is homogeneous
thus  dom f is with_common_domain by FUNCOP_1:13; :: according to MARGREL1:def_21 ::_thesis: verum
end;
end;

registration
let X be non empty set ;
cluster non empty Relation-like X * -defined X -valued Function-like homogeneous quasi_total for Element of K32(K33((X *),X));
existence
 ex b1 being PartFunc of (X *),X st
( not b1 is empty & b1 is homogeneous & b1 is quasi_total )
proof
set n = the Element of NAT ;
set y = the Element of X;
reconsider Z = the Element of NAT -tuples_on X as non empty Subset of (X *) by Th10;
reconsider f = Z --> the Element of X as PartFunc of (X *),X ;
take  f ; ::_thesis: ( not f is empty & f is homogeneous & f is quasi_total )
thus  not f is empty ; ::_thesis: ( f is homogeneous & f is quasi_total )
A1: dom f = Z by FUNCOP_1:13;
thus  f is homogeneous ::_thesis: f is quasi_total
proof
thus  dom f is with_common_domain by A1; :: according to MARGREL1:def_21 ::_thesis: verum
end;
let p, q be FinSequence of X; :: according to MARGREL1:def_22 ::_thesis: ( not len p = len q or not p in dom f or q in dom f )
assume that
A2: len p = len q and
A3: p in dom f ; ::_thesis: q in dom f
 len q = the Element of NAT by A1, A2, A3, CARD_1:def_7;
then  q is Element of Z by FINSEQ_2:92;
hence  q in dom f by A1; ::_thesis: verum
end;
end;

registration
cluster non empty Relation-like NAT * -defined Function-like to-naturals homogeneous len-total for set ;
existence
 ex b1 being NAT * -defined Function st
( not b1 is empty & b1 is homogeneous & b1 is to-naturals & b1 is len-total )
proof
reconsider Z = 0 -tuples_on NAT as non empty Subset of (NAT *) by Th10;
set f = Z --> 0;
A1: dom (Z --> 0) = Z by FUNCOP_1:13;
reconsider f = Z --> 0 as NAT * -defined Function ;
A2: f is len-total
proof
let x, y be FinSequence of NAT ; :: according to COMPUT_1:def_2 ::_thesis: ( len x = len y & x in dom f implies y in dom f )
assume that
A3: len x = len y and
A4: x in dom f ; ::_thesis: y in dom f
A5: y is Element of (len y) -tuples_on NAT by FINSEQ_2:92;
 len x = 0 by A1, A4;
hence  y in dom f by A1, A3, A5; ::_thesis: verum
end;
take  f ; ::_thesis: ( not f is empty & f is homogeneous & f is to-naturals & f is len-total )
 f is homogeneous
proof
thus  dom f is with_common_domain by FUNCOP_1:13; :: according to MARGREL1:def_21 ::_thesis: verum
end;
hence  ( not f is empty & f is homogeneous & f is to-naturals & f is len-total ) by A2; ::_thesis: verum
end;
end;

registration
cluster Function-like  -> NAT * -defined to-naturals for Element of K32(K33((NAT *),NAT));
coherence
 for b1 being PartFunc of (NAT *),NAT holds
( b1 is to-naturals & b1 is NAT * -defined ) ;
end;

registration
cluster Function-like quasi_total  ->  len-total for Element of K32(K33((NAT *),NAT));
coherence
 for b1 being PartFunc of (NAT *),NAT st b1 is quasi_total holds
b1 is len-total
proof
let f be PartFunc of (NAT *),NAT; ::_thesis: ( f is quasi_total implies f is len-total )
assume A1: f is quasi_total ; ::_thesis: f is len-total
let x, y be FinSequence of NAT ; :: according to COMPUT_1:def_2 ::_thesis: ( len x = len y & x in dom f implies y in dom f )
assume that
A2: len x = len y and
A3: x in dom f ; ::_thesis: y in dom f
thus  y in dom f by A1, A2, A3, MARGREL1:def_22; ::_thesis: verum
end;
end;

theorem Th17: :: COMPUT_1:17
for g being NAT * -defined to-naturals len-total Function holds g is quasi_total PartFunc of (NAT *),NAT
proof
let g be NAT * -defined to-naturals len-total Function; ::_thesis: g is quasi_total PartFunc of (NAT *),NAT
A1: rng g c= NAT by VALUED_0:def_6;
 dom g c= NAT * ;
then reconsider g9 = g as PartFunc of (NAT *),NAT by A1, RELSET_1:4;
 for x, y being FinSequence of NAT st len x = len y & x in dom g holds
y in dom g by Def2;
then  g9 is quasi_total by MARGREL1:def_22;
hence  g is quasi_total PartFunc of (NAT *),NAT ; ::_thesis: verum
end;

theorem Th18: :: COMPUT_1:18
arity {} = 0
proof
 for x being FinSequence holds not x in dom {} ;
hence  arity {} = 0 by MARGREL1:def_25; ::_thesis: verum
end;

theorem Th19: :: COMPUT_1:19
for f being homogeneous Relation st dom f = {{}} holds
arity f = 0
proof
let f be homogeneous Relation; ::_thesis: ( dom f = {{}} implies arity f = 0 )
assume  dom f = {{}} ; ::_thesis: arity f = 0
then  {} in dom f by TARSKI:def_1;
hence  arity f = 0 by CARD_1:27, MARGREL1:def_25; ::_thesis: verum
end;

theorem Th20: :: COMPUT_1:20
for X, Y being set
for f being homogeneous PartFunc of (X *),Y holds dom f c= (arity f) -tuples_on X
proof
let X, Y be set ; ::_thesis: for f being homogeneous PartFunc of (X *),Y holds dom f c= (arity f) -tuples_on X
let f be homogeneous PartFunc of (X *),Y; ::_thesis: dom f c= (arity f) -tuples_on X
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom f or x in (arity f) -tuples_on X )
assume A1: x in dom f ; ::_thesis: x in (arity f) -tuples_on X
percases ( X is empty or not X is empty ) ;
supposeA2: X is empty ; ::_thesis: x in (arity f) -tuples_on X
then  x = <*> (X *) by A1, FUNCT_7:17, TARSKI:def_1;
then A3: arity f = len (<*> (X *)) by A1, MARGREL1:def_25;
 0 -tuples_on X = {{}} by Th5;
hence  x in (arity f) -tuples_on X by A1, A2, A3, FUNCT_7:17; ::_thesis: verum
end;
suppose not X is empty ; ::_thesis: x in (arity f) -tuples_on X
then reconsider X9 = X as non empty set ;
reconsider x9 = x as FinSequence of X9 by A1, FINSEQ_1:def_11;
 len x9 = arity f by A1, MARGREL1:def_25;
then  x9 is Element of (arity f) -tuples_on X9 by FINSEQ_2:92;
hence  x in (arity f) -tuples_on X ; ::_thesis: verum
end;
end;
end;

theorem Th21: :: COMPUT_1:21
for f being NAT * -defined homogeneous Function holds dom f c= (arity f) -tuples_on NAT
proof
let f be NAT * -defined homogeneous Function; ::_thesis: dom f c= (arity f) -tuples_on NAT
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom f or x in (arity f) -tuples_on NAT )
assume A1: x in dom f ; ::_thesis: x in (arity f) -tuples_on NAT
reconsider x9 = x as FinSequence of NAT by A1, FINSEQ_1:def_11;
 len x9 = arity f by A1, MARGREL1:def_25;
then  x9 is Element of (arity f) -tuples_on NAT by FINSEQ_2:92;
hence  x in (arity f) -tuples_on NAT ; ::_thesis: verum
end;

Lm1:  for D being non empty set
for f being homogeneous PartFunc of (D *),D holds
( ( f is quasi_total & not f is empty ) iff dom f = (arity f) -tuples_on D )
proof
let D be non empty set ; ::_thesis: for f being homogeneous PartFunc of (D *),D holds
( ( f is quasi_total & not f is empty ) iff dom f = (arity f) -tuples_on D )
set X = D;
let f be homogeneous PartFunc of (D *),D; ::_thesis: ( ( f is quasi_total & not f is empty ) iff dom f = (arity f) -tuples_on D )
A1: dom f c= (arity f) -tuples_on D by Th20;
hereby  ::_thesis: ( dom f = (arity f) -tuples_on D implies ( f is quasi_total & not f is empty ) )
assume  ( f is quasi_total & not f is empty ) ; ::_thesis: dom f = (arity f) -tuples_on D
then reconsider f9 = f as non empty homogeneous quasi_total PartFunc of (D *),D ;
consider x being set  such that
A2: x in dom f9 by XBOOLE_0:def_1;
reconsider x9 = x as FinSequence of D by A2, FINSEQ_1:def_11;
A3: len x9 = arity f by A2, MARGREL1:def_25;
now__::_thesis:_for_z_being_set_holds_
(_(_z_in_dom_f_implies_z_in_(arity_f)_-tuples_on_D_)_&_(_z_in_(arity_f)_-tuples_on_D_implies_z_in_dom_f_)_)
let z be set ; ::_thesis: ( ( z in dom f implies z in (arity f) -tuples_on D ) & ( z in (arity f) -tuples_on D implies z in dom f ) )
thus  ( z in dom f implies z in (arity f) -tuples_on D ) by A1; ::_thesis: ( z in (arity f) -tuples_on D implies z in dom f )
assume  z in (arity f) -tuples_on D ; ::_thesis: z in dom f
then reconsider z9 = z as Element of (arity f) -tuples_on D ;
 len z9 = arity f by CARD_1:def_7;
hence  z in dom f by A2, A3, MARGREL1:def_22; ::_thesis: verum
end;
hence  dom f = (arity f) -tuples_on D by TARSKI:1; ::_thesis: verum
end;
assume A4: dom f = (arity f) -tuples_on D ; ::_thesis: ( f is quasi_total & not f is empty )
thus  f is quasi_total ::_thesis: not f is empty
proof
let x, y be FinSequence of D; :: according to MARGREL1:def_22 ::_thesis: ( not len x = len y or not x in dom f or y in dom f )
assume that
A5: len x = len y and
A6: x in dom f ; ::_thesis: y in dom f
 len x = arity f by A4, A6, CARD_1:def_7;
then  y is Element of (arity f) -tuples_on D by A5, FINSEQ_2:92;
hence  y in dom f by A4; ::_thesis: verum
end;
thus  not f is empty by A4; ::_thesis: verum
end;

theorem Th22: :: COMPUT_1:22
for X being set
for f being homogeneous PartFunc of (X *),X holds
( ( f is quasi_total & not f is empty ) iff dom f = (arity f) -tuples_on X )
proof
let X be set ; ::_thesis: for f being homogeneous PartFunc of (X *),X holds
( ( f is quasi_total & not f is empty ) iff dom f = (arity f) -tuples_on X )
let f be homogeneous PartFunc of (X *),X; ::_thesis: ( ( f is quasi_total & not f is empty ) iff dom f = (arity f) -tuples_on X )
percases ( X is empty or not X is empty ) ;
suppose X is empty ; ::_thesis: ( ( f is quasi_total & not f is empty ) iff dom f = (arity f) -tuples_on X )
then  f = {} ;
hence  ( ( f is quasi_total & not f is empty ) iff dom f = (arity f) -tuples_on X ) by Th5, Th18; ::_thesis: verum
end;
suppose not X is empty ; ::_thesis: ( ( f is quasi_total & not f is empty ) iff dom f = (arity f) -tuples_on X )
hence  ( ( f is quasi_total & not f is empty ) iff dom f = (arity f) -tuples_on X ) by Lm1; ::_thesis: verum
end;
end;
end;

theorem Th23: :: COMPUT_1:23
for f being NAT * -defined to-naturals homogeneous Function holds
( ( f is len-total & not f is empty ) iff dom f = (arity f) -tuples_on NAT )
proof
let f be NAT * -defined to-naturals homogeneous Function; ::_thesis: ( ( f is len-total & not f is empty ) iff dom f = (arity f) -tuples_on NAT )
A1: dom f c= (arity f) -tuples_on NAT by Th21;
hereby  ::_thesis: ( dom f = (arity f) -tuples_on NAT implies ( f is len-total & not f is empty ) )
assume  ( f is len-total & not f is empty ) ; ::_thesis: dom f = (arity f) -tuples_on NAT
then reconsider f9 = f as non empty homogeneous quasi_total PartFunc of (NAT *),NAT by Th17;
consider x being set  such that
A2: x in dom f9 by XBOOLE_0:def_1;
reconsider x9 = x as FinSequence of NAT by A2, FINSEQ_1:def_11;
A3: len x9 = arity f by A2, MARGREL1:def_25;
now__::_thesis:_for_z_being_set_holds_
(_(_z_in_dom_f_implies_z_in_(arity_f)_-tuples_on_NAT_)_&_(_z_in_(arity_f)_-tuples_on_NAT_implies_z_in_dom_f_)_)
let z be set ; ::_thesis: ( ( z in dom f implies z in (arity f) -tuples_on NAT ) & ( z in (arity f) -tuples_on NAT implies z in dom f ) )
thus  ( z in dom f implies z in (arity f) -tuples_on NAT ) by A1; ::_thesis: ( z in (arity f) -tuples_on NAT implies z in dom f )
assume  z in (arity f) -tuples_on NAT ; ::_thesis: z in dom f
then reconsider z9 = z as Element of (arity f) -tuples_on NAT ;
 len z9 = arity f by CARD_1:def_7;
hence  z in dom f by A2, A3, MARGREL1:def_22; ::_thesis: verum
end;
hence  dom f = (arity f) -tuples_on NAT by TARSKI:1; ::_thesis: verum
end;
assume A4: dom f = (arity f) -tuples_on NAT ; ::_thesis: ( f is len-total & not f is empty )
thus  f is len-total ::_thesis: not f is empty
proof
let x, y be FinSequence of NAT ; :: according to COMPUT_1:def_2 ::_thesis: ( len x = len y & x in dom f implies y in dom f )
assume that
A5: len x = len y and
A6: x in dom f ; ::_thesis: y in dom f
 len x = arity f by A4, A6, CARD_1:def_7;
then  y is Element of (arity f) -tuples_on NAT by A5, FINSEQ_2:92;
hence  y in dom f by A4; ::_thesis: verum
end;
thus  not f is empty by A4; ::_thesis: verum
end;

theorem :: COMPUT_1:24
for D being non empty set
for f being non empty homogeneous PartFunc of (D *),D
for n being Element of NAT st dom f c= n -tuples_on D holds
arity f = n
proof
let D be non empty set ; ::_thesis: for f being non empty homogeneous PartFunc of (D *),D
for n being Element of NAT st dom f c= n -tuples_on D holds
arity f = n
let f be non empty homogeneous PartFunc of (D *),D; ::_thesis: for n being Element of NAT st dom f c= n -tuples_on D holds
arity f = n
let n be Element of NAT ; ::_thesis: ( dom f c= n -tuples_on D implies arity f = n )
consider x being set  such that
A1: x in dom f by XBOOLE_0:def_1;
assume A2: dom f c= n -tuples_on D ; ::_thesis: arity f = n
then A3: for x being FinSequence st x in dom f holds
n = len x by CARD_1:def_7;
reconsider x = x as Element of n -tuples_on D by A2, A1;
 x in dom f by A1;
hence  arity f = n by A3, MARGREL1:def_25; ::_thesis: verum
end;

theorem Th25: :: COMPUT_1:25
for D being non empty set
for f being homogeneous PartFunc of (D *),D
for n being Element of NAT st dom f = n -tuples_on D holds
arity f = n
proof
let D be non empty set ; ::_thesis: for f being homogeneous PartFunc of (D *),D
for n being Element of NAT st dom f = n -tuples_on D holds
arity f = n
let f be homogeneous PartFunc of (D *),D; ::_thesis: for n being Element of NAT st dom f = n -tuples_on D holds
arity f = n
let n be Element of NAT ; ::_thesis: ( dom f = n -tuples_on D implies arity f = n )
consider x being set  such that
A1: x in n -tuples_on D by XBOOLE_0:def_1;
reconsider x = x as Element of n -tuples_on D by A1;
assume A2: dom f = n -tuples_on D ; ::_thesis: arity f = n
then A3: for x being FinSequence st x in dom f holds
len x = n by CARD_1:def_7;
 x in dom f by A2;
hence  arity f = n by A3, MARGREL1:def_25; ::_thesis: verum
end;

definition
let R be Relation;
attrR is with_the_same_arity  means :Def3: :: COMPUT_1:def 3
 for f, g being Function st f in rng R & g in rng R holds
( ( not f is empty or g is empty or dom g = {{}} ) & ( not f is empty & not g is empty implies ex n being Element of NAT ex X being non empty set st
( dom f c= n -tuples_on X & dom g c= n -tuples_on X ) ) );
end;

:: deftheorem Def3 defines with_the_same_arity COMPUT_1:def_3_:_
 for R being Relation holds
( R is with_the_same_arity iff for f, g being Function st f in rng R & g in rng R holds
( ( not f is empty or g is empty or dom g = {{}} ) & ( not f is empty & not g is empty implies ex n being Element of NAT ex X being non empty set st
( dom f c= n -tuples_on X & dom g c= n -tuples_on X ) ) ) );

registration
cluster empty Relation-like  ->  with_the_same_arity for set ;
coherence
 for b1 being Relation st b1 is empty holds
b1 is with_the_same_arity
proof
let R be Relation; ::_thesis: ( R is empty implies R is with_the_same_arity )
assume A1: R is empty ; ::_thesis: R is with_the_same_arity
let f, g be Function; :: according to COMPUT_1:def_3 ::_thesis: ( f in rng R & g in rng R implies ( ( not f is empty or g is empty or dom g = {{}} ) & ( not f is empty & not g is empty implies ex n being Element of NAT ex X being non empty set st
( dom f c= n -tuples_on X & dom g c= n -tuples_on X ) ) ) )
thus  ( f in rng R & g in rng R implies ( ( not f is empty or g is empty or dom g = {{}} ) & ( not f is empty & not g is empty implies ex n being Element of NAT ex X being non empty set st
( dom f c= n -tuples_on X & dom g c= n -tuples_on X ) ) ) ) by A1; ::_thesis: verum
end;
end;

registration
let X be set ;
cluster Relation-like NAT -defined X -valued Function-like finite countable FinSequence-like FinSubsequence-like with_the_same_arity for FinSequence of X;
existence
 ex b1 being FinSequence of X st b1 is with_the_same_arity
proof
take  <*> X ; ::_thesis: <*> X is with_the_same_arity
thus  <*> X is with_the_same_arity ; ::_thesis: verum
end;
cluster Relation-like NAT -defined X -valued Function-like finite countable FinSequence-like FinSubsequence-like with_the_same_arity for Element of X * ;
existence
 ex b1 being Element of X * st b1 is with_the_same_arity
proof
reconsider p = <*> X as Element of X * by FINSEQ_1:def_11;
take  p ; ::_thesis: p is with_the_same_arity
thus  p is with_the_same_arity ; ::_thesis: verum
end;
end;

definition
let F be with_the_same_arity Relation;
func arity F ->  Element of NAT  means :Def4: :: COMPUT_1:def 4
 for f being homogeneous Function st f in rng F holds
it = arity f if  ex f being homogeneous Function st f in rng F
otherwise it = 0 ;
existence
 ( ( ex f being homogeneous Function st f in rng F implies ex b1 being Element of NAT st
for f being homogeneous Function st f in rng F holds
b1 = arity f ) & ( ( for f being homogeneous Function holds not f in rng F ) implies ex b1 being Element of NAT st b1 = 0 ) )
proof
hereby  ::_thesis: ( ( for f being homogeneous Function holds not f in rng F ) implies ex b1 being Element of NAT st b1 = 0 )
given f being homogeneous Function such that A1: f in rng F ; ::_thesis: ex i being Element of NAT st
for g being homogeneous Function st g in rng F holds
b3 = arity b4
take i = arity f; ::_thesis: for g being homogeneous Function st g in rng F holds
b2 = arity b3
let g be homogeneous Function; ::_thesis: ( g in rng F implies b1 = arity b2 )
assume A2: g in rng F ; ::_thesis: b1 = arity b2
percases ( f is empty or g is empty or ( not f is empty & not g is empty ) ) ;
supposeA3: f is empty ; ::_thesis: b1 = arity b2
thus  i = arity g ::_thesis: verum
proof
percases ( g is empty or dom g = {{}} ) by A1, A2, A3, Def3;
suppose g is empty ; ::_thesis: i = arity g
hence  i = arity g by A3; ::_thesis: verum
end;
suppose dom g = {{}} ; ::_thesis: i = arity g
hence  i = arity g by A3, Th18, Th19; ::_thesis: verum
end;
end;
end;
end;
supposeA4: g is empty ; ::_thesis: b1 = arity b2
thus  i = arity g ::_thesis: verum
proof
percases ( f is empty or dom f = {{}} ) by A1, A2, A4, Def3;
suppose f is empty ; ::_thesis: i = arity g
hence  i = arity g by A4; ::_thesis: verum
end;
suppose dom f = {{}} ; ::_thesis: i = arity g
hence  i = arity g by A4, Th18, Th19; ::_thesis: verum
end;
end;
end;
end;
supposeA5: ( not f is empty & not g is empty ) ; ::_thesis: b1 = arity b2
set a = the Element of dom f;
 dom f <> {} by A5;
then A6: the Element of dom f in dom f ;
consider n being Element of NAT , X being non empty set  such that
A7: dom f c= n -tuples_on X and
A8: dom g c= n -tuples_on X by A1, A2, A5, Def3;
reconsider a = the Element of dom f as Element of n -tuples_on X by A7, A6;
A9: arity f = len a by A5, MARGREL1:def_25
.= n by CARD_1:def_7 ;
set a = the Element of dom g;
 dom g <> {} by A5;
then  the Element of dom g in dom g ;
then reconsider a = the Element of dom g as Element of n -tuples_on X by A8;
 arity g = len a by A5, MARGREL1:def_25;
hence  i = arity g by A9, CARD_1:def_7; ::_thesis: verum
end;
end;
end;
thus  ( ( for f being homogeneous Function holds not f in rng F ) implies ex b1 being Element of NAT st b1 = 0 ) ; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Element of NAT holds
( ( ex f being homogeneous Function st f in rng F & ( for f being homogeneous Function st f in rng F holds
b1 = arity f ) & ( for f being homogeneous Function st f in rng F holds
b2 = arity f ) implies b1 = b2 ) & ( ( for f being homogeneous Function holds not f in rng F ) & b1 = 0 & b2 = 0 implies b1 = b2 ) )
proof
let i1, i2 be Element of NAT ; ::_thesis: ( ( ex f being homogeneous Function st f in rng F & ( for f being homogeneous Function st f in rng F holds
i1 = arity f ) & ( for f being homogeneous Function st f in rng F holds
i2 = arity f ) implies i1 = i2 ) & ( ( for f being homogeneous Function holds not f in rng F ) & i1 = 0 & i2 = 0 implies i1 = i2 ) )
hereby  ::_thesis: ( ( for f being homogeneous Function holds not f in rng F ) & i1 = 0 & i2 = 0 implies i1 = i2 )
given f being homogeneous Function such that A10: f in rng F ; ::_thesis: ( ( for f being homogeneous Function st f in rng F holds
i1 = arity f ) & ( for f being homogeneous Function st f in rng F holds
i2 = arity f ) implies i1 = i2 )
assume  for f being homogeneous Function st f in rng F holds
i1 = arity f ; ::_thesis: ( ( for f being homogeneous Function st f in rng F holds
i2 = arity f ) implies i1 = i2 )
then  i1 = arity f by A10;
hence  ( ( for f being homogeneous Function st f in rng F holds
i2 = arity f ) implies i1 = i2 ) by A10; ::_thesis: verum
end;
thus  ( ( for f being homogeneous Function holds not f in rng F ) & i1 = 0 & i2 = 0 implies i1 = i2 ) ; ::_thesis: verum
end;
correctness
consistency
for b1 being Element of NAT holds verum;
;
end;

:: deftheorem Def4 defines arity COMPUT_1:def_4_:_
 for F being with_the_same_arity Relation
for b2 being Element of NAT holds
( ( ex f being homogeneous Function st f in rng F implies ( b2 = arity F iff for f being homogeneous Function st f in rng F holds
b2 = arity f ) ) & ( ( for f being homogeneous Function holds not f in rng F ) implies ( b2 = arity F iff b2 = 0 ) ) );

theorem :: COMPUT_1:26
for F being with_the_same_arity FinSequence st len F = 0 holds
arity F = 0
proof
let F be with_the_same_arity FinSequence; ::_thesis: ( len F = 0 implies arity F = 0 )
assume  len F = 0 ; ::_thesis: arity F = 0
then  F = {} ;
then  for f being homogeneous Function holds not f in rng F ;
hence  arity F = 0 by Def4; ::_thesis: verum
end;

definition
let X be set ;
func HFuncs X ->  PFUNC_DOMAIN of X * ,X equals :: COMPUT_1:def 5
 {  f where f is Element of PFuncs ((X *),X) : f is homogeneous  }  ;
coherence
  {  f where f is Element of PFuncs ((X *),X) : f is homogeneous  }  is PFUNC_DOMAIN of X * ,X
proof
set H =  {  f where f is Element of PFuncs ((X *),X) : f is homogeneous  }  ;
 ( not  {  f where f is Element of PFuncs ((X *),X) : f is homogeneous  }  is empty &  {  f where f is Element of PFuncs ((X *),X) : f is homogeneous  }  is functional )
proof
 {} is PartFunc of (X *),X by RELSET_1:12;
then  {} in PFuncs ((X *),X) by PARTFUN1:45;
then  {} in  {  f where f is Element of PFuncs ((X *),X) : f is homogeneous  }  ;
hence  not  {  f where f is Element of PFuncs ((X *),X) : f is homogeneous  }  is empty ; ::_thesis:  {  f where f is Element of PFuncs ((X *),X) : f is homogeneous  }  is functional
let x be set ; :: according to FUNCT_1:def_13 ::_thesis: ( not x in  {  f where f is Element of PFuncs ((X *),X) : f is homogeneous  }  or x is set )
assume  x in  {  f where f is Element of PFuncs ((X *),X) : f is homogeneous  }  ; ::_thesis: x is set
then  ex g being Element of PFuncs ((X *),X) st
( x = g & g is homogeneous ) ;
hence  x is set ; ::_thesis: verum
end;
then reconsider H =  {  f where f is Element of PFuncs ((X *),X) : f is homogeneous  }  as non empty functional set ;
now__::_thesis:_for_f_being_Element_of_H_holds_f_is_PartFunc_of_(X_*),X
let f be Element of H; ::_thesis: f is PartFunc of (X *),X
 f in H ;
then  ex g being Element of PFuncs ((X *),X) st
( f = g & g is homogeneous ) ;
hence  f is PartFunc of (X *),X ; ::_thesis: verum
end;
hence   {  f where f is Element of PFuncs ((X *),X) : f is homogeneous  }  is PFUNC_DOMAIN of X * ,X by RFUNCT_3:def_3; ::_thesis: verum
end;
end;

:: deftheorem  defines HFuncs COMPUT_1:def_5_:_
 for X being set holds HFuncs X =  {  f where f is Element of PFuncs ((X *),X) : f is homogeneous  }  ;

theorem :: COMPUT_1:27
for X being set holds {} in HFuncs X
proof
let X be set ; ::_thesis: {} in HFuncs X
set f = {} ;
reconsider f = {} as PartFunc of (X *),X by XBOOLE_1:2;
reconsider f = f as Element of PFuncs ((X *),X) by PARTFUN1:45;
 f in  {  g where g is Element of PFuncs ((X *),X) : g is homogeneous  }  ;
hence  {} in HFuncs X ; ::_thesis: verum
end;

registration
let X be non empty set ;
cluster non empty Relation-like X * -defined X -valued Function-like homogeneous quasi_total for Element of HFuncs X;
existence
 ex b1 being Element of HFuncs X st
( not b1 is empty & b1 is homogeneous & b1 is quasi_total )
proof
set p = <*> X;
set x = the Element of X;
 <*> X in X * by FINSEQ_1:def_11;
then reconsider Y = {(<*> X)} as Subset of (X *) by ZFMISC_1:31;
A1: Y --> the Element of X in PFuncs ((X *),X) ;
 (<*> X) .--> the Element of X is homogeneous ;
then  {(<*> X)} --> the Element of X in  {  f where f is Element of PFuncs ((X *),X) : f is homogeneous  }  by A1;
then reconsider f = (<*> X) .--> the Element of X as Element of HFuncs X ;
take  f ; ::_thesis: ( not f is empty & f is homogeneous & f is quasi_total )
A2: dom f = {(<*> X)} by FUNCOP_1:13;
thus  ( not f is empty & f is homogeneous ) ; ::_thesis: f is quasi_total
let a, b be FinSequence of X; :: according to MARGREL1:def_22 ::_thesis: ( not len a = len b or not a in dom f or b in dom f )
assume A3: len a = len b ; ::_thesis: ( not a in dom f or b in dom f )
assume  a in dom f ; ::_thesis: b in dom f
then  a = <*> X by A2, TARSKI:def_1;
then  b = {} by A3;
hence  b in dom f by A2, TARSKI:def_1; ::_thesis: verum
end;
end;

registration
let X be set ;
cluster  ->  homogeneous for Element of HFuncs X;
coherence
 for b1 being Element of HFuncs X holds b1 is homogeneous
proof
let f be Element of HFuncs X; ::_thesis: f is homogeneous
 f in HFuncs X ;
then  ex g being Element of PFuncs ((X *),X) st
( f = g & g is homogeneous ) ;
hence  f is homogeneous ; ::_thesis: verum
end;
end;

registration
let X be non empty set ;
let S be non empty Subset of (HFuncs X);
cluster  ->  homogeneous for Element of S;
coherence
 for b1 being Element of S holds b1 is homogeneous ;
end;

theorem Th28: :: COMPUT_1:28
for f being NAT * -defined to-naturals homogeneous Function holds f is Element of HFuncs NAT
proof
let f be NAT * -defined to-naturals homogeneous Function; ::_thesis: f is Element of HFuncs NAT
A1: rng f c= NAT by VALUED_0:def_6;
 dom f c= NAT * ;
then  f is PartFunc of (NAT *),NAT by A1, RELSET_1:4;
then reconsider f9 = f as Element of PFuncs ((NAT *),NAT) by PARTFUN1:45;
 f9 in  {  f1 where f1 is Element of PFuncs ((NAT *),NAT) : f1 is homogeneous  }  ;
hence  f is Element of HFuncs NAT ; ::_thesis: verum
end;

theorem Th29: :: COMPUT_1:29
for f being NAT * -defined to-naturals homogeneous len-total Function holds f is quasi_total Element of HFuncs NAT
proof
let f be NAT * -defined to-naturals homogeneous len-total Function; ::_thesis: f is quasi_total Element of HFuncs NAT
reconsider f9 = f as Element of HFuncs NAT by Th28;
 f9 is quasi_total
proof
let x, y be FinSequence of NAT ; :: according to MARGREL1:def_22 ::_thesis: ( not len x = len y or not x in dom f9 or y in dom f9 )
assume that
A1: len x = len y and
A2: x in dom f9 ; ::_thesis: y in dom f9
thus  y in dom f9 by A1, A2, Def2; ::_thesis: verum
end;
hence  f is quasi_total Element of HFuncs NAT ; ::_thesis: verum
end;

theorem Th30: :: COMPUT_1:30
for X being non empty set
for F being Relation st rng F c= HFuncs X & ( for f, g being homogeneous Function st f in rng F & g in rng F holds
arity f = arity g ) holds
F is with_the_same_arity
proof
let X be non empty set ; ::_thesis: for F being Relation st rng F c= HFuncs X & ( for f, g being homogeneous Function st f in rng F & g in rng F holds
arity f = arity g ) holds
F is with_the_same_arity
let R be Relation; ::_thesis: ( rng R c= HFuncs X & ( for f, g being homogeneous Function st f in rng R & g in rng R holds
arity f = arity g ) implies R is with_the_same_arity )
assume that
A1: rng R c= HFuncs X and
A2: for f, g being homogeneous Function st f in rng R & g in rng R holds
arity f = arity g ; ::_thesis: R is with_the_same_arity
let f, g be Function; :: according to COMPUT_1:def_3 ::_thesis: ( f in rng R & g in rng R implies ( ( not f is empty or g is empty or dom g = {{}} ) & ( not f is empty & not g is empty implies ex n being Element of NAT ex X being non empty set st
( dom f c= n -tuples_on X & dom g c= n -tuples_on X ) ) ) )
assume that
A3: f in rng R and
A4: g in rng R ; ::_thesis: ( ( not f is empty or g is empty or dom g = {{}} ) & ( not f is empty & not g is empty implies ex n being Element of NAT ex X being non empty set st
( dom f c= n -tuples_on X & dom g c= n -tuples_on X ) ) )
reconsider f9 = f, g9 = g as Element of HFuncs X by A1, A3, A4;
A5: arity f9 = arity g9 by A2, A3, A4;
hereby  ::_thesis: ( not f is empty & not g is empty implies ex n being Element of NAT ex X being non empty set st
( dom f c= n -tuples_on X & dom g c= n -tuples_on X ) )
assume  f is empty ; ::_thesis: ( g is empty or dom g = {{}} )
then  dom g9 c= 0 -tuples_on X by A5, Th18, Th20;
then  dom g9 c= {(<*> X)} by FINSEQ_2:94;
hence  ( g is empty or dom g = {{}} ) by ZFMISC_1:33; ::_thesis: verum
end;
assume that
A6: not f is empty and
 not g is empty ; ::_thesis: ex n being Element of NAT ex X being non empty set st
( dom f c= n -tuples_on X & dom g c= n -tuples_on X )
reconsider f9 = f9 as non empty Element of HFuncs X by A6;
take  arity f9 ; ::_thesis: ex X being non empty set st
( dom f c= (arity f9) -tuples_on X & dom g c= (arity f9) -tuples_on X )
take  X ; ::_thesis: ( dom f c= (arity f9) -tuples_on X & dom g c= (arity f9) -tuples_on X )
thus  ( dom f c= (arity f9) -tuples_on X & dom g c= (arity f9) -tuples_on X ) by A5, Th20; ::_thesis: verum
end;

definition
let n, m be Element of NAT ;
funcn const m ->  NAT * -defined to-naturals homogeneous Function equals :: COMPUT_1:def 6
(n -tuples_on NAT) --> m;
coherence
 (n -tuples_on NAT) --> m is NAT * -defined to-naturals homogeneous Function
proof
set X = NAT ;
set f = (n -tuples_on NAT) --> m;
A1: n -tuples_on NAT c= NAT * by FINSEQ_2:142;
A2: dom ((n -tuples_on NAT) --> m) = n -tuples_on NAT by FUNCOP_1:13;
 (n -tuples_on NAT) --> m is homogeneous
proof
thus  dom ((n -tuples_on NAT) --> m) is with_common_domain ; :: according to MARGREL1:def_21 ::_thesis: verum
end;
hence  (n -tuples_on NAT) --> m is NAT * -defined to-naturals homogeneous Function by A1, A2, RELAT_1:def_18; ::_thesis: verum
end;
end;

:: deftheorem  defines const COMPUT_1:def_6_:_
 for n, m being Element of NAT holds n const m = (n -tuples_on NAT) --> m;

theorem Th31: :: COMPUT_1:31
for n, m being Element of NAT holds n const m in HFuncs NAT
proof
let n, m be Element of NAT ; ::_thesis: n const m in HFuncs NAT
set X = NAT ;
set f = n const m;
A1: rng (n const m) c= NAT by VALUED_0:def_6;
 dom (n const m) c= NAT * ;
then  n const m is PartFunc of (NAT *),NAT by A1, RELSET_1:4;
then  n const m is Element of PFuncs ((NAT *),NAT) by PARTFUN1:45;
hence  n const m in HFuncs NAT ; ::_thesis: verum
end;

registration
let n, m be Element of NAT ;
clustern const m ->  non empty NAT * -defined to-naturals homogeneous len-total ;
coherence
 ( n const m is len-total & not n const m is empty )
proof
set X = NAT ;
A1: dom (n const m) = n -tuples_on NAT by FUNCOP_1:13;
 n const m is len-total
proof
let x, y be FinSequence of NAT ; :: according to COMPUT_1:def_2 ::_thesis: ( len x = len y & x in dom (n const m) implies y in dom (n const m) )
assume that
A2: len x = len y and
A3: x in dom (n const m) ; ::_thesis: y in dom (n const m)
 len x = n by A1, A3, CARD_1:def_7;
then  y is Element of n -tuples_on NAT by A2, FINSEQ_2:92;
hence  y in dom (n const m) by A1; ::_thesis: verum
end;
hence  ( n const m is len-total & not n const m is empty ) ; ::_thesis: verum
end;
end;

theorem Th32: :: COMPUT_1:32
for n, m being Element of NAT holds arity (n const m) = n
proof
let n, m be Element of NAT ; ::_thesis: arity (n const m) = n
set X = NAT ;
set d = the Element of n -tuples_on NAT;
A1: dom (n const m) = n -tuples_on NAT by FUNCOP_1:13;
then A2: for x being FinSequence st x in dom (n const m) holds
n = len x by CARD_1:def_7;
 the Element of n -tuples_on NAT in dom (n const m) by A1;
hence  arity (n const m) = n by A2, MARGREL1:def_25; ::_thesis: verum
end;

registration
cluster  -> NAT * -defined to-naturals homogeneous for Element of HFuncs NAT;
coherence
 for b1 being Element of HFuncs NAT holds
( b1 is homogeneous & b1 is to-naturals & b1 is NAT * -defined ) ;
end;

definition
let n, i be Element of NAT ;
funcn succ i ->  NAT * -defined to-naturals homogeneous Function means :Def7: :: COMPUT_1:def 7
( dom it = n -tuples_on NAT & ( for p being Element of n -tuples_on NAT holds it . p = (p /. i) + 1 ) );
existence
 ex b1 being NAT * -defined to-naturals homogeneous Function st
( dom b1 = n -tuples_on NAT & ( for p being Element of n -tuples_on NAT holds b1 . p = (p /. i) + 1 ) )
proof
deffunc H1( Element of NAT * ) ->  Element of NAT = ($1 /. i) + 1;
defpred S1[ set ] means $1 in n -tuples_on NAT;
consider f being PartFunc of (NAT *),NAT such that
A1: for d being Element of NAT * holds
( d in dom f iff S1[d] ) and
A2: for d being Element of NAT * st d in dom f holds
f /. d = H1(d) from PARTFUN2:sch_2();
A3: n -tuples_on NAT c= NAT * by FINSEQ_2:142;
then A4: for x being set holds
( x in dom f iff x in n -tuples_on NAT ) by A1;
then A5: dom f = n -tuples_on NAT by TARSKI:1;
reconsider f = f as Element of PFuncs ((NAT *),NAT) by PARTFUN1:45;
 f is homogeneous
proof
thus  dom f is with_common_domain by A5; :: according to MARGREL1:def_21 ::_thesis: verum
end;
then  f in  {  g where g is Element of PFuncs ((NAT *),NAT) : g is homogeneous  }  ;
then reconsider f = f as Element of HFuncs NAT ;
take  f ; ::_thesis: ( dom f = n -tuples_on NAT & ( for p being Element of n -tuples_on NAT holds f . p = (p /. i) + 1 ) )
thus  dom f = n -tuples_on NAT by A4, TARSKI:1; ::_thesis: for p being Element of n -tuples_on NAT holds f . p = (p /. i) + 1
let p be Element of n -tuples_on NAT; ::_thesis: f . p = (p /. i) + 1
 p in n -tuples_on NAT ;
then reconsider p9 = p as Element of NAT * by A3;
thus f . p = f /. p9 by A5, PARTFUN1:def_6
.= (p /. i) + 1 by A2, A5 ; ::_thesis: verum
end;
uniqueness
 for b1, b2 being NAT * -defined to-naturals homogeneous Function st dom b1 = n -tuples_on NAT & ( for p being Element of n -tuples_on NAT holds b1 . p = (p /. i) + 1 ) & dom b2 = n -tuples_on NAT & ( for p being Element of n -tuples_on NAT holds b2 . p = (p /. i) + 1 ) holds
b1 = b2
proof
let it1, it2 be NAT * -defined to-naturals homogeneous Function; ::_thesis: ( dom it1 = n -tuples_on NAT & ( for p being Element of n -tuples_on NAT holds it1 . p = (p /. i) + 1 ) & dom it2 = n -tuples_on NAT & ( for p being Element of n -tuples_on NAT holds it2 . p = (p /. i) + 1 ) implies it1 = it2 )
assume that
A6: dom it1 = n -tuples_on NAT and
A7: for p being Element of n -tuples_on NAT holds it1 . p = (p /. i) + 1 and
A8: dom it2 = n -tuples_on NAT and
A9: for p being Element of n -tuples_on NAT holds it2 . p = (p /. i) + 1 ; ::_thesis: it1 = it2
now__::_thesis:_for_x_being_set_st_x_in_n_-tuples_on_NAT_holds_
it1_._x_=_it2_._x
let x be set ; ::_thesis: ( x in n -tuples_on NAT implies it1 . x = it2 . x )
assume  x in n -tuples_on NAT ; ::_thesis: it1 . x = it2 . x
then reconsider x9 = x as Element of n -tuples_on NAT ;
thus it1 . x = (x9 /. i) + 1 by A7
.= it2 . x by A9 ; ::_thesis: verum
end;
hence  it1 = it2 by A6, A8, FUNCT_1:2; ::_thesis: verum
end;
end;

:: deftheorem Def7 defines succ COMPUT_1:def_7_:_
 for n, i being Element of NAT
for b3 being NAT * -defined to-naturals homogeneous Function holds
( b3 = n succ i iff ( dom b3 = n -tuples_on NAT & ( for p being Element of n -tuples_on NAT holds b3 . p = (p /. i) + 1 ) ) );

theorem Th33: :: COMPUT_1:33
for n, i being Element of NAT holds n succ i in HFuncs NAT
proof
let n, i be Element of NAT ; ::_thesis: n succ i in HFuncs NAT
set X = NAT ;
set f = n succ i;
A1: rng (n succ i) c= NAT by VALUED_0:def_6;
 dom (n succ i) c= NAT * ;
then  n succ i is PartFunc of (NAT *),NAT by A1, RELSET_1:4;
then  n succ i is Element of PFuncs ((NAT *),NAT) by PARTFUN1:45;
hence  n succ i in HFuncs NAT ; ::_thesis: verum
end;

registration
let n, i be Element of NAT ;
clustern succ i ->  non empty NAT * -defined to-naturals homogeneous len-total ;
coherence
 ( n succ i is len-total & not n succ i is empty )
proof
A1: dom (n succ i) = n -tuples_on NAT by Def7;
 n succ i is len-total
proof
let x, y be FinSequence of NAT ; :: according to COMPUT_1:def_2 ::_thesis: ( len x = len y & x in dom (n succ i) implies y in dom (n succ i) )
assume that
A2: len x = len y and
A3: x in dom (n succ i) ; ::_thesis: y in dom (n succ i)
 len x = n by A1, A3, CARD_1:def_7;
then  y is Element of n -tuples_on NAT by A2, FINSEQ_2:92;
hence  y in dom (n succ i) by A1; ::_thesis: verum
end;
hence  ( n succ i is len-total & not n succ i is empty ) by A1; ::_thesis: verum
end;
end;

theorem Th34: :: COMPUT_1:34
for n, i being Element of NAT holds arity (n succ i) = n
proof
let n, i be Element of NAT ; ::_thesis: arity (n succ i) = n
consider d being set  such that
A1: d in dom (n succ i) by XBOOLE_0:def_1;
reconsider d = d as Element of n -tuples_on NAT by A1, Def7;
 dom (n succ i) = n -tuples_on NAT by Def7;
then A2: for y being FinSequence st y in dom (n succ i) holds
n = len y by CARD_1:def_7;
 d in dom (n succ i) by A1;
hence  arity (n succ i) = n by A2, MARGREL1:def_25; ::_thesis: verum
end;

definition
let n, i be Element of NAT ;
funcn proj i ->  NAT * -defined to-naturals homogeneous Function equals :: COMPUT_1:def 8
 proj ((n |-> NAT),i);
correctness
coherence
proj ((n |-> NAT),i) is NAT * -defined to-naturals homogeneous Function;
proof
A1: dom (proj ((n |-> NAT),i)) = product (n |-> NAT) by CARD_3:def_16
.= n -tuples_on NAT by FINSEQ_3:131 ;
now__::_thesis:_proj_((n_|->_NAT),i)_is_Element_of_HFuncs_NAT
set P = proj ((n |-> NAT),i);
A2: rng (proj ((n |-> NAT),i)) c= NAT
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (proj ((n |-> NAT),i)) or x in NAT )
assume  x in rng (proj ((n |-> NAT),i)) ; ::_thesis: x in NAT
then consider d being set  such that
A3: d in dom (proj ((n |-> NAT),i)) and
A4: x = (proj ((n |-> NAT),i)) . d by FUNCT_1:def_3;
reconsider d = d as Element of n -tuples_on NAT by A1, A3;
reconsider d = d as FinSequence of NAT ;
 (proj ((n |-> NAT),i)) . d = d . i by A3, CARD_3:def_16;
hence  x in NAT by A4; ::_thesis: verum
end;
A5: proj ((n |-> NAT),i) is homogeneous
proof
thus  dom (proj ((n |-> NAT),i)) is with_common_domain ; :: according to MARGREL1:def_21 ::_thesis: verum
end;
reconsider P = proj ((n |-> NAT),i) as PartFunc of (NAT *),NAT by A1, A2, FINSEQ_2:142, RELSET_1:4;
 P is Element of PFuncs ((NAT *),NAT) by PARTFUN1:45;
then  P in HFuncs NAT by A5;
hence  proj ((n |-> NAT),i) is Element of HFuncs NAT ; ::_thesis: verum
end;
hence  proj ((n |-> NAT),i) is NAT * -defined to-naturals homogeneous Function ; ::_thesis: verum
end;
end;

:: deftheorem  defines proj COMPUT_1:def_8_:_
 for n, i being Element of NAT holds n proj i = proj ((n |-> NAT),i);

theorem Th35: :: COMPUT_1:35
for n, i being Element of NAT holds n proj i in HFuncs NAT
proof
let n, i be Element of NAT ; ::_thesis: n proj i in HFuncs NAT
set X = NAT ;
set f = n proj i;
A1: rng (n proj i) c= NAT by VALUED_0:def_6;
 dom (n proj i) c= NAT * ;
then  n proj i is PartFunc of (NAT *),NAT by A1, RELSET_1:4;
then  n proj i is Element of PFuncs ((NAT *),NAT) by PARTFUN1:45;
hence  n proj i in HFuncs NAT ; ::_thesis: verum
end;

theorem Th36: :: COMPUT_1:36
for n, i being Element of NAT holds
( dom (n proj i) = n -tuples_on NAT & ( 1 <= i & i <= n implies rng (n proj i) = NAT ) )
proof
let n, i be Element of NAT ; ::_thesis: ( dom (n proj i) = n -tuples_on NAT & ( 1 <= i & i <= n implies rng (n proj i) = NAT ) )
thus A1: dom (n proj i) = product (n |-> NAT) by CARD_3:def_16
.= n -tuples_on NAT by FINSEQ_3:131 ; ::_thesis: ( 1 <= i & i <= n implies rng (n proj i) = NAT )
assume that
A2: 1 <= i and
A3: i <= n ; ::_thesis: rng (n proj i) = NAT
now__::_thesis:_for_x_being_set_holds_
(_(_x_in_rng_(n_proj_i)_implies_x_in_NAT_)_&_(_x_in_NAT_implies_x_in_rng_(n_proj_i)_)_)
let x be set ; ::_thesis: ( ( x in rng (n proj i) implies x in NAT ) & ( x in NAT implies x in rng (n proj i) ) )
hereby  ::_thesis: ( x in NAT implies x in rng (n proj i) )
assume  x in rng (n proj i) ; ::_thesis: x in NAT
then consider d being set  such that
 d in dom (n proj i) and
A4: x = (n proj i) . d by FUNCT_1:def_3;
thus  x in NAT by A4; ::_thesis: verum
end;
assume  x in NAT ; ::_thesis: x in rng (n proj i)
then reconsider x9 = x as Element of NAT ;
reconsider d = n |-> x9 as FinSequence of NAT ;
 i in Seg n by A2, A3, FINSEQ_1:1;
then A5: d . i = x9 by FUNCOP_1:7;
 (n proj i) . d = d . i by A1, CARD_3:def_16;
hence  x in rng (n proj i) by A1, A5, FUNCT_1:def_3; ::_thesis: verum
end;
hence  rng (n proj i) = NAT by TARSKI:1; ::_thesis: verum
end;

registration
let n, i be Element of NAT ;
clustern proj i ->  non empty NAT * -defined to-naturals homogeneous len-total ;
coherence
 ( n proj i is len-total & not n proj i is empty )
proof
A1: dom (n proj i) = n -tuples_on NAT by Th36;
 n proj i is len-total
proof
let x, y be FinSequence of NAT ; :: according to COMPUT_1:def_2 ::_thesis: ( len x = len y & x in dom (n proj i) implies y in dom (n proj i) )
assume that
A2: len x = len y and
A3: x in dom (n proj i) ; ::_thesis: y in dom (n proj i)
 len x = n by A1, A3, CARD_1:def_7;
then  y is Element of n -tuples_on NAT by A2, FINSEQ_2:92;
hence  y in dom (n proj i) by A1; ::_thesis: verum
end;
hence  ( n proj i is len-total & not n proj i is empty ) by A1; ::_thesis: verum
end;
end;

theorem Th37: :: COMPUT_1:37
for n, i being Element of NAT holds arity (n proj i) = n
proof
let n, i be Element of NAT ; ::_thesis: arity (n proj i) = n
consider d being set  such that
A1: d in n -tuples_on NAT by XBOOLE_0:def_1;
reconsider d = d as Element of n -tuples_on NAT by A1;
A2: dom (n proj i) = n -tuples_on NAT by Th36;
then A3: for x being FinSequence st x in dom (n proj i) holds
n = len x by CARD_1:def_7;
 d in dom (n proj i) by A2;
hence  arity (n proj i) = n by A3, MARGREL1:def_25; ::_thesis: verum
end;

theorem Th38: :: COMPUT_1:38
for n, i being Element of NAT
for t being Element of n -tuples_on NAT holds (n proj i) . t = t . i
proof
let n, i be Element of NAT ; ::_thesis: for t being Element of n -tuples_on NAT holds (n proj i) . t = t . i
let t be Element of n -tuples_on NAT; ::_thesis: (n proj i) . t = t . i
 dom (n proj i) = n -tuples_on NAT by Th36;
hence  (n proj i) . t = t . i by CARD_3:def_16; ::_thesis: verum
end;

registration
let X be set ;
cluster HFuncs X ->  functional ;
coherence
 HFuncs X is functional ;
end;

theorem Th39: :: COMPUT_1:39
for n being Element of NAT
for D, E being non empty set
for F being Function of D,(HFuncs E) st rng F is compatible & ( for x being Element of D holds dom (F . x) c= n -tuples_on E ) holds
ex f being Element of HFuncs E st
( f = Union F & dom f c= n -tuples_on E )
proof
let n be Element of NAT ; ::_thesis: for D, E being non empty set
for F being Function of D,(HFuncs E) st rng F is compatible & ( for x being Element of D holds dom (F . x) c= n -tuples_on E ) holds
ex f being Element of HFuncs E st
( f = Union F & dom f c= n -tuples_on E )
let D, E be non empty set ; ::_thesis: for F being Function of D,(HFuncs E) st rng F is compatible & ( for x being Element of D holds dom (F . x) c= n -tuples_on E ) holds
ex f being Element of HFuncs E st
( f = Union F & dom f c= n -tuples_on E )
set X = D;
set Y = E;
let F be Function of D,(HFuncs E); ::_thesis: ( rng F is compatible & ( for x being Element of D holds dom (F . x) c= n -tuples_on E ) implies ex f being Element of HFuncs E st
( f = Union F & dom f c= n -tuples_on E ) )
assume A1: rng F is compatible ; ::_thesis: ( ex x being Element of D st not dom (F . x) c= n -tuples_on E or ex f being Element of HFuncs E st
( f = Union F & dom f c= n -tuples_on E ) )
assume A2: for x being Element of D holds dom (F . x) c= n -tuples_on E ; ::_thesis: ex f being Element of HFuncs E st
( f = Union F & dom f c= n -tuples_on E )
A3: rng F is functional
proof
let x be set ; :: according to FUNCT_1:def_13 ::_thesis: ( not x in rng F or x is set )
A4: rng F c= HFuncs E by RELAT_1:def_19;
assume  x in rng F ; ::_thesis: x is set
hence  x is set by A4; ::_thesis: verum
end;
then reconsider rngF = rng F as non empty functional compatible set by A1;
set D =  {  (dom g) where g is Element of rngF : verum  }  ;
reconsider f = Union F as Function by A1, A3;
A5: rng f c= E
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng f or y in E )
assume  y in rng f ; ::_thesis: y in E
then consider x being set  such that
A6: x in dom f and
A7: f . x = y by FUNCT_1:def_3;
 x in union  {  (dom g) where g is Element of rngF : verum  }  by A6, Th12;
then consider d being set  such that
A8: x in d and
A9: d in  {  (dom g) where g is Element of rngF : verum  }  by TARSKI:def_4;
consider g being Element of rngF such that
A10: d = dom g and
 verum by A9;
A11: g in rngF ;
 rng F c= HFuncs E by RELAT_1:def_19;
then reconsider g = g as Element of HFuncs E by A11;
A12: g . x in rng g by A8, A10, FUNCT_1:3;
A13: rng g c= E by RELAT_1:def_19;
 f . x = g . x by A8, A10, Th13;
hence  y in E by A7, A13, A12; ::_thesis: verum
end;
A14: dom f c= n -tuples_on E
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom f or x in n -tuples_on E )
assume  x in dom f ; ::_thesis: x in n -tuples_on E
then  x in union  {  (dom g) where g is Element of rngF : verum  }  by Th12;
then consider d being set  such that
A15: x in d and
A16: d in  {  (dom g) where g is Element of rngF : verum  }  by TARSKI:def_4;
consider g being Element of rngF such that
A17: d = dom g and
 verum by A16;
consider e being set  such that
A18: e in dom F and
A19: F . e = g by FUNCT_1:def_3;
reconsider e = e as Element of D by A18;
 dom (F . e) c= n -tuples_on E by A2;
hence  x in n -tuples_on E by A15, A17, A19; ::_thesis: verum
end;
 n -tuples_on E c= E * by FINSEQ_2:142;
then  dom f c= E * by A14, XBOOLE_1:1;
then reconsider f = f as PartFunc of (E *),E by A5, RELSET_1:4;
reconsider f = f as Element of PFuncs ((E *),E) by PARTFUN1:45;
 f is homogeneous
proof
thus  dom f is with_common_domain by A14; :: according to MARGREL1:def_21 ::_thesis: verum
end;
then  f in  {  g where g is Element of PFuncs ((E *),E) : g is homogeneous  }  ;
then reconsider f = Union F as Element of HFuncs E ;
take  f ; ::_thesis: ( f = Union F & dom f c= n -tuples_on E )
thus  f = Union F ; ::_thesis: dom f c= n -tuples_on E
thus  dom f c= n -tuples_on E by A14; ::_thesis: verum
end;

theorem :: COMPUT_1:40
for D being non empty set
for F being Function of NAT,(HFuncs D) st ( for i being Element of NAT holds F . i c= F . (i + 1) ) holds
Union F in HFuncs D
proof
let D be non empty set ; ::_thesis: for F being Function of NAT,(HFuncs D) st ( for i being Element of NAT holds F . i c= F . (i + 1) ) holds
Union F in HFuncs D
set X = D;
let F be Function of NAT,(HFuncs D); ::_thesis: ( ( for i being Element of NAT holds F . i c= F . (i + 1) ) implies Union F in HFuncs D )
assume A1: for i being Element of NAT holds F . i c= F . (i + 1) ; ::_thesis: Union F in HFuncs D
A2: now__::_thesis:_for_n,_m_being_Element_of_NAT_st_n_<=_m_holds_
F_._n_c=_F_._m
let n be Element of NAT ; ::_thesis: for m being Element of NAT st n <= m holds
F . n c= F . m
defpred S1[ Element of NAT ] means F . n c= F . (n + $1);
let m be Element of NAT ; ::_thesis: ( n <= m implies F . n c= F . m )
A3: now__::_thesis:_for_i_being_Element_of_NAT_st_S1[i]_holds_
S1[i_+_1]
let i be Element of NAT ; ::_thesis: ( S1[i] implies S1[i + 1] )
assume A4: S1[i] ; ::_thesis: S1[i + 1]
 F . (n + i) c= F . ((n + i) + 1) by A1;
hence  S1[i + 1] by A4, XBOOLE_1:1; ::_thesis: verum
end;
A5: S1[ 0 ] ;
A6: for i being Element of NAT holds S1[i] from NAT_1:sch_1(A5, A3);
assume  n <= m ; ::_thesis: F . n c= F . m
then consider i being Nat such that
A7: m = n + i by NAT_1:10;
 i in NAT by ORDINAL1:def_12;
hence  F . n c= F . m by A6, A7; ::_thesis: verum
end;
reconsider Y = rng F as non empty Subset of (HFuncs D) by RELAT_1:def_19;
A8: Y is compatible
proof
let f, g be Function; :: according to COMPUT_1:def_1 ::_thesis: ( f in Y & g in Y implies f tolerates g )
assume  f in Y ; ::_thesis: ( not g in Y or f tolerates g )
then consider i being set  such that
A9: i in dom F and
A10: f = F . i by FUNCT_1:def_3;
assume  g in Y ; ::_thesis: f tolerates g
then consider j being set  such that
A11: j in dom F and
A12: g = F . j by FUNCT_1:def_3;
reconsider i = i, j = j as Element of NAT by A9, A11;
 ( i <= j or j <= i ) ;
hence  f tolerates g by A2, A10, A12, PARTFUN1:54; ::_thesis: verum
end;
 Y is PartFunc-set of D * ,D
proof
let f be Element of Y; :: according to RFUNCT_3:def_3 ::_thesis: f is Element of K32(K33((D *),D))
 f is Element of HFuncs D ;
hence  f is Element of K32(K33((D *),D)) ; ::_thesis: verum
end;
then  Union F is PartFunc of (D *),D by A8, Th15;
then reconsider f = Union F as Element of PFuncs ((D *),D) by PARTFUN1:45;
A13: dom f = union  {  (dom g) where g is Element of Y : verum  }  by A8, Th12;
 f is homogeneous
proof
thus  dom f is with_common_domain :: according to MARGREL1:def_21 ::_thesis: verum
proof
let x, y be FinSequence; :: according to MARGREL1:def_1 ::_thesis: ( not x in dom f or not y in dom f or len x = len y )
assume  x in dom f ; ::_thesis: ( not y in dom f or len x = len y )
then consider A1 being set  such that
A14: x in A1 and
A15: A1 in  {  (dom g) where g is Element of Y : verum  }  by A13, TARSKI:def_4;
consider g1 being Element of Y such that
A16: A1 = dom g1 by A15;
assume  y in dom f ; ::_thesis: len x = len y
then consider A2 being set  such that
A17: y in A2 and
A18: A2 in  {  (dom g) where g is Element of Y : verum  }  by A13, TARSKI:def_4;
consider g2 being Element of Y such that
A19: A2 = dom g2 by A18;
consider i1 being set  such that
A20: i1 in dom F and
A21: g1 = F . i1 by FUNCT_1:def_3;
consider i2 being set  such that
A22: i2 in dom F and
A23: g2 = F . i2 by FUNCT_1:def_3;
reconsider i1 = i1, i2 = i2 as Element of NAT by A20, A22;
 ( i1 <= i2 or i2 <= i1 ) ;
then  ( g1 c= g2 or g2 c= g1 ) by A2, A21, A23;
then  ( dom g1 c= dom g2 or dom g2 c= dom g1 ) by GRFUNC_1:2;
then  dom x = dom y by A14, A16, A17, A19, CARD_3:def_10;
hence  len x = len y by FINSEQ_3:29; ::_thesis: verum
end;
end;
hence  Union F in HFuncs D ; ::_thesis: verum
end;

theorem Th41: :: COMPUT_1:41
for D being non empty set
for F being with_the_same_arity FinSequence of HFuncs D holds dom <:F:> c= (arity F) -tuples_on D
proof
let D be non empty set ; ::_thesis: for F being with_the_same_arity FinSequence of HFuncs D holds dom <:F:> c= (arity F) -tuples_on D
set X = D;
let F be with_the_same_arity FinSequence of HFuncs D; ::_thesis: dom <:F:> c= (arity F) -tuples_on D
thus  dom <:F:> c= (arity F) -tuples_on D ::_thesis: verum
proof
A1: dom <:F:> = meet (doms F) by FUNCT_6:29
.= meet (rng (doms F)) by FUNCT_6:def_4 ;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom <:F:> or x in (arity F) -tuples_on D )
assume A2: x in dom <:F:> ; ::_thesis: x in (arity F) -tuples_on D
consider y being set  such that
A3: y in rng (doms F) by A2, A1, SETFAM_1:1, XBOOLE_0:def_1;
A4: x in y by A2, A1, A3, SETFAM_1:def_1;
consider z being set  such that
A5: z in dom (doms F) and
A6: (doms F) . z = y by A3, FUNCT_1:def_3;
A7: dom (doms F) = F " (SubFuncs (rng F)) by FUNCT_6:def_2;
then  z in dom F by A5, FUNCT_6:19;
then A8: F . z in rng F by FUNCT_1:3;
 rng F c= HFuncs D by RELAT_1:def_19;
then reconsider Fz = F . z as Element of HFuncs D by A8;
A9: dom Fz c= (arity Fz) -tuples_on D by Th20;
 (doms F) . z = dom Fz by A7, A5, FUNCT_6:def_2;
then  x in (arity Fz) -tuples_on D by A6, A4, A9;
hence  x in (arity F) -tuples_on D by A8, Def4; ::_thesis: verum
end;
end;

registration
let X be non empty set ;
let F be with_the_same_arity FinSequence of HFuncs X;
cluster<:F:> ->  homogeneous ;
coherence
 <:F:> is homogeneous
proof
 dom <:F:> c= (arity F) -tuples_on X by Th41;
hence  dom <:F:> is with_common_domain ; :: according to MARGREL1:def_21 ::_thesis: verum
end;
end;

theorem Th42: :: COMPUT_1:42
for D being non empty set
for f being Element of HFuncs D
for F being with_the_same_arity FinSequence of HFuncs D holds
( dom (f * <:F:>) c= (arity F) -tuples_on D & rng (f * <:F:>) c= D & f * <:F:> in HFuncs D )
proof
let D be non empty set ; ::_thesis: for f being Element of HFuncs D
for F being with_the_same_arity FinSequence of HFuncs D holds
( dom (f * <:F:>) c= (arity F) -tuples_on D & rng (f * <:F:>) c= D & f * <:F:> in HFuncs D )
set X = D;
let f be Element of HFuncs D; ::_thesis: for F being with_the_same_arity FinSequence of HFuncs D holds
( dom (f * <:F:>) c= (arity F) -tuples_on D & rng (f * <:F:>) c= D & f * <:F:> in HFuncs D )
let F be with_the_same_arity FinSequence of HFuncs D; ::_thesis: ( dom (f * <:F:>) c= (arity F) -tuples_on D & rng (f * <:F:>) c= D & f * <:F:> in HFuncs D )
A1: dom (f * <:F:>) c= dom <:F:> by RELAT_1:25;
A2: (arity F) -tuples_on D c= D * by FINSEQ_2:142;
 dom <:F:> c= (arity F) -tuples_on D by Th41;
hence  dom (f * <:F:>) c= (arity F) -tuples_on D by A1, XBOOLE_1:1; ::_thesis: ( rng (f * <:F:>) c= D & f * <:F:> in HFuncs D )
then A3: dom (f * <:F:>) c= D * by A2, XBOOLE_1:1;
thus  rng (f * <:F:>) c= D by RELAT_1:def_19; ::_thesis: f * <:F:> in HFuncs D
then  f * <:F:> is PartFunc of (D *),D by A3, RELSET_1:4;
then  f * <:F:> in PFuncs ((D *),D) by PARTFUN1:45;
hence  f * <:F:> in HFuncs D ; ::_thesis: verum
end;

definition
let X, Y be non empty set ;
let P be PFUNC_DOMAIN of X,Y;
let S be non empty Subset of P;
:: original: Element
redefine mode Element of S ->  Element of P;
coherence
 for b1 being Element of S holds b1 is Element of P
proof
let f be Element of S; ::_thesis: f is Element of P
thus  f is Element of P ; ::_thesis: verum
end;
end;

registration
let f be NAT * -defined homogeneous Function;
cluster<*f*> ->  with_the_same_arity ;
coherence
 <*f*> is with_the_same_arity
proof
let h, g be Function; :: according to COMPUT_1:def_3 ::_thesis: ( h in rng <*f*> & g in rng <*f*> implies ( ( not h is empty or g is empty or dom g = {{}} ) & ( not h is empty & not g is empty implies ex n being Element of NAT ex X being non empty set st
( dom h c= n -tuples_on X & dom g c= n -tuples_on X ) ) ) )
assume that
A1: h in rng <*f*> and
A2: g in rng <*f*> ; ::_thesis: ( ( not h is empty or g is empty or dom g = {{}} ) & ( not h is empty & not g is empty implies ex n being Element of NAT ex X being non empty set st
( dom h c= n -tuples_on X & dom g c= n -tuples_on X ) ) )
A3: rng <*f*> = {f} by FINSEQ_1:39;
then A4: h = f by A1, TARSKI:def_1;
hence  ( not h is empty or g is empty or dom g = {{}} ) by A2, A3, TARSKI:def_1; ::_thesis: ( not h is empty & not g is empty implies ex n being Element of NAT ex X being non empty set st
( dom h c= n -tuples_on X & dom g c= n -tuples_on X ) )
assume that
 not h is empty and
 not g is empty ; ::_thesis: ex n being Element of NAT ex X being non empty set st
( dom h c= n -tuples_on X & dom g c= n -tuples_on X )
take  arity f ; ::_thesis: ex X being non empty set st
( dom h c= (arity f) -tuples_on X & dom g c= (arity f) -tuples_on X )
take  NAT ; ::_thesis: ( dom h c= (arity f) -tuples_on NAT & dom g c= (arity f) -tuples_on NAT )
 g = f by A2, A3, TARSKI:def_1;
hence  ( dom h c= (arity f) -tuples_on NAT & dom g c= (arity f) -tuples_on NAT ) by A4, Th21; ::_thesis: verum
end;
end;

theorem :: COMPUT_1:43
for f being NAT * -defined to-naturals homogeneous Function holds arity <*f*> = arity f
proof
let f be NAT * -defined to-naturals homogeneous Function; ::_thesis: arity <*f*> = arity f
 rng <*f*> = {f} by FINSEQ_1:38;
then  f in rng <*f*> by TARSKI:def_1;
hence  arity <*f*> = arity f by Def4; ::_thesis: verum
end;

theorem Th44: :: COMPUT_1:44
for f, g being non empty Element of HFuncs NAT
for F being with_the_same_arity FinSequence of HFuncs NAT st g = f * <:F:> holds
arity g = arity F
proof
let f, g be non empty Element of HFuncs NAT; ::_thesis: for F being with_the_same_arity FinSequence of HFuncs NAT st g = f * <:F:> holds
arity g = arity F
let F be with_the_same_arity FinSequence of HFuncs NAT; ::_thesis: ( g = f * <:F:> implies arity g = arity F )
assume  g = f * <:F:> ; ::_thesis: arity g = arity F
then A1: dom g c= (arity F) -tuples_on NAT by Th42;
consider x being set  such that
A2: x in dom g by XBOOLE_0:def_1;
reconsider x = x as Element of (arity F) -tuples_on NAT by A1, A2;
 len x = arity F by CARD_1:def_7;
hence  arity g = arity F by A2, MARGREL1:def_25; ::_thesis: verum
end;

theorem Th45: :: COMPUT_1:45
for D being non empty set
for f being non empty quasi_total Element of HFuncs D
for F being with_the_same_arity FinSequence of HFuncs D st arity f = len F & not F is empty & ( for h being Element of HFuncs D st h in rng F holds
( h is quasi_total & not h is empty ) ) holds
( f * <:F:> is non empty quasi_total Element of HFuncs D & dom (f * <:F:>) = (arity F) -tuples_on D )
proof
let D be non empty set ; ::_thesis: for f being non empty quasi_total Element of HFuncs D
for F being with_the_same_arity FinSequence of HFuncs D st arity f = len F & not F is empty & ( for h being Element of HFuncs D st h in rng F holds
( h is quasi_total & not h is empty ) ) holds
( f * <:F:> is non empty quasi_total Element of HFuncs D & dom (f * <:F:>) = (arity F) -tuples_on D )
set X = D;
let f be non empty quasi_total Element of HFuncs D; ::_thesis: for F being with_the_same_arity FinSequence of HFuncs D st arity f = len F & not F is empty & ( for h being Element of HFuncs D st h in rng F holds
( h is quasi_total & not h is empty ) ) holds
( f * <:F:> is non empty quasi_total Element of HFuncs D & dom (f * <:F:>) = (arity F) -tuples_on D )
let F be with_the_same_arity FinSequence of HFuncs D; ::_thesis: ( arity f = len F & not F is empty & ( for h being Element of HFuncs D st h in rng F holds
( h is quasi_total & not h is empty ) ) implies ( f * <:F:> is non empty quasi_total Element of HFuncs D & dom (f * <:F:>) = (arity F) -tuples_on D ) )
assume that
A1: arity f = len F and
A2: not F is empty and
A3: for h being Element of HFuncs D st h in rng F holds
( h is quasi_total & not h is empty ) ; ::_thesis: ( f * <:F:> is non empty quasi_total Element of HFuncs D & dom (f * <:F:>) = (arity F) -tuples_on D )
set n = arity F;
set fF = f * <:F:>;
A4: dom (f * <:F:>) c= (arity F) -tuples_on D by Th42;
A5: (arity F) -tuples_on D c= dom (f * <:F:>)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (arity F) -tuples_on D or x in dom (f * <:F:>) )
A6: product (rngs F) c= (len F) -tuples_on D
proof
let p be set ; :: according to TARSKI:def_3 ::_thesis: ( not p in product (rngs F) or p in (len F) -tuples_on D )
A7: dom (rngs F) = F " (SubFuncs (rng F)) by FUNCT_6:def_3;
assume  p in product (rngs F) ; ::_thesis: p in (len F) -tuples_on D
then consider g being Function such that
A8: p = g and
A9: dom g = dom (rngs F) and
A10: for x being set st x in dom (rngs F) holds
g . x in (rngs F) . x by CARD_3:def_5;
now__::_thesis:_for_x_being_set_holds_
(_(_x_in_F_"_(SubFuncs_(rng_F))_implies_x_in_Seg_(len_F)_)_&_(_x_in_Seg_(len_F)_implies_x_in_F_"_(SubFuncs_(rng_F))_)_)
let x be set ; ::_thesis: ( ( x in F " (SubFuncs (rng F)) implies x in Seg (len F) ) & ( x in Seg (len F) implies x in F " (SubFuncs (rng F)) ) )
hereby  ::_thesis: ( x in Seg (len F) implies x in F " (SubFuncs (rng F)) )
assume  x in F " (SubFuncs (rng F)) ; ::_thesis: x in Seg (len F)
then  x in dom F by FUNCT_6:19;
hence  x in Seg (len F) by FINSEQ_1:def_3; ::_thesis: verum
end;
assume  x in Seg (len F) ; ::_thesis: x in F " (SubFuncs (rng F))
then A11: x in dom F by FINSEQ_1:def_3;
then  F . x in rng F by FUNCT_1:3;
hence  x in F " (SubFuncs (rng F)) by A11, FUNCT_6:19; ::_thesis: verum
end;
then A12: F " (SubFuncs (rng F)) = Seg (len F) by TARSKI:1;
then reconsider g = g as FinSequence by A9, A7, FINSEQ_1:def_2;
 rng g c= D
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng g or x in D )
assume  x in rng g ; ::_thesis: x in D
then consider d being set  such that
A13: d in dom g and
A14: g . d = x by FUNCT_1:def_3;
A15: g . d in (rngs F) . d by A9, A10, A13;
reconsider d = d as Element of NAT by A13;
 dom F = Seg (len F) by FINSEQ_1:def_3;
then reconsider Fd = F . d as Element of HFuncs D by A9, A7, A12, A13, FINSEQ_2:11;
A16: rng Fd c= D by RELAT_1:def_19;
 (rngs F) . d = rng Fd by A9, A7, A13, FUNCT_6:def_3;
hence  x in D by A14, A15, A16; ::_thesis: verum
end;
then reconsider g = g as FinSequence of D by FINSEQ_1:def_4;
 len g = len F by A9, A7, A12, FINSEQ_1:def_3;
then  p is Element of (len F) -tuples_on D by A8, FINSEQ_2:92;
hence  p in (len F) -tuples_on D ; ::_thesis: verum
end;
 rng <:F:> c= product (rngs F) by FUNCT_6:29;
then A17: rng <:F:> c= (len F) -tuples_on D by A6, XBOOLE_1:1;
A18: dom f = (arity f) -tuples_on D by Th22;
A19: (arity F) -tuples_on D c= dom <:F:>
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (arity F) -tuples_on D or x in dom <:F:> )
A20: dom (doms F) = F " (SubFuncs (rng F)) by FUNCT_6:def_2;
assume A21: x in (arity F) -tuples_on D ; ::_thesis: x in dom <:F:>
A22: now__::_thesis:_for_y_being_set_st_y_in_rng_(doms_F)_holds_
x_in_y
let y be set ; ::_thesis: ( y in rng (doms F) implies x in y )
assume  y in rng (doms F) ; ::_thesis: x in y
then consider w being set  such that
A23: w in dom (doms F) and
A24: (doms F) . w = y by FUNCT_1:def_3;
A25: w in dom F by A20, A23, FUNCT_6:19;
then reconsider w = w as Element of NAT ;
reconsider Fw = F . w as Element of HFuncs D by A25, FINSEQ_2:11;
A26: (doms F) . w = dom Fw by A20, A23, FUNCT_6:def_2;
A27: Fw in rng F by A25, FUNCT_1:3;
then  ( not Fw is empty & Fw is quasi_total ) by A3;
then  dom Fw = (arity Fw) -tuples_on D by Th22;
hence  x in y by A21, A24, A26, A27, Def4; ::_thesis: verum
end;
consider z being set  such that
A28: z in dom F by A2, XBOOLE_0:def_1;
 F . z in rng F by A28, FUNCT_1:3;
then  z in dom (doms F) by A28, A20, FUNCT_6:19;
then A29: rng (doms F) <> {} by RELAT_1:42;
dom <:F:> = meet (doms F) by FUNCT_6:29
.= meet (rng (doms F)) by FUNCT_6:def_4 ;
hence  x in dom <:F:> by A29, A22, SETFAM_1:def_1; ::_thesis: verum
end;
assume A30: x in (arity F) -tuples_on D ; ::_thesis: x in dom (f * <:F:>)
then  <:F:> . x in rng <:F:> by A19, FUNCT_1:3;
hence  x in dom (f * <:F:>) by A1, A30, A19, A18, A17, FUNCT_1:11; ::_thesis: verum
end;
then A31: dom (f * <:F:>) = (arity F) -tuples_on D by A4, XBOOLE_0:def_10;
A32: rng (f * <:F:>) c= D by Th42;
 (arity F) -tuples_on D c= D * by FINSEQ_2:142;
then  dom (f * <:F:>) c= D * by A4, XBOOLE_1:1;
then  f * <:F:> is Relation of (D *),D by A32, RELSET_1:4;
then  f * <:F:> is Element of PFuncs ((D *),D) by PARTFUN1:45;
then  f * <:F:> in HFuncs D ;
then reconsider fF = f * <:F:> as Element of HFuncs D ;
 fF is quasi_total
proof
let x, y be FinSequence of D; :: according to MARGREL1:def_22 ::_thesis: ( not len x = len y or not x in dom fF or y in dom fF )
assume that
A33: len x = len y and
A34: x in dom fF ; ::_thesis: y in dom fF
 len x = arity F by A4, A34, CARD_1:def_7;
then  y is Element of (arity F) -tuples_on D by A33, FINSEQ_2:92;
hence  y in dom fF by A31; ::_thesis: verum
end;
hence  f * <:F:> is non empty quasi_total Element of HFuncs D by A5; ::_thesis: dom (f * <:F:>) = (arity F) -tuples_on D
thus  dom (f * <:F:>) = (arity F) -tuples_on D by A4, A5, XBOOLE_0:def_10; ::_thesis: verum
end;

theorem Th46: :: COMPUT_1:46
for D being non empty set
for f being quasi_total Element of HFuncs D
for F being with_the_same_arity FinSequence of HFuncs D st arity f = len F & ( for h being Element of HFuncs D st h in rng F holds
h is quasi_total ) holds
f * <:F:> is quasi_total Element of HFuncs D
proof
let D be non empty set ; ::_thesis: for f being quasi_total Element of HFuncs D
for F being with_the_same_arity FinSequence of HFuncs D st arity f = len F & ( for h being Element of HFuncs D st h in rng F holds
h is quasi_total ) holds
f * <:F:> is quasi_total Element of HFuncs D
set X = D;
let f be quasi_total Element of HFuncs D; ::_thesis: for F being with_the_same_arity FinSequence of HFuncs D st arity f = len F & ( for h being Element of HFuncs D st h in rng F holds
h is quasi_total ) holds
f * <:F:> is quasi_total Element of HFuncs D
let F be with_the_same_arity FinSequence of HFuncs D; ::_thesis: ( arity f = len F & ( for h being Element of HFuncs D st h in rng F holds
h is quasi_total ) implies f * <:F:> is quasi_total Element of HFuncs D )
assume that
A1: arity f = len F and
A2: for h being Element of HFuncs D st h in rng F holds
h is quasi_total ; ::_thesis: f * <:F:> is quasi_total Element of HFuncs D
reconsider g = {} as PartFunc of (D *),D by RELSET_1:12;
A3: g is quasi_total
proof
let x, y be FinSequence of D; :: according to MARGREL1:def_22 ::_thesis: ( not len x = len y or not x in dom g or y in dom g )
assume that
 len x = len y and
A4: x in dom g ; ::_thesis: y in dom g
thus  y in dom g by A4; ::_thesis: verum
end;
 g in PFuncs ((D *),D) by PARTFUN1:45;
then A5: g in  {  h where h is Element of PFuncs ((D *),D) : h is homogeneous  }  ;
percases ( f = {} or F = {} or ex h being set st
( h in rng F & h = {} ) or ( F <> {} & f <> {} & ( for h being set st h in rng F holds
h <> {} ) ) ) ;
suppose f = {} ; ::_thesis: f * <:F:> is quasi_total Element of HFuncs D
hence  f * <:F:> is quasi_total Element of HFuncs D ; ::_thesis: verum
end;
suppose F = {} ; ::_thesis: f * <:F:> is quasi_total Element of HFuncs D
hence  f * <:F:> is quasi_total Element of HFuncs D by A5, A3, FUNCT_6:40; ::_thesis: verum
end;
suppose ex h being set st
( h in rng F & h = {} ) ; ::_thesis: f * <:F:> is quasi_total Element of HFuncs D
then  <:F:> = {} by Th7;
hence  f * <:F:> is quasi_total Element of HFuncs D by A5, A3; ::_thesis: verum
end;
supposeA6: ( F <> {} & f <> {} & ( for h being set st h in rng F holds
h <> {} ) ) ; ::_thesis: f * <:F:> is quasi_total Element of HFuncs D
then  for h being Element of HFuncs D st h in rng F holds
( h is quasi_total & not h is empty ) by A2;
hence  f * <:F:> is quasi_total Element of HFuncs D by A1, A6, Th45; ::_thesis: verum
end;
end;
end;

theorem Th47: :: COMPUT_1:47
for D being non empty set
for f, g being non empty quasi_total Element of HFuncs D st arity f = 0 & arity g = 0 & f . {} = g . {} holds
f = g
proof
let D be non empty set ; ::_thesis: for f, g being non empty quasi_total Element of HFuncs D st arity f = 0 & arity g = 0 & f . {} = g . {} holds
f = g
set X = D;
let f, g be non empty quasi_total Element of HFuncs D; ::_thesis: ( arity f = 0 & arity g = 0 & f . {} = g . {} implies f = g )
assume that
A1: arity f = 0 and
A2: arity g = 0 and
A3: f . {} = g . {} ; ::_thesis: f = g
now__::_thesis:_(_dom_f_=_{(<*>_D)}_&_dom_g_=_{(<*>_D)}_&_(_for_x_being_set_st_x_in_{(<*>_D)}_holds_
f_._x_=_g_._x_)_)
thus  dom f = 0 -tuples_on D by A1, Th22
.= {(<*> D)} by FINSEQ_2:94 ; ::_thesis: ( dom g = {(<*> D)} & ( for x being set st x in {(<*> D)} holds
f . x = g . x ) )
thus  dom g = 0 -tuples_on D by A2, Th22
.= {(<*> D)} by FINSEQ_2:94 ; ::_thesis: for x being set st x in {(<*> D)} holds
f . x = g . x
let x be set ; ::_thesis: ( x in {(<*> D)} implies f . x = g . x )
assume  x in {(<*> D)} ; ::_thesis: f . x = g . x
then  x = {} by TARSKI:def_1;
hence  f . x = g . x by A3; ::_thesis: verum
end;
hence  f = g by FUNCT_1:2; ::_thesis: verum
end;

theorem Th48: :: COMPUT_1:48
for f, g being non empty NAT * -defined to-naturals homogeneous len-total Function st arity f = 0 & arity g = 0 & f . {} = g . {} holds
f = g
proof
let f, g be non empty NAT * -defined to-naturals homogeneous len-total Function; ::_thesis: ( arity f = 0 & arity g = 0 & f . {} = g . {} implies f = g )
assume that
A1: arity f = 0 and
A2: arity g = 0 and
A3: f . {} = g . {} ; ::_thesis: f = g
A4: g is non empty quasi_total Element of HFuncs NAT by Th29;
 f is non empty quasi_total Element of HFuncs NAT by Th29;
hence  f = g by A1, A2, A3, A4, Th47; ::_thesis: verum
end;

begin


definition
let g, f1, f2 be NAT * -defined to-naturals homogeneous Function;
let i be Element of NAT ;
predg is_primitive-recursively_expressed_by f1,f2,i means :Def9: :: COMPUT_1:def 9
 ex n being Element of NAT st
( dom g c= n -tuples_on NAT & i >= 1 & i <= n & (arity f1) + 1 = n & n + 1 = arity f2 & ( for p being FinSequence of NAT st len p = n holds
( ( p +* (i,0) in dom g implies Del (p,i) in dom f1 ) & ( Del (p,i) in dom f1 implies p +* (i,0) in dom g ) & ( p +* (i,0) in dom g implies g . (p +* (i,0)) = f1 . (Del (p,i)) ) & ( for n being Element of NAT holds
( ( p +* (i,(n + 1)) in dom g implies ( p +* (i,n) in dom g & (p +* (i,n)) ^ <*(g . (p +* (i,n)))*> in dom f2 ) ) & ( p +* (i,n) in dom g & (p +* (i,n)) ^ <*(g . (p +* (i,n)))*> in dom f2 implies p +* (i,(n + 1)) in dom g ) & ( p +* (i,(n + 1)) in dom g implies g . (p +* (i,(n + 1))) = f2 . ((p +* (i,n)) ^ <*(g . (p +* (i,n)))*>) ) ) ) ) ) );
end;

:: deftheorem Def9 defines is_primitive-recursively_expressed_by COMPUT_1:def_9_:_
 for g, f1, f2 being NAT * -defined to-naturals homogeneous Function
for i being Element of NAT holds
( g is_primitive-recursively_expressed_by f1,f2,i iff ex n being Element of NAT st
( dom g c= n -tuples_on NAT & i >= 1 & i <= n & (arity f1) + 1 = n & n + 1 = arity f2 & ( for p being FinSequence of NAT st len p = n holds
( ( p +* (i,0) in dom g implies Del (p,i) in dom f1 ) & ( Del (p,i) in dom f1 implies p +* (i,0) in dom g ) & ( p +* (i,0) in dom g implies g . (p +* (i,0)) = f1 . (Del (p,i)) ) & ( for n being Element of NAT holds
( ( p +* (i,(n + 1)) in dom g implies ( p +* (i,n) in dom g & (p +* (i,n)) ^ <*(g . (p +* (i,n)))*> in dom f2 ) ) & ( p +* (i,n) in dom g & (p +* (i,n)) ^ <*(g . (p +* (i,n)))*> in dom f2 implies p +* (i,(n + 1)) in dom g ) & ( p +* (i,(n + 1)) in dom g implies g . (p +* (i,(n + 1))) = f2 . ((p +* (i,n)) ^ <*(g . (p +* (i,n)))*>) ) ) ) ) ) ) );

defpred S1[ Element of NAT , Element of HFuncs NAT, Element of HFuncs NAT, FinSequence of NAT , Element of NAT , homogeneous Function] means ( ( $5 in dom $4 & $4 +* ($5,$1) in dom $2 & ($4 +* ($5,$1)) ^ <*($2 . ($4 +* ($5,$1)))*> in dom $6 implies $3 = $2 +* (($4 +* ($5,($1 + 1))) .--> ($6 . (($4 +* ($5,$1)) ^ <*($2 . ($4 +* ($5,$1)))*>))) ) & ( ( not $5 in dom $4 or not $4 +* ($5,$1) in dom $2 or not ($4 +* ($5,$1)) ^ <*($2 . ($4 +* ($5,$1)))*> in dom $6 ) implies $3 = $2 ) );

definition
let f1, f2 be NAT * -defined to-naturals homogeneous Function;
let i be Element of NAT ;
let p be FinSequence of NAT ;
func primrec (f1,f2,i,p) ->  Element of HFuncs NAT means :Def10: :: COMPUT_1:def 10
 ex F being Function of NAT,(HFuncs NAT) st
( it = F . (p /. i) & ( i in dom p & Del (p,i) in dom f1 implies F . 0 = (p +* (i,0)) .--> (f1 . (Del (p,i))) ) & ( ( not i in dom p or not Del (p,i) in dom f1 ) implies F . 0 = {} ) & ( for m being Element of NAT holds
( ( i in dom p & p +* (i,m) in dom (F . m) & (p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*> in dom f2 implies F . (m + 1) = (F . m) +* ((p +* (i,(m + 1))) .--> (f2 . ((p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*>))) ) & ( ( not i in dom p or not p +* (i,m) in dom (F . m) or not (p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*> in dom f2 ) implies F . (m + 1) = F . m ) ) ) );
existence
 ex b1 being Element of HFuncs NAT ex F being Function of NAT,(HFuncs NAT) st
( b1 = F . (p /. i) & ( i in dom p & Del (p,i) in dom f1 implies F . 0 = (p +* (i,0)) .--> (f1 . (Del (p,i))) ) & ( ( not i in dom p or not Del (p,i) in dom f1 ) implies F . 0 = {} ) & ( for m being Element of NAT holds
( ( i in dom p & p +* (i,m) in dom (F . m) & (p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*> in dom f2 implies F . (m + 1) = (F . m) +* ((p +* (i,(m + 1))) .--> (f2 . ((p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*>))) ) & ( ( not i in dom p or not p +* (i,m) in dom (F . m) or not (p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*> in dom f2 ) implies F . (m + 1) = F . m ) ) ) )
proof
reconsider ff1 = f1, ff2 = f2 as Element of HFuncs NAT by Th28;
set n = len p;
reconsider z = ff1 . (Del (p,i)) as Element of NAT ;
defpred S2[ Element of NAT , Element of HFuncs NAT, Element of HFuncs NAT] means S1[$1,$2,$3,p,i,ff2];
 p +* (i,0) in NAT * by FINSEQ_1:def_11;
then reconsider Ap = {(p +* (i,0))} as non empty Subset of (NAT *) by ZFMISC_1:31;
 {} is PartFunc of (NAT *),NAT by RELSET_1:12;
then  {} in PFuncs ((NAT *),NAT) by PARTFUN1:45;
then A1: {} in HFuncs NAT ;
 Ap --> z = (p +* (i,0)) .--> z ;
then  Ap --> z in HFuncs NAT ;
then reconsider t = Ap --> z, e = {} as Element of HFuncs NAT by A1;
 ( ( i in dom p & Del (p,i) in dom f1 ) or not i in dom p or not Del (p,i) in dom f1 ) ;
then consider A being Element of HFuncs NAT such that
A2: ( i in dom p & Del (p,i) in dom f1 implies A = t ) and
A3: ( ( not i in dom p or not Del (p,i) in dom f1 ) implies A = e ) ;
A4: for m being Element of NAT
for x being Element of HFuncs NAT ex y being Element of HFuncs NAT st S2[m,x,y]
proof
let m be Element of NAT ; ::_thesis: for x being Element of HFuncs NAT ex y being Element of HFuncs NAT st S2[m,x,y]
let x be Element of HFuncs NAT; ::_thesis: ex y being Element of HFuncs NAT st S2[m,x,y]
set f = x +* ({(p +* (i,(m + 1)))} --> (f2 . ((p +* (i,m)) ^ <*(x . (p +* (i,m)))*>)));
reconsider z = ff2 . ((p +* (i,m)) ^ <*(x . (p +* (i,m)))*>) as Element of NAT ;
 ( ( i in dom p & p +* (i,m) in dom x & (p +* (i,m)) ^ <*(x . (p +* (i,m)))*> in dom f2 ) or not i in dom p or not p +* (i,m) in dom x or not (p +* (i,m)) ^ <*(x . (p +* (i,m)))*> in dom f2 ) ;
then consider y being Function such that
A5: ( ( i in dom p & p +* (i,m) in dom x & (p +* (i,m)) ^ <*(x . (p +* (i,m)))*> in dom f2 & y = x +* ({(p +* (i,(m + 1)))} --> (f2 . ((p +* (i,m)) ^ <*(x . (p +* (i,m)))*>))) ) or ( ( not i in dom p or not p +* (i,m) in dom x or not (p +* (i,m)) ^ <*(x . (p +* (i,m)))*> in dom f2 ) & y = x ) ) ;
 p +* (i,(m + 1)) in NAT * by FINSEQ_1:def_11;
then reconsider B = {(p +* (i,(m + 1)))} as Subset of (NAT *) by ZFMISC_1:31;
 dom (B --> z) = B by FUNCOP_1:13;
then A6: dom (x +* ({(p +* (i,(m + 1)))} --> (f2 . ((p +* (i,m)) ^ <*(x . (p +* (i,m)))*>)))) = (dom x) \/ B by FUNCT_4:def_1;
 dom p = dom (p +* (i,(m + 1))) by FUNCT_7:30;
then  len (p +* (i,(m + 1))) = len p by FINSEQ_3:29;
then  p +* (i,(m + 1)) is Element of (len p) -tuples_on NAT by FINSEQ_2:92;
then A7: B c= (len p) -tuples_on NAT by ZFMISC_1:31;
A8: x +* ({(p +* (i,(m + 1)))} --> (f2 . ((p +* (i,m)) ^ <*(x . (p +* (i,m)))*>))) = x +* (B --> z) ;
now__::_thesis:_(_p_+*_(i,m)_in_dom_x_implies_x_+*_({(p_+*_(i,(m_+_1)))}_-->_(f2_._((p_+*_(i,m))_^_<*(x_._(p_+*_(i,m)))*>)))_in_HFuncs_NAT_)
assume A9: p +* (i,m) in dom x ; ::_thesis: x +* ({(p +* (i,(m + 1)))} --> (f2 . ((p +* (i,m)) ^ <*(x . (p +* (i,m)))*>))) in HFuncs NAT
 dom x c= (len p) -tuples_on NAT
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in dom x or a in (len p) -tuples_on NAT )
A10: dom p = dom (p +* (i,m)) by FUNCT_7:30;
assume A11: a in dom x ; ::_thesis: a in (len p) -tuples_on NAT
then reconsider q = a as Element of NAT * ;
 len q = len (p +* (i,m)) by A9, A11, MARGREL1:def_1;
then  len q = len p by A10, FINSEQ_3:29;
hence  a in (len p) -tuples_on NAT ; ::_thesis: verum
end;
then A12: x +* ({(p +* (i,(m + 1)))} --> (f2 . ((p +* (i,m)) ^ <*(x . (p +* (i,m)))*>))) is homogeneous by A7, A6, Th16, XBOOLE_1:8;
 x +* ({(p +* (i,(m + 1)))} --> (f2 . ((p +* (i,m)) ^ <*(x . (p +* (i,m)))*>))) in PFuncs ((NAT *),NAT) by A8, PARTFUN1:45;
hence  x +* ({(p +* (i,(m + 1)))} --> (f2 . ((p +* (i,m)) ^ <*(x . (p +* (i,m)))*>))) in HFuncs NAT by A12; ::_thesis: verum
end;
then reconsider y = y as Element of HFuncs NAT by A5;
take  y ; ::_thesis: S2[m,x,y]
thus  ( i in dom p & p +* (i,m) in dom x & (p +* (i,m)) ^ <*(x . (p +* (i,m)))*> in dom ff2 implies y = x +* ((p +* (i,(m + 1))) .--> (ff2 . ((p +* (i,m)) ^ <*(x . (p +* (i,m)))*>))) ) by A5; ::_thesis: ( ( not i in dom p or not p +* (i,m) in dom x or not (p +* (i,m)) ^ <*(x . (p +* (i,m)))*> in dom ff2 ) implies y = x )
thus  ( ( not i in dom p or not p +* (i,m) in dom x or not (p +* (i,m)) ^ <*(x . (p +* (i,m)))*> in dom ff2 ) implies y = x ) by A5; ::_thesis: verum
end;
consider FF being Function of NAT,(HFuncs NAT) such that
A13: FF . 0 = A and
A14: for m being Element of NAT holds S2[m,FF . m,FF . (m + 1)] from RECDEF_1:sch_2(A4);
take  FF . (p /. i) ; ::_thesis: ex F being Function of NAT,(HFuncs NAT) st
( FF . (p /. i) = F . (p /. i) & ( i in dom p & Del (p,i) in dom f1 implies F . 0 = (p +* (i,0)) .--> (f1 . (Del (p,i))) ) & ( ( not i in dom p or not Del (p,i) in dom f1 ) implies F . 0 = {} ) & ( for m being Element of NAT holds
( ( i in dom p & p +* (i,m) in dom (F . m) & (p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*> in dom f2 implies F . (m + 1) = (F . m) +* ((p +* (i,(m + 1))) .--> (f2 . ((p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*>))) ) & ( ( not i in dom p or not p +* (i,m) in dom (F . m) or not (p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*> in dom f2 ) implies F . (m + 1) = F . m ) ) ) )
take  FF ; ::_thesis: ( FF . (p /. i) = FF . (p /. i) & ( i in dom p & Del (p,i) in dom f1 implies FF . 0 = (p +* (i,0)) .--> (f1 . (Del (p,i))) ) & ( ( not i in dom p or not Del (p,i) in dom f1 ) implies FF . 0 = {} ) & ( for m being Element of NAT holds
( ( i in dom p & p +* (i,m) in dom (FF . m) & (p +* (i,m)) ^ <*((FF . m) . (p +* (i,m)))*> in dom f2 implies FF . (m + 1) = (FF . m) +* ((p +* (i,(m + 1))) .--> (f2 . ((p +* (i,m)) ^ <*((FF . m) . (p +* (i,m)))*>))) ) & ( ( not i in dom p or not p +* (i,m) in dom (FF . m) or not (p +* (i,m)) ^ <*((FF . m) . (p +* (i,m)))*> in dom f2 ) implies FF . (m + 1) = FF . m ) ) ) )
thus  ( FF . (p /. i) = FF . (p /. i) & ( i in dom p & Del (p,i) in dom f1 implies FF . 0 = (p +* (i,0)) .--> (f1 . (Del (p,i))) ) & ( ( not i in dom p or not Del (p,i) in dom f1 ) implies FF . 0 = {} ) & ( for m being Element of NAT holds
( ( i in dom p & p +* (i,m) in dom (FF . m) & (p +* (i,m)) ^ <*((FF . m) . (p +* (i,m)))*> in dom f2 implies FF . (m + 1) = (FF . m) +* ((p +* (i,(m + 1))) .--> (f2 . ((p +* (i,m)) ^ <*((FF . m) . (p +* (i,m)))*>))) ) & ( ( not i in dom p or not p +* (i,m) in dom (FF . m) or not (p +* (i,m)) ^ <*((FF . m) . (p +* (i,m)))*> in dom f2 ) implies FF . (m + 1) = FF . m ) ) ) ) by A2, A3, A13, A14; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Element of HFuncs NAT st ex F being Function of NAT,(HFuncs NAT) st
( b1 = F . (p /. i) & ( i in dom p & Del (p,i) in dom f1 implies F . 0 = (p +* (i,0)) .--> (f1 . (Del (p,i))) ) & ( ( not i in dom p or not Del (p,i) in dom f1 ) implies F . 0 = {} ) & ( for m being Element of NAT holds
( ( i in dom p & p +* (i,m) in dom (F . m) & (p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*> in dom f2 implies F . (m + 1) = (F . m) +* ((p +* (i,(m + 1))) .--> (f2 . ((p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*>))) ) & ( ( not i in dom p or not p +* (i,m) in dom (F . m) or not (p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*> in dom f2 ) implies F . (m + 1) = F . m ) ) ) ) & ex F being Function of NAT,(HFuncs NAT) st
( b2 = F . (p /. i) & ( i in dom p & Del (p,i) in dom f1 implies F . 0 = (p +* (i,0)) .--> (f1 . (Del (p,i))) ) & ( ( not i in dom p or not Del (p,i) in dom f1 ) implies F . 0 = {} ) & ( for m being Element of NAT holds
( ( i in dom p & p +* (i,m) in dom (F . m) & (p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*> in dom f2 implies F . (m + 1) = (F . m) +* ((p +* (i,(m + 1))) .--> (f2 . ((p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*>))) ) & ( ( not i in dom p or not p +* (i,m) in dom (F . m) or not (p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*> in dom f2 ) implies F . (m + 1) = F . m ) ) ) ) holds
b1 = b2
proof
reconsider ff2 = f2 as Element of HFuncs NAT by Th28;
let g1, g2 be Element of HFuncs NAT; ::_thesis: ( ex F being Function of NAT,(HFuncs NAT) st
( g1 = F . (p /. i) & ( i in dom p & Del (p,i) in dom f1 implies F . 0 = (p +* (i,0)) .--> (f1 . (Del (p,i))) ) & ( ( not i in dom p or not Del (p,i) in dom f1 ) implies F . 0 = {} ) & ( for m being Element of NAT holds
( ( i in dom p & p +* (i,m) in dom (F . m) & (p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*> in dom f2 implies F . (m + 1) = (F . m) +* ((p +* (i,(m + 1))) .--> (f2 . ((p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*>))) ) & ( ( not i in dom p or not p +* (i,m) in dom (F . m) or not (p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*> in dom f2 ) implies F . (m + 1) = F . m ) ) ) ) & ex F being Function of NAT,(HFuncs NAT) st
( g2 = F . (p /. i) & ( i in dom p & Del (p,i) in dom f1 implies F . 0 = (p +* (i,0)) .--> (f1 . (Del (p,i))) ) & ( ( not i in dom p or not Del (p,i) in dom f1 ) implies F . 0 = {} ) & ( for m being Element of NAT holds
( ( i in dom p & p +* (i,m) in dom (F . m) & (p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*> in dom f2 implies F . (m + 1) = (F . m) +* ((p +* (i,(m + 1))) .--> (f2 . ((p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*>))) ) & ( ( not i in dom p or not p +* (i,m) in dom (F . m) or not (p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*> in dom f2 ) implies F . (m + 1) = F . m ) ) ) ) implies g1 = g2 )
given F1 being Function of NAT,(HFuncs NAT) such that A15: g1 = F1 . (p /. i) and
A16: ( i in dom p & Del (p,i) in dom f1 implies F1 . 0 = (p +* (i,0)) .--> (f1 . (Del (p,i))) ) and
A17: ( ( not i in dom p or not Del (p,i) in dom f1 ) implies F1 . 0 = {} ) and
A18: for m being Element of NAT holds S1[m,F1 . m,F1 . (m + 1),p,i,f2] ; ::_thesis: ( for F being Function of NAT,(HFuncs NAT) holds
( not g2 = F . (p /. i) or ( i in dom p & Del (p,i) in dom f1 & not F . 0 = (p +* (i,0)) .--> (f1 . (Del (p,i))) ) or ( ( not i in dom p or not Del (p,i) in dom f1 ) & not F . 0 = {} ) or ex m being Element of NAT st
( ( i in dom p & p +* (i,m) in dom (F . m) & (p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*> in dom f2 implies F . (m + 1) = (F . m) +* ((p +* (i,(m + 1))) .--> (f2 . ((p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*>))) ) implies ( ( not i in dom p or not p +* (i,m) in dom (F . m) or not (p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*> in dom f2 ) & not F . (m + 1) = F . m ) ) ) or g1 = g2 )
given F2 being Function of NAT,(HFuncs NAT) such that A19: g2 = F2 . (p /. i) and
A20: ( i in dom p & Del (p,i) in dom f1 implies F2 . 0 = (p +* (i,0)) .--> (f1 . (Del (p,i))) ) and
A21: ( ( not i in dom p or not Del (p,i) in dom f1 ) implies F2 . 0 = {} ) and
A22: for m being Element of NAT holds S1[m,F2 . m,F2 . (m + 1),p,i,f2] ; ::_thesis: g1 = g2
defpred S2[ Element of NAT ] means F1 . $1 = F2 . $1;
A23: now__::_thesis:_for_m_being_Element_of_NAT_st_S2[m]_holds_
S2[m_+_1]
let m be Element of NAT ; ::_thesis: ( S2[m] implies S2[m + 1] )
assume  S2[m] ; ::_thesis: S2[m + 1]
then A24: S1[m,F1 . m,F2 . (m + 1),p,i,ff2] by A22;
 S1[m,F1 . m,F1 . (m + 1),p,i,f2] by A18;
hence  S2[m + 1] by A24; ::_thesis: verum
end;
A25: S2[ 0 ] by A16, A17, A20, A21;
 for m being Element of NAT holds S2[m] from NAT_1:sch_1(A25, A23);
hence  g1 = g2 by A15, A19; ::_thesis: verum
end;
end;

:: deftheorem Def10 defines primrec COMPUT_1:def_10_:_
 for f1, f2 being NAT * -defined to-naturals homogeneous Function
for i being Element of NAT
for p being FinSequence of NAT
for b5 being Element of HFuncs NAT holds
( b5 = primrec (f1,f2,i,p) iff ex F being Function of NAT,(HFuncs NAT) st
( b5 = F . (p /. i) & ( i in dom p & Del (p,i) in dom f1 implies F . 0 = (p +* (i,0)) .--> (f1 . (Del (p,i))) ) & ( ( not i in dom p or not Del (p,i) in dom f1 ) implies F . 0 = {} ) & ( for m being Element of NAT holds
( ( i in dom p & p +* (i,m) in dom (F . m) & (p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*> in dom f2 implies F . (m + 1) = (F . m) +* ((p +* (i,(m + 1))) .--> (f2 . ((p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*>))) ) & ( ( not i in dom p or not p +* (i,m) in dom (F . m) or not (p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*> in dom f2 ) implies F . (m + 1) = F . m ) ) ) ) );

theorem Th49: :: COMPUT_1:49
for i being Element of NAT
for e1, e2 being NAT * -defined to-naturals homogeneous Function
for p, q being FinSequence of NAT st q in dom (primrec (e1,e2,i,p)) holds
ex k being Element of NAT st q = p +* (i,k)
proof
let i be Element of NAT ; ::_thesis: for e1, e2 being NAT * -defined to-naturals homogeneous Function
for p, q being FinSequence of NAT st q in dom (primrec (e1,e2,i,p)) holds
ex k being Element of NAT st q = p +* (i,k)
let e1, e2 be NAT * -defined to-naturals homogeneous Function; ::_thesis: for p, q being FinSequence of NAT st q in dom (primrec (e1,e2,i,p)) holds
ex k being Element of NAT st q = p +* (i,k)
set f1 = e1;
set f2 = e2;
let p, q be FinSequence of NAT ; ::_thesis: ( q in dom (primrec (e1,e2,i,p)) implies ex k being Element of NAT st q = p +* (i,k) )
assume A1: q in dom (primrec (e1,e2,i,p)) ; ::_thesis: ex k being Element of NAT st q = p +* (i,k)
consider F being Function of NAT,(HFuncs NAT) such that
A2: primrec (e1,e2,i,p) = F . (p /. i) and
A3: ( i in dom p & Del (p,i) in dom e1 implies F . 0 = (p +* (i,0)) .--> (e1 . (Del (p,i))) ) and
A4: ( ( not i in dom p or not Del (p,i) in dom e1 ) implies F . 0 = {} ) and
A5: for m being Element of NAT holds S1[m,F . m,F . (m + 1),p,i,e2] by Def10;
defpred S2[ Element of NAT ] means ( q in dom (F . $1) implies ex k being Element of NAT st q = p +* (i,k) );
A6: for m being Element of NAT st S2[m] holds
S2[m + 1]
proof
let m be Element of NAT ; ::_thesis: ( S2[m] implies S2[m + 1] )
assume that
A7: S2[m] and
A8: q in dom (F . (m + 1)) ; ::_thesis: ex k being Element of NAT st q = p +* (i,k)
percases ( ( i in dom p & p +* (i,m) in dom (F . m) & (p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*> in dom e2 ) or not i in dom p or not p +* (i,m) in dom (F . m) or not (p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*> in dom e2 ) ;
suppose ( i in dom p & p +* (i,m) in dom (F . m) & (p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*> in dom e2 ) ; ::_thesis: ex k being Element of NAT st q = p +* (i,k)
then  F . (m + 1) = (F . m) +* ((p +* (i,(m + 1))) .--> (e2 . ((p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*>))) by A5;
then  dom (F . (m + 1)) = (dom (F . m)) \/ (dom ((p +* (i,(m + 1))) .--> (e2 . ((p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*>)))) by FUNCT_4:def_1;
then A9: dom (F . (m + 1)) = (dom (F . m)) \/ {(p +* (i,(m + 1)))} by FUNCOP_1:13;
thus  ex k being Element of NAT st q = p +* (i,k) ::_thesis: verum
proof
percases ( q in dom (F . m) or q in {(p +* (i,(m + 1)))} ) by A8, A9, XBOOLE_0:def_3;
suppose q in dom (F . m) ; ::_thesis: ex k being Element of NAT st q = p +* (i,k)
hence  ex k being Element of NAT st q = p +* (i,k) by A7; ::_thesis: verum
end;
suppose q in {(p +* (i,(m + 1)))} ; ::_thesis: ex k being Element of NAT st q = p +* (i,k)
then  q = p +* (i,(m + 1)) by TARSKI:def_1;
hence  ex k being Element of NAT st q = p +* (i,k) ; ::_thesis: verum
end;
end;
end;
end;
suppose ( not i in dom p or not p +* (i,m) in dom (F . m) or not (p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*> in dom e2 ) ; ::_thesis: ex k being Element of NAT st q = p +* (i,k)
hence  ex k being Element of NAT st q = p +* (i,k) by A5, A7, A8; ::_thesis: verum
end;
end;
end;
A10: S2[ 0 ]
proof
assume A11: q in dom (F . 0) ; ::_thesis: ex k being Element of NAT st q = p +* (i,k)
percases ( ( i in dom p & Del (p,i) in dom e1 ) or not i in dom p or not Del (p,i) in dom e1 ) ;
suppose ( i in dom p & Del (p,i) in dom e1 ) ; ::_thesis: ex k being Element of NAT st q = p +* (i,k)
then  dom (F . 0) = {(p +* (i,0))} by A3, FUNCOP_1:13;
then  p +* (i,0) = q by A11, TARSKI:def_1;
hence  ex k being Element of NAT st q = p +* (i,k) ; ::_thesis: verum
end;
suppose ( not i in dom p or not Del (p,i) in dom e1 ) ; ::_thesis: ex k being Element of NAT st q = p +* (i,k)
hence  ex k being Element of NAT st q = p +* (i,k) by A4, A11; ::_thesis: verum
end;
end;
end;
 for n being Element of NAT holds S2[n] from NAT_1:sch_1(A10, A6);
hence  ex k being Element of NAT st q = p +* (i,k) by A1, A2; ::_thesis: verum
end;

theorem Th50: :: COMPUT_1:50
for i being Element of NAT
for e1, e2 being NAT * -defined to-naturals homogeneous Function
for p being FinSequence of NAT st not i in dom p holds
primrec (e1,e2,i,p) = {}
proof
let i be Element of NAT ; ::_thesis: for e1, e2 being NAT * -defined to-naturals homogeneous Function
for p being FinSequence of NAT st not i in dom p holds
primrec (e1,e2,i,p) = {}
let e1, e2 be NAT * -defined to-naturals homogeneous Function; ::_thesis: for p being FinSequence of NAT st not i in dom p holds
primrec (e1,e2,i,p) = {}
set f1 = e1;
set f2 = e2;
let p be FinSequence of NAT ; ::_thesis: ( not i in dom p implies primrec (e1,e2,i,p) = {} )
consider F being Function of NAT,(HFuncs NAT) such that
A1: primrec (e1,e2,i,p) = F . (p /. i) and
 ( i in dom p & Del (p,i) in dom e1 implies F . 0 = (p +* (i,0)) .--> (e1 . (Del (p,i))) ) and
A2: ( ( not i in dom p or not Del (p,i) in dom e1 ) implies F . 0 = {} ) and
A3: for m being Element of NAT holds S1[m,F . m,F . (m + 1),p,i,e2] by Def10;
defpred S2[ Element of NAT ] means F . $1 = {} ;
assume A4: not i in dom p ; ::_thesis: primrec (e1,e2,i,p) = {}
then A5: for m being Element of NAT st S2[m] holds
S2[m + 1] by A3;
A6: S2[ 0 ] by A4, A2;
 for m being Element of NAT holds S2[m] from NAT_1:sch_1(A6, A5);
hence  primrec (e1,e2,i,p) = {} by A1; ::_thesis: verum
end;

theorem Th51: :: COMPUT_1:51
for i being Element of NAT
for e1, e2 being NAT * -defined to-naturals homogeneous Function
for p, q being FinSequence of NAT holds primrec (e1,e2,i,p) tolerates primrec (e1,e2,i,q)
proof
let i be Element of NAT ; ::_thesis: for e1, e2 being NAT * -defined to-naturals homogeneous Function
for p, q being FinSequence of NAT holds primrec (e1,e2,i,p) tolerates primrec (e1,e2,i,q)
let e1, e2 be NAT * -defined to-naturals homogeneous Function; ::_thesis: for p, q being FinSequence of NAT holds primrec (e1,e2,i,p) tolerates primrec (e1,e2,i,q)
set f1 = e1;
set f2 = e2;
let p1, p2 be FinSequence of NAT ; ::_thesis: primrec (e1,e2,i,p1) tolerates primrec (e1,e2,i,p2)
percases ( not i in dom p1 or not i in dom p2 or ( i in dom p1 & i in dom p2 ) ) ;
suppose ( not i in dom p1 or not i in dom p2 ) ; ::_thesis: primrec (e1,e2,i,p1) tolerates primrec (e1,e2,i,p2)
then  ( primrec (e1,e2,i,p1) = {} or primrec (e1,e2,i,p2) = {} ) by Th50;
hence  primrec (e1,e2,i,p1) tolerates primrec (e1,e2,i,p2) by PARTFUN1:59; ::_thesis: verum
end;
supposeA1: ( i in dom p1 & i in dom p2 ) ; ::_thesis: primrec (e1,e2,i,p1) tolerates primrec (e1,e2,i,p2)
reconsider m = p1 /. i, n = p2 /. i as Element of NAT ;
consider F2 being Function of NAT,(HFuncs NAT) such that
A2: primrec (e1,e2,i,p2) = F2 . (p2 /. i) and
A3: ( i in dom p2 & Del (p2,i) in dom e1 implies F2 . 0 = (p2 +* (i,0)) .--> (e1 . (Del (p2,i))) ) and
A4: ( ( not i in dom p2 or not Del (p2,i) in dom e1 ) implies F2 . 0 = {} ) and
A5: for m being Element of NAT holds S1[m,F2 . m,F2 . (m + 1),p2,i,e2] by Def10;
consider F1 being Function of NAT,(HFuncs NAT) such that
A6: primrec (e1,e2,i,p1) = F1 . (p1 /. i) and
A7: ( i in dom p1 & Del (p1,i) in dom e1 implies F1 . 0 = (p1 +* (i,0)) .--> (e1 . (Del (p1,i))) ) and
A8: ( ( not i in dom p1 or not Del (p1,i) in dom e1 ) implies F1 . 0 = {} ) and
A9: for m being Element of NAT holds S1[m,F1 . m,F1 . (m + 1),p1,i,e2] by Def10;
defpred S2[ Element of NAT ] means  for x being set st x in dom (F1 . $1) holds
ex n being Element of NAT st
( x = p1 +* (i,n) & n <= $1 );
A10: now__::_thesis:_for_m_being_Element_of_NAT_st_S2[m]_holds_
S2[m_+_1]
let m be Element of NAT ; ::_thesis: ( S2[m] implies S2[m + 1] )
assume A11: S2[m] ; ::_thesis: S2[m + 1]
thus  S2[m + 1] ::_thesis: verum
proof
set p1c = (p1 +* (i,m)) ^ <*((F1 . m) . (p1 +* (i,m)))*>;
let x be set ; ::_thesis: ( x in dom (F1 . (m + 1)) implies ex n being Element of NAT st
( x = p1 +* (i,n) & n <= m + 1 ) )
assume A12: x in dom (F1 . (m + 1)) ; ::_thesis: ex n being Element of NAT st
( x = p1 +* (i,n) & n <= m + 1 )
A13: m <= m + 1 by NAT_1:11;
percases ( ( i in dom p1 & p1 +* (i,m) in dom (F1 . m) & (p1 +* (i,m)) ^ <*((F1 . m) . (p1 +* (i,m)))*> in dom e2 ) or not i in dom p1 or not p1 +* (i,m) in dom (F1 . m) or not (p1 +* (i,m)) ^ <*((F1 . m) . (p1 +* (i,m)))*> in dom e2 ) ;
suppose ( i in dom p1 & p1 +* (i,m) in dom (F1 . m) & (p1 +* (i,m)) ^ <*((F1 . m) . (p1 +* (i,m)))*> in dom e2 ) ; ::_thesis: ex n being Element of NAT st
( x = p1 +* (i,n) & n <= m + 1 )
then  F1 . (m + 1) = (F1 . m) +* ((p1 +* (i,(m + 1))) .--> (e2 . ((p1 +* (i,m)) ^ <*((F1 . m) . (p1 +* (i,m)))*>))) by A9;
then  dom (F1 . (m + 1)) = (dom (F1 . m)) \/ (dom ((p1 +* (i,(m + 1))) .--> (e2 . ((p1 +* (i,m)) ^ <*((F1 . m) . (p1 +* (i,m)))*>)))) by FUNCT_4:def_1
.= (dom (F1 . m)) \/ {(p1 +* (i,(m + 1)))} by FUNCOP_1:13 ;
then A14: ( x in dom (F1 . m) or x in {(p1 +* (i,(m + 1)))} ) by A12, XBOOLE_0:def_3;
thus  ex n being Element of NAT st
( x = p1 +* (i,n) & n <= m + 1 ) ::_thesis: verum
proof
percases ( x in dom (F1 . m) or x = p1 +* (i,(m + 1)) ) by A14, TARSKI:def_1;
suppose x in dom (F1 . m) ; ::_thesis: ex n being Element of NAT st
( x = p1 +* (i,n) & n <= m + 1 )
then  ex n being Element of NAT st
( x = p1 +* (i,n) & n <= m ) by A11;
hence  ex n being Element of NAT st
( x = p1 +* (i,n) & n <= m + 1 ) by A13, XXREAL_0:2; ::_thesis: verum
end;
suppose x = p1 +* (i,(m + 1)) ; ::_thesis: ex n being Element of NAT st
( x = p1 +* (i,n) & n <= m + 1 )
hence  ex n being Element of NAT st
( x = p1 +* (i,n) & n <= m + 1 ) ; ::_thesis: verum
end;
end;
end;
end;
suppose ( not i in dom p1 or not p1 +* (i,m) in dom (F1 . m) or not (p1 +* (i,m)) ^ <*((F1 . m) . (p1 +* (i,m)))*> in dom e2 ) ; ::_thesis: ex n being Element of NAT st
( x = p1 +* (i,n) & n <= m + 1 )
then  F1 . (m + 1) = F1 . m by A9;
then  ex n being Element of NAT st
( x = p1 +* (i,n) & n <= m ) by A11, A12;
hence  ex n being Element of NAT st
( x = p1 +* (i,n) & n <= m + 1 ) by A13, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
end;
A15: now__::_thesis:_(_p1_+*_(i,0)_=_p2_+*_(i,0)_implies_for_m_being_Element_of_NAT_holds_S3[m]_)
defpred S3[ Element of NAT ] means F1 . $1 = F2 . $1;
assume A16: p1 +* (i,0) = p2 +* (i,0) ; ::_thesis: for m being Element of NAT holds S3[m]
A17: S3[ 0 ]
proof
percases ( ( i in dom p1 & Del (p1,i) in dom e1 ) or not i in dom p1 or not Del (p1,i) in dom e1 ) ;
suppose ( i in dom p1 & Del (p1,i) in dom e1 ) ; ::_thesis: S3[ 0 ]
hence  S3[ 0 ] by A1, A7, A3, A16, Th4; ::_thesis: verum
end;
suppose ( not i in dom p1 or not Del (p1,i) in dom e1 ) ; ::_thesis: S3[ 0 ]
hence  S3[ 0 ] by A1, A8, A4, A16, Th4; ::_thesis: verum
end;
end;
end;
A18: now__::_thesis:_for_m_being_Element_of_NAT_st_S3[m]_holds_
S3[m_+_1]
let m be Element of NAT ; ::_thesis: ( S3[m] implies S3[m + 1] )
assume A19: S3[m] ; ::_thesis: S3[m + 1]
A20: S1[m,F1 . m,F1 . (m + 1),p1,i,e2] by A9;
A21: S1[m,F2 . m,F2 . (m + 1),p2,i,e2] by A5;
 p1 +* (i,m) = p2 +* (i,m) by A16, Th2;
hence  S3[m + 1] by A1, A19, A20, A21, Th2; ::_thesis: verum
end;
thus  for m being Element of NAT holds S3[m] from NAT_1:sch_1(A17, A18); ::_thesis: verum
end;
A22: S2[ 0 ]
proof
let x be set ; ::_thesis: ( x in dom (F1 . 0) implies ex n being Element of NAT st
( x = p1 +* (i,n) & n <= 0 ) )
assume A23: x in dom (F1 . 0) ; ::_thesis: ex n being Element of NAT st
( x = p1 +* (i,n) & n <= 0 )
 dom (F1 . 0) = {(p1 +* (i,0))} by A7, A8, A23, FUNCOP_1:13;
then  x = p1 +* (i,0) by A23, TARSKI:def_1;
hence  ex n being Element of NAT st
( x = p1 +* (i,n) & n <= 0 ) ; ::_thesis: verum
end;
A24: for m being Element of NAT holds S2[m] from NAT_1:sch_1(A22, A10);
A25: for m, n being Element of NAT holds F1 . m c= F1 . (m + n)
proof
let m be Element of NAT ; ::_thesis: for n being Element of NAT holds F1 . m c= F1 . (m + n)
defpred S3[ Element of NAT ] means F1 . m c= F1 . (m + $1);
A26: now__::_thesis:_for_n_being_Element_of_NAT_st_S3[n]_holds_
S3[n_+_1]
let n be Element of NAT ; ::_thesis: ( S3[n] implies S3[n + 1] )
assume A27: S3[n] ; ::_thesis: S3[n + 1]
set k = m + n;
 F1 . (m + n) c= F1 . ((m + n) + 1)
proof
set p1c = (p1 +* (i,(m + n))) ^ <*((F1 . (m + n)) . (p1 +* (i,(m + n))))*>;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in F1 . (m + n) or x in F1 . ((m + n) + 1) )
assume A28: x in F1 . (m + n) ; ::_thesis: x in F1 . ((m + n) + 1)
percases ( ( i in dom p1 & p1 +* (i,(m + n)) in dom (F1 . (m + n)) & (p1 +* (i,(m + n))) ^ <*((F1 . (m + n)) . (p1 +* (i,(m + n))))*> in dom e2 ) or not i in dom p1 or not p1 +* (i,(m + n)) in dom (F1 . (m + n)) or not (p1 +* (i,(m + n))) ^ <*((F1 . (m + n)) . (p1 +* (i,(m + n))))*> in dom e2 ) ;
supposeA29: ( i in dom p1 & p1 +* (i,(m + n)) in dom (F1 . (m + n)) & (p1 +* (i,(m + n))) ^ <*((F1 . (m + n)) . (p1 +* (i,(m + n))))*> in dom e2 ) ; ::_thesis: x in F1 . ((m + n) + 1)
A30: dom (F1 . (m + n)) misses dom ((p1 +* (i,((m + n) + 1))) .--> (e2 . ((p1 +* (i,(m + n))) ^ <*((F1 . (m + n)) . (p1 +* (i,(m + n))))*>)))
proof
assume  not dom (F1 . (m + n)) misses dom ((p1 +* (i,((m + n) + 1))) .--> (e2 . ((p1 +* (i,(m + n))) ^ <*((F1 . (m + n)) . (p1 +* (i,(m + n))))*>))) ; ::_thesis: contradiction
then consider x being set  such that
A31: x in (dom (F1 . (m + n))) /\ (dom ({(p1 +* (i,((m + n) + 1)))} --> (e2 . ((p1 +* (i,(m + n))) ^ <*((F1 . (m + n)) . (p1 +* (i,(m + n))))*>)))) by XBOOLE_0:4;
 x in dom (F1 . (m + n)) by A31, XBOOLE_0:def_4;
then consider n1 being Element of NAT  such that
A32: x = p1 +* (i,n1) and
A33: n1 <= m + n by A24;
A34: (m + n) + 1 = (p1 +* (i,((m + n) + 1))) . i by A1, FUNCT_7:31;
A35: x = p1 +* (i,((m + n) + 1)) by A31, TARSKI:def_1;
 n1 = (p1 +* (i,n1)) . i by A1, FUNCT_7:31;
hence  contradiction by A35, A32, A33, A34, NAT_1:13; ::_thesis: verum
end;
 F1 . ((m + n) + 1) = (F1 . (m + n)) +* ((p1 +* (i,((m + n) + 1))) .--> (e2 . ((p1 +* (i,(m + n))) ^ <*((F1 . (m + n)) . (p1 +* (i,(m + n))))*>))) by A9, A29;
then  F1 . (m + n) c= F1 . ((m + n) + 1) by A30, FUNCT_4:32;
hence  x in F1 . ((m + n) + 1) by A28; ::_thesis: verum
end;
suppose ( not i in dom p1 or not p1 +* (i,(m + n)) in dom (F1 . (m + n)) or not (p1 +* (i,(m + n))) ^ <*((F1 . (m + n)) . (p1 +* (i,(m + n))))*> in dom e2 ) ; ::_thesis: x in F1 . ((m + n) + 1)
hence  x in F1 . ((m + n) + 1) by A9, A28; ::_thesis: verum
end;
end;
end;
hence  S3[n + 1] by A27, XBOOLE_1:1; ::_thesis: verum
end;
A36: S3[ 0 ] ;
thus  for n being Element of NAT holds S3[n] from NAT_1:sch_1(A36, A26); ::_thesis: verum
end;
defpred S3[ Element of NAT ] means  for x being set st x in dom (F2 . $1) holds
ex n being Element of NAT st
( x = p2 +* (i,n) & n <= $1 );
A37: now__::_thesis:_for_m_being_Element_of_NAT_st_S3[m]_holds_
S3[m_+_1]
let m be Element of NAT ; ::_thesis: ( S3[m] implies S3[m + 1] )
assume A38: S3[m] ; ::_thesis: S3[m + 1]
thus  S3[m + 1] ::_thesis: verum
proof
set p2c = (p2 +* (i,m)) ^ <*((F2 . m) . (p2 +* (i,m)))*>;
let x be set ; ::_thesis: ( x in dom (F2 . (m + 1)) implies ex n being Element of NAT st
( x = p2 +* (i,n) & n <= m + 1 ) )
assume A39: x in dom (F2 . (m + 1)) ; ::_thesis: ex n being Element of NAT st
( x = p2 +* (i,n) & n <= m + 1 )
A40: m <= m + 1 by NAT_1:11;
percases ( ( i in dom p2 & p2 +* (i,m) in dom (F2 . m) & (p2 +* (i,m)) ^ <*((F2 . m) . (p2 +* (i,m)))*> in dom e2 ) or not i in dom p2 or not p2 +* (i,m) in dom (F2 . m) or not (p2 +* (i,m)) ^ <*((F2 . m) . (p2 +* (i,m)))*> in dom e2 ) ;
suppose ( i in dom p2 & p2 +* (i,m) in dom (F2 . m) & (p2 +* (i,m)) ^ <*((F2 . m) . (p2 +* (i,m)))*> in dom e2 ) ; ::_thesis: ex n being Element of NAT st
( x = p2 +* (i,n) & n <= m + 1 )
then  F2 . (m + 1) = (F2 . m) +* ((p2 +* (i,(m + 1))) .--> (e2 . ((p2 +* (i,m)) ^ <*((F2 . m) . (p2 +* (i,m)))*>))) by A5;
then  dom (F2 . (m + 1)) = (dom (F2 . m)) \/ (dom ((p2 +* (i,(m + 1))) .--> (e2 . ((p2 +* (i,m)) ^ <*((F2 . m) . (p2 +* (i,m)))*>)))) by FUNCT_4:def_1
.= (dom (F2 . m)) \/ {(p2 +* (i,(m + 1)))} by FUNCOP_1:13 ;
then A41: ( x in dom (F2 . m) or x in {(p2 +* (i,(m + 1)))} ) by A39, XBOOLE_0:def_3;
thus  ex n being Element of NAT st
( x = p2 +* (i,n) & n <= m + 1 ) ::_thesis: verum
proof
percases ( x in dom (F2 . m) or x = p2 +* (i,(m + 1)) ) by A41, TARSKI:def_1;
suppose x in dom (F2 . m) ; ::_thesis: ex n being Element of NAT st
( x = p2 +* (i,n) & n <= m + 1 )
then  ex n being Element of NAT st
( x = p2 +* (i,n) & n <= m ) by A38;
hence  ex n being Element of NAT st
( x = p2 +* (i,n) & n <= m + 1 ) by A40, XXREAL_0:2; ::_thesis: verum
end;
suppose x = p2 +* (i,(m + 1)) ; ::_thesis: ex n being Element of NAT st
( x = p2 +* (i,n) & n <= m + 1 )
hence  ex n being Element of NAT st
( x = p2 +* (i,n) & n <= m + 1 ) ; ::_thesis: verum
end;
end;
end;
end;
suppose ( not i in dom p2 or not p2 +* (i,m) in dom (F2 . m) or not (p2 +* (i,m)) ^ <*((F2 . m) . (p2 +* (i,m)))*> in dom e2 ) ; ::_thesis: ex n being Element of NAT st
( x = p2 +* (i,n) & n <= m + 1 )
then  F2 . (m + 1) = F2 . m by A5;
then  ex n being Element of NAT st
( x = p2 +* (i,n) & n <= m ) by A38, A39;
hence  ex n being Element of NAT st
( x = p2 +* (i,n) & n <= m + 1 ) by A40, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
end;
A42: S3[ 0 ]
proof
let x be set ; ::_thesis: ( x in dom (F2 . 0) implies ex n being Element of NAT st
( x = p2 +* (i,n) & n <= 0 ) )
assume A43: x in dom (F2 . 0) ; ::_thesis: ex n being Element of NAT st
( x = p2 +* (i,n) & n <= 0 )
 dom (F2 . 0) = {(p2 +* (i,0))} by A3, A4, A43, FUNCOP_1:13;
then  x = p2 +* (i,0) by A43, TARSKI:def_1;
hence  ex n being Element of NAT st
( x = p2 +* (i,n) & n <= 0 ) ; ::_thesis: verum
end;
A44: for m being Element of NAT holds S3[m] from NAT_1:sch_1(A42, A37);
A45: for m, n being Element of NAT holds F2 . m c= F2 . (m + n)
proof
let m be Element of NAT ; ::_thesis: for n being Element of NAT holds F2 . m c= F2 . (m + n)
defpred S4[ Element of NAT ] means F2 . m c= F2 . (m + $1);
A46: now__::_thesis:_for_n_being_Element_of_NAT_st_S4[n]_holds_
S4[n_+_1]
let n be Element of NAT ; ::_thesis: ( S4[n] implies S4[n + 1] )
assume A47: S4[n] ; ::_thesis: S4[n + 1]
set k = m + n;
 F2 . (m + n) c= F2 . ((m + n) + 1)
proof
set p2c = (p2 +* (i,(m + n))) ^ <*((F2 . (m + n)) . (p2 +* (i,(m + n))))*>;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in F2 . (m + n) or x in F2 . ((m + n) + 1) )
assume A48: x in F2 . (m + n) ; ::_thesis: x in F2 . ((m + n) + 1)
percases ( ( i in dom p2 & p2 +* (i,(m + n)) in dom (F2 . (m + n)) & (p2 +* (i,(m + n))) ^ <*((F2 . (m + n)) . (p2 +* (i,(m + n))))*> in dom e2 ) or not i in dom p2 or not p2 +* (i,(m + n)) in dom (F2 . (m + n)) or not (p2 +* (i,(m + n))) ^ <*((F2 . (m + n)) . (p2 +* (i,(m + n))))*> in dom e2 ) ;
supposeA49: ( i in dom p2 & p2 +* (i,(m + n)) in dom (F2 . (m + n)) & (p2 +* (i,(m + n))) ^ <*((F2 . (m + n)) . (p2 +* (i,(m + n))))*> in dom e2 ) ; ::_thesis: x in F2 . ((m + n) + 1)
A50: dom (F2 . (m + n)) misses dom ((p2 +* (i,((m + n) + 1))) .--> (e2 . ((p2 +* (i,(m + n))) ^ <*((F2 . (m + n)) . (p2 +* (i,(m + n))))*>)))
proof
assume  not dom (F2 . (m + n)) misses dom ((p2 +* (i,((m + n) + 1))) .--> (e2 . ((p2 +* (i,(m + n))) ^ <*((F2 . (m + n)) . (p2 +* (i,(m + n))))*>))) ; ::_thesis: contradiction
then consider x being set  such that
A51: x in (dom (F2 . (m + n))) /\ (dom ({(p2 +* (i,((m + n) + 1)))} --> (e2 . ((p2 +* (i,(m + n))) ^ <*((F2 . (m + n)) . (p2 +* (i,(m + n))))*>)))) by XBOOLE_0:4;
 x in dom (F2 . (m + n)) by A51, XBOOLE_0:def_4;
then consider n2 being Element of NAT  such that
A52: x = p2 +* (i,n2) and
A53: n2 <= m + n by A44;
A54: (m + n) + 1 = (p2 +* (i,((m + n) + 1))) . i by A1, FUNCT_7:31;
A55: x = p2 +* (i,((m + n) + 1)) by A51, TARSKI:def_1;
 n2 = (p2 +* (i,n2)) . i by A1, FUNCT_7:31;
hence  contradiction by A55, A52, A53, A54, NAT_1:13; ::_thesis: verum
end;
 F2 . ((m + n) + 1) = (F2 . (m + n)) +* ((p2 +* (i,((m + n) + 1))) .--> (e2 . ((p2 +* (i,(m + n))) ^ <*((F2 . (m + n)) . (p2 +* (i,(m + n))))*>))) by A5, A49;
then  F2 . (m + n) c= F2 . ((m + n) + 1) by A50, FUNCT_4:32;
hence  x in F2 . ((m + n) + 1) by A48; ::_thesis: verum
end;
suppose ( not i in dom p2 or not p2 +* (i,(m + n)) in dom (F2 . (m + n)) or not (p2 +* (i,(m + n))) ^ <*((F2 . (m + n)) . (p2 +* (i,(m + n))))*> in dom e2 ) ; ::_thesis: x in F2 . ((m + n) + 1)
hence  x in F2 . ((m + n) + 1) by A5, A48; ::_thesis: verum
end;
end;
end;
hence  S4[n + 1] by A47, XBOOLE_1:1; ::_thesis: verum
end;
A56: S4[ 0 ] ;
thus  for n being Element of NAT holds S4[n] from NAT_1:sch_1(A56, A46); ::_thesis: verum
end;
reconsider F1m = F1 . m, F1n = F1 . n, F2m = F2 . m, F2n = F2 . n as Element of HFuncs NAT ;
A57: now__::_thesis:_(_p1_+*_(i,0)_<>_p2_+*_(i,0)_implies_for_m_being_Element_of_NAT_holds_not_(dom_(F1_._m))_/\_(dom_(F2_._m))_<>_{}_)
assume A58: p1 +* (i,0) <> p2 +* (i,0) ; ::_thesis: for m being Element of NAT holds not (dom (F1 . m)) /\ (dom (F2 . m)) <> {}
let m be Element of NAT ; ::_thesis: not (dom (F1 . m)) /\ (dom (F2 . m)) <> {}
assume  (dom (F1 . m)) /\ (dom (F2 . m)) <> {} ; ::_thesis: contradiction
then consider x being set  such that
A59: x in (dom (F1 . m)) /\ (dom (F2 . m)) by XBOOLE_0:def_1;
 x in dom (F2 . m) by A59, XBOOLE_0:def_4;
then A60: ex n2 being Element of NAT st
( x = p2 +* (i,n2) & n2 <= m ) by A44;
 x in dom (F1 . m) by A59, XBOOLE_0:def_4;
then  ex n1 being Element of NAT st
( x = p1 +* (i,n1) & n1 <= m ) by A24;
hence  contradiction by A58, A60, Th2; ::_thesis: verum
end;
thus  primrec (e1,e2,i,p1) tolerates primrec (e1,e2,i,p2) ::_thesis: verum
proof
percases ( m <= n or m >= n ) ;
suppose m <= n ; ::_thesis: primrec (e1,e2,i,p1) tolerates primrec (e1,e2,i,p2)
then consider k being Nat such that
A61: n = m + k by NAT_1:10;
reconsider k = k as Element of NAT by ORDINAL1:def_12;
A62: n = m + k by A61;
thus  primrec (e1,e2,i,p1) tolerates primrec (e1,e2,i,p2) ::_thesis: verum
proof
percases ( F1n = F2n or (dom F1n) /\ (dom F2n) = {} ) by A15, A57;
suppose F1n = F2n ; ::_thesis: primrec (e1,e2,i,p1) tolerates primrec (e1,e2,i,p2)
hence  primrec (e1,e2,i,p1) tolerates primrec (e1,e2,i,p2) by A6, A2, A25, A62, PARTFUN1:58; ::_thesis: verum
end;
suppose (dom F1n) /\ (dom F2n) = {} ; ::_thesis: primrec (e1,e2,i,p1) tolerates primrec (e1,e2,i,p2)
then  dom F1n misses dom F2n by XBOOLE_0:def_7;
then  F1n tolerates F2n by PARTFUN1:56;
hence  primrec (e1,e2,i,p1) tolerates primrec (e1,e2,i,p2) by A6, A2, A25, A62, PARTFUN1:58; ::_thesis: verum
end;
end;
end;
end;
suppose m >= n ; ::_thesis: primrec (e1,e2,i,p1) tolerates primrec (e1,e2,i,p2)
then consider k being Nat such that
A63: m = n + k by NAT_1:10;
reconsider k = k as Element of NAT by ORDINAL1:def_12;
A64: m = n + k by A63;
thus  primrec (e1,e2,i,p1) tolerates primrec (e1,e2,i,p2) ::_thesis: verum
proof
percases ( F1m = F2m or (dom F1m) /\ (dom F2m) = {} ) by A15, A57;
suppose F1m = F2m ; ::_thesis: primrec (e1,e2,i,p1) tolerates primrec (e1,e2,i,p2)
hence  primrec (e1,e2,i,p1) tolerates primrec (e1,e2,i,p2) by A6, A2, A45, A64, PARTFUN1:58; ::_thesis: verum
end;
suppose (dom F1m) /\ (dom F2m) = {} ; ::_thesis: primrec (e1,e2,i,p1) tolerates primrec (e1,e2,i,p2)
then  dom F1m misses dom F2m by XBOOLE_0:def_7;
then  F1m tolerates F2m by PARTFUN1:56;
hence  primrec (e1,e2,i,p1) tolerates primrec (e1,e2,i,p2) by A6, A2, A45, A64, PARTFUN1:58; ::_thesis: verum
end;
end;
end;
end;
end;
end;
end;
end;
end;

theorem Th52: :: COMPUT_1:52
for i being Element of NAT
for e1, e2 being NAT * -defined to-naturals homogeneous Function
for p being FinSequence of NAT holds dom (primrec (e1,e2,i,p)) c= (1 + (arity e1)) -tuples_on NAT
proof
let i be Element of NAT ; ::_thesis: for e1, e2 being NAT * -defined to-naturals homogeneous Function
for p being FinSequence of NAT holds dom (primrec (e1,e2,i,p)) c= (1 + (arity e1)) -tuples_on NAT
let e1, e2 be NAT * -defined to-naturals homogeneous Function; ::_thesis: for p being FinSequence of NAT holds dom (primrec (e1,e2,i,p)) c= (1 + (arity e1)) -tuples_on NAT
set f1 = e1;
set f2 = e2;
let p be FinSequence of NAT ; ::_thesis: dom (primrec (e1,e2,i,p)) c= (1 + (arity e1)) -tuples_on NAT
percases ( i in dom p or not i in dom p ) ;
supposeA1: i in dom p ; ::_thesis: dom (primrec (e1,e2,i,p)) c= (1 + (arity e1)) -tuples_on NAT
consider F being Function of NAT,(HFuncs NAT) such that
A2: primrec (e1,e2,i,p) = F . (p /. i) and
A3: ( i in dom p & Del (p,i) in dom e1 implies F . 0 = (p +* (i,0)) .--> (e1 . (Del (p,i))) ) and
A4: ( ( not i in dom p or not Del (p,i) in dom e1 ) implies F . 0 = {} ) and
A5: for m being Element of NAT holds S1[m,F . m,F . (m + 1),p,i,e2] by Def10;
defpred S2[ Element of NAT ] means  dom (F . $1) c= (1 + (arity e1)) -tuples_on NAT;
A6: now__::_thesis:_for_m_being_Element_of_NAT_st_S2[m]_holds_
S2[m_+_1]
let m be Element of NAT ; ::_thesis: ( S2[m] implies S2[b1 + 1] )
assume A7: S2[m] ; ::_thesis: S2[b1 + 1]
set pc = (p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*>;
percases ( ( p +* (i,m) in dom (F . m) & (p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*> in dom e2 ) or not p +* (i,m) in dom (F . m) or not (p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*> in dom e2 ) ;
supposeA8: ( p +* (i,m) in dom (F . m) & (p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*> in dom e2 ) ; ::_thesis: S2[b1 + 1]
dom (p +* (i,m)) = dom p by FUNCT_7:30
.= dom (p +* (i,(m + 1))) by FUNCT_7:30 ;
then A9: len (p +* (i,m)) = len (p +* (i,(m + 1))) by FINSEQ_3:29;
 len (p +* (i,m)) = 1 + (arity e1) by A7, A8, CARD_1:def_7;
then  p +* (i,(m + 1)) is Element of (1 + (arity e1)) -tuples_on NAT by A9, FINSEQ_2:92;
then A10: {(p +* (i,(m + 1)))} c= (1 + (arity e1)) -tuples_on NAT by ZFMISC_1:31;
 F . (m + 1) = (F . m) +* ((p +* (i,(m + 1))) .--> (e2 . ((p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*>))) by A1, A5, A8;
then  dom (F . (m + 1)) = (dom (F . m)) \/ (dom ((p +* (i,(m + 1))) .--> (e2 . ((p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*>)))) by FUNCT_4:def_1;
then  dom (F . (m + 1)) = (dom (F . m)) \/ {(p +* (i,(m + 1)))} by FUNCOP_1:13;
hence  S2[m + 1] by A7, A10, XBOOLE_1:8; ::_thesis: verum
end;
suppose ( not p +* (i,m) in dom (F . m) or not (p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*> in dom e2 ) ; ::_thesis: S2[b1 + 1]
hence  S2[m + 1] by A5, A7; ::_thesis: verum
end;
end;
end;
A11: S2[ 0 ]
proof
percases ( Del (p,i) in dom e1 or not Del (p,i) in dom e1 ) ;
supposeA12: Del (p,i) in dom e1 ; ::_thesis: S2[ 0 ]
 dom e1 c= (arity e1) -tuples_on NAT by Th21;
then A13: len (Del (p,i)) = arity e1 by A12, CARD_1:def_7;
A14: dom p = dom (p +* (i,0)) by FUNCT_7:30;
 p <> <*> NAT by A1;
then consider n being Nat such that
A15: len p = n + 1 by NAT_1:6;
 len (Del (p,i)) = n by A1, A15, FINSEQ_3:109;
then  len (p +* (i,0)) = 1 + (arity e1) by A13, A15, A14, FINSEQ_3:29;
then A16: p +* (i,0) is Element of (1 + (arity e1)) -tuples_on NAT by FINSEQ_2:92;
 dom (F . 0) = {(p +* (i,0))} by A1, A3, A12, FUNCOP_1:13;
hence  S2[ 0 ] by A16, ZFMISC_1:31; ::_thesis: verum
end;
suppose not Del (p,i) in dom e1 ; ::_thesis: S2[ 0 ]
hence  S2[ 0 ] by A4, RELAT_1:38, XBOOLE_1:2; ::_thesis: verum
end;
end;
end;
 for m being Element of NAT holds S2[m] from NAT_1:sch_1(A11, A6);
hence  dom (primrec (e1,e2,i,p)) c= (1 + (arity e1)) -tuples_on NAT by A2; ::_thesis: verum
end;
suppose not i in dom p ; ::_thesis: dom (primrec (e1,e2,i,p)) c= (1 + (arity e1)) -tuples_on NAT
then  primrec (e1,e2,i,p) = {} by Th50;
hence  dom (primrec (e1,e2,i,p)) c= (1 + (arity e1)) -tuples_on NAT by RELAT_1:38, XBOOLE_1:2; ::_thesis: verum
end;
end;
end;

theorem Th53: :: COMPUT_1:53
for i being Element of NAT
for e1, e2 being NAT * -defined to-naturals homogeneous Function
for p being FinSequence of NAT st e1 is empty holds
primrec (e1,e2,i,p) is empty
proof
let i be Element of NAT ; ::_thesis: for e1, e2 being NAT * -defined to-naturals homogeneous Function
for p being FinSequence of NAT st e1 is empty holds
primrec (e1,e2,i,p) is empty
let e1, e2 be NAT * -defined to-naturals homogeneous Function; ::_thesis: for p being FinSequence of NAT st e1 is empty holds
primrec (e1,e2,i,p) is empty
set f1 = e1;
set f2 = e2;
let p be FinSequence of NAT ; ::_thesis: ( e1 is empty implies primrec (e1,e2,i,p) is empty )
consider F being Function of NAT,(HFuncs NAT) such that
A1: primrec (e1,e2,i,p) = F . (p /. i) and
 ( i in dom p & Del (p,i) in dom e1 implies F . 0 = (p +* (i,0)) .--> (e1 . (Del (p,i))) ) and
A2: ( ( not i in dom p or not Del (p,i) in dom e1 ) implies F . 0 = {} ) and
A3: for m being Element of NAT holds
( ( i in dom p & p +* (i,m) in dom (F . m) & (p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*> in dom e2 implies F . (m + 1) = (F . m) +* ((p +* (i,(m + 1))) .--> (e2 . ((p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*>))) ) & ( ( not i in dom p or not p +* (i,m) in dom (F . m) or not (p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*> in dom e2 ) implies F . (m + 1) = F . m ) ) by Def10;
defpred S2[ Element of NAT ] means F . $1 = {} ;
A4: for k being Element of NAT st S2[k] holds
S2[k + 1] by A3, RELAT_1:38;
assume  e1 is empty ; ::_thesis: primrec (e1,e2,i,p) is empty
then A5: S2[ 0 ] by A2;
 for k being Element of NAT holds S2[k] from NAT_1:sch_1(A5, A4);
hence  primrec (e1,e2,i,p) is empty by A1; ::_thesis: verum
end;

theorem Th54: :: COMPUT_1:54
for i being Element of NAT
for f1, f2 being non empty NAT * -defined to-naturals homogeneous Function
for p being Element of ((arity f1) + 1) -tuples_on NAT st f1 is len-total & f2 is len-total & (arity f1) + 2 = arity f2 & 1 <= i & i <= 1 + (arity f1) holds
p in dom (primrec (f1,f2,i,p))
proof
let i be Element of NAT ; ::_thesis: for f1, f2 being non empty NAT * -defined to-naturals homogeneous Function
for p being Element of ((arity f1) + 1) -tuples_on NAT st f1 is len-total & f2 is len-total & (arity f1) + 2 = arity f2 & 1 <= i & i <= 1 + (arity f1) holds
p in dom (primrec (f1,f2,i,p))
let f1, f2 be non empty NAT * -defined to-naturals homogeneous Function; ::_thesis: for p being Element of ((arity f1) + 1) -tuples_on NAT st f1 is len-total & f2 is len-total & (arity f1) + 2 = arity f2 & 1 <= i & i <= 1 + (arity f1) holds
p in dom (primrec (f1,f2,i,p))
let p be Element of ((arity f1) + 1) -tuples_on NAT; ::_thesis: ( f1 is len-total & f2 is len-total & (arity f1) + 2 = arity f2 & 1 <= i & i <= 1 + (arity f1) implies p in dom (primrec (f1,f2,i,p)) )
assume that
A1: f1 is len-total and
A2: f2 is len-total and
A3: (arity f1) + 2 = arity f2 and
A4: 1 <= i and
A5: i <= 1 + (arity f1) ; ::_thesis: p in dom (primrec (f1,f2,i,p))
A6: len p = 1 + (arity f1) by CARD_1:def_7;
then A7: i in dom p by A4, A5, FINSEQ_3:25;
then A8: p /. i = p . i by PARTFUN1:def_6;
consider F being Function of NAT,(HFuncs NAT) such that
A9: primrec (f1,f2,i,p) = F . (p /. i) and
A10: ( i in dom p & Del (p,i) in dom f1 implies F . 0 = (p +* (i,0)) .--> (f1 . (Del (p,i))) ) and
 ( ( not i in dom p or not Del (p,i) in dom f1 ) implies F . 0 = {} ) and
A11: for m being Element of NAT holds
( ( i in dom p & p +* (i,m) in dom (F . m) & (p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*> in dom f2 implies F . (m + 1) = (F . m) +* ((p +* (i,(m + 1))) .--> (f2 . ((p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*>))) ) & ( ( not i in dom p or not p +* (i,m) in dom (F . m) or not (p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*> in dom f2 ) implies F . (m + 1) = F . m ) ) by Def10;
defpred S2[ Element of NAT ] means p +* (i,$1) in dom (F . $1);
A12: p +* (i,(p . i)) = p by FUNCT_7:35;
A13: now__::_thesis:_for_m_being_Element_of_NAT_st_S2[m]_holds_
S2[m_+_1]
let m be Element of NAT ; ::_thesis: ( S2[m] implies S2[m + 1] )
assume A14: S2[m] ; ::_thesis: S2[m + 1]
reconsider aa = (F . m) . (p +* (i,m)) as Element of NAT ;
set pc = (p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*>;
reconsider p2 = <*aa*> as FinSequence of NAT ;
reconsider p1 = (p +* (i,m)) ^ p2 as FinSequence of NAT ;
A15: len p2 = 1 by CARD_1:def_7;
 len (p +* (i,m)) = 1 + (arity f1) by CARD_1:def_7;
then  len p1 = ((arity f1) + 1) + 1 by A15, FINSEQ_1:22
.= arity f2 by A3 ;
then A16: p1 is Element of (arity f2) -tuples_on NAT by FINSEQ_2:92;
 p +* (i,(m + 1)) in {(p +* (i,(m + 1)))} by TARSKI:def_1;
then A17: p +* (i,(m + 1)) in dom ({(p +* (i,(m + 1)))} --> (f2 . ((p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*>))) by FUNCOP_1:13;
 dom f2 = (arity f2) -tuples_on NAT by A2, Th23;
then  F . (m + 1) = (F . m) +* ((p +* (i,(m + 1))) .--> (f2 . ((p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*>))) by A7, A11, A14, A16;
then  dom (F . (m + 1)) = (dom (F . m)) \/ (dom ({(p +* (i,(m + 1)))} --> (f2 . ((p +* (i,m)) ^ <*((F . m) . (p +* (i,m)))*>)))) by FUNCT_4:def_1;
hence  S2[m + 1] by A17, XBOOLE_0:def_3; ::_thesis: verum
end;
A18: now__::_thesis:_S2[_0_]
reconsider Dpi = Del (p,i) as FinSequence of NAT by FINSEQ_3:105;
reconsider Dpi9 = Dpi as Element of (len Dpi) -tuples_on NAT by FINSEQ_2:92;
A19: dom f1 = (arity f1) -tuples_on NAT by A1, Th23;
 len (Del (p,i)) = arity f1 by A6, A7, FINSEQ_3:109;
then  Dpi9 in dom f1 by A19;
then  dom (F . 0) = {(p +* (i,0))} by A4, A5, A6, A10, FINSEQ_3:25, FUNCOP_1:13;
hence  S2[ 0 ] by TARSKI:def_1; ::_thesis: verum
end;
 for m being Element of NAT holds S2[m] from NAT_1:sch_1(A18, A13);
hence  p in dom (primrec (f1,f2,i,p)) by A8, A9, A12; ::_thesis: verum
end;

definition
let f1, f2 be NAT * -defined to-naturals homogeneous Function;
let i be Element of NAT ;
func primrec (f1,f2,i) ->  Element of HFuncs NAT means :Def11: :: COMPUT_1:def 11
 ex G being Function of (((arity f1) + 1) -tuples_on NAT),(HFuncs NAT) st
( it = Union G & ( for p being Element of ((arity f1) + 1) -tuples_on NAT holds G . p = primrec (f1,f2,i,p) ) );
existence
 ex b1 being Element of HFuncs NAT ex G being Function of (((arity f1) + 1) -tuples_on NAT),(HFuncs NAT) st
( b1 = Union G & ( for p being Element of ((arity f1) + 1) -tuples_on NAT holds G . p = primrec (f1,f2,i,p) ) )
proof
deffunc H1( FinSequence of NAT ) ->  Element of HFuncs NAT = primrec (f1,f2,i,$1);
reconsider ff1 = f1, ff2 = f2 as Element of HFuncs NAT by Th28;
set n = (arity f1) + 1;
consider G being Function of (((arity f1) + 1) -tuples_on NAT),(HFuncs NAT) such that
A1: for p being Element of ((arity f1) + 1) -tuples_on NAT holds G . p = H1(p) from FUNCT_2:sch_4();
reconsider Y = rng G as non empty Subset of (HFuncs NAT) by RELAT_1:def_19;
 Y is PartFunc-set of NAT * , NAT
proof
let f be Element of Y; :: according to RFUNCT_3:def_3 ::_thesis: f is Element of K32(K33((NAT *),NAT))
thus  f is Element of K32(K33((NAT *),NAT)) ; ::_thesis: verum
end;
then reconsider Y = Y as PFUNC_DOMAIN of NAT * , NAT ;
A2: Y is compatible
proof
let f, g be Function; :: according to COMPUT_1:def_1 ::_thesis: ( f in Y & g in Y implies f tolerates g )
assume  f in Y ; ::_thesis: ( not g in Y or f tolerates g )
then consider x being set  such that
A3: x in dom G and
A4: f = G . x by FUNCT_1:def_3;
assume  g in Y ; ::_thesis: f tolerates g
then consider y being set  such that
A5: y in dom G and
A6: g = G . y by FUNCT_1:def_3;
reconsider x = x, y = y as Element of ((arity f1) + 1) -tuples_on NAT by A3, A5;
A7: g = primrec (ff1,ff2,i,y) by A1, A6;
 f = primrec (ff1,ff2,i,x) by A1, A4;
hence  f tolerates g by A7, Th51; ::_thesis: verum
end;
now__::_thesis:_for_x_being_Element_of_((arity_f1)_+_1)_-tuples_on_NAT_holds_dom_(G_._x)_c=_((arity_f1)_+_1)_-tuples_on_NAT
let x be Element of ((arity f1) + 1) -tuples_on NAT; ::_thesis: dom (G . x) c= ((arity f1) + 1) -tuples_on NAT
 G . x = primrec (ff1,ff2,i,x) by A1;
hence  dom (G . x) c= ((arity f1) + 1) -tuples_on NAT by Th52; ::_thesis: verum
end;
then consider g being Element of HFuncs NAT such that
A8: g = Union G and
 dom g c= ((arity f1) + 1) -tuples_on NAT by A2, Th39;
take  g ; ::_thesis: ex G being Function of (((arity f1) + 1) -tuples_on NAT),(HFuncs NAT) st
( g = Union G & ( for p being Element of ((arity f1) + 1) -tuples_on NAT holds G . p = primrec (f1,f2,i,p) ) )
take  G ; ::_thesis: ( g = Union G & ( for p being Element of ((arity f1) + 1) -tuples_on NAT holds G . p = primrec (f1,f2,i,p) ) )
thus  ( g = Union G & ( for p being Element of ((arity f1) + 1) -tuples_on NAT holds G . p = primrec (f1,f2,i,p) ) ) by A1, A8; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Element of HFuncs NAT st ex G being Function of (((arity f1) + 1) -tuples_on NAT),(HFuncs NAT) st
( b1 = Union G & ( for p being Element of ((arity f1) + 1) -tuples_on NAT holds G . p = primrec (f1,f2,i,p) ) ) & ex G being Function of (((arity f1) + 1) -tuples_on NAT),(HFuncs NAT) st
( b2 = Union G & ( for p being Element of ((arity f1) + 1) -tuples_on NAT holds G . p = primrec (f1,f2,i,p) ) ) holds
b1 = b2
proof
let g1, g2 be Element of HFuncs NAT; ::_thesis: ( ex G being Function of (((arity f1) + 1) -tuples_on NAT),(HFuncs NAT) st
( g1 = Union G & ( for p being Element of ((arity f1) + 1) -tuples_on NAT holds G . p = primrec (f1,f2,i,p) ) ) & ex G being Function of (((arity f1) + 1) -tuples_on NAT),(HFuncs NAT) st
( g2 = Union G & ( for p being Element of ((arity f1) + 1) -tuples_on NAT holds G . p = primrec (f1,f2,i,p) ) ) implies g1 = g2 )
given G1 being Function of (((arity f1) + 1) -tuples_on NAT),(HFuncs NAT) such that A9: g1 = Union G1 and
A10: for p being Element of ((arity f1) + 1) -tuples_on NAT holds G1 . p = primrec (f1,f2,i,p) ; ::_thesis: ( for G being Function of (((arity f1) + 1) -tuples_on NAT),(HFuncs NAT) holds
( not g2 = Union G or ex p being Element of ((arity f1) + 1) -tuples_on NAT st not G . p = primrec (f1,f2,i,p) ) or g1 = g2 )
given G2 being Function of (((arity f1) + 1) -tuples_on NAT),(HFuncs NAT) such that A11: g2 = Union G2 and
A12: for p being Element of ((arity f1) + 1) -tuples_on NAT holds G2 . p = primrec (f1,f2,i,p) ; ::_thesis: g1 = g2
now__::_thesis:_for_p_being_Element_of_((arity_f1)_+_1)_-tuples_on_NAT_holds_G1_._p_=_G2_._p
let p be Element of ((arity f1) + 1) -tuples_on NAT; ::_thesis: G1 . p = G2 . p
thus G1 . p = primrec (f1,f2,i,p) by A10
.= G2 . p by A12 ; ::_thesis: verum
end;
hence  g1 = g2 by A9, A11, FUNCT_2:63; ::_thesis: verum
end;
end;

:: deftheorem Def11 defines primrec COMPUT_1:def_11_:_
 for f1, f2 being NAT * -defined to-naturals homogeneous Function
for i being Element of NAT
for b4 being Element of HFuncs NAT holds
( b4 = primrec (f1,f2,i) iff ex G being Function of (((arity f1) + 1) -tuples_on NAT),(HFuncs NAT) st
( b4 = Union G & ( for p being Element of ((arity f1) + 1) -tuples_on NAT holds G . p = primrec (f1,f2,i,p) ) ) );

theorem Th55: :: COMPUT_1:55
for i being Element of NAT
for e1, e2 being NAT * -defined to-naturals homogeneous Function st e1 is empty holds
primrec (e1,e2,i) is empty
proof
let i be Element of NAT ; ::_thesis: for e1, e2 being NAT * -defined to-naturals homogeneous Function st e1 is empty holds
primrec (e1,e2,i) is empty
let e1, e2 be NAT * -defined to-naturals homogeneous Function; ::_thesis: ( e1 is empty implies primrec (e1,e2,i) is empty )
set f1 = e1;
set f2 = e2;
assume A1: e1 is empty ; ::_thesis: primrec (e1,e2,i) is empty
consider G being Function of (((arity e1) + 1) -tuples_on NAT),(HFuncs NAT) such that
A2: primrec (e1,e2,i) = Union G and
A3: for p being Element of ((arity e1) + 1) -tuples_on NAT holds G . p = primrec (e1,e2,i,p) by Def11;
A4: dom G = ((arity e1) + 1) -tuples_on NAT by FUNCT_2:def_1;
now__::_thesis:_for_y_being_set_holds_
(_(_y_in_rng_G_implies_y_in_{{}}_)_&_(_y_in_{{}}_implies_y_in_rng_G_)_)
set p = the Element of ((arity e1) + 1) -tuples_on NAT;
let y be set ; ::_thesis: ( ( y in rng G implies y in {{}} ) & ( y in {{}} implies y in rng G ) )
hereby  ::_thesis: ( y in {{}} implies y in rng G )
assume  y in rng G ; ::_thesis: y in {{}}
then consider x being set  such that
A5: x in dom G and
A6: G . x = y by FUNCT_1:def_3;
reconsider p = x as Element of ((arity e1) + 1) -tuples_on NAT by A5;
G . p = primrec (e1,e2,i,p) by A3
.= {} by A1, Th53 ;
hence  y in {{}} by A6, TARSKI:def_1; ::_thesis: verum
end;
assume  y in {{}} ; ::_thesis: y in rng G
then A7: y = {} by TARSKI:def_1;
G . the Element of ((arity e1) + 1) -tuples_on NAT = primrec (e1,e2,i, the Element of ((arity e1) + 1) -tuples_on NAT) by A3
.= {} by A1, Th53 ;
hence  y in rng G by A4, A7, FUNCT_1:3; ::_thesis: verum
end;
then A8: rng G = {{}} by TARSKI:1;
Union G = union (rng G)
.= {} by A8, ZFMISC_1:25 ;
hence  primrec (e1,e2,i) is empty by A2; ::_thesis: verum
end;

theorem Th56: :: COMPUT_1:56
for i being Element of NAT
for f1, f2 being non empty NAT * -defined to-naturals homogeneous Function holds dom (primrec (f1,f2,i)) c= ((arity f1) + 1) -tuples_on NAT
proof
let i be Element of NAT ; ::_thesis: for f1, f2 being non empty NAT * -defined to-naturals homogeneous Function holds dom (primrec (f1,f2,i)) c= ((arity f1) + 1) -tuples_on NAT
let f1, f2 be non empty NAT * -defined to-naturals homogeneous Function; ::_thesis: dom (primrec (f1,f2,i)) c= ((arity f1) + 1) -tuples_on NAT
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom (primrec (f1,f2,i)) or x in ((arity f1) + 1) -tuples_on NAT )
assume A1: x in dom (primrec (f1,f2,i)) ; ::_thesis: x in ((arity f1) + 1) -tuples_on NAT
consider G being Function of (((arity f1) + 1) -tuples_on NAT),(HFuncs NAT) such that
A2: primrec (f1,f2,i) = Union G and
A3: for p being Element of ((arity f1) + 1) -tuples_on NAT holds G . p = primrec (f1,f2,i,p) by Def11;
A4: rng G is compatible
proof
let f, g be Function; :: according to COMPUT_1:def_1 ::_thesis: ( f in rng G & g in rng G implies f tolerates g )
assume that
A5: f in rng G and
A6: g in rng G ; ::_thesis: f tolerates g
consider fx being set  such that
A7: fx in dom G and
A8: f = G . fx by A5, FUNCT_1:def_3;
reconsider fx = fx as Element of ((arity f1) + 1) -tuples_on NAT by A7;
consider gx being set  such that
A9: gx in dom G and
A10: g = G . gx by A6, FUNCT_1:def_3;
reconsider gx = gx as Element of ((arity f1) + 1) -tuples_on NAT by A9;
A11: G . gx = primrec (f1,f2,i,gx) by A3;
 G . fx = primrec (f1,f2,i,fx) by A3;
hence  f tolerates g by A8, A10, A11, Th51; ::_thesis: verum
end;
now__::_thesis:_for_x_being_Element_of_((arity_f1)_+_1)_-tuples_on_NAT_holds_dom_(G_._x)_c=_((arity_f1)_+_1)_-tuples_on_NAT
let x be Element of ((arity f1) + 1) -tuples_on NAT; ::_thesis: dom (G . x) c= ((arity f1) + 1) -tuples_on NAT
 G . x = primrec (f1,f2,i,x) by A3;
hence  dom (G . x) c= ((arity f1) + 1) -tuples_on NAT by Th52; ::_thesis: verum
end;
then  ex F being Element of HFuncs NAT st
( F = Union G & dom F c= ((arity f1) + 1) -tuples_on NAT ) by A4, Th39;
hence  x in ((arity f1) + 1) -tuples_on NAT by A1, A2; ::_thesis: verum
end;

theorem Th57: :: COMPUT_1:57
for i being Element of NAT
for f1, f2 being non empty NAT * -defined to-naturals homogeneous Function st f1 is len-total & f2 is len-total & (arity f1) + 2 = arity f2 & 1 <= i & i <= 1 + (arity f1) holds
( dom (primrec (f1,f2,i)) = ((arity f1) + 1) -tuples_on NAT & arity (primrec (f1,f2,i)) = (arity f1) + 1 )
proof
let i be Element of NAT ; ::_thesis: for f1, f2 being non empty NAT * -defined to-naturals homogeneous Function st f1 is len-total & f2 is len-total & (arity f1) + 2 = arity f2 & 1 <= i & i <= 1 + (arity f1) holds
( dom (primrec (f1,f2,i)) = ((arity f1) + 1) -tuples_on NAT & arity (primrec (f1,f2,i)) = (arity f1) + 1 )
let f1, f2 be non empty NAT * -defined to-naturals homogeneous Function; ::_thesis: ( f1 is len-total & f2 is len-total & (arity f1) + 2 = arity f2 & 1 <= i & i <= 1 + (arity f1) implies ( dom (primrec (f1,f2,i)) = ((arity f1) + 1) -tuples_on NAT & arity (primrec (f1,f2,i)) = (arity f1) + 1 ) )
assume that
A1: f1 is len-total and
A2: f2 is len-total and
A3: (arity f1) + 2 = arity f2 and
A4: 1 <= i and
A5: i <= 1 + (arity f1) ; ::_thesis: ( dom (primrec (f1,f2,i)) = ((arity f1) + 1) -tuples_on NAT & arity (primrec (f1,f2,i)) = (arity f1) + 1 )
set n = (arity f1) + 1;
A6: ((arity f1) + 1) -tuples_on NAT c= dom (primrec (f1,f2,i))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ((arity f1) + 1) -tuples_on NAT or x in dom (primrec (f1,f2,i)) )
assume  x in ((arity f1) + 1) -tuples_on NAT ; ::_thesis: x in dom (primrec (f1,f2,i))
then reconsider x9 = x as Element of ((arity f1) + 1) -tuples_on NAT ;
consider G being Function of (((arity f1) + 1) -tuples_on NAT),(HFuncs NAT) such that
A7: primrec (f1,f2,i) = Union G and
A8: for p being Element of ((arity f1) + 1) -tuples_on NAT holds G . p = primrec (f1,f2,i,p) by Def11;
reconsider rngG = rng G as non empty functional compatible set by A7, Th11;
A9: dom (union rngG) = union  {  (dom f) where f is Element of rngG : verum  }  by Th12;
 dom G = ((arity f1) + 1) -tuples_on NAT by FUNCT_2:def_1;
then  G . x9 in rng G by FUNCT_1:3;
then A10: dom (G . x9) in  {  (dom f) where f is Element of rngG : verum  }  ;
A11: x9 in dom (primrec (f1,f2,i,x9)) by A1, A2, A3, A4, A5, Th54;
 G . x9 = primrec (f1,f2,i,x9) by A8;
hence  x in dom (primrec (f1,f2,i)) by A7, A11, A9, A10, TARSKI:def_4; ::_thesis: verum
end;
 dom (primrec (f1,f2,i)) c= ((arity f1) + 1) -tuples_on NAT by Th56;
hence  dom (primrec (f1,f2,i)) = ((arity f1) + 1) -tuples_on NAT by A6, XBOOLE_0:def_10; ::_thesis: arity (primrec (f1,f2,i)) = (arity f1) + 1
hence  arity (primrec (f1,f2,i)) = (arity f1) + 1 by Th25; ::_thesis: verum
end;

Lm2: now__::_thesis:_for_f1,_f2_being_non_empty_NAT_*_-defined_to-naturals_homogeneous_Function
for_p_being_Element_of_((arity_f1)_+_1)_-tuples_on_NAT
for_i,_m_being_Element_of_NAT_
for_F1,_F_being_Function_of_NAT,(HFuncs_NAT)_st_(_i_in_dom_(p_+*_(i,(m_+_1)))_&_Del_((p_+*_(i,(m_+_1))),i)_in_dom_f1_implies_F1_._0_=_{((p_+*_(i,(m_+_1)))_+*_(i,0))}_-->_(f1_._(Del_((p_+*_(i,(m_+_1))),i)))_)_&_(_(_not_i_in_dom_(p_+*_(i,(m_+_1)))_or_not_Del_((p_+*_(i,(m_+_1))),i)_in_dom_f1_)_implies_F1_._0_=_{}_)_&_(_for_M_being_Element_of_NAT_holds_S1[M,F1_._M,F1_._(M_+_1),p_+*_(i,(m_+_1)),i,f2]_)_&_(_i_in_dom_(p_+*_(i,m))_&_Del_((p_+*_(i,m)),i)_in_dom_f1_implies_F_._0_=_{((p_+*_(i,m))_+*_(i,0))}_-->_(f1_._(Del_((p_+*_(i,m)),i)))_)_&_(_(_not_i_in_dom_(p_+*_(i,m))_or_not_Del_((p_+*_(i,m)),i)_in_dom_f1_)_implies_F_._0_=_{}_)_&_(_for_M_being_Element_of_NAT_holds_S1[M,F_._M,F_._(M_+_1),p_+*_(i,m),i,f2]_)_holds_
F1_=_F
let f1, f2 be non empty NAT * -defined to-naturals homogeneous Function; ::_thesis: for p being Element of ((arity f1) + 1) -tuples_on NAT
for i, m being Element of NAT
for F1, F being Function of NAT,(HFuncs NAT) st ( i in dom (p +* (i,(m + 1))) & Del ((p +* (i,(m + 1))),i) in dom f1 implies F1 . 0 = {((p +* (i,(m + 1))) +* (i,0))} --> (f1 . (Del ((p +* (i,(m + 1))),i))) ) & ( ( not i in dom (p +* (i,(m + 1))) or not Del ((p +* (i,(m + 1))),i) in dom f1 ) implies F1 . 0 = {} ) & ( for M being Element of NAT holds S1[M,F1 . M,F1 . (M + 1),p +* (i,(m + 1)),i,f2] ) & ( i in dom (p +* (i,m)) & Del ((p +* (i,m)),i) in dom f1 implies F . 0 = {((p +* (i,m)) +* (i,0))} --> (f1 . (Del ((p +* (i,m)),i))) ) & ( ( not i in dom (p +* (i,m)) or not Del ((p +* (i,m)),i) in dom f1 ) implies F . 0 = {} ) & ( for M being Element of NAT holds S1[M,F . M,F . (M + 1),p +* (i,m),i,f2] ) holds
F1 = F
let p be Element of ((arity f1) + 1) -tuples_on NAT; ::_thesis: for i, m being Element of NAT
for F1, F being Function of NAT,(HFuncs NAT) st ( i in dom (p +* (i,(m + 1))) & Del ((p +* (i,(m + 1))),i) in dom f1 implies F1 . 0 = {((p +* (i,(m + 1))) +* (i,0))} --> (f1 . (Del ((p +* (i,(m + 1))),i))) ) & ( ( not i in dom (p +* (i,(m + 1))) or not Del ((p +* (i,(m + 1))),i) in dom f1 ) implies F1 . 0 = {} ) & ( for M being Element of NAT holds S1[M,F1 . M,F1 . (M + 1),p +* (i,(m + 1)),i,f2] ) & ( i in dom (p +* (i,m)) & Del ((p +* (i,m)),i) in dom f1 implies F . 0 = {((p +* (i,m)) +* (i,0))} --> (f1 . (Del ((p +* (i,m)),i))) ) & ( ( not i in dom (p +* (i,m)) or not Del ((p +* (i,m)),i) in dom f1 ) implies F . 0 = {} ) & ( for M being Element of NAT holds S1[M,F . M,F . (M + 1),p +* (i,m),i,f2] ) holds
F1 = F
let i, m be Element of NAT ; ::_thesis: for F1, F being Function of NAT,(HFuncs NAT) st ( i in dom (p +* (i,(m + 1))) & Del ((p +* (i,(m + 1))),i) in dom f1 implies F1 . 0 = {((p +* (i,(m + 1))) +* (i,0))} --> (f1 . (Del ((p +* (i,(m + 1))),i))) ) & ( ( not i in dom (p +* (i,(m + 1))) or not Del ((p +* (i,(m + 1))),i) in dom f1 ) implies F1 . 0 = {} ) & ( for M being Element of NAT holds S1[M,F1 . M,F1 . (M + 1),p +* (i,(m + 1)),i,f2] ) & ( i in dom (p +* (i,m)) & Del ((p +* (i,m)),i) in dom f1 implies F . 0 = {((p +* (i,m)) +* (i,0))} --> (f1 . (Del ((p +* (i,m)),i))) ) & ( ( not i in dom (p +* (i,m)) or not Del ((p +* (i,m)),i) in dom f1 ) implies F . 0 = {} ) & ( for M being Element of NAT holds S1[M,F . M,F . (M + 1),p +* (i,m),i,f2] ) holds
F1 = F
set pm1 = p +* (i,(m + 1));
set pm = p +* (i,m);
let F1, F be Function of NAT,(HFuncs NAT); ::_thesis: ( ( i in dom (p +* (i,(m + 1))) & Del ((p +* (i,(m + 1))),i) in dom f1 implies F1 . 0 = {((p +* (i,(m + 1))) +* (i,0))} --> (f1 . (Del ((p +* (i,(m + 1))),i))) ) & ( ( not i in dom (p +* (i,(m + 1))) or not Del ((p +* (i,(m + 1))),i) in dom f1 ) implies F1 . 0 = {} ) & ( for M being Element of NAT holds S1[M,F1 . M,F1 . (M + 1),p +* (i,(m + 1)),i,f2] ) & ( i in dom (p +* (i,m)) & Del ((p +* (i,m)),i) in dom f1 implies F . 0 = {((p +* (i,m)) +* (i,0))} --> (f1 . (Del ((p +* (i,m)),i))) ) & ( ( not i in dom (p +* (i,m)) or not Del ((p +* (i,m)),i) in dom f1 ) implies F . 0 = {} ) & ( for M being Element of NAT holds S1[M,F . M,F . (M + 1),p +* (i,m),i,f2] ) implies F1 = F )
assume that
A1: ( i in dom (p +* (i,(m + 1))) & Del ((p +* (i,(m + 1))),i) in dom f1 implies F1 . 0 = {((p +* (i,(m + 1))) +* (i,0))} --> (f1 . (Del ((p +* (i,(m + 1))),i))) ) and
A2: ( ( not i in dom (p +* (i,(m + 1))) or not Del ((p +* (i,(m + 1))),i) in dom f1 ) implies F1 . 0 = {} ) and
A3: for M being Element of NAT holds S1[M,F1 . M,F1 . (M + 1),p +* (i,(m + 1)),i,f2] and
A4: ( i in dom (p +* (i,m)) & Del ((p +* (i,m)),i) in dom f1 implies F . 0 = {((p +* (i,m)) +* (i,0))} --> (f1 . (Del ((p +* (i,m)),i))) ) and
A5: ( ( not i in dom (p +* (i,m)) or not Del ((p +* (i,m)),i) in dom f1 ) implies F . 0 = {} ) and
A6: for M being Element of NAT holds S1[M,F . M,F . (M + 1),p +* (i,m),i,f2] ; ::_thesis: F1 = F
A7: dom p = dom (p +* (i,m)) by FUNCT_7:30;
A8: (p +* (i,m)) +* (i,0) = p +* (i,0) by FUNCT_7:34
.= (p +* (i,(m + 1))) +* (i,0) by FUNCT_7:34 ;
A9: dom p = dom (p +* (i,(m + 1))) by FUNCT_7:30;
A10: Del ((p +* (i,m)),i) = Del (p,i) by Th3
.= Del ((p +* (i,(m + 1))),i) by Th3 ;
 for x being set st x in NAT holds
F . x = F1 . x
proof
let x be set ; ::_thesis: ( x in NAT implies F . x = F1 . x )
defpred S2[ Element of NAT ] means F . $1 = F1 . $1;
A11: for k being Element of NAT st S2[k] holds
S2[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S2[k] implies S2[k + 1] )
assume A12: S2[k] ; ::_thesis: S2[k + 1]
A13: (p +* (i,m)) +* (i,(k + 1)) = p +* (i,(k + 1)) by FUNCT_7:34
.= (p +* (i,(m + 1))) +* (i,(k + 1)) by FUNCT_7:34 ;
A14: (p +* (i,m)) +* (i,k) = p +* (i,k) by FUNCT_7:34
.= (p +* (i,(m + 1))) +* (i,k) by FUNCT_7:34 ;
percases ( ( i in dom (p +* (i,m)) & (p +* (i,m)) +* (i,k) in dom (F . k) & ((p +* (i,m)) +* (i,k)) ^ <*((F . k) . ((p +* (i,m)) +* (i,k)))*> in dom f2 ) or not i in dom (p +* (i,m)) or not (p +* (i,m)) +* (i,k) in dom (F . k) or not ((p +* (i,m)) +* (i,k)) ^ <*((F . k) . ((p +* (i,m)) +* (i,k)))*> in dom f2 ) ;
supposeA15: ( i in dom (p +* (i,m)) & (p +* (i,m)) +* (i,k) in dom (F . k) & ((p +* (i,m)) +* (i,k)) ^ <*((F . k) . ((p +* (i,m)) +* (i,k)))*> in dom f2 ) ; ::_thesis: S2[k + 1]
hence F . (k + 1) = (F . k) +* (((p +* (i,m)) +* (i,(k + 1))) .--> (f2 . (((p +* (i,m)) +* (i,k)) ^ <*((F . k) . ((p +* (i,m)) +* (i,k)))*>))) by A6
.= F1 . (k + 1) by A3, A7, A9, A12, A14, A13, A15 ;
::_thesis: verum
end;
supposeA16: ( not i in dom (p +* (i,m)) or not (p +* (i,m)) +* (i,k) in dom (F . k) or not ((p +* (i,m)) +* (i,k)) ^ <*((F . k) . ((p +* (i,m)) +* (i,k)))*> in dom f2 ) ; ::_thesis: S2[k + 1]
hence F . (k + 1) = F . k by A6
.= F1 . (k + 1) by A3, A7, A9, A12, A14, A16 ;
::_thesis: verum
end;
end;
end;
A17: S2[ 0 ] by A1, A2, A4, A5, A7, A8, A10, FUNCT_7:30;
A18: for k being Element of NAT holds S2[k] from NAT_1:sch_1(A17, A11);
assume  x in NAT ; ::_thesis: F . x = F1 . x
hence  F . x = F1 . x by A18; ::_thesis: verum
end;
hence  F1 = F by FUNCT_2:12; ::_thesis: verum
end;

Lm3: now__::_thesis:_for_f1,_f2_being_non_empty_NAT_*_-defined_to-naturals_homogeneous_Function
for_p_being_Element_of_((arity_f1)_+_1)_-tuples_on_NAT
for_i,_m_being_Element_of_NAT_st_i_in_dom_p_holds_
for_F_being_Function_of_NAT,(HFuncs_NAT)_st_(_i_in_dom_(p_+*_(i,(m_+_1)))_&_Del_((p_+*_(i,(m_+_1))),i)_in_dom_f1_implies_F_._0_=_{((p_+*_(i,(m_+_1)))_+*_(i,0))}_-->_(f1_._(Del_((p_+*_(i,(m_+_1))),i)))_)_&_(_(_not_i_in_dom_(p_+*_(i,(m_+_1)))_or_not_Del_((p_+*_(i,(m_+_1))),i)_in_dom_f1_)_implies_F_._0_=_{}_)_&_(_for_M_being_Element_of_NAT_holds_S1[M,F_._M,F_._(M_+_1),p_+*_(i,(m_+_1)),i,f2]_)_holds_
(_(F_._(m_+_1))_._(p_+*_(i,m))_=_(F_._m)_._(p_+*_(i,m))_&_not_p_+*_(i,(m_+_1))_in_dom_(F_._m)_)
let f1, f2 be non empty NAT * -defined to-naturals homogeneous Function; ::_thesis: for p being Element of ((arity f1) + 1) -tuples_on NAT
for i, m being Element of NAT st i in dom p holds
for F being Function of NAT,(HFuncs NAT) st ( i in dom (p +* (i,(m + 1))) & Del ((p +* (i,(m + 1))),i) in dom f1 implies F . 0 = {((p +* (i,(m + 1))) +* (i,0))} --> (f1 . (Del ((p +* (i,(m + 1))),i))) ) & ( ( not i in dom (p +* (i,(m + 1))) or not Del ((p +* (i,(m + 1))),i) in dom f1 ) implies F . 0 = {} ) & ( for M being Element of NAT holds S1[M,F . M,F . (M + 1),p +* (i,(m + 1)),i,f2] ) holds
( (F . (m + 1)) . (p +* (i,m)) = (F . m) . (p +* (i,m)) & not p +* (i,(m + 1)) in dom (F . m) )
let p be Element of ((arity f1) + 1) -tuples_on NAT; ::_thesis: for i, m being Element of NAT st i in dom p holds
for F being Function of NAT,(HFuncs NAT) st ( i in dom (p +* (i,(m + 1))) & Del ((p +* (i,(m + 1))),i) in dom f1 implies F . 0 = {((p +* (i,(m + 1))) +* (i,0))} --> (f1 . (Del ((p +* (i,(m + 1))),i))) ) & ( ( not i in dom (p +* (i,(m + 1))) or not Del ((p +* (i,(m + 1))),i) in dom f1 ) implies F . 0 = {} ) & ( for M being Element of NAT holds S1[M,F . M,F . (M + 1),p +* (i,(m + 1)),i,f2] ) holds
( (F . (m + 1)) . (p +* (i,m)) = (F . m) . (p +* (i,m)) & not p +* (i,(m + 1)) in dom (F . m) )
let i, m be Element of NAT ; ::_thesis: ( i in dom p implies for F being Function of NAT,(HFuncs NAT) st ( i in dom (p +* (i,(m + 1))) & Del ((p +* (i,(m + 1))),i) in dom f1 implies F . 0 = {((p +* (i,(m + 1))) +* (i,0))} --> (f1 . (Del ((p +* (i,(m + 1))),i))) ) & ( ( not i in dom (p +* (i,(m + 1))) or not Del ((p +* (i,(m + 1))),i) in dom f1 ) implies F . 0 = {} ) & ( for M being Element of NAT holds S1[M,F . M,F . (M + 1),p +* (i,(m + 1)),i,f2] ) holds
( (F . (m + 1)) . (p +* (i,m)) = (F . m) . (p +* (i,m)) & not p +* (i,(m + 1)) in dom (F . m) ) )
assume A1: i in dom p ; ::_thesis: for F being Function of NAT,(HFuncs NAT) st ( i in dom (p +* (i,(m + 1))) & Del ((p +* (i,(m + 1))),i) in dom f1 implies F . 0 = {((p +* (i,(m + 1))) +* (i,0))} --> (f1 . (Del ((p +* (i,(m + 1))),i))) ) & ( ( not i in dom (p +* (i,(m + 1))) or not Del ((p +* (i,(m + 1))),i) in dom f1 ) implies F . 0 = {} ) & ( for M being Element of NAT holds S1[M,F . M,F . (M + 1),p +* (i,(m + 1)),i,f2] ) holds
( (F . (m + 1)) . (p +* (i,m)) = (F . m) . (p +* (i,m)) & not p +* (i,(m + 1)) in dom (F . m) )
set pm = p +* (i,m);
set pm1 = p +* (i,(m + 1));
let F be Function of NAT,(HFuncs NAT); ::_thesis: ( ( i in dom (p +* (i,(m + 1))) & Del ((p +* (i,(m + 1))),i) in dom f1 implies F . 0 = {((p +* (i,(m + 1))) +* (i,0))} --> (f1 . (Del ((p +* (i,(m + 1))),i))) ) & ( ( not i in dom (p +* (i,(m + 1))) or not Del ((p +* (i,(m + 1))),i) in dom f1 ) implies F . 0 = {} ) & ( for M being Element of NAT holds S1[M,F . M,F . (M + 1),p +* (i,(m + 1)),i,f2] ) implies ( (F . (m + 1)) . (p +* (i,m)) = (F . m) . (p +* (i,m)) & not p +* (i,(m + 1)) in dom (F . m) ) )
assume that
A2: ( i in dom (p +* (i,(m + 1))) & Del ((p +* (i,(m + 1))),i) in dom f1 implies F . 0 = {((p +* (i,(m + 1))) +* (i,0))} --> (f1 . (Del ((p +* (i,(m + 1))),i))) ) and
A3: ( ( not i in dom (p +* (i,(m + 1))) or not Del ((p +* (i,(m + 1))),i) in dom f1 ) implies F . 0 = {} ) and
A4: for M being Element of NAT holds S1[M,F . M,F . (M + 1),p +* (i,(m + 1)),i,f2] ; ::_thesis: ( (F . (m + 1)) . (p +* (i,m)) = (F . m) . (p +* (i,m)) & not p +* (i,(m + 1)) in dom (F . m) )
thus  (F . (m + 1)) . (p +* (i,m)) = (F . m) . (p +* (i,m)) ::_thesis: not p +* (i,(m + 1)) in dom (F . m)
proof
percases ( ( i in dom (p +* (i,(m + 1))) & (p +* (i,(m + 1))) +* (i,m) in dom (F . m) & ((p +* (i,(m + 1))) +* (i,m)) ^ <*((F . m) . ((p +* (i,(m + 1))) +* (i,m)))*> in dom f2 ) or not i in dom (p +* (i,(m + 1))) or not (p +* (i,(m + 1))) +* (i,m) in dom (F . m) or not ((p +* (i,(m + 1))) +* (i,m)) ^ <*((F . m) . ((p +* (i,(m + 1))) +* (i,m)))*> in dom f2 ) ;
supposeA5: ( i in dom (p +* (i,(m + 1))) & (p +* (i,(m + 1))) +* (i,m) in dom (F . m) & ((p +* (i,(m + 1))) +* (i,m)) ^ <*((F . m) . ((p +* (i,(m + 1))) +* (i,m)))*> in dom f2 ) ; ::_thesis: (F . (m + 1)) . (p +* (i,m)) = (F . m) . (p +* (i,m))
A6: p +* (i,(m + 1)) = (p +* (i,(m + 1))) +* (i,(m + 1)) by FUNCT_7:34;
A7: (p +* (i,(m + 1))) . i = m + 1 by A1, FUNCT_7:31;
 (p +* (i,m)) . i = m by A1, FUNCT_7:31;
then  p +* (i,m) <> p +* (i,(m + 1)) by A7;
then A8: not p +* (i,m) in dom ({((p +* (i,(m + 1))) +* (i,(m + 1)))} --> (f2 . (((p +* (i,(m + 1))) +* (i,m)) ^ <*((F . m) . ((p +* (i,(m + 1))) +* (i,m)))*>))) by A6, TARSKI:def_1;
 F . (m + 1) = (F . m) +* (((p +* (i,(m + 1))) +* (i,(m + 1))) .--> (f2 . (((p +* (i,(m + 1))) +* (i,m)) ^ <*((F . m) . ((p +* (i,(m + 1))) +* (i,m)))*>))) by A4, A5;
hence  (F . (m + 1)) . (p +* (i,m)) = (F . m) . (p +* (i,m)) by A8, FUNCT_4:11; ::_thesis: verum
end;
suppose ( not i in dom (p +* (i,(m + 1))) or not (p +* (i,(m + 1))) +* (i,m) in dom (F . m) or not ((p +* (i,(m + 1))) +* (i,m)) ^ <*((F . m) . ((p +* (i,(m + 1))) +* (i,m)))*> in dom f2 ) ; ::_thesis: (F . (m + 1)) . (p +* (i,m)) = (F . m) . (p +* (i,m))
hence  (F . (m + 1)) . (p +* (i,m)) = (F . m) . (p +* (i,m)) by A4; ::_thesis: verum
end;
end;
end;
A9: for m, k being Element of NAT st p +* (i,k) in dom (F . m) holds
k <= m
proof
defpred S2[ Element of NAT ] means  for k being Element of NAT st p +* (i,k) in dom (F . $1) holds
k <= $1;
A10: for m being Element of NAT st S2[m] holds
S2[m + 1]
proof
let m be Element of NAT ; ::_thesis: ( S2[m] implies S2[m + 1] )
assume A11: S2[m] ; ::_thesis: S2[m + 1]
let k be Element of NAT ; ::_thesis: ( p +* (i,k) in dom (F . (m + 1)) implies k <= m + 1 )
assume A12: p +* (i,k) in dom (F . (m + 1)) ; ::_thesis: k <= m + 1
percases ( ( i in dom (p +* (i,(m + 1))) & (p +* (i,(m + 1))) +* (i,m) in dom (F . m) & ((p +* (i,(m + 1))) +* (i,m)) ^ <*((F . m) . ((p +* (i,(m + 1))) +* (i,m)))*> in dom f2 ) or not i in dom (p +* (i,(m + 1))) or not (p +* (i,(m + 1))) +* (i,m) in dom (F . m) or not ((p +* (i,(m + 1))) +* (i,m)) ^ <*((F . m) . ((p +* (i,(m + 1))) +* (i,m)))*> in dom f2 ) ;
suppose ( i in dom (p +* (i,(m + 1))) & (p +* (i,(m + 1))) +* (i,m) in dom (F . m) & ((p +* (i,(m + 1))) +* (i,m)) ^ <*((F . m) . ((p +* (i,(m + 1))) +* (i,m)))*> in dom f2 ) ; ::_thesis: k <= m + 1
then  F . (m + 1) = (F . m) +* (((p +* (i,(m + 1))) +* (i,(m + 1))) .--> (f2 . (((p +* (i,(m + 1))) +* (i,m)) ^ <*((F . m) . ((p +* (i,(m + 1))) +* (i,m)))*>))) by A4;
then  dom (F . (m + 1)) = (dom (F . m)) \/ (dom ({((p +* (i,(m + 1))) +* (i,(m + 1)))} --> (f2 . (((p +* (i,(m + 1))) +* (i,m)) ^ <*((F . m) . ((p +* (i,(m + 1))) +* (i,m)))*>)))) by FUNCT_4:def_1;
then A13: dom (F . (m + 1)) = (dom (F . m)) \/ {((p +* (i,(m + 1))) +* (i,(m + 1)))} by FUNCOP_1:13;
thus  k <= m + 1 ::_thesis: verum
proof
percases ( p +* (i,k) in dom (F . m) or p +* (i,k) in {((p +* (i,(m + 1))) +* (i,(m + 1)))} ) by A12, A13, XBOOLE_0:def_3;
supposeA14: p +* (i,k) in dom (F . m) ; ::_thesis: k <= m + 1
A15: m <= m + 1 by NAT_1:11;
 k <= m by A11, A14;
hence  k <= m + 1 by A15, XXREAL_0:2; ::_thesis: verum
end;
suppose p +* (i,k) in {((p +* (i,(m + 1))) +* (i,(m + 1)))} ; ::_thesis: k <= m + 1
then A16: p +* (i,k) = (p +* (i,(m + 1))) +* (i,(m + 1)) by TARSKI:def_1
.= p +* (i,(m + 1)) by FUNCT_7:34 ;
k = (p +* (i,k)) . i by A1, FUNCT_7:31
.= m + 1 by A1, A16, FUNCT_7:31 ;
hence  k <= m + 1 ; ::_thesis: verum
end;
end;
end;
end;
suppose ( not i in dom (p +* (i,(m + 1))) or not (p +* (i,(m + 1))) +* (i,m) in dom (F . m) or not ((p +* (i,(m + 1))) +* (i,m)) ^ <*((F . m) . ((p +* (i,(m + 1))) +* (i,m)))*> in dom f2 ) ; ::_thesis: k <= m + 1
then  F . (m + 1) = F . m by A4;
then A17: k <= m by A11, A12;
 m <= m + 1 by NAT_1:11;
hence  k <= m + 1 by A17, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
A18: S2[ 0 ]
proof
let k be Element of NAT ; ::_thesis: ( p +* (i,k) in dom (F . 0) implies k <= 0 )
assume A19: p +* (i,k) in dom (F . 0) ; ::_thesis: k <= 0
percases ( ( i in dom (p +* (i,(m + 1))) & Del ((p +* (i,(m + 1))),i) in dom f1 ) or not i in dom (p +* (i,(m + 1))) or not Del ((p +* (i,(m + 1))),i) in dom f1 ) ;
suppose ( i in dom (p +* (i,(m + 1))) & Del ((p +* (i,(m + 1))),i) in dom f1 ) ; ::_thesis: k <= 0
then  dom (F . 0) = {((p +* (i,(m + 1))) +* (i,0))} by A2, FUNCOP_1:13;
then A20: p +* (i,k) = (p +* (i,(m + 1))) +* (i,0) by A19, TARSKI:def_1
.= p +* (i,0) by FUNCT_7:34 ;
k = (p +* (i,k)) . i by A1, FUNCT_7:31
.= 0 by A1, A20, FUNCT_7:31 ;
hence  k <= 0 ; ::_thesis: verum
end;
suppose ( not i in dom (p +* (i,(m + 1))) or not Del ((p +* (i,(m + 1))),i) in dom f1 ) ; ::_thesis: k <= 0
hence  k <= 0 by A3, A19; ::_thesis: verum
end;
end;
end;
thus  for n being Element of NAT holds S2[n] from NAT_1:sch_1(A18, A10); ::_thesis: verum
end;
thus  not p +* (i,(m + 1)) in dom (F . m) ::_thesis: verum
proof
assume  p +* (i,(m + 1)) in dom (F . m) ; ::_thesis: contradiction
then  m + 1 <= m by A9;
hence  contradiction by NAT_1:13; ::_thesis: verum
end;
end;

definition
let n be Nat;
let p be Element of n -tuples_on NAT;
let i, k be Element of NAT ;
:: original: +*
redefine funcp +* (i,k) ->  Element of n -tuples_on NAT;
coherence
 p +* (i,k) is Element of n -tuples_on NAT
proof
 dom (p +* (i,k)) = dom p by FUNCT_7:30;
then  len (p +* (i,k)) = len p by FINSEQ_3:29
.= n by CARD_1:def_7 ;
hence  p +* (i,k) is Element of n -tuples_on NAT by FINSEQ_2:92; ::_thesis: verum
end;
end;

Lm4: now__::_thesis:_for_f1,_f2_being_non_empty_NAT_*_-defined_to-naturals_homogeneous_Function
for_p_being_Element_of_((arity_f1)_+_1)_-tuples_on_NAT
for_i,_m_being_Element_of_NAT_st_i_in_dom_p_holds_
for_G_being_Function_of_(((arity_f1)_+_1)_-tuples_on_NAT),(HFuncs_NAT)_st_(_for_p_being_Element_of_((arity_f1)_+_1)_-tuples_on_NAT_holds_G_._p_=_primrec_(f1,f2,i,p)_)_holds_
for_k,_n_being_Element_of_NAT_holds_dom_(G_._(p_+*_(i,k)))_c=_dom_(G_._(p_+*_(i,(k_+_n))))
let f1, f2 be non empty NAT * -defined to-naturals homogeneous Function; ::_thesis: for p being Element of ((arity f1) + 1) -tuples_on NAT
for i, m being Element of NAT st i in dom p holds
for G being Function of (((arity f1) + 1) -tuples_on NAT),(HFuncs NAT) st ( for p being Element of ((arity f1) + 1) -tuples_on NAT holds G . p = primrec (f1,f2,i,p) ) holds
for k, n being Element of NAT holds dom (G . (p +* (i,k))) c= dom (G . (p +* (i,(k + n))))
let p be Element of ((arity f1) + 1) -tuples_on NAT; ::_thesis: for i, m being Element of NAT st i in dom p holds
for G being Function of (((arity f1) + 1) -tuples_on NAT),(HFuncs NAT) st ( for p being Element of ((arity f1) + 1) -tuples_on NAT holds G . p = primrec (f1,f2,i,p) ) holds
for k, n being Element of NAT holds dom (G . (p +* (i,k))) c= dom (G . (p +* (i,(k + n))))
let i, m be Element of NAT ; ::_thesis: ( i in dom p implies for G being Function of (((arity f1) + 1) -tuples_on NAT),(HFuncs NAT) st ( for p being Element of ((arity f1) + 1) -tuples_on NAT holds G . p = primrec (f1,f2,i,p) ) holds
for k, n being Element of NAT holds dom (G . (p +* (i,k))) c= dom (G . (p +* (i,(k + n)))) )
assume A1: i in dom p ; ::_thesis: for G being Function of (((arity f1) + 1) -tuples_on NAT),(HFuncs NAT) st ( for p being Element of ((arity f1) + 1) -tuples_on NAT holds G . p = primrec (f1,f2,i,p) ) holds
for k, n being Element of NAT holds dom (G . (p +* (i,k))) c= dom (G . (p +* (i,(k + n))))
let G be Function of (((arity f1) + 1) -tuples_on NAT),(HFuncs NAT); ::_thesis: ( ( for p being Element of ((arity f1) + 1) -tuples_on NAT holds G . p = primrec (f1,f2,i,p) ) implies for k, n being Element of NAT holds dom (G . (p +* (i,k))) c= dom (G . (p +* (i,(k + n)))) )
assume A2: for p being Element of ((arity f1) + 1) -tuples_on NAT holds G . p = primrec (f1,f2,i,p) ; ::_thesis: for k, n being Element of NAT holds dom (G . (p +* (i,k))) c= dom (G . (p +* (i,(k + n))))
thus  for k, n being Element of NAT holds dom (G . (p +* (i,k))) c= dom (G . (p +* (i,(k + n)))) ::_thesis: verum
proof
let k be Element of NAT ; ::_thesis: for n being Element of NAT holds dom (G . (p +* (i,k))) c= dom (G . (p +* (i,(k + n))))
set pk = p +* (i,k);
defpred S2[ Element of NAT ] means  dom (G . (p +* (i,k))) c= dom (G . (p +* (i,(k + $1))));
A3: now__::_thesis:_for_n_being_Element_of_NAT_st_S2[n]_holds_
S2[n_+_1]
let n be Element of NAT ; ::_thesis: ( S2[n] implies S2[b1 + 1] )
assume A4: S2[n] ; ::_thesis: S2[b1 + 1]
set m = k + n;
set pkn1 = p +* (i,((k + n) + 1));
set pkn = p +* (i,(k + n));
consider F being Function of NAT,(HFuncs NAT) such that
A5: primrec (f1,f2,i,(p +* (i,(k + n)))) = F . ((p +* (i,(k + n))) /. i) and
A6: ( i in dom (p +* (i,(k + n))) & Del ((p +* (i,(k + n))),i) in dom f1 implies F . 0 = ((p +* (i,(k + n))) +* (i,0)) .--> (f1 . (Del ((p +* (i,(k + n))),i))) ) and
A7: ( ( not i in dom (p +* (i,(k + n))) or not Del ((p +* (i,(k + n))),i) in dom f1 ) implies F . 0 = {} ) and
A8: for M being Element of NAT holds S1[M,F . M,F . (M + 1),p +* (i,(k + n)),i,f2] by Def10;
 dom p = dom (p +* (i,((k + n) + 1))) by FUNCT_7:30;
then A9: (p +* (i,((k + n) + 1))) /. i = (p +* (i,((k + n) + 1))) . i by A1, PARTFUN1:def_6
.= (k + n) + 1 by A1, FUNCT_7:31 ;
consider F1 being Function of NAT,(HFuncs NAT) such that
A10: primrec (f1,f2,i,(p +* (i,((k + n) + 1)))) = F1 . ((p +* (i,((k + n) + 1))) /. i) and
A11: ( i in dom (p +* (i,((k + n) + 1))) & Del ((p +* (i,((k + n) + 1))),i) in dom f1 implies F1 . 0 = ((p +* (i,((k + n) + 1))) +* (i,0)) .--> (f1 . (Del ((p +* (i,((k + n) + 1))),i))) ) and
A12: ( ( not i in dom (p +* (i,((k + n) + 1))) or not Del ((p +* (i,((k + n) + 1))),i) in dom f1 ) implies F1 . 0 = {} ) and
A13: for M being Element of NAT holds S1[M,F1 . M,F1 . (M + 1),p +* (i,((k + n) + 1)),i,f2] by Def10;
 F1 = F by A6, A7, A8, A11, A12, A13, Lm2;
then A14: G . (p +* (i,(k + (n + 1)))) = F . ((k + n) + 1) by A2, A10, A9;
 dom p = dom (p +* (i,(k + n))) by FUNCT_7:30;
then (p +* (i,(k + n))) /. i = (p +* (i,(k + n))) . i by A1, PARTFUN1:def_6
.= k + n by A1, FUNCT_7:31 ;
then A15: G . (p +* (i,(k + n))) = F . (k + n) by A2, A5;
percases ( ( i in dom (p +* (i,(k + n))) & (p +* (i,(k + n))) +* (i,(k + n)) in dom (F . (k + n)) & ((p +* (i,(k + n))) +* (i,(k + n))) ^ <*((F . (k + n)) . ((p +* (i,(k + n))) +* (i,(k + n))))*> in dom f2 ) or not i in dom (p +* (i,(k + n))) or not (p +* (i,(k + n))) +* (i,(k + n)) in dom (F . (k + n)) or not ((p +* (i,(k + n))) +* (i,(k + n))) ^ <*((F . (k + n)) . ((p +* (i,(k + n))) +* (i,(k + n))))*> in dom f2 ) ;
suppose ( i in dom (p +* (i,(k + n))) & (p +* (i,(k + n))) +* (i,(k + n)) in dom (F . (k + n)) & ((p +* (i,(k + n))) +* (i,(k + n))) ^ <*((F . (k + n)) . ((p +* (i,(k + n))) +* (i,(k + n))))*> in dom f2 ) ; ::_thesis: S2[b1 + 1]
then  F . ((k + n) + 1) = (F . (k + n)) +* (((p +* (i,(k + n))) +* (i,((k + n) + 1))) .--> (f2 . (((p +* (i,(k + n))) +* (i,(k + n))) ^ <*((F . (k + n)) . ((p +* (i,(k + n))) +* (i,(k + n))))*>))) by A8;
then  dom (F . ((k + n) + 1)) = (dom (F . (k + n))) \/ (dom ({((p +* (i,(k + n))) +* (i,((k + n) + 1)))} --> (f2 . (((p +* (i,(k + n))) +* (i,(k + n))) ^ <*((F . (k + n)) . ((p +* (i,(k + n))) +* (i,(k + n))))*>)))) by FUNCT_4:def_1;
then  dom (F . (k + n)) c= dom (F . ((k + n) + 1)) by XBOOLE_1:7;
hence  S2[n + 1] by A4, A14, A15, XBOOLE_1:1; ::_thesis: verum
end;
suppose ( not i in dom (p +* (i,(k + n))) or not (p +* (i,(k + n))) +* (i,(k + n)) in dom (F . (k + n)) or not ((p +* (i,(k + n))) +* (i,(k + n))) ^ <*((F . (k + n)) . ((p +* (i,(k + n))) +* (i,(k + n))))*> in dom f2 ) ; ::_thesis: S2[b1 + 1]
hence  S2[n + 1] by A4, A8, A14, A15; ::_thesis: verum
end;
end;
end;
A16: S2[ 0 ] ;
thus  for n being Element of NAT holds S2[n] from NAT_1:sch_1(A16, A3); ::_thesis: verum
end;
end;

Lm5: now__::_thesis:_for_f1,_f2_being_non_empty_NAT_*_-defined_to-naturals_homogeneous_Function
for_p_being_Element_of_((arity_f1)_+_1)_-tuples_on_NAT
for_i,_m_being_Element_of_NAT_st_i_in_dom_p_holds_
for_G_being_Function_of_(((arity_f1)_+_1)_-tuples_on_NAT),(HFuncs_NAT)_st_(_for_p_being_Element_of_((arity_f1)_+_1)_-tuples_on_NAT_holds_G_._p_=_primrec_(f1,f2,i,p)_)_holds_
for_k,_n_being_Element_of_NAT_st_not_p_+*_(i,k)_in_dom_(G_._(p_+*_(i,k)))_holds_
not_p_+*_(i,k)_in_dom_(G_._(p_+*_(i,(k_+_n))))
let f1, f2 be non empty NAT * -defined to-naturals homogeneous Function; ::_thesis: for p being Element of ((arity f1) + 1) -tuples_on NAT
for i, m being Element of NAT st i in dom p holds
for G being Function of (((arity f1) + 1) -tuples_on NAT),(HFuncs NAT) st ( for p being Element of ((arity f1) + 1) -tuples_on NAT holds G . p = primrec (f1,f2,i,p) ) holds
for k, n being Element of NAT st not p +* (i,k) in dom (G . (p +* (i,k))) holds
not p +* (i,k) in dom (G . (p +* (i,(k + n))))
let p be Element of ((arity f1) + 1) -tuples_on NAT; ::_thesis: for i, m being Element of NAT st i in dom p holds
for G being Function of (((arity f1) + 1) -tuples_on NAT),(HFuncs NAT) st ( for p being Element of ((arity f1) + 1) -tuples_on NAT holds G . p = primrec (f1,f2,i,p) ) holds
for k, n being Element of NAT st not p +* (i,k) in dom (G . (p +* (i,k))) holds
not p +* (i,k) in dom (G . (p +* (i,(k + n))))
let i, m be Element of NAT ; ::_thesis: ( i in dom p implies for G being Function of (((arity f1) + 1) -tuples_on NAT),(HFuncs NAT) st ( for p being Element of ((arity f1) + 1) -tuples_on NAT holds G . p = primrec (f1,f2,i,p) ) holds
for k, n being Element of NAT st not p +* (i,k) in dom (G . (p +* (i,k))) holds
not p +* (i,k) in dom (G . (p +* (i,(k + n)))) )
assume A1: i in dom p ; ::_thesis: for G being Function of (((arity f1) + 1) -tuples_on NAT),(HFuncs NAT) st ( for p being Element of ((arity f1) + 1) -tuples_on NAT holds G . p = primrec (f1,f2,i,p) ) holds
for k, n being Element of NAT st not p +* (i,k) in dom (G . (p +* (i,k))) holds
not p +* (i,k) in dom (G . (p +* (i,(k + n))))
let G be Function of (((arity f1) + 1) -tuples_on NAT),(HFuncs NAT); ::_thesis: ( ( for p being Element of ((arity f1) + 1) -tuples_on NAT holds G . p = primrec (f1,f2,i,p) ) implies for k, n being Element of NAT st not p +* (i,k) in dom (G . (p +* (i,k))) holds
not p +* (i,k) in dom (G . (p +* (i,(k + n)))) )
assume A2: for p being Element of ((arity f1) + 1) -tuples_on NAT holds G . p = primrec (f1,f2,i,p) ; ::_thesis: for k, n being Element of NAT st not p +* (i,k) in dom (G . (p +* (i,k))) holds
not p +* (i,k) in dom (G . (p +* (i,(k + n))))
thus  for k, n being Element of NAT st not p +* (i,k) in dom (G . (p +* (i,k))) holds
not p +* (i,k) in dom (G . (p +* (i,(k + n)))) ::_thesis: verum
proof
let k be Element of NAT ; ::_thesis: for n being Element of NAT st not p +* (i,k) in dom (G . (p +* (i,k))) holds
not p +* (i,k) in dom (G . (p +* (i,(k + n))))
set pk = p +* (i,k);
defpred S2[ Element of NAT ] means ( not p +* (i,k) in dom (G . (p +* (i,k))) implies not p +* (i,k) in dom (G . (p +* (i,(k + $1)))) );
A3: for n being Element of NAT st S2[n] holds
S2[n + 1]
proof
let n be Element of NAT ; ::_thesis: ( S2[n] implies S2[n + 1] )
assume that
A4: S2[n] and
A5: not p +* (i,k) in dom (G . (p +* (i,k))) ; ::_thesis: not p +* (i,k) in dom (G . (p +* (i,(k + (n + 1)))))
set m = k + n;
set pkn = p +* (i,(k + n));
consider F being Function of NAT,(HFuncs NAT) such that
A6: primrec (f1,f2,i,(p +* (i,(k + n)))) = F . ((p +* (i,(k + n))) /. i) and
A7: ( i in dom (p +* (i,(k + n))) & Del ((p +* (i,(k + n))),i) in dom f1 implies F . 0 = ((p +* (i,(k + n))) +* (i,0)) .--> (f1 . (Del ((p +* (i,(k + n))),i))) ) and
A8: ( ( not i in dom (p +* (i,(k + n))) or not Del ((p +* (i,(k + n))),i) in dom f1 ) implies F . 0 = {} ) and
A9: for M being Element of NAT holds S1[M,F . M,F . (M + 1),p +* (i,(k + n)),i,f2] by Def10;
A10: dom p = dom (p +* (i,(k + n))) by FUNCT_7:30;
then (p +* (i,(k + n))) /. i = (p +* (i,(k + n))) . i by A1, PARTFUN1:def_6
.= k + n by A1, FUNCT_7:31 ;
then A11: not p +* (i,k) in dom (F . (k + n)) by A2, A4, A5, A6;
set pkn1 = p +* (i,((k + n) + 1));
consider F1 being Function of NAT,(HFuncs NAT) such that
A12: primrec (f1,f2,i,(p +* (i,((k + n) + 1)))) = F1 . ((p +* (i,((k + n) + 1))) /. i) and
A13: ( i in dom (p +* (i,((k + n) + 1))) & Del ((p +* (i,((k + n) + 1))),i) in dom f1 implies F1 . 0 = ((p +* (i,((k + n) + 1))) +* (i,0)) .--> (f1 . (Del ((p +* (i,((k + n) + 1))),i))) ) and
A14: ( ( not i in dom (p +* (i,((k + n) + 1))) or not Del ((p +* (i,((k + n) + 1))),i) in dom f1 ) implies F1 . 0 = {} ) and
A15: for M being Element of NAT holds S1[M,F1 . M,F1 . (M + 1),p +* (i,((k + n) + 1)),i,f2] by Def10;
 dom p = dom (p +* (i,((k + n) + 1))) by FUNCT_7:30;
then A16: (p +* (i,((k + n) + 1))) /. i = (p +* (i,((k + n) + 1))) . i by A1, PARTFUN1:def_6
.= (k + n) + 1 by A1, FUNCT_7:31 ;
 F1 = F by A7, A8, A9, A13, A14, A15, Lm2;
then A17: G . (p +* (i,(k + (n + 1)))) = F . ((k + n) + 1) by A2, A12, A16;
percases ( ( i in dom (p +* (i,(k + n))) & (p +* (i,(k + n))) +* (i,(k + n)) in dom (F . (k + n)) & ((p +* (i,(k + n))) +* (i,(k + n))) ^ <*((F . (k + n)) . ((p +* (i,(k + n))) +* (i,(k + n))))*> in dom f2 ) or not i in dom (p +* (i,(k + n))) or not (p +* (i,(k + n))) +* (i,(k + n)) in dom (F . (k + n)) or not ((p +* (i,(k + n))) +* (i,(k + n))) ^ <*((F . (k + n)) . ((p +* (i,(k + n))) +* (i,(k + n))))*> in dom f2 ) ;
suppose ( i in dom (p +* (i,(k + n))) & (p +* (i,(k + n))) +* (i,(k + n)) in dom (F . (k + n)) & ((p +* (i,(k + n))) +* (i,(k + n))) ^ <*((F . (k + n)) . ((p +* (i,(k + n))) +* (i,(k + n))))*> in dom f2 ) ; ::_thesis: not p +* (i,k) in dom (G . (p +* (i,(k + (n + 1)))))
then  F . ((k + n) + 1) = (F . (k + n)) +* (((p +* (i,(k + n))) +* (i,((k + n) + 1))) .--> (f2 . (((p +* (i,(k + n))) +* (i,(k + n))) ^ <*((F . (k + n)) . ((p +* (i,(k + n))) +* (i,(k + n))))*>))) by A9;
then  dom (F . ((k + n) + 1)) = (dom (F . (k + n))) \/ (dom ({((p +* (i,(k + n))) +* (i,((k + n) + 1)))} --> (f2 . (((p +* (i,(k + n))) +* (i,(k + n))) ^ <*((F . (k + n)) . ((p +* (i,(k + n))) +* (i,(k + n))))*>)))) by FUNCT_4:def_1;
then A18: dom (F . ((k + n) + 1)) = (dom (F . (k + n))) \/ {((p +* (i,(k + n))) +* (i,((k + n) + 1)))} by FUNCOP_1:13;
 k <= k + n by NAT_1:11;
then A19: k <> (k + n) + 1 by NAT_1:13;
A20: (p +* (i,k)) . i = k by A1, FUNCT_7:31;
 ((p +* (i,(k + n))) +* (i,((k + n) + 1))) . i = (k + n) + 1 by A1, A10, FUNCT_7:31;
then  not p +* (i,k) in {((p +* (i,(k + n))) +* (i,((k + n) + 1)))} by A19, A20, TARSKI:def_1;
hence  not p +* (i,k) in dom (G . (p +* (i,(k + (n + 1))))) by A11, A17, A18, XBOOLE_0:def_3; ::_thesis: verum
end;
suppose ( not i in dom (p +* (i,(k + n))) or not (p +* (i,(k + n))) +* (i,(k + n)) in dom (F . (k + n)) or not ((p +* (i,(k + n))) +* (i,(k + n))) ^ <*((F . (k + n)) . ((p +* (i,(k + n))) +* (i,(k + n))))*> in dom f2 ) ; ::_thesis: not p +* (i,k) in dom (G . (p +* (i,(k + (n + 1)))))
hence  not p +* (i,k) in dom (G . (p +* (i,(k + (n + 1))))) by A9, A11, A17; ::_thesis: verum
end;
end;
end;
A21: S2[ 0 ] ;
thus  for n being Element of NAT holds S2[n] from NAT_1:sch_1(A21, A3); ::_thesis: verum
end;
end;

Lm6:  for i, m being Element of NAT
for f1, f2 being non empty NAT * -defined to-naturals homogeneous Function
for p being Element of ((arity f1) + 1) -tuples_on NAT st i in dom p holds
( ( p +* (i,0) in dom (primrec (f1,f2,i)) implies Del (p,i) in dom f1 ) & ( Del (p,i) in dom f1 implies p +* (i,0) in dom (primrec (f1,f2,i)) ) & ( p +* (i,0) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,0)) = f1 . (Del (p,i)) ) & ( p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) implies ( p +* (i,m) in dom (primrec (f1,f2,i)) & (p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*> in dom f2 ) ) & ( p +* (i,m) in dom (primrec (f1,f2,i)) & (p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*> in dom f2 implies p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) ) & ( p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,(m + 1))) = f2 . ((p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*>) ) )
proof
let i, m be Element of NAT ; ::_thesis: for f1, f2 being non empty NAT * -defined to-naturals homogeneous Function
for p being Element of ((arity f1) + 1) -tuples_on NAT st i in dom p holds
( ( p +* (i,0) in dom (primrec (f1,f2,i)) implies Del (p,i) in dom f1 ) & ( Del (p,i) in dom f1 implies p +* (i,0) in dom (primrec (f1,f2,i)) ) & ( p +* (i,0) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,0)) = f1 . (Del (p,i)) ) & ( p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) implies ( p +* (i,m) in dom (primrec (f1,f2,i)) & (p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*> in dom f2 ) ) & ( p +* (i,m) in dom (primrec (f1,f2,i)) & (p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*> in dom f2 implies p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) ) & ( p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,(m + 1))) = f2 . ((p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*>) ) )
let f1, f2 be non empty NAT * -defined to-naturals homogeneous Function; ::_thesis: for p being Element of ((arity f1) + 1) -tuples_on NAT st i in dom p holds
( ( p +* (i,0) in dom (primrec (f1,f2,i)) implies Del (p,i) in dom f1 ) & ( Del (p,i) in dom f1 implies p +* (i,0) in dom (primrec (f1,f2,i)) ) & ( p +* (i,0) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,0)) = f1 . (Del (p,i)) ) & ( p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) implies ( p +* (i,m) in dom (primrec (f1,f2,i)) & (p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*> in dom f2 ) ) & ( p +* (i,m) in dom (primrec (f1,f2,i)) & (p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*> in dom f2 implies p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) ) & ( p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,(m + 1))) = f2 . ((p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*>) ) )
let p be Element of ((arity f1) + 1) -tuples_on NAT; ::_thesis: ( i in dom p implies ( ( p +* (i,0) in dom (primrec (f1,f2,i)) implies Del (p,i) in dom f1 ) & ( Del (p,i) in dom f1 implies p +* (i,0) in dom (primrec (f1,f2,i)) ) & ( p +* (i,0) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,0)) = f1 . (Del (p,i)) ) & ( p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) implies ( p +* (i,m) in dom (primrec (f1,f2,i)) & (p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*> in dom f2 ) ) & ( p +* (i,m) in dom (primrec (f1,f2,i)) & (p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*> in dom f2 implies p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) ) & ( p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,(m + 1))) = f2 . ((p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*>) ) ) )
set p0 = p +* (i,0);
consider G being Function of (((arity f1) + 1) -tuples_on NAT),(HFuncs NAT) such that
A1: primrec (f1,f2,i) = Union G and
A2: for p being Element of ((arity f1) + 1) -tuples_on NAT holds G . p = primrec (f1,f2,i,p) by Def11;
reconsider rngG = rng G as functional compatible set by A1, Th11;
assume A3: i in dom p ; ::_thesis: ( ( p +* (i,0) in dom (primrec (f1,f2,i)) implies Del (p,i) in dom f1 ) & ( Del (p,i) in dom f1 implies p +* (i,0) in dom (primrec (f1,f2,i)) ) & ( p +* (i,0) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,0)) = f1 . (Del (p,i)) ) & ( p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) implies ( p +* (i,m) in dom (primrec (f1,f2,i)) & (p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*> in dom f2 ) ) & ( p +* (i,m) in dom (primrec (f1,f2,i)) & (p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*> in dom f2 implies p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) ) & ( p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,(m + 1))) = f2 . ((p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*>) ) )
A4: now__::_thesis:_for_k_being_Element_of_NAT_st_p_+*_(i,k)_in_dom_(primrec_(f1,f2,i))_holds_
p_+*_(i,k)_in_dom_(G_._(p_+*_(i,k)))
let k be Element of NAT ; ::_thesis: ( p +* (i,k) in dom (primrec (f1,f2,i)) implies p +* (i,k) in dom (G . (p +* (i,k))) )
assume that
A5: p +* (i,k) in dom (primrec (f1,f2,i)) and
A6: not p +* (i,k) in dom (G . (p +* (i,k))) ; ::_thesis: contradiction
 union rngG <> {} by A1, A5;
then reconsider rngG = rng G as non empty functional compatible set ;
set pk = p +* (i,k);
 dom (union rngG) = union  {  (dom f) where f is Element of rngG : verum  }  by Th12;
then consider X being set  such that
A7: p +* (i,k) in X and
A8: X in  {  (dom f) where f is Element of rngG : verum  }  by A1, A5, TARSKI:def_4;
consider f being Element of rngG such that
A9: X = dom f and
 verum by A8;
consider pp being set  such that
A10: pp in dom G and
A11: f = G . pp by FUNCT_1:def_3;
reconsider pp = pp as Element of ((arity f1) + 1) -tuples_on NAT by A10;
 G . pp = primrec (f1,f2,i,pp) by A2;
then A12: ex m being Element of NAT st p +* (i,k) = pp +* (i,m) by A7, A9, A11, Th49;
set ppi = pp . i;
A13: p +* (i,(pp . i)) = (p +* (i,k)) +* (i,(pp . i)) by FUNCT_7:34
.= pp +* (i,(pp . i)) by A12, FUNCT_7:34
.= pp by FUNCT_7:35 ;
percases ( k = pp . i or pp . i < k or pp . i > k ) by XXREAL_0:1;
suppose k = pp . i ; ::_thesis: contradiction
hence  contradiction by A6, A7, A9, A11, A13; ::_thesis: verum
end;
suppose pp . i < k ; ::_thesis: contradiction
then consider m being Nat such that
A14: k = (pp . i) + m by NAT_1:10;
reconsider m = m as Element of NAT by ORDINAL1:def_12;
 k = (pp . i) + m by A14;
then  dom (G . pp) c= dom (G . (p +* (i,k))) by A3, A2, A13, Lm4;
hence  contradiction by A6, A7, A9, A11; ::_thesis: verum
end;
suppose pp . i > k ; ::_thesis: contradiction
then consider m being Nat such that
A15: pp . i = k + m by NAT_1:10;
reconsider m = m as Element of NAT by ORDINAL1:def_12;
 pp . i = k + m by A15;
hence  contradiction by A3, A2, A6, A7, A9, A11, A13, Lm5; ::_thesis: verum
end;
end;
end;
A16: dom p = dom (p +* (i,0)) by FUNCT_7:30;
A17: now__::_thesis:_(_(_p_+*_(i,0)_in_dom_(G_._(p_+*_(i,0)))_implies_Del_(p,i)_in_dom_f1_)_&_(_Del_(p,i)_in_dom_f1_implies_p_+*_(i,0)_in_dom_(G_._(p_+*_(i,0)))_)_&_(_p_+*_(i,0)_in_dom_(G_._(p_+*_(i,0)))_implies_(G_._(p_+*_(i,0)))_._(p_+*_(i,0))_=_f1_._(Del_(p,i))_)_)
A18: p +* (i,0) in {(p +* (i,0))} by TARSKI:def_1;
A19: (p +* (i,0)) /. i = (p +* (i,0)) . i by A3, A16, PARTFUN1:def_6
.= 0 by A3, FUNCT_7:31 ;
consider F being Function of NAT,(HFuncs NAT) such that
A20: primrec (f1,f2,i,(p +* (i,0))) = F . ((p +* (i,0)) /. i) and
A21: ( i in dom (p +* (i,0)) & Del ((p +* (i,0)),i) in dom f1 implies F . 0 = ((p +* (i,0)) +* (i,0)) .--> (f1 . (Del ((p +* (i,0)),i))) ) and
A22: ( ( not i in dom (p +* (i,0)) or not Del ((p +* (i,0)),i) in dom f1 ) implies F . 0 = {} ) and
 for m being Element of NAT holds S1[m,F . m,F . (m + 1),p +* (i,0),i,f2] by Def10;
A23: G . (p +* (i,0)) = primrec (f1,f2,i,(p +* (i,0))) by A2;
thus  ( p +* (i,0) in dom (G . (p +* (i,0))) iff Del (p,i) in dom f1 ) ::_thesis: ( p +* (i,0) in dom (G . (p +* (i,0))) implies (G . (p +* (i,0))) . (p +* (i,0)) = f1 . (Del (p,i)) )
proof
thus  ( p +* (i,0) in dom (G . (p +* (i,0))) implies Del (p,i) in dom f1 ) by A2, A20, A22, A19, Th3, RELAT_1:38; ::_thesis: ( Del (p,i) in dom f1 implies p +* (i,0) in dom (G . (p +* (i,0))) )
assume  Del (p,i) in dom f1 ; ::_thesis: p +* (i,0) in dom (G . (p +* (i,0)))
then  dom (F . 0) = {((p +* (i,0)) +* (i,0))} by A3, A21, Th3, FUNCOP_1:13, FUNCT_7:30
.= {(p +* (i,0))} by FUNCT_7:34 ;
hence  p +* (i,0) in dom (G . (p +* (i,0))) by A23, A20, A19, TARSKI:def_1; ::_thesis: verum
end;
assume  p +* (i,0) in dom (G . (p +* (i,0))) ; ::_thesis: (G . (p +* (i,0))) . (p +* (i,0)) = f1 . (Del (p,i))
then  F . 0 = {(p +* (i,0))} --> (f1 . (Del ((p +* (i,0)),i))) by A2, A20, A21, A22, A19, FUNCT_7:34, RELAT_1:38;
then  F . 0 = {(p +* (i,0))} --> (f1 . (Del (p,i))) by Th3;
hence  (G . (p +* (i,0))) . (p +* (i,0)) = f1 . (Del (p,i)) by A23, A20, A19, A18, FUNCOP_1:7; ::_thesis: verum
end;
set pm1 = p +* (i,(m + 1));
set pm = p +* (i,m);
set pc = <*((G . (p +* (i,(m + 1)))) . ((p +* (i,(m + 1))) +* (i,m)))*>;
A24: dom G = ((arity f1) + 1) -tuples_on NAT by FUNCT_2:def_1;
then A25: G . (p +* (i,(m + 1))) in rng G by FUNCT_1:def_3;
reconsider rngG = rngG as non empty functional compatible set ;
A26: G . (p +* (i,0)) in rng G by A24, FUNCT_1:def_3;
thus  ( p +* (i,0) in dom (primrec (f1,f2,i)) iff Del (p,i) in dom f1 ) ::_thesis: ( ( p +* (i,0) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,0)) = f1 . (Del (p,i)) ) & ( p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) implies ( p +* (i,m) in dom (primrec (f1,f2,i)) & (p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*> in dom f2 ) ) & ( p +* (i,m) in dom (primrec (f1,f2,i)) & (p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*> in dom f2 implies p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) ) & ( p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,(m + 1))) = f2 . ((p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*>) ) )
proof
thus  ( p +* (i,0) in dom (primrec (f1,f2,i)) implies Del (p,i) in dom f1 ) by A4, A17; ::_thesis: ( Del (p,i) in dom f1 implies p +* (i,0) in dom (primrec (f1,f2,i)) )
assume A27: Del (p,i) in dom f1 ; ::_thesis: p +* (i,0) in dom (primrec (f1,f2,i))
 dom (G . (p +* (i,0))) in  {  (dom f) where f is Element of rngG : verum  }  by A26;
then  p +* (i,0) in union  {  (dom f) where f is Element of rngG : verum  }  by A17, A27, TARSKI:def_4;
hence  p +* (i,0) in dom (primrec (f1,f2,i)) by A1, Th12; ::_thesis: verum
end;
hereby  ::_thesis: ( ( p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) implies ( p +* (i,m) in dom (primrec (f1,f2,i)) & (p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*> in dom f2 ) ) & ( p +* (i,m) in dom (primrec (f1,f2,i)) & (p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*> in dom f2 implies p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) ) & ( p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,(m + 1))) = f2 . ((p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*>) ) )
assume A28: p +* (i,0) in dom (primrec (f1,f2,i)) ; ::_thesis: (primrec (f1,f2,i)) . (p +* (i,0)) = f1 . (Del (p,i))
then  p +* (i,0) in dom (G . (p +* (i,0))) by A4;
then  (union rngG) . (p +* (i,0)) = (G . (p +* (i,0))) . (p +* (i,0)) by A26, Th13;
hence  (primrec (f1,f2,i)) . (p +* (i,0)) = f1 . (Del (p,i)) by A1, A4, A17, A28; ::_thesis: verum
end;
A29: dom p = dom (p +* (i,(m + 1))) by FUNCT_7:30;
A30: (p +* (i,(m + 1))) +* (i,(m + 1)) = p +* (i,(m + 1)) by FUNCT_7:34;
A31: (p +* (i,(m + 1))) +* (i,m) = p +* (i,m) by FUNCT_7:34;
A32: dom p = dom (p +* (i,m)) by FUNCT_7:30;
A33: now__::_thesis:_(_(_p_+*_(i,(m_+_1))_in_dom_(G_._(p_+*_(i,(m_+_1))))_implies_(_p_+*_(i,m)_in_dom_(G_._(p_+*_(i,m)))_&_(p_+*_(i,m))_^_<*((G_._(p_+*_(i,(m_+_1))))_._((p_+*_(i,(m_+_1)))_+*_(i,m)))*>_in_dom_f2_)_)_&_(_p_+*_(i,m)_in_dom_(G_._(p_+*_(i,m)))_&_(p_+*_(i,m))_^_<*((G_._(p_+*_(i,(m_+_1))))_._((p_+*_(i,(m_+_1)))_+*_(i,m)))*>_in_dom_f2_implies_p_+*_(i,(m_+_1))_in_dom_(G_._(p_+*_(i,(m_+_1))))_)_&_(_p_+*_(i,(m_+_1))_in_dom_(G_._(p_+*_(i,(m_+_1))))_implies_(G_._(p_+*_(i,(m_+_1))))_._(p_+*_(i,(m_+_1)))_=_f2_._((p_+*_(i,m))_^_<*((G_._(p_+*_(i,(m_+_1))))_._((p_+*_(i,(m_+_1)))_+*_(i,m)))*>)_)_)
A34: (p +* (i,m)) /. i = (p +* (i,m)) . i by A3, A32, PARTFUN1:def_6
.= m by A3, FUNCT_7:31 ;
consider F being Function of NAT,(HFuncs NAT) such that
A35: primrec (f1,f2,i,(p +* (i,m))) = F . ((p +* (i,m)) /. i) and
A36: ( i in dom (p +* (i,m)) & Del ((p +* (i,m)),i) in dom f1 implies F . 0 = ((p +* (i,m)) +* (i,0)) .--> (f1 . (Del ((p +* (i,m)),i))) ) and
A37: ( ( not i in dom (p +* (i,m)) or not Del ((p +* (i,m)),i) in dom f1 ) implies F . 0 = {} ) and
A38: for M being Element of NAT holds S1[M,F . M,F . (M + 1),p +* (i,m),i,f2] by Def10;
A39: G . (p +* (i,(m + 1))) = primrec (f1,f2,i,(p +* (i,(m + 1)))) by A2;
consider F1 being Function of NAT,(HFuncs NAT) such that
A40: primrec (f1,f2,i,(p +* (i,(m + 1)))) = F1 . ((p +* (i,(m + 1))) /. i) and
A41: ( i in dom (p +* (i,(m + 1))) & Del ((p +* (i,(m + 1))),i) in dom f1 implies F1 . 0 = ((p +* (i,(m + 1))) +* (i,0)) .--> (f1 . (Del ((p +* (i,(m + 1))),i))) ) and
A42: ( ( not i in dom (p +* (i,(m + 1))) or not Del ((p +* (i,(m + 1))),i) in dom f1 ) implies F1 . 0 = {} ) and
A43: for M being Element of NAT holds S1[M,F1 . M,F1 . (M + 1),p +* (i,(m + 1)),i,f2] by Def10;
A44: (F1 . (m + 1)) . (p +* (i,m)) = (F1 . m) . (p +* (i,m)) by A3, A41, A42, A43, Lm3;
A45: p +* (i,(m + 1)) in {(p +* (i,(m + 1)))} by TARSKI:def_1;
then A46: p +* (i,(m + 1)) in dom ({(p +* (i,(m + 1)))} --> (f2 . ((p +* (i,m)) ^ <*((F1 . m) . (p +* (i,m)))*>))) by FUNCOP_1:13;
A47: G . (p +* (i,m)) = primrec (f1,f2,i,(p +* (i,m))) by A2;
A48: (p +* (i,(m + 1))) /. i = (p +* (i,(m + 1))) . i by A3, A29, PARTFUN1:def_6
.= m + 1 by A3, FUNCT_7:31 ;
A49: F1 . m = F . m by A41, A42, A43, A36, A37, A38, Lm2;
A50: not p +* (i,(m + 1)) in dom (F1 . m) by A3, A41, A42, A43, Lm3;
thus A51: ( p +* (i,(m + 1)) in dom (G . (p +* (i,(m + 1)))) iff ( p +* (i,m) in dom (G . (p +* (i,m))) & (p +* (i,m)) ^ <*((G . (p +* (i,(m + 1)))) . ((p +* (i,(m + 1))) +* (i,m)))*> in dom f2 ) ) ::_thesis: ( p +* (i,(m + 1)) in dom (G . (p +* (i,(m + 1)))) implies (G . (p +* (i,(m + 1)))) . (p +* (i,(m + 1))) = f2 . ((p +* (i,m)) ^ <*((G . (p +* (i,(m + 1)))) . ((p +* (i,(m + 1))) +* (i,m)))*>) )
proof
hereby  ::_thesis: ( p +* (i,m) in dom (G . (p +* (i,m))) & (p +* (i,m)) ^ <*((G . (p +* (i,(m + 1)))) . ((p +* (i,(m + 1))) +* (i,m)))*> in dom f2 implies p +* (i,(m + 1)) in dom (G . (p +* (i,(m + 1)))) )
assume A52: p +* (i,(m + 1)) in dom (G . (p +* (i,(m + 1)))) ; ::_thesis: ( p +* (i,m) in dom (G . (p +* (i,m))) & (p +* (i,m)) ^ <*((G . (p +* (i,(m + 1)))) . ((p +* (i,(m + 1))) +* (i,m)))*> in dom f2 )
then A53: p +* (i,(m + 1)) in dom (F1 . (m + 1)) by A2, A40, A48;
assume A54: ( not p +* (i,m) in dom (G . (p +* (i,m))) or not (p +* (i,m)) ^ <*((G . (p +* (i,(m + 1)))) . ((p +* (i,(m + 1))) +* (i,m)))*> in dom f2 ) ; ::_thesis: contradiction
percases ( not p +* (i,m) in dom (G . (p +* (i,m))) or not (p +* (i,m)) ^ <*((G . (p +* (i,(m + 1)))) . ((p +* (i,(m + 1))) +* (i,m)))*> in dom f2 ) by A54;
suppose not p +* (i,m) in dom (G . (p +* (i,m))) ; ::_thesis: contradiction
then  not p +* (i,m) in dom (F1 . m) by A2, A35, A34, A49;
hence  contradiction by A31, A43, A50, A53; ::_thesis: verum
end;
suppose not (p +* (i,m)) ^ <*((G . (p +* (i,(m + 1)))) . ((p +* (i,(m + 1))) +* (i,m)))*> in dom f2 ; ::_thesis: contradiction
hence  contradiction by A31, A39, A40, A43, A48, A44, A50, A52; ::_thesis: verum
end;
end;
end;
assume that
A55: p +* (i,m) in dom (G . (p +* (i,m))) and
A56: (p +* (i,m)) ^ <*((G . (p +* (i,(m + 1)))) . ((p +* (i,(m + 1))) +* (i,m)))*> in dom f2 ; ::_thesis: p +* (i,(m + 1)) in dom (G . (p +* (i,(m + 1))))
 p +* (i,(m + 1)) in {(p +* (i,(m + 1)))} by TARSKI:def_1;
then  p +* (i,(m + 1)) in dom ({(p +* (i,(m + 1)))} --> (f2 . ((p +* (i,m)) ^ <*((F1 . m) . (p +* (i,m)))*>))) by FUNCOP_1:13;
then A57: p +* (i,(m + 1)) in (dom (F1 . m)) \/ (dom ({(p +* (i,(m + 1)))} --> (f2 . ((p +* (i,m)) ^ <*((F1 . m) . (p +* (i,m)))*>)))) by XBOOLE_0:def_3;
 F1 . (m + 1) = (F1 . m) +* ((p +* (i,(m + 1))) .--> (f2 . ((p +* (i,m)) ^ <*((F1 . m) . (p +* (i,m)))*>))) by A3, A29, A31, A30, A39, A47, A40, A43, A35, A48, A34, A49, A44, A55, A56;
hence  p +* (i,(m + 1)) in dom (G . (p +* (i,(m + 1)))) by A39, A40, A48, A57, FUNCT_4:def_1; ::_thesis: verum
end;
assume A58: p +* (i,(m + 1)) in dom (G . (p +* (i,(m + 1)))) ; ::_thesis: (G . (p +* (i,(m + 1)))) . (p +* (i,(m + 1))) = f2 . ((p +* (i,m)) ^ <*((G . (p +* (i,(m + 1)))) . ((p +* (i,(m + 1))) +* (i,m)))*>)
then  (p +* (i,m)) ^ <*((F1 . m) . (p +* (i,m)))*> in dom f2 by A31, A39, A40, A43, A48, A51;
then  F1 . (m + 1) = (F1 . m) +* ((p +* (i,(m + 1))) .--> (f2 . ((p +* (i,m)) ^ <*((F1 . m) . (p +* (i,m)))*>))) by A3, A29, A31, A30, A47, A43, A35, A34, A49, A51, A58;
hence (G . (p +* (i,(m + 1)))) . (p +* (i,(m + 1))) = ({(p +* (i,(m + 1)))} --> (f2 . ((p +* (i,m)) ^ <*((F1 . m) . (p +* (i,m)))*>))) . (p +* (i,(m + 1))) by A39, A40, A48, A46, FUNCT_4:13
.= f2 . ((p +* (i,m)) ^ <*((G . (p +* (i,(m + 1)))) . ((p +* (i,(m + 1))) +* (i,m)))*>) by A31, A39, A40, A48, A44, A45, FUNCOP_1:7 ;
::_thesis: verum
end;
thus  ( p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) iff ( p +* (i,m) in dom (primrec (f1,f2,i)) & (p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*> in dom f2 ) ) ::_thesis: ( p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,(m + 1))) = f2 . ((p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*>) )
proof
hereby  ::_thesis: ( p +* (i,m) in dom (primrec (f1,f2,i)) & (p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*> in dom f2 implies p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) )
assume A59: p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) ; ::_thesis: ( p +* (i,m) in dom (primrec (f1,f2,i)) & (p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*> in dom f2 )
 G . (p +* (i,m)) in rng G by A24, FUNCT_1:def_3;
then  dom (G . (p +* (i,m))) in  {  (dom f) where f is Element of rngG : verum  }  ;
then  p +* (i,m) in union  {  (dom f) where f is Element of rngG : verum  }  by A4, A33, A59, TARSKI:def_4;
hence  p +* (i,m) in dom (primrec (f1,f2,i)) by A1, Th12; ::_thesis: (p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*> in dom f2
A60: G . (p +* (i,(m + 1))) in rng G by A24, FUNCT_1:def_3;
 dom (G . (p +* (i,m))) c= dom (G . (p +* (i,(m + 1)))) by A3, A2, Lm4;
then  (union rngG) . (p +* (i,m)) = (G . (p +* (i,(m + 1)))) . (p +* (i,m)) by A4, A33, A59, A60, Th13;
hence  (p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*> in dom f2 by A1, A4, A33, A59, FUNCT_7:34; ::_thesis: verum
end;
assume that
A61: p +* (i,m) in dom (primrec (f1,f2,i)) and
A62: (p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*> in dom f2 ; ::_thesis: p +* (i,(m + 1)) in dom (primrec (f1,f2,i))
A63: G . (p +* (i,(m + 1))) in rng G by A24, FUNCT_1:def_3;
 G . (p +* (i,(m + 1))) in rng G by A24, FUNCT_1:def_3;
then A64: dom (G . (p +* (i,(m + 1)))) in  {  (dom f) where f is Element of rngG : verum  }  ;
A65: dom (G . (p +* (i,m))) c= dom (G . (p +* (i,(m + 1)))) by A3, A2, Lm4;
 p +* (i,m) in dom (G . (p +* (i,m))) by A4, A61;
then  p +* (i,(m + 1)) in union  {  (dom f) where f is Element of rngG : verum  }  by A1, A31, A33, A62, A64, A63, A65, Th13, TARSKI:def_4;
hence  p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) by A1, Th12; ::_thesis: verum
end;
assume A66: p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) ; ::_thesis: (primrec (f1,f2,i)) . (p +* (i,(m + 1))) = f2 . ((p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*>)
A67: dom (G . (p +* (i,m))) c= dom (G . (p +* (i,(m + 1)))) by A3, A2, Lm4;
then A68: (union rngG) . (p +* (i,(m + 1))) = (G . (p +* (i,(m + 1)))) . (p +* (i,(m + 1))) by A4, A66, A25, Th13;
 (union rngG) . (p +* (i,m)) = (G . (p +* (i,(m + 1)))) . (p +* (i,m)) by A4, A33, A66, A25, A67, Th13;
hence  (primrec (f1,f2,i)) . (p +* (i,(m + 1))) = f2 . ((p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*>) by A1, A4, A33, A66, A68, FUNCT_7:34; ::_thesis: verum
end;

theorem :: COMPUT_1:58
for i being Element of NAT
for f1, f2 being non empty NAT * -defined to-naturals homogeneous Function
for p being Element of ((arity f1) + 1) -tuples_on NAT st i in dom p holds
( p +* (i,0) in dom (primrec (f1,f2,i)) iff Del (p,i) in dom f1 ) by Lm6;

theorem :: COMPUT_1:59
for i being Element of NAT
for f1, f2 being non empty NAT * -defined to-naturals homogeneous Function
for p being Element of ((arity f1) + 1) -tuples_on NAT st i in dom p & p +* (i,0) in dom (primrec (f1,f2,i)) holds
(primrec (f1,f2,i)) . (p +* (i,0)) = f1 . (Del (p,i)) by Lm6;

theorem Th60: :: COMPUT_1:60
for i being Element of NAT
for f1, f2 being non empty NAT * -defined to-naturals homogeneous Function
for p being Element of ((arity f1) + 1) -tuples_on NAT st i in dom p & f1 is len-total holds
(primrec (f1,f2,i)) . (p +* (i,0)) = f1 . (Del (p,i))
proof
let i be Element of NAT ; ::_thesis: for f1, f2 being non empty NAT * -defined to-naturals homogeneous Function
for p being Element of ((arity f1) + 1) -tuples_on NAT st i in dom p & f1 is len-total holds
(primrec (f1,f2,i)) . (p +* (i,0)) = f1 . (Del (p,i))
let f1, f2 be non empty NAT * -defined to-naturals homogeneous Function; ::_thesis: for p being Element of ((arity f1) + 1) -tuples_on NAT st i in dom p & f1 is len-total holds
(primrec (f1,f2,i)) . (p +* (i,0)) = f1 . (Del (p,i))
let p be Element of ((arity f1) + 1) -tuples_on NAT; ::_thesis: ( i in dom p & f1 is len-total implies (primrec (f1,f2,i)) . (p +* (i,0)) = f1 . (Del (p,i)) )
assume that
A1: i in dom p and
A2: f1 is len-total ; ::_thesis: (primrec (f1,f2,i)) . (p +* (i,0)) = f1 . (Del (p,i))
A3: Del (p,i) is FinSequence of NAT by FINSEQ_3:105;
 len p = (arity f1) + 1 by CARD_1:def_7;
then  len (Del (p,i)) = arity f1 by A1, FINSEQ_3:109;
then A4: Del (p,i) is Element of (arity f1) -tuples_on NAT by A3, FINSEQ_2:92;
 dom f1 = (arity f1) -tuples_on NAT by A2, Th23;
then  p +* (i,0) in dom (primrec (f1,f2,i)) by A1, A4, Lm6;
hence  (primrec (f1,f2,i)) . (p +* (i,0)) = f1 . (Del (p,i)) by A1, Lm6; ::_thesis: verum
end;

theorem :: COMPUT_1:61
for i, m being Element of NAT
for f1, f2 being non empty NAT * -defined to-naturals homogeneous Function
for p being Element of ((arity f1) + 1) -tuples_on NAT st i in dom p holds
( p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) iff ( p +* (i,m) in dom (primrec (f1,f2,i)) & (p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*> in dom f2 ) ) by Lm6;

theorem :: COMPUT_1:62
for i, m being Element of NAT
for f1, f2 being non empty NAT * -defined to-naturals homogeneous Function
for p being Element of ((arity f1) + 1) -tuples_on NAT st i in dom p & p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) holds
(primrec (f1,f2,i)) . (p +* (i,(m + 1))) = f2 . ((p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*>) by Lm6;

theorem Th63: :: COMPUT_1:63
for i, m being Element of NAT
for f1, f2 being non empty NAT * -defined to-naturals homogeneous Function
for p being Element of ((arity f1) + 1) -tuples_on NAT st f1 is len-total & f2 is len-total & (arity f1) + 2 = arity f2 & 1 <= i & i <= 1 + (arity f1) holds
(primrec (f1,f2,i)) . (p +* (i,(m + 1))) = f2 . ((p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*>)
proof
let i, m be Element of NAT ; ::_thesis: for f1, f2 being non empty NAT * -defined to-naturals homogeneous Function
for p being Element of ((arity f1) + 1) -tuples_on NAT st f1 is len-total & f2 is len-total & (arity f1) + 2 = arity f2 & 1 <= i & i <= 1 + (arity f1) holds
(primrec (f1,f2,i)) . (p +* (i,(m + 1))) = f2 . ((p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*>)
let f1, f2 be non empty NAT * -defined to-naturals homogeneous Function; ::_thesis: for p being Element of ((arity f1) + 1) -tuples_on NAT st f1 is len-total & f2 is len-total & (arity f1) + 2 = arity f2 & 1 <= i & i <= 1 + (arity f1) holds
(primrec (f1,f2,i)) . (p +* (i,(m + 1))) = f2 . ((p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*>)
let p be Element of ((arity f1) + 1) -tuples_on NAT; ::_thesis: ( f1 is len-total & f2 is len-total & (arity f1) + 2 = arity f2 & 1 <= i & i <= 1 + (arity f1) implies (primrec (f1,f2,i)) . (p +* (i,(m + 1))) = f2 . ((p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*>) )
assume that
A1: f1 is len-total and
A2: f2 is len-total and
A3: (arity f1) + 2 = arity f2 and
A4: 1 <= i and
A5: i <= 1 + (arity f1) ; ::_thesis: (primrec (f1,f2,i)) . (p +* (i,(m + 1))) = f2 . ((p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*>)
 len p = (arity f1) + 1 by CARD_1:def_7;
then A6: i in dom p by A4, A5, FINSEQ_3:25;
 p +* (i,(m + 1)) in ((arity f1) + 1) -tuples_on NAT ;
then  p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) by A1, A2, A3, A4, A5, Th57;
hence  (primrec (f1,f2,i)) . (p +* (i,(m + 1))) = f2 . ((p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*>) by A6, Lm6; ::_thesis: verum
end;

theorem Th64: :: COMPUT_1:64
for i being Element of NAT
for f1, f2 being non empty NAT * -defined to-naturals homogeneous Function st (arity f1) + 2 = arity f2 & 1 <= i & i <= (arity f1) + 1 holds
primrec (f1,f2,i) is_primitive-recursively_expressed_by f1,f2,i
proof
let i be Element of NAT ; ::_thesis: for f1, f2 being non empty NAT * -defined to-naturals homogeneous Function st (arity f1) + 2 = arity f2 & 1 <= i & i <= (arity f1) + 1 holds
primrec (f1,f2,i) is_primitive-recursively_expressed_by f1,f2,i
let f1, f2 be non empty NAT * -defined to-naturals homogeneous Function; ::_thesis: ( (arity f1) + 2 = arity f2 & 1 <= i & i <= (arity f1) + 1 implies primrec (f1,f2,i) is_primitive-recursively_expressed_by f1,f2,i )
assume that
A1: (arity f1) + 2 = arity f2 and
A2: 1 <= i and
A3: i <= (arity f1) + 1 ; ::_thesis: primrec (f1,f2,i) is_primitive-recursively_expressed_by f1,f2,i
take n = (arity f1) + 1; :: according to COMPUT_1:def_9 ::_thesis: ( dom (primrec (f1,f2,i)) c= n -tuples_on NAT & i >= 1 & i <= n & (arity f1) + 1 = n & n + 1 = arity f2 & ( for p being FinSequence of NAT st len p = n holds
( ( p +* (i,0) in dom (primrec (f1,f2,i)) implies Del (p,i) in dom f1 ) & ( Del (p,i) in dom f1 implies p +* (i,0) in dom (primrec (f1,f2,i)) ) & ( p +* (i,0) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,0)) = f1 . (Del (p,i)) ) & ( for n being Element of NAT holds
( ( p +* (i,(n + 1)) in dom (primrec (f1,f2,i)) implies ( p +* (i,n) in dom (primrec (f1,f2,i)) & (p +* (i,n)) ^ <*((primrec (f1,f2,i)) . (p +* (i,n)))*> in dom f2 ) ) & ( p +* (i,n) in dom (primrec (f1,f2,i)) & (p +* (i,n)) ^ <*((primrec (f1,f2,i)) . (p +* (i,n)))*> in dom f2 implies p +* (i,(n + 1)) in dom (primrec (f1,f2,i)) ) & ( p +* (i,(n + 1)) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,(n + 1))) = f2 . ((p +* (i,n)) ^ <*((primrec (f1,f2,i)) . (p +* (i,n)))*>) ) ) ) ) ) )
set g = primrec (f1,f2,i);
thus  dom (primrec (f1,f2,i)) c= n -tuples_on NAT by Th56; ::_thesis: ( i >= 1 & i <= n & (arity f1) + 1 = n & n + 1 = arity f2 & ( for p being FinSequence of NAT st len p = n holds
( ( p +* (i,0) in dom (primrec (f1,f2,i)) implies Del (p,i) in dom f1 ) & ( Del (p,i) in dom f1 implies p +* (i,0) in dom (primrec (f1,f2,i)) ) & ( p +* (i,0) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,0)) = f1 . (Del (p,i)) ) & ( for n being Element of NAT holds
( ( p +* (i,(n + 1)) in dom (primrec (f1,f2,i)) implies ( p +* (i,n) in dom (primrec (f1,f2,i)) & (p +* (i,n)) ^ <*((primrec (f1,f2,i)) . (p +* (i,n)))*> in dom f2 ) ) & ( p +* (i,n) in dom (primrec (f1,f2,i)) & (p +* (i,n)) ^ <*((primrec (f1,f2,i)) . (p +* (i,n)))*> in dom f2 implies p +* (i,(n + 1)) in dom (primrec (f1,f2,i)) ) & ( p +* (i,(n + 1)) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,(n + 1))) = f2 . ((p +* (i,n)) ^ <*((primrec (f1,f2,i)) . (p +* (i,n)))*>) ) ) ) ) ) )
thus  ( i >= 1 & i <= n ) by A2, A3; ::_thesis: ( (arity f1) + 1 = n & n + 1 = arity f2 & ( for p being FinSequence of NAT st len p = n holds
( ( p +* (i,0) in dom (primrec (f1,f2,i)) implies Del (p,i) in dom f1 ) & ( Del (p,i) in dom f1 implies p +* (i,0) in dom (primrec (f1,f2,i)) ) & ( p +* (i,0) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,0)) = f1 . (Del (p,i)) ) & ( for n being Element of NAT holds
( ( p +* (i,(n + 1)) in dom (primrec (f1,f2,i)) implies ( p +* (i,n) in dom (primrec (f1,f2,i)) & (p +* (i,n)) ^ <*((primrec (f1,f2,i)) . (p +* (i,n)))*> in dom f2 ) ) & ( p +* (i,n) in dom (primrec (f1,f2,i)) & (p +* (i,n)) ^ <*((primrec (f1,f2,i)) . (p +* (i,n)))*> in dom f2 implies p +* (i,(n + 1)) in dom (primrec (f1,f2,i)) ) & ( p +* (i,(n + 1)) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,(n + 1))) = f2 . ((p +* (i,n)) ^ <*((primrec (f1,f2,i)) . (p +* (i,n)))*>) ) ) ) ) ) )
thus  (arity f1) + 1 = n ; ::_thesis: ( n + 1 = arity f2 & ( for p being FinSequence of NAT st len p = n holds
( ( p +* (i,0) in dom (primrec (f1,f2,i)) implies Del (p,i) in dom f1 ) & ( Del (p,i) in dom f1 implies p +* (i,0) in dom (primrec (f1,f2,i)) ) & ( p +* (i,0) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,0)) = f1 . (Del (p,i)) ) & ( for n being Element of NAT holds
( ( p +* (i,(n + 1)) in dom (primrec (f1,f2,i)) implies ( p +* (i,n) in dom (primrec (f1,f2,i)) & (p +* (i,n)) ^ <*((primrec (f1,f2,i)) . (p +* (i,n)))*> in dom f2 ) ) & ( p +* (i,n) in dom (primrec (f1,f2,i)) & (p +* (i,n)) ^ <*((primrec (f1,f2,i)) . (p +* (i,n)))*> in dom f2 implies p +* (i,(n + 1)) in dom (primrec (f1,f2,i)) ) & ( p +* (i,(n + 1)) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,(n + 1))) = f2 . ((p +* (i,n)) ^ <*((primrec (f1,f2,i)) . (p +* (i,n)))*>) ) ) ) ) ) )
thus  n + 1 = arity f2 by A1; ::_thesis: for p being FinSequence of NAT st len p = n holds
( ( p +* (i,0) in dom (primrec (f1,f2,i)) implies Del (p,i) in dom f1 ) & ( Del (p,i) in dom f1 implies p +* (i,0) in dom (primrec (f1,f2,i)) ) & ( p +* (i,0) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,0)) = f1 . (Del (p,i)) ) & ( for n being Element of NAT holds
( ( p +* (i,(n + 1)) in dom (primrec (f1,f2,i)) implies ( p +* (i,n) in dom (primrec (f1,f2,i)) & (p +* (i,n)) ^ <*((primrec (f1,f2,i)) . (p +* (i,n)))*> in dom f2 ) ) & ( p +* (i,n) in dom (primrec (f1,f2,i)) & (p +* (i,n)) ^ <*((primrec (f1,f2,i)) . (p +* (i,n)))*> in dom f2 implies p +* (i,(n + 1)) in dom (primrec (f1,f2,i)) ) & ( p +* (i,(n + 1)) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,(n + 1))) = f2 . ((p +* (i,n)) ^ <*((primrec (f1,f2,i)) . (p +* (i,n)))*>) ) ) ) )
let p be FinSequence of NAT ; ::_thesis: ( len p = n implies ( ( p +* (i,0) in dom (primrec (f1,f2,i)) implies Del (p,i) in dom f1 ) & ( Del (p,i) in dom f1 implies p +* (i,0) in dom (primrec (f1,f2,i)) ) & ( p +* (i,0) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,0)) = f1 . (Del (p,i)) ) & ( for n being Element of NAT holds
( ( p +* (i,(n + 1)) in dom (primrec (f1,f2,i)) implies ( p +* (i,n) in dom (primrec (f1,f2,i)) & (p +* (i,n)) ^ <*((primrec (f1,f2,i)) . (p +* (i,n)))*> in dom f2 ) ) & ( p +* (i,n) in dom (primrec (f1,f2,i)) & (p +* (i,n)) ^ <*((primrec (f1,f2,i)) . (p +* (i,n)))*> in dom f2 implies p +* (i,(n + 1)) in dom (primrec (f1,f2,i)) ) & ( p +* (i,(n + 1)) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,(n + 1))) = f2 . ((p +* (i,n)) ^ <*((primrec (f1,f2,i)) . (p +* (i,n)))*>) ) ) ) ) )
assume A4: len p = n ; ::_thesis: ( ( p +* (i,0) in dom (primrec (f1,f2,i)) implies Del (p,i) in dom f1 ) & ( Del (p,i) in dom f1 implies p +* (i,0) in dom (primrec (f1,f2,i)) ) & ( p +* (i,0) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,0)) = f1 . (Del (p,i)) ) & ( for n being Element of NAT holds
( ( p +* (i,(n + 1)) in dom (primrec (f1,f2,i)) implies ( p +* (i,n) in dom (primrec (f1,f2,i)) & (p +* (i,n)) ^ <*((primrec (f1,f2,i)) . (p +* (i,n)))*> in dom f2 ) ) & ( p +* (i,n) in dom (primrec (f1,f2,i)) & (p +* (i,n)) ^ <*((primrec (f1,f2,i)) . (p +* (i,n)))*> in dom f2 implies p +* (i,(n + 1)) in dom (primrec (f1,f2,i)) ) & ( p +* (i,(n + 1)) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,(n + 1))) = f2 . ((p +* (i,n)) ^ <*((primrec (f1,f2,i)) . (p +* (i,n)))*>) ) ) ) )
then A5: i in dom p by A2, A3, FINSEQ_3:25;
A6: p is Element of n -tuples_on NAT by A4, FINSEQ_2:92;
hence  ( p +* (i,0) in dom (primrec (f1,f2,i)) iff Del (p,i) in dom f1 ) by A5, Lm6; ::_thesis: ( p +* (i,0) in dom (primrec (f1,f2,i)) iff (primrec (f1,f2,i)) . (p +* (i,0)) = f1 . (Del (p,i)) )
thus  ( p +* (i,0) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,0)) = f1 . (Del (p,i)) ) by A6, A5, Lm6; ::_thesis: for n being Element of NAT holds
( ( p +* (i,(n + 1)) in dom (primrec (f1,f2,i)) implies ( p +* (i,n) in dom (primrec (f1,f2,i)) & (p +* (i,n)) ^ <*((primrec (f1,f2,i)) . (p +* (i,n)))*> in dom f2 ) ) & ( p +* (i,n) in dom (primrec (f1,f2,i)) & (p +* (i,n)) ^ <*((primrec (f1,f2,i)) . (p +* (i,n)))*> in dom f2 implies p +* (i,(n + 1)) in dom (primrec (f1,f2,i)) ) & ( p +* (i,(n + 1)) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,(n + 1))) = f2 . ((p +* (i,n)) ^ <*((primrec (f1,f2,i)) . (p +* (i,n)))*>) ) )
let m be Element of NAT ; ::_thesis: ( ( p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) implies ( p +* (i,m) in dom (primrec (f1,f2,i)) & (p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*> in dom f2 ) ) & ( p +* (i,m) in dom (primrec (f1,f2,i)) & (p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*> in dom f2 implies p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) ) & ( p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,(m + 1))) = f2 . ((p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*>) ) )
thus  ( p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) iff ( p +* (i,m) in dom (primrec (f1,f2,i)) & (p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*> in dom f2 ) ) by A6, A5, Lm6; ::_thesis: ( p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,(m + 1))) = f2 . ((p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*>) )
thus  ( p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) implies (primrec (f1,f2,i)) . (p +* (i,(m + 1))) = f2 . ((p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*>) ) by A6, A5, Lm6; ::_thesis: verum
end;

theorem :: COMPUT_1:65
for i being Element of NAT
for f1, f2 being non empty NAT * -defined to-naturals homogeneous Function st 1 <= i & i <= (arity f1) + 1 holds
for g being Element of HFuncs NAT st g is_primitive-recursively_expressed_by f1,f2,i holds
g = primrec (f1,f2,i)
proof
let i be Element of NAT ; ::_thesis: for f1, f2 being non empty NAT * -defined to-naturals homogeneous Function st 1 <= i & i <= (arity f1) + 1 holds
for g being Element of HFuncs NAT st g is_primitive-recursively_expressed_by f1,f2,i holds
g = primrec (f1,f2,i)
let f1, f2 be non empty NAT * -defined to-naturals homogeneous Function; ::_thesis: ( 1 <= i & i <= (arity f1) + 1 implies for g being Element of HFuncs NAT st g is_primitive-recursively_expressed_by f1,f2,i holds
g = primrec (f1,f2,i) )
assume that
A1: i >= 1 and
A2: i <= (arity f1) + 1 ; ::_thesis: for g being Element of HFuncs NAT st g is_primitive-recursively_expressed_by f1,f2,i holds
g = primrec (f1,f2,i)
let g be Element of HFuncs NAT; ::_thesis: ( g is_primitive-recursively_expressed_by f1,f2,i implies g = primrec (f1,f2,i) )
set n = (arity f1) + 1;
set h = primrec (f1,f2,i);
given n9 being Element of NAT  such that A3: dom g c= n9 -tuples_on NAT and
 i >= 1 and
 i <= n9 and
A4: (arity f1) + 1 = n9 and
 n9 + 1 = arity f2 and
A5: for p being FinSequence of NAT st len p = n9 holds
( ( p +* (i,0) in dom g implies Del (p,i) in dom f1 ) & ( Del (p,i) in dom f1 implies p +* (i,0) in dom g ) & ( p +* (i,0) in dom g implies g . (p +* (i,0)) = f1 . (Del (p,i)) ) & ( for n being Element of NAT holds
( ( p +* (i,(n + 1)) in dom g implies ( p +* (i,n) in dom g & (p +* (i,n)) ^ <*(g . (p +* (i,n)))*> in dom f2 ) ) & ( p +* (i,n) in dom g & (p +* (i,n)) ^ <*(g . (p +* (i,n)))*> in dom f2 implies p +* (i,(n + 1)) in dom g ) & ( p +* (i,(n + 1)) in dom g implies g . (p +* (i,(n + 1))) = f2 . ((p +* (i,n)) ^ <*(g . (p +* (i,n)))*>) ) ) ) ) ; :: according to COMPUT_1:def_9 ::_thesis: g = primrec (f1,f2,i)
A6: now__::_thesis:_for_p_being_Element_of_((arity_f1)_+_1)_-tuples_on_NAT_holds_
(_(_p_in_dom_g_implies_p_in_dom_(primrec_(f1,f2,i))_)_&_(_p_in_dom_(primrec_(f1,f2,i))_implies_p_in_dom_g_)_&_(_p_in_dom_g_implies_g_._p_=_(primrec_(f1,f2,i))_._p_)_)
let p be Element of ((arity f1) + 1) -tuples_on NAT; ::_thesis: ( ( p in dom g implies p in dom (primrec (f1,f2,i)) ) & ( p in dom (primrec (f1,f2,i)) implies p in dom g ) & ( p in dom g implies g . p = (primrec (f1,f2,i)) . p ) )
defpred S2[ Element of NAT ] means ( ( p +* (i,$1) in dom g implies p +* (i,$1) in dom (primrec (f1,f2,i)) ) & ( p +* (i,$1) in dom (primrec (f1,f2,i)) implies p +* (i,$1) in dom g ) & ( p +* (i,$1) in dom g implies g . (p +* (i,$1)) = (primrec (f1,f2,i)) . (p +* (i,$1)) ) );
set k = p /. i;
A7: p = p +* (i,(p /. i)) by FUNCT_7:38;
A8: len p = (arity f1) + 1 by CARD_1:def_7;
then A9: i in dom p by A1, A2, FINSEQ_3:25;
A10: for m being Element of NAT st S2[m] holds
S2[m + 1]
proof
let m be Element of NAT ; ::_thesis: ( S2[m] implies S2[m + 1] )
assume that
A11: ( p +* (i,m) in dom g iff p +* (i,m) in dom (primrec (f1,f2,i)) ) and
A12: ( p +* (i,m) in dom g implies g . (p +* (i,m)) = (primrec (f1,f2,i)) . (p +* (i,m)) ) ; ::_thesis: S2[m + 1]
 ( p +* (i,(m + 1)) in dom g iff ( p +* (i,m) in dom g & (p +* (i,m)) ^ <*(g . (p +* (i,m)))*> in dom f2 ) ) by A4, A5, A8;
hence  ( p +* (i,(m + 1)) in dom g iff p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) ) by A9, A11, A12, Lm6; ::_thesis: ( p +* (i,(m + 1)) in dom g implies g . (p +* (i,(m + 1))) = (primrec (f1,f2,i)) . (p +* (i,(m + 1))) )
A13: ( p +* (i,(m + 1)) in dom (primrec (f1,f2,i)) iff ( p +* (i,m) in dom (primrec (f1,f2,i)) & (p +* (i,m)) ^ <*((primrec (f1,f2,i)) . (p +* (i,m)))*> in dom f2 ) ) by A9, Lm6;
assume A14: p +* (i,(m + 1)) in dom g ; ::_thesis: g . (p +* (i,(m + 1))) = (primrec (f1,f2,i)) . (p +* (i,(m + 1)))
then  g . (p +* (i,(m + 1))) = f2 . ((p +* (i,m)) ^ <*(g . (p +* (i,m)))*>) by A4, A5, A8;
hence  g . (p +* (i,(m + 1))) = (primrec (f1,f2,i)) . (p +* (i,(m + 1))) by A4, A5, A8, A9, A11, A12, A13, A14, Lm6; ::_thesis: verum
end;
A15: ( p +* (i,0) in dom (primrec (f1,f2,i)) iff Del (p,i) in dom f1 ) by A9, Lm6;
then  ( p +* (i,0) in dom g implies ( g . (p +* (i,0)) = f1 . (Del (p,i)) & (primrec (f1,f2,i)) . (p +* (i,0)) = f1 . (Del (p,i)) ) ) by A4, A5, A8, A9, Lm6;
then A16: S2[ 0 ] by A4, A5, A8, A15;
 for m being Element of NAT holds S2[m] from NAT_1:sch_1(A16, A10);
hence  ( ( p in dom g implies p in dom (primrec (f1,f2,i)) ) & ( p in dom (primrec (f1,f2,i)) implies p in dom g ) & ( p in dom g implies g . p = (primrec (f1,f2,i)) . p ) ) by A7; ::_thesis: verum
end;
A17: dom (primrec (f1,f2,i)) c= ((arity f1) + 1) -tuples_on NAT by Th56;
then A18: dom g = dom (primrec (f1,f2,i)) by A3, A4, A6, SUBSET_1:3;
 for x being set st x in dom (primrec (f1,f2,i)) holds
g . x = (primrec (f1,f2,i)) . x by A17, A6;
hence  g = primrec (f1,f2,i) by A18, FUNCT_1:2; ::_thesis: verum
end;

begin

definition
let X be set ;
attrX is composition_closed  means :Def12: :: COMPUT_1:def 12
 for f being Element of HFuncs NAT
for F being with_the_same_arity FinSequence of HFuncs NAT st f in X & arity f = len F & rng F c= X holds
f * <:F:> in X;
attrX is primitive-recursion_closed  means :Def13: :: COMPUT_1:def 13
 for g, f1, f2 being Element of HFuncs NAT
for i being Element of NAT st g is_primitive-recursively_expressed_by f1,f2,i & f1 in X & f2 in X holds
g in X;
end;

:: deftheorem Def12 defines composition_closed COMPUT_1:def_12_:_
 for X being set holds
( X is composition_closed iff for f being Element of HFuncs NAT
for F being with_the_same_arity FinSequence of HFuncs NAT st f in X & arity f = len F & rng F c= X holds
f * <:F:> in X );

:: deftheorem Def13 defines primitive-recursion_closed COMPUT_1:def_13_:_
 for X being set holds
( X is primitive-recursion_closed iff for g, f1, f2 being Element of HFuncs NAT
for i being Element of NAT st g is_primitive-recursively_expressed_by f1,f2,i & f1 in X & f2 in X holds
g in X );

definition
let X be set ;
attrX is primitive-recursively_closed  means :Def14: :: COMPUT_1:def 14
( 0 const 0 in X & 1 succ 1 in X & ( for n, i being Element of NAT st 1 <= i & i <= n holds
n proj i in X ) & X is composition_closed & X is primitive-recursion_closed );
end;

:: deftheorem Def14 defines primitive-recursively_closed COMPUT_1:def_14_:_
 for X being set holds
( X is primitive-recursively_closed iff ( 0 const 0 in X & 1 succ 1 in X & ( for n, i being Element of NAT st 1 <= i & i <= n holds
n proj i in X ) & X is composition_closed & X is primitive-recursion_closed ) );

theorem Th66: :: COMPUT_1:66
HFuncs NAT is primitive-recursively_closed
proof
set X = HFuncs NAT;
thus  ( 0 const 0 in HFuncs NAT & 1 succ 1 in HFuncs NAT & ( for n, i being Element of NAT st 1 <= i & i <= n holds
n proj i in HFuncs NAT ) ) by Th31, Th33, Th35; :: according to COMPUT_1:def_14 ::_thesis: ( HFuncs NAT is composition_closed & HFuncs NAT is primitive-recursion_closed )
thus  for f being Element of HFuncs NAT
for F being with_the_same_arity FinSequence of HFuncs NAT st f in HFuncs NAT & arity f = len F & rng F c= HFuncs NAT holds
f * <:F:> in HFuncs NAT by Th42; :: according to COMPUT_1:def_12 ::_thesis: HFuncs NAT is primitive-recursion_closed
let g be Element of HFuncs NAT; :: according to COMPUT_1:def_13 ::_thesis: for f1, f2 being Element of HFuncs NAT
for i being Element of NAT st g is_primitive-recursively_expressed_by f1,f2,i & f1 in HFuncs NAT & f2 in HFuncs NAT holds
g in HFuncs NAT
thus  for f1, f2 being Element of HFuncs NAT
for i being Element of NAT st g is_primitive-recursively_expressed_by f1,f2,i & f1 in HFuncs NAT & f2 in HFuncs NAT holds
g in HFuncs NAT ; ::_thesis: verum
end;

registration
cluster non empty functional primitive-recursively_closed for Element of K32((HFuncs NAT));
existence
 ex b1 being Subset of (HFuncs NAT) st
( b1 is primitive-recursively_closed & not b1 is empty )
proof
 HFuncs NAT c= HFuncs NAT ;
then reconsider X = HFuncs NAT as non empty Subset of (HFuncs NAT) ;
take  X ; ::_thesis: ( X is primitive-recursively_closed & not X is empty )
thus  ( X is primitive-recursively_closed & not X is empty ) by Th66; ::_thesis: verum
end;
end;

 Lm7:  for X being non empty set
for n, i being Element of NAT st 1 <= i & i <= n holds
for x being Element of X
for p being Element of n -tuples_on X holds p +* (i,x) in n -tuples_on X
;

theorem Th67: :: COMPUT_1:67
for i being Element of NAT
for e1, e2 being NAT * -defined to-naturals homogeneous Function
for g being Element of HFuncs NAT st e1 = {} & g is_primitive-recursively_expressed_by e1,e2,i holds
g = {}
proof
let i be Element of NAT ; ::_thesis: for e1, e2 being NAT * -defined to-naturals homogeneous Function
for g being Element of HFuncs NAT st e1 = {} & g is_primitive-recursively_expressed_by e1,e2,i holds
g = {}
let e1, e2 be NAT * -defined to-naturals homogeneous Function; ::_thesis: for g being Element of HFuncs NAT st e1 = {} & g is_primitive-recursively_expressed_by e1,e2,i holds
g = {}
set f1 = e1;
set f2 = e2;
let g be Element of HFuncs NAT; ::_thesis: ( e1 = {} & g is_primitive-recursively_expressed_by e1,e2,i implies g = {} )
assume A1: e1 = {} ; ::_thesis: ( not g is_primitive-recursively_expressed_by e1,e2,i or g = {} )
assume  g is_primitive-recursively_expressed_by e1,e2,i ; ::_thesis: g = {}
then consider n being Element of NAT  such that
A2: dom g c= n -tuples_on NAT and
 i >= 1 and
 i <= n and
 (arity e1) + 1 = n and
 n + 1 = arity e2 and
A3: for p being FinSequence of NAT st len p = n holds
( ( p +* (i,0) in dom g implies Del (p,i) in dom e1 ) & ( Del (p,i) in dom e1 implies p +* (i,0) in dom g ) & ( p +* (i,0) in dom g implies g . (p +* (i,0)) = e1 . (Del (p,i)) ) & ( for n being Element of NAT holds
( ( p +* (i,(n + 1)) in dom g implies ( p +* (i,n) in dom g & (p +* (i,n)) ^ <*(g . (p +* (i,n)))*> in dom e2 ) ) & ( p +* (i,n) in dom g & (p +* (i,n)) ^ <*(g . (p +* (i,n)))*> in dom e2 implies p +* (i,(n + 1)) in dom g ) & ( p +* (i,(n + 1)) in dom g implies g . (p +* (i,(n + 1))) = e2 . ((p +* (i,n)) ^ <*(g . (p +* (i,n)))*>) ) ) ) ) by Def9;
A4: now__::_thesis:_for_y_being_Element_of_n_-tuples_on_NAT
for_k_being_Element_of_NAT_holds_S2[k]
let y be Element of n -tuples_on NAT; ::_thesis: for k being Element of NAT holds S2[k]
defpred S2[ Element of NAT ] means  not y +* (i,$1) in dom g;
A5: len y = n by CARD_1:def_7;
then A6: for k being Element of NAT st S2[k] holds
S2[k + 1] by A3;
A7: S2[ 0 ] by A1, A3, A5;
thus  for k being Element of NAT holds S2[k] from NAT_1:sch_1(A7, A6); ::_thesis: verum
end;
assume  g <> {} ; ::_thesis: contradiction
then consider x being set  such that
A8: x in dom g by XBOOLE_0:def_1;
reconsider x = x as Element of n -tuples_on NAT by A2, A8;
set xi = x . i;
 x = x +* (i,(x . i)) by FUNCT_7:35;
hence  contradiction by A4, A8; ::_thesis: verum
end;

theorem Th68: :: COMPUT_1:68
for g being Element of HFuncs NAT
for f1, f2 being quasi_total Element of HFuncs NAT
for i being Element of NAT st g is_primitive-recursively_expressed_by f1,f2,i holds
( g is quasi_total & ( not f1 is empty implies not g is empty ) )
proof
let g be Element of HFuncs NAT; ::_thesis: for f1, f2 being quasi_total Element of HFuncs NAT
for i being Element of NAT st g is_primitive-recursively_expressed_by f1,f2,i holds
( g is quasi_total & ( not f1 is empty implies not g is empty ) )
let f1, f2 be quasi_total Element of HFuncs NAT; ::_thesis: for i being Element of NAT st g is_primitive-recursively_expressed_by f1,f2,i holds
( g is quasi_total & ( not f1 is empty implies not g is empty ) )
let i be Element of NAT ; ::_thesis: ( g is_primitive-recursively_expressed_by f1,f2,i implies ( g is quasi_total & ( not f1 is empty implies not g is empty ) ) )
assume A1: g is_primitive-recursively_expressed_by f1,f2,i ; ::_thesis: ( g is quasi_total & ( not f1 is empty implies not g is empty ) )
then consider n being Element of NAT  such that
A2: dom g c= n -tuples_on NAT and
A3: i >= 1 and
A4: i <= n and
A5: (arity f1) + 1 = n and
A6: n + 1 = arity f2 and
A7: for p being FinSequence of NAT st len p = n holds
( ( p +* (i,0) in dom g implies Del (p,i) in dom f1 ) & ( Del (p,i) in dom f1 implies p +* (i,0) in dom g ) & ( p +* (i,0) in dom g implies g . (p +* (i,0)) = f1 . (Del (p,i)) ) & ( for n being Element of NAT holds
( ( p +* (i,(n + 1)) in dom g implies ( p +* (i,n) in dom g & (p +* (i,n)) ^ <*(g . (p +* (i,n)))*> in dom f2 ) ) & ( p +* (i,n) in dom g & (p +* (i,n)) ^ <*(g . (p +* (i,n)))*> in dom f2 implies p +* (i,(n + 1)) in dom g ) & ( p +* (i,(n + 1)) in dom g implies g . (p +* (i,(n + 1))) = f2 . ((p +* (i,n)) ^ <*(g . (p +* (i,n)))*>) ) ) ) ) by Def9;
reconsider f29 = f2 as non empty quasi_total Element of HFuncs NAT by A6, Th18;
percases ( f1 is empty or not f1 is empty ) ;
suppose f1 is empty ; ::_thesis: ( g is quasi_total & ( not f1 is empty implies not g is empty ) )
hence  ( g is quasi_total & ( not f1 is empty implies not g is empty ) ) by A1, Th67; ::_thesis: verum
end;
suppose not f1 is empty ; ::_thesis: ( g is quasi_total & ( not f1 is empty implies not g is empty ) )
then A8: dom f1 = (arity f1) -tuples_on NAT by Th22;
A9: g is quasi_total
proof
let x, y be FinSequence of NAT ; :: according to MARGREL1:def_22 ::_thesis: ( not len x = len y or not x in dom g or y in dom g )
assume that
A10: len x = len y and
A11: x in dom g ; ::_thesis: y in dom g
defpred S2[ Element of NAT ] means y +* (i,$1) in dom g;
A12: len x = n by A2, A11, CARD_1:def_7;
A13: now__::_thesis:_for_k_being_Element_of_NAT_st_S2[k]_holds_
S2[k_+_1]
let k be Element of NAT ; ::_thesis: ( S2[k] implies S2[k + 1] )
assume A14: S2[k] ; ::_thesis: S2[k + 1]
A15: dom f29 = (arity f2) -tuples_on NAT by Th22;
len ((y +* (i,k)) ^ <*(g . (y +* (i,k)))*>) = (len (y +* (i,k))) + (len <*(g . (y +* (i,k)))*>) by FINSEQ_1:22
.= n + (len <*(g . (y +* (i,k)))*>) by A10, A12, FUNCT_7:97
.= n + 1 by FINSEQ_1:39 ;
then  (y +* (i,k)) ^ <*(g . (y +* (i,k)))*> is Element of dom f2 by A6, A15, FINSEQ_2:92;
then  (y +* (i,k)) ^ <*(g . (y +* (i,k)))*> in dom f29 ;
hence  S2[k + 1] by A7, A10, A12, A14; ::_thesis: verum
end;
 y is Element of (len y) -tuples_on NAT by FINSEQ_2:92;
then  Del (y,i) in (arity f1) -tuples_on NAT by A3, A4, A5, A10, A12, Th9;
then A16: S2[ 0 ] by A7, A8, A10, A12;
 for k being Element of NAT holds S2[k] from NAT_1:sch_1(A16, A13);
then A17: y +* (i,(y /. i)) in dom g ;
 i in dom y by A3, A4, A10, A12, FINSEQ_3:25;
then  y . i = y /. i by PARTFUN1:def_6;
hence  y in dom g by A17, FUNCT_7:35; ::_thesis: verum
end;
consider pp being set  such that
A18: pp in n -tuples_on NAT by XBOOLE_0:def_1;
 pp is Element of n -tuples_on NAT by A18;
then reconsider p = pp as FinSequence of NAT ;
A19: len p = n by A18, CARD_1:def_7;
 Del (p,i) in (arity f1) -tuples_on NAT by A3, A4, A5, A18, Th9;
hence  ( g is quasi_total & ( not f1 is empty implies not g is empty ) ) by A7, A19, A8, A9; ::_thesis: verum
end;
end;
end;

theorem Th69: :: COMPUT_1:69
for n, c being Element of NAT
for P being non empty primitive-recursively_closed Subset of (HFuncs NAT) holds n const c in P
proof
let n, c be Element of NAT ; ::_thesis: for P being non empty primitive-recursively_closed Subset of (HFuncs NAT) holds n const c in P
let P be non empty primitive-recursively_closed Subset of (HFuncs NAT); ::_thesis: n const c in P
defpred S2[ Element of NAT ] means 0 const $1 in P;
defpred S3[ Element of NAT ] means  for c being Element of NAT holds $1 const c in P;
A1: P is composition_closed by Def14;
A2: for i being Element of NAT st S2[i] holds
S2[i + 1]
proof
reconsider 1succ1 = 1 succ 1 as quasi_total Element of HFuncs NAT by Th29;
let i be Element of NAT ; ::_thesis: ( S2[i] implies S2[i + 1] )
A3: 1 succ 1 in P by Def14;
A4: <*> NAT is Element of 0 -tuples_on NAT by FINSEQ_2:131;
then A5: (0 const i) . {} = i by FUNCOP_1:7;
reconsider 0consti = 0 const i as Element of HFuncs NAT by Th28;
set F = <*(0 const i)*>;
 <*0consti*> is FinSequence of HFuncs NAT ;
then reconsider F = <*(0 const i)*> as with_the_same_arity FinSequence of HFuncs NAT ;
assume  0 const i in P ; ::_thesis: S2[i + 1]
then A6: {(0 const i)} c= P by ZFMISC_1:31;
A7: arity (1 succ 1) = 1 by Th34
.= len F by FINSEQ_1:39 ;
A8: rng F = {(0 const i)} by FINSEQ_1:39;
now__::_thesis:_for_h_being_Element_of_HFuncs_NAT_st_h_in_rng_F_holds_
h_is_quasi_total
let h be Element of HFuncs NAT; ::_thesis: ( h in rng F implies h is quasi_total )
assume  h in rng F ; ::_thesis: h is quasi_total
then  h = 0 const i by A8, TARSKI:def_1;
hence  h is quasi_total by Th29; ::_thesis: verum
end;
then reconsider g = 1succ1 * <:F:> as quasi_total Element of HFuncs NAT by A7, Th46;
A9: arity (0 const (i + 1)) = 0 by Th32;
A10: (0 const (i + 1)) . {} = i + 1 by A4, FUNCOP_1:7;
A11: dom (1 succ 1) = 1 -tuples_on NAT by Def7;
A12: arity (0 const i) = 0 by Th32;
then A13: dom (0 const i) = 0 -tuples_on NAT by Th23;
then A14: <:F:> . {} = <*i*> by A4, A5, FINSEQ_3:141;
reconsider ii = <*i*> as Element of 1 -tuples_on NAT by FINSEQ_2:131;
A15: dom <:<*(0 const i)*>:> = dom (0 const i) by FINSEQ_3:141;
then  g . {} = (1 succ 1) . (<:F:> . {}) by A4, A13, FUNCT_1:13;
then A16: g . {} = (ii /. 1) + 1 by A14, Def7
.= i + 1 by FINSEQ_4:16 ;
A17: dom (0 const i) = 0 -tuples_on NAT by A12, Th23;
then  <:<*(0 const i)*>:> . {} = ii by A4, A5, FINSEQ_3:141;
then A18: {} in dom g by A4, A15, A17, A11, FUNCT_1:11;
 0 const i in rng F by A8, TARSKI:def_1;
then  arity F = 0 by A12, Def4;
then  arity g = 0 by A18, Th44, RELAT_1:38;
then  0 const (i + 1) = (1 succ 1) * <:<*(0 const i)*>:> by A9, A10, A18, A16, Th48, RELAT_1:38;
hence  S2[i + 1] by A1, A6, A8, A3, A7, Def12; ::_thesis: verum
end;
A19: P is primitive-recursion_closed by Def14;
A20: now__::_thesis:_for_n_being_Element_of_NAT_st_S3[n]_holds_
S3[n_+_1]
let n be Element of NAT ; ::_thesis: ( S3[n] implies S3[n + 1] )
assume A21: S3[n] ; ::_thesis: S3[n + 1]
thus  S3[n + 1] ::_thesis: verum
proof
let i be Element of NAT ; ::_thesis: (n + 1) const i in P
set g = (n + 1) const i;
set f1 = n const i;
set j = n + 1;
set f2 = (n + 2) proj (n + 2);
A22: dom ((n + 1) const i) = (n + 1) -tuples_on NAT by FUNCOP_1:13;
A23: n + (1 + 1) = (n + 1) + 1 ;
then  1 <= n + 2 by NAT_1:11;
then A24: (n + 2) proj (n + 2) in P by Def14;
A25: arity ((n + 2) proj (n + 2)) = n + 2 by Th37;
A26: dom ((n + 2) proj (n + 2)) = (n + 2) -tuples_on NAT by Th36;
A27: arity (n const i) = n by Th32;
A28: dom (n const i) = n -tuples_on NAT by FUNCOP_1:13;
A29: arity ((n + 1) const i) = n + 1 by Th32;
A30: (n + 1) const i is_primitive-recursively_expressed_by n const i,(n + 2) proj (n + 2),n + 1
proof
take m = arity ((n + 1) const i); :: according to COMPUT_1:def_9 ::_thesis: ( dom ((n + 1) const i) c= m -tuples_on NAT & n + 1 >= 1 & n + 1 <= m & (arity (n const i)) + 1 = m & m + 1 = arity ((n + 2) proj (n + 2)) & ( for p being FinSequence of NAT st len p = m holds
( ( p +* ((n + 1),0) in dom ((n + 1) const i) implies Del (p,(n + 1)) in dom (n const i) ) & ( Del (p,(n + 1)) in dom (n const i) implies p +* ((n + 1),0) in dom ((n + 1) const i) ) & ( p +* ((n + 1),0) in dom ((n + 1) const i) implies ((n + 1) const i) . (p +* ((n + 1),0)) = (n const i) . (Del (p,(n + 1))) ) & ( for n being Element of NAT holds
( ( p +* ((n + 1),(n + 1)) in dom ((n + 1) const i) implies ( p +* ((n + 1),n) in dom ((n + 1) const i) & (p +* ((n + 1),n)) ^ <*(((n + 1) const i) . (p +* ((n + 1),n)))*> in dom ((n + 2) proj (n + 2)) ) ) & ( p +* ((n + 1),n) in dom ((n + 1) const i) & (p +* ((n + 1),n)) ^ <*(((n + 1) const i) . (p +* ((n + 1),n)))*> in dom ((n + 2) proj (n + 2)) implies p +* ((n + 1),(n + 1)) in dom ((n + 1) const i) ) & ( p +* ((n + 1),(n + 1)) in dom ((n + 1) const i) implies ((n + 1) const i) . (p +* ((n + 1),(n + 1))) = ((n + 2) proj (n + 2)) . ((p +* ((n + 1),n)) ^ <*(((n + 1) const i) . (p +* ((n + 1),n)))*>) ) ) ) ) ) )
thus  dom ((n + 1) const i) c= m -tuples_on NAT by Th21; ::_thesis: ( n + 1 >= 1 & n + 1 <= m & (arity (n const i)) + 1 = m & m + 1 = arity ((n + 2) proj (n + 2)) & ( for p being FinSequence of NAT st len p = m holds
( ( p +* ((n + 1),0) in dom ((n + 1) const i) implies Del (p,(n + 1)) in dom (n const i) ) & ( Del (p,(n + 1)) in dom (n const i) implies p +* ((n + 1),0) in dom ((n + 1) const i) ) & ( p +* ((n + 1),0) in dom ((n + 1) const i) implies ((n + 1) const i) . (p +* ((n + 1),0)) = (n const i) . (Del (p,(n + 1))) ) & ( for n being Element of NAT holds
( ( p +* ((n + 1),(n + 1)) in dom ((n + 1) const i) implies ( p +* ((n + 1),n) in dom ((n + 1) const i) & (p +* ((n + 1),n)) ^ <*(((n + 1) const i) . (p +* ((n + 1),n)))*> in dom ((n + 2) proj (n + 2)) ) ) & ( p +* ((n + 1),n) in dom ((n + 1) const i) & (p +* ((n + 1),n)) ^ <*(((n + 1) const i) . (p +* ((n + 1),n)))*> in dom ((n + 2) proj (n + 2)) implies p +* ((n + 1),(n + 1)) in dom ((n + 1) const i) ) & ( p +* ((n + 1),(n + 1)) in dom ((n + 1) const i) implies ((n + 1) const i) . (p +* ((n + 1),(n + 1))) = ((n + 2) proj (n + 2)) . ((p +* ((n + 1),n)) ^ <*(((n + 1) const i) . (p +* ((n + 1),n)))*>) ) ) ) ) ) )
thus  ( n + 1 >= 1 & n + 1 <= m ) by Th32, NAT_1:11; ::_thesis: ( (arity (n const i)) + 1 = m & m + 1 = arity ((n + 2) proj (n + 2)) & ( for p being FinSequence of NAT st len p = m holds
( ( p +* ((n + 1),0) in dom ((n + 1) const i) implies Del (p,(n + 1)) in dom (n const i) ) & ( Del (p,(n + 1)) in dom (n const i) implies p +* ((n + 1),0) in dom ((n + 1) const i) ) & ( p +* ((n + 1),0) in dom ((n + 1) const i) implies ((n + 1) const i) . (p +* ((n + 1),0)) = (n const i) . (Del (p,(n + 1))) ) & ( for n being Element of NAT holds
( ( p +* ((n + 1),(n + 1)) in dom ((n + 1) const i) implies ( p +* ((n + 1),n) in dom ((n + 1) const i) & (p +* ((n + 1),n)) ^ <*(((n + 1) const i) . (p +* ((n + 1),n)))*> in dom ((n + 2) proj (n + 2)) ) ) & ( p +* ((n + 1),n) in dom ((n + 1) const i) & (p +* ((n + 1),n)) ^ <*(((n + 1) const i) . (p +* ((n + 1),n)))*> in dom ((n + 2) proj (n + 2)) implies p +* ((n + 1),(n + 1)) in dom ((n + 1) const i) ) & ( p +* ((n + 1),(n + 1)) in dom ((n + 1) const i) implies ((n + 1) const i) . (p +* ((n + 1),(n + 1))) = ((n + 2) proj (n + 2)) . ((p +* ((n + 1),n)) ^ <*(((n + 1) const i) . (p +* ((n + 1),n)))*>) ) ) ) ) ) )
thus  ( (arity (n const i)) + 1 = m & m + 1 = arity ((n + 2) proj (n + 2)) ) by A27, A25, A23, Th32; ::_thesis: for p being FinSequence of NAT st len p = m holds
( ( p +* ((n + 1),0) in dom ((n + 1) const i) implies Del (p,(n + 1)) in dom (n const i) ) & ( Del (p,(n + 1)) in dom (n const i) implies p +* ((n + 1),0) in dom ((n + 1) const i) ) & ( p +* ((n + 1),0) in dom ((n + 1) const i) implies ((n + 1) const i) . (p +* ((n + 1),0)) = (n const i) . (Del (p,(n + 1))) ) & ( for n being Element of NAT holds
( ( p +* ((n + 1),(n + 1)) in dom ((n + 1) const i) implies ( p +* ((n + 1),n) in dom ((n + 1) const i) & (p +* ((n + 1),n)) ^ <*(((n + 1) const i) . (p +* ((n + 1),n)))*> in dom ((n + 2) proj (n + 2)) ) ) & ( p +* ((n + 1),n) in dom ((n + 1) const i) & (p +* ((n + 1),n)) ^ <*(((n + 1) const i) . (p +* ((n + 1),n)))*> in dom ((n + 2) proj (n + 2)) implies p +* ((n + 1),(n + 1)) in dom ((n + 1) const i) ) & ( p +* ((n + 1),(n + 1)) in dom ((n + 1) const i) implies ((n + 1) const i) . (p +* ((n + 1),(n + 1))) = ((n + 2) proj (n + 2)) . ((p +* ((n + 1),n)) ^ <*(((n + 1) const i) . (p +* ((n + 1),n)))*>) ) ) ) )
let p be FinSequence of NAT ; ::_thesis: ( len p = m implies ( ( p +* ((n + 1),0) in dom ((n + 1) const i) implies Del (p,(n + 1)) in dom (n const i) ) & ( Del (p,(n + 1)) in dom (n const i) implies p +* ((n + 1),0) in dom ((n + 1) const i) ) & ( p +* ((n + 1),0) in dom ((n + 1) const i) implies ((n + 1) const i) . (p +* ((n + 1),0)) = (n const i) . (Del (p,(n + 1))) ) & ( for n being Element of NAT holds
( ( p +* ((n + 1),(n + 1)) in dom ((n + 1) const i) implies ( p +* ((n + 1),n) in dom ((n + 1) const i) & (p +* ((n + 1),n)) ^ <*(((n + 1) const i) . (p +* ((n + 1),n)))*> in dom ((n + 2) proj (n + 2)) ) ) & ( p +* ((n + 1),n) in dom ((n + 1) const i) & (p +* ((n + 1),n)) ^ <*(((n + 1) const i) . (p +* ((n + 1),n)))*> in dom ((n + 2) proj (n + 2)) implies p +* ((n + 1),(n + 1)) in dom ((n + 1) const i) ) & ( p +* ((n + 1),(n + 1)) in dom ((n + 1) const i) implies ((n + 1) const i) . (p +* ((n + 1),(n + 1))) = ((n + 2) proj (n + 2)) . ((p +* ((n + 1),n)) ^ <*(((n + 1) const i) . (p +* ((n + 1),n)))*>) ) ) ) ) )
assume  len p = m ; ::_thesis: ( ( p +* ((n + 1),0) in dom ((n + 1) const i) implies Del (p,(n + 1)) in dom (n const i) ) & ( Del (p,(n + 1)) in dom (n const i) implies p +* ((n + 1),0) in dom ((n + 1) const i) ) & ( p +* ((n + 1),0) in dom ((n + 1) const i) implies ((n + 1) const i) . (p +* ((n + 1),0)) = (n const i) . (Del (p,(n + 1))) ) & ( for n being Element of NAT holds
( ( p +* ((n + 1),(n + 1)) in dom ((n + 1) const i) implies ( p +* ((n + 1),n) in dom ((n + 1) const i) & (p +* ((n + 1),n)) ^ <*(((n + 1) const i) . (p +* ((n + 1),n)))*> in dom ((n + 2) proj (n + 2)) ) ) & ( p +* ((n + 1),n) in dom ((n + 1) const i) & (p +* ((n + 1),n)) ^ <*(((n + 1) const i) . (p +* ((n + 1),n)))*> in dom ((n + 2) proj (n + 2)) implies p +* ((n + 1),(n + 1)) in dom ((n + 1) const i) ) & ( p +* ((n + 1),(n + 1)) in dom ((n + 1) const i) implies ((n + 1) const i) . (p +* ((n + 1),(n + 1))) = ((n + 2) proj (n + 2)) . ((p +* ((n + 1),n)) ^ <*(((n + 1) const i) . (p +* ((n + 1),n)))*>) ) ) ) )
then A31: p is Element of (n + 1) -tuples_on NAT by A29, FINSEQ_2:92;
A32: n + 1 >= 1 by NAT_1:11;
hence  ( p +* ((n + 1),0) in dom ((n + 1) const i) implies Del (p,(n + 1)) in dom (n const i) ) by A28, A31, Th9; ::_thesis: ( ( Del (p,(n + 1)) in dom (n const i) implies p +* ((n + 1),0) in dom ((n + 1) const i) ) & ( p +* ((n + 1),0) in dom ((n + 1) const i) implies ((n + 1) const i) . (p +* ((n + 1),0)) = (n const i) . (Del (p,(n + 1))) ) & ( for n being Element of NAT holds
( ( p +* ((n + 1),(n + 1)) in dom ((n + 1) const i) implies ( p +* ((n + 1),n) in dom ((n + 1) const i) & (p +* ((n + 1),n)) ^ <*(((n + 1) const i) . (p +* ((n + 1),n)))*> in dom ((n + 2) proj (n + 2)) ) ) & ( p +* ((n + 1),n) in dom ((n + 1) const i) & (p +* ((n + 1),n)) ^ <*(((n + 1) const i) . (p +* ((n + 1),n)))*> in dom ((n + 2) proj (n + 2)) implies p +* ((n + 1),(n + 1)) in dom ((n + 1) const i) ) & ( p +* ((n + 1),(n + 1)) in dom ((n + 1) const i) implies ((n + 1) const i) . (p +* ((n + 1),(n + 1))) = ((n + 2) proj (n + 2)) . ((p +* ((n + 1),n)) ^ <*(((n + 1) const i) . (p +* ((n + 1),n)))*>) ) ) ) )
thus  ( Del (p,(n + 1)) in dom (n const i) implies p +* ((n + 1),0) in dom ((n + 1) const i) ) by A22, A31, A32, Lm7; ::_thesis: ( p +* ((n + 1),0) in dom ((n + 1) const i) iff ((n + 1) const i) . (p +* ((n + 1),0)) = (n const i) . (Del (p,(n + 1))) )
 (n const i) . (Del (p,(n + 1))) = i by A31, A32, Th9, FUNCOP_1:7;
hence  ( p +* ((n + 1),0) in dom ((n + 1) const i) implies ((n + 1) const i) . (p +* ((n + 1),0)) = (n const i) . (Del (p,(n + 1))) ) by A31, A32, Lm7, FUNCOP_1:7; ::_thesis: for n being Element of NAT holds
( ( p +* ((n + 1),(n + 1)) in dom ((n + 1) const i) implies ( p +* ((n + 1),n) in dom ((n + 1) const i) & (p +* ((n + 1),n)) ^ <*(((n + 1) const i) . (p +* ((n + 1),n)))*> in dom ((n + 2) proj (n + 2)) ) ) & ( p +* ((n + 1),n) in dom ((n + 1) const i) & (p +* ((n + 1),n)) ^ <*(((n + 1) const i) . (p +* ((n + 1),n)))*> in dom ((n + 2) proj (n + 2)) implies p +* ((n + 1),(n + 1)) in dom ((n + 1) const i) ) & ( p +* ((n + 1),(n + 1)) in dom ((n + 1) const i) implies ((n + 1) const i) . (p +* ((n + 1),(n + 1))) = ((n + 2) proj (n + 2)) . ((p +* ((n + 1),n)) ^ <*(((n + 1) const i) . (p +* ((n + 1),n)))*>) ) )
let m be Element of NAT ; ::_thesis: ( ( p +* ((n + 1),(m + 1)) in dom ((n + 1) const i) implies ( p +* ((n + 1),m) in dom ((n + 1) const i) & (p +* ((n + 1),m)) ^ <*(((n + 1) const i) . (p +* ((n + 1),m)))*> in dom ((n + 2) proj (n + 2)) ) ) & ( p +* ((n + 1),m) in dom ((n + 1) const i) & (p +* ((n + 1),m)) ^ <*(((n + 1) const i) . (p +* ((n + 1),m)))*> in dom ((n + 2) proj (n + 2)) implies p +* ((n + 1),(m + 1)) in dom ((n + 1) const i) ) & ( p +* ((n + 1),(m + 1)) in dom ((n + 1) const i) implies ((n + 1) const i) . (p +* ((n + 1),(m + 1))) = ((n + 2) proj (n + 2)) . ((p +* ((n + 1),m)) ^ <*(((n + 1) const i) . (p +* ((n + 1),m)))*>) ) )
A33: p +* ((n + 1),m) in (n + 1) -tuples_on NAT by A31, A32, Lm7;
then A34: (p +* ((n + 1),m)) ^ <*i*> is Tuple of n + 2, NAT by A23, FINSEQ_2:107;
then A35: (p +* ((n + 1),m)) ^ <*i*> is Element of (n + 2) -tuples_on NAT by FINSEQ_2:131;
hereby  ::_thesis: ( p +* ((n + 1),(m + 1)) in dom ((n + 1) const i) implies ((n + 1) const i) . (p +* ((n + 1),(m + 1))) = ((n + 2) proj (n + 2)) . ((p +* ((n + 1),m)) ^ <*(((n + 1) const i) . (p +* ((n + 1),m)))*>) )
hereby  ::_thesis: ( p +* ((n + 1),m) in dom ((n + 1) const i) & (p +* ((n + 1),m)) ^ <*(((n + 1) const i) . (p +* ((n + 1),m)))*> in dom ((n + 2) proj (n + 2)) implies p +* ((n + 1),(m + 1)) in dom ((n + 1) const i) )
assume  p +* ((n + 1),(m + 1)) in dom ((n + 1) const i) ; ::_thesis: ( p +* ((n + 1),m) in dom ((n + 1) const i) & (p +* ((n + 1),m)) ^ <*(((n + 1) const i) . (p +* ((n + 1),m)))*> in dom ((n + 2) proj (n + 2)) )
thus  p +* ((n + 1),m) in dom ((n + 1) const i) by A22, A31, A32, Lm7; ::_thesis: (p +* ((n + 1),m)) ^ <*(((n + 1) const i) . (p +* ((n + 1),m)))*> in dom ((n + 2) proj (n + 2))
 ((n + 1) const i) . (p +* ((n + 1),m)) = i by A31, A32, Lm7, FUNCOP_1:7;
hence  (p +* ((n + 1),m)) ^ <*(((n + 1) const i) . (p +* ((n + 1),m)))*> in dom ((n + 2) proj (n + 2)) by A26, A34, FINSEQ_2:131; ::_thesis: verum
end;
thus  ( p +* ((n + 1),m) in dom ((n + 1) const i) & (p +* ((n + 1),m)) ^ <*(((n + 1) const i) . (p +* ((n + 1),m)))*> in dom ((n + 2) proj (n + 2)) implies p +* ((n + 1),(m + 1)) in dom ((n + 1) const i) ) by A22, A31, A32, Lm7; ::_thesis: verum
end;
assume  p +* ((n + 1),(m + 1)) in dom ((n + 1) const i) ; ::_thesis: ((n + 1) const i) . (p +* ((n + 1),(m + 1))) = ((n + 2) proj (n + 2)) . ((p +* ((n + 1),m)) ^ <*(((n + 1) const i) . (p +* ((n + 1),m)))*>)
 len (p +* ((n + 1),m)) = n + 1 by A33, CARD_1:def_7;
then A36: ((p +* ((n + 1),m)) ^ <*i*>) . ((n + 1) + 1) = i by FINSEQ_1:42;
thus ((n + 1) const i) . (p +* ((n + 1),(m + 1))) = i by A31, A32, Lm7, FUNCOP_1:7
.= ((n + 2) proj (n + 2)) . ((p +* ((n + 1),m)) ^ <*i*>) by A36, A35, Th38
.= ((n + 2) proj (n + 2)) . ((p +* ((n + 1),m)) ^ <*(((n + 1) const i) . (p +* ((n + 1),m)))*>) by A31, A32, Lm7, FUNCOP_1:7 ; ::_thesis: verum
end;
A37: n const i in P by A21;
 (n + 1) const i is Element of HFuncs NAT by Th28;
hence  (n + 1) const i in P by A19, A30, A37, A24, Def13; ::_thesis: verum
end;
end;
A38: S2[ 0 ] by Def14;
 for i being Element of NAT holds S2[i] from NAT_1:sch_1(A38, A2);
then A39: S3[ 0 ] ;
 for n being Element of NAT holds S3[n] from NAT_1:sch_1(A39, A20);
hence  n const c in P ; ::_thesis: verum
end;

theorem Th70: :: COMPUT_1:70
for i, n being Element of NAT
for P being non empty primitive-recursively_closed Subset of (HFuncs NAT) st 1 <= i & i <= n holds
n succ i in P
proof
let i, n be Element of NAT ; ::_thesis: for P being non empty primitive-recursively_closed Subset of (HFuncs NAT) st 1 <= i & i <= n holds
n succ i in P
let P be non empty primitive-recursively_closed Subset of (HFuncs NAT); ::_thesis: ( 1 <= i & i <= n implies n succ i in P )
A1: 1 succ 1 in P by Def14;
A2: arity (1 succ 1) = 1 by Th34
.= len <*(n proj i)*> by FINSEQ_1:39 ;
reconsider nproji = n proj i as Element of HFuncs NAT by Th28;
assume that
A3: 1 <= i and
A4: i <= n ; ::_thesis: n succ i in P
A5: <*nproji*> is with_the_same_arity FinSequence of HFuncs NAT ;
now__::_thesis:_(_dom_(n_succ_i)_=_n_-tuples_on_NAT_&_dom_((1_succ_1)_*_<:<*(n_proj_i)*>:>)_=_n_-tuples_on_NAT_&_(_for_x_being_set_st_x_in_n_-tuples_on_NAT_holds_
(n_succ_i)_._x_=_((1_succ_1)_*_<:<*(n_proj_i)*>:>)_._x_)_)
 rng (n proj i) = NAT by A3, A4, Th36;
then A6: rng <:<*(n proj i)*>:> = 1 -tuples_on NAT by Th8;
thus  dom (n succ i) = n -tuples_on NAT by Def7; ::_thesis: ( dom ((1 succ 1) * <:<*(n proj i)*>:>) = n -tuples_on NAT & ( for x being set st x in n -tuples_on NAT holds
(n succ i) . x = ((1 succ 1) * <:<*(n proj i)*>:>) . x ) )
A7: dom (n proj i) = n -tuples_on NAT by Th36;
then A8: dom <:<*(n proj i)*>:> = n -tuples_on NAT by FINSEQ_3:141;
 dom (1 succ 1) = 1 -tuples_on NAT by Def7;
hence  dom ((1 succ 1) * <:<*(n proj i)*>:>) = n -tuples_on NAT by A8, A6, RELAT_1:27; ::_thesis: for x being set st x in n -tuples_on NAT holds
(n succ i) . x = ((1 succ 1) * <:<*(n proj i)*>:>) . x
let x be set ; ::_thesis: ( x in n -tuples_on NAT implies (n succ i) . x = ((1 succ 1) * <:<*(n proj i)*>:>) . x )
assume  x in n -tuples_on NAT ; ::_thesis: (n succ i) . x = ((1 succ 1) * <:<*(n proj i)*>:>) . x
then reconsider x9 = x as Element of n -tuples_on NAT ;
set xi = x9 . i;
 len x9 = n by CARD_1:def_7;
then A9: i in dom x9 by A3, A4, FINSEQ_3:25;
A10: (n succ i) . x = (x9 /. i) + 1 by Def7
.= (x9 . i) + 1 by A9, PARTFUN1:def_6 ;
reconsider ii = <*(x9 . i)*> as Element of 1 -tuples_on NAT by FINSEQ_2:131;
((1 succ 1) * <:<*(n proj i)*>:>) . x9 = (1 succ 1) . (<:<*(n proj i)*>:> . x9) by A8, FUNCT_1:13
.= (1 succ 1) . <*((n proj i) . x9)*> by A7, FINSEQ_3:141
.= (1 succ 1) . <*(x9 . i)*> by Th38
.= (ii /. 1) + 1 by Def7
.= (x9 . i) + 1 by FINSEQ_4:16 ;
hence  (n succ i) . x = ((1 succ 1) * <:<*(n proj i)*>:>) . x by A10; ::_thesis: verum
end;
then A11: n succ i = (1 succ 1) * <:<*(n proj i)*>:> by FUNCT_1:2;
A12: rng <*(n proj i)*> c= P
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng <*(n proj i)*> or x in P )
assume  x in rng <*(n proj i)*> ; ::_thesis: x in P
then  x in {(n proj i)} by FINSEQ_1:39;
then  x = n proj i by TARSKI:def_1;
hence  x in P by A3, A4, Def14; ::_thesis: verum
end;
 P is composition_closed by Def14;
hence  n succ i in P by A11, A1, A2, A12, A5, Def12; ::_thesis: verum
end;

theorem Th71: :: COMPUT_1:71
for P being non empty primitive-recursively_closed Subset of (HFuncs NAT) holds {} in P
proof
let P be non empty primitive-recursively_closed Subset of (HFuncs NAT); ::_thesis: {} in P
reconsider F = {} as with_the_same_arity Element of (HFuncs NAT) * by FINSEQ_1:49;
set f = 0 const 0;
A1: rng {} c= P by XBOOLE_1:2;
A2: arity (0 const 0) = 0 by Th32;
A3: P is composition_closed by Def14;
 0 const 0 in P by Th69;
then  (0 const 0) * <:F:> in P by A2, A1, A3, Def12, CARD_1:27;
hence  {} in P by FUNCT_6:40; ::_thesis: verum
end;

theorem Th72: :: COMPUT_1:72
for P being non empty primitive-recursively_closed Subset of (HFuncs NAT)
for f being Element of P
for F being with_the_same_arity FinSequence of P st arity f = len F holds
f * <:F:> in P
proof
let P be non empty primitive-recursively_closed Subset of (HFuncs NAT); ::_thesis: for f being Element of P
for F being with_the_same_arity FinSequence of P st arity f = len F holds
f * <:F:> in P
let f be Element of P; ::_thesis: for F being with_the_same_arity FinSequence of P st arity f = len F holds
f * <:F:> in P
let F be with_the_same_arity FinSequence of P; ::_thesis: ( arity f = len F implies f * <:F:> in P )
assume A1: arity f = len F ; ::_thesis: f * <:F:> in P
A2: rng F c= P by FINSEQ_1:def_4;
percases ( f is empty or not f is empty ) ;
suppose f is empty ; ::_thesis: f * <:F:> in P
then  f * <:F:> = {} ;
hence  f * <:F:> in P by Th71; ::_thesis: verum
end;
suppose not f is empty ; ::_thesis: f * <:F:> in P
then reconsider f9 = f as non empty Element of HFuncs NAT ;
A3: P * c= (HFuncs NAT) * by FINSEQ_1:62;
 F in P * by FINSEQ_1:def_11;
then reconsider F9 = F as with_the_same_arity Element of (HFuncs NAT) * by A3;
 P is composition_closed by Def14;
then  f9 * <:F9:> in P by A1, A2, Def12;
hence  f * <:F:> in P ; ::_thesis: verum
end;
end;
end;

theorem Th73: :: COMPUT_1:73
for P being non empty primitive-recursively_closed Subset of (HFuncs NAT)
for f1, f2 being Element of P st (arity f1) + 2 = arity f2 holds
for i being Element of NAT st 1 <= i & i <= (arity f1) + 1 holds
primrec (f1,f2,i) in P
proof
let P be non empty primitive-recursively_closed Subset of (HFuncs NAT); ::_thesis: for f1, f2 being Element of P st (arity f1) + 2 = arity f2 holds
for i being Element of NAT st 1 <= i & i <= (arity f1) + 1 holds
primrec (f1,f2,i) in P
let f1, f2 be Element of P; ::_thesis: ( (arity f1) + 2 = arity f2 implies for i being Element of NAT st 1 <= i & i <= (arity f1) + 1 holds
primrec (f1,f2,i) in P )
assume A1: (arity f1) + 2 = arity f2 ; ::_thesis: for i being Element of NAT st 1 <= i & i <= (arity f1) + 1 holds
primrec (f1,f2,i) in P
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= (arity f1) + 1 implies primrec (f1,f2,i) in P )
assume that
A2: 1 <= i and
A3: i <= (arity f1) + 1 ; ::_thesis: primrec (f1,f2,i) in P
A4: P is primitive-recursion_closed by Def14;
percases ( f1 is empty or not f1 is empty ) ;
suppose f1 is empty ; ::_thesis: primrec (f1,f2,i) in P
then  primrec (f1,f2,i) is empty by Th55;
hence  primrec (f1,f2,i) in P by Th71; ::_thesis: verum
end;
suppose not f1 is empty ; ::_thesis: primrec (f1,f2,i) in P
then  primrec (f1,f2,i) is_primitive-recursively_expressed_by f1,f2,i by A1, A2, A3, Th18, Th64;
hence  primrec (f1,f2,i) in P by A4, Def13; ::_thesis: verum
end;
end;
end;

definition
func PrimRec  ->  Subset of (HFuncs NAT) equals :: COMPUT_1:def 15
 meet  {  R where R is Subset of (HFuncs NAT) : R is primitive-recursively_closed  }  ;
coherence
 meet  {  R where R is Subset of (HFuncs NAT) : R is primitive-recursively_closed  }  is Subset of (HFuncs NAT)
proof
set S =  {  R where R is Subset of (HFuncs NAT) : R is primitive-recursively_closed  }  ;
set T = meet  {  R where R is Subset of (HFuncs NAT) : R is primitive-recursively_closed  }  ;
A1: {(HFuncs NAT)} c= bool (HFuncs NAT) by ZFMISC_1:68;
 HFuncs NAT in {(HFuncs NAT)} by TARSKI:def_1;
then A2: HFuncs NAT in  {  R where R is Subset of (HFuncs NAT) : R is primitive-recursively_closed  }  by A1, Th66;
 meet  {  R where R is Subset of (HFuncs NAT) : R is primitive-recursively_closed  }  c= HFuncs NAT
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in meet  {  R where R is Subset of (HFuncs NAT) : R is primitive-recursively_closed  }  or x in HFuncs NAT )
assume  x in meet  {  R where R is Subset of (HFuncs NAT) : R is primitive-recursively_closed  }  ; ::_thesis: x in HFuncs NAT
hence  x in HFuncs NAT by A2, SETFAM_1:def_1; ::_thesis: verum
end;
hence  meet  {  R where R is Subset of (HFuncs NAT) : R is primitive-recursively_closed  }  is Subset of (HFuncs NAT) ; ::_thesis: verum
end;
end;

:: deftheorem  defines PrimRec COMPUT_1:def_15_:_
 PrimRec = meet  {  R where R is Subset of (HFuncs NAT) : R is primitive-recursively_closed  }  ;

theorem Th74: :: COMPUT_1:74
for X being Subset of (HFuncs NAT) st X is primitive-recursively_closed holds
PrimRec c= X
proof
let X be Subset of (HFuncs NAT); ::_thesis: ( X is primitive-recursively_closed implies PrimRec c= X )
set S =  {  R where R is Subset of (HFuncs NAT) : R is primitive-recursively_closed  }  ;
assume  X is primitive-recursively_closed ; ::_thesis: PrimRec c= X
then A1: X in  {  R where R is Subset of (HFuncs NAT) : R is primitive-recursively_closed  }  ;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in PrimRec or x in X )
assume  x in PrimRec ; ::_thesis: x in X
hence  x in X by A1, SETFAM_1:def_1; ::_thesis: verum
end;

registration
cluster PrimRec  ->  non empty primitive-recursively_closed ;
coherence
 ( not PrimRec is empty & PrimRec is primitive-recursively_closed )
proof
set S =  {  R where R is Subset of (HFuncs NAT) : R is primitive-recursively_closed  }  ;
A1: {(HFuncs NAT)} c= bool (HFuncs NAT) by ZFMISC_1:68;
 HFuncs NAT in {(HFuncs NAT)} by TARSKI:def_1;
then A2: HFuncs NAT in  {  R where R is Subset of (HFuncs NAT) : R is primitive-recursively_closed  }  by A1, Th66;
A3: now__::_thesis:_for_Y_being_set_st_Y_in__{__R_where_R_is_Subset_of_(HFuncs_NAT)_:_R_is_primitive-recursively_closed__}__holds_
0_const_0_in_Y
let Y be set ; ::_thesis: ( Y in  {  R where R is Subset of (HFuncs NAT) : R is primitive-recursively_closed  }  implies 0 const 0 in Y )
assume  Y in  {  R where R is Subset of (HFuncs NAT) : R is primitive-recursively_closed  }  ; ::_thesis: 0 const 0 in Y
then  ex R being Subset of (HFuncs NAT) st
( R = Y & R is primitive-recursively_closed ) ;
hence  0 const 0 in Y by Def14; ::_thesis: verum
end;
hence  not PrimRec is empty by A2, SETFAM_1:def_1; ::_thesis: PrimRec is primitive-recursively_closed
thus  PrimRec is primitive-recursively_closed ::_thesis: verum
proof
thus  0 const 0 in PrimRec by A2, A3, SETFAM_1:def_1; :: according to COMPUT_1:def_14 ::_thesis: ( 1 succ 1 in PrimRec & ( for n, i being Element of NAT st 1 <= i & i <= n holds
n proj i in PrimRec ) & PrimRec is composition_closed & PrimRec is primitive-recursion_closed )
now__::_thesis:_for_Y_being_set_st_Y_in__{__R_where_R_is_Subset_of_(HFuncs_NAT)_:_R_is_primitive-recursively_closed__}__holds_
1_succ_1_in_Y
let Y be set ; ::_thesis: ( Y in  {  R where R is Subset of (HFuncs NAT) : R is primitive-recursively_closed  }  implies 1 succ 1 in Y )
assume  Y in  {  R where R is Subset of (HFuncs NAT) : R is primitive-recursively_closed  }  ; ::_thesis: 1 succ 1 in Y
then  ex R being Subset of (HFuncs NAT) st
( R = Y & R is primitive-recursively_closed ) ;
hence  1 succ 1 in Y by Def14; ::_thesis: verum
end;
hence  1 succ 1 in PrimRec by A2, SETFAM_1:def_1; ::_thesis: ( ( for n, i being Element of NAT st 1 <= i & i <= n holds
n proj i in PrimRec ) & PrimRec is composition_closed & PrimRec is primitive-recursion_closed )
hereby  ::_thesis: ( PrimRec is composition_closed & PrimRec is primitive-recursion_closed )
let n, i be Element of NAT ; ::_thesis: ( 1 <= i & i <= n implies n proj i in PrimRec )
assume that
A4: 1 <= i and
A5: i <= n ; ::_thesis: n proj i in PrimRec
now__::_thesis:_for_Y_being_set_st_Y_in__{__R_where_R_is_Subset_of_(HFuncs_NAT)_:_R_is_primitive-recursively_closed__}__holds_
n_proj_i_in_Y
let Y be set ; ::_thesis: ( Y in  {  R where R is Subset of (HFuncs NAT) : R is primitive-recursively_closed  }  implies n proj i in Y )
assume  Y in  {  R where R is Subset of (HFuncs NAT) : R is primitive-recursively_closed  }  ; ::_thesis: n proj i in Y
then  ex R being Subset of (HFuncs NAT) st
( R = Y & R is primitive-recursively_closed ) ;
hence  n proj i in Y by A4, A5, Def14; ::_thesis: verum
end;
hence  n proj i in PrimRec by A2, SETFAM_1:def_1; ::_thesis: verum
end;
hereby  :: according to COMPUT_1:def_12 ::_thesis: PrimRec is primitive-recursion_closed
let f be Element of HFuncs NAT; ::_thesis: for F being with_the_same_arity FinSequence of HFuncs NAT st f in PrimRec & arity f = len F & rng F c= PrimRec holds
f * <:F:> in PrimRec
let F be with_the_same_arity FinSequence of HFuncs NAT; ::_thesis: ( f in PrimRec & arity f = len F & rng F c= PrimRec implies f * <:F:> in PrimRec )
assume that
A6: f in PrimRec and
A7: arity f = len F and
A8: rng F c= PrimRec ; ::_thesis: f * <:F:> in PrimRec
now__::_thesis:_for_Y_being_set_st_Y_in__{__R_where_R_is_Subset_of_(HFuncs_NAT)_:_R_is_primitive-recursively_closed__}__holds_
f_*_<:F:>_in_Y
let Y be set ; ::_thesis: ( Y in  {  R where R is Subset of (HFuncs NAT) : R is primitive-recursively_closed  }  implies f * <:F:> in Y )
assume A9: Y in  {  R where R is Subset of (HFuncs NAT) : R is primitive-recursively_closed  }  ; ::_thesis: f * <:F:> in Y
then consider R being Subset of (HFuncs NAT) such that
A10: R = Y and
A11: R is primitive-recursively_closed ;
A12: R is composition_closed by A11, Def14;
A13: PrimRec c= R by A9, A10, SETFAM_1:3;
then  rng F c= R by A8, XBOOLE_1:1;
hence  f * <:F:> in Y by A6, A7, A10, A12, A13, Def12; ::_thesis: verum
end;
hence  f * <:F:> in PrimRec by A2, SETFAM_1:def_1; ::_thesis: verum
end;
hereby  :: according to COMPUT_1:def_13 ::_thesis: verum
let g, f1, f2 be Element of HFuncs NAT; ::_thesis: for i being Element of NAT st g is_primitive-recursively_expressed_by f1,f2,i & f1 in PrimRec & f2 in PrimRec holds
g in PrimRec
let i be Element of NAT ; ::_thesis: ( g is_primitive-recursively_expressed_by f1,f2,i & f1 in PrimRec & f2 in PrimRec implies g in PrimRec )
assume that
A14: g is_primitive-recursively_expressed_by f1,f2,i and
A15: f1 in PrimRec and
A16: f2 in PrimRec ; ::_thesis: g in PrimRec
now__::_thesis:_for_Y_being_set_st_Y_in__{__R_where_R_is_Subset_of_(HFuncs_NAT)_:_R_is_primitive-recursively_closed__}__holds_
g_in_Y
let Y be set ; ::_thesis: ( Y in  {  R where R is Subset of (HFuncs NAT) : R is primitive-recursively_closed  }  implies g in Y )
assume A17: Y in  {  R where R is Subset of (HFuncs NAT) : R is primitive-recursively_closed  }  ; ::_thesis: g in Y
then consider R being Subset of (HFuncs NAT) such that
A18: R = Y and
A19: R is primitive-recursively_closed ;
A20: f2 in R by A16, A17, A18, SETFAM_1:def_1;
A21: R is primitive-recursion_closed by A19, Def14;
 f1 in R by A15, A17, A18, SETFAM_1:def_1;
hence  g in Y by A14, A18, A21, A20, Def13; ::_thesis: verum
end;
hence  g in PrimRec by A2, SETFAM_1:def_1; ::_thesis: verum
end;
end;
end;
end;

registration
cluster  ->  homogeneous for Element of PrimRec ;
coherence
 for b1 being Element of PrimRec holds b1 is homogeneous ;
end;

definition
let x be set ;
attrx is primitive-recursive  means :Def16: :: COMPUT_1:def 16
x in PrimRec ;
end;

:: deftheorem Def16 defines primitive-recursive COMPUT_1:def_16_:_
 for x being set holds
( x is primitive-recursive iff x in PrimRec );

registration
cluster primitive-recursive  ->  Relation-like Function-like for set ;
coherence
 for b1 being set st b1 is primitive-recursive holds
( b1 is Relation-like & b1 is Function-like )
proof
let x be set ; ::_thesis: ( x is primitive-recursive implies ( x is Relation-like & x is Function-like ) )
assume  x in PrimRec ; :: according to COMPUT_1:def_16 ::_thesis: ( x is Relation-like & x is Function-like )
hence  ( x is Relation-like & x is Function-like ) ; ::_thesis: verum
end;
end;

registration
cluster Relation-like primitive-recursive  -> NAT * -defined to-naturals homogeneous for set ;
coherence
 for b1 being Relation st b1 is primitive-recursive holds
( b1 is homogeneous & b1 is to-naturals & b1 is NAT * -defined )
proof
let x be Relation; ::_thesis: ( x is primitive-recursive implies ( x is homogeneous & x is to-naturals & x is NAT * -defined ) )
assume  x in PrimRec ; :: according to COMPUT_1:def_16 ::_thesis: ( x is homogeneous & x is to-naturals & x is NAT * -defined )
hence  ( x is homogeneous & x is to-naturals & x is NAT * -defined ) ; ::_thesis: verum
end;
end;

registration
cluster  ->  primitive-recursive for Element of PrimRec ;
coherence
 for b1 being Element of PrimRec holds b1 is primitive-recursive by Def16;
end;

registration
cluster Relation-like Function-like primitive-recursive for set ;
existence
 ex b1 being Function st b1 is primitive-recursive
proof
set x = the Element of PrimRec ;
take  the Element of PrimRec ; ::_thesis: the Element of PrimRec is primitive-recursive
thus  the Element of PrimRec is primitive-recursive ; ::_thesis: verum
end;
cluster Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued to-naturals homogeneous primitive-recursive for Element of HFuncs NAT;
existence
 ex b1 being Element of HFuncs NAT st b1 is primitive-recursive
proof
set x = the Element of PrimRec ;
take  the Element of PrimRec ; ::_thesis: the Element of PrimRec is primitive-recursive
thus  the Element of PrimRec is primitive-recursive ; ::_thesis: verum
end;
end;

definition
func initial-funcs  ->  Subset of (HFuncs NAT) equals :: COMPUT_1:def 17
{(0 const 0),(1 succ 1)} \/  {  (n proj i) where n, i is Element of NAT : ( 1 <= i & i <= n )  }  ;
coherence
 {(0 const 0),(1 succ 1)} \/  {  (n proj i) where n, i is Element of NAT : ( 1 <= i & i <= n )  }  is Subset of (HFuncs NAT)
proof
set Q1 = {(0 const 0),(1 succ 1)};
set Q2 =  {  (n proj i) where n, i is Element of NAT : ( 1 <= i & i <= n )  }  ;
A1:  {  (n proj i) where n, i is Element of NAT : ( 1 <= i & i <= n )  }  c= HFuncs NAT
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  (n proj i) where n, i is Element of NAT : ( 1 <= i & i <= n )  }  or x in HFuncs NAT )
assume  x in  {  (n proj i) where n, i is Element of NAT : ( 1 <= i & i <= n )  }  ; ::_thesis: x in HFuncs NAT
then  ex n, i being Element of NAT st
( x = n proj i & 1 <= i & i <= n ) ;
hence  x in HFuncs NAT by Th35; ::_thesis: verum
end;
 {(0 const 0),(1 succ 1)} c= HFuncs NAT
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(0 const 0),(1 succ 1)} or x in HFuncs NAT )
assume  x in {(0 const 0),(1 succ 1)} ; ::_thesis: x in HFuncs NAT
then  ( x = 0 const 0 or x = 1 succ 1 ) by TARSKI:def_2;
hence  x in HFuncs NAT by Th31, Th33; ::_thesis: verum
end;
hence  {(0 const 0),(1 succ 1)} \/  {  (n proj i) where n, i is Element of NAT : ( 1 <= i & i <= n )  }  is Subset of (HFuncs NAT) by A1, XBOOLE_1:8; ::_thesis: verum
end;
let Q be Subset of (HFuncs NAT);
func PR-closure Q ->  Subset of (HFuncs NAT) equals :: COMPUT_1:def 18
Q \/  {  g where g is Element of HFuncs NAT : ex f1, f2 being Element of HFuncs NAT ex i being Element of NAT st
( f1 in Q & f2 in Q & g is_primitive-recursively_expressed_by f1,f2,i )  }  ;
coherence
 Q \/  {  g where g is Element of HFuncs NAT : ex f1, f2 being Element of HFuncs NAT ex i being Element of NAT st
( f1 in Q & f2 in Q & g is_primitive-recursively_expressed_by f1,f2,i )  }  is Subset of (HFuncs NAT)
proof
set Q1 =  {  g where g is Element of HFuncs NAT : ex f1, f2 being Element of HFuncs NAT ex i being Element of NAT st
( f1 in Q & f2 in Q & g is_primitive-recursively_expressed_by f1,f2,i )  }  ;
  {  g where g is Element of HFuncs NAT : ex f1, f2 being Element of HFuncs NAT ex i being Element of NAT st
( f1 in Q & f2 in Q & g is_primitive-recursively_expressed_by f1,f2,i )  }  c= HFuncs NAT
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  g where g is Element of HFuncs NAT : ex f1, f2 being Element of HFuncs NAT ex i being Element of NAT st
( f1 in Q & f2 in Q & g is_primitive-recursively_expressed_by f1,f2,i )  }  or x in HFuncs NAT )
assume  x in  {  g where g is Element of HFuncs NAT : ex f1, f2 being Element of HFuncs NAT ex i being Element of NAT st
( f1 in Q & f2 in Q & g is_primitive-recursively_expressed_by f1,f2,i )  }  ; ::_thesis: x in HFuncs NAT
then  ex g being Element of HFuncs NAT st
( x = g & ex f1, f2 being Element of HFuncs NAT ex i being Element of NAT st
( f1 in Q & f2 in Q & g is_primitive-recursively_expressed_by f1,f2,i ) ) ;
hence  x in HFuncs NAT ; ::_thesis: verum
end;
hence  Q \/  {  g where g is Element of HFuncs NAT : ex f1, f2 being Element of HFuncs NAT ex i being Element of NAT st
( f1 in Q & f2 in Q & g is_primitive-recursively_expressed_by f1,f2,i )  }  is Subset of (HFuncs NAT) by XBOOLE_1:8; ::_thesis: verum
end;
func composition-closure Q ->  Subset of (HFuncs NAT) equals :: COMPUT_1:def 19
Q \/  {  (f * <:F:>) where f is Element of HFuncs NAT, F is with_the_same_arity Element of (HFuncs NAT) * : ( f in Q & arity f = len F & rng F c= Q )  }  ;
coherence
 Q \/  {  (f * <:F:>) where f is Element of HFuncs NAT, F is with_the_same_arity Element of (HFuncs NAT) * : ( f in Q & arity f = len F & rng F c= Q )  }  is Subset of (HFuncs NAT)
proof
set Q1 =  {  (f * <:F:>) where f is Element of HFuncs NAT, F is with_the_same_arity Element of (HFuncs NAT) * : ( f in Q & arity f = len F & rng F c= Q )  }  ;
  {  (f * <:F:>) where f is Element of HFuncs NAT, F is with_the_same_arity Element of (HFuncs NAT) * : ( f in Q & arity f = len F & rng F c= Q )  }  c= HFuncs NAT
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  (f * <:F:>) where f is Element of HFuncs NAT, F is with_the_same_arity Element of (HFuncs NAT) * : ( f in Q & arity f = len F & rng F c= Q )  }  or x in HFuncs NAT )
assume  x in  {  (f * <:F:>) where f is Element of HFuncs NAT, F is with_the_same_arity Element of (HFuncs NAT) * : ( f in Q & arity f = len F & rng F c= Q )  }  ; ::_thesis: x in HFuncs NAT
then  ex f being Element of HFuncs NAT ex F being with_the_same_arity Element of (HFuncs NAT) * st
( x = f * <:F:> & f in Q & arity f = len F & rng F c= Q ) ;
hence  x in HFuncs NAT by Th42; ::_thesis: verum
end;
hence  Q \/  {  (f * <:F:>) where f is Element of HFuncs NAT, F is with_the_same_arity Element of (HFuncs NAT) * : ( f in Q & arity f = len F & rng F c= Q )  }  is Subset of (HFuncs NAT) by XBOOLE_1:8; ::_thesis: verum
end;
end;

:: deftheorem  defines initial-funcs COMPUT_1:def_17_:_
 initial-funcs = {(0 const 0),(1 succ 1)} \/  {  (n proj i) where n, i is Element of NAT : ( 1 <= i & i <= n )  }  ;

:: deftheorem  defines PR-closure COMPUT_1:def_18_:_
 for Q being Subset of (HFuncs NAT) holds PR-closure Q = Q \/  {  g where g is Element of HFuncs NAT : ex f1, f2 being Element of HFuncs NAT ex i being Element of NAT st
( f1 in Q & f2 in Q & g is_primitive-recursively_expressed_by f1,f2,i )  }  ;

:: deftheorem  defines composition-closure COMPUT_1:def_19_:_
 for Q being Subset of (HFuncs NAT) holds composition-closure Q = Q \/  {  (f * <:F:>) where f is Element of HFuncs NAT, F is with_the_same_arity Element of (HFuncs NAT) * : ( f in Q & arity f = len F & rng F c= Q )  }  ;

definition
func PrimRec-Approximation  ->  Function of NAT,(bool (HFuncs NAT)) means :Def20: :: COMPUT_1:def 20
( it . 0 = initial-funcs & ( for m being Nat holds it . (m + 1) = (PR-closure (it . m)) \/ (composition-closure (it . m)) ) );
existence
 ex b1 being Function of NAT,(bool (HFuncs NAT)) st
( b1 . 0 = initial-funcs & ( for m being Nat holds b1 . (m + 1) = (PR-closure (b1 . m)) \/ (composition-closure (b1 . m)) ) )
proof
deffunc H1( set , Subset of (HFuncs NAT)) ->  Element of K32((HFuncs NAT)) = (PR-closure $2) \/ (composition-closure $2);
consider A being Function of NAT,(bool (HFuncs NAT)) such that
A1: A . 0 = initial-funcs and
A2: for m being Nat holds A . (m + 1) = H1(m,A . m) from NAT_1:sch_12();
take  A ; ::_thesis: ( A . 0 = initial-funcs & ( for m being Nat holds A . (m + 1) = (PR-closure (A . m)) \/ (composition-closure (A . m)) ) )
thus  A . 0 = initial-funcs by A1; ::_thesis: for m being Nat holds A . (m + 1) = (PR-closure (A . m)) \/ (composition-closure (A . m))
let m be Nat; ::_thesis: A . (m + 1) = (PR-closure (A . m)) \/ (composition-closure (A . m))
thus  A . (m + 1) = (PR-closure (A . m)) \/ (composition-closure (A . m)) by A2; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Function of NAT,(bool (HFuncs NAT)) st b1 . 0 = initial-funcs & ( for m being Nat holds b1 . (m + 1) = (PR-closure (b1 . m)) \/ (composition-closure (b1 . m)) ) & b2 . 0 = initial-funcs & ( for m being Nat holds b2 . (m + 1) = (PR-closure (b2 . m)) \/ (composition-closure (b2 . m)) ) holds
b1 = b2
proof
deffunc H1( set , Subset of (HFuncs NAT)) ->  Element of K32((HFuncs NAT)) = (PR-closure $2) \/ (composition-closure $2);
let it1, it2 be Function of NAT,(bool (HFuncs NAT)); ::_thesis: ( it1 . 0 = initial-funcs & ( for m being Nat holds it1 . (m + 1) = (PR-closure (it1 . m)) \/ (composition-closure (it1 . m)) ) & it2 . 0 = initial-funcs & ( for m being Nat holds it2 . (m + 1) = (PR-closure (it2 . m)) \/ (composition-closure (it2 . m)) ) implies it1 = it2 )
assume that
A3: it1 . 0 = initial-funcs and
A4: for m being Nat holds it1 . (m + 1) = (PR-closure (it1 . m)) \/ (composition-closure (it1 . m)) and
A5: it2 . 0 = initial-funcs and
A6: for m being Nat holds it2 . (m + 1) = (PR-closure (it2 . m)) \/ (composition-closure (it2 . m)) ; ::_thesis: it1 = it2
A7: for m being Nat holds it1 . (m + 1) = H1(m,it1 . m) by A4;
A8: for m being Nat holds it2 . (m + 1) = H1(m,it2 . m) by A6;
A9: it2 . 0 = initial-funcs by A5;
A10: it1 . 0 = initial-funcs by A3;
thus  it1 = it2 from NAT_1:sch_16(A10, A7, A9, A8); ::_thesis: verum
end;
end;

:: deftheorem Def20 defines PrimRec-Approximation COMPUT_1:def_20_:_
 for b1 being Function of NAT,(bool (HFuncs NAT)) holds
( b1 = PrimRec-Approximation iff ( b1 . 0 = initial-funcs & ( for m being Nat holds b1 . (m + 1) = (PR-closure (b1 . m)) \/ (composition-closure (b1 . m)) ) ) );

theorem Th75: :: COMPUT_1:75
for m, n being Element of NAT st m <= n holds
PrimRec-Approximation . m c= PrimRec-Approximation . n
proof
let m, n be Element of NAT ; ::_thesis: ( m <= n implies PrimRec-Approximation . m c= PrimRec-Approximation . n )
set prd = PrimRec-Approximation ;
defpred S2[ Element of NAT ] means ( m <= $1 implies PrimRec-Approximation . m c= PrimRec-Approximation . $1 );
A1: for n being Element of NAT st S2[n] holds
S2[n + 1]
proof
let n be Element of NAT ; ::_thesis: ( S2[n] implies S2[n + 1] )
assume that
A2: S2[n] and
A3: m <= n + 1 ; ::_thesis: PrimRec-Approximation . m c= PrimRec-Approximation . (n + 1)
 PrimRec-Approximation . (n + 1) = (PR-closure (PrimRec-Approximation . n)) \/ (composition-closure (PrimRec-Approximation . n)) by Def20;
then A4: PR-closure (PrimRec-Approximation . n) c= PrimRec-Approximation . (n + 1) by XBOOLE_1:7;
 PrimRec-Approximation . n c= PR-closure (PrimRec-Approximation . n) by XBOOLE_1:7;
then A5: PrimRec-Approximation . n c= PrimRec-Approximation . (n + 1) by A4, XBOOLE_1:1;
percases ( m < n + 1 or m = n + 1 ) by A3, XXREAL_0:1;
suppose m < n + 1 ; ::_thesis: PrimRec-Approximation . m c= PrimRec-Approximation . (n + 1)
hence  PrimRec-Approximation . m c= PrimRec-Approximation . (n + 1) by A2, A5, NAT_1:13, XBOOLE_1:1; ::_thesis: verum
end;
suppose m = n + 1 ; ::_thesis: PrimRec-Approximation . m c= PrimRec-Approximation . (n + 1)
hence  PrimRec-Approximation . m c= PrimRec-Approximation . (n + 1) ; ::_thesis: verum
end;
end;
end;
A6: S2[ 0 ] ;
 for n being Element of NAT holds S2[n] from NAT_1:sch_1(A6, A1);
hence  ( m <= n implies PrimRec-Approximation . m c= PrimRec-Approximation . n ) ; ::_thesis: verum
end;

theorem Th76: :: COMPUT_1:76
Union PrimRec-Approximation is primitive-recursively_closed
proof
set PROJ =  {  (n proj i) where n, i is Element of NAT : ( 1 <= i & i <= n )  }  ;
set prd = PrimRec-Approximation ;
set UP = Union PrimRec-Approximation;
A1: dom PrimRec-Approximation = NAT by FUNCT_2:def_1;
A2: PrimRec-Approximation . 0 = {(0 const 0),(1 succ 1)} \/  {  (n proj i) where n, i is Element of NAT : ( 1 <= i & i <= n )  }  by Def20;
 0 const 0 in {(0 const 0),(1 succ 1)} by TARSKI:def_2;
then  0 const 0 in PrimRec-Approximation . 0 by A2, XBOOLE_0:def_3;
hence  0 const 0 in Union PrimRec-Approximation by A1, CARD_5:2; :: according to COMPUT_1:def_14 ::_thesis: ( 1 succ 1 in Union PrimRec-Approximation & ( for n, i being Element of NAT st 1 <= i & i <= n holds
n proj i in Union PrimRec-Approximation ) & Union PrimRec-Approximation is composition_closed & Union PrimRec-Approximation is primitive-recursion_closed )
 1 succ 1 in {(0 const 0),(1 succ 1)} by TARSKI:def_2;
then  1 succ 1 in PrimRec-Approximation . 0 by A2, XBOOLE_0:def_3;
hence  1 succ 1 in Union PrimRec-Approximation by A1, CARD_5:2; ::_thesis: ( ( for n, i being Element of NAT st 1 <= i & i <= n holds
n proj i in Union PrimRec-Approximation ) & Union PrimRec-Approximation is composition_closed & Union PrimRec-Approximation is primitive-recursion_closed )
hereby  ::_thesis: ( Union PrimRec-Approximation is composition_closed & Union PrimRec-Approximation is primitive-recursion_closed )
let n, i be Element of NAT ; ::_thesis: ( 1 <= i & i <= n implies n proj i in Union PrimRec-Approximation )
assume that
A3: 1 <= i and
A4: i <= n ; ::_thesis: n proj i in Union PrimRec-Approximation
 n proj i in  {  (n proj i) where n, i is Element of NAT : ( 1 <= i & i <= n )  }  by A3, A4;
then  n proj i in PrimRec-Approximation . 0 by A2, XBOOLE_0:def_3;
hence  n proj i in Union PrimRec-Approximation by A1, CARD_5:2; ::_thesis: verum
end;
hereby  :: according to COMPUT_1:def_12 ::_thesis: Union PrimRec-Approximation is primitive-recursion_closed
deffunc H1( Element of NAT ) ->  set =  {  (f * <:F:>) where f is Element of HFuncs NAT, F is with_the_same_arity Element of (HFuncs NAT) * : ( f in PrimRec-Approximation . $1 & arity f = len F & rng F c= PrimRec-Approximation . $1 )  }  ;
let f be Element of HFuncs NAT; ::_thesis: for F being with_the_same_arity FinSequence of HFuncs NAT st f in Union PrimRec-Approximation & arity f = len F & rng F c= Union PrimRec-Approximation holds
b2 * <:b3:> in Union PrimRec-Approximation
let F be with_the_same_arity FinSequence of HFuncs NAT; ::_thesis: ( f in Union PrimRec-Approximation & arity f = len F & rng F c= Union PrimRec-Approximation implies b1 * <:b2:> in Union PrimRec-Approximation )
assume that
A5: f in Union PrimRec-Approximation and
A6: arity f = len F and
A7: rng F c= Union PrimRec-Approximation ; ::_thesis: b1 * <:b2:> in Union PrimRec-Approximation
consider kf being set  such that
A8: kf in dom PrimRec-Approximation and
A9: f in PrimRec-Approximation . kf by A5, CARD_5:2;
reconsider kf = kf as Element of NAT by A8;
percases ( arity f = 0 or arity f <> 0 ) ;
suppose arity f = 0 ; ::_thesis: b1 * <:b2:> in Union PrimRec-Approximation
then  F = {} by A6;
then A10: rng F c= PrimRec-Approximation . kf by RELAT_1:38, XBOOLE_1:2;
 F is with_the_same_arity Element of (HFuncs NAT) * by FINSEQ_1:def_11;
then  f * <:F:> in H1(kf) by A6, A9, A10;
then  f * <:F:> in (PrimRec-Approximation . kf) \/ H1(kf) by XBOOLE_0:def_3;
then  f * <:F:> in (PR-closure (PrimRec-Approximation . kf)) \/ (composition-closure (PrimRec-Approximation . kf)) by XBOOLE_0:def_3;
then  f * <:F:> in PrimRec-Approximation . (kf + 1) by Def20;
hence  f * <:F:> in Union PrimRec-Approximation by A1, CARD_5:2; ::_thesis: verum
end;
supposeA11: arity f <> 0 ; ::_thesis: b1 * <:b2:> in Union PrimRec-Approximation
defpred S2[ set , set ] means  ex k being Element of NAT st
( F . $1 in PrimRec-Approximation . k & $2 = k );
A12: for x being set st x in Seg (len F) holds
ex y being Element of NAT st S2[x,y]
proof
let x be set ; ::_thesis: ( x in Seg (len F) implies ex y being Element of NAT st S2[x,y] )
assume  x in Seg (len F) ; ::_thesis: ex y being Element of NAT st S2[x,y]
then  x in dom F by FINSEQ_1:def_3;
then  F . x in rng F by FUNCT_1:def_3;
then consider k being set  such that
A13: k in dom PrimRec-Approximation and
A14: F . x in PrimRec-Approximation . k by A7, CARD_5:2;
reconsider k = k as Element of NAT by A13;
take  k ; ::_thesis: S2[x,k]
take k1 = k; ::_thesis: ( F . x in PrimRec-Approximation . k1 & k = k1 )
thus  F . x in PrimRec-Approximation . k1 by A14; ::_thesis: k = k1
thus  k = k1 ; ::_thesis: verum
end;
consider fKF being Function of (Seg (len F)),NAT such that
A15: for x being set st x in Seg (len F) holds
S2[x,fKF . x] from MONOID_1:sch_1(A12);
set KF = rng fKF;
reconsider KF = rng fKF as non empty finite Subset of NAT by A6, A11, RELAT_1:def_19;
set K = max KF;
set k = max (kf,(max KF));
reconsider k = max (kf,(max KF)) as Element of NAT by ORDINAL1:def_12;
A16: dom fKF = Seg (len F) by FUNCT_2:def_1;
A17: rng F c= PrimRec-Approximation . k
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng F or x in PrimRec-Approximation . k )
A18: max KF <= k by XXREAL_0:25;
assume  x in rng F ; ::_thesis: x in PrimRec-Approximation . k
then consider d being set  such that
A19: d in dom F and
A20: x = F . d by FUNCT_1:def_3;
A21: d in Seg (len F) by A19, FINSEQ_1:def_3;
then consider m being Element of NAT  such that
A22: F . d in PrimRec-Approximation . m and
A23: fKF . d = m by A15;
 m in KF by A16, A21, A23, FUNCT_1:3;
then  m <= max KF by XXREAL_2:def_8;
then  PrimRec-Approximation . m c= PrimRec-Approximation . k by A18, Th75, XXREAL_0:2;
hence  x in PrimRec-Approximation . k by A20, A22; ::_thesis: verum
end;
A24: F is with_the_same_arity Element of (HFuncs NAT) * by FINSEQ_1:def_11;
 PrimRec-Approximation . kf c= PrimRec-Approximation . k by Th75, XXREAL_0:25;
then  f * <:F:> in H1(k) by A6, A9, A17, A24;
then  f * <:F:> in (PrimRec-Approximation . k) \/ H1(k) by XBOOLE_0:def_3;
then  f * <:F:> in (PR-closure (PrimRec-Approximation . k)) \/ (composition-closure (PrimRec-Approximation . k)) by XBOOLE_0:def_3;
then  f * <:F:> in PrimRec-Approximation . (k + 1) by Def20;
hence  f * <:F:> in Union PrimRec-Approximation by A1, CARD_5:2; ::_thesis: verum
end;
end;
end;
deffunc H1( Element of NAT ) ->  set =  {  g where g is Element of HFuncs NAT : ex f1, f2 being Element of HFuncs NAT ex i being Element of NAT st
( f1 in PrimRec-Approximation . $1 & f2 in PrimRec-Approximation . $1 & g is_primitive-recursively_expressed_by f1,f2,i )  }  ;
let g, f1, f2 be Element of HFuncs NAT; :: according to COMPUT_1:def_13 ::_thesis: for i being Element of NAT st g is_primitive-recursively_expressed_by f1,f2,i & f1 in Union PrimRec-Approximation & f2 in Union PrimRec-Approximation holds
g in Union PrimRec-Approximation
let i be Element of NAT ; ::_thesis: ( g is_primitive-recursively_expressed_by f1,f2,i & f1 in Union PrimRec-Approximation & f2 in Union PrimRec-Approximation implies g in Union PrimRec-Approximation )
assume that
A25: g is_primitive-recursively_expressed_by f1,f2,i and
A26: f1 in Union PrimRec-Approximation and
A27: f2 in Union PrimRec-Approximation ; ::_thesis: g in Union PrimRec-Approximation
consider k2 being set  such that
A28: k2 in dom PrimRec-Approximation and
A29: f2 in PrimRec-Approximation . k2 by A27, CARD_5:2;
reconsider k2 = k2 as Element of NAT by A28;
consider k1 being set  such that
A30: k1 in dom PrimRec-Approximation and
A31: f1 in PrimRec-Approximation . k1 by A26, CARD_5:2;
reconsider k1 = k1 as Element of NAT by A30;
percases ( k1 <= k2 or k2 <= k1 ) ;
suppose k1 <= k2 ; ::_thesis: g in Union PrimRec-Approximation
then  PrimRec-Approximation . k1 c= PrimRec-Approximation . k2 by Th75;
then  g in H1(k2) by A25, A31, A29;
then  g in (PrimRec-Approximation . k2) \/ H1(k2) by XBOOLE_0:def_3;
then  g in (PR-closure (PrimRec-Approximation . k2)) \/ (composition-closure (PrimRec-Approximation . k2)) by XBOOLE_0:def_3;
then  g in PrimRec-Approximation . (k2 + 1) by Def20;
hence  g in Union PrimRec-Approximation by A1, CARD_5:2; ::_thesis: verum
end;
suppose k2 <= k1 ; ::_thesis: g in Union PrimRec-Approximation
then  PrimRec-Approximation . k2 c= PrimRec-Approximation . k1 by Th75;
then  g in H1(k1) by A25, A31, A29;
then  g in (PrimRec-Approximation . k1) \/ H1(k1) by XBOOLE_0:def_3;
then  g in (PR-closure (PrimRec-Approximation . k1)) \/ (composition-closure (PrimRec-Approximation . k1)) by XBOOLE_0:def_3;
then  g in PrimRec-Approximation . (k1 + 1) by Def20;
hence  g in Union PrimRec-Approximation by A1, CARD_5:2; ::_thesis: verum
end;
end;
end;

theorem Th77: :: COMPUT_1:77
PrimRec = Union PrimRec-Approximation
proof
set prd = PrimRec-Approximation ;
defpred S2[ Element of NAT ] means PrimRec-Approximation . $1 c= PrimRec ;
A1: now__::_thesis:_for_m_being_Element_of_NAT_st_S2[m]_holds_
S2[m_+_1]
let m be Element of NAT ; ::_thesis: ( S2[m] implies S2[m + 1] )
assume A2: S2[m] ; ::_thesis: S2[m + 1]
thus  S2[m + 1] ::_thesis: verum
proof
set mcocl =  {  (f * <:F:>) where f is Element of HFuncs NAT, F is with_the_same_arity Element of (HFuncs NAT) * : ( f in PrimRec-Approximation . m & arity f = len F & rng F c= PrimRec-Approximation . m )  }  ;
set mprcl =  {  g where g is Element of HFuncs NAT : ex f1, f2 being Element of HFuncs NAT ex i being Element of NAT st
( f1 in PrimRec-Approximation . m & f2 in PrimRec-Approximation . m & g is_primitive-recursively_expressed_by f1,f2,i )  }  ;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in PrimRec-Approximation . (m + 1) or x in PrimRec )
A3: PrimRec-Approximation . (m + 1) = (PR-closure (PrimRec-Approximation . m)) \/ (composition-closure (PrimRec-Approximation . m)) by Def20;
assume A4: x in PrimRec-Approximation . (m + 1) ; ::_thesis: x in PrimRec
percases ( x in PR-closure (PrimRec-Approximation . m) or x in composition-closure (PrimRec-Approximation . m) ) by A4, A3, XBOOLE_0:def_3;
supposeA5: x in PR-closure (PrimRec-Approximation . m) ; ::_thesis: x in PrimRec
thus  x in PrimRec ::_thesis: verum
proof
percases ( x in PrimRec-Approximation . m or x in  {  g where g is Element of HFuncs NAT : ex f1, f2 being Element of HFuncs NAT ex i being Element of NAT st
( f1 in PrimRec-Approximation . m & f2 in PrimRec-Approximation . m & g is_primitive-recursively_expressed_by f1,f2,i )  }  ) by A5, XBOOLE_0:def_3;
suppose x in PrimRec-Approximation . m ; ::_thesis: x in PrimRec
hence  x in PrimRec by A2; ::_thesis: verum
end;
supposeA6: x in  {  g where g is Element of HFuncs NAT : ex f1, f2 being Element of HFuncs NAT ex i being Element of NAT st
( f1 in PrimRec-Approximation . m & f2 in PrimRec-Approximation . m & g is_primitive-recursively_expressed_by f1,f2,i )  }  ; ::_thesis: x in PrimRec
A7: PrimRec is primitive-recursion_closed by Def14;
 ex g being Element of HFuncs NAT st
( x = g & ex f1, f2 being Element of HFuncs NAT ex i being Element of NAT st
( f1 in PrimRec-Approximation . m & f2 in PrimRec-Approximation . m & g is_primitive-recursively_expressed_by f1,f2,i ) ) by A6;
hence  x in PrimRec by A2, A7, Def13; ::_thesis: verum
end;
end;
end;
end;
supposeA8: x in composition-closure (PrimRec-Approximation . m) ; ::_thesis: x in PrimRec
thus  x in PrimRec ::_thesis: verum
proof
percases ( x in PrimRec-Approximation . m or x in  {  (f * <:F:>) where f is Element of HFuncs NAT, F is with_the_same_arity Element of (HFuncs NAT) * : ( f in PrimRec-Approximation . m & arity f = len F & rng F c= PrimRec-Approximation . m )  }  ) by A8, XBOOLE_0:def_3;
suppose x in PrimRec-Approximation . m ; ::_thesis: x in PrimRec
hence  x in PrimRec by A2; ::_thesis: verum
end;
supposeA9: x in  {  (f * <:F:>) where f is Element of HFuncs NAT, F is with_the_same_arity Element of (HFuncs NAT) * : ( f in PrimRec-Approximation . m & arity f = len F & rng F c= PrimRec-Approximation . m )  }  ; ::_thesis: x in PrimRec
A10: PrimRec is composition_closed by Def14;
consider f being Element of HFuncs NAT, F being with_the_same_arity Element of (HFuncs NAT) *  such that
A11: x = f * <:F:> and
A12: f in PrimRec-Approximation . m and
A13: arity f = len F and
A14: rng F c= PrimRec-Approximation . m by A9;
 rng F c= PrimRec by A2, A14, XBOOLE_1:1;
hence  x in PrimRec by A2, A11, A12, A13, A10, Def12; ::_thesis: verum
end;
end;
end;
end;
end;
end;
end;
A15: S2[ 0 ]
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in PrimRec-Approximation . 0 or x in PrimRec )
assume A16: x in PrimRec-Approximation . 0 ; ::_thesis: x in PrimRec
 PrimRec-Approximation . 0 = {(0 const 0),(1 succ 1)} \/  {  (n proj i) where n, i is Element of NAT : ( 1 <= i & i <= n )  }  by Def20;
then A17: ( x in {(0 const 0),(1 succ 1)} or x in  {  (n proj i) where n, i is Element of NAT : ( 1 <= i & i <= n )  }  ) by A16, XBOOLE_0:def_3;
percases ( x = 0 const 0 or x = 1 succ 1 or x in  {  (n proj i) where n, i is Element of NAT : ( 1 <= i & i <= n )  }  ) by A17, TARSKI:def_2;
suppose x = 0 const 0 ; ::_thesis: x in PrimRec
hence  x in PrimRec by Def14; ::_thesis: verum
end;
suppose x = 1 succ 1 ; ::_thesis: x in PrimRec
hence  x in PrimRec by Def14; ::_thesis: verum
end;
suppose x in  {  (n proj i) where n, i is Element of NAT : ( 1 <= i & i <= n )  }  ; ::_thesis: x in PrimRec
then  ex n, i being Element of NAT st
( x = n proj i & 1 <= i & i <= n ) ;
hence  x in PrimRec by Def14; ::_thesis: verum
end;
end;
end;
A18: for k being Element of NAT holds S2[k] from NAT_1:sch_1(A15, A1);
A19: Union PrimRec-Approximation c= PrimRec
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Union PrimRec-Approximation or x in PrimRec )
assume that
A20: x in Union PrimRec-Approximation and
A21: not x in PrimRec ; ::_thesis: contradiction
consider X being set  such that
A22: x in X and
A23: X in rng PrimRec-Approximation by A20, TARSKI:def_4;
consider m being set  such that
A24: m in dom PrimRec-Approximation and
A25: PrimRec-Approximation . m = X by A23, FUNCT_1:def_3;
reconsider m = m as Element of NAT by A24;
 PrimRec-Approximation . m c= PrimRec by A18;
hence  contradiction by A21, A22, A25; ::_thesis: verum
end;
 PrimRec c= Union PrimRec-Approximation by Th74, Th76;
hence  PrimRec = Union PrimRec-Approximation by A19, XBOOLE_0:def_10; ::_thesis: verum
end;

theorem Th78: :: COMPUT_1:78
for m being Element of NAT
for f being Element of HFuncs NAT st f in PrimRec-Approximation . m holds
f is quasi_total
proof
let m be Element of NAT ; ::_thesis: for f being Element of HFuncs NAT st f in PrimRec-Approximation . m holds
f is quasi_total
defpred S2[ Element of NAT ] means  for f being Element of HFuncs NAT st f in PrimRec-Approximation . $1 holds
f is quasi_total ;
set prd = PrimRec-Approximation ;
A1: for m being Element of NAT st S2[m] holds
S2[m + 1]
proof
let m be Element of NAT ; ::_thesis: ( S2[m] implies S2[m + 1] )
assume A2: S2[m] ; ::_thesis: S2[m + 1]
let f be Element of HFuncs NAT; ::_thesis: ( f in PrimRec-Approximation . (m + 1) implies f is quasi_total )
assume  f in PrimRec-Approximation . (m + 1) ; ::_thesis: f is quasi_total
then A3: f in (PR-closure (PrimRec-Approximation . m)) \/ (composition-closure (PrimRec-Approximation . m)) by Def20;
percases ( f in PR-closure (PrimRec-Approximation . m) or f in composition-closure (PrimRec-Approximation . m) ) by A3, XBOOLE_0:def_3;
supposeA4: f in PR-closure (PrimRec-Approximation . m) ; ::_thesis: f is quasi_total
thus  f is quasi_total ::_thesis: verum
proof
percases ( f in PrimRec-Approximation . m or f in  {  g where g is Element of HFuncs NAT : ex f1, f2 being Element of HFuncs NAT ex i being Element of NAT st
( f1 in PrimRec-Approximation . m & f2 in PrimRec-Approximation . m & g is_primitive-recursively_expressed_by f1,f2,i )  }  ) by A4, XBOOLE_0:def_3;
suppose f in PrimRec-Approximation . m ; ::_thesis: f is quasi_total
hence  f is quasi_total by A2; ::_thesis: verum
end;
suppose f in  {  g where g is Element of HFuncs NAT : ex f1, f2 being Element of HFuncs NAT ex i being Element of NAT st
( f1 in PrimRec-Approximation . m & f2 in PrimRec-Approximation . m & g is_primitive-recursively_expressed_by f1,f2,i )  }  ; ::_thesis: f is quasi_total
then consider g being Element of HFuncs NAT such that
A5: f = g and
A6: ex f1, f2 being Element of HFuncs NAT ex i being Element of NAT st
( f1 in PrimRec-Approximation . m & f2 in PrimRec-Approximation . m & g is_primitive-recursively_expressed_by f1,f2,i ) ;
consider f1, f2 being Element of HFuncs NAT, i being Element of NAT  such that
A7: f1 in PrimRec-Approximation . m and
A8: f2 in PrimRec-Approximation . m and
A9: g is_primitive-recursively_expressed_by f1,f2,i by A6;
A10: f2 is quasi_total by A2, A8;
 f1 is quasi_total by A2, A7;
hence  f is quasi_total by A5, A9, A10, Th68; ::_thesis: verum
end;
end;
end;
end;
supposeA11: f in composition-closure (PrimRec-Approximation . m) ; ::_thesis: f is quasi_total
thus  f is quasi_total ::_thesis: verum
proof
percases ( f in PrimRec-Approximation . m or f in  {  (h * <:F:>) where h is Element of HFuncs NAT, F is with_the_same_arity Element of (HFuncs NAT) * : ( h in PrimRec-Approximation . m & arity h = len F & rng F c= PrimRec-Approximation . m )  }  ) by A11, XBOOLE_0:def_3;
suppose f in PrimRec-Approximation . m ; ::_thesis: f is quasi_total
hence  f is quasi_total by A2; ::_thesis: verum
end;
suppose f in  {  (h * <:F:>) where h is Element of HFuncs NAT, F is with_the_same_arity Element of (HFuncs NAT) * : ( h in PrimRec-Approximation . m & arity h = len F & rng F c= PrimRec-Approximation . m )  }  ; ::_thesis: f is quasi_total
then consider h being Element of HFuncs NAT, F being with_the_same_arity Element of (HFuncs NAT) *  such that
A12: f = h * <:F:> and
A13: h in PrimRec-Approximation . m and
A14: arity h = len F and
A15: rng F c= PrimRec-Approximation . m ;
A16: for h being Element of HFuncs NAT st h in rng F holds
h is quasi_total by A2, A15;
 h is quasi_total by A2, A13;
hence  f is quasi_total by A12, A14, A16, Th46; ::_thesis: verum
end;
end;
end;
end;
end;
end;
A17: S2[ 0 ]
proof
let f be Element of HFuncs NAT; ::_thesis: ( f in PrimRec-Approximation . 0 implies f is quasi_total )
assume  f in PrimRec-Approximation . 0 ; ::_thesis: f is quasi_total
then  f in initial-funcs by Def20;
then A18: ( f in {(0 const 0),(1 succ 1)} or f in  {  (n proj i) where n, i is Element of NAT : ( 1 <= i & i <= n )  }  ) by XBOOLE_0:def_3;
percases ( f = 0 const 0 or f = 1 succ 1 or f in  {  (n proj i) where n, i is Element of NAT : ( 1 <= i & i <= n )  }  ) by A18, TARSKI:def_2;
suppose f = 0 const 0 ; ::_thesis: f is quasi_total
hence  f is quasi_total by Th29; ::_thesis: verum
end;
suppose f = 1 succ 1 ; ::_thesis: f is quasi_total
hence  f is quasi_total by Th29; ::_thesis: verum
end;
suppose f in  {  (n proj i) where n, i is Element of NAT : ( 1 <= i & i <= n )  }  ; ::_thesis: f is quasi_total
then  ex n, i being Element of NAT st
( f = n proj i & 1 <= i & i <= n ) ;
hence  f is quasi_total by Th29; ::_thesis: verum
end;
end;
end;
 for m being Element of NAT holds S2[m] from NAT_1:sch_1(A17, A1);
hence  for f being Element of HFuncs NAT st f in PrimRec-Approximation . m holds
f is quasi_total ; ::_thesis: verum
end;

registration
cluster  ->  homogeneous quasi_total for Element of PrimRec ;
coherence
 for b1 being Element of PrimRec holds
( b1 is quasi_total & b1 is homogeneous )
proof
set prd = PrimRec-Approximation ;
let f be Element of PrimRec ; ::_thesis: ( f is quasi_total & f is homogeneous )
consider X being set  such that
A1: f in X and
A2: X in rng PrimRec-Approximation by Th77, TARSKI:def_4;
 ex m being set st
( m in dom PrimRec-Approximation & PrimRec-Approximation . m = X ) by A2, FUNCT_1:def_3;
hence  ( f is quasi_total & f is homogeneous ) by A1, Th78; ::_thesis: verum
end;
end;

registration
cluster primitive-recursive  ->  quasi_total for Element of HFuncs NAT;
coherence
 for b1 being Element of HFuncs NAT st b1 is primitive-recursive holds
b1 is quasi_total
proof
let f be Element of HFuncs NAT; ::_thesis: ( f is primitive-recursive implies f is quasi_total )
assume  f in PrimRec ; :: according to COMPUT_1:def_16 ::_thesis: f is quasi_total
hence  f is quasi_total ; ::_thesis: verum
end;
end;

registration
cluster Relation-like NAT * -defined Function-like primitive-recursive  -> NAT * -defined len-total for set ;
coherence
 for b1 being NAT * -defined Function st b1 is primitive-recursive holds
b1 is len-total
proof
let x be NAT * -defined Function; ::_thesis: ( x is primitive-recursive implies x is len-total )
assume  x in PrimRec ; :: according to COMPUT_1:def_16 ::_thesis: x is len-total
then  x is Element of PrimRec ;
hence  x is len-total ; ::_thesis: verum
end;
cluster non empty Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued to-naturals homogeneous quasi_total primitive-recursive for Element of PrimRec ;
existence
 not for b1 being Element of PrimRec holds b1 is empty
proof
 0 const 0 in PrimRec by Th69;
hence  not for b1 being Element of PrimRec holds b1 is empty ; ::_thesis: verum
end;
end;

begin

definition
let n be set ;
let f be homogeneous Relation;
attrf is n -ary  means :Def21: :: COMPUT_1:def 21
 arity f = n;
end;

:: deftheorem Def21 defines -ary COMPUT_1:def_21_:_
 for n being set
for f being homogeneous Relation holds
( f is n -ary iff arity f = n );

registration
cluster Relation-like Function-like homogeneous 1 -ary  ->  non empty homogeneous for set ;
coherence
 for b1 being homogeneous Function st b1 is 1 -ary holds
not b1 is empty
proof
let f be homogeneous Function; ::_thesis: ( f is 1 -ary implies not f is empty )
assume  arity f = 1 ; :: according to COMPUT_1:def_21 ::_thesis: not f is empty
then  ex x being FinSequence st x in dom f by MARGREL1:def_25;
hence  not f is empty ; ::_thesis: verum
end;
cluster Relation-like Function-like homogeneous 2 -ary  ->  non empty homogeneous for set ;
coherence
 for b1 being homogeneous Function st b1 is 2 -ary holds
not b1 is empty
proof
let f be homogeneous Function; ::_thesis: ( f is 2 -ary implies not f is empty )
assume  arity f = 2 ; :: according to COMPUT_1:def_21 ::_thesis: not f is empty
then  ex x being FinSequence st x in dom f by MARGREL1:def_25;
hence  not f is empty ; ::_thesis: verum
end;
cluster Relation-like Function-like homogeneous 3 -ary  ->  non empty homogeneous for set ;
coherence
 for b1 being homogeneous Function st b1 is 3 -ary holds
not b1 is empty
proof
let f be homogeneous Function; ::_thesis: ( f is 3 -ary implies not f is empty )
assume  arity f = 3 ; :: according to COMPUT_1:def_21 ::_thesis: not f is empty
then  ex x being FinSequence st x in dom f by MARGREL1:def_25;
hence  not f is empty ; ::_thesis: verum
end;
end;

registration
let n be non empty Element of NAT ;
clustern proj 1 -> NAT * -defined to-naturals homogeneous primitive-recursive ;
coherence
 n proj 1 is primitive-recursive
proof
 1 <= n by NAT_1:14;
then  n proj 1 in PrimRec by Def14;
hence  n proj 1 is primitive-recursive ; ::_thesis: verum
end;
end;

registration
cluster2 proj 2 -> NAT * -defined to-naturals homogeneous primitive-recursive ;
coherence
 2 proj 2 is primitive-recursive
proof
 2 proj 2 in PrimRec by Def14;
hence  2 proj 2 is primitive-recursive ; ::_thesis: verum
end;
cluster1 succ 1 -> NAT * -defined to-naturals homogeneous primitive-recursive ;
coherence
 1 succ 1 is primitive-recursive
proof
 1 succ 1 in PrimRec by Th70;
hence  1 succ 1 is primitive-recursive ; ::_thesis: verum
end;
cluster3 succ 3 -> NAT * -defined to-naturals homogeneous primitive-recursive ;
coherence
 3 succ 3 is primitive-recursive
proof
 3 succ 3 in PrimRec by Th70;
hence  3 succ 3 is primitive-recursive ; ::_thesis: verum
end;
let i be Element of NAT ;
cluster0 const i -> NAT * -defined to-naturals homogeneous 0 -ary ;
coherence
 0 const i is 0 -ary
proof
thus  arity (0 const i) = 0 by Th32; :: according to COMPUT_1:def_21 ::_thesis: verum
end;
cluster1 const i -> NAT * -defined to-naturals homogeneous 1 -ary ;
coherence
 1 const i is 1 -ary
proof
thus  arity (1 const i) = 1 by Th32; :: according to COMPUT_1:def_21 ::_thesis: verum
end;
cluster2 const i -> NAT * -defined to-naturals homogeneous 2 -ary ;
coherence
 2 const i is 2 -ary
proof
thus  arity (2 const i) = 2 by Th32; :: according to COMPUT_1:def_21 ::_thesis: verum
end;
cluster3 const i -> NAT * -defined to-naturals homogeneous 3 -ary ;
coherence
 3 const i is 3 -ary
proof
thus  arity (3 const i) = 3 by Th32; :: according to COMPUT_1:def_21 ::_thesis: verum
end;
cluster1 proj i -> NAT * -defined to-naturals homogeneous 1 -ary ;
coherence
 1 proj i is 1 -ary
proof
thus  arity (1 proj i) = 1 by Th37; :: according to COMPUT_1:def_21 ::_thesis: verum
end;
cluster2 proj i -> NAT * -defined to-naturals homogeneous 2 -ary ;
coherence
 2 proj i is 2 -ary
proof
thus  arity (2 proj i) = 2 by Th37; :: according to COMPUT_1:def_21 ::_thesis: verum
end;
cluster3 proj i -> NAT * -defined to-naturals homogeneous 3 -ary ;
coherence
 3 proj i is 3 -ary
proof
thus  arity (3 proj i) = 3 by Th37; :: according to COMPUT_1:def_21 ::_thesis: verum
end;
cluster1 succ i -> NAT * -defined to-naturals homogeneous 1 -ary ;
coherence
 1 succ i is 1 -ary
proof
thus  arity (1 succ i) = 1 by Th34; :: according to COMPUT_1:def_21 ::_thesis: verum
end;
cluster2 succ i -> NAT * -defined to-naturals homogeneous 2 -ary ;
coherence
 2 succ i is 2 -ary
proof
thus  arity (2 succ i) = 2 by Th34; :: according to COMPUT_1:def_21 ::_thesis: verum
end;
cluster3 succ i -> NAT * -defined to-naturals homogeneous 3 -ary ;
coherence
 3 succ i is 3 -ary
proof
thus  arity (3 succ i) = 3 by Th34; :: according to COMPUT_1:def_21 ::_thesis: verum
end;
let j be Element of NAT ;
clusteri const j -> NAT * -defined to-naturals homogeneous primitive-recursive ;
coherence
 i const j is primitive-recursive
proof
thus  i const j in PrimRec by Th69; :: according to COMPUT_1:def_16 ::_thesis: verum
end;
end;

registration
cluster non empty Relation-like Function-like homogeneous primitive-recursive 0 -ary for set ;
existence
 ex b1 being homogeneous Function st
( b1 is 0 -ary & b1 is primitive-recursive & not b1 is empty ) by Def14;
cluster Relation-like Function-like homogeneous primitive-recursive 1 -ary for set ;
existence
 ex b1 being homogeneous Function st
( b1 is 1 -ary & b1 is primitive-recursive ) by Def14;
cluster Relation-like Function-like homogeneous primitive-recursive 2 -ary for set ;
existence
 ex b1 being homogeneous Function st
( b1 is 2 -ary & b1 is primitive-recursive )
proof
take f = 2 proj 1; ::_thesis: ( f is 2 -ary & f is primitive-recursive )
thus  f is 2 -ary ; ::_thesis: f is primitive-recursive
thus  f in PrimRec by Def14; :: according to COMPUT_1:def_16 ::_thesis: verum
end;
cluster Relation-like Function-like homogeneous primitive-recursive 3 -ary for set ;
existence
 ex b1 being homogeneous Function st
( b1 is 3 -ary & b1 is primitive-recursive )
proof
take f = 3 proj 1; ::_thesis: ( f is 3 -ary & f is primitive-recursive )
thus  f is 3 -ary ; ::_thesis: f is primitive-recursive
thus  f in PrimRec by Def14; :: according to COMPUT_1:def_16 ::_thesis: verum
end;
end;

registration
cluster non empty Relation-like NAT * -defined Function-like to-naturals homogeneous len-total 0 -ary for set ;
existence
 ex b1 being NAT * -defined homogeneous Function st
( not b1 is empty & b1 is 0 -ary & b1 is len-total & b1 is to-naturals )
proof
 0 const 0 is 0 -ary ;
hence  ex b1 being NAT * -defined homogeneous Function st
( not b1 is empty & b1 is 0 -ary & b1 is len-total & b1 is to-naturals ) ; ::_thesis: verum
end;
cluster non empty Relation-like NAT * -defined Function-like to-naturals homogeneous len-total 1 -ary for set ;
existence
 ex b1 being NAT * -defined homogeneous Function st
( not b1 is empty & b1 is 1 -ary & b1 is len-total & b1 is to-naturals )
proof
 1 const 0 is 1 -ary ;
hence  ex b1 being NAT * -defined homogeneous Function st
( not b1 is empty & b1 is 1 -ary & b1 is len-total & b1 is to-naturals ) ; ::_thesis: verum
end;
cluster non empty Relation-like NAT * -defined Function-like to-naturals homogeneous len-total 2 -ary for set ;
existence
 ex b1 being NAT * -defined homogeneous Function st
( not b1 is empty & b1 is 2 -ary & b1 is len-total & b1 is to-naturals )
proof
 2 const 0 is 2 -ary ;
hence  ex b1 being NAT * -defined homogeneous Function st
( not b1 is empty & b1 is 2 -ary & b1 is len-total & b1 is to-naturals ) ; ::_thesis: verum
end;
cluster non empty Relation-like NAT * -defined Function-like to-naturals homogeneous len-total 3 -ary for set ;
existence
 ex b1 being NAT * -defined homogeneous Function st
( not b1 is empty & b1 is 3 -ary & b1 is len-total & b1 is to-naturals )
proof
 3 const 0 is 3 -ary ;
hence  ex b1 being NAT * -defined homogeneous Function st
( not b1 is empty & b1 is 3 -ary & b1 is len-total & b1 is to-naturals ) ; ::_thesis: verum
end;
end;

registration
let f be non empty primitive-recursive 0 -ary Function;
let g be primitive-recursive 2 -ary Function;
cluster primrec (f,g,1) ->  primitive-recursive 1 -ary ;
coherence
 ( primrec (f,g,1) is primitive-recursive & primrec (f,g,1) is 1 -ary )
proof
A1: g in PrimRec by Def16;
A2: arity f = 0 by Def21;
 f is Element of PrimRec by Def16;
then  dom f = 0 -tuples_on NAT by A2, Th22
.= {(<*> NAT)} by FINSEQ_2:94 ;
then A3: {} in dom f by TARSKI:def_1;
A4: f in PrimRec by Def16;
A5: 1 <= 0 + 1 ;
 arity g = 0 + 2 by Def21;
hence  primrec (f,g,1) in PrimRec by A5, A2, A4, A1, Th73; :: according to COMPUT_1:def_16 ::_thesis: primrec (f,g,1) is 1 -ary
reconsider ii = <*0*> as Element of 1 -tuples_on NAT by FINSEQ_2:131;
 dom <*0*> = {1} by FINSEQ_1:2, FINSEQ_1:38;
then A6: 1 in dom <*0*> by TARSKI:def_1;
 Del (<*0*>,1) = {} by WSIERP_1:19;
then  ii +* (1,0) in dom (primrec (f,g,1)) by A5, A2, A3, A6, Lm6;
then A7: <*0*> in dom (primrec (f,g,1)) by FUNCT_7:95;
 len <*0*> = 1 by FINSEQ_1:39;
hence  arity (primrec (f,g,1)) = 1 by A7, MARGREL1:def_25; :: according to COMPUT_1:def_21 ::_thesis: verum
end;
end;

registration
let f be primitive-recursive 1 -ary Function;
let g be primitive-recursive 3 -ary Function;
cluster primrec (f,g,1) ->  primitive-recursive 2 -ary ;
coherence
 ( primrec (f,g,1) is primitive-recursive & primrec (f,g,1) is 2 -ary )
proof
A1: g in PrimRec by Def16;
A2: 1 <= 1 + 1 ;
A3: arity g = 1 + 2 by Def21;
A4: arity f = 1 by Def21;
 f in PrimRec by Def16;
hence  primrec (f,g,1) in PrimRec by A2, A4, A3, A1, Th73; :: according to COMPUT_1:def_16 ::_thesis: primrec (f,g,1) is 2 -ary
thus  arity (primrec (f,g,1)) = 2 by A2, A4, A3, Th57; :: according to COMPUT_1:def_21 ::_thesis: verum
end;
cluster primrec (f,g,2) ->  primitive-recursive 2 -ary ;
coherence
 ( primrec (f,g,2) is primitive-recursive & primrec (f,g,2) is 2 -ary )
proof
A5: g in PrimRec by Def16;
A6: 2 <= 1 + 1 ;
A7: arity g = 1 + 2 by Def21;
A8: arity f = 1 by Def21;
 f in PrimRec by Def16;
hence  primrec (f,g,2) in PrimRec by A6, A8, A7, A5, Th73; :: according to COMPUT_1:def_16 ::_thesis: primrec (f,g,2) is 2 -ary
thus  arity (primrec (f,g,2)) = 2 by A6, A8, A7, Th57; :: according to COMPUT_1:def_21 ::_thesis: verum
end;
end;

theorem Th79: :: COMPUT_1:79
for i being Element of NAT
for f1 being NAT * -defined to-naturals homogeneous len-total 1 -ary Function
for f2 being non empty NAT * -defined to-naturals homogeneous Function holds (primrec (f1,f2,2)) . <*i,0*> = f1 . <*i*>
proof
let i be Element of NAT ; ::_thesis: for f1 being NAT * -defined to-naturals homogeneous len-total 1 -ary Function
for f2 being non empty NAT * -defined to-naturals homogeneous Function holds (primrec (f1,f2,2)) . <*i,0*> = f1 . <*i*>
let f1 be NAT * -defined to-naturals homogeneous len-total 1 -ary Function; ::_thesis: for f2 being non empty NAT * -defined to-naturals homogeneous Function holds (primrec (f1,f2,2)) . <*i,0*> = f1 . <*i*>
let f2 be non empty NAT * -defined to-naturals homogeneous Function; ::_thesis: (primrec (f1,f2,2)) . <*i,0*> = f1 . <*i*>
 arity f1 = 1 by Def21;
then reconsider p = <*i,0*> as Element of ((arity f1) + 1) -tuples_on NAT by FINSEQ_2:101;
 len p = 2 by FINSEQ_1:44;
then A1: 2 in dom p by FINSEQ_3:25;
 p +* (2,0) = p by Th1;
hence (primrec (f1,f2,2)) . <*i,0*> = f1 . (Del (p,2)) by A1, Th60
.= f1 . <*i*> by WSIERP_1:19 ;
::_thesis: verum
end;

theorem Th80: :: COMPUT_1:80
for f1, f2 being non empty NAT * -defined to-naturals homogeneous Function st f1 is len-total & arity f1 = 0 holds
(primrec (f1,f2,1)) . <*0*> = f1 . {}
proof
let f1, f2 be non empty NAT * -defined to-naturals homogeneous Function; ::_thesis: ( f1 is len-total & arity f1 = 0 implies (primrec (f1,f2,1)) . <*0*> = f1 . {} )
assume that
A1: f1 is len-total and
A2: arity f1 = 0 ; ::_thesis: (primrec (f1,f2,1)) . <*0*> = f1 . {}
reconsider p = <*0*> as Element of ((arity f1) + 1) -tuples_on NAT by A2, FINSEQ_2:131;
 len p = 1 by FINSEQ_1:39;
then A3: 1 in dom p by FINSEQ_3:25;
 p +* (1,0) = p by FUNCT_7:95;
hence (primrec (f1,f2,1)) . <*0*> = f1 . (Del (p,1)) by A1, A3, Th60
.= f1 . {} by WSIERP_1:19 ;
::_thesis: verum
end;

theorem Th81: :: COMPUT_1:81
for i, j being Element of NAT
for f1 being NAT * -defined to-naturals homogeneous len-total 1 -ary Function
for f2 being NAT * -defined to-naturals homogeneous len-total 3 -ary Function holds (primrec (f1,f2,2)) . <*i,(j + 1)*> = f2 . <*i,j,((primrec (f1,f2,2)) . <*i,j*>)*>
proof
let i, j be Element of NAT ; ::_thesis: for f1 being NAT * -defined to-naturals homogeneous len-total 1 -ary Function
for f2 being NAT * -defined to-naturals homogeneous len-total 3 -ary Function holds (primrec (f1,f2,2)) . <*i,(j + 1)*> = f2 . <*i,j,((primrec (f1,f2,2)) . <*i,j*>)*>
let f1 be NAT * -defined to-naturals homogeneous len-total 1 -ary Function; ::_thesis: for f2 being NAT * -defined to-naturals homogeneous len-total 3 -ary Function holds (primrec (f1,f2,2)) . <*i,(j + 1)*> = f2 . <*i,j,((primrec (f1,f2,2)) . <*i,j*>)*>
let f2 be NAT * -defined to-naturals homogeneous len-total 3 -ary Function; ::_thesis: (primrec (f1,f2,2)) . <*i,(j + 1)*> = f2 . <*i,j,((primrec (f1,f2,2)) . <*i,j*>)*>
A1: arity f1 = 1 by Def21;
then reconsider p = <*i,j*> as Element of ((arity f1) + 1) -tuples_on NAT by FINSEQ_2:101;
A2: p +* (2,(j + 1)) = <*i,(j + 1)*> by Th1;
A3: p +* (2,j) = <*i,j*> by Th1;
 (arity f1) + 2 = arity f2 by A1, Def21;
hence (primrec (f1,f2,2)) . <*i,(j + 1)*> = f2 . ((p +* (2,j)) ^ <*((primrec (f1,f2,2)) . (p +* (2,j)))*>) by A1, A2, Th63
.= f2 . <*i,j,((primrec (f1,f2,2)) . <*i,j*>)*> by A3, FINSEQ_1:43 ;
::_thesis: verum
end;

theorem Th82: :: COMPUT_1:82
for i being Element of NAT
for f1, f2 being non empty NAT * -defined to-naturals homogeneous Function st f1 is len-total & f2 is len-total & arity f1 = 0 & arity f2 = 2 holds
(primrec (f1,f2,1)) . <*(i + 1)*> = f2 . <*i,((primrec (f1,f2,1)) . <*i*>)*>
proof
let i be Element of NAT ; ::_thesis: for f1, f2 being non empty NAT * -defined to-naturals homogeneous Function st f1 is len-total & f2 is len-total & arity f1 = 0 & arity f2 = 2 holds
(primrec (f1,f2,1)) . <*(i + 1)*> = f2 . <*i,((primrec (f1,f2,1)) . <*i*>)*>
let f1, f2 be non empty NAT * -defined to-naturals homogeneous Function; ::_thesis: ( f1 is len-total & f2 is len-total & arity f1 = 0 & arity f2 = 2 implies (primrec (f1,f2,1)) . <*(i + 1)*> = f2 . <*i,((primrec (f1,f2,1)) . <*i*>)*> )
assume that
A1: f1 is len-total and
A2: f2 is len-total and
A3: arity f1 = 0 and
A4: arity f2 = 2 ; ::_thesis: (primrec (f1,f2,1)) . <*(i + 1)*> = f2 . <*i,((primrec (f1,f2,1)) . <*i*>)*>
reconsider p = <*i*> as Element of ((arity f1) + 1) -tuples_on NAT by A3, FINSEQ_2:131;
A5: p +* (1,(i + 1)) = <*(i + 1)*> by FUNCT_7:95;
A6: p +* (1,i) = <*i*> by FUNCT_7:95;
 (arity f1) + 2 = arity f2 by A3, A4;
hence (primrec (f1,f2,1)) . <*(i + 1)*> = f2 . ((p +* (1,i)) ^ <*((primrec (f1,f2,1)) . (p +* (1,i)))*>) by A1, A2, A3, A5, Th63
.= f2 . <*i,((primrec (f1,f2,1)) . <*i*>)*> by A6, FINSEQ_1:def_9 ;
::_thesis: verum
end;

Lm8: now__::_thesis:_for_g_being_non_empty_homogeneous_quasi_total_PartFunc_of_(NAT_*),NAT_st_arity_g_=_2_holds_
(_dom_<:<*(3_proj_1),(3_proj_3)*>:>_=_(dom_(3_proj_1))_/\_(dom_(3_proj_3))_&_dom_<:<*(3_proj_1),(3_proj_3)*>:>_=_3_-tuples_on_NAT_&_dom_(g_*_<:<*(3_proj_1),(3_proj_3)*>:>)_=_3_-tuples_on_NAT_&_ex_G_being_homogeneous_PartFunc_of_(NAT_*),NAT_st_
(_G_=_g_*_<:<*(3_proj_1),(3_proj_3)*>:>_&_G_is_Element_of_HFuncs_NAT_&_arity_G_=_3_&_G_is_quasi_total_&_not_G_is_empty_)_)
reconsider z3 = <*0,0,0*> as FinSequence of NAT ;
let g be non empty homogeneous quasi_total PartFunc of (NAT *),NAT; ::_thesis: ( arity g = 2 implies ( dom <:<*(3 proj 1),(3 proj 3)*>:> = (dom (3 proj 1)) /\ (dom (3 proj 3)) & dom <:<*(3 proj 1),(3 proj 3)*>:> = 3 -tuples_on NAT & dom (g * <:<*(3 proj 1),(3 proj 3)*>:>) = 3 -tuples_on NAT & ex G being homogeneous PartFunc of (NAT *),NAT st
( G = g * <:<*(3 proj 1),(3 proj 3)*>:> & G is Element of HFuncs NAT & arity G = 3 & G is quasi_total & not G is empty ) ) )
set G = g * <:<*(3 proj 1),(3 proj 3)*>:>;
A1: rng (g * <:<*(3 proj 1),(3 proj 3)*>:>) c= NAT by RELAT_1:def_19;
assume A2: arity g = 2 ; ::_thesis: ( dom <:<*(3 proj 1),(3 proj 3)*>:> = (dom (3 proj 1)) /\ (dom (3 proj 3)) & dom <:<*(3 proj 1),(3 proj 3)*>:> = 3 -tuples_on NAT & dom (g * <:<*(3 proj 1),(3 proj 3)*>:>) = 3 -tuples_on NAT & ex G being homogeneous PartFunc of (NAT *),NAT st
( G = g * <:<*(3 proj 1),(3 proj 3)*>:> & G is Element of HFuncs NAT & arity G = 3 & G is quasi_total & not G is empty ) )
then A3: dom g = 2 -tuples_on NAT by Th22;
thus A4: dom <:<*(3 proj 1),(3 proj 3)*>:> = (dom (3 proj 1)) /\ (dom (3 proj 3)) by FINSEQ_3:142; ::_thesis: ( dom <:<*(3 proj 1),(3 proj 3)*>:> = 3 -tuples_on NAT & dom (g * <:<*(3 proj 1),(3 proj 3)*>:>) = 3 -tuples_on NAT & ex G being homogeneous PartFunc of (NAT *),NAT st
( G = g * <:<*(3 proj 1),(3 proj 3)*>:> & G is Element of HFuncs NAT & arity G = 3 & G is quasi_total & not G is empty ) )
hence A5: dom <:<*(3 proj 1),(3 proj 3)*>:> = (3 -tuples_on NAT) /\ (dom (3 proj 3)) by Th36
.= (3 -tuples_on NAT) /\ (3 -tuples_on NAT) by Th36
.= 3 -tuples_on NAT ;
::_thesis: ( dom (g * <:<*(3 proj 1),(3 proj 3)*>:>) = 3 -tuples_on NAT & ex G being homogeneous PartFunc of (NAT *),NAT st
( G = g * <:<*(3 proj 1),(3 proj 3)*>:> & G is Element of HFuncs NAT & arity G = 3 & G is quasi_total & not G is empty ) )
now__::_thesis:_for_x_being_set_holds_
(_(_x_in_rng_<:<*(3_proj_1),(3_proj_3)*>:>_implies_x_in_dom_g_)_&_(_x_in_dom_g_implies_x_in_rng_<:<*(3_proj_1),(3_proj_3)*>:>_)_)
set f = <*(3 proj 1),(3 proj 3)*>;
let x be set ; ::_thesis: ( ( x in rng <:<*(3 proj 1),(3 proj 3)*>:> implies x in dom g ) & ( x in dom g implies x in rng <:<*(3 proj 1),(3 proj 3)*>:> ) )
set F = <:<*(3 proj 1),(3 proj 3)*>:>;
A6: product (rngs <*(3 proj 1),(3 proj 3)*>) = product <*(rng (3 proj 1)),(rng (3 proj 3))*> by FINSEQ_3:133
.= product <*NAT,(rng (3 proj 3))*> by Th36
.= product <*NAT,NAT*> by Th36
.= 2 -tuples_on NAT by FINSEQ_3:128 ;
hereby  ::_thesis: ( x in dom g implies x in rng <:<*(3 proj 1),(3 proj 3)*>:> )
A7: rng <:<*(3 proj 1),(3 proj 3)*>:> c= product (rngs <*(3 proj 1),(3 proj 3)*>) by FUNCT_6:29;
assume  x in rng <:<*(3 proj 1),(3 proj 3)*>:> ; ::_thesis: x in dom g
hence  x in dom g by A3, A6, A7; ::_thesis: verum
end;
assume  x in dom g ; ::_thesis: x in rng <:<*(3 proj 1),(3 proj 3)*>:>
then  x is Element of 2 -tuples_on NAT by A2, Th22;
then consider d1, d2 being Element of NAT  such that
A8: x = <*d1,d2*> by FINSEQ_2:100;
reconsider x9 = <*d1,0,d2*> as Element of 3 -tuples_on NAT by FINSEQ_2:104;
<:<*(3 proj 1),(3 proj 3)*>:> . x9 = <*((3 proj 1) . x9),((3 proj 3) . x9)*> by A4, A5, FINSEQ_3:142
.= <*(x9 . 1),((3 proj 3) . x9)*> by Th38
.= <*(x9 . 1),(x9 . 3)*> by Th38
.= <*d1,(x9 . 3)*> by FINSEQ_1:45
.= x by A8, FINSEQ_1:45 ;
hence  x in rng <:<*(3 proj 1),(3 proj 3)*>:> by A5, FUNCT_1:def_3; ::_thesis: verum
end;
then  rng <:<*(3 proj 1),(3 proj 3)*>:> = dom g by TARSKI:1;
hence A9: dom (g * <:<*(3 proj 1),(3 proj 3)*>:>) = 3 -tuples_on NAT by A5, RELAT_1:27; ::_thesis: ex G being homogeneous PartFunc of (NAT *),NAT st
( G = g * <:<*(3 proj 1),(3 proj 3)*>:> & G is Element of HFuncs NAT & arity G = 3 & G is quasi_total & not G is empty )
then reconsider G = g * <:<*(3 proj 1),(3 proj 3)*>:> as PartFunc of (NAT *),NAT by A1, FINSEQ_2:142, RELSET_1:4;
reconsider G = G as homogeneous PartFunc of (NAT *),NAT by A9, MARGREL1:def_21;
take G = G; ::_thesis: ( G = g * <:<*(3 proj 1),(3 proj 3)*>:> & G is Element of HFuncs NAT & arity G = 3 & G is quasi_total & not G is empty )
thus  G = g * <:<*(3 proj 1),(3 proj 3)*>:> ; ::_thesis: ( G is Element of HFuncs NAT & arity G = 3 & G is quasi_total & not G is empty )
 G is Element of PFuncs ((NAT *),NAT) by PARTFUN1:45;
then  G in HFuncs NAT ;
hence  G is Element of HFuncs NAT ; ::_thesis: ( arity G = 3 & G is quasi_total & not G is empty )
 len z3 = 3 by FINSEQ_1:45;
then A10: z3 is Element of 3 -tuples_on NAT by FINSEQ_2:92;
 for x being FinSequence st x in dom G holds
3 = len x by A9, CARD_1:def_7;
hence  arity G = 3 by A10, A9, MARGREL1:def_25; ::_thesis: ( G is quasi_total & not G is empty )
hence  ( G is quasi_total & not G is empty ) by A9, Th22; ::_thesis: verum
end;

definition
let g be Function;
func (1,2)->(1,?,2) g ->  Function equals :: COMPUT_1:def 22
g * <:<*(3 proj 1),(3 proj 3)*>:>;
coherence
 g * <:<*(3 proj 1),(3 proj 3)*>:> is Function ;
end;

:: deftheorem  defines (1,2)->(1,?,2) COMPUT_1:def_22_:_
 for g being Function holds (1,2)->(1,?,2) g = g * <:<*(3 proj 1),(3 proj 3)*>:>;

registration
let g be NAT * -defined to-naturals Function;
cluster (1,2)->(1,?,2) g -> NAT * -defined to-naturals ;
coherence
 ( (1,2)->(1,?,2) g is to-naturals & (1,2)->(1,?,2) g is NAT * -defined )
proof
set G = (1,2)->(1,?,2) g;
A1: 3 -tuples_on NAT c= NAT * by FINSEQ_2:142;
 dom <:<*(3 proj 1),(3 proj 3)*>:> = (dom (3 proj 1)) /\ (dom (3 proj 3)) by FINSEQ_3:142;
then A2: dom <:<*(3 proj 1),(3 proj 3)*>:> = (3 -tuples_on NAT) /\ (dom (3 proj 3)) by Th36
.= (3 -tuples_on NAT) /\ (3 -tuples_on NAT) by Th36
.= 3 -tuples_on NAT ;
 dom ((1,2)->(1,?,2) g) c= dom <:<*(3 proj 1),(3 proj 3)*>:> by RELAT_1:25;
then  dom ((1,2)->(1,?,2) g) c= NAT * by A2, A1, XBOOLE_1:1;
hence  ( (1,2)->(1,?,2) g is to-naturals & (1,2)->(1,?,2) g is NAT * -defined ) by RELAT_1:def_18; ::_thesis: verum
end;
end;

registration
let g be homogeneous Function;
cluster (1,2)->(1,?,2) g ->  homogeneous ;
coherence
 (1,2)->(1,?,2) g is homogeneous
proof
set G = (1,2)->(1,?,2) g;
 dom <:<*(3 proj 1),(3 proj 3)*>:> = (dom (3 proj 1)) /\ (dom (3 proj 3)) by FINSEQ_3:142;
then  dom <:<*(3 proj 1),(3 proj 3)*>:> = (3 -tuples_on NAT) /\ (dom (3 proj 3)) by Th36
.= (3 -tuples_on NAT) /\ (3 -tuples_on NAT) by Th36
.= 3 -tuples_on NAT ;
then  dom ((1,2)->(1,?,2) g) c= 3 -tuples_on NAT by RELAT_1:25;
hence  dom ((1,2)->(1,?,2) g) is with_common_domain ; :: according to MARGREL1:def_21 ::_thesis: verum
end;
end;

registration
let g be NAT * -defined to-naturals homogeneous len-total 2 -ary Function;
cluster (1,2)->(1,?,2) g ->  non empty len-total 3 -ary ;
coherence
 ( not (1,2)->(1,?,2) g is empty & (1,2)->(1,?,2) g is 3 -ary & (1,2)->(1,?,2) g is len-total )
proof
A1: g is non empty homogeneous quasi_total PartFunc of (NAT *),NAT by Th17;
 arity g = 2 by Def21;
then consider G being homogeneous PartFunc of (NAT *),NAT such that
A2: G = g * <:<*(3 proj 1),(3 proj 3)*>:> and
 G is Element of HFuncs NAT and
A3: arity G = 3 and
A4: ( G is quasi_total & not G is empty ) by A1, Lm8;
reconsider G9 = G as non empty homogeneous quasi_total PartFunc of (NAT *),NAT by A4;
 G9 is non empty NAT * -defined to-naturals homogeneous len-total 3 -ary Function by A3, Def21;
hence  ( not (1,2)->(1,?,2) g is empty & (1,2)->(1,?,2) g is 3 -ary & (1,2)->(1,?,2) g is len-total ) by A2; ::_thesis: verum
end;
end;

theorem Th83: :: COMPUT_1:83
for i, j, k being Element of NAT
for f being NAT * -defined to-naturals homogeneous len-total 2 -ary Function holds ((1,2)->(1,?,2) f) . <*i,j,k*> = f . <*i,k*>
proof
let i, j, k be Element of NAT ; ::_thesis: for f being NAT * -defined to-naturals homogeneous len-total 2 -ary Function holds ((1,2)->(1,?,2) f) . <*i,j,k*> = f . <*i,k*>
let f be NAT * -defined to-naturals homogeneous len-total 2 -ary Function; ::_thesis: ((1,2)->(1,?,2) f) . <*i,j,k*> = f . <*i,k*>
reconsider ff = f as non empty homogeneous quasi_total PartFunc of (NAT *),NAT by Th17;
reconsider ijk = <*i,j,k*> as Element of 3 -tuples_on NAT by FINSEQ_2:104;
A1: arity ff = 2 by Def21;
then A2: dom <:<*(3 proj 1),(3 proj 3)*>:> = (dom (3 proj 1)) /\ (dom (3 proj 3)) by Lm8;
A3: dom <:<*(3 proj 1),(3 proj 3)*>:> = 3 -tuples_on NAT by A1, Lm8;
 dom (ff * <:<*(3 proj 1),(3 proj 3)*>:>) = 3 -tuples_on NAT by A1, Lm8;
hence ((1,2)->(1,?,2) f) . <*i,j,k*> = f . (<:<*(3 proj 1),(3 proj 3)*>:> . ijk) by FUNCT_1:12
.= f . <*((3 proj 1) . ijk),((3 proj 3) . ijk)*> by A2, A3, FINSEQ_3:142
.= f . <*(ijk . 1),((3 proj 3) . ijk)*> by Th38
.= f . <*(ijk . 1),(ijk . 3)*> by Th38
.= f . <*i,(ijk . 3)*> by FINSEQ_1:45
.= f . <*i,k*> by FINSEQ_1:45 ;
::_thesis: verum
end;

theorem Th84: :: COMPUT_1:84
for g being primitive-recursive 2 -ary Function holds (1,2)->(1,?,2) g in PrimRec
proof
A1: 3 proj 3 in PrimRec by Def14;
A2: 2 -tuples_on PrimRec c= PrimRec * by FINSEQ_2:142;
 3 proj 1 in PrimRec by Def14;
then  <*(3 proj 1),(3 proj 3)*> in 2 -tuples_on PrimRec by A1, FINSEQ_2:101;
then reconsider F = <*(3 proj 1),(3 proj 3)*> as Element of PrimRec * by A2;
 F is with_the_same_arity
proof
let f, g be Function; :: according to COMPUT_1:def_3 ::_thesis: ( f in rng F & g in rng F implies ( ( not f is empty or g is empty or dom g = {{}} ) & ( not f is empty & not g is empty implies ex n being Element of NAT ex X being non empty set st
( dom f c= n -tuples_on X & dom g c= n -tuples_on X ) ) ) )
assume that
A3: f in rng F and
A4: g in rng F ; ::_thesis: ( ( not f is empty or g is empty or dom g = {{}} ) & ( not f is empty & not g is empty implies ex n being Element of NAT ex X being non empty set st
( dom f c= n -tuples_on X & dom g c= n -tuples_on X ) ) )
A5: rng F = {(3 proj 1),(3 proj 3)} by FINSEQ_2:127;
hence  ( not f is empty or g is empty or dom g = {{}} ) by A3, TARSKI:def_2; ::_thesis: ( not f is empty & not g is empty implies ex n being Element of NAT ex X being non empty set st
( dom f c= n -tuples_on X & dom g c= n -tuples_on X ) )
assume that
 not f is empty and
 not g is empty ; ::_thesis: ex n being Element of NAT ex X being non empty set st
( dom f c= n -tuples_on X & dom g c= n -tuples_on X )
take  3 ; ::_thesis: ex X being non empty set st
( dom f c= 3 -tuples_on X & dom g c= 3 -tuples_on X )
take  NAT ; ::_thesis: ( dom f c= 3 -tuples_on NAT & dom g c= 3 -tuples_on NAT )
 ( f = 3 proj 1 or f = 3 proj 3 ) by A3, A5, TARSKI:def_2;
hence  dom f c= 3 -tuples_on NAT by Th36; ::_thesis: dom g c= 3 -tuples_on NAT
 ( g = 3 proj 1 or g = 3 proj 3 ) by A4, A5, TARSKI:def_2;
hence  dom g c= 3 -tuples_on NAT by Th36; ::_thesis: verum
end;
then reconsider F = F as with_the_same_arity Element of PrimRec * ;
let g be primitive-recursive 2 -ary Function; ::_thesis: (1,2)->(1,?,2) g in PrimRec
 arity g = 2 by Def21;
then A6: arity g = len F by FINSEQ_1:44;
 g is Element of PrimRec by Def16;
hence  (1,2)->(1,?,2) g in PrimRec by A6, Th72; ::_thesis: verum
end;

registration
let f be homogeneous primitive-recursive 2 -ary Function;
cluster (1,2)->(1,?,2) f ->  primitive-recursive 3 -ary ;
coherence
 ( (1,2)->(1,?,2) f is primitive-recursive & (1,2)->(1,?,2) f is 3 -ary )
proof
thus  (1,2)->(1,?,2) f in PrimRec by Th84; :: according to COMPUT_1:def_16 ::_thesis: (1,2)->(1,?,2) f is 3 -ary
thus  arity ((1,2)->(1,?,2) f) = 3 by Def21; :: according to COMPUT_1:def_21 ::_thesis: verum
end;
end;

definition
func [+]  ->  primitive-recursive 2 -ary Function equals :: COMPUT_1:def 23
 primrec ((1 proj 1),(3 succ 3),2);
coherence
 primrec ((1 proj 1),(3 succ 3),2) is primitive-recursive 2 -ary Function ;
end;

:: deftheorem  defines [+] COMPUT_1:def_23_:_
 [+] = primrec ((1 proj 1),(3 succ 3),2);

theorem Th85: :: COMPUT_1:85
for i, j being Element of NAT holds [+] . <*i,j*> = i + j
proof
let i, j be Element of NAT ; ::_thesis: [+] . <*i,j*> = i + j
reconsider q = <*i*> as Element of 1 -tuples_on NAT by FINSEQ_2:131;
defpred S2[ Element of NAT ] means [+] . <*i,$1*> = i + $1;
A1: now__::_thesis:_for_j_being_Element_of_NAT_st_S2[j]_holds_
S2[j_+_1]
let j be Element of NAT ; ::_thesis: ( S2[j] implies S2[j + 1] )
reconsider r = <*i,j,(i + j)*> as Element of 3 -tuples_on NAT by FINSEQ_2:104;
assume  S2[j] ; ::_thesis: S2[j + 1]
then [+] . <*i,(j + 1)*> = (3 succ 3) . r by Th81
.= (r /. 3) + 1 by Def7
.= (i + j) + 1 by FINSEQ_4:18
.= i + (j + 1) ;
hence  S2[j + 1] ; ::_thesis: verum
end;
[+] . <*i,0*> = (1 proj 1) . q by Th79
.= q . 1 by Th38
.= i + 0 by FINSEQ_1:40 ;
then A2: S2[ 0 ] ;
 for j being Element of NAT holds S2[j] from NAT_1:sch_1(A2, A1);
hence  [+] . <*i,j*> = i + j ; ::_thesis: verum
end;

definition
func [*]  ->  primitive-recursive 2 -ary Function equals :: COMPUT_1:def 24
 primrec ((1 const 0),((1,2)->(1,?,2) [+]),2);
coherence
 primrec ((1 const 0),((1,2)->(1,?,2) [+]),2) is primitive-recursive 2 -ary Function ;
end;

:: deftheorem  defines [*] COMPUT_1:def_24_:_
 [*] = primrec ((1 const 0),((1,2)->(1,?,2) [+]),2);

theorem Th86: :: COMPUT_1:86
for i, j being Element of NAT holds [*] . <*i,j*> = i * j
proof
let i be Element of NAT ; ::_thesis: for j being Element of NAT holds [*] . <*i,j*> = i * j
defpred S2[ Element of NAT ] means [*] . <*i,$1*> = i * $1;
A1: now__::_thesis:_for_j_being_Element_of_NAT_st_S2[j]_holds_
S2[j_+_1]
let j be Element of NAT ; ::_thesis: ( S2[j] implies S2[j + 1] )
assume  S2[j] ; ::_thesis: S2[j + 1]
then [*] . <*i,(j + 1)*> = ((1,2)->(1,?,2) [+]) . <*i,j,(i * j)*> by Th81
.= [+] . <*i,(i * j)*> by Th83
.= (i * 1) + (i * j) by Th85
.= i * (j + 1) ;
hence  S2[j + 1] ; ::_thesis: verum
end;
reconsider ii = <*i*> as Element of 1 -tuples_on NAT by FINSEQ_2:98;
[*] . <*i,0*> = (1 const 0) . ii by Th79
.= i * 0 by FUNCOP_1:7 ;
then A2: S2[ 0 ] ;
thus  for i being Element of NAT holds S2[i] from NAT_1:sch_1(A2, A1); ::_thesis: verum
end;

registration
let g, h be homogeneous primitive-recursive 2 -ary Function;
cluster<*g,h*> ->  with_the_same_arity ;
coherence
 <*g,h*> is with_the_same_arity
proof
reconsider g = g, h = h as Element of PrimRec by Def16;
A1: rng <*g,h*> = {g,h} by FINSEQ_2:127;
A2: now__::_thesis:_for_f1,_f2_being_homogeneous_Function_st_f1_in_rng_<*g,h*>_&_f2_in_rng_<*g,h*>_holds_
arity_f1_=_arity_f2
let f1, f2 be homogeneous Function; ::_thesis: ( f1 in rng <*g,h*> & f2 in rng <*g,h*> implies arity f1 = arity f2 )
assume that
A3: f1 in rng <*g,h*> and
A4: f2 in rng <*g,h*> ; ::_thesis: arity f1 = arity f2
 ( f1 = g or f1 = h ) by A1, A3, TARSKI:def_2;
then A5: arity f1 = 2 by Def21;
 ( f2 = g or f2 = h ) by A1, A4, TARSKI:def_2;
hence  arity f1 = arity f2 by A5, Def21; ::_thesis: verum
end;
 rng <*g,h*> c= PrimRec by FINSEQ_1:def_4;
hence  <*g,h*> is with_the_same_arity by A2, Th30, XBOOLE_1:1; ::_thesis: verum
end;
end;

registration
let f, g, h be primitive-recursive 2 -ary Function;
cluster<:<*g,h*>:> * f ->  primitive-recursive ;
coherence
 f * <:<*g,h*>:> is primitive-recursive
proof
reconsider g9 = g, h9 = h as Element of PrimRec by Def16;
A1: f in PrimRec by Def16;
A2: rng <*g9,h9*> c= PrimRec by FINSEQ_1:def_4;
then  rng <*g,h*> c= HFuncs NAT by XBOOLE_1:1;
then reconsider F = <*g,h*> as with_the_same_arity FinSequence of HFuncs NAT by FINSEQ_1:def_4;
A3: PrimRec is composition_closed by Def14;
A4: arity f = 2 by Def21;
 len F = 2 by FINSEQ_1:44;
hence  f * <:<*g,h*>:> in PrimRec by A2, A1, A3, A4, Def12; :: according to COMPUT_1:def_16 ::_thesis: verum
end;
end;

registration
let f, g, h be primitive-recursive 2 -ary Function;
cluster<:<*g,h*>:> * f -> 2 -ary ;
coherence
 f * <:<*g,h*>:> is 2 -ary
proof
set x = the Element of 2 -tuples_on NAT;
reconsider f9 = f, fgh = f * <:<*g,h*>:>, g9 = g, h9 = h as Element of PrimRec by Def16;
A1: f9 = f ;
 rng <*g9,h9*> c= PrimRec by FINSEQ_1:def_4;
then  rng <*g,h*> c= HFuncs NAT by XBOOLE_1:1;
then reconsider F = <*g,h*> as with_the_same_arity FinSequence of HFuncs NAT by FINSEQ_1:def_4;
A2: dom <:F:> = (dom g) /\ (dom h) by FINSEQ_3:142;
A3: arity g = 2 by Def21;
 rng F = {g,h} by FINSEQ_2:127;
then  g in rng F by TARSKI:def_2;
then A4: arity F = 2 by A3, Def4;
 arity f = 2 by Def21;
then A5: dom f9 = 2 -tuples_on NAT by Lm1;
 arity h = 2 by Def21;
then A6: dom h9 = 2 -tuples_on NAT by Lm1;
A7: dom g9 = 2 -tuples_on NAT by A3, Lm1;
then  <:F:> . the Element of 2 -tuples_on NAT = <*(g9 . the Element of 2 -tuples_on NAT),(h9 . the Element of 2 -tuples_on NAT)*> by A6, A2, FINSEQ_3:142;
then  <:F:> . the Element of 2 -tuples_on NAT is Element of 2 -tuples_on NAT by FINSEQ_2:101;
then  not fgh is empty by A5, A7, A6, A2, FUNCT_1:11, RELAT_1:38;
hence  arity (f * <:<*g,h*>:>) = 2 by A1, A4, Th44; :: according to COMPUT_1:def_21 ::_thesis: verum
end;
end;

registration
let f be primitive-recursive 1 -ary Function;
let g be primitive-recursive Function;
cluster<:<*g*>:> * f ->  primitive-recursive ;
coherence
 f * <:<*g*>:> is primitive-recursive
proof
reconsider g9 = g as Element of PrimRec by Def16;
A1: f in PrimRec by Def16;
A2: rng <*g9*> c= PrimRec by FINSEQ_1:def_4;
then  rng <*g*> c= HFuncs NAT by XBOOLE_1:1;
then reconsider F = <*g9*> as with_the_same_arity FinSequence of HFuncs NAT by FINSEQ_1:def_4;
A3: PrimRec is composition_closed by Def14;
A4: arity f = 1 by Def21;
 len F = 1 by FINSEQ_1:39;
hence  f * <:<*g*>:> in PrimRec by A2, A1, A3, A4, Def12; :: according to COMPUT_1:def_16 ::_thesis: verum
end;
end;

registration
let f be primitive-recursive 1 -ary Function;
let g be primitive-recursive 2 -ary Function;
cluster<:<*g*>:> * f -> 2 -ary ;
coherence
 f * <:<*g*>:> is 2 -ary
proof
set x = the Element of 2 -tuples_on NAT;
reconsider f9 = f, fg = f * <:<*g*>:>, g9 = g as Element of PrimRec by Def16;
A1: f9 = f ;
 rng <*g9*> c= PrimRec by FINSEQ_1:def_4;
then  rng <*g*> c= HFuncs NAT by XBOOLE_1:1;
then reconsider F = <*g9*> as with_the_same_arity FinSequence of HFuncs NAT by FINSEQ_1:def_4;
A2: dom <:F:> = dom g by FINSEQ_3:141;
 arity f = 1 by Def21;
then A3: dom f9 = 1 -tuples_on NAT by Lm1;
A4: arity g = 2 by Def21;
 rng F = {g} by FINSEQ_1:39;
then  g in rng F by TARSKI:def_1;
then A5: arity F = 2 by A4, Def4;
A6: dom g9 = 2 -tuples_on NAT by A4, Lm1;
then  <:F:> . the Element of 2 -tuples_on NAT = <*(g9 . the Element of 2 -tuples_on NAT)*> by FINSEQ_3:141;
then  <:F:> . the Element of 2 -tuples_on NAT in 1 -tuples_on NAT by FINSEQ_2:98;
then  not fg is empty by A6, A2, A3, FUNCT_1:11, RELAT_1:38;
hence  arity (f * <:<*g*>:>) = 2 by A1, A5, Th44; :: according to COMPUT_1:def_21 ::_thesis: verum
end;
end;

definition
func [!]  ->  primitive-recursive 1 -ary Function equals :: COMPUT_1:def 25
 primrec ((0 const 1),([*] * <:<*((1 succ 1) * <:<*(2 proj 1)*>:>),(2 proj 2)*>:>),1);
coherence
 primrec ((0 const 1),([*] * <:<*((1 succ 1) * <:<*(2 proj 1)*>:>),(2 proj 2)*>:>),1) is primitive-recursive 1 -ary Function ;
end;

:: deftheorem  defines [!] COMPUT_1:def_25_:_
 [!] = primrec ((0 const 1),([*] * <:<*((1 succ 1) * <:<*(2 proj 1)*>:>),(2 proj 2)*>:>),1);

scheme :: COMPUT_1:sch 1
Primrec1{ F1() ->  NAT * -defined to-naturals homogeneous len-total 1 -ary Function, F2() ->  NAT * -defined to-naturals homogeneous len-total 2 -ary Function, F3( set ) ->  Element of NAT , F4( set , set ) ->  Element of NAT } :
 for i, j being Element of NAT holds (F1() * <:<*F2()*>:>) . <*i,j*> = F3(F4(i,j))
provided
A1: for i being Element of NAT holds F1() . <*i*> = F3(i) and
A2: for i, j being Element of NAT holds F2() . <*i,j*> = F4(i,j)
proof
let i, j be Element of NAT ; ::_thesis: (F1() * <:<*F2()*>:>) . <*i,j*> = F3(F4(i,j))
 arity F2() = 2 by Def21;
then A3: dom F2() = 2 -tuples_on NAT by Th23;
 dom <:<*F2()*>:> = dom F2() by FINSEQ_3:141;
hence (F1() * <:<*F2()*>:>) . <*i,j*> = F1() . (<:<*F2()*>:> . <*i,j*>) by A3, FINSEQ_2:101, FUNCT_1:13
.= F1() . <*(F2() . <*i,j*>)*> by A3, FINSEQ_2:101, FINSEQ_3:141
.= F1() . <*F4(i,j)*> by A2
.= F3(F4(i,j)) by A1 ;
::_thesis: verum
end;

scheme :: COMPUT_1:sch 2
Primrec2{ F1() ->  NAT * -defined to-naturals homogeneous len-total 2 -ary Function, F2() ->  NAT * -defined to-naturals homogeneous len-total 2 -ary Function, F3() ->  NAT * -defined to-naturals homogeneous len-total 2 -ary Function, F4( set , set ) ->  Element of NAT , F5( set , set ) ->  Element of NAT , F6( set , set ) ->  Element of NAT } :
 for i, j being Element of NAT holds (F1() * <:<*F2(),F3()*>:>) . <*i,j*> = F4(F5(i,j),F6(i,j))
provided
A1: for i, j being Element of NAT holds F1() . <*i,j*> = F4(i,j) and
A2: for i, j being Element of NAT holds F2() . <*i,j*> = F5(i,j) and
A3: for i, j being Element of NAT holds F3() . <*i,j*> = F6(i,j)
proof
let i, j be Element of NAT ; ::_thesis: (F1() * <:<*F2(),F3()*>:>) . <*i,j*> = F4(F5(i,j),F6(i,j))
 arity F2() = 2 by Def21;
then A4: dom F2() = 2 -tuples_on NAT by Th23;
 arity F3() = 2 by Def21;
then A5: dom F3() = 2 -tuples_on NAT by Th23;
A6: dom <:<*F2(),F3()*>:> = (dom F2()) /\ (dom F3()) by FINSEQ_3:142;
hence (F1() * <:<*F2(),F3()*>:>) . <*i,j*> = F1() . (<:<*F2(),F3()*>:> . <*i,j*>) by A4, A5, FINSEQ_2:101, FUNCT_1:13
.= F1() . <*(F2() . <*i,j*>),(F3() . <*i,j*>)*> by A6, A4, A5, FINSEQ_2:101, FINSEQ_3:142
.= F1() . <*F5(i,j),(F3() . <*i,j*>)*> by A2
.= F1() . <*F5(i,j),F6(i,j)*> by A3
.= F4(F5(i,j),F6(i,j)) by A1 ;
::_thesis: verum
end;

theorem :: COMPUT_1:87
for i being Element of NAT holds [!] . <*i*> = i !
proof
let i be Element of NAT ; ::_thesis: [!] . <*i*> = i !
defpred S2[ Element of NAT ] means [!] . <*$1*> = $1 ! ;
deffunc H1( Element of NAT , Element of NAT ) ->  Element of NAT = $2;
deffunc H2( Element of NAT , Element of NAT ) ->  Element of NAT = $1 * $2;
deffunc H3( Element of NAT , Element of NAT ) ->  Element of NAT = $1;
deffunc H4( Element of NAT ) ->  Element of NAT = $1 + 1;
set g = [*] * <:<*((1 succ 1) * <:<*(2 proj 1)*>:>),(2 proj 2)*>:>;
deffunc H5( Element of NAT , Element of NAT ) ->  Element of NAT = H4(H3($1,$2));
A1: for i, j being Element of NAT holds (2 proj 1) . <*i,j*> = H3(i,j)
proof
let i, j be Element of NAT ; ::_thesis: (2 proj 1) . <*i,j*> = H3(i,j)
reconsider ij = <*i,j*> as Element of 2 -tuples_on NAT by FINSEQ_2:101;
thus (2 proj 1) . <*i,j*> = ij . 1 by Th38
.= i by FINSEQ_1:44 ; ::_thesis: verum
end;
A2: for i being Element of NAT holds (1 succ 1) . <*i*> = H4(i)
proof
let i be Element of NAT ; ::_thesis: (1 succ 1) . <*i*> = H4(i)
reconsider ij = <*i*> as Element of 1 -tuples_on NAT by FINSEQ_2:131;
thus (1 succ 1) . <*i*> = (ij /. 1) + 1 by Def7
.= i + 1 by FINSEQ_4:16 ; ::_thesis: verum
end;
 for i, j being Element of NAT holds ((1 succ 1) * <:<*(2 proj 1)*>:>) . <*i,j*> = H4(H3(i,j)) from COMPUT_1:sch_1(A2, A1);
then A3: for i, j being Element of NAT holds ((1 succ 1) * <:<*(2 proj 1)*>:>) . <*i,j*> = H5(i,j) ;
A4: for i, j being Element of NAT holds (2 proj 2) . <*i,j*> = H1(i,j)
proof
let i, j be Element of NAT ; ::_thesis: (2 proj 2) . <*i,j*> = H1(i,j)
reconsider ij = <*i,j*> as Element of 2 -tuples_on NAT by FINSEQ_2:101;
thus (2 proj 2) . <*i,j*> = ij . 2 by Th38
.= j by FINSEQ_1:44 ; ::_thesis: verum
end;
A5: for i, j being Element of NAT holds [*] . <*i,j*> = H2(i,j) by Th86;
A6: for i, j being Element of NAT holds ([*] * <:<*((1 succ 1) * <:<*(2 proj 1)*>:>),(2 proj 2)*>:>) . <*i,j*> = H2(H5(i,j),H1(i,j)) from COMPUT_1:sch_2(A5, A3, A4);
A7: arity (0 const 1) = 0 by Th32;
A8: arity ([*] * <:<*((1 succ 1) * <:<*(2 proj 1)*>:>),(2 proj 2)*>:>) = 2 by Def21;
A9: now__::_thesis:_for_i_being_Element_of_NAT_st_S2[i]_holds_
S2[i_+_1]
let i be Element of NAT ; ::_thesis: ( S2[i] implies S2[i + 1] )
reconsider ie = i ! as Element of NAT ;
assume  S2[i] ; ::_thesis: S2[i + 1]
then [!] . <*(i + 1)*> = ([*] * <:<*((1 succ 1) * <:<*(2 proj 1)*>:>),(2 proj 2)*>:>) . <*i,ie*> by A8, A7, Th82
.= (i + 1) * ie by A6
.= (i + 1) ! by NEWTON:15 ;
hence  S2[i + 1] ; ::_thesis: verum
end;
 0 -tuples_on NAT = {{}} by Th5;
then A10: {} in 0 -tuples_on NAT by TARSKI:def_1;
[!] . <*0*> = (0 const 1) . {} by A7, Th80
.= 0 ! by A10, FUNCOP_1:7, NEWTON:12 ;
then A11: S2[ 0 ] ;
 for i being Element of NAT holds S2[i] from NAT_1:sch_1(A11, A9);
hence  [!] . <*i*> = i ! ; ::_thesis: verum
end;

definition
func [^]  ->  primitive-recursive 2 -ary Function equals :: COMPUT_1:def 26
 primrec ((1 const 1),((1,2)->(1,?,2) [*]),2);
coherence
 primrec ((1 const 1),((1,2)->(1,?,2) [*]),2) is primitive-recursive 2 -ary Function ;
end;

:: deftheorem  defines [^] COMPUT_1:def_26_:_
 [^] = primrec ((1 const 1),((1,2)->(1,?,2) [*]),2);

theorem :: COMPUT_1:88
for i, j being Element of NAT holds [^] . <*i,j*> = i |^ j
proof
let i, j be Element of NAT ; ::_thesis: [^] . <*i,j*> = i |^ j
defpred S2[ Element of NAT ] means [^] . <*i,$1*> = i |^ $1;
A1: now__::_thesis:_for_j_being_Element_of_NAT_st_S2[j]_holds_
S2[j_+_1]
let j be Element of NAT ; ::_thesis: ( S2[j] implies S2[j + 1] )
reconsider ij = i |^ j as Element of NAT ;
assume  S2[j] ; ::_thesis: S2[j + 1]
then [^] . <*i,(j + 1)*> = ((1,2)->(1,?,2) [*]) . <*i,j,ij*> by Th81
.= [*] . <*i,ij*> by Th83
.= i * ij by Th86
.= i |^ (j + 1) by NEWTON:6 ;
hence  S2[j + 1] ; ::_thesis: verum
end;
reconsider ii = <*i*> as Element of 1 -tuples_on NAT by FINSEQ_2:131;
[^] . <*i,0*> = (1 const 1) . ii by Th79
.= 1 by FUNCOP_1:7
.= i |^ 0 by NEWTON:4 ;
then A2: S2[ 0 ] ;
 for j being Element of NAT holds S2[j] from NAT_1:sch_1(A2, A1);
hence  [^] . <*i,j*> = i |^ j ; ::_thesis: verum
end;

definition
func [pred]  ->  primitive-recursive 1 -ary Function equals :: COMPUT_1:def 27
 primrec ((0 const 0),(2 proj 1),1);
coherence
 primrec ((0 const 0),(2 proj 1),1) is primitive-recursive 1 -ary Function ;
end;

:: deftheorem  defines [pred] COMPUT_1:def_27_:_
 [pred] = primrec ((0 const 0),(2 proj 1),1);

theorem Th89: :: COMPUT_1:89
for i being Element of NAT holds
( [pred] . <*0*> = 0 & [pred] . <*(i + 1)*> = i )
proof
let i be Element of NAT ; ::_thesis: ( [pred] . <*0*> = 0 & [pred] . <*(i + 1)*> = i )
 0 -tuples_on NAT = {{}} by Th5;
then A1: {} in 0 -tuples_on NAT by TARSKI:def_1;
defpred S2[ Element of NAT ] means [pred] . <*($1 + 1)*> = $1;
reconsider p0 = <*0,0*> as Element of 2 -tuples_on NAT by FINSEQ_2:101;
A2: arity (0 const 0) = 0 by Th32;
A3: arity (2 proj 1) = 2 by Th37;
A4: now__::_thesis:_for_i_being_Element_of_NAT_st_S2[i]_holds_
S2[i_+_1]
let i be Element of NAT ; ::_thesis: ( S2[i] implies S2[i + 1] )
reconsider p0 = <*(i + 1),i*> as Element of 2 -tuples_on NAT by FINSEQ_2:101;
assume  S2[i] ; ::_thesis: S2[i + 1]
then [pred] . <*((i + 1) + 1)*> = (2 proj 1) . p0 by A2, A3, Th82
.= <*(i + 1),i*> . 1 by Th38
.= i + 1 by FINSEQ_1:44 ;
hence  S2[i + 1] ; ::_thesis: verum
end;
thus [pred] . <*0*> = (0 const 0) . {} by A2, Th80
.= 0 by A1, FUNCOP_1:7 ; ::_thesis: [pred] . <*(i + 1)*> = i
then [pred] . <*(0 + 1)*> = (2 proj 1) . p0 by A2, A3, Th82
.= <*0,0*> . 1 by Th38
.= 0 by FINSEQ_1:44 ;
then A5: S2[ 0 ] ;
 for i being Element of NAT holds S2[i] from NAT_1:sch_1(A5, A4);
hence  [pred] . <*(i + 1)*> = i ; ::_thesis: verum
end;

definition
func [-]  ->  primitive-recursive 2 -ary Function equals :: COMPUT_1:def 28
 primrec ((1 proj 1),((1,2)->(1,?,2) ([pred] * <:<*(2 proj 2)*>:>)),2);
coherence
 primrec ((1 proj 1),((1,2)->(1,?,2) ([pred] * <:<*(2 proj 2)*>:>)),2) is primitive-recursive 2 -ary Function ;
end;

:: deftheorem  defines [-] COMPUT_1:def_28_:_
 [-] = primrec ((1 proj 1),((1,2)->(1,?,2) ([pred] * <:<*(2 proj 2)*>:>)),2);

theorem :: COMPUT_1:90
for i, j being Element of NAT holds [-] . <*i,j*> = i -' j
proof
let i, j be Element of NAT ; ::_thesis: [-] . <*i,j*> = i -' j
set F = <*(2 proj 2)*>;
set g = [pred] * <:<*(2 proj 2)*>:>;
 rng <*(2 proj 2)*> c= PrimRec
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng <*(2 proj 2)*> or x in PrimRec )
assume  x in rng <*(2 proj 2)*> ; ::_thesis: x in PrimRec
then  x in {(2 proj 2)} by FINSEQ_1:39;
then  x = 2 proj 2 by TARSKI:def_1;
hence  x in PrimRec by Def14; ::_thesis: verum
end;
then reconsider F = <*(2 proj 2)*> as with_the_same_arity FinSequence of PrimRec by FINSEQ_1:def_4;
defpred S2[ Element of NAT ] means [-] . <*i,$1*> = i -' $1;
A1: for i, j being Element of NAT holds
( ([pred] * <:<*(2 proj 2)*>:>) . <*i,0*> = 0 & ([pred] * <:<*(2 proj 2)*>:>) . <*i,(j + 1)*> = j )
proof
let i, j be Element of NAT ; ::_thesis: ( ([pred] * <:<*(2 proj 2)*>:>) . <*i,0*> = 0 & ([pred] * <:<*(2 proj 2)*>:>) . <*i,(j + 1)*> = j )
reconsider i0 = <*i,0*> as Element of 2 -tuples_on NAT by FINSEQ_2:101;
reconsider ij = <*i,(j + 1)*> as Element of 2 -tuples_on NAT by FINSEQ_2:101;
A2: dom (2 proj 2) = 2 -tuples_on NAT by Th36;
A3: dom <:F:> = dom (2 proj 2) by FINSEQ_3:141;
hence ([pred] * <:<*(2 proj 2)*>:>) . <*i,0*> = [pred] . (<:F:> . i0) by A2, FUNCT_1:13
.= [pred] . <*((2 proj 2) . i0)*> by A2, FINSEQ_3:141
.= [pred] . <*(i0 . 2)*> by Th38
.= 0 by Th89, FINSEQ_1:44 ;
::_thesis: ([pred] * <:<*(2 proj 2)*>:>) . <*i,(j + 1)*> = j
thus ([pred] * <:<*(2 proj 2)*>:>) . <*i,(j + 1)*> = [pred] . (<:F:> . ij) by A3, A2, FUNCT_1:13
.= [pred] . <*((2 proj 2) . ij)*> by A2, FINSEQ_3:141
.= [pred] . <*(ij . 2)*> by Th38
.= [pred] . <*(j + 1)*> by FINSEQ_1:44
.= j by Th89 ; ::_thesis: verum
end;
A4: now__::_thesis:_for_j_being_Element_of_NAT_st_S2[j]_holds_
S2[j_+_1]
let j be Element of NAT ; ::_thesis: ( S2[j] implies S2[j + 1] )
assume A5: S2[j] ; ::_thesis: S2[j + 1]
A6: now__::_thesis:_([pred]_*_<:<*(2_proj_2)*>:>)_._<*i,(i_-'_j)*>_=_i_-'_(j_+_1)
percases ( i -' j = 0 or ex k being Nat st i -' j = k + 1 ) by NAT_1:6;
supposeA7: i -' j = 0 ; ::_thesis: ([pred] * <:<*(2 proj 2)*>:>) . <*i,(i -' j)*> = i -' (j + 1)
then  i <= j by NAT_D:36;
then  i < j + 1 by NAT_1:13;
then A8: i - (j + 1) < 0 by XREAL_1:49;
thus ([pred] * <:<*(2 proj 2)*>:>) . <*i,(i -' j)*> = 0 by A1, A7
.= i -' (j + 1) by A8, XREAL_0:def_2 ; ::_thesis: verum
end;
suppose ex k being Nat st i -' j = k + 1 ; ::_thesis: ([pred] * <:<*(2 proj 2)*>:>) . <*i,(i -' j)*> = i -' (j + 1)
then consider k being Nat such that
A9: i -' j = k + 1 ;
reconsider k = k as Element of NAT by ORDINAL1:def_12;
 i - j = k + 1 by A9, XREAL_0:def_2;
then A10: i - (j + 1) = k ;
thus ([pred] * <:<*(2 proj 2)*>:>) . <*i,(i -' j)*> = k by A1, A9
.= i -' (j + 1) by A10, XREAL_0:def_2 ; ::_thesis: verum
end;
end;
end;
[-] . <*i,(j + 1)*> = ((1,2)->(1,?,2) ([pred] * <:<*(2 proj 2)*>:>)) . <*i,j,([-] . <*i,j*>)*> by Th81
.= i -' (j + 1) by A5, A6, Th83 ;
hence  S2[j + 1] ; ::_thesis: verum
end;
reconsider ii = <*i*> as Element of 1 -tuples_on NAT by FINSEQ_2:98;
[-] . <*i,0*> = (1 proj 1) . <*i*> by Th79
.= ii . 1 by Th38
.= i by FINSEQ_1:40
.= (i + 0) -' 0 by NAT_D:34
.= i -' 0 ;
then A11: S2[ 0 ] ;
 for j being Element of NAT holds S2[j] from NAT_1:sch_1(A11, A4);
hence  [-] . <*i,j*> = i -' j ; ::_thesis: verum
end;