:: AOFA_I00 semantic presentation

begin

theorem Th1: :: AOFA_I00:1
for x, y, z, a, b, c being set st a <> b & b <> c & c <> a holds
ex f being Function st
( f . a = x & f . b = y & f . c = z )
proof
let x, y, z, a, b, c be set ; ::_thesis: ( a <> b & b <> c & c <> a implies ex f being Function st
( f . a = x & f . b = y & f . c = z ) )
assume that
A1: a <> b and
A2: b <> c and
A3: c <> a ; ::_thesis: ex f being Function st
( f . a = x & f . b = y & f . c = z )
set fb = b .--> y;
A4: dom (b .--> y) = {b} by FUNCOP_1:13;
A5: a nin dom (b .--> y) by A1, TARSKI:def_1;
A6: b in dom (b .--> y) by A4, TARSKI:def_1;
set fc = c .--> z;
set fa = a .--> x;
take f = ((a .--> x) +* (b .--> y)) +* (c .--> z); ::_thesis: ( f . a = x & f . b = y & f . c = z )
A7: dom (c .--> z) = {c} by FUNCOP_1:13;
 a nin dom (c .--> z) by A3, TARSKI:def_1;
hence f . a = ((a .--> x) +* (b .--> y)) . a by FUNCT_4:11
.= (a .--> x) . a by A5, FUNCT_4:11
.= x by FUNCOP_1:72 ;
::_thesis: ( f . b = y & f . c = z )
 b nin dom (c .--> z) by A2, TARSKI:def_1;
hence f . b = ((a .--> x) +* (b .--> y)) . b by FUNCT_4:11
.= (b .--> y) . b by A6, FUNCT_4:13
.= y by FUNCOP_1:72 ;
::_thesis: f . c = z
 c in dom (c .--> z) by A7, TARSKI:def_1;
hence f . c = (c .--> z) . c by FUNCT_4:13
.= z by FUNCOP_1:72 ;
::_thesis: verum
end;

definition
let F be non empty functional set ;
let x, f be set ;
funcF \ (x,f) ->  Subset of F equals :: AOFA_I00:def 1
 {  g where g is Element of F : g . x <> f  }  ;
coherence
  {  g where g is Element of F : g . x <> f  }  is Subset of F
proof
set X =  {  g where g is Element of F : g . x <> f  }  ;
  {  g where g is Element of F : g . x <> f  }  c= F
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in  {  g where g is Element of F : g . x <> f  }  or a in F )
assume  a in  {  g where g is Element of F : g . x <> f  }  ; ::_thesis: a in F
then  ex g being Element of F st
( g = a & g . x <> f ) ;
hence  a in F ; ::_thesis: verum
end;
hence   {  g where g is Element of F : g . x <> f  }  is Subset of F ; ::_thesis: verum
end;
end;

:: deftheorem  defines \ AOFA_I00:def_1_:_
 for F being non empty functional set
for x, f being set holds F \ (x,f) =  {  g where g is Element of F : g . x <> f  }  ;

theorem Th2: :: AOFA_I00:2
for F being non empty functional set
for x, y being set
for g being Element of F holds
( g in F \ (x,y) iff g . x <> y )
proof
let F be non empty functional set ; ::_thesis: for x, y being set
for g being Element of F holds
( g in F \ (x,y) iff g . x <> y )
let x, y be set ; ::_thesis: for g being Element of F holds
( g in F \ (x,y) iff g . x <> y )
let g be Element of F; ::_thesis: ( g in F \ (x,y) iff g . x <> y )
 ( g in F \ (x,y) iff ex f being Element of F st
( g = f & f . x <> y ) ) ;
hence  ( g in F \ (x,y) iff g . x <> y ) ; ::_thesis: verum
end;

definition
let X be set ;
let Y, Z be set ;
let f be Function of [:(Funcs (X,INT)),Y:],Z;
mode Variable of f ->  Element of X means :Def2: :: AOFA_I00:def 2
verum;
existence
 ex b1 being Element of X st verum ;
end;

:: deftheorem Def2 defines Variable AOFA_I00:def_2_:_
 for X, Y, Z being set
for f being Function of [:(Funcs (X,INT)),Y:],Z
for b5 being Element of X holds
( b5 is Variable of f iff verum );

notation
let f be real-valued Function;
let x be real number ;
synonym f * x for x (#) f;
end;

definition
let t1, t2 be INT -valued Function;
funct1 div t2 ->  INT -valued Function means :Def3: :: AOFA_I00:def 3
( dom it = (dom t1) /\ (dom t2) & ( for s being set st s in dom it holds
it . s = (t1 . s) div (t2 . s) ) );
correctness
existence
ex b1 being INT -valued Function st
( dom b1 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b1 holds
b1 . s = (t1 . s) div (t2 . s) ) );
uniqueness
for b1, b2 being INT -valued Function st dom b1 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b1 holds
b1 . s = (t1 . s) div (t2 . s) ) & dom b2 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b2 holds
b2 . s = (t1 . s) div (t2 . s) ) holds
b1 = b2;
proof
deffunc H1( set ) ->  set = (t1 . $1) div (t2 . $1);
consider f being Function such that
A1: ( dom f = (dom t1) /\ (dom t2) & ( for x being set st x in (dom t1) /\ (dom t2) holds
f . x = H1(x) ) ) from FUNCT_1:sch_3();
 f is INT -valued
proof
let y be set ; :: according to TARSKI:def_3,RELAT_1:def_19 ::_thesis: ( not y in proj2 f or y in INT )
assume  y in rng f ; ::_thesis: y in INT
then consider x being set  such that
A2: x in dom f and
A3: f . x = y by FUNCT_1:def_3;
 f . x = H1(x) by A1, A2;
hence  y in INT by A3, INT_1:def_2; ::_thesis: verum
end;
hence  ex f being INT -valued Function st
( dom f = (dom t1) /\ (dom t2) & ( for x being set st x in dom f holds
f . x = H1(x) ) ) by A1; ::_thesis: for b1, b2 being INT -valued Function st dom b1 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b1 holds
b1 . s = (t1 . s) div (t2 . s) ) & dom b2 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b2 holds
b2 . s = (t1 . s) div (t2 . s) ) holds
b1 = b2
let f1, f2 be INT -valued Function; ::_thesis: ( dom f1 = (dom t1) /\ (dom t2) & ( for s being set st s in dom f1 holds
f1 . s = (t1 . s) div (t2 . s) ) & dom f2 = (dom t1) /\ (dom t2) & ( for s being set st s in dom f2 holds
f2 . s = (t1 . s) div (t2 . s) ) implies f1 = f2 )
assume that
A4: dom f1 = (dom t1) /\ (dom t2) and
A5: for s being set st s in dom f1 holds
f1 . s = H1(s) and
A6: dom f2 = (dom t1) /\ (dom t2) and
A7: for s being set st s in dom f2 holds
f2 . s = H1(s) ; ::_thesis: f1 = f2
now__::_thesis:_for_s_being_set_st_s_in_(dom_t1)_/\_(dom_t2)_holds_
f1_._s_=_f2_._s
let s be set ; ::_thesis: ( s in (dom t1) /\ (dom t2) implies f1 . s = f2 . s )
assume A8: s in (dom t1) /\ (dom t2) ; ::_thesis: f1 . s = f2 . s
hence f1 . s = H1(s) by A4, A5
.= f2 . s by A6, A7, A8 ;
::_thesis: verum
end;
hence  f1 = f2 by A4, A6, FUNCT_1:2; ::_thesis: verum
end;
funct1 mod t2 ->  INT -valued Function means :Def4: :: AOFA_I00:def 4
( dom it = (dom t1) /\ (dom t2) & ( for s being set st s in dom it holds
it . s = (t1 . s) mod (t2 . s) ) );
correctness
existence
ex b1 being INT -valued Function st
( dom b1 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b1 holds
b1 . s = (t1 . s) mod (t2 . s) ) );
uniqueness
for b1, b2 being INT -valued Function st dom b1 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b1 holds
b1 . s = (t1 . s) mod (t2 . s) ) & dom b2 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b2 holds
b2 . s = (t1 . s) mod (t2 . s) ) holds
b1 = b2;
proof
deffunc H1( set ) ->  set = (t1 . $1) mod (t2 . $1);
consider f being Function such that
A9: ( dom f = (dom t1) /\ (dom t2) & ( for x being set st x in (dom t1) /\ (dom t2) holds
f . x = H1(x) ) ) from FUNCT_1:sch_3();
 f is INT -valued
proof
let y be set ; :: according to TARSKI:def_3,RELAT_1:def_19 ::_thesis: ( not y in proj2 f or y in INT )
assume  y in rng f ; ::_thesis: y in INT
then consider x being set  such that
A10: x in dom f and
A11: f . x = y by FUNCT_1:def_3;
 f . x = H1(x) by A9, A10;
hence  y in INT by A11, INT_1:def_2; ::_thesis: verum
end;
hence  ex f being INT -valued Function st
( dom f = (dom t1) /\ (dom t2) & ( for x being set st x in dom f holds
f . x = H1(x) ) ) by A9; ::_thesis: for b1, b2 being INT -valued Function st dom b1 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b1 holds
b1 . s = (t1 . s) mod (t2 . s) ) & dom b2 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b2 holds
b2 . s = (t1 . s) mod (t2 . s) ) holds
b1 = b2
let f1, f2 be INT -valued Function; ::_thesis: ( dom f1 = (dom t1) /\ (dom t2) & ( for s being set st s in dom f1 holds
f1 . s = (t1 . s) mod (t2 . s) ) & dom f2 = (dom t1) /\ (dom t2) & ( for s being set st s in dom f2 holds
f2 . s = (t1 . s) mod (t2 . s) ) implies f1 = f2 )
assume that
A12: dom f1 = (dom t1) /\ (dom t2) and
A13: for s being set st s in dom f1 holds
f1 . s = H1(s) and
A14: dom f2 = (dom t1) /\ (dom t2) and
A15: for s being set st s in dom f2 holds
f2 . s = H1(s) ; ::_thesis: f1 = f2
now__::_thesis:_for_s_being_set_st_s_in_(dom_t1)_/\_(dom_t2)_holds_
f1_._s_=_f2_._s
let s be set ; ::_thesis: ( s in (dom t1) /\ (dom t2) implies f1 . s = f2 . s )
assume A16: s in (dom t1) /\ (dom t2) ; ::_thesis: f1 . s = f2 . s
hence f1 . s = H1(s) by A12, A13
.= f2 . s by A14, A15, A16 ;
::_thesis: verum
end;
hence  f1 = f2 by A12, A14, FUNCT_1:2; ::_thesis: verum
end;
func leq (t1,t2) ->  INT -valued Function means :Def5: :: AOFA_I00:def 5
( dom it = (dom t1) /\ (dom t2) & ( for s being set st s in dom it holds
it . s = IFGT ((t1 . s),(t2 . s),0,1) ) );
correctness
existence
ex b1 being INT -valued Function st
( dom b1 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b1 holds
b1 . s = IFGT ((t1 . s),(t2 . s),0,1) ) );
uniqueness
for b1, b2 being INT -valued Function st dom b1 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b1 holds
b1 . s = IFGT ((t1 . s),(t2 . s),0,1) ) & dom b2 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b2 holds
b2 . s = IFGT ((t1 . s),(t2 . s),0,1) ) holds
b1 = b2;
proof
deffunc H1( set ) ->  set = IFGT ((t1 . $1),(t2 . $1),0,1);
consider f being Function such that
A17: ( dom f = (dom t1) /\ (dom t2) & ( for x being set st x in (dom t1) /\ (dom t2) holds
f . x = H1(x) ) ) from FUNCT_1:sch_3();
 f is INT -valued
proof
let y be set ; :: according to TARSKI:def_3,RELAT_1:def_19 ::_thesis: ( not y in proj2 f or y in INT )
assume  y in rng f ; ::_thesis: y in INT
then consider x being set  such that
A18: x in dom f and
A19: f . x = y by FUNCT_1:def_3;
 f . x = H1(x) by A17, A18;
hence  y in INT by A19, INT_1:def_2; ::_thesis: verum
end;
hence  ex f being INT -valued Function st
( dom f = (dom t1) /\ (dom t2) & ( for x being set st x in dom f holds
f . x = H1(x) ) ) by A17; ::_thesis: for b1, b2 being INT -valued Function st dom b1 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b1 holds
b1 . s = IFGT ((t1 . s),(t2 . s),0,1) ) & dom b2 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b2 holds
b2 . s = IFGT ((t1 . s),(t2 . s),0,1) ) holds
b1 = b2
let f1, f2 be INT -valued Function; ::_thesis: ( dom f1 = (dom t1) /\ (dom t2) & ( for s being set st s in dom f1 holds
f1 . s = IFGT ((t1 . s),(t2 . s),0,1) ) & dom f2 = (dom t1) /\ (dom t2) & ( for s being set st s in dom f2 holds
f2 . s = IFGT ((t1 . s),(t2 . s),0,1) ) implies f1 = f2 )
assume that
A20: dom f1 = (dom t1) /\ (dom t2) and
A21: for s being set st s in dom f1 holds
f1 . s = H1(s) and
A22: dom f2 = (dom t1) /\ (dom t2) and
A23: for s being set st s in dom f2 holds
f2 . s = H1(s) ; ::_thesis: f1 = f2
now__::_thesis:_for_s_being_set_st_s_in_(dom_t1)_/\_(dom_t2)_holds_
f1_._s_=_f2_._s
let s be set ; ::_thesis: ( s in (dom t1) /\ (dom t2) implies f1 . s = f2 . s )
assume A24: s in (dom t1) /\ (dom t2) ; ::_thesis: f1 . s = f2 . s
hence f1 . s = H1(s) by A20, A21
.= f2 . s by A22, A23, A24 ;
::_thesis: verum
end;
hence  f1 = f2 by A20, A22, FUNCT_1:2; ::_thesis: verum
end;
func gt (t1,t2) ->  INT -valued Function means :Def6: :: AOFA_I00:def 6
( dom it = (dom t1) /\ (dom t2) & ( for s being set st s in dom it holds
it . s = IFGT ((t1 . s),(t2 . s),1,0) ) );
correctness
existence
ex b1 being INT -valued Function st
( dom b1 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b1 holds
b1 . s = IFGT ((t1 . s),(t2 . s),1,0) ) );
uniqueness
for b1, b2 being INT -valued Function st dom b1 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b1 holds
b1 . s = IFGT ((t1 . s),(t2 . s),1,0) ) & dom b2 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b2 holds
b2 . s = IFGT ((t1 . s),(t2 . s),1,0) ) holds
b1 = b2;
proof
deffunc H1( set ) ->  set = IFGT ((t1 . $1),(t2 . $1),1,0);
consider f being Function such that
A25: ( dom f = (dom t1) /\ (dom t2) & ( for x being set st x in (dom t1) /\ (dom t2) holds
f . x = H1(x) ) ) from FUNCT_1:sch_3();
 f is INT -valued
proof
let y be set ; :: according to TARSKI:def_3,RELAT_1:def_19 ::_thesis: ( not y in proj2 f or y in INT )
assume  y in rng f ; ::_thesis: y in INT
then consider x being set  such that
A26: x in dom f and
A27: f . x = y by FUNCT_1:def_3;
 f . x = H1(x) by A25, A26;
hence  y in INT by A27, INT_1:def_2; ::_thesis: verum
end;
hence  ex f being INT -valued Function st
( dom f = (dom t1) /\ (dom t2) & ( for x being set st x in dom f holds
f . x = H1(x) ) ) by A25; ::_thesis: for b1, b2 being INT -valued Function st dom b1 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b1 holds
b1 . s = IFGT ((t1 . s),(t2 . s),1,0) ) & dom b2 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b2 holds
b2 . s = IFGT ((t1 . s),(t2 . s),1,0) ) holds
b1 = b2
let f1, f2 be INT -valued Function; ::_thesis: ( dom f1 = (dom t1) /\ (dom t2) & ( for s being set st s in dom f1 holds
f1 . s = IFGT ((t1 . s),(t2 . s),1,0) ) & dom f2 = (dom t1) /\ (dom t2) & ( for s being set st s in dom f2 holds
f2 . s = IFGT ((t1 . s),(t2 . s),1,0) ) implies f1 = f2 )
assume that
A28: dom f1 = (dom t1) /\ (dom t2) and
A29: for s being set st s in dom f1 holds
f1 . s = H1(s) and
A30: dom f2 = (dom t1) /\ (dom t2) and
A31: for s being set st s in dom f2 holds
f2 . s = H1(s) ; ::_thesis: f1 = f2
now__::_thesis:_for_s_being_set_st_s_in_(dom_t1)_/\_(dom_t2)_holds_
f1_._s_=_f2_._s
let s be set ; ::_thesis: ( s in (dom t1) /\ (dom t2) implies f1 . s = f2 . s )
assume A32: s in (dom t1) /\ (dom t2) ; ::_thesis: f1 . s = f2 . s
hence f1 . s = H1(s) by A28, A29
.= f2 . s by A30, A31, A32 ;
::_thesis: verum
end;
hence  f1 = f2 by A28, A30, FUNCT_1:2; ::_thesis: verum
end;
func eq (t1,t2) ->  INT -valued Function means :Def7: :: AOFA_I00:def 7
( dom it = (dom t1) /\ (dom t2) & ( for s being set st s in dom it holds
it . s = IFEQ ((t1 . s),(t2 . s),1,0) ) );
correctness
existence
ex b1 being INT -valued Function st
( dom b1 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b1 holds
b1 . s = IFEQ ((t1 . s),(t2 . s),1,0) ) );
uniqueness
for b1, b2 being INT -valued Function st dom b1 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b1 holds
b1 . s = IFEQ ((t1 . s),(t2 . s),1,0) ) & dom b2 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b2 holds
b2 . s = IFEQ ((t1 . s),(t2 . s),1,0) ) holds
b1 = b2;
proof
deffunc H1( set ) ->  Element of NAT = IFEQ ((t1 . $1),(t2 . $1),1,0);
consider f being Function such that
A33: ( dom f = (dom t1) /\ (dom t2) & ( for x being set st x in (dom t1) /\ (dom t2) holds
f . x = H1(x) ) ) from FUNCT_1:sch_3();
 f is INT -valued
proof
let y be set ; :: according to TARSKI:def_3,RELAT_1:def_19 ::_thesis: ( not y in proj2 f or y in INT )
assume  y in rng f ; ::_thesis: y in INT
then consider x being set  such that
A34: x in dom f and
A35: f . x = y by FUNCT_1:def_3;
 f . x = H1(x) by A33, A34;
hence  y in INT by A35, INT_1:def_2; ::_thesis: verum
end;
hence  ex f being INT -valued Function st
( dom f = (dom t1) /\ (dom t2) & ( for x being set st x in dom f holds
f . x = H1(x) ) ) by A33; ::_thesis: for b1, b2 being INT -valued Function st dom b1 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b1 holds
b1 . s = IFEQ ((t1 . s),(t2 . s),1,0) ) & dom b2 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b2 holds
b2 . s = IFEQ ((t1 . s),(t2 . s),1,0) ) holds
b1 = b2
let f1, f2 be INT -valued Function; ::_thesis: ( dom f1 = (dom t1) /\ (dom t2) & ( for s being set st s in dom f1 holds
f1 . s = IFEQ ((t1 . s),(t2 . s),1,0) ) & dom f2 = (dom t1) /\ (dom t2) & ( for s being set st s in dom f2 holds
f2 . s = IFEQ ((t1 . s),(t2 . s),1,0) ) implies f1 = f2 )
assume that
A36: dom f1 = (dom t1) /\ (dom t2) and
A37: for s being set st s in dom f1 holds
f1 . s = H1(s) and
A38: dom f2 = (dom t1) /\ (dom t2) and
A39: for s being set st s in dom f2 holds
f2 . s = H1(s) ; ::_thesis: f1 = f2
now__::_thesis:_for_s_being_set_st_s_in_(dom_t1)_/\_(dom_t2)_holds_
f1_._s_=_f2_._s
let s be set ; ::_thesis: ( s in (dom t1) /\ (dom t2) implies f1 . s = f2 . s )
assume A40: s in (dom t1) /\ (dom t2) ; ::_thesis: f1 . s = f2 . s
hence f1 . s = H1(s) by A36, A37
.= f2 . s by A38, A39, A40 ;
::_thesis: verum
end;
hence  f1 = f2 by A36, A38, FUNCT_1:2; ::_thesis: verum
end;
end;

:: deftheorem Def3 defines div AOFA_I00:def_3_:_
 for t1, t2, b3 being INT -valued Function holds
( b3 = t1 div t2 iff ( dom b3 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b3 holds
b3 . s = (t1 . s) div (t2 . s) ) ) );

:: deftheorem Def4 defines mod AOFA_I00:def_4_:_
 for t1, t2, b3 being INT -valued Function holds
( b3 = t1 mod t2 iff ( dom b3 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b3 holds
b3 . s = (t1 . s) mod (t2 . s) ) ) );

:: deftheorem Def5 defines leq AOFA_I00:def_5_:_
 for t1, t2, b3 being INT -valued Function holds
( b3 = leq (t1,t2) iff ( dom b3 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b3 holds
b3 . s = IFGT ((t1 . s),(t2 . s),0,1) ) ) );

:: deftheorem Def6 defines gt AOFA_I00:def_6_:_
 for t1, t2, b3 being INT -valued Function holds
( b3 = gt (t1,t2) iff ( dom b3 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b3 holds
b3 . s = IFGT ((t1 . s),(t2 . s),1,0) ) ) );

:: deftheorem Def7 defines eq AOFA_I00:def_7_:_
 for t1, t2, b3 being INT -valued Function holds
( b3 = eq (t1,t2) iff ( dom b3 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b3 holds
b3 . s = IFEQ ((t1 . s),(t2 . s),1,0) ) ) );

definition
let X be non empty set ;
let f be Function of X,INT;
let x be integer number ;
:: original: +
redefine funcf + x ->  Function of X,INT means :Def8: :: AOFA_I00:def 8
 for s being Element of X holds it . s = (f . s) + x;
compatibility
 for b1 being Function of X,INT holds
( b1 = x + f iff for s being Element of X holds b1 . s = (f . s) + x )
proof
let ti be Function of X,INT; ::_thesis: ( ti = x + f iff for s being Element of X holds ti . s = (f . s) + x )
A1: dom f = X by FUNCT_2:def_1;
 dom (f + x) = dom f by VALUED_1:def_2;
hence  ( ti = f + x implies for s being Element of X holds ti . s = (f . s) + x ) by A1, VALUED_1:def_2; ::_thesis: ( ( for s being Element of X holds ti . s = (f . s) + x ) implies ti = x + f )
A2: dom ti = X by FUNCT_2:def_1;
assume  for s being Element of X holds ti . s = (f . s) + x ; ::_thesis: ti = x + f
then  for s being set st s in X holds
ti . s = x + (f . s) ;
hence  ti = x + f by A1, A2, VALUED_1:def_2; ::_thesis: verum
end;
coherence
 x + f is Function of X,INT
proof
 x + f is Function of X,INT ;
hence  x + f is Function of X,INT ; ::_thesis: verum
end;
:: original: -
redefine funcf - x ->  Function of X,INT means :: AOFA_I00:def 9
 for s being Element of X holds it . s = (f . s) - x;
compatibility
 for b1 being Function of X,INT holds
( b1 = f - x iff for s being Element of X holds b1 . s = (f . s) - x )
proof
let ti be Function of X,INT; ::_thesis: ( ti = f - x iff for s being Element of X holds ti . s = (f . s) - x )
A3: dom ti = X by FUNCT_2:def_1;
A4: dom f = X by FUNCT_2:def_1;
hence  ( ti = f - x implies for s being Element of X holds ti . s = (f . s) - x ) by VALUED_1:3; ::_thesis: ( ( for s being Element of X holds ti . s = (f . s) - x ) implies ti = f - x )
assume A5: for s being Element of X holds ti . s = (f . s) - x ; ::_thesis: ti = f - x
A6: now__::_thesis:_for_s_being_set_st_s_in_dom_ti_holds_
ti_._s_=_(f_-_x)_._s
let s be set ; ::_thesis: ( s in dom ti implies ti . s = (f - x) . s )
assume A7: s in dom ti ; ::_thesis: ti . s = (f - x) . s
hence ti . s = (f . s) - x by A5
.= (f - x) . s by A4, A7, VALUED_1:3 ;
::_thesis: verum
end;
 dom (f - x) = dom f by VALUED_1:3;
hence  ti = f - x by A4, A3, A6, FUNCT_1:2; ::_thesis: verum
end;
coherence
 f - x is Function of X,INT
proof
thus  f - x is Function of X,INT ; ::_thesis: verum
end;
:: original: *
redefine funcf * x ->  Function of X,INT means :Def10: :: AOFA_I00:def 10
 for s being Element of X holds it . s = (f . s) * x;
compatibility
 for b1 being Function of X,INT holds
( b1 = f * x iff for s being Element of X holds b1 . s = (f . s) * x )
proof
let ti be Function of X,INT; ::_thesis: ( ti = f * x iff for s being Element of X holds ti . s = (f . s) * x )
A8: dom (f * x) = dom f by VALUED_1:def_5;
A9: dom f = X by FUNCT_2:def_1;
hence  ( ti = f * x implies for s being Element of X holds ti . s = (f . s) * x ) by A8, VALUED_1:def_5; ::_thesis: ( ( for s being Element of X holds ti . s = (f . s) * x ) implies ti = f * x )
A10: dom ti = X by FUNCT_2:def_1;
assume  for s being Element of X holds ti . s = (f . s) * x ; ::_thesis: ti = f * x
then  for s being set st s in dom (f * x) holds
ti . s = x * (f . s) ;
hence  ti = f * x by A8, A9, A10, VALUED_1:def_5; ::_thesis: verum
end;
coherence
 f * x is Function of X,INT
proof
 x (#) f is Function of X,INT ;
hence  f * x is Function of X,INT ; ::_thesis: verum
end;
end;

:: deftheorem Def8 defines + AOFA_I00:def_8_:_
 for X being non empty set
for f being Function of X,INT
for x being integer number
for b4 being Function of X,INT holds
( b4 = f + x iff for s being Element of X holds b4 . s = (f . s) + x );

:: deftheorem  defines - AOFA_I00:def_9_:_
 for X being non empty set
for f being Function of X,INT
for x being integer number
for b4 being Function of X,INT holds
( b4 = f - x iff for s being Element of X holds b4 . s = (f . s) - x );

:: deftheorem Def10 defines * AOFA_I00:def_10_:_
 for X being non empty set
for f being Function of X,INT
for x being integer number
for b4 being Function of X,INT holds
( b4 = f * x iff for s being Element of X holds b4 . s = (f . s) * x );

definition
let X be set ;
let f, g be Function of X,INT;
:: original: div
redefine funcf div g ->  Function of X,INT;
coherence
 f div g is Function of X,INT
proof
A1: dom g = X by FUNCT_2:def_1;
A2: dom (f div g) = (dom f) /\ (dom g) by Def3;
A3: dom f = X by FUNCT_2:def_1;
 rng (f div g) c= INT ;
hence  f div g is Function of X,INT by A2, A3, A1, FUNCT_2:2; ::_thesis: verum
end;
:: original: mod
redefine funcf mod g ->  Function of X,INT;
coherence
 f mod g is Function of X,INT
proof
A4: dom g = X by FUNCT_2:def_1;
A5: dom (f mod g) = (dom f) /\ (dom g) by Def4;
A6: dom f = X by FUNCT_2:def_1;
 rng (f mod g) c= INT ;
hence  f mod g is Function of X,INT by A5, A6, A4, FUNCT_2:2; ::_thesis: verum
end;
:: original: leq
redefine func leq (f,g) ->  Function of X,INT;
coherence
 leq (f,g) is Function of X,INT
proof
A7: dom g = X by FUNCT_2:def_1;
A8: dom (leq (f,g)) = (dom f) /\ (dom g) by Def5;
A9: dom f = X by FUNCT_2:def_1;
 rng (leq (f,g)) c= INT ;
hence  leq (f,g) is Function of X,INT by A8, A9, A7, FUNCT_2:2; ::_thesis: verum
end;
:: original: gt
redefine func gt (f,g) ->  Function of X,INT;
coherence
 gt (f,g) is Function of X,INT
proof
A10: dom g = X by FUNCT_2:def_1;
A11: dom (gt (f,g)) = (dom f) /\ (dom g) by Def6;
A12: dom f = X by FUNCT_2:def_1;
 rng (gt (f,g)) c= INT ;
hence  gt (f,g) is Function of X,INT by A11, A12, A10, FUNCT_2:2; ::_thesis: verum
end;
:: original: eq
redefine func eq (f,g) ->  Function of X,INT;
coherence
 eq (f,g) is Function of X,INT
proof
A13: dom g = X by FUNCT_2:def_1;
A14: dom (eq (f,g)) = (dom f) /\ (dom g) by Def7;
A15: dom f = X by FUNCT_2:def_1;
 rng (eq (f,g)) c= INT ;
hence  eq (f,g) is Function of X,INT by A14, A15, A13, FUNCT_2:2; ::_thesis: verum
end;
end;

definition
let X be non empty set ;
let t1, t2 be Function of X,INT;
:: original: -
redefine funct1 - t2 ->  Function of X,INT means :Def11: :: AOFA_I00:def 11
 for s being Element of X holds it . s = (t1 . s) - (t2 . s);
compatibility
 for b1 being Function of X,INT holds
( b1 = t1 - t2 iff for s being Element of X holds b1 . s = (t1 . s) - (t2 . s) )
proof
let ti be Function of X,INT; ::_thesis: ( ti = t1 - t2 iff for s being Element of X holds ti . s = (t1 . s) - (t2 . s) )
A1: dom t1 = X by FUNCT_2:def_1;
thus  ( ti = t1 - t2 implies for s being Element of X holds ti . s = (t1 . s) - (t2 . s) ) by VALUED_1:15; ::_thesis: ( ( for s being Element of X holds ti . s = (t1 . s) - (t2 . s) ) implies ti = t1 - t2 )
A2: dom ti = X by FUNCT_2:def_1;
assume A3: for s being Element of X holds ti . s = (t1 . s) - (t2 . s) ; ::_thesis: ti = t1 - t2
A4: now__::_thesis:_for_s_being_set_st_s_in_X_holds_
ti_._s_=_(t1_-_t2)_._s
let s be set ; ::_thesis: ( s in X implies ti . s = (t1 - t2) . s )
assume A5: s in X ; ::_thesis: ti . s = (t1 - t2) . s
hence ti . s = (t1 . s) - (t2 . s) by A3
.= (t1 - t2) . s by A5, VALUED_1:15 ;
::_thesis: verum
end;
A6: dom t2 = X by FUNCT_2:def_1;
 dom (t1 - t2) = (dom t1) /\ (dom t2) by VALUED_1:12;
hence  ti = t1 - t2 by A1, A6, A2, A4, FUNCT_1:2; ::_thesis: verum
end;
coherence
 t1 - t2 is Function of X,INT
proof
thus  t1 - t2 is Function of X,INT ; ::_thesis: verum
end;
:: original: +
redefine funct1 + t2 ->  Function of X,INT means :: AOFA_I00:def 12
 for s being Element of X holds it . s = (t1 . s) + (t2 . s);
compatibility
 for b1 being Function of X,INT holds
( b1 = t1 + t2 iff for s being Element of X holds b1 . s = (t1 . s) + (t2 . s) )
proof
let ti be Function of X,INT; ::_thesis: ( ti = t1 + t2 iff for s being Element of X holds ti . s = (t1 . s) + (t2 . s) )
A7: dom t1 = X by FUNCT_2:def_1;
A8: dom t2 = X by FUNCT_2:def_1;
A9: dom (t1 + t2) = (dom t1) /\ (dom t2) by VALUED_1:def_1;
hence  ( ti = t1 + t2 implies for s being Element of X holds ti . s = (t1 . s) + (t2 . s) ) by A7, A8, VALUED_1:def_1; ::_thesis: ( ( for s being Element of X holds ti . s = (t1 . s) + (t2 . s) ) implies ti = t1 + t2 )
A10: dom ti = X by FUNCT_2:def_1;
assume  for s being Element of X holds ti . s = (t1 . s) + (t2 . s) ; ::_thesis: ti = t1 + t2
then  for s being set st s in X holds
ti . s = (t1 . s) + (t2 . s) ;
hence  ti = t1 + t2 by A9, A7, A8, A10, VALUED_1:def_1; ::_thesis: verum
end;
coherence
 t1 + t2 is Function of X,INT
proof
thus  t1 + t2 is Function of X,INT ; ::_thesis: verum
end;
end;

:: deftheorem Def11 defines - AOFA_I00:def_11_:_
 for X being non empty set
for t1, t2, b4 being Function of X,INT holds
( b4 = t1 - t2 iff for s being Element of X holds b4 . s = (t1 . s) - (t2 . s) );

:: deftheorem  defines + AOFA_I00:def_12_:_
 for X being non empty set
for t1, t2, b4 being Function of X,INT holds
( b4 = t1 + t2 iff for s being Element of X holds b4 . s = (t1 . s) + (t2 . s) );

registration
let A be non empty set ;
let B be infinite set ;
cluster Funcs (A,B) ->  infinite ;
coherence
 not Funcs (A,B) is finite
proof
A1: card (card B) = card B ;
 card A = card (card A) ;
then A2: card (Funcs (A,B)) = exp ((card B),(card A)) by A1, FUNCT_5:61;
set a = the Element of A;
A3: card { the Element of A} = 1 by CARD_1:30;
 { the Element of A} c= A by ZFMISC_1:31;
then  1 c= card A by A3, CARD_1:11;
then  exp ((card B),1) c= card (Funcs (A,B)) by A2, CARD_2:93;
hence  not Funcs (A,B) is finite by CARD_2:27; ::_thesis: verum
end;
end;

definition
let N be set ;
let v, f be Function;
funcv ** (f,N) ->  Function means :Def13: :: AOFA_I00:def 13
( ex Y being set st
( ( for y being set holds
( y in Y iff ex h being Function st
( h in dom v & y in rng h ) ) ) & ( for a being set holds
( a in dom it iff ( a in Funcs (N,Y) & ex g being Function st
( a = g & g * f in dom v ) ) ) ) ) & ( for g being Function st g in dom it holds
it . g = v . (g * f) ) );
uniqueness
 for b1, b2 being Function st ex Y being set st
( ( for y being set holds
( y in Y iff ex h being Function st
( h in dom v & y in rng h ) ) ) & ( for a being set holds
( a in dom b1 iff ( a in Funcs (N,Y) & ex g being Function st
( a = g & g * f in dom v ) ) ) ) ) & ( for g being Function st g in dom b1 holds
b1 . g = v . (g * f) ) & ex Y being set st
( ( for y being set holds
( y in Y iff ex h being Function st
( h in dom v & y in rng h ) ) ) & ( for a being set holds
( a in dom b2 iff ( a in Funcs (N,Y) & ex g being Function st
( a = g & g * f in dom v ) ) ) ) ) & ( for g being Function st g in dom b2 holds
b2 . g = v . (g * f) ) holds
b1 = b2
proof
let F1, F2 be Function; ::_thesis: ( ex Y being set st
( ( for y being set holds
( y in Y iff ex h being Function st
( h in dom v & y in rng h ) ) ) & ( for a being set holds
( a in dom F1 iff ( a in Funcs (N,Y) & ex g being Function st
( a = g & g * f in dom v ) ) ) ) ) & ( for g being Function st g in dom F1 holds
F1 . g = v . (g * f) ) & ex Y being set st
( ( for y being set holds
( y in Y iff ex h being Function st
( h in dom v & y in rng h ) ) ) & ( for a being set holds
( a in dom F2 iff ( a in Funcs (N,Y) & ex g being Function st
( a = g & g * f in dom v ) ) ) ) ) & ( for g being Function st g in dom F2 holds
F2 . g = v . (g * f) ) implies F1 = F2 )
given Y1 being set  such that A1: for y being set holds
( y in Y1 iff ex h being Function st
( h in dom v & y in rng h ) ) and
A2: for a being set holds
( a in dom F1 iff ( a in Funcs (N,Y1) & ex g being Function st
( a = g & g * f in dom v ) ) ) ; ::_thesis: ( ex g being Function st
( g in dom F1 & not F1 . g = v . (g * f) ) or for Y being set holds
( ex y being set st
( ( y in Y implies ex h being Function st
( h in dom v & y in rng h ) ) implies ( ex h being Function st
( h in dom v & y in rng h ) & not y in Y ) ) or ex a being set st
( ( a in dom F2 implies ( a in Funcs (N,Y) & ex g being Function st
( a = g & g * f in dom v ) ) ) implies ( a in Funcs (N,Y) & ex g being Function st
( a = g & g * f in dom v ) & not a in dom F2 ) ) ) or ex g being Function st
( g in dom F2 & not F2 . g = v . (g * f) ) or F1 = F2 )
assume A3: for g being Function st g in dom F1 holds
F1 . g = v . (g * f) ; ::_thesis: ( for Y being set holds
( ex y being set st
( ( y in Y implies ex h being Function st
( h in dom v & y in rng h ) ) implies ( ex h being Function st
( h in dom v & y in rng h ) & not y in Y ) ) or ex a being set st
( ( a in dom F2 implies ( a in Funcs (N,Y) & ex g being Function st
( a = g & g * f in dom v ) ) ) implies ( a in Funcs (N,Y) & ex g being Function st
( a = g & g * f in dom v ) & not a in dom F2 ) ) ) or ex g being Function st
( g in dom F2 & not F2 . g = v . (g * f) ) or F1 = F2 )
given Y2 being set  such that A4: for y being set holds
( y in Y2 iff ex h being Function st
( h in dom v & y in rng h ) ) and
A5: for a being set holds
( a in dom F2 iff ( a in Funcs (N,Y2) & ex g being Function st
( a = g & g * f in dom v ) ) ) ; ::_thesis: ( ex g being Function st
( g in dom F2 & not F2 . g = v . (g * f) ) or F1 = F2 )
now__::_thesis:_for_a_being_set_holds_
(_a_in_Y1_iff_a_in_Y2_)
let a be set ; ::_thesis: ( a in Y1 iff a in Y2 )
 ( a in Y1 iff ex h being Function st
( h in dom v & a in rng h ) ) by A1;
hence  ( a in Y1 iff a in Y2 ) by A4; ::_thesis: verum
end;
then A6: Y1 = Y2 by TARSKI:1;
now__::_thesis:_for_a_being_set_holds_
(_a_in_dom_F1_iff_a_in_dom_F2_)
let a be set ; ::_thesis: ( a in dom F1 iff a in dom F2 )
 ( a in dom F1 iff ( a in Funcs (N,Y1) & ex g being Function st
( a = g & g * f in dom v ) ) ) by A2;
hence  ( a in dom F1 iff a in dom F2 ) by A5, A6; ::_thesis: verum
end;
then A7: dom F1 = dom F2 by TARSKI:1;
assume A8: for g being Function st g in dom F2 holds
F2 . g = v . (g * f) ; ::_thesis: F1 = F2
now__::_thesis:_for_a_being_set_st_a_in_dom_F1_holds_
F1_._a_=_F2_._a
let a be set ; ::_thesis: ( a in dom F1 implies F1 . a = F2 . a )
assume A9: a in dom F1 ; ::_thesis: F1 . a = F2 . a
then consider g being Function such that
A10: a = g and
 g * f in dom v by A2;
thus F1 . a = v . (g * f) by A3, A9, A10
.= F2 . a by A8, A7, A9, A10 ; ::_thesis: verum
end;
hence  F1 = F2 by A7, FUNCT_1:2; ::_thesis: verum
end;
existence
 ex b1 being Function st
( ex Y being set st
( ( for y being set holds
( y in Y iff ex h being Function st
( h in dom v & y in rng h ) ) ) & ( for a being set holds
( a in dom b1 iff ( a in Funcs (N,Y) & ex g being Function st
( a = g & g * f in dom v ) ) ) ) ) & ( for g being Function st g in dom b1 holds
b1 . g = v . (g * f) ) )
proof
defpred S1[ set , set ] means  ex g being Function st
( g = $1 & $2 = v . (g * f) );
defpred S2[ set ] means  ex g being Function st
( g = $1 & g * f in dom v );
defpred S3[ set , set ] means  ex g being Function st
( $1 = g & $2 = rng g );
A11: for x, y, z being set st S3[x,y] & S3[x,z] holds
y = z ;
consider X being set  such that
A12: for x being set holds
( x in X iff ex y being set st
( y in dom v & S3[y,x] ) ) from TARSKI:sch_1(A11);
set Y = union X;
consider D being set  such that
A13: for x being set holds
( x in D iff ( x in Funcs (N,(union X)) & S2[x] ) ) from XBOOLE_0:sch_1();
A14: for x being set st x in D holds
ex y being set st S1[x,y]
proof
let x be set ; ::_thesis: ( x in D implies ex y being set st S1[x,y] )
assume  x in D ; ::_thesis: ex y being set st S1[x,y]
then consider y being Function such that
A15: y = x and
 y * f in dom v by A13;
take  v . (y * f) ; ::_thesis: S1[x,v . (y * f)]
thus  S1[x,v . (y * f)] by A15; ::_thesis: verum
end;
consider F being Function such that
A16: ( dom F = D & ( for g being set st g in D holds
S1[g,F . g] ) ) from CLASSES1:sch_1(A14);
take  F ; ::_thesis: ( ex Y being set st
( ( for y being set holds
( y in Y iff ex h being Function st
( h in dom v & y in rng h ) ) ) & ( for a being set holds
( a in dom F iff ( a in Funcs (N,Y) & ex g being Function st
( a = g & g * f in dom v ) ) ) ) ) & ( for g being Function st g in dom F holds
F . g = v . (g * f) ) )
hereby  ::_thesis: for g being Function st g in dom F holds
F . g = v . (g * f)
take Y = union X; ::_thesis: ( ( for y being set holds
( ( y in Y implies ex h being Function st
( h in dom v & y in rng h ) ) & ( ex h being Function st
( h in dom v & y in rng h ) implies y in Y ) ) ) & ( for a being set holds
( a in dom F iff ( a in Funcs (N,Y) & ex g being Function st
( a = g & g * f in dom v ) ) ) ) )
hereby  ::_thesis: for a being set holds
( a in dom F iff ( a in Funcs (N,Y) & ex g being Function st
( a = g & g * f in dom v ) ) )
let y be set ; ::_thesis: ( ( y in Y implies ex h being Function st
( h in dom v & y in rng h ) ) & ( ex h being Function st
( h in dom v & y in rng h ) implies y in Y ) )
hereby  ::_thesis: ( ex h being Function st
( h in dom v & y in rng h ) implies y in Y )
assume  y in Y ; ::_thesis: ex h being Function st
( h in dom v & y in rng h )
then consider a being set  such that
A17: y in a and
A18: a in X by TARSKI:def_4;
 ex z being set st
( z in dom v & S3[z,a] ) by A12, A18;
hence  ex h being Function st
( h in dom v & y in rng h ) by A17; ::_thesis: verum
end;
given h being Function such that A19: h in dom v and
A20: y in rng h ; ::_thesis: y in Y
 rng h in X by A12, A19;
hence  y in Y by A20, TARSKI:def_4; ::_thesis: verum
end;
let a be set ; ::_thesis: ( a in dom F iff ( a in Funcs (N,Y) & ex g being Function st
( a = g & g * f in dom v ) ) )
thus  ( a in dom F iff ( a in Funcs (N,Y) & ex g being Function st
( a = g & g * f in dom v ) ) ) by A13, A16; ::_thesis: verum
end;
let g be Function; ::_thesis: ( g in dom F implies F . g = v . (g * f) )
assume  g in dom F ; ::_thesis: F . g = v . (g * f)
then  S1[g,F . g] by A16;
hence  F . g = v . (g * f) ; ::_thesis: verum
end;
end;

:: deftheorem Def13 defines ** AOFA_I00:def_13_:_
 for N being set
for v, f, b4 being Function holds
( b4 = v ** (f,N) iff ( ex Y being set st
( ( for y being set holds
( y in Y iff ex h being Function st
( h in dom v & y in rng h ) ) ) & ( for a being set holds
( a in dom b4 iff ( a in Funcs (N,Y) & ex g being Function st
( a = g & g * f in dom v ) ) ) ) ) & ( for g being Function st g in dom b4 holds
b4 . g = v . (g * f) ) ) );

definition
let X, Y, Z, N be non empty set ;
let v be Element of Funcs ((Funcs (X,Y)),Z);
let f be Function of X,N;
:: original: **
redefine funcv ** (f,N) ->  Element of Funcs ((Funcs (N,Y)),Z);
coherence
 v ** (f,N) is Element of Funcs ((Funcs (N,Y)),Z)
proof
consider Y0 being set  such that
A1: for y being set holds
( y in Y0 iff ex h being Function st
( h in dom v & y in rng h ) ) and
A2: for a being set holds
( a in dom (v ** (f,N)) iff ( a in Funcs (N,Y0) & ex g being Function st
( a = g & g * f in dom v ) ) ) by Def13;
A3: dom v = Funcs (X,Y) by FUNCT_2:def_1;
A4: Y0 = Y
proof
thus  Y0 c= Y :: according to XBOOLE_0:def_10 ::_thesis: Y c= Y0
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in Y0 or y in Y )
assume  y in Y0 ; ::_thesis: y in Y
then consider h being Function such that
A5: h in dom v and
A6: y in rng h by A1;
 rng h c= Y by A5, FUNCT_2:92;
hence  y in Y by A6; ::_thesis: verum
end;
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in Y or y in Y0 )
assume  y in Y ; ::_thesis: y in Y0
then reconsider y = y as Element of Y ;
reconsider h = X --> y as Function of X,Y ;
A7: rng h = {y} by FUNCOP_1:8;
A8: h in Funcs (X,Y) by FUNCT_2:8;
 y in {y} by TARSKI:def_1;
hence  y in Y0 by A3, A1, A7, A8; ::_thesis: verum
end;
A9: dom (v ** (f,N)) = Funcs (N,Y)
proof
thus  dom (v ** (f,N)) c= Funcs (N,Y) :: according to XBOOLE_0:def_10 ::_thesis: Funcs (N,Y) c= dom (v ** (f,N))
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in dom (v ** (f,N)) or a in Funcs (N,Y) )
thus  ( not a in dom (v ** (f,N)) or a in Funcs (N,Y) ) by A2, A4; ::_thesis: verum
end;
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in Funcs (N,Y) or a in dom (v ** (f,N)) )
assume A10: a in Funcs (N,Y) ; ::_thesis: a in dom (v ** (f,N))
then reconsider g = a as Function of N,Y by FUNCT_2:66;
 g * f in Funcs (X,Y) by FUNCT_2:8;
hence  a in dom (v ** (f,N)) by A3, A2, A4, A10; ::_thesis: verum
end;
 rng (v ** (f,N)) c= Z
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in rng (v ** (f,N)) or a in Z )
assume  a in rng (v ** (f,N)) ; ::_thesis: a in Z
then consider g being set  such that
A11: g in dom (v ** (f,N)) and
A12: a = (v ** (f,N)) . g by FUNCT_1:def_3;
reconsider g = g as Element of Funcs (N,Y) by A9, A11;
reconsider gf = g * f as Element of Funcs (X,Y) by FUNCT_2:8;
 a = v . gf by A11, A12, Def13;
hence  a in Z ; ::_thesis: verum
end;
hence  v ** (f,N) is Element of Funcs ((Funcs (N,Y)),Z) by A9, FUNCT_2:def_2; ::_thesis: verum
end;
end;

theorem Th3: :: AOFA_I00:3
for f1, f2, g being Function st rng g c= dom f2 holds
(f1 +* f2) * g = f2 * g
proof
let f1, f2, g be Function; ::_thesis: ( rng g c= dom f2 implies (f1 +* f2) * g = f2 * g )
assume A1: rng g c= dom f2 ; ::_thesis: (f1 +* f2) * g = f2 * g
A2: now__::_thesis:_for_x_being_set_st_x_in_dom_g_holds_
((f1_+*_f2)_*_g)_._x_=_(f2_*_g)_._x
let x be set ; ::_thesis: ( x in dom g implies ((f1 +* f2) * g) . x = (f2 * g) . x )
assume A3: x in dom g ; ::_thesis: ((f1 +* f2) * g) . x = (f2 * g) . x
then A4: (f2 * g) . x = f2 . (g . x) by FUNCT_1:13;
A5: g . x in rng g by A3, FUNCT_1:3;
 ((f1 +* f2) * g) . x = (f1 +* f2) . (g . x) by A3, FUNCT_1:13;
hence  ((f1 +* f2) * g) . x = (f2 * g) . x by A1, A4, A5, FUNCT_4:13; ::_thesis: verum
end;
 dom (f1 +* f2) = (dom f1) \/ (dom f2) by FUNCT_4:def_1;
then A6: dom ((f1 +* f2) * g) = dom g by A1, RELAT_1:27, XBOOLE_1:10;
 dom (f2 * g) = dom g by A1, RELAT_1:27;
hence  (f1 +* f2) * g = f2 * g by A6, A2, FUNCT_1:2; ::_thesis: verum
end;

theorem Th4: :: AOFA_I00:4
for X, N, I being non empty set
for s being Function of X,I
for c being Function of X,N st c is one-to-one holds
for n being Element of I holds (N --> n) +* (s * (c ")) is Function of N,I
proof
let X, N, I be non empty set ; ::_thesis: for s being Function of X,I
for c being Function of X,N st c is one-to-one holds
for n being Element of I holds (N --> n) +* (s * (c ")) is Function of N,I
let s be Function of X,I; ::_thesis: for c being Function of X,N st c is one-to-one holds
for n being Element of I holds (N --> n) +* (s * (c ")) is Function of N,I
let c be Function of X,N; ::_thesis: ( c is one-to-one implies for n being Element of I holds (N --> n) +* (s * (c ")) is Function of N,I )
assume  c is one-to-one ; ::_thesis: for n being Element of I holds (N --> n) +* (s * (c ")) is Function of N,I
then A1: dom (c ") = rng c by FUNCT_1:33;
let n be Element of I; ::_thesis: (N --> n) +* (s * (c ")) is Function of N,I
set f = N --> n;
set g = s * (c ");
A2: dom (s * (c ")) c= dom (c ") by RELAT_1:25;
 rng (s * (c ")) c= rng s by RELAT_1:26;
then  rng (s * (c ")) c= I by XBOOLE_1:1;
then A3: (rng (N --> n)) \/ (rng (s * (c "))) c= I by XBOOLE_1:8;
A4: dom (N --> n) = N by FUNCOP_1:13;
A5: rng ((N --> n) +* (s * (c "))) c= (rng (N --> n)) \/ (rng (s * (c "))) by FUNCT_4:17;
 dom ((N --> n) +* (s * (c "))) = (dom (N --> n)) \/ (dom (s * (c "))) by FUNCT_4:def_1;
then  dom ((N --> n) +* (s * (c "))) = N by A2, A4, A1, XBOOLE_1:1, XBOOLE_1:12;
hence  (N --> n) +* (s * (c ")) is Function of N,I by A5, A3, FUNCT_2:2, XBOOLE_1:1; ::_thesis: verum
end;

theorem Th5: :: AOFA_I00:5
for N, X, I being non empty set
for v1, v2 being Function st dom v1 = dom v2 & dom v1 = Funcs (X,I) holds
for f being Function of X,N st f is one-to-one & v1 ** (f,N) = v2 ** (f,N) holds
v1 = v2
proof
let N, X, I be non empty set ; ::_thesis: for v1, v2 being Function st dom v1 = dom v2 & dom v1 = Funcs (X,I) holds
for f being Function of X,N st f is one-to-one & v1 ** (f,N) = v2 ** (f,N) holds
v1 = v2
let v1, v2 be Function; ::_thesis: ( dom v1 = dom v2 & dom v1 = Funcs (X,I) implies for f being Function of X,N st f is one-to-one & v1 ** (f,N) = v2 ** (f,N) holds
v1 = v2 )
assume that
A1: dom v1 = dom v2 and
A2: dom v1 = Funcs (X,I) ; ::_thesis: for f being Function of X,N st f is one-to-one & v1 ** (f,N) = v2 ** (f,N) holds
v1 = v2
reconsider rv1 = rng v1, rv2 = rng v2 as non empty set by A1, A2, RELAT_1:42;
reconsider Z = rv1 \/ rv2 as non empty set ;
A3: rng v2 c= Z by XBOOLE_1:7;
 rng v1 c= Z by XBOOLE_1:7;
then reconsider f1 = v1, f2 = v2 as Element of Funcs ((Funcs (X,I)),Z) by A1, A2, A3, FUNCT_2:def_2;
let f be Function of X,N; ::_thesis: ( f is one-to-one & v1 ** (f,N) = v2 ** (f,N) implies v1 = v2 )
assume that
A4: f is one-to-one and
A5: v1 ** (f,N) = v2 ** (f,N) ; ::_thesis: v1 = v2
A6: dom (f2 ** (f,N)) = Funcs (N,I) by FUNCT_2:def_1;
A7: dom (f1 ** (f,N)) = Funcs (N,I) by FUNCT_2:def_1;
now__::_thesis:_for_a_being_set_st_a_in_Funcs_(X,I)_holds_
v1_._a_=_v2_._a
set i = the Element of I;
let a be set ; ::_thesis: ( a in Funcs (X,I) implies v1 . a = v2 . a )
A8: dom f = X by FUNCT_2:def_1;
assume  a in Funcs (X,I) ; ::_thesis: v1 . a = v2 . a
then reconsider h = a as Element of Funcs (X,I) ;
set g = (N --> the Element of I) +* (h * (f "));
A9: dom h = X by FUNCT_2:def_1;
 rng (f ") = dom f by A4, FUNCT_1:33;
then  dom (h * (f ")) = dom (f ") by A9, RELAT_1:27
.= rng f by A4, FUNCT_1:33 ;
then A10: ((N --> the Element of I) +* (h * (f "))) * f = (h * (f ")) * f by Th3
.= h * ((f ") * f) by RELAT_1:36
.= h * (id X) by A4, A8, FUNCT_1:39
.= a by A9, RELAT_1:52 ;
 (N --> the Element of I) +* (h * (f ")) is Function of N,I by A4, Th4;
then A11: (N --> the Element of I) +* (h * (f ")) in Funcs (N,I) by FUNCT_2:8;
then  (v1 ** (f,N)) . ((N --> the Element of I) +* (h * (f "))) = v1 . a by A7, A10, Def13;
hence  v1 . a = v2 . a by A5, A6, A11, A10, Def13; ::_thesis: verum
end;
hence  v1 = v2 by A1, A2, FUNCT_1:2; ::_thesis: verum
end;

registration
let X be set ;
cluster Relation-like X -defined card X -valued Function-like one-to-one quasi_total onto for Element of bool [:X,(card X):];
existence
 ex b1 being Function of X,(card X) st
( b1 is one-to-one & b1 is onto )
proof
 X, card X are_equipotent by CARD_1:def_2;
then consider f being Function such that
A1: f is one-to-one and
A2: dom f = X and
A3: rng f = card X by WELLORD2:def_4;
reconsider f = f as Function of X,(card X) by A2, A3, FUNCT_2:2;
take  f ; ::_thesis: ( f is one-to-one & f is onto )
thus  f is one-to-one by A1; ::_thesis: f is onto
thus  rng f = card X by A3; :: according to FUNCT_2:def_3 ::_thesis: verum
end;
cluster Relation-like card X -defined X -valued Function-like one-to-one quasi_total onto for Element of bool [:(card X),X:];
existence
 ex b1 being Function of (card X),X st
( b1 is one-to-one & b1 is onto )
proof
 X, card X are_equipotent by CARD_1:def_2;
then consider f being Function such that
A4: f is one-to-one and
A5: dom f = card X and
A6: rng f = X by WELLORD2:def_4;
reconsider f = f as Function of (card X),X by A5, A6, FUNCT_2:2;
take  f ; ::_thesis: ( f is one-to-one & f is onto )
thus  f is one-to-one by A4; ::_thesis: f is onto
thus  rng f = X by A6; :: according to FUNCT_2:def_3 ::_thesis: verum
end;
end;

definition
let X be set ;
mode Enumeration of X is one-to-one onto Function of X,(card X);
mode Denumeration of X is one-to-one onto Function of (card X),X;
end;

theorem Th6: :: AOFA_I00:6
for X being set
for f being Function holds
( f is Enumeration of X iff ( dom f = X & rng f = card X & f is one-to-one ) )
proof
let X be set ; ::_thesis: for f being Function holds
( f is Enumeration of X iff ( dom f = X & rng f = card X & f is one-to-one ) )
let f be Function; ::_thesis: ( f is Enumeration of X iff ( dom f = X & rng f = card X & f is one-to-one ) )
 ( card X = {} implies X = {} ) ;
hence  ( f is Enumeration of X iff ( dom f = X & rng f = card X & f is one-to-one ) ) by FUNCT_2:2, FUNCT_2:def_1, FUNCT_2:def_3; ::_thesis: verum
end;

theorem Th7: :: AOFA_I00:7
for X being set
for f being Function holds
( f is Denumeration of X iff ( dom f = card X & rng f = X & f is one-to-one ) )
proof
let X be set ; ::_thesis: for f being Function holds
( f is Denumeration of X iff ( dom f = card X & rng f = X & f is one-to-one ) )
let f be Function; ::_thesis: ( f is Denumeration of X iff ( dom f = card X & rng f = X & f is one-to-one ) )
 ( X = {} implies card X = {} ) ;
hence  ( f is Denumeration of X iff ( dom f = card X & rng f = X & f is one-to-one ) ) by FUNCT_2:2, FUNCT_2:def_1, FUNCT_2:def_3; ::_thesis: verum
end;

theorem Th8: :: AOFA_I00:8
for X being non empty set
for x, y being Element of X
for f being Enumeration of X holds (f +* (x,(f . y))) +* (y,(f . x)) is Enumeration of X
proof
let X be non empty set ; ::_thesis: for x, y being Element of X
for f being Enumeration of X holds (f +* (x,(f . y))) +* (y,(f . x)) is Enumeration of X
let x, y be Element of X; ::_thesis: for f being Enumeration of X holds (f +* (x,(f . y))) +* (y,(f . x)) is Enumeration of X
let f be Enumeration of X; ::_thesis: (f +* (x,(f . y))) +* (y,(f . x)) is Enumeration of X
set g = (f +* (x,(f . y))) +* (y,(f . x));
set A = dom ((f +* (x,(f . y))) +* (y,(f . x)));
A1: dom (f +* (x,(f . y))) = dom f by FUNCT_7:30;
A2: dom ((f +* (x,(f . y))) +* (y,(f . x))) = dom (f +* (x,(f . y))) by FUNCT_7:30;
A3: dom f = X by Th6;
A4: rng f = card X by Th6;
A5: rng ((f +* (x,(f . y))) +* (y,(f . x))) = rng f
proof
 {(f . x)} c= rng f by A4, ZFMISC_1:31;
then A6: (rng f) \/ {(f . x)} = rng f by XBOOLE_1:12;
A7: rng ((f +* (x,(f . y))) +* (y,(f . x))) c= (rng (f +* (x,(f . y)))) \/ {(f . x)} by FUNCT_7:100;
 {(f . y)} c= rng f by A4, ZFMISC_1:31;
then  (rng f) \/ {(f . y)} = rng f by XBOOLE_1:12;
then  (rng (f +* (x,(f . y)))) \/ {(f . x)} c= rng f by A6, FUNCT_7:100, XBOOLE_1:9;
hence  rng ((f +* (x,(f . y))) +* (y,(f . x))) c= rng f by A7, XBOOLE_1:1; :: according to XBOOLE_0:def_10 ::_thesis: rng f c= rng ((f +* (x,(f . y))) +* (y,(f . x)))
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in rng f or z in rng ((f +* (x,(f . y))) +* (y,(f . x))) )
assume  z in rng f ; ::_thesis: z in rng ((f +* (x,(f . y))) +* (y,(f . x)))
then consider a being set  such that
A8: a in dom f and
A9: z = f . a by FUNCT_1:def_3;
percases ( ( a <> x & a <> y ) or a = x or a = y ) ;
supposeA10: ( a <> x & a <> y ) ; ::_thesis: z in rng ((f +* (x,(f . y))) +* (y,(f . x)))
then A11: ((f +* (x,(f . y))) +* (y,(f . x))) . a = (f +* (x,(f . y))) . a by FUNCT_7:32;
 (f +* (x,(f . y))) . a = z by A9, A10, FUNCT_7:32;
hence  z in rng ((f +* (x,(f . y))) +* (y,(f . x))) by A1, A2, A8, A11, FUNCT_1:def_3; ::_thesis: verum
end;
suppose a = x ; ::_thesis: z in rng ((f +* (x,(f . y))) +* (y,(f . x)))
then  ((f +* (x,(f . y))) +* (y,(f . x))) . y = z by A1, A3, A9, FUNCT_7:31;
hence  z in rng ((f +* (x,(f . y))) +* (y,(f . x))) by A1, A2, A3, FUNCT_1:def_3; ::_thesis: verum
end;
supposeA12: a = y ; ::_thesis: z in rng ((f +* (x,(f . y))) +* (y,(f . x)))
then A13: ( x <> y implies ((f +* (x,(f . y))) +* (y,(f . x))) . x = (f +* (x,z)) . x ) by A9, FUNCT_7:32;
A14: (f +* (x,z)) . x = z by A3, FUNCT_7:31;
 ( x = y implies ((f +* (x,(f . y))) +* (y,(f . x))) . x = z ) by A1, A3, A9, A12, FUNCT_7:31;
hence  z in rng ((f +* (x,(f . y))) +* (y,(f . x))) by A1, A2, A3, A14, A13, FUNCT_1:def_3; ::_thesis: verum
end;
end;
end;
 (f +* (x,(f . y))) +* (y,(f . x)) is one-to-one
proof
let a, b be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not a in proj1 ((f +* (x,(f . y))) +* (y,(f . x))) or not b in proj1 ((f +* (x,(f . y))) +* (y,(f . x))) or not ((f +* (x,(f . y))) +* (y,(f . x))) . a = ((f +* (x,(f . y))) +* (y,(f . x))) . b or a = b )
A15: ( a <> y implies ((f +* (x,(f . y))) +* (y,(f . x))) . a = (f +* (x,(f . y))) . a ) by FUNCT_7:32;
A16: ( a <> x implies (f +* (x,(f . y))) . a = f . a ) by FUNCT_7:32;
A17: ( b = y implies ((f +* (x,(f . y))) +* (y,(f . x))) . b = f . x ) by A1, A3, FUNCT_7:31;
A18: ( b <> y implies ((f +* (x,(f . y))) +* (y,(f . x))) . b = (f +* (x,(f . y))) . b ) by FUNCT_7:32;
A19: ( b = x implies (f +* (x,(f . y))) . b = f . y ) by A3, FUNCT_7:31;
A20: ( a = x implies (f +* (x,(f . y))) . a = f . y ) by A3, FUNCT_7:31;
A21: ( b <> x implies (f +* (x,(f . y))) . b = f . b ) by FUNCT_7:32;
 ( a = y implies ((f +* (x,(f . y))) +* (y,(f . x))) . a = f . x ) by A1, A3, FUNCT_7:31;
hence  ( not a in proj1 ((f +* (x,(f . y))) +* (y,(f . x))) or not b in proj1 ((f +* (x,(f . y))) +* (y,(f . x))) or not ((f +* (x,(f . y))) +* (y,(f . x))) . a = ((f +* (x,(f . y))) +* (y,(f . x))) . b or a = b ) by A2, A3, A15, A20, A16, A17, A18, A19, A21, FUNCT_1:def_4; ::_thesis: verum
end;
hence  (f +* (x,(f . y))) +* (y,(f . x)) is Enumeration of X by A1, A2, A3, A4, A5, Th6; ::_thesis: verum
end;

theorem :: AOFA_I00:9
for X being non empty set
for x being Element of X ex f being Enumeration of X st f . x = 0
proof
let X be non empty set ; ::_thesis: for x being Element of X ex f being Enumeration of X st f . x = 0
let x be Element of X; ::_thesis: ex f being Enumeration of X st f . x = 0
set f = the Enumeration of X;
A1: 0 in card X by ORDINAL3:8;
A2: rng the Enumeration of X = card X by Th6;
 dom the Enumeration of X = X by Th6;
then consider y being set  such that
A3: y in X and
A4: 0 = the Enumeration of X . y by A1, A2, FUNCT_1:def_3;
reconsider y = y as Element of X by A3;
reconsider g = ( the Enumeration of X +* (y,( the Enumeration of X . x))) +* (x,0) as Enumeration of X by A4, Th8;
take  g ; ::_thesis: g . x = 0
 dom the Enumeration of X = X by Th6;
then  dom ( the Enumeration of X +* (y,( the Enumeration of X . x))) = X by FUNCT_7:30;
hence  g . x = 0 by FUNCT_7:31; ::_thesis: verum
end;

theorem :: AOFA_I00:10
for X being non empty set
for f being Denumeration of X holds f . 0 in X by FUNCT_2:5, ORDINAL3:8;

theorem Th11: :: AOFA_I00:11
for X being countable set
for f being Enumeration of X holds rng f c= NAT
proof
let X be countable set ; ::_thesis: for f being Enumeration of X holds rng f c= NAT
let f be Enumeration of X; ::_thesis: rng f c= NAT
 card X c= NAT by CARD_3:def_14;
hence  rng f c= NAT by Th6; ::_thesis: verum
end;

definition
let X be set ;
let f be Enumeration of X;
:: original: "
redefine funcf "  ->  Denumeration of X;
coherence
 f " is Denumeration of X
proof
 rng f = card X by Th6;
then A1: dom (f ") = card X by FUNCT_1:33;
 dom f = X by Th6;
then  rng (f ") = X by FUNCT_1:33;
hence  f " is Denumeration of X by A1, Th7; ::_thesis: verum
end;
end;

definition
let X be set ;
let f be Denumeration of X;
:: original: "
redefine funcf "  ->  Enumeration of X;
coherence
 f " is Enumeration of X
proof
 rng f = X by Th7;
then A1: dom (f ") = X by FUNCT_1:33;
 dom f = card X by Th7;
then  rng (f ") = card X by FUNCT_1:33;
hence  f " is Enumeration of X by A1, Th6; ::_thesis: verum
end;
end;

theorem :: AOFA_I00:12
for n, m being Nat holds 0 to_power (n + m) = (0 to_power n) * (0 to_power m)
proof
let n, m be Nat; ::_thesis: 0 to_power (n + m) = (0 to_power n) * (0 to_power m)
percases ( n > 0 or m > 0 or ( n = 0 & m = 0 ) ) ;
supposeA1: ( n > 0 or m > 0 ) ; ::_thesis: 0 to_power (n + m) = (0 to_power n) * (0 to_power m)
then  ( 0 to_power n = 0 or 0 to_power m = 0 ) by POWER:def_2;
hence  0 to_power (n + m) = (0 to_power n) * (0 to_power m) by A1, POWER:def_2; ::_thesis: verum
end;
supposeA2: ( n = 0 & m = 0 ) ; ::_thesis: 0 to_power (n + m) = (0 to_power n) * (0 to_power m)
then  0 to_power (n + m) = 1 by POWER:24;
hence  0 to_power (n + m) = (0 to_power n) * (0 to_power m) by A2; ::_thesis: verum
end;
end;
end;

theorem :: AOFA_I00:13
for x being real number
for n, m being Nat holds (x to_power n) to_power m = x to_power (n * m) by NEWTON:9;

begin

definition
let X be non empty set ;
mode INT-Variable of X is Function of (Funcs (X,INT)),X;
mode INT-Expression of X is Function of (Funcs (X,INT)),INT;
mode INT-Array of X is Function of INT,X;
end;

definition
let A be preIfWhileAlgebra;
let I be Element of A;
let X be non empty set ;
let T be Subset of (Funcs (X,INT));
let f be ExecutionFunction of A, Funcs (X,INT),T;
predI is_assignment_wrt A,X,f means :Def14: :: AOFA_I00:def 14
( I in ElementaryInstructions A & ex v being INT-Variable of X ex t being INT-Expression of X st
for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) );
end;

:: deftheorem Def14 defines is_assignment_wrt AOFA_I00:def_14_:_
 for A being preIfWhileAlgebra
for I being Element of A
for X being non empty set
for T being Subset of (Funcs (X,INT))
for f being ExecutionFunction of A, Funcs (X,INT),T holds
( I is_assignment_wrt A,X,f iff ( I in ElementaryInstructions A & ex v being INT-Variable of X ex t being INT-Expression of X st
for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) );

definition
let A be preIfWhileAlgebra;
let X be non empty set ;
let T be Subset of (Funcs (X,INT));
let f be ExecutionFunction of A, Funcs (X,INT),T;
let v be INT-Variable of X;
let t be INT-Expression of X;
predv,t form_assignment_wrt f means :Def15: :: AOFA_I00:def 15
 ex I being Element of A st
( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) );
end;

:: deftheorem Def15 defines form_assignment_wrt AOFA_I00:def_15_:_
 for A being preIfWhileAlgebra
for X being non empty set
for T being Subset of (Funcs (X,INT))
for f being ExecutionFunction of A, Funcs (X,INT),T
for v being INT-Variable of X
for t being INT-Expression of X holds
( v,t form_assignment_wrt f iff ex I being Element of A st
( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) ) );

definition
let A be preIfWhileAlgebra;
let X be non empty set ;
let T be Subset of (Funcs (X,INT));
let f be ExecutionFunction of A, Funcs (X,INT),T;
assume B1: ex I being Element of A st I is_assignment_wrt A,X,f ;
mode INT-Variable of A,f ->  INT-Variable of X means :: AOFA_I00:def 16
 ex t being INT-Expression of X st it,t form_assignment_wrt f;
existence
 ex b1 being INT-Variable of X ex t being INT-Expression of X st b1,t form_assignment_wrt f
proof
consider I being Element of A such that
A1: I is_assignment_wrt A,X,f by B1;
consider v being INT-Variable of X, t being INT-Expression of X such that
A2: for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) by A1, Def14;
take  v ; ::_thesis: ex t being INT-Expression of X st v,t form_assignment_wrt f
take  t ; ::_thesis: v,t form_assignment_wrt f
take  I ; :: according to AOFA_I00:def_15 ::_thesis: ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) )
thus  ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) ) by A1, A2, Def14; ::_thesis: verum
end;
end;

:: deftheorem  defines INT-Variable AOFA_I00:def_16_:_
 for A being preIfWhileAlgebra
for X being non empty set
for T being Subset of (Funcs (X,INT))
for f being ExecutionFunction of A, Funcs (X,INT),T st ex I being Element of A st I is_assignment_wrt A,X,f holds
for b5 being INT-Variable of X holds
( b5 is INT-Variable of A,f iff ex t being INT-Expression of X st b5,t form_assignment_wrt f );

definition
let A be preIfWhileAlgebra;
let X be non empty set ;
let T be Subset of (Funcs (X,INT));
let f be ExecutionFunction of A, Funcs (X,INT),T;
assume B1: ex I being Element of A st I is_assignment_wrt A,X,f ;
mode INT-Expression of A,f ->  INT-Expression of X means :: AOFA_I00:def 17
 ex v being INT-Variable of X st v,it form_assignment_wrt f;
existence
 ex b1 being INT-Expression of X ex v being INT-Variable of X st v,b1 form_assignment_wrt f
proof
consider I being Element of A such that
A1: I is_assignment_wrt A,X,f by B1;
consider v being INT-Variable of X, t being INT-Expression of X such that
A2: for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) by A1, Def14;
take  t ; ::_thesis: ex v being INT-Variable of X st v,t form_assignment_wrt f
take  v ; ::_thesis: v,t form_assignment_wrt f
take  I ; :: according to AOFA_I00:def_15 ::_thesis: ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) )
thus  ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) ) by A1, A2, Def14; ::_thesis: verum
end;
end;

:: deftheorem  defines INT-Expression AOFA_I00:def_17_:_
 for A being preIfWhileAlgebra
for X being non empty set
for T being Subset of (Funcs (X,INT))
for f being ExecutionFunction of A, Funcs (X,INT),T st ex I being Element of A st I is_assignment_wrt A,X,f holds
for b5 being INT-Expression of X holds
( b5 is INT-Expression of A,f iff ex v being INT-Variable of X st v,b5 form_assignment_wrt f );

definition
let X, Y be non empty set ;
let f be Element of Funcs (X,Y);
let x be Element of X;
:: original: .
redefine funcf . x ->  Element of Y;
coherence
 f . x is Element of Y
proof
 f . x in rng f by FUNCT_2:4;
hence  f . x is Element of Y ; ::_thesis: verum
end;
end;

definition
let X be non empty set ;
let x be Element of X;
func . x ->  INT-Expression of X means :Def18: :: AOFA_I00:def 18
 for s being Element of Funcs (X,INT) holds it . s = s . x;
correctness
existence
ex b1 being INT-Expression of X st
for s being Element of Funcs (X,INT) holds b1 . s = s . x;
uniqueness
for b1, b2 being INT-Expression of X st ( for s being Element of Funcs (X,INT) holds b1 . s = s . x ) & ( for s being Element of Funcs (X,INT) holds b2 . s = s . x ) holds
b1 = b2;
proof
deffunc H1( Element of Funcs (X,INT)) ->  Element of INT = $1 . x;
thus  ex f being Function of (Funcs (X,INT)),INT st
for x being Element of Funcs (X,INT) holds f . x = H1(x) from FUNCT_2:sch_4(); ::_thesis: for b1, b2 being INT-Expression of X st ( for s being Element of Funcs (X,INT) holds b1 . s = s . x ) & ( for s being Element of Funcs (X,INT) holds b2 . s = s . x ) holds
b1 = b2
let f1, f2 be INT-Expression of X; ::_thesis: ( ( for s being Element of Funcs (X,INT) holds f1 . s = s . x ) & ( for s being Element of Funcs (X,INT) holds f2 . s = s . x ) implies f1 = f2 )
assume that
A1: for s being Element of Funcs (X,INT) holds f1 . s = s . x and
A2: for s being Element of Funcs (X,INT) holds f2 . s = s . x ; ::_thesis: f1 = f2
now__::_thesis:_for_s_being_Element_of_Funcs_(X,INT)_holds_f1_._s_=_f2_._s
let s be Element of Funcs (X,INT); ::_thesis: f1 . s = f2 . s
thus f1 . s = s . x by A1
.= f2 . s by A2 ; ::_thesis: verum
end;
hence  f1 = f2 by FUNCT_2:63; ::_thesis: verum
end;
end;

:: deftheorem Def18 defines . AOFA_I00:def_18_:_
 for X being non empty set
for x being Element of X
for b3 being INT-Expression of X holds
( b3 = . x iff for s being Element of Funcs (X,INT) holds b3 . s = s . x );

definition
let X be non empty set ;
let v be INT-Variable of X;
func . v ->  INT-Expression of X means :Def19: :: AOFA_I00:def 19
 for s being Element of Funcs (X,INT) holds it . s = s . (v . s);
correctness
existence
ex b1 being INT-Expression of X st
for s being Element of Funcs (X,INT) holds b1 . s = s . (v . s);
uniqueness
for b1, b2 being INT-Expression of X st ( for s being Element of Funcs (X,INT) holds b1 . s = s . (v . s) ) & ( for s being Element of Funcs (X,INT) holds b2 . s = s . (v . s) ) holds
b1 = b2;
proof
deffunc H1( Element of Funcs (X,INT)) ->  Element of INT = $1 . (v . $1);
thus  ex f being Function of (Funcs (X,INT)),INT st
for x being Element of Funcs (X,INT) holds f . x = H1(x) from FUNCT_2:sch_4(); ::_thesis: for b1, b2 being INT-Expression of X st ( for s being Element of Funcs (X,INT) holds b1 . s = s . (v . s) ) & ( for s being Element of Funcs (X,INT) holds b2 . s = s . (v . s) ) holds
b1 = b2
let f1, f2 be INT-Expression of X; ::_thesis: ( ( for s being Element of Funcs (X,INT) holds f1 . s = s . (v . s) ) & ( for s being Element of Funcs (X,INT) holds f2 . s = s . (v . s) ) implies f1 = f2 )
assume that
A1: for s being Element of Funcs (X,INT) holds f1 . s = s . (v . s) and
A2: for s being Element of Funcs (X,INT) holds f2 . s = s . (v . s) ; ::_thesis: f1 = f2
now__::_thesis:_for_s_being_Element_of_Funcs_(X,INT)_holds_f1_._s_=_f2_._s
let s be Element of Funcs (X,INT); ::_thesis: f1 . s = f2 . s
thus f1 . s = s . (v . s) by A1
.= f2 . s by A2 ; ::_thesis: verum
end;
hence  f1 = f2 by FUNCT_2:63; ::_thesis: verum
end;
end;

:: deftheorem Def19 defines . AOFA_I00:def_19_:_
 for X being non empty set
for v being INT-Variable of X
for b3 being INT-Expression of X holds
( b3 = . v iff for s being Element of Funcs (X,INT) holds b3 . s = s . (v . s) );

definition
let X be non empty set ;
let x be Element of X;
func ^ x ->  INT-Variable of X equals :: AOFA_I00:def 20
(Funcs (X,INT)) --> x;
correctness
coherence
(Funcs (X,INT)) --> x is INT-Variable of X;
;
end;

:: deftheorem  defines ^ AOFA_I00:def_20_:_
 for X being non empty set
for x being Element of X holds ^ x = (Funcs (X,INT)) --> x;

theorem :: AOFA_I00:14
for X being non empty set
for x being Element of X holds . x = . (^ x)
proof
let X be non empty set ; ::_thesis: for x being Element of X holds . x = . (^ x)
let x be Element of X; ::_thesis: . x = . (^ x)
 for s being Element of Funcs (X,INT) holds (. (^ x)) . s = s . x
proof
let s be Element of Funcs (X,INT); ::_thesis: (. (^ x)) . s = s . x
thus (. (^ x)) . s = s . ((^ x) . s) by Def19
.= s . x by FUNCOP_1:7 ; ::_thesis: verum
end;
hence  . x = . (^ x) by Def18; ::_thesis: verum
end;

definition
let X be non empty set ;
let i be integer number ;
func . (i,X) ->  INT-Expression of X equals :: AOFA_I00:def 21
(Funcs (X,INT)) --> i;
correctness
coherence
(Funcs (X,INT)) --> i is INT-Expression of X;
proof
 i in INT by INT_1:def_2;
hence  (Funcs (X,INT)) --> i is INT-Expression of X by FUNCOP_1:45; ::_thesis: verum
end;
end;

:: deftheorem  defines . AOFA_I00:def_21_:_
 for X being non empty set
for i being integer number holds . (i,X) = (Funcs (X,INT)) --> i;

theorem :: AOFA_I00:15
for X being non empty set
for t being INT-Expression of X holds
( t + (. (0,X)) = t & t (#) (. (1,X)) = t )
proof
let X be non empty set ; ::_thesis: for t being INT-Expression of X holds
( t + (. (0,X)) = t & t (#) (. (1,X)) = t )
let t be INT-Expression of X; ::_thesis: ( t + (. (0,X)) = t & t (#) (. (1,X)) = t )
now__::_thesis:_for_s_being_Element_of_Funcs_(X,INT)_holds_(t_+_(._(0,X)))_._s_=_t_._s
let s be Element of Funcs (X,INT); ::_thesis: (t + (. (0,X))) . s = t . s
A1: (. (0,X)) . s = 0 by FUNCOP_1:7;
 dom (t + (. (0,X))) = Funcs (X,INT) by FUNCT_2:def_1;
hence (t + (. (0,X))) . s = (t . s) + ((. (0,X)) . s) by VALUED_1:def_1
.= t . s by A1 ;
::_thesis: verum
end;
hence  t + (. (0,X)) = t by FUNCT_2:63; ::_thesis: t (#) (. (1,X)) = t
now__::_thesis:_for_s_being_Element_of_Funcs_(X,INT)_holds_(t_(#)_(._(1,X)))_._s_=_t_._s
let s be Element of Funcs (X,INT); ::_thesis: (t (#) (. (1,X))) . s = t . s
 dom (t (#) (. (1,X))) = Funcs (X,INT) by FUNCT_2:def_1;
hence (t (#) (. (1,X))) . s = (t . s) * ((. (1,X)) . s) by VALUED_1:def_4
.= (t . s) * 1 by FUNCOP_1:7
.= t . s ;
::_thesis: verum
end;
hence  t (#) (. (1,X)) = t by FUNCT_2:63; ::_thesis: verum
end;

definition
let A be preIfWhileAlgebra;
let X be non empty set ;
let T be Subset of (Funcs (X,INT));
let f be ExecutionFunction of A, Funcs (X,INT),T;
attrf is Euclidean  means :Def22: :: AOFA_I00:def 22
( ( for v being INT-Variable of A,f
for t being INT-Expression of A,f holds v,t form_assignment_wrt f ) & ( for i being integer number holds . (i,X) is INT-Expression of A,f ) & ( for v being INT-Variable of A,f holds . v is INT-Expression of A,f ) & ( for x being Element of X holds ^ x is INT-Variable of A,f ) & ex a being INT-Array of X st
( a | (card X) is one-to-one & ( for t being INT-Expression of A,f holds a * t is INT-Variable of A,f ) ) & ( for t being INT-Expression of A,f holds - t is INT-Expression of A,f ) & ( for t1, t2 being INT-Expression of A,f holds
( t1 (#) t2 is INT-Expression of A,f & t1 + t2 is INT-Expression of A,f & t1 div t2 is INT-Expression of A,f & t1 mod t2 is INT-Expression of A,f & leq (t1,t2) is INT-Expression of A,f & gt (t1,t2) is INT-Expression of A,f ) ) );
end;

:: deftheorem Def22 defines Euclidean AOFA_I00:def_22_:_
 for A being preIfWhileAlgebra
for X being non empty set
for T being Subset of (Funcs (X,INT))
for f being ExecutionFunction of A, Funcs (X,INT),T holds
( f is Euclidean iff ( ( for v being INT-Variable of A,f
for t being INT-Expression of A,f holds v,t form_assignment_wrt f ) & ( for i being integer number holds . (i,X) is INT-Expression of A,f ) & ( for v being INT-Variable of A,f holds . v is INT-Expression of A,f ) & ( for x being Element of X holds ^ x is INT-Variable of A,f ) & ex a being INT-Array of X st
( a | (card X) is one-to-one & ( for t being INT-Expression of A,f holds a * t is INT-Variable of A,f ) ) & ( for t being INT-Expression of A,f holds - t is INT-Expression of A,f ) & ( for t1, t2 being INT-Expression of A,f holds
( t1 (#) t2 is INT-Expression of A,f & t1 + t2 is INT-Expression of A,f & t1 div t2 is INT-Expression of A,f & t1 mod t2 is INT-Expression of A,f & leq (t1,t2) is INT-Expression of A,f & gt (t1,t2) is INT-Expression of A,f ) ) ) );

definition
let A be preIfWhileAlgebra;
attrA is Euclidean  means :Def23: :: AOFA_I00:def 23
 for X being non empty countable set
for T being Subset of (Funcs (X,INT)) ex f being ExecutionFunction of A, Funcs (X,INT),T st f is Euclidean ;
end;

:: deftheorem Def23 defines Euclidean AOFA_I00:def_23_:_
 for A being preIfWhileAlgebra holds
( A is Euclidean iff for X being non empty countable set
for T being Subset of (Funcs (X,INT)) ex f being ExecutionFunction of A, Funcs (X,INT),T st f is Euclidean );

definition
func INT-ElemIns  ->  infinite disjoint_with_NAT set  equals :: AOFA_I00:def 24
[:(Funcs ((Funcs (NAT,INT)),NAT)),(Funcs ((Funcs (NAT,INT)),INT)):];
coherence
 [:(Funcs ((Funcs (NAT,INT)),NAT)),(Funcs ((Funcs (NAT,INT)),INT)):] is infinite disjoint_with_NAT set ;
end;

:: deftheorem  defines INT-ElemIns AOFA_I00:def_24_:_
 INT-ElemIns = [:(Funcs ((Funcs (NAT,INT)),NAT)),(Funcs ((Funcs (NAT,INT)),INT)):];

definition
mode INT-Exec  ->  ExecutionFunction of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns), Funcs (NAT,INT),(Funcs (NAT,INT)) \ (0,0) means :Def25: :: AOFA_I00:def 25
 for s being Element of Funcs (NAT,INT)
for v being Element of Funcs ((Funcs (NAT,INT)),NAT)
for e being Element of Funcs ((Funcs (NAT,INT)),INT) holds it . (s,(root-tree [v,e])) = s +* ((v . s),(e . s));
existence
 ex b1 being ExecutionFunction of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns), Funcs (NAT,INT),(Funcs (NAT,INT)) \ (0,0) st
for s being Element of Funcs (NAT,INT)
for v being Element of Funcs ((Funcs (NAT,INT)),NAT)
for e being Element of Funcs ((Funcs (NAT,INT)),INT) holds b1 . (s,(root-tree [v,e])) = s +* ((v . s),(e . s))
proof
reconsider i0 = 0 as Element of INT by INT_1:def_1;
set Q = Funcs (NAT,INT);
set T = (Funcs (NAT,INT)) \ (0,0);
set S = ECIW-signature ;
set G = INT-ElemIns ;
set A = FreeUnivAlgNSG (ECIW-signature,INT-ElemIns);
A1: Terminals (DTConUA (ECIW-signature,INT-ElemIns)) = INT-ElemIns by FREEALG:3;
reconsider q0 = NAT --> i0 as Element of Funcs (NAT,INT) by FUNCT_2:8;
defpred S1[ set , set ] means  ex s being Element of Funcs (NAT,INT) ex v being Element of Funcs ((Funcs (NAT,INT)),NAT) ex e being Element of Funcs ((Funcs (NAT,INT)),INT) st
( $1 = [s,(root-tree [v,e])] & $2 = s +* ((v . s),(e . s)) );
A2: ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) = FreeGenSetNSG (ECIW-signature,INT-ElemIns) by AOFA_000:70;
A3: for x being set st x in [:(Funcs (NAT,INT)),(ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns))):] holds
ex y being set st
( y in Funcs (NAT,INT) & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in [:(Funcs (NAT,INT)),(ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns))):] implies ex y being set st
( y in Funcs (NAT,INT) & S1[x,y] ) )
assume  x in [:(Funcs (NAT,INT)),(ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns))):] ; ::_thesis: ex y being set st
( y in Funcs (NAT,INT) & S1[x,y] )
then consider s, I being set  such that
A4: s in Funcs (NAT,INT) and
A5: I in ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) and
A6: x = [s,I] by ZFMISC_1:def_2;
reconsider s = s as Element of Funcs (NAT,INT) by A4;
consider a being Symbol of (DTConUA (ECIW-signature,INT-ElemIns)) such that
A7: I = root-tree a and
A8: a in Terminals (DTConUA (ECIW-signature,INT-ElemIns)) by A2, A5;
consider v, e being set  such that
A9: v in Funcs ((Funcs (NAT,INT)),NAT) and
A10: e in Funcs ((Funcs (NAT,INT)),INT) and
A11: a = [v,e] by A1, A8, ZFMISC_1:def_2;
reconsider e = e as Element of Funcs ((Funcs (NAT,INT)),INT) by A10;
reconsider v = v as Element of Funcs ((Funcs (NAT,INT)),NAT) by A9;
take y = s +* ((v . s),(e . s)); ::_thesis: ( y in Funcs (NAT,INT) & S1[x,y] )
thus  y in Funcs (NAT,INT) by FUNCT_2:8; ::_thesis: S1[x,y]
take  s ; ::_thesis: ex v being Element of Funcs ((Funcs (NAT,INT)),NAT) ex e being Element of Funcs ((Funcs (NAT,INT)),INT) st
( x = [s,(root-tree [v,e])] & y = s +* ((v . s),(e . s)) )
take  v ; ::_thesis: ex e being Element of Funcs ((Funcs (NAT,INT)),INT) st
( x = [s,(root-tree [v,e])] & y = s +* ((v . s),(e . s)) )
take  e ; ::_thesis: ( x = [s,(root-tree [v,e])] & y = s +* ((v . s),(e . s)) )
thus  ( x = [s,(root-tree [v,e])] & y = s +* ((v . s),(e . s)) ) by A6, A7, A11; ::_thesis: verum
end;
consider g being Function such that
A12: ( dom g = [:(Funcs (NAT,INT)),(ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns))):] & rng g c= Funcs (NAT,INT) ) and
A13: for x being set st x in [:(Funcs (NAT,INT)),(ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns))):] holds
S1[x,g . x] from FUNCT_1:sch_5(A3);
reconsider g = g as Function of [:(Funcs (NAT,INT)),(ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns))):],(Funcs (NAT,INT)) by A12, FUNCT_2:2;
consider f being ExecutionFunction of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns), Funcs (NAT,INT),(Funcs (NAT,INT)) \ (0,0) such that
A14: f | [:(Funcs (NAT,INT)),(ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns))):] = g and
 for s being Element of Funcs (NAT,INT)
for C, I being Element of (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) st not f iteration_terminates_for I \; C,f . (s,C) holds
f . (s,(while (C,I))) = q0 by AOFA_000:91;
take  f ; ::_thesis: for s being Element of Funcs (NAT,INT)
for v being Element of Funcs ((Funcs (NAT,INT)),NAT)
for e being Element of Funcs ((Funcs (NAT,INT)),INT) holds f . (s,(root-tree [v,e])) = s +* ((v . s),(e . s))
let s be Element of Funcs (NAT,INT); ::_thesis: for v being Element of Funcs ((Funcs (NAT,INT)),NAT)
for e being Element of Funcs ((Funcs (NAT,INT)),INT) holds f . (s,(root-tree [v,e])) = s +* ((v . s),(e . s))
let v be Element of Funcs ((Funcs (NAT,INT)),NAT); ::_thesis: for e being Element of Funcs ((Funcs (NAT,INT)),INT) holds f . (s,(root-tree [v,e])) = s +* ((v . s),(e . s))
let e be Element of Funcs ((Funcs (NAT,INT)),INT); ::_thesis: f . (s,(root-tree [v,e])) = s +* ((v . s),(e . s))
set I = root-tree [v,e];
 [v,e] in INT-ElemIns by ZFMISC_1:87;
then  root-tree [v,e] in ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) by A2, A1;
then A15: [s,(root-tree [v,e])] in [:(Funcs (NAT,INT)),(ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns))):] by ZFMISC_1:87;
then consider s9 being Element of Funcs (NAT,INT), v9 being Element of Funcs ((Funcs (NAT,INT)),NAT), e9 being Element of Funcs ((Funcs (NAT,INT)),INT) such that
A16: [s,(root-tree [v,e])] = [s9,(root-tree [v9,e9])] and
A17: g . [s,(root-tree [v,e])] = s9 +* ((v9 . s9),(e9 . s9)) by A13;
 root-tree [v,e] = root-tree [v9,e9] by A16, XTUPLE_0:1;
then A18: [v,e] = [v9,e9] by TREES_4:4;
then A19: v = v9 by XTUPLE_0:1;
A20: e = e9 by A18, XTUPLE_0:1;
 s = s9 by A16, XTUPLE_0:1;
hence  f . (s,(root-tree [v,e])) = s +* ((v . s),(e . s)) by A14, A15, A17, A19, A20, FUNCT_1:49; ::_thesis: verum
end;
end;

:: deftheorem Def25 defines INT-Exec AOFA_I00:def_25_:_
 for b1 being ExecutionFunction of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns), Funcs (NAT,INT),(Funcs (NAT,INT)) \ (0,0) holds
( b1 is INT-Exec iff for s being Element of Funcs (NAT,INT)
for v being Element of Funcs ((Funcs (NAT,INT)),NAT)
for e being Element of Funcs ((Funcs (NAT,INT)),INT) holds b1 . (s,(root-tree [v,e])) = s +* ((v . s),(e . s)) );

definition
let X be non empty set ;
func INT-ElemIns X ->  infinite disjoint_with_NAT set  equals :: AOFA_I00:def 26
[:(Funcs ((Funcs (X,INT)),X)),(Funcs ((Funcs (X,INT)),INT)):];
coherence
 [:(Funcs ((Funcs (X,INT)),X)),(Funcs ((Funcs (X,INT)),INT)):] is infinite disjoint_with_NAT set ;
end;

:: deftheorem  defines INT-ElemIns AOFA_I00:def_26_:_
 for X being non empty set holds INT-ElemIns X = [:(Funcs ((Funcs (X,INT)),X)),(Funcs ((Funcs (X,INT)),INT)):];

definition
let X be non empty set ;
let x be Element of X;
mode INT-Exec of x ->  ExecutionFunction of FreeUnivAlgNSG (ECIW-signature,(INT-ElemIns X)), Funcs (X,INT),(Funcs (X,INT)) \ (x,0) means :: AOFA_I00:def 27
 for s being Element of Funcs (X,INT)
for v being Element of Funcs ((Funcs (X,INT)),X)
for e being Element of Funcs ((Funcs (X,INT)),INT) holds it . (s,(root-tree [v,e])) = s +* ((v . s),(e . s));
existence
 ex b1 being ExecutionFunction of FreeUnivAlgNSG (ECIW-signature,(INT-ElemIns X)), Funcs (X,INT),(Funcs (X,INT)) \ (x,0) st
for s being Element of Funcs (X,INT)
for v being Element of Funcs ((Funcs (X,INT)),X)
for e being Element of Funcs ((Funcs (X,INT)),INT) holds b1 . (s,(root-tree [v,e])) = s +* ((v . s),(e . s))
proof
reconsider i0 = 0 as Element of INT by INT_1:def_1;
set Q = Funcs (X,INT);
set T = (Funcs (X,INT)) \ (x,0);
set S = ECIW-signature ;
set G = INT-ElemIns X;
set A = FreeUnivAlgNSG (ECIW-signature,(INT-ElemIns X));
A1: Terminals (DTConUA (ECIW-signature,(INT-ElemIns X))) = INT-ElemIns X by FREEALG:3;
reconsider q0 = X --> i0 as Element of Funcs (X,INT) by FUNCT_2:8;
defpred S1[ set , set ] means  ex s being Element of Funcs (X,INT) ex v being Element of Funcs ((Funcs (X,INT)),X) ex e being Element of Funcs ((Funcs (X,INT)),INT) st
( $1 = [s,(root-tree [v,e])] & $2 = s +* ((v . s),(e . s)) );
A2: ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,(INT-ElemIns X))) = FreeGenSetNSG (ECIW-signature,(INT-ElemIns X)) by AOFA_000:70;
A3: for x being set st x in [:(Funcs (X,INT)),(ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,(INT-ElemIns X)))):] holds
ex y being set st
( y in Funcs (X,INT) & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in [:(Funcs (X,INT)),(ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,(INT-ElemIns X)))):] implies ex y being set st
( y in Funcs (X,INT) & S1[x,y] ) )
assume  x in [:(Funcs (X,INT)),(ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,(INT-ElemIns X)))):] ; ::_thesis: ex y being set st
( y in Funcs (X,INT) & S1[x,y] )
then consider s, I being set  such that
A4: s in Funcs (X,INT) and
A5: I in ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,(INT-ElemIns X))) and
A6: x = [s,I] by ZFMISC_1:def_2;
reconsider s = s as Element of Funcs (X,INT) by A4;
consider a being Symbol of (DTConUA (ECIW-signature,(INT-ElemIns X))) such that
A7: I = root-tree a and
A8: a in Terminals (DTConUA (ECIW-signature,(INT-ElemIns X))) by A2, A5;
consider v, e being set  such that
A9: v in Funcs ((Funcs (X,INT)),X) and
A10: e in Funcs ((Funcs (X,INT)),INT) and
A11: a = [v,e] by A1, A8, ZFMISC_1:def_2;
reconsider e = e as Element of Funcs ((Funcs (X,INT)),INT) by A10;
reconsider v = v as Element of Funcs ((Funcs (X,INT)),X) by A9;
take y = s +* ((v . s),(e . s)); ::_thesis: ( y in Funcs (X,INT) & S1[x,y] )
thus  y in Funcs (X,INT) by FUNCT_2:8; ::_thesis: S1[x,y]
take  s ; ::_thesis: ex v being Element of Funcs ((Funcs (X,INT)),X) ex e being Element of Funcs ((Funcs (X,INT)),INT) st
( x = [s,(root-tree [v,e])] & y = s +* ((v . s),(e . s)) )
take  v ; ::_thesis: ex e being Element of Funcs ((Funcs (X,INT)),INT) st
( x = [s,(root-tree [v,e])] & y = s +* ((v . s),(e . s)) )
take  e ; ::_thesis: ( x = [s,(root-tree [v,e])] & y = s +* ((v . s),(e . s)) )
thus  ( x = [s,(root-tree [v,e])] & y = s +* ((v . s),(e . s)) ) by A6, A7, A11; ::_thesis: verum
end;
consider g being Function such that
A12: ( dom g = [:(Funcs (X,INT)),(ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,(INT-ElemIns X)))):] & rng g c= Funcs (X,INT) ) and
A13: for x being set st x in [:(Funcs (X,INT)),(ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,(INT-ElemIns X)))):] holds
S1[x,g . x] from FUNCT_1:sch_5(A3);
reconsider g = g as Function of [:(Funcs (X,INT)),(ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,(INT-ElemIns X)))):],(Funcs (X,INT)) by A12, FUNCT_2:2;
consider f being ExecutionFunction of FreeUnivAlgNSG (ECIW-signature,(INT-ElemIns X)), Funcs (X,INT),(Funcs (X,INT)) \ (x,0) such that
A14: f | [:(Funcs (X,INT)),(ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,(INT-ElemIns X)))):] = g and
 for s being Element of Funcs (X,INT)
for C, I being Element of (FreeUnivAlgNSG (ECIW-signature,(INT-ElemIns X))) st not f iteration_terminates_for I \; C,f . (s,C) holds
f . (s,(while (C,I))) = q0 by AOFA_000:91;
take  f ; ::_thesis: for s being Element of Funcs (X,INT)
for v being Element of Funcs ((Funcs (X,INT)),X)
for e being Element of Funcs ((Funcs (X,INT)),INT) holds f . (s,(root-tree [v,e])) = s +* ((v . s),(e . s))
let s be Element of Funcs (X,INT); ::_thesis: for v being Element of Funcs ((Funcs (X,INT)),X)
for e being Element of Funcs ((Funcs (X,INT)),INT) holds f . (s,(root-tree [v,e])) = s +* ((v . s),(e . s))
let v be Element of Funcs ((Funcs (X,INT)),X); ::_thesis: for e being Element of Funcs ((Funcs (X,INT)),INT) holds f . (s,(root-tree [v,e])) = s +* ((v . s),(e . s))
let e be Element of Funcs ((Funcs (X,INT)),INT); ::_thesis: f . (s,(root-tree [v,e])) = s +* ((v . s),(e . s))
set I = root-tree [v,e];
 [v,e] in INT-ElemIns X by ZFMISC_1:87;
then  root-tree [v,e] in ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,(INT-ElemIns X))) by A2, A1;
then A15: [s,(root-tree [v,e])] in [:(Funcs (X,INT)),(ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,(INT-ElemIns X)))):] by ZFMISC_1:87;
then consider s9 being Element of Funcs (X,INT), v9 being Element of Funcs ((Funcs (X,INT)),X), e9 being Element of Funcs ((Funcs (X,INT)),INT) such that
A16: [s,(root-tree [v,e])] = [s9,(root-tree [v9,e9])] and
A17: g . [s,(root-tree [v,e])] = s9 +* ((v9 . s9),(e9 . s9)) by A13;
 root-tree [v,e] = root-tree [v9,e9] by A16, XTUPLE_0:1;
then A18: [v,e] = [v9,e9] by TREES_4:4;
then A19: v = v9 by XTUPLE_0:1;
A20: e = e9 by A18, XTUPLE_0:1;
 s = s9 by A16, XTUPLE_0:1;
hence  f . (s,(root-tree [v,e])) = s +* ((v . s),(e . s)) by A14, A15, A17, A19, A20, FUNCT_1:49; ::_thesis: verum
end;
end;

:: deftheorem  defines INT-Exec AOFA_I00:def_27_:_
 for X being non empty set
for x being Element of X
for b3 being ExecutionFunction of FreeUnivAlgNSG (ECIW-signature,(INT-ElemIns X)), Funcs (X,INT),(Funcs (X,INT)) \ (x,0) holds
( b3 is INT-Exec of x iff for s being Element of Funcs (X,INT)
for v being Element of Funcs ((Funcs (X,INT)),X)
for e being Element of Funcs ((Funcs (X,INT)),INT) holds b3 . (s,(root-tree [v,e])) = s +* ((v . s),(e . s)) );

definition
let X be non empty set ;
let T be Subset of (Funcs (X,INT));
let c be Enumeration of X;
assume A1: rng c c= NAT ;
mode INT-Exec of c,T ->  ExecutionFunction of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns), Funcs (X,INT),T means :Def28: :: AOFA_I00:def 28
 for s being Element of Funcs (X,INT)
for v being Element of Funcs ((Funcs (X,INT)),X)
for e being Element of Funcs ((Funcs (X,INT)),INT) holds it . (s,(root-tree [((c * v) ** (c,NAT)),(e ** (c,NAT))])) = s +* ((v . s),(e . s));
existence
 ex b1 being ExecutionFunction of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns), Funcs (X,INT),T st
for s being Element of Funcs (X,INT)
for v being Element of Funcs ((Funcs (X,INT)),X)
for e being Element of Funcs ((Funcs (X,INT)),INT) holds b1 . (s,(root-tree [((c * v) ** (c,NAT)),(e ** (c,NAT))])) = s +* ((v . s),(e . s))
proof
 dom c = X by Th6;
then reconsider c9 = c as Function of X,NAT by A1, FUNCT_2:2;
reconsider i0 = 0 as Element of INT by INT_1:def_1;
set Q = Funcs (X,INT);
set S = ECIW-signature ;
set G = INT-ElemIns ;
set A = FreeUnivAlgNSG (ECIW-signature,INT-ElemIns);
reconsider q0 = X --> i0 as Element of Funcs (X,INT) by FUNCT_2:8;
A1: Terminals (DTConUA (ECIW-signature,INT-ElemIns)) = INT-ElemIns by FREEALG:3;
defpred S1[ set , set ] means  ex s being Element of Funcs (X,INT) st
( $1 `1 = s & ( ( $2 = s & ( for v being Element of Funcs ((Funcs (X,INT)),X)
for e being Element of Funcs ((Funcs (X,INT)),INT) holds not $1 `2 = root-tree [((c * v) ** (c9,NAT)),(e ** (c9,NAT))] ) ) or ex v being Element of Funcs ((Funcs (X,INT)),X) ex e being Element of Funcs ((Funcs (X,INT)),INT) st
( $1 `2 = root-tree [((c * v) ** (c9,NAT)),(e ** (c9,NAT))] & $2 = s +* ((v . s),(e . s)) ) ) );
A2: ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) = FreeGenSetNSG (ECIW-signature,INT-ElemIns) by AOFA_000:70;
A3: for x being set st x in [:(Funcs (X,INT)),(ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns))):] holds
ex y being set st
( y in Funcs (X,INT) & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in [:(Funcs (X,INT)),(ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns))):] implies ex y being set st
( y in Funcs (X,INT) & S1[x,y] ) )
assume  x in [:(Funcs (X,INT)),(ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns))):] ; ::_thesis: ex y being set st
( y in Funcs (X,INT) & S1[x,y] )
then consider s, I being set  such that
A4: s in Funcs (X,INT) and
A5: I in ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) and
A6: x = [s,I] by ZFMISC_1:def_2;
A7: x `1 = s by A6, MCART_1:7;
reconsider s = s as Element of Funcs (X,INT) by A4;
A8: x `2 = I by A6, MCART_1:7;
consider a being Symbol of (DTConUA (ECIW-signature,INT-ElemIns)) such that
A9: I = root-tree a and
 a in Terminals (DTConUA (ECIW-signature,INT-ElemIns)) by A2, A5;
percases ( ex v being Element of Funcs ((Funcs (X,INT)),X) ex e being Element of Funcs ((Funcs (X,INT)),INT) st a = [((c * v) ** (c9,NAT)),(e ** (c9,NAT))] or for v being Element of Funcs ((Funcs (X,INT)),X)
for e being Element of Funcs ((Funcs (X,INT)),INT) holds not a = [((c * v) ** (c9,NAT)),(e ** (c9,NAT))] ) ;
suppose ex v being Element of Funcs ((Funcs (X,INT)),X) ex e being Element of Funcs ((Funcs (X,INT)),INT) st a = [((c * v) ** (c9,NAT)),(e ** (c9,NAT))] ; ::_thesis: ex y being set st
( y in Funcs (X,INT) & S1[x,y] )
then consider v being Element of Funcs ((Funcs (X,INT)),X), e being Element of Funcs ((Funcs (X,INT)),INT) such that
A10: a = [((c * v) ** (c9,NAT)),(e ** (c9,NAT))] ;
take y = s +* ((v . s),(e . s)); ::_thesis: ( y in Funcs (X,INT) & S1[x,y] )
thus  y in Funcs (X,INT) by FUNCT_2:8; ::_thesis: S1[x,y]
thus  S1[x,y] by A7, A8, A9, A10; ::_thesis: verum
end;
supposeA11: for v being Element of Funcs ((Funcs (X,INT)),X)
for e being Element of Funcs ((Funcs (X,INT)),INT) holds not a = [((c * v) ** (c9,NAT)),(e ** (c9,NAT))] ; ::_thesis: ex y being set st
( y in Funcs (X,INT) & S1[x,y] )
take y = s; ::_thesis: ( y in Funcs (X,INT) & S1[x,y] )
thus  y in Funcs (X,INT) ; ::_thesis: S1[x,y]
 for v being Element of Funcs ((Funcs (X,INT)),X)
for e being Element of Funcs ((Funcs (X,INT)),INT) holds not x `2 = root-tree [((c * v) ** (c9,NAT)),(e ** (c9,NAT))] by A8, A9, A11, TREES_4:4;
hence  S1[x,y] by A7; ::_thesis: verum
end;
end;
end;
consider g being Function such that
A12: ( dom g = [:(Funcs (X,INT)),(ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns))):] & rng g c= Funcs (X,INT) ) and
A13: for x being set st x in [:(Funcs (X,INT)),(ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns))):] holds
S1[x,g . x] from FUNCT_1:sch_5(A3);
reconsider g = g as Function of [:(Funcs (X,INT)),(ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns))):],(Funcs (X,INT)) by A12, FUNCT_2:2;
consider f being ExecutionFunction of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns), Funcs (X,INT),T such that
A14: f | [:(Funcs (X,INT)),(ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns))):] = g and
 for s being Element of Funcs (X,INT)
for C, I being Element of (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) st not f iteration_terminates_for I \; C,f . (s,C) holds
f . (s,(while (C,I))) = q0 by AOFA_000:91;
take  f ; ::_thesis: for s being Element of Funcs (X,INT)
for v being Element of Funcs ((Funcs (X,INT)),X)
for e being Element of Funcs ((Funcs (X,INT)),INT) holds f . (s,(root-tree [((c * v) ** (c,NAT)),(e ** (c,NAT))])) = s +* ((v . s),(e . s))
let s be Element of Funcs (X,INT); ::_thesis: for v being Element of Funcs ((Funcs (X,INT)),X)
for e being Element of Funcs ((Funcs (X,INT)),INT) holds f . (s,(root-tree [((c * v) ** (c,NAT)),(e ** (c,NAT))])) = s +* ((v . s),(e . s))
let v be Element of Funcs ((Funcs (X,INT)),X); ::_thesis: for e being Element of Funcs ((Funcs (X,INT)),INT) holds f . (s,(root-tree [((c * v) ** (c,NAT)),(e ** (c,NAT))])) = s +* ((v . s),(e . s))
let e be Element of Funcs ((Funcs (X,INT)),INT); ::_thesis: f . (s,(root-tree [((c * v) ** (c,NAT)),(e ** (c,NAT))])) = s +* ((v . s),(e . s))
reconsider vv = v as Function of (Funcs (X,INT)),X ;
reconsider cv = c9 * vv as Element of Funcs ((Funcs (X,INT)),NAT) by FUNCT_2:8;
set v0 = cv ** (c9,NAT);
set e0 = e ** (c9,NAT);
set I = root-tree [(cv ** (c9,NAT)),(e ** (c9,NAT))];
 [(cv ** (c9,NAT)),(e ** (c9,NAT))] in INT-ElemIns by ZFMISC_1:87;
then  root-tree [(cv ** (c9,NAT)),(e ** (c9,NAT))] in ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) by A2, A1;
then A15: [s,(root-tree [(cv ** (c9,NAT)),(e ** (c9,NAT))])] in [:(Funcs (X,INT)),(ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns))):] by ZFMISC_1:87;
then  S1[[s,(root-tree [(cv ** (c9,NAT)),(e ** (c9,NAT))])],g . [s,(root-tree [(cv ** (c9,NAT)),(e ** (c9,NAT))])]] by A13;
then consider v9 being Element of Funcs ((Funcs (X,INT)),X), e9 being Element of Funcs ((Funcs (X,INT)),INT) such that
A16: [s,(root-tree [(cv ** (c9,NAT)),(e ** (c9,NAT))])] `2 = root-tree [((c * v9) ** (c9,NAT)),(e9 ** (c9,NAT))] and
A17: g . [s,(root-tree [(cv ** (c9,NAT)),(e ** (c9,NAT))])] = s +* ((v9 . s),(e9 . s)) ;
A18: dom (c9 * v9) = Funcs (X,INT) by FUNCT_2:def_1;
A19: dom e = Funcs (X,INT) by FUNCT_2:def_1;
A20: dom v = Funcs (X,INT) by FUNCT_2:def_1;
A21: dom cv = Funcs (X,INT) by FUNCT_2:def_1;
A22: dom c9 = X by FUNCT_2:def_1;
A23: rng v c= X ;
A24: dom e9 = Funcs (X,INT) by FUNCT_2:def_1;
A25: dom v9 = Funcs (X,INT) by FUNCT_2:def_1;
A26: [(cv ** (c9,NAT)),(e ** (c9,NAT))] = [((c * v9) ** (c9,NAT)),(e9 ** (c9,NAT))] by A16, TREES_4:4;
then  cv ** (c9,NAT) = (c * v9) ** (c9,NAT) by XTUPLE_0:1;
then A27: cv = c * v9 by A21, A18, Th5;
 e ** (c9,NAT) = e9 ** (c9,NAT) by A26, XTUPLE_0:1;
then A28: e = e9 by A19, A24, Th5;
 rng v9 c= X ;
then  v = v9 by A23, A22, A25, A20, A27, FUNCT_1:27;
hence  f . (s,(root-tree [((c * v) ** (c,NAT)),(e ** (c,NAT))])) = s +* ((v . s),(e . s)) by A14, A15, A17, A28, FUNCT_1:49; ::_thesis: verum
end;
end;

:: deftheorem Def28 defines INT-Exec AOFA_I00:def_28_:_
 for X being non empty set
for T being Subset of (Funcs (X,INT))
for c being Enumeration of X st rng c c= NAT holds
for b4 being ExecutionFunction of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns), Funcs (X,INT),T holds
( b4 is INT-Exec of c,T iff for s being Element of Funcs (X,INT)
for v being Element of Funcs ((Funcs (X,INT)),X)
for e being Element of Funcs ((Funcs (X,INT)),INT) holds b4 . (s,(root-tree [((c * v) ** (c,NAT)),(e ** (c,NAT))])) = s +* ((v . s),(e . s)) );

theorem Th16: :: AOFA_I00:16
for f being INT-Exec
for v being INT-Variable of NAT
for t being INT-Expression of NAT holds v,t form_assignment_wrt f
proof
let f be INT-Exec ; ::_thesis: for v being INT-Variable of NAT
for t being INT-Expression of NAT holds v,t form_assignment_wrt f
set S = ECIW-signature ;
set G = INT-ElemIns ;
set X = NAT ;
set A = FreeUnivAlgNSG (ECIW-signature,INT-ElemIns);
let v be INT-Variable of NAT; ::_thesis: for t being INT-Expression of NAT holds v,t form_assignment_wrt f
let t be INT-Expression of NAT; ::_thesis: v,t form_assignment_wrt f
reconsider v9 = v as Element of Funcs ((Funcs (NAT,INT)),NAT) by FUNCT_2:8;
reconsider t9 = t as Element of Funcs ((Funcs (NAT,INT)),INT) by FUNCT_2:8;
A1: Terminals (DTConUA (ECIW-signature,INT-ElemIns)) = INT-ElemIns by FREEALG:3;
A2: [v9,t9] in INT-ElemIns by ZFMISC_1:87;
A3: ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) = FreeGenSetNSG (ECIW-signature,INT-ElemIns) by AOFA_000:70;
then  root-tree [v9,t9] in ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) by A1, A2;
then reconsider I = root-tree [v9,t9] as Element of (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) ;
take  I ; :: according to AOFA_I00:def_15 ::_thesis: ( I in ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) & ( for s being Element of Funcs (NAT,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) )
thus  I in ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) by A3, A1, A2; ::_thesis: for s being Element of Funcs (NAT,INT) holds f . (s,I) = s +* ((v . s),(t . s))
thus  for s being Element of Funcs (NAT,INT) holds f . (s,I) = s +* ((v . s),(t . s)) by Def25; ::_thesis: verum
end;

theorem Th17: :: AOFA_I00:17
for f being INT-Exec
for v being INT-Variable of NAT holds v is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
proof
set t = the INT-Expression of NAT;
let f be INT-Exec ; ::_thesis: for v being INT-Variable of NAT holds v is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
set S = ECIW-signature ;
set G = INT-ElemIns ;
set X = NAT ;
set A = FreeUnivAlgNSG (ECIW-signature,INT-ElemIns);
let v be INT-Variable of NAT; ::_thesis: v is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
A1: Terminals (DTConUA (ECIW-signature,INT-ElemIns)) = INT-ElemIns by FREEALG:3;
reconsider t9 = the INT-Expression of NAT as Element of Funcs ((Funcs (NAT,INT)),INT) by FUNCT_2:8;
reconsider v9 = v as Element of Funcs ((Funcs (NAT,INT)),NAT) by FUNCT_2:8;
A2: [v9,t9] in INT-ElemIns by ZFMISC_1:87;
A3: ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) = FreeGenSetNSG (ECIW-signature,INT-ElemIns) by AOFA_000:70;
then  root-tree [v9,t9] in ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) by A1, A2;
then reconsider I = root-tree [v9,t9] as Element of (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) ;
hereby  :: according to AOFA_I00:def_16 ::_thesis: ex t being INT-Expression of NAT st v,t form_assignment_wrt f
take I = I; ::_thesis: I is_assignment_wrt FreeUnivAlgNSG (ECIW-signature,INT-ElemIns), NAT ,f
thus  I is_assignment_wrt FreeUnivAlgNSG (ECIW-signature,INT-ElemIns), NAT ,f ::_thesis: verum
proof
thus  I in ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) by A3, A1, A2; :: according to AOFA_I00:def_14 ::_thesis: ex v being INT-Variable of NAT ex t being INT-Expression of NAT st
for s being Element of Funcs (NAT,INT) holds f . (s,I) = s +* ((v . s),(t . s))
take  v ; ::_thesis: ex t being INT-Expression of NAT st
for s being Element of Funcs (NAT,INT) holds f . (s,I) = s +* ((v . s),(t . s))
take  the INT-Expression of NAT ; ::_thesis: for s being Element of Funcs (NAT,INT) holds f . (s,I) = s +* ((v . s),( the INT-Expression of NAT . s))
thus  for s being Element of Funcs (NAT,INT) holds f . (s,I) = s +* ((v . s),( the INT-Expression of NAT . s)) by Def25; ::_thesis: verum
end;
end;
take  the INT-Expression of NAT ; ::_thesis: v, the INT-Expression of NAT form_assignment_wrt f
thus  v, the INT-Expression of NAT form_assignment_wrt f by Th16; ::_thesis: verum
end;

theorem Th18: :: AOFA_I00:18
for f being INT-Exec
for t being INT-Expression of NAT holds t is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
proof
set v = the INT-Variable of NAT;
let f be INT-Exec ; ::_thesis: for t being INT-Expression of NAT holds t is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
set S = ECIW-signature ;
set G = INT-ElemIns ;
set X = NAT ;
set A = FreeUnivAlgNSG (ECIW-signature,INT-ElemIns);
let t be INT-Expression of NAT; ::_thesis: t is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
A1: Terminals (DTConUA (ECIW-signature,INT-ElemIns)) = INT-ElemIns by FREEALG:3;
reconsider t9 = t as Element of Funcs ((Funcs (NAT,INT)),INT) by FUNCT_2:8;
reconsider v9 = the INT-Variable of NAT as Element of Funcs ((Funcs (NAT,INT)),NAT) by FUNCT_2:8;
A2: [v9,t9] in INT-ElemIns by ZFMISC_1:87;
A3: ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) = FreeGenSetNSG (ECIW-signature,INT-ElemIns) by AOFA_000:70;
then  root-tree [v9,t9] in ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) by A1, A2;
then reconsider I = root-tree [v9,t9] as Element of (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) ;
hereby  :: according to AOFA_I00:def_17 ::_thesis: ex v being INT-Variable of NAT st v,t form_assignment_wrt f
take I = I; ::_thesis: I is_assignment_wrt FreeUnivAlgNSG (ECIW-signature,INT-ElemIns), NAT ,f
thus  I is_assignment_wrt FreeUnivAlgNSG (ECIW-signature,INT-ElemIns), NAT ,f ::_thesis: verum
proof
thus  I in ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) by A3, A1, A2; :: according to AOFA_I00:def_14 ::_thesis: ex v being INT-Variable of NAT ex t being INT-Expression of NAT st
for s being Element of Funcs (NAT,INT) holds f . (s,I) = s +* ((v . s),(t . s))
take  the INT-Variable of NAT ; ::_thesis: ex t being INT-Expression of NAT st
for s being Element of Funcs (NAT,INT) holds f . (s,I) = s +* (( the INT-Variable of NAT . s),(t . s))
take  t ; ::_thesis: for s being Element of Funcs (NAT,INT) holds f . (s,I) = s +* (( the INT-Variable of NAT . s),(t . s))
thus  for s being Element of Funcs (NAT,INT) holds f . (s,I) = s +* (( the INT-Variable of NAT . s),(t . s)) by Def25; ::_thesis: verum
end;
end;
take  the INT-Variable of NAT ; ::_thesis: the INT-Variable of NAT,t form_assignment_wrt f
thus  the INT-Variable of NAT,t form_assignment_wrt f by Th16; ::_thesis: verum
end;

registration
cluster  ->  Euclidean for INT-Exec ;
coherence
 for b1 being INT-Exec holds b1 is Euclidean
proof
set X = NAT ;
set a = (INT --> 0) +* (id NAT);
A1: dom (id NAT) = NAT by RELAT_1:45;
A2: INT \/ NAT = INT by NUMBERS:17, XBOOLE_1:12;
 dom (INT --> 0) = INT by FUNCOP_1:13;
then A3: dom ((INT --> 0) +* (id NAT)) = INT by A1, A2, FUNCT_4:def_1;
A4: rng (id NAT) = NAT by RELAT_1:45;
 {0} c= NAT by ZFMISC_1:31;
then A5: {0} \/ NAT = NAT by XBOOLE_1:12;
 rng (INT --> 0) = {0} by FUNCOP_1:8;
then  rng ((INT --> 0) +* (id NAT)) c= NAT by A4, A5, FUNCT_4:17;
then reconsider a = (INT --> 0) +* (id NAT) as INT-Array of NAT by A3, FUNCT_2:2;
let f be INT-Exec ; ::_thesis: f is Euclidean
set S = ECIW-signature ;
set G = INT-ElemIns ;
set A = FreeUnivAlgNSG (ECIW-signature,INT-ElemIns);
thus  for v being INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
for t being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds v,t form_assignment_wrt f by Th16; :: according to AOFA_I00:def_22 ::_thesis: ( ( for i being integer number holds . (i,NAT) is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) & ( for v being INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds . v is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) & ( for x being Element of NAT holds ^ x is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) & ex a being INT-Array of NAT st
( a | (card NAT) is one-to-one & ( for t being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds a * t is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) ) & ( for t being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds - t is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) & ( for t1, t2 being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds
( t1 (#) t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 + t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 div t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 mod t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & leq (t1,t2) is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & gt (t1,t2) is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) ) )
thus  for i being integer number holds . (i,NAT) is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f by Th18; ::_thesis: ( ( for v being INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds . v is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) & ( for x being Element of NAT holds ^ x is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) & ex a being INT-Array of NAT st
( a | (card NAT) is one-to-one & ( for t being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds a * t is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) ) & ( for t being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds - t is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) & ( for t1, t2 being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds
( t1 (#) t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 + t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 div t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 mod t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & leq (t1,t2) is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & gt (t1,t2) is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) ) )
thus  for v being INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds . v is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f by Th18; ::_thesis: ( ( for x being Element of NAT holds ^ x is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) & ex a being INT-Array of NAT st
( a | (card NAT) is one-to-one & ( for t being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds a * t is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) ) & ( for t being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds - t is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) & ( for t1, t2 being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds
( t1 (#) t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 + t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 div t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 mod t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & leq (t1,t2) is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & gt (t1,t2) is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) ) )
thus  for x being Element of NAT holds ^ x is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f by Th17; ::_thesis: ( ex a being INT-Array of NAT st
( a | (card NAT) is one-to-one & ( for t being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds a * t is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) ) & ( for t being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds - t is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) & ( for t1, t2 being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds
( t1 (#) t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 + t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 div t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 mod t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & leq (t1,t2) is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & gt (t1,t2) is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) ) )
hereby  ::_thesis: ( ( for t being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds - t is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) & ( for t1, t2 being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds
( t1 (#) t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 + t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 div t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 mod t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & leq (t1,t2) is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & gt (t1,t2) is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) ) )
take a = a; ::_thesis: ( a | (card NAT) is one-to-one & ( for t being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds a * t is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) )
 dom (id NAT) = NAT by RELAT_1:45;
hence  a | (card NAT) is one-to-one by CARD_1:47, FUNCT_4:23; ::_thesis: for t being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds a * t is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
thus  for t being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds a * t is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f by Th17; ::_thesis: verum
end;
thus  ( ( for t being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds - t is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) & ( for t1, t2 being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds
( t1 (#) t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 + t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 div t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 mod t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & leq (t1,t2) is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & gt (t1,t2) is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) ) ) by Th18; ::_thesis: verum
end;
end;

theorem Th19: :: AOFA_I00:19
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for c being Enumeration of X
for f being INT-Exec of c,T
for v being INT-Variable of X
for t being INT-Expression of X holds v,t form_assignment_wrt f
proof
set S = ECIW-signature ;
set G = INT-ElemIns ;
let X be non empty countable set ; ::_thesis: for T being Subset of (Funcs (X,INT))
for c being Enumeration of X
for f being INT-Exec of c,T
for v being INT-Variable of X
for t being INT-Expression of X holds v,t form_assignment_wrt f
let T be Subset of (Funcs (X,INT)); ::_thesis: for c being Enumeration of X
for f being INT-Exec of c,T
for v being INT-Variable of X
for t being INT-Expression of X holds v,t form_assignment_wrt f
let c be Enumeration of X; ::_thesis: for f being INT-Exec of c,T
for v being INT-Variable of X
for t being INT-Expression of X holds v,t form_assignment_wrt f
set A = FreeUnivAlgNSG (ECIW-signature,INT-ElemIns);
let f be INT-Exec of c,T; ::_thesis: for v being INT-Variable of X
for t being INT-Expression of X holds v,t form_assignment_wrt f
let v be INT-Variable of X; ::_thesis: for t being INT-Expression of X holds v,t form_assignment_wrt f
let t be INT-Expression of X; ::_thesis: v,t form_assignment_wrt f
reconsider v9 = v as Element of Funcs ((Funcs (X,INT)),X) by FUNCT_2:8;
reconsider t9 = t as Element of Funcs ((Funcs (X,INT)),INT) by FUNCT_2:8;
A1: ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) = FreeGenSetNSG (ECIW-signature,INT-ElemIns) by AOFA_000:70;
A2: rng c c= NAT by Th11;
 dom c = X by Th6;
then reconsider c9 = c as Function of X,NAT by A2, FUNCT_2:2;
reconsider cv = c9 * v as Element of Funcs ((Funcs (X,INT)),NAT) by FUNCT_2:8;
set v1 = cv ** (c9,NAT);
set t1 = t9 ** (c9,NAT);
A3: Terminals (DTConUA (ECIW-signature,INT-ElemIns)) = INT-ElemIns by FREEALG:3;
A4: [(cv ** (c9,NAT)),(t9 ** (c9,NAT))] in INT-ElemIns by ZFMISC_1:87;
then  root-tree [(cv ** (c9,NAT)),(t9 ** (c9,NAT))] in ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) by A1, A3;
then reconsider I = root-tree [(cv ** (c9,NAT)),(t9 ** (c9,NAT))] as Element of (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) ;
take  I ; :: according to AOFA_I00:def_15 ::_thesis: ( I in ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) )
 for s being Element of Funcs (X,INT) holds f . (s,(root-tree [((c * v9) ** (c,NAT)),(t9 ** (c,NAT))])) = s +* ((v9 . s),(t9 . s)) by A2, Def28;
hence  ( I in ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) ) by A1, A3, A4; ::_thesis: verum
end;

theorem Th20: :: AOFA_I00:20
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for c being Enumeration of X
for f being INT-Exec of c,T
for v being INT-Variable of X holds v is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
proof
set S = ECIW-signature ;
set G = INT-ElemIns ;
let X be non empty countable set ; ::_thesis: for T being Subset of (Funcs (X,INT))
for c being Enumeration of X
for f being INT-Exec of c,T
for v being INT-Variable of X holds v is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
let T be Subset of (Funcs (X,INT)); ::_thesis: for c being Enumeration of X
for f being INT-Exec of c,T
for v being INT-Variable of X holds v is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
let c be Enumeration of X; ::_thesis: for f being INT-Exec of c,T
for v being INT-Variable of X holds v is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
set A = FreeUnivAlgNSG (ECIW-signature,INT-ElemIns);
let f be INT-Exec of c,T; ::_thesis: for v being INT-Variable of X holds v is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
let v be INT-Variable of X; ::_thesis: v is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
set t = the INT-Expression of X;
A1: ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) = FreeGenSetNSG (ECIW-signature,INT-ElemIns) by AOFA_000:70;
A2: rng c c= NAT by Th11;
 dom c = X by Th6;
then reconsider c9 = c as Function of X,NAT by A2, FUNCT_2:2;
reconsider cv = c9 * v as Element of Funcs ((Funcs (X,INT)),NAT) by FUNCT_2:8;
reconsider v9 = v as Element of Funcs ((Funcs (X,INT)),X) by FUNCT_2:8;
reconsider t9 = the INT-Expression of X as Element of Funcs ((Funcs (X,INT)),INT) by FUNCT_2:8;
A3: Terminals (DTConUA (ECIW-signature,INT-ElemIns)) = INT-ElemIns by FREEALG:3;
set v1 = cv ** (c9,NAT);
set t1 = t9 ** (c9,NAT);
A4: [(cv ** (c9,NAT)),(t9 ** (c9,NAT))] in INT-ElemIns by ZFMISC_1:87;
then  root-tree [(cv ** (c9,NAT)),(t9 ** (c9,NAT))] in ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) by A1, A3;
then reconsider I = root-tree [(cv ** (c9,NAT)),(t9 ** (c9,NAT))] as Element of (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) ;
hereby  :: according to AOFA_I00:def_16 ::_thesis: ex t being INT-Expression of X st v,t form_assignment_wrt f
take I = I; ::_thesis: I is_assignment_wrt FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),X,f
thus  I is_assignment_wrt FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),X,f ::_thesis: verum
proof
thus  I in ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) by A1, A3, A4; :: according to AOFA_I00:def_14 ::_thesis: ex v being INT-Variable of X ex t being INT-Expression of X st
for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s))
take  v ; ::_thesis: ex t being INT-Expression of X st
for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s))
take  the INT-Expression of X ; ::_thesis: for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),( the INT-Expression of X . s))
 for s being Element of Funcs (X,INT) holds f . (s,(root-tree [((c * v9) ** (c,NAT)),(t9 ** (c,NAT))])) = s +* ((v9 . s),(t9 . s)) by A2, Def28;
hence  for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),( the INT-Expression of X . s)) ; ::_thesis: verum
end;
end;
take  the INT-Expression of X ; ::_thesis: v, the INT-Expression of X form_assignment_wrt f
thus  v, the INT-Expression of X form_assignment_wrt f by Th19; ::_thesis: verum
end;

theorem Th21: :: AOFA_I00:21
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for c being Enumeration of X
for f being INT-Exec of c,T
for t being INT-Expression of X holds t is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
proof
set S = ECIW-signature ;
set G = INT-ElemIns ;
let X be non empty countable set ; ::_thesis: for T being Subset of (Funcs (X,INT))
for c being Enumeration of X
for f being INT-Exec of c,T
for t being INT-Expression of X holds t is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
let T be Subset of (Funcs (X,INT)); ::_thesis: for c being Enumeration of X
for f being INT-Exec of c,T
for t being INT-Expression of X holds t is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
let c be Enumeration of X; ::_thesis: for f being INT-Exec of c,T
for t being INT-Expression of X holds t is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
set A = FreeUnivAlgNSG (ECIW-signature,INT-ElemIns);
let f be INT-Exec of c,T; ::_thesis: for t being INT-Expression of X holds t is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
set v = the INT-Variable of X;
let t be INT-Expression of X; ::_thesis: t is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
A1: ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) = FreeGenSetNSG (ECIW-signature,INT-ElemIns) by AOFA_000:70;
A2: rng c c= NAT by Th11;
 dom c = X by Th6;
then reconsider c9 = c as Function of X,NAT by A2, FUNCT_2:2;
reconsider cv = c9 * the INT-Variable of X as Element of Funcs ((Funcs (X,INT)),NAT) by FUNCT_2:8;
reconsider v9 = the INT-Variable of X as Element of Funcs ((Funcs (X,INT)),X) by FUNCT_2:8;
reconsider t9 = t as Element of Funcs ((Funcs (X,INT)),INT) by FUNCT_2:8;
A3: Terminals (DTConUA (ECIW-signature,INT-ElemIns)) = INT-ElemIns by FREEALG:3;
set v1 = cv ** (c9,NAT);
set t1 = t9 ** (c9,NAT);
A4: [(cv ** (c9,NAT)),(t9 ** (c9,NAT))] in INT-ElemIns by ZFMISC_1:87;
then  root-tree [(cv ** (c9,NAT)),(t9 ** (c9,NAT))] in ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) by A1, A3;
then reconsider I = root-tree [(cv ** (c9,NAT)),(t9 ** (c9,NAT))] as Element of (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) ;
hereby  :: according to AOFA_I00:def_17 ::_thesis: ex v being INT-Variable of X st v,t form_assignment_wrt f
take I = I; ::_thesis: I is_assignment_wrt FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),X,f
thus  I is_assignment_wrt FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),X,f ::_thesis: verum
proof
thus  I in ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) by A1, A3, A4; :: according to AOFA_I00:def_14 ::_thesis: ex v being INT-Variable of X ex t being INT-Expression of X st
for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s))
take  the INT-Variable of X ; ::_thesis: ex t being INT-Expression of X st
for s being Element of Funcs (X,INT) holds f . (s,I) = s +* (( the INT-Variable of X . s),(t . s))
take  t ; ::_thesis: for s being Element of Funcs (X,INT) holds f . (s,I) = s +* (( the INT-Variable of X . s),(t . s))
 for s being Element of Funcs (X,INT) holds f . (s,(root-tree [((c * v9) ** (c,NAT)),(t9 ** (c,NAT))])) = s +* ((v9 . s),(t9 . s)) by A2, Def28;
hence  for s being Element of Funcs (X,INT) holds f . (s,I) = s +* (( the INT-Variable of X . s),(t . s)) ; ::_thesis: verum
end;
end;
take  the INT-Variable of X ; ::_thesis: the INT-Variable of X,t form_assignment_wrt f
thus  the INT-Variable of X,t form_assignment_wrt f by Th19; ::_thesis: verum
end;

registration
let X be non empty countable set ;
let T be Subset of (Funcs (X,INT));
let c be Enumeration of X;
cluster  ->  Euclidean for INT-Exec of c,T;
coherence
 for b1 being INT-Exec of c,T holds b1 is Euclidean
proof
A1: rng c c= NAT by Th11;
 card X = rng c by Th6;
then A2: INT \/ (card X) = INT by A1, NUMBERS:17, XBOOLE_1:1, XBOOLE_1:12;
set x = (c ") . 0;
set a = (INT --> ((c ") . 0)) +* (c ");
A3: dom (INT --> ((c ") . 0)) = INT by FUNCOP_1:13;
A4: rng (INT --> ((c ") . 0)) = {((c ") . 0)} by FUNCOP_1:8;
 (c ") . 0 in X by FUNCT_2:5, ORDINAL3:8;
then  {((c ") . 0)} c= X by ZFMISC_1:31;
then A5: {((c ") . 0)} \/ X = X by XBOOLE_1:12;
 rng (c ") = X by Th7;
then A6: rng ((INT --> ((c ") . 0)) +* (c ")) c= X by A4, A5, FUNCT_4:17;
 dom (c ") = card X by Th7;
then  dom ((INT --> ((c ") . 0)) +* (c ")) = INT by A3, A2, FUNCT_4:def_1;
then reconsider a = (INT --> ((c ") . 0)) +* (c ") as INT-Array of X by A6, FUNCT_2:2;
let f be INT-Exec of c,T; ::_thesis: f is Euclidean
set S = ECIW-signature ;
set G = INT-ElemIns ;
set A = FreeUnivAlgNSG (ECIW-signature,INT-ElemIns);
thus  for v being INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
for t being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds v,t form_assignment_wrt f by Th19; :: according to AOFA_I00:def_22 ::_thesis: ( ( for i being integer number holds . (i,X) is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) & ( for v being INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds . v is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) & ( for x being Element of X holds ^ x is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) & ex a being INT-Array of X st
( a | (card X) is one-to-one & ( for t being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds a * t is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) ) & ( for t being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds - t is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) & ( for t1, t2 being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds
( t1 (#) t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 + t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 div t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 mod t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & leq (t1,t2) is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & gt (t1,t2) is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) ) )
thus  for i being integer number holds . (i,X) is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f by Th21; ::_thesis: ( ( for v being INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds . v is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) & ( for x being Element of X holds ^ x is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) & ex a being INT-Array of X st
( a | (card X) is one-to-one & ( for t being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds a * t is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) ) & ( for t being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds - t is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) & ( for t1, t2 being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds
( t1 (#) t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 + t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 div t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 mod t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & leq (t1,t2) is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & gt (t1,t2) is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) ) )
thus  for v being INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds . v is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f by Th21; ::_thesis: ( ( for x being Element of X holds ^ x is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) & ex a being INT-Array of X st
( a | (card X) is one-to-one & ( for t being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds a * t is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) ) & ( for t being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds - t is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) & ( for t1, t2 being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds
( t1 (#) t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 + t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 div t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 mod t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & leq (t1,t2) is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & gt (t1,t2) is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) ) )
thus  for x being Element of X holds ^ x is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f by Th20; ::_thesis: ( ex a being INT-Array of X st
( a | (card X) is one-to-one & ( for t being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds a * t is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) ) & ( for t being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds - t is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) & ( for t1, t2 being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds
( t1 (#) t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 + t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 div t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 mod t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & leq (t1,t2) is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & gt (t1,t2) is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) ) )
hereby  ::_thesis: ( ( for t being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds - t is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) & ( for t1, t2 being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds
( t1 (#) t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 + t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 div t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 mod t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & leq (t1,t2) is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & gt (t1,t2) is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) ) )
take a = a; ::_thesis: ( a | (card X) is one-to-one & ( for t being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds a * t is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) )
 dom (c ") = card X by Th7;
hence  a | (card X) is one-to-one by FUNCT_4:23; ::_thesis: for t being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds a * t is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
thus  for t being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds a * t is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f by Th20; ::_thesis: verum
end;
thus  ( ( for t being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds - t is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) & ( for t1, t2 being INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f holds
( t1 (#) t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 + t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 div t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & t1 mod t2 is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & leq (t1,t2) is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f & gt (t1,t2) is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f ) ) ) by Th21; ::_thesis: verum
end;
end;

registration
cluster FreeUnivAlgNSG (ECIW-signature,INT-ElemIns) ->  Euclidean ;
coherence
 FreeUnivAlgNSG (ECIW-signature,INT-ElemIns) is Euclidean
proof
let X be non empty countable set ; :: according to AOFA_I00:def_23 ::_thesis: for T being Subset of (Funcs (X,INT)) ex f being ExecutionFunction of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns), Funcs (X,INT),T st f is Euclidean
let T be Subset of (Funcs (X,INT)); ::_thesis: ex f being ExecutionFunction of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns), Funcs (X,INT),T st f is Euclidean
set c = the Enumeration of X;
set f = the INT-Exec of the Enumeration of X,T;
take  the INT-Exec of the Enumeration of X,T ; ::_thesis: the INT-Exec of the Enumeration of X,T is Euclidean
thus  the INT-Exec of the Enumeration of X,T is Euclidean ; ::_thesis: verum
end;
end;

registration
cluster non empty V157() V158() V159() with_empty-instruction with_catenation with_if-instruction with_while-instruction non degenerated Euclidean for L9();
existence
 ex b1 being preIfWhileAlgebra st
( b1 is Euclidean & not b1 is degenerated )
proof
take  FreeUnivAlgNSG (ECIW-signature,INT-ElemIns) ; ::_thesis: ( FreeUnivAlgNSG (ECIW-signature,INT-ElemIns) is Euclidean & not FreeUnivAlgNSG (ECIW-signature,INT-ElemIns) is degenerated )
thus  ( FreeUnivAlgNSG (ECIW-signature,INT-ElemIns) is Euclidean & not FreeUnivAlgNSG (ECIW-signature,INT-ElemIns) is degenerated ) ; ::_thesis: verum
end;
end;

registration
let A be Euclidean preIfWhileAlgebra;
let X be non empty countable set ;
let T be Subset of (Funcs (X,INT));
cluster non empty Relation-like [:(Funcs (X,INT)), the carrier of A:] -defined Funcs (X,INT) -valued Function-like V29([:(Funcs (X,INT)), the carrier of A:]) quasi_total Function-yielding V89() complying_with_empty-instruction complying_with_catenation Euclidean for ExecutionFunction of A, Funcs (X,INT),T;
existence
 ex b1 being ExecutionFunction of A, Funcs (X,INT),T st b1 is Euclidean by Def23;
end;


definition
let A be Euclidean preIfWhileAlgebra;
let X be non empty countable set ;
let T be Subset of (Funcs (X,INT));
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T;
let t be INT-Expression of A,f;
:: original: -
redefine func - t ->  INT-Expression of A,f;
coherence
 - t is INT-Expression of A,f by Def22;
end;

definition
let A be Euclidean preIfWhileAlgebra;
let X be non empty countable set ;
let T be Subset of (Funcs (X,INT));
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T;
let t be INT-Expression of A,f;
let i be integer number ;
:: original: +
redefine funct + i ->  INT-Expression of A,f;
coherence
 i + t is INT-Expression of A,f
proof
A1: dom (t + (. (i,X))) = Funcs (X,INT) by FUNCT_2:def_1;
A2: now__::_thesis:_for_a_being_set_st_a_in_Funcs_(X,INT)_holds_
(t_+_(._(i,X)))_._a_=_i_+_(t_._a)
let a be set ; ::_thesis: ( a in Funcs (X,INT) implies (t + (. (i,X))) . a = i + (t . a) )
assume  a in Funcs (X,INT) ; ::_thesis: (t + (. (i,X))) . a = i + (t . a)
then reconsider s = a as Element of Funcs (X,INT) ;
thus (t + (. (i,X))) . a = (t . s) + ((. (i,X)) . s) by A1, VALUED_1:def_1
.= i + (t . a) by FUNCOP_1:7 ; ::_thesis: verum
end;
 . (i,X) is INT-Expression of A,f by Def22;
then A3: t + (. (i,X)) is INT-Expression of A,f by Def22;
 dom t = Funcs (X,INT) by FUNCT_2:def_1;
hence  i + t is INT-Expression of A,f by A3, A1, A2, VALUED_1:def_2; ::_thesis: verum
end;
:: original: -
redefine funct - i ->  INT-Expression of A,f;
coherence
 t - i is INT-Expression of A,f
proof
 . (i,X) is INT-Expression of A,f by Def22;
then  - (. (i,X)) is INT-Expression of A,f by Def22;
then A4: t + (- (. (i,X))) is INT-Expression of A,f by Def22;
A5: dom t = Funcs (X,INT) by FUNCT_2:def_1;
A6: dom (t + (- (. (i,X)))) = Funcs (X,INT) by FUNCT_2:def_1;
A7: now__::_thesis:_for_a_being_set_st_a_in_Funcs_(X,INT)_holds_
(t_+_(-_(._(i,X))))_._a_=_(t_-_i)_._a
let a be set ; ::_thesis: ( a in Funcs (X,INT) implies (t + (- (. (i,X)))) . a = (t - i) . a )
assume  a in Funcs (X,INT) ; ::_thesis: (t + (- (. (i,X)))) . a = (t - i) . a
then reconsider s = a as Element of Funcs (X,INT) ;
thus (t + (- (. (i,X)))) . a = (t . s) + ((- (. (i,X))) . s) by A6, VALUED_1:def_1
.= (t . s) + (- ((. (i,X)) . s)) by VALUED_1:8
.= (t . s) - i by FUNCOP_1:7
.= (t - i) . a by A5, VALUED_1:3 ; ::_thesis: verum
end;
 dom (t - i) = dom t by VALUED_1:3;
hence  t - i is INT-Expression of A,f by A4, A6, A5, A7, FUNCT_1:2; ::_thesis: verum
end;
:: original: *
redefine funct * i ->  INT-Expression of A,f;
coherence
 t * i is INT-Expression of A,f
proof
A8: dom (t (#) (. (i,X))) = Funcs (X,INT) by FUNCT_2:def_1;
A9: now__::_thesis:_for_a_being_set_st_a_in_Funcs_(X,INT)_holds_
(t_(#)_(._(i,X)))_._a_=_i_*_(t_._a)
let a be set ; ::_thesis: ( a in Funcs (X,INT) implies (t (#) (. (i,X))) . a = i * (t . a) )
assume  a in Funcs (X,INT) ; ::_thesis: (t (#) (. (i,X))) . a = i * (t . a)
then reconsider s = a as Element of Funcs (X,INT) ;
thus (t (#) (. (i,X))) . a = (t . s) * ((. (i,X)) . s) by A8, VALUED_1:def_4
.= i * (t . a) by FUNCOP_1:7 ; ::_thesis: verum
end;
 . (i,X) is INT-Expression of A,f by Def22;
then A10: t (#) (. (i,X)) is INT-Expression of A,f by Def22;
 dom t = Funcs (X,INT) by FUNCT_2:def_1;
hence  t * i is INT-Expression of A,f by A10, A8, A9, VALUED_1:def_5; ::_thesis: verum
end;
end;

definition
let A be Euclidean preIfWhileAlgebra;
let X be non empty countable set ;
let T be Subset of (Funcs (X,INT));
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T;
let t1, t2 be INT-Expression of A,f;
:: original: -
redefine funct1 - t2 ->  INT-Expression of A,f;
coherence
 t1 - t2 is INT-Expression of A,f
proof
 - t2 is INT-Expression of A,f ;
hence  t1 - t2 is INT-Expression of A,f by Def22; ::_thesis: verum
end;
:: original: +
redefine funct1 + t2 ->  INT-Expression of A,f;
coherence
 t1 + t2 is INT-Expression of A,f by Def22;
:: original: (#)
redefine funct1 (#) t2 ->  INT-Expression of A,f;
coherence
 t1 (#) t2 is INT-Expression of A,f by Def22;
end;

definition
let A be Euclidean preIfWhileAlgebra;
let X be non empty countable set ;
let T be Subset of (Funcs (X,INT));
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T;
let t1, t2 be INT-Expression of A,f;
:: original: div
redefine funct1 div t2 ->  INT-Expression of A,f means :Def29: :: AOFA_I00:def 29
 for s being Element of Funcs (X,INT) holds it . s = (t1 . s) div (t2 . s);
coherence
 t1 div t2 is INT-Expression of A,f by Def22;
compatibility
 for b1 being INT-Expression of A,f holds
( b1 = t1 div t2 iff for s being Element of Funcs (X,INT) holds b1 . s = (t1 . s) div (t2 . s) )
proof
let ti be INT-Expression of A,f; ::_thesis: ( ti = t1 div t2 iff for s being Element of Funcs (X,INT) holds ti . s = (t1 . s) div (t2 . s) )
A1: dom t1 = Funcs (X,INT) by FUNCT_2:def_1;
A2: dom t2 = Funcs (X,INT) by FUNCT_2:def_1;
A3: dom (t1 div t2) = (dom t1) /\ (dom t2) by Def3;
hence  ( ti = t1 div t2 implies for s being Element of Funcs (X,INT) holds ti . s = (t1 . s) div (t2 . s) ) by A1, A2, Def3; ::_thesis: ( ( for s being Element of Funcs (X,INT) holds ti . s = (t1 . s) div (t2 . s) ) implies ti = t1 div t2 )
A4: dom ti = Funcs (X,INT) by FUNCT_2:def_1;
assume  for s being Element of Funcs (X,INT) holds ti . s = (t1 . s) div (t2 . s) ; ::_thesis: ti = t1 div t2
then  for s being set st s in Funcs (X,INT) holds
ti . s = (t1 . s) div (t2 . s) ;
hence  ti = t1 div t2 by A3, A1, A2, A4, Def3; ::_thesis: verum
end;
:: original: mod
redefine funct1 mod t2 ->  INT-Expression of A,f means :Def30: :: AOFA_I00:def 30
 for s being Element of Funcs (X,INT) holds it . s = (t1 . s) mod (t2 . s);
coherence
 t1 mod t2 is INT-Expression of A,f by Def22;
compatibility
 for b1 being INT-Expression of A,f holds
( b1 = t1 mod t2 iff for s being Element of Funcs (X,INT) holds b1 . s = (t1 . s) mod (t2 . s) )
proof
let ti be INT-Expression of A,f; ::_thesis: ( ti = t1 mod t2 iff for s being Element of Funcs (X,INT) holds ti . s = (t1 . s) mod (t2 . s) )
A5: dom t1 = Funcs (X,INT) by FUNCT_2:def_1;
A6: dom t2 = Funcs (X,INT) by FUNCT_2:def_1;
A7: dom (t1 mod t2) = (dom t1) /\ (dom t2) by Def4;
hence  ( ti = t1 mod t2 implies for s being Element of Funcs (X,INT) holds ti . s = (t1 . s) mod (t2 . s) ) by A5, A6, Def4; ::_thesis: ( ( for s being Element of Funcs (X,INT) holds ti . s = (t1 . s) mod (t2 . s) ) implies ti = t1 mod t2 )
A8: dom ti = Funcs (X,INT) by FUNCT_2:def_1;
assume  for s being Element of Funcs (X,INT) holds ti . s = (t1 . s) mod (t2 . s) ; ::_thesis: ti = t1 mod t2
then  for s being set st s in Funcs (X,INT) holds
ti . s = (t1 . s) mod (t2 . s) ;
hence  ti = t1 mod t2 by A7, A5, A6, A8, Def4; ::_thesis: verum
end;
:: original: leq
redefine func leq (t1,t2) ->  INT-Expression of A,f means :Def31: :: AOFA_I00:def 31
 for s being Element of Funcs (X,INT) holds it . s = IFGT ((t1 . s),(t2 . s),0,1);
compatibility
 for b1 being INT-Expression of A,f holds
( b1 = leq (t1,t2) iff for s being Element of Funcs (X,INT) holds b1 . s = IFGT ((t1 . s),(t2 . s),0,1) )
proof
let ti be INT-Expression of A,f; ::_thesis: ( ti = leq (t1,t2) iff for s being Element of Funcs (X,INT) holds ti . s = IFGT ((t1 . s),(t2 . s),0,1) )
A9: dom t1 = Funcs (X,INT) by FUNCT_2:def_1;
A10: dom t2 = Funcs (X,INT) by FUNCT_2:def_1;
A11: dom (leq (t1,t2)) = (dom t1) /\ (dom t2) by Def5;
hence  ( ti = leq (t1,t2) implies for s being Element of Funcs (X,INT) holds ti . s = IFGT ((t1 . s),(t2 . s),0,1) ) by A9, A10, Def5; ::_thesis: ( ( for s being Element of Funcs (X,INT) holds ti . s = IFGT ((t1 . s),(t2 . s),0,1) ) implies ti = leq (t1,t2) )
A12: dom ti = Funcs (X,INT) by FUNCT_2:def_1;
assume  for s being Element of Funcs (X,INT) holds ti . s = IFGT ((t1 . s),(t2 . s),0,1) ; ::_thesis: ti = leq (t1,t2)
then  for s being set st s in Funcs (X,INT) holds
ti . s = IFGT ((t1 . s),(t2 . s),0,1) ;
hence  ti = leq (t1,t2) by A11, A9, A10, A12, Def5; ::_thesis: verum
end;
coherence
 leq (t1,t2) is INT-Expression of A,f by Def22;
:: original: gt
redefine func gt (t1,t2) ->  INT-Expression of A,f means :Def32: :: AOFA_I00:def 32
 for s being Element of Funcs (X,INT) holds it . s = IFGT ((t1 . s),(t2 . s),1,0);
coherence
 gt (t1,t2) is INT-Expression of A,f by Def22;
compatibility
 for b1 being INT-Expression of A,f holds
( b1 = gt (t1,t2) iff for s being Element of Funcs (X,INT) holds b1 . s = IFGT ((t1 . s),(t2 . s),1,0) )
proof
let ti be INT-Expression of A,f; ::_thesis: ( ti = gt (t1,t2) iff for s being Element of Funcs (X,INT) holds ti . s = IFGT ((t1 . s),(t2 . s),1,0) )
A13: dom t1 = Funcs (X,INT) by FUNCT_2:def_1;
A14: dom t2 = Funcs (X,INT) by FUNCT_2:def_1;
A15: dom (gt (t1,t2)) = (dom t1) /\ (dom t2) by Def6;
hence  ( ti = gt (t1,t2) implies for s being Element of Funcs (X,INT) holds ti . s = IFGT ((t1 . s),(t2 . s),1,0) ) by A13, A14, Def6; ::_thesis: ( ( for s being Element of Funcs (X,INT) holds ti . s = IFGT ((t1 . s),(t2 . s),1,0) ) implies ti = gt (t1,t2) )
A16: dom ti = Funcs (X,INT) by FUNCT_2:def_1;
assume  for s being Element of Funcs (X,INT) holds ti . s = IFGT ((t1 . s),(t2 . s),1,0) ; ::_thesis: ti = gt (t1,t2)
then  for s being set st s in Funcs (X,INT) holds
ti . s = IFGT ((t1 . s),(t2 . s),1,0) ;
hence  ti = gt (t1,t2) by A15, A13, A14, A16, Def6; ::_thesis: verum
end;
end;

:: deftheorem Def29 defines div AOFA_I00:def_29_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for t1, t2, b7 being INT-Expression of A,f holds
( b7 = t1 div t2 iff for s being Element of Funcs (X,INT) holds b7 . s = (t1 . s) div (t2 . s) );

:: deftheorem Def30 defines mod AOFA_I00:def_30_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for t1, t2, b7 being INT-Expression of A,f holds
( b7 = t1 mod t2 iff for s being Element of Funcs (X,INT) holds b7 . s = (t1 . s) mod (t2 . s) );

:: deftheorem Def31 defines leq AOFA_I00:def_31_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for t1, t2, b7 being INT-Expression of A,f holds
( b7 = leq (t1,t2) iff for s being Element of Funcs (X,INT) holds b7 . s = IFGT ((t1 . s),(t2 . s),0,1) );

:: deftheorem Def32 defines gt AOFA_I00:def_32_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for t1, t2, b7 being INT-Expression of A,f holds
( b7 = gt (t1,t2) iff for s being Element of Funcs (X,INT) holds b7 . s = IFGT ((t1 . s),(t2 . s),1,0) );

definition
let A be Euclidean preIfWhileAlgebra;
let X be non empty countable set ;
let T be Subset of (Funcs (X,INT));
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T;
let t1, t2 be INT-Expression of A,f;
:: original: eq
redefine func eq (t1,t2) ->  INT-Expression of A,f means :: AOFA_I00:def 33
 for s being Element of Funcs (X,INT) holds it . s = IFEQ ((t1 . s),(t2 . s),1,0);
compatibility
 for b1 being INT-Expression of A,f holds
( b1 = eq (t1,t2) iff for s being Element of Funcs (X,INT) holds b1 . s = IFEQ ((t1 . s),(t2 . s),1,0) )
proof
let ti be INT-Expression of A,f; ::_thesis: ( ti = eq (t1,t2) iff for s being Element of Funcs (X,INT) holds ti . s = IFEQ ((t1 . s),(t2 . s),1,0) )
A1: dom t1 = Funcs (X,INT) by FUNCT_2:def_1;
A2: dom t2 = Funcs (X,INT) by FUNCT_2:def_1;
A3: dom (eq (t1,t2)) = (dom t1) /\ (dom t2) by Def7;
hence  ( ti = eq (t1,t2) implies for s being Element of Funcs (X,INT) holds ti . s = IFEQ ((t1 . s),(t2 . s),1,0) ) by A1, A2, Def7; ::_thesis: ( ( for s being Element of Funcs (X,INT) holds ti . s = IFEQ ((t1 . s),(t2 . s),1,0) ) implies ti = eq (t1,t2) )
A4: dom ti = Funcs (X,INT) by FUNCT_2:def_1;
assume  for s being Element of Funcs (X,INT) holds ti . s = IFEQ ((t1 . s),(t2 . s),1,0) ; ::_thesis: ti = eq (t1,t2)
then  for s being set st s in Funcs (X,INT) holds
ti . s = IFEQ ((t1 . s),(t2 . s),1,0) ;
hence  ti = eq (t1,t2) by A3, A1, A2, A4, Def7; ::_thesis: verum
end;
coherence
 eq (t1,t2) is INT-Expression of A,f
proof
reconsider l1 = leq (t1,t2), l2 = leq (t2,t1) as INT-Expression of A,f ;
A5: dom (l1 (#) l2) = Funcs (X,INT) by FUNCT_2:def_1;
A6: dom (eq (t1,t2)) = Funcs (X,INT) by FUNCT_2:def_1;
now__::_thesis:_for_a_being_set_st_a_in_Funcs_(X,INT)_holds_
(eq_(t1,t2))_._a_=_(l1_(#)_l2)_._a
let a be set ; ::_thesis: ( a in Funcs (X,INT) implies (eq (t1,t2)) . a = (l1 (#) l2) . a )
assume  a in Funcs (X,INT) ; ::_thesis: (eq (t1,t2)) . a = (l1 (#) l2) . a
then reconsider s = a as Element of Funcs (X,INT) ;
A7: ( t1 . s = t2 . s or t1 . s < t2 . s or t2 . s < t1 . s ) by XXREAL_0:1;
A8: l2 . s = IFGT ((t2 . s),(t1 . s),0,1) by Def31;
A9: (l1 (#) l2) . s = (l1 . s) * (l2 . s) by A5, VALUED_1:def_4;
A10: l1 . s = IFGT ((t1 . s),(t2 . s),0,1) by Def31;
 (eq (t1,t2)) . s = IFEQ ((t1 . s),(t2 . s),1,0) by A6, Def7;
then  ( ( (eq (t1,t2)) . s = 1 & l1 . s = 1 & l2 . s = 1 ) or ( (eq (t1,t2)) . s = 0 & l1 . s = 0 & l2 . s = 1 ) or ( (eq (t1,t2)) . s = 0 & l1 . s = 1 & l2 . s = 0 ) ) by A8, A10, A7, FUNCOP_1:def_8, XXREAL_0:def_11;
hence  (eq (t1,t2)) . a = (l1 (#) l2) . a by A9; ::_thesis: verum
end;
hence  eq (t1,t2) is INT-Expression of A,f by A6, A5, FUNCT_1:2; ::_thesis: verum
end;
end;

:: deftheorem  defines eq AOFA_I00:def_33_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for t1, t2, b7 being INT-Expression of A,f holds
( b7 = eq (t1,t2) iff for s being Element of Funcs (X,INT) holds b7 . s = IFEQ ((t1 . s),(t2 . s),1,0) );

definition
let A be Euclidean preIfWhileAlgebra;
let X be non empty countable set ;
let T be Subset of (Funcs (X,INT));
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T;
let v be INT-Variable of A,f;
func . v ->  INT-Expression of A,f equals :: AOFA_I00:def 34
 . v;
coherence
 . v is INT-Expression of A,f by Def22;
end;

:: deftheorem  defines . AOFA_I00:def_34_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for v being INT-Variable of A,f holds . v = . v;

definition
let A be Euclidean preIfWhileAlgebra;
let X be non empty countable set ;
let T be Subset of (Funcs (X,INT));
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T;
let x be Element of X;
funcx ^ (A,f) ->  INT-Variable of A,f equals :: AOFA_I00:def 35
 ^ x;
coherence
 ^ x is INT-Variable of A,f by Def22;
end;

:: deftheorem  defines ^ AOFA_I00:def_35_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Element of X holds x ^ (A,f) = ^ x;

notation
let A be Euclidean preIfWhileAlgebra;
let X be non empty countable set ;
let T be Subset of (Funcs (X,INT));
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T;
let x be Variable of f;
synonym  ^ x for x ^ (A,f);
end;

definition
let A be Euclidean preIfWhileAlgebra;
let X be non empty countable set ;
let T be Subset of (Funcs (X,INT));
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T;
let x be Variable of f;
func . x ->  INT-Expression of A,f equals :: AOFA_I00:def 36
 . (^ x);
coherence
 . (^ x) is INT-Expression of A,f ;
end;

:: deftheorem  defines . AOFA_I00:def_36_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f holds . x = . (^ x);

theorem Th22: :: AOFA_I00:22
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for s being Element of Funcs (X,INT) holds (. x) . s = s . x
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for s being Element of Funcs (X,INT) holds (. x) . s = s . x
let X be non empty countable set ; ::_thesis: for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for s being Element of Funcs (X,INT) holds (. x) . s = s . x
let T be Subset of (Funcs (X,INT)); ::_thesis: for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for s being Element of Funcs (X,INT) holds (. x) . s = s . x
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T; ::_thesis: for x being Variable of f
for s being Element of Funcs (X,INT) holds (. x) . s = s . x
let x be Variable of f; ::_thesis: for s being Element of Funcs (X,INT) holds (. x) . s = s . x
let s be Element of Funcs (X,INT); ::_thesis: (. x) . s = s . x
thus (. x) . s = s . ((x ^ (A,f)) . s) by Def19
.= s . x by FUNCOP_1:7 ; ::_thesis: verum
end;

definition
let A be Euclidean preIfWhileAlgebra;
let X be non empty countable set ;
let T be Subset of (Funcs (X,INT));
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T;
let i be integer number ;
func . (i,A,f) ->  INT-Expression of A,f equals :: AOFA_I00:def 37
 . (i,X);
coherence
 . (i,X) is INT-Expression of A,f by Def22;
end;

:: deftheorem  defines . AOFA_I00:def_37_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for i being integer number holds . (i,A,f) = . (i,X);

definition
let A be Euclidean preIfWhileAlgebra;
let X be non empty countable set ;
let T be Subset of (Funcs (X,INT));
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T;
let v be INT-Variable of A,f;
let t be INT-Expression of A,f;
funcv := t ->  Element of A equals :: AOFA_I00:def 38
 choose  {  I where I is Element of A : ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) )  }  ;
coherence
 choose  {  I where I is Element of A : ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) )  }  is Element of A
proof
set Y =  {  I where I is Element of A : ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) )  }  ;
A1:  {  I where I is Element of A : ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) )  }  c= the carrier of A
proof
let i be set ; :: according to TARSKI:def_3 ::_thesis: ( not i in  {  I where I is Element of A : ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) )  }  or i in the carrier of A )
assume  i in  {  I where I is Element of A : ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) )  }  ; ::_thesis: i in the carrier of A
then  ex I being Element of A st
( i = I & I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) ) ;
hence  i in the carrier of A ; ::_thesis: verum
end;
 v,t form_assignment_wrt f by Def22;
then consider I0 being Element of A such that
A2: I0 in ElementaryInstructions A and
A3: for s being Element of Funcs (X,INT) holds f . (s,I0) = s +* ((v . s),(t . s)) by Def15;
 I0 in  {  I where I is Element of A : ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) )  }  by A2, A3;
then  choose  {  I where I is Element of A : ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) )  }  in  {  I where I is Element of A : ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) )  }  ;
hence  choose  {  I where I is Element of A : ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) )  }  is Element of A by A1; ::_thesis: verum
end;
end;

:: deftheorem  defines := AOFA_I00:def_38_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for v being INT-Variable of A,f
for t being INT-Expression of A,f holds v := t = choose  {  I where I is Element of A : ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) )  }  ;

theorem Th23: :: AOFA_I00:23
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for v being INT-Variable of A,f
for t being INT-Expression of A,f holds v := t in ElementaryInstructions A
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for v being INT-Variable of A,f
for t being INT-Expression of A,f holds v := t in ElementaryInstructions A
let X be non empty countable set ; ::_thesis: for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for v being INT-Variable of A,f
for t being INT-Expression of A,f holds v := t in ElementaryInstructions A
let T be Subset of (Funcs (X,INT)); ::_thesis: for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for v being INT-Variable of A,f
for t being INT-Expression of A,f holds v := t in ElementaryInstructions A
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T; ::_thesis: for v being INT-Variable of A,f
for t being INT-Expression of A,f holds v := t in ElementaryInstructions A
let v be INT-Variable of A,f; ::_thesis: for t being INT-Expression of A,f holds v := t in ElementaryInstructions A
let t be INT-Expression of A,f; ::_thesis: v := t in ElementaryInstructions A
set Y =  {  I where I is Element of A : ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) )  }  ;
 v,t form_assignment_wrt f by Def22;
then consider I0 being Element of A such that
A1: I0 in ElementaryInstructions A and
A2: for s being Element of Funcs (X,INT) holds f . (s,I0) = s +* ((v . s),(t . s)) by Def15;
 I0 in  {  I where I is Element of A : ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) )  }  by A1, A2;
then  v := t in  {  I where I is Element of A : ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) )  }  ;
then  ex I being Element of A st
( v := t = I & I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) ) ;
hence  v := t in ElementaryInstructions A ; ::_thesis: verum
end;

registration
let A be Euclidean preIfWhileAlgebra;
let X be non empty countable set ;
let T be Subset of (Funcs (X,INT));
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T;
let v be INT-Variable of A,f;
let t be INT-Expression of A,f;
clusterv := t ->  absolutely-terminating ;
coherence
 v := t is absolutely-terminating by Th23, AOFA_000:95;
end;

definition
let A be Euclidean preIfWhileAlgebra;
let X be non empty countable set ;
let T be Subset of (Funcs (X,INT));
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T;
let v be INT-Variable of A,f;
let t be INT-Expression of A,f;
funcv += t ->  absolutely-terminating Element of A equals :: AOFA_I00:def 39
v := ((. v) + t);
coherence
 v := ((. v) + t) is absolutely-terminating Element of A ;
funcv *= t ->  absolutely-terminating Element of A equals :: AOFA_I00:def 40
v := ((. v) (#) t);
coherence
 v := ((. v) (#) t) is absolutely-terminating Element of A ;
end;

:: deftheorem  defines += AOFA_I00:def_39_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for v being INT-Variable of A,f
for t being INT-Expression of A,f holds v += t = v := ((. v) + t);

:: deftheorem  defines *= AOFA_I00:def_40_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for v being INT-Variable of A,f
for t being INT-Expression of A,f holds v *= t = v := ((. v) (#) t);

definition
let A be Euclidean preIfWhileAlgebra;
let X be non empty countable set ;
let T be Subset of (Funcs (X,INT));
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T;
let x be Element of X;
let t be INT-Expression of A,f;
funcx := t ->  absolutely-terminating Element of A equals :: AOFA_I00:def 41
(x ^ (A,f)) := t;
correctness
coherence
(x ^ (A,f)) := t is absolutely-terminating Element of A;
;
end;

:: deftheorem  defines := AOFA_I00:def_41_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Element of X
for t being INT-Expression of A,f holds x := t = (x ^ (A,f)) := t;

definition
let A be Euclidean preIfWhileAlgebra;
let X be non empty countable set ;
let T be Subset of (Funcs (X,INT));
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T;
let x be Element of X;
let y be Variable of f;
funcx := y ->  absolutely-terminating Element of A equals :: AOFA_I00:def 42
x := (. y);
correctness
coherence
x := (. y) is absolutely-terminating Element of A;
;
end;

:: deftheorem  defines := AOFA_I00:def_42_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Element of X
for y being Variable of f holds x := y = x := (. y);

definition
let A be Euclidean preIfWhileAlgebra;
let X be non empty countable set ;
let T be Subset of (Funcs (X,INT));
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T;
let x be Element of X;
let v be INT-Variable of A,f;
funcx := v ->  absolutely-terminating Element of A equals :: AOFA_I00:def 43
x := (. v);
correctness
coherence
x := (. v) is absolutely-terminating Element of A;
;
end;

:: deftheorem  defines := AOFA_I00:def_43_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Element of X
for v being INT-Variable of A,f holds x := v = x := (. v);

definition
let A be Euclidean preIfWhileAlgebra;
let X be non empty countable set ;
let T be Subset of (Funcs (X,INT));
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T;
let v, w be INT-Variable of A,f;
funcv := w ->  absolutely-terminating Element of A equals :: AOFA_I00:def 44
v := (. w);
correctness
coherence
v := (. w) is absolutely-terminating Element of A;
;
end;

:: deftheorem  defines := AOFA_I00:def_44_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for v, w being INT-Variable of A,f holds v := w = v := (. w);

definition
let A be Euclidean preIfWhileAlgebra;
let X be non empty countable set ;
let T be Subset of (Funcs (X,INT));
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T;
let x be Variable of f;
let i be integer number ;
funcx := i ->  absolutely-terminating Element of A equals :: AOFA_I00:def 45
x := (. (i,A,f));
correctness
coherence
x := (. (i,A,f)) is absolutely-terminating Element of A;
;
end;

:: deftheorem  defines := AOFA_I00:def_45_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for i being integer number holds x := i = x := (. (i,A,f));

definition
let A be Euclidean preIfWhileAlgebra;
let X be non empty countable set ;
let T be Subset of (Funcs (X,INT));
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T;
let v1, v2 be INT-Variable of A,f;
let x be Variable of f;
func swap (v1,x,v2) ->  absolutely-terminating Element of A equals :: AOFA_I00:def 46
((x := v1) \; (v1 := v2)) \; (v2 := (. x));
coherence
 ((x := v1) \; (v1 := v2)) \; (v2 := (. x)) is absolutely-terminating Element of A ;
end;

:: deftheorem  defines swap AOFA_I00:def_46_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for v1, v2 being INT-Variable of A,f
for x being Variable of f holds swap (v1,x,v2) = ((x := v1) \; (v1 := v2)) \; (v2 := (. x));

definition
let A be Euclidean preIfWhileAlgebra;
let X be non empty countable set ;
let T be Subset of (Funcs (X,INT));
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T;
let x be Variable of f;
let t be INT-Expression of A,f;
funcx += t ->  absolutely-terminating Element of A equals :: AOFA_I00:def 47
x := ((. x) + t);
correctness
coherence
x := ((. x) + t) is absolutely-terminating Element of A;
;
funcx *= t ->  absolutely-terminating Element of A equals :: AOFA_I00:def 48
x := ((. x) (#) t);
correctness
coherence
x := ((. x) (#) t) is absolutely-terminating Element of A;
;
funcx %= t ->  absolutely-terminating Element of A equals :: AOFA_I00:def 49
x := ((. x) mod t);
correctness
coherence
x := ((. x) mod t) is absolutely-terminating Element of A;
;
funcx /= t ->  absolutely-terminating Element of A equals :: AOFA_I00:def 50
x := ((. x) div t);
correctness
coherence
x := ((. x) div t) is absolutely-terminating Element of A;
;
end;

:: deftheorem  defines += AOFA_I00:def_47_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds x += t = x := ((. x) + t);

:: deftheorem  defines *= AOFA_I00:def_48_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds x *= t = x := ((. x) (#) t);

:: deftheorem  defines %= AOFA_I00:def_49_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds x %= t = x := ((. x) mod t);

:: deftheorem  defines /= AOFA_I00:def_50_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds x /= t = x := ((. x) div t);

definition
let A be Euclidean preIfWhileAlgebra;
let X be non empty countable set ;
let T be Subset of (Funcs (X,INT));
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T;
let x be Variable of f;
let i be integer number ;
funcx += i ->  absolutely-terminating Element of A equals :: AOFA_I00:def 51
x := ((. x) + i);
correctness
coherence
x := ((. x) + i) is absolutely-terminating Element of A;
;
funcx *= i ->  absolutely-terminating Element of A equals :: AOFA_I00:def 52
x := ((. x) * i);
correctness
coherence
x := ((. x) * i) is absolutely-terminating Element of A;
;
funcx %= i ->  absolutely-terminating Element of A equals :: AOFA_I00:def 53
x := ((. x) mod (. (i,A,f)));
correctness
coherence
x := ((. x) mod (. (i,A,f))) is absolutely-terminating Element of A;
;
funcx /= i ->  absolutely-terminating Element of A equals :: AOFA_I00:def 54
x := ((. x) div (. (i,A,f)));
correctness
coherence
x := ((. x) div (. (i,A,f))) is absolutely-terminating Element of A;
;
funcx div i ->  INT-Expression of A,f equals :: AOFA_I00:def 55
(. x) div (. (i,A,f));
correctness
coherence
(. x) div (. (i,A,f)) is INT-Expression of A,f;
;
end;

:: deftheorem  defines += AOFA_I00:def_51_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for i being integer number holds x += i = x := ((. x) + i);

:: deftheorem  defines *= AOFA_I00:def_52_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for i being integer number holds x *= i = x := ((. x) * i);

:: deftheorem  defines %= AOFA_I00:def_53_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for i being integer number holds x %= i = x := ((. x) mod (. (i,A,f)));

:: deftheorem  defines /= AOFA_I00:def_54_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for i being integer number holds x /= i = x := ((. x) div (. (i,A,f)));

:: deftheorem  defines div AOFA_I00:def_55_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for i being integer number holds x div i = (. x) div (. (i,A,f));

definition
let A be Euclidean preIfWhileAlgebra;
let X be non empty countable set ;
let T be Subset of (Funcs (X,INT));
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T;
let v be INT-Variable of A,f;
let i be integer number ;
funcv := i ->  absolutely-terminating Element of A equals :: AOFA_I00:def 56
v := (. (i,A,f));
correctness
coherence
v := (. (i,A,f)) is absolutely-terminating Element of A;
;
end;

:: deftheorem  defines := AOFA_I00:def_56_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for v being INT-Variable of A,f
for i being integer number holds v := i = v := (. (i,A,f));

definition
let A be Euclidean preIfWhileAlgebra;
let X be non empty countable set ;
let T be Subset of (Funcs (X,INT));
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T;
let v be INT-Variable of A,f;
let i be integer number ;
funcv += i ->  absolutely-terminating Element of A equals :: AOFA_I00:def 57
v := ((. v) + i);
correctness
coherence
v := ((. v) + i) is absolutely-terminating Element of A;
;
funcv *= i ->  absolutely-terminating Element of A equals :: AOFA_I00:def 58
v := ((. v) * i);
correctness
coherence
v := ((. v) * i) is absolutely-terminating Element of A;
;
end;

:: deftheorem  defines += AOFA_I00:def_57_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for v being INT-Variable of A,f
for i being integer number holds v += i = v := ((. v) + i);

:: deftheorem  defines *= AOFA_I00:def_58_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for v being INT-Variable of A,f
for i being integer number holds v *= i = v := ((. v) * i);

definition
let A be Euclidean preIfWhileAlgebra;
let X be non empty countable set ;
let b be Element of X;
let g be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0);
let t1 be INT-Expression of A,g;
funct1 is_odd  ->  absolutely-terminating Element of A equals :: AOFA_I00:def 59
b := (t1 mod (. (2,A,g)));
coherence
 b := (t1 mod (. (2,A,g))) is absolutely-terminating Element of A ;
funct1 is_even  ->  absolutely-terminating Element of A equals :: AOFA_I00:def 60
b := ((t1 + 1) mod (. (2,A,g)));
coherence
 b := ((t1 + 1) mod (. (2,A,g))) is absolutely-terminating Element of A ;
let t2 be INT-Expression of A,g;
funct1 leq t2 ->  absolutely-terminating Element of A equals :: AOFA_I00:def 61
b := (leq (t1,t2));
coherence
 b := (leq (t1,t2)) is absolutely-terminating Element of A ;
funct1 gt t2 ->  absolutely-terminating Element of A equals :: AOFA_I00:def 62
b := (gt (t1,t2));
coherence
 b := (gt (t1,t2)) is absolutely-terminating Element of A ;
funct1 eq t2 ->  absolutely-terminating Element of A equals :: AOFA_I00:def 63
b := (eq (t1,t2));
coherence
 b := (eq (t1,t2)) is absolutely-terminating Element of A ;
end;

:: deftheorem  defines is_odd AOFA_I00:def_59_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for t1 being INT-Expression of A,g holds t1 is_odd = b := (t1 mod (. (2,A,g)));

:: deftheorem  defines is_even AOFA_I00:def_60_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for t1 being INT-Expression of A,g holds t1 is_even = b := ((t1 + 1) mod (. (2,A,g)));

:: deftheorem  defines leq AOFA_I00:def_61_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for t1, t2 being INT-Expression of A,g holds t1 leq t2 = b := (leq (t1,t2));

:: deftheorem  defines gt AOFA_I00:def_62_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for t1, t2 being INT-Expression of A,g holds t1 gt t2 = b := (gt (t1,t2));

:: deftheorem  defines eq AOFA_I00:def_63_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for t1, t2 being INT-Expression of A,g holds t1 eq t2 = b := (eq (t1,t2));

notation
let A be Euclidean preIfWhileAlgebra;
let X be non empty countable set ;
let b be Element of X;
let g be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0);
let t1, t2 be INT-Expression of A,g;
synonym t2 geq t1 for t1 leq t2;
synonym t2 lt t1 for t1 gt t2;
end;

definition
let A be Euclidean preIfWhileAlgebra;
let X be non empty countable set ;
let b be Element of X;
let g be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0);
let v1, v2 be INT-Variable of A,g;
funcv1 leq v2 ->  absolutely-terminating Element of A equals :: AOFA_I00:def 64
(. v1) leq (. v2);
coherence
 (. v1) leq (. v2) is absolutely-terminating Element of A ;
funcv1 gt v2 ->  absolutely-terminating Element of A equals :: AOFA_I00:def 65
(. v1) gt (. v2);
coherence
 (. v1) gt (. v2) is absolutely-terminating Element of A ;
end;

:: deftheorem  defines leq AOFA_I00:def_64_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for v1, v2 being INT-Variable of A,g holds v1 leq v2 = (. v1) leq (. v2);

:: deftheorem  defines gt AOFA_I00:def_65_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for v1, v2 being INT-Variable of A,g holds v1 gt v2 = (. v1) gt (. v2);

notation
let A be Euclidean preIfWhileAlgebra;
let X be non empty countable set ;
let b be Element of X;
let g be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0);
let v1, v2 be INT-Variable of A,g;
synonym v2 geq v1 for v1 leq v2;
synonym v2 lt v1 for v1 gt v2;
end;

definition
let A be Euclidean preIfWhileAlgebra;
let X be non empty countable set ;
let b be Element of X;
let g be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0);
let x1 be Variable of g;
funcx1 is_odd  ->  absolutely-terminating Element of A equals :: AOFA_I00:def 66
(. x1) is_odd ;
coherence
 (. x1) is_odd is absolutely-terminating Element of A ;
funcx1 is_even  ->  absolutely-terminating Element of A equals :: AOFA_I00:def 67
(. x1) is_even ;
coherence
 (. x1) is_even is absolutely-terminating Element of A ;
let x2 be Variable of g;
funcx1 leq x2 ->  absolutely-terminating Element of A equals :: AOFA_I00:def 68
(. x1) leq (. x2);
coherence
 (. x1) leq (. x2) is absolutely-terminating Element of A ;
funcx1 gt x2 ->  absolutely-terminating Element of A equals :: AOFA_I00:def 69
(. x1) gt (. x2);
coherence
 (. x1) gt (. x2) is absolutely-terminating Element of A ;
end;

:: deftheorem  defines is_odd AOFA_I00:def_66_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x1 being Variable of g holds x1 is_odd = (. x1) is_odd ;

:: deftheorem  defines is_even AOFA_I00:def_67_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x1 being Variable of g holds x1 is_even = (. x1) is_even ;

:: deftheorem  defines leq AOFA_I00:def_68_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x1, x2 being Variable of g holds x1 leq x2 = (. x1) leq (. x2);

:: deftheorem  defines gt AOFA_I00:def_69_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x1, x2 being Variable of g holds x1 gt x2 = (. x1) gt (. x2);

notation
let A be Euclidean preIfWhileAlgebra;
let X be non empty countable set ;
let b be Element of X;
let g be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0);
let x1, x2 be Variable of g;
synonym x2 geq x1 for x1 leq x2;
synonym x2 lt x1 for x1 gt x2;
end;

definition
let A be Euclidean preIfWhileAlgebra;
let X be non empty countable set ;
let b be Element of X;
let g be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0);
let x be Variable of g;
let i be integer number ;
funcx leq i ->  absolutely-terminating Element of A equals :: AOFA_I00:def 70
(. x) leq (. (i,A,g));
coherence
 (. x) leq (. (i,A,g)) is absolutely-terminating Element of A ;
funcx geq i ->  absolutely-terminating Element of A equals :: AOFA_I00:def 71
(. x) geq (. (i,A,g));
coherence
 (. x) geq (. (i,A,g)) is absolutely-terminating Element of A ;
funcx gt i ->  absolutely-terminating Element of A equals :: AOFA_I00:def 72
(. x) gt (. (i,A,g));
coherence
 (. x) gt (. (i,A,g)) is absolutely-terminating Element of A ;
funcx lt i ->  absolutely-terminating Element of A equals :: AOFA_I00:def 73
(. x) lt (. (i,A,g));
coherence
 (. x) lt (. (i,A,g)) is absolutely-terminating Element of A ;
funcx / i ->  INT-Expression of A,g equals :: AOFA_I00:def 74
(. x) div (. (i,A,g));
coherence
 (. x) div (. (i,A,g)) is INT-Expression of A,g ;
end;

:: deftheorem  defines leq AOFA_I00:def_70_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g
for i being integer number holds x leq i = (. x) leq (. (i,A,g));

:: deftheorem  defines geq AOFA_I00:def_71_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g
for i being integer number holds x geq i = (. x) geq (. (i,A,g));

:: deftheorem  defines gt AOFA_I00:def_72_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g
for i being integer number holds x gt i = (. x) gt (. (i,A,g));

:: deftheorem  defines lt AOFA_I00:def_73_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g
for i being integer number holds x lt i = (. x) lt (. (i,A,g));

:: deftheorem  defines / AOFA_I00:def_74_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g
for i being integer number holds x / i = (. x) div (. (i,A,g));

definition
let A be Euclidean preIfWhileAlgebra;
let X be non empty countable set ;
let T be Subset of (Funcs (X,INT));
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T;
let x1, x2 be Variable of f;
funcx1 += x2 ->  absolutely-terminating Element of A equals :: AOFA_I00:def 75
x1 += (. x2);
coherence
 x1 += (. x2) is absolutely-terminating Element of A ;
funcx1 *= x2 ->  absolutely-terminating Element of A equals :: AOFA_I00:def 76
x1 *= (. x2);
coherence
 x1 *= (. x2) is absolutely-terminating Element of A ;
funcx1 %= x2 ->  absolutely-terminating Element of A equals :: AOFA_I00:def 77
x1 := ((. x1) mod (. x2));
coherence
 x1 := ((. x1) mod (. x2)) is absolutely-terminating Element of A ;
funcx1 /= x2 ->  absolutely-terminating Element of A equals :: AOFA_I00:def 78
x1 := ((. x1) div (. x2));
coherence
 x1 := ((. x1) div (. x2)) is absolutely-terminating Element of A ;
funcx1 + x2 ->  INT-Expression of A,f equals :: AOFA_I00:def 79
(. x1) + (. x2);
coherence
 (. x1) + (. x2) is INT-Expression of A,f ;
funcx1 * x2 ->  INT-Expression of A,f equals :: AOFA_I00:def 80
(. x1) (#) (. x2);
coherence
 (. x1) (#) (. x2) is INT-Expression of A,f ;
funcx1 mod x2 ->  INT-Expression of A,f equals :: AOFA_I00:def 81
(. x1) mod (. x2);
coherence
 (. x1) mod (. x2) is INT-Expression of A,f ;
funcx1 div x2 ->  INT-Expression of A,f equals :: AOFA_I00:def 82
(. x1) div (. x2);
coherence
 (. x1) div (. x2) is INT-Expression of A,f ;
end;

:: deftheorem  defines += AOFA_I00:def_75_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x1, x2 being Variable of f holds x1 += x2 = x1 += (. x2);

:: deftheorem  defines *= AOFA_I00:def_76_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x1, x2 being Variable of f holds x1 *= x2 = x1 *= (. x2);

:: deftheorem  defines %= AOFA_I00:def_77_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x1, x2 being Variable of f holds x1 %= x2 = x1 := ((. x1) mod (. x2));

:: deftheorem  defines /= AOFA_I00:def_78_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x1, x2 being Variable of f holds x1 /= x2 = x1 := ((. x1) div (. x2));

:: deftheorem  defines + AOFA_I00:def_79_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x1, x2 being Variable of f holds x1 + x2 = (. x1) + (. x2);

:: deftheorem  defines * AOFA_I00:def_80_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x1, x2 being Variable of f holds x1 * x2 = (. x1) (#) (. x2);

:: deftheorem  defines mod AOFA_I00:def_81_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x1, x2 being Variable of f holds x1 mod x2 = (. x1) mod (. x2);

:: deftheorem  defines div AOFA_I00:def_82_:_
 for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x1, x2 being Variable of f holds x1 div x2 = (. x1) div (. x2);


theorem Th24: :: AOFA_I00:24
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for v being INT-Variable of A,f
for t being INT-Expression of A,f holds
( (f . (s,(v := t))) . (v . s) = t . s & ( for z being Element of X st z <> v . s holds
(f . (s,(v := t))) . z = s . z ) )
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for v being INT-Variable of A,f
for t being INT-Expression of A,f holds
( (f . (s,(v := t))) . (v . s) = t . s & ( for z being Element of X st z <> v . s holds
(f . (s,(v := t))) . z = s . z ) )
let X be non empty countable set ; ::_thesis: for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for v being INT-Variable of A,f
for t being INT-Expression of A,f holds
( (f . (s,(v := t))) . (v . s) = t . s & ( for z being Element of X st z <> v . s holds
(f . (s,(v := t))) . z = s . z ) )
let s be Element of Funcs (X,INT); ::_thesis: for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for v being INT-Variable of A,f
for t being INT-Expression of A,f holds
( (f . (s,(v := t))) . (v . s) = t . s & ( for z being Element of X st z <> v . s holds
(f . (s,(v := t))) . z = s . z ) )
let T be Subset of (Funcs (X,INT)); ::_thesis: for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for v being INT-Variable of A,f
for t being INT-Expression of A,f holds
( (f . (s,(v := t))) . (v . s) = t . s & ( for z being Element of X st z <> v . s holds
(f . (s,(v := t))) . z = s . z ) )
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T; ::_thesis: for v being INT-Variable of A,f
for t being INT-Expression of A,f holds
( (f . (s,(v := t))) . (v . s) = t . s & ( for z being Element of X st z <> v . s holds
(f . (s,(v := t))) . z = s . z ) )
let v be INT-Variable of A,f; ::_thesis: for t being INT-Expression of A,f holds
( (f . (s,(v := t))) . (v . s) = t . s & ( for z being Element of X st z <> v . s holds
(f . (s,(v := t))) . z = s . z ) )
let t be INT-Expression of A,f; ::_thesis: ( (f . (s,(v := t))) . (v . s) = t . s & ( for z being Element of X st z <> v . s holds
(f . (s,(v := t))) . z = s . z ) )
set Y =  {  I where I is Element of A : ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) )  }  ;
 v,t form_assignment_wrt f by Def22;
then consider I0 being Element of A such that
A1: I0 in ElementaryInstructions A and
A2: for s being Element of Funcs (X,INT) holds f . (s,I0) = s +* ((v . s),(t . s)) by Def15;
 I0 in  {  I where I is Element of A : ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) )  }  by A1, A2;
then  v := t in  {  I where I is Element of A : ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) )  }  ;
then  ex I being Element of A st
( v := t = I & I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) ) ;
then A3: f . (s,(v := t)) = s +* ((v . s),(t . s)) ;
 dom s = X by FUNCT_2:def_1;
hence  ( (f . (s,(v := t))) . (v . s) = t . s & ( for z being Element of X st z <> v . s holds
(f . (s,(v := t))) . z = s . z ) ) by A3, FUNCT_7:31, FUNCT_7:32; ::_thesis: verum
end;

theorem Th25: :: AOFA_I00:25
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for i being integer number holds
( (f . (s,(x := i))) . x = i & ( for z being Element of X st z <> x holds
(f . (s,(x := i))) . z = s . z ) )
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for i being integer number holds
( (f . (s,(x := i))) . x = i & ( for z being Element of X st z <> x holds
(f . (s,(x := i))) . z = s . z ) )
let X be non empty countable set ; ::_thesis: for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for i being integer number holds
( (f . (s,(x := i))) . x = i & ( for z being Element of X st z <> x holds
(f . (s,(x := i))) . z = s . z ) )
let s be Element of Funcs (X,INT); ::_thesis: for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for i being integer number holds
( (f . (s,(x := i))) . x = i & ( for z being Element of X st z <> x holds
(f . (s,(x := i))) . z = s . z ) )
let T be Subset of (Funcs (X,INT)); ::_thesis: for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for i being integer number holds
( (f . (s,(x := i))) . x = i & ( for z being Element of X st z <> x holds
(f . (s,(x := i))) . z = s . z ) )
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T; ::_thesis: for x being Variable of f
for i being integer number holds
( (f . (s,(x := i))) . x = i & ( for z being Element of X st z <> x holds
(f . (s,(x := i))) . z = s . z ) )
let x be Variable of f; ::_thesis: for i being integer number holds
( (f . (s,(x := i))) . x = i & ( for z being Element of X st z <> x holds
(f . (s,(x := i))) . z = s . z ) )
let i be integer number ; ::_thesis: ( (f . (s,(x := i))) . x = i & ( for z being Element of X st z <> x holds
(f . (s,(x := i))) . z = s . z ) )
A1: (. (i,A,f)) . s = i by FUNCOP_1:7;
 (^ x) . s = x by FUNCOP_1:7;
hence  ( (f . (s,(x := i))) . x = i & ( for z being Element of X st z <> x holds
(f . (s,(x := i))) . z = s . z ) ) by A1, Th24; ::_thesis: verum
end;

theorem Th26: :: AOFA_I00:26
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x := t))) . x = t . s & ( for z being Element of X st z <> x holds
(f . (s,(x := t))) . z = s . z ) )
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x := t))) . x = t . s & ( for z being Element of X st z <> x holds
(f . (s,(x := t))) . z = s . z ) )
let X be non empty countable set ; ::_thesis: for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x := t))) . x = t . s & ( for z being Element of X st z <> x holds
(f . (s,(x := t))) . z = s . z ) )
let s be Element of Funcs (X,INT); ::_thesis: for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x := t))) . x = t . s & ( for z being Element of X st z <> x holds
(f . (s,(x := t))) . z = s . z ) )
let T be Subset of (Funcs (X,INT)); ::_thesis: for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x := t))) . x = t . s & ( for z being Element of X st z <> x holds
(f . (s,(x := t))) . z = s . z ) )
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T; ::_thesis: for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x := t))) . x = t . s & ( for z being Element of X st z <> x holds
(f . (s,(x := t))) . z = s . z ) )
let x be Variable of f; ::_thesis: for t being INT-Expression of A,f holds
( (f . (s,(x := t))) . x = t . s & ( for z being Element of X st z <> x holds
(f . (s,(x := t))) . z = s . z ) )
let t be INT-Expression of A,f; ::_thesis: ( (f . (s,(x := t))) . x = t . s & ( for z being Element of X st z <> x holds
(f . (s,(x := t))) . z = s . z ) )
 (^ x) . s = x by FUNCOP_1:7;
hence  ( (f . (s,(x := t))) . x = t . s & ( for z being Element of X st z <> x holds
(f . (s,(x := t))) . z = s . z ) ) by Th24; ::_thesis: verum
end;

theorem Th27: :: AOFA_I00:27
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x, y being Variable of f holds
( (f . (s,(x := y))) . x = s . y & ( for z being Element of X st z <> x holds
(f . (s,(x := y))) . z = s . z ) )
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x, y being Variable of f holds
( (f . (s,(x := y))) . x = s . y & ( for z being Element of X st z <> x holds
(f . (s,(x := y))) . z = s . z ) )
let X be non empty countable set ; ::_thesis: for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x, y being Variable of f holds
( (f . (s,(x := y))) . x = s . y & ( for z being Element of X st z <> x holds
(f . (s,(x := y))) . z = s . z ) )
let s be Element of Funcs (X,INT); ::_thesis: for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x, y being Variable of f holds
( (f . (s,(x := y))) . x = s . y & ( for z being Element of X st z <> x holds
(f . (s,(x := y))) . z = s . z ) )
let T be Subset of (Funcs (X,INT)); ::_thesis: for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x, y being Variable of f holds
( (f . (s,(x := y))) . x = s . y & ( for z being Element of X st z <> x holds
(f . (s,(x := y))) . z = s . z ) )
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T; ::_thesis: for x, y being Variable of f holds
( (f . (s,(x := y))) . x = s . y & ( for z being Element of X st z <> x holds
(f . (s,(x := y))) . z = s . z ) )
let x, y be Variable of f; ::_thesis: ( (f . (s,(x := y))) . x = s . y & ( for z being Element of X st z <> x holds
(f . (s,(x := y))) . z = s . z ) )
 (. y) . s = s . y by Th22;
hence  ( (f . (s,(x := y))) . x = s . y & ( for z being Element of X st z <> x holds
(f . (s,(x := y))) . z = s . z ) ) by Th26; ::_thesis: verum
end;

theorem Th28: :: AOFA_I00:28
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for i being integer number
for x being Variable of f holds
( (f . (s,(x += i))) . x = (s . x) + i & ( for z being Element of X st z <> x holds
(f . (s,(x += i))) . z = s . z ) )
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for i being integer number
for x being Variable of f holds
( (f . (s,(x += i))) . x = (s . x) + i & ( for z being Element of X st z <> x holds
(f . (s,(x += i))) . z = s . z ) )
let X be non empty countable set ; ::_thesis: for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for i being integer number
for x being Variable of f holds
( (f . (s,(x += i))) . x = (s . x) + i & ( for z being Element of X st z <> x holds
(f . (s,(x += i))) . z = s . z ) )
let s be Element of Funcs (X,INT); ::_thesis: for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for i being integer number
for x being Variable of f holds
( (f . (s,(x += i))) . x = (s . x) + i & ( for z being Element of X st z <> x holds
(f . (s,(x += i))) . z = s . z ) )
let T be Subset of (Funcs (X,INT)); ::_thesis: for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for i being integer number
for x being Variable of f holds
( (f . (s,(x += i))) . x = (s . x) + i & ( for z being Element of X st z <> x holds
(f . (s,(x += i))) . z = s . z ) )
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T; ::_thesis: for i being integer number
for x being Variable of f holds
( (f . (s,(x += i))) . x = (s . x) + i & ( for z being Element of X st z <> x holds
(f . (s,(x += i))) . z = s . z ) )
let i be integer number ; ::_thesis: for x being Variable of f holds
( (f . (s,(x += i))) . x = (s . x) + i & ( for z being Element of X st z <> x holds
(f . (s,(x += i))) . z = s . z ) )
let x be Variable of f; ::_thesis: ( (f . (s,(x += i))) . x = (s . x) + i & ( for z being Element of X st z <> x holds
(f . (s,(x += i))) . z = s . z ) )
A1: (^ x) . s = x by FUNCOP_1:7;
A2: (. (^ x)) . s = s . ((^ x) . s) by Def19;
 ((. x) + i) . s = ((. x) . s) + i by Def8;
hence  ( (f . (s,(x += i))) . x = (s . x) + i & ( for z being Element of X st z <> x holds
(f . (s,(x += i))) . z = s . z ) ) by A1, A2, Th24; ::_thesis: verum
end;

theorem Th29: :: AOFA_I00:29
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x += t))) . x = (s . x) + (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x += t))) . z = s . z ) )
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x += t))) . x = (s . x) + (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x += t))) . z = s . z ) )
let X be non empty countable set ; ::_thesis: for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x += t))) . x = (s . x) + (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x += t))) . z = s . z ) )
let s be Element of Funcs (X,INT); ::_thesis: for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x += t))) . x = (s . x) + (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x += t))) . z = s . z ) )
let T be Subset of (Funcs (X,INT)); ::_thesis: for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x += t))) . x = (s . x) + (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x += t))) . z = s . z ) )
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T; ::_thesis: for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x += t))) . x = (s . x) + (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x += t))) . z = s . z ) )
let x be Variable of f; ::_thesis: for t being INT-Expression of A,f holds
( (f . (s,(x += t))) . x = (s . x) + (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x += t))) . z = s . z ) )
let t be INT-Expression of A,f; ::_thesis: ( (f . (s,(x += t))) . x = (s . x) + (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x += t))) . z = s . z ) )
A1: (^ x) . s = x by FUNCOP_1:7;
 dom ((. x) + t) = Funcs (X,INT) by FUNCT_2:def_1;
then A2: ((. x) + t) . s = ((. x) . s) + (t . s) by VALUED_1:def_1;
 (. x) . s = s . x by Th22;
hence  ( (f . (s,(x += t))) . x = (s . x) + (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x += t))) . z = s . z ) ) by A1, A2, Th24; ::_thesis: verum
end;

theorem Th30: :: AOFA_I00:30
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x, y being Variable of f holds
( (f . (s,(x += y))) . x = (s . x) + (s . y) & ( for z being Element of X st z <> x holds
(f . (s,(x += y))) . z = s . z ) )
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x, y being Variable of f holds
( (f . (s,(x += y))) . x = (s . x) + (s . y) & ( for z being Element of X st z <> x holds
(f . (s,(x += y))) . z = s . z ) )
let X be non empty countable set ; ::_thesis: for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x, y being Variable of f holds
( (f . (s,(x += y))) . x = (s . x) + (s . y) & ( for z being Element of X st z <> x holds
(f . (s,(x += y))) . z = s . z ) )
let s be Element of Funcs (X,INT); ::_thesis: for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x, y being Variable of f holds
( (f . (s,(x += y))) . x = (s . x) + (s . y) & ( for z being Element of X st z <> x holds
(f . (s,(x += y))) . z = s . z ) )
let T be Subset of (Funcs (X,INT)); ::_thesis: for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x, y being Variable of f holds
( (f . (s,(x += y))) . x = (s . x) + (s . y) & ( for z being Element of X st z <> x holds
(f . (s,(x += y))) . z = s . z ) )
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T; ::_thesis: for x, y being Variable of f holds
( (f . (s,(x += y))) . x = (s . x) + (s . y) & ( for z being Element of X st z <> x holds
(f . (s,(x += y))) . z = s . z ) )
let x, y be Variable of f; ::_thesis: ( (f . (s,(x += y))) . x = (s . x) + (s . y) & ( for z being Element of X st z <> x holds
(f . (s,(x += y))) . z = s . z ) )
 (. y) . s = s . y by Th22;
hence  ( (f . (s,(x += y))) . x = (s . x) + (s . y) & ( for z being Element of X st z <> x holds
(f . (s,(x += y))) . z = s . z ) ) by Th29; ::_thesis: verum
end;

theorem Th31: :: AOFA_I00:31
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for i being integer number
for x being Variable of f holds
( (f . (s,(x *= i))) . x = (s . x) * i & ( for z being Element of X st z <> x holds
(f . (s,(x *= i))) . z = s . z ) )
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for i being integer number
for x being Variable of f holds
( (f . (s,(x *= i))) . x = (s . x) * i & ( for z being Element of X st z <> x holds
(f . (s,(x *= i))) . z = s . z ) )
let X be non empty countable set ; ::_thesis: for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for i being integer number
for x being Variable of f holds
( (f . (s,(x *= i))) . x = (s . x) * i & ( for z being Element of X st z <> x holds
(f . (s,(x *= i))) . z = s . z ) )
let s be Element of Funcs (X,INT); ::_thesis: for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for i being integer number
for x being Variable of f holds
( (f . (s,(x *= i))) . x = (s . x) * i & ( for z being Element of X st z <> x holds
(f . (s,(x *= i))) . z = s . z ) )
let T be Subset of (Funcs (X,INT)); ::_thesis: for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for i being integer number
for x being Variable of f holds
( (f . (s,(x *= i))) . x = (s . x) * i & ( for z being Element of X st z <> x holds
(f . (s,(x *= i))) . z = s . z ) )
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T; ::_thesis: for i being integer number
for x being Variable of f holds
( (f . (s,(x *= i))) . x = (s . x) * i & ( for z being Element of X st z <> x holds
(f . (s,(x *= i))) . z = s . z ) )
let i be integer number ; ::_thesis: for x being Variable of f holds
( (f . (s,(x *= i))) . x = (s . x) * i & ( for z being Element of X st z <> x holds
(f . (s,(x *= i))) . z = s . z ) )
let x be Variable of f; ::_thesis: ( (f . (s,(x *= i))) . x = (s . x) * i & ( for z being Element of X st z <> x holds
(f . (s,(x *= i))) . z = s . z ) )
A1: (^ x) . s = x by FUNCOP_1:7;
A2: (. (^ x)) . s = s . ((^ x) . s) by Def19;
 ((. x) * i) . s = ((. x) . s) * i by Def10;
hence  ( (f . (s,(x *= i))) . x = (s . x) * i & ( for z being Element of X st z <> x holds
(f . (s,(x *= i))) . z = s . z ) ) by A1, A2, Th24; ::_thesis: verum
end;

theorem Th32: :: AOFA_I00:32
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x *= t))) . x = (s . x) * (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x *= t))) . z = s . z ) )
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x *= t))) . x = (s . x) * (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x *= t))) . z = s . z ) )
let X be non empty countable set ; ::_thesis: for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x *= t))) . x = (s . x) * (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x *= t))) . z = s . z ) )
let s be Element of Funcs (X,INT); ::_thesis: for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x *= t))) . x = (s . x) * (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x *= t))) . z = s . z ) )
let T be Subset of (Funcs (X,INT)); ::_thesis: for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x *= t))) . x = (s . x) * (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x *= t))) . z = s . z ) )
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T; ::_thesis: for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x *= t))) . x = (s . x) * (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x *= t))) . z = s . z ) )
let x be Variable of f; ::_thesis: for t being INT-Expression of A,f holds
( (f . (s,(x *= t))) . x = (s . x) * (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x *= t))) . z = s . z ) )
let t be INT-Expression of A,f; ::_thesis: ( (f . (s,(x *= t))) . x = (s . x) * (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x *= t))) . z = s . z ) )
A1: (^ x) . s = x by FUNCOP_1:7;
 dom ((. x) (#) t) = Funcs (X,INT) by FUNCT_2:def_1;
then A2: ((. x) (#) t) . s = ((. x) . s) * (t . s) by VALUED_1:def_4;
 (. x) . s = s . x by Th22;
hence  ( (f . (s,(x *= t))) . x = (s . x) * (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x *= t))) . z = s . z ) ) by A1, A2, Th24; ::_thesis: verum
end;

theorem Th33: :: AOFA_I00:33
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x, y being Variable of f holds
( (f . (s,(x *= y))) . x = (s . x) * (s . y) & ( for z being Element of X st z <> x holds
(f . (s,(x *= y))) . z = s . z ) )
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x, y being Variable of f holds
( (f . (s,(x *= y))) . x = (s . x) * (s . y) & ( for z being Element of X st z <> x holds
(f . (s,(x *= y))) . z = s . z ) )
let X be non empty countable set ; ::_thesis: for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x, y being Variable of f holds
( (f . (s,(x *= y))) . x = (s . x) * (s . y) & ( for z being Element of X st z <> x holds
(f . (s,(x *= y))) . z = s . z ) )
let s be Element of Funcs (X,INT); ::_thesis: for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x, y being Variable of f holds
( (f . (s,(x *= y))) . x = (s . x) * (s . y) & ( for z being Element of X st z <> x holds
(f . (s,(x *= y))) . z = s . z ) )
let T be Subset of (Funcs (X,INT)); ::_thesis: for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x, y being Variable of f holds
( (f . (s,(x *= y))) . x = (s . x) * (s . y) & ( for z being Element of X st z <> x holds
(f . (s,(x *= y))) . z = s . z ) )
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T; ::_thesis: for x, y being Variable of f holds
( (f . (s,(x *= y))) . x = (s . x) * (s . y) & ( for z being Element of X st z <> x holds
(f . (s,(x *= y))) . z = s . z ) )
let x, y be Variable of f; ::_thesis: ( (f . (s,(x *= y))) . x = (s . x) * (s . y) & ( for z being Element of X st z <> x holds
(f . (s,(x *= y))) . z = s . z ) )
A1: dom ((. x) (#) (. y)) = Funcs (X,INT) by FUNCT_2:def_1;
 (^ x) . s = x by FUNCOP_1:7;
hence (f . (s,(x *= y))) . x = ((. x) (#) (. y)) . s by Th24
.= ((. x) . s) * ((. y) . s) by A1, VALUED_1:def_4
.= (s . x) * ((. y) . s) by Th22
.= (s . x) * (s . y) by Th22 ;
::_thesis: for z being Element of X st z <> x holds
(f . (s,(x *= y))) . z = s . z
thus  for z being Element of X st z <> x holds
(f . (s,(x *= y))) . z = s . z by Th26; ::_thesis: verum
end;

theorem Th34: :: AOFA_I00:34
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g
for i being integer number holds
( ( s . x <= i implies (g . (s,(x leq i))) . b = 1 ) & ( s . x > i implies (g . (s,(x leq i))) . b = 0 ) & ( s . x >= i implies (g . (s,(x geq i))) . b = 1 ) & ( s . x < i implies (g . (s,(x geq i))) . b = 0 ) & ( for z being Element of X st z <> b holds
( (g . (s,(x leq i))) . z = s . z & (g . (s,(x geq i))) . z = s . z ) ) )
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g
for i being integer number holds
( ( s . x <= i implies (g . (s,(x leq i))) . b = 1 ) & ( s . x > i implies (g . (s,(x leq i))) . b = 0 ) & ( s . x >= i implies (g . (s,(x geq i))) . b = 1 ) & ( s . x < i implies (g . (s,(x geq i))) . b = 0 ) & ( for z being Element of X st z <> b holds
( (g . (s,(x leq i))) . z = s . z & (g . (s,(x geq i))) . z = s . z ) ) )
let X be non empty countable set ; ::_thesis: for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g
for i being integer number holds
( ( s . x <= i implies (g . (s,(x leq i))) . b = 1 ) & ( s . x > i implies (g . (s,(x leq i))) . b = 0 ) & ( s . x >= i implies (g . (s,(x geq i))) . b = 1 ) & ( s . x < i implies (g . (s,(x geq i))) . b = 0 ) & ( for z being Element of X st z <> b holds
( (g . (s,(x leq i))) . z = s . z & (g . (s,(x geq i))) . z = s . z ) ) )
let s be Element of Funcs (X,INT); ::_thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g
for i being integer number holds
( ( s . x <= i implies (g . (s,(x leq i))) . b = 1 ) & ( s . x > i implies (g . (s,(x leq i))) . b = 0 ) & ( s . x >= i implies (g . (s,(x geq i))) . b = 1 ) & ( s . x < i implies (g . (s,(x geq i))) . b = 0 ) & ( for z being Element of X st z <> b holds
( (g . (s,(x leq i))) . z = s . z & (g . (s,(x geq i))) . z = s . z ) ) )
let b be Element of X; ::_thesis: for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g
for i being integer number holds
( ( s . x <= i implies (g . (s,(x leq i))) . b = 1 ) & ( s . x > i implies (g . (s,(x leq i))) . b = 0 ) & ( s . x >= i implies (g . (s,(x geq i))) . b = 1 ) & ( s . x < i implies (g . (s,(x geq i))) . b = 0 ) & ( for z being Element of X st z <> b holds
( (g . (s,(x leq i))) . z = s . z & (g . (s,(x geq i))) . z = s . z ) ) )
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); ::_thesis: for x being Variable of f
for i being integer number holds
( ( s . x <= i implies (f . (s,(x leq i))) . b = 1 ) & ( s . x > i implies (f . (s,(x leq i))) . b = 0 ) & ( s . x >= i implies (f . (s,(x geq i))) . b = 1 ) & ( s . x < i implies (f . (s,(x geq i))) . b = 0 ) & ( for z being Element of X st z <> b holds
( (f . (s,(x leq i))) . z = s . z & (f . (s,(x geq i))) . z = s . z ) ) )
reconsider b9 = b as Variable of f by Def2;
let x be Variable of f; ::_thesis: for i being integer number holds
( ( s . x <= i implies (f . (s,(x leq i))) . b = 1 ) & ( s . x > i implies (f . (s,(x leq i))) . b = 0 ) & ( s . x >= i implies (f . (s,(x geq i))) . b = 1 ) & ( s . x < i implies (f . (s,(x geq i))) . b = 0 ) & ( for z being Element of X st z <> b holds
( (f . (s,(x leq i))) . z = s . z & (f . (s,(x geq i))) . z = s . z ) ) )
let i be integer number ; ::_thesis: ( ( s . x <= i implies (f . (s,(x leq i))) . b = 1 ) & ( s . x > i implies (f . (s,(x leq i))) . b = 0 ) & ( s . x >= i implies (f . (s,(x geq i))) . b = 1 ) & ( s . x < i implies (f . (s,(x geq i))) . b = 0 ) & ( for z being Element of X st z <> b holds
( (f . (s,(x leq i))) . z = s . z & (f . (s,(x geq i))) . z = s . z ) ) )
set v = ^ b9;
A1: (. (i,A,f)) . s = i by FUNCOP_1:7;
A2: (^ b9) . s = b by FUNCOP_1:7;
A3: ( (. x) . s < i implies IFGT (i,((. x) . s),0,1) = 0 ) by XXREAL_0:def_11;
A4: ( (. x) . s >= i implies IFGT (i,((. x) . s),0,1) = 1 ) by XXREAL_0:def_11;
A5: (leq ((. (i,A,f)),(. x))) . s = IFGT (((. (i,A,f)) . s),((. x) . s),0,1) by Def31;
A6: ( (. x) . s > i implies IFGT (((. x) . s),i,0,1) = 0 ) by XXREAL_0:def_11;
A7: (leq ((. x),(. (i,A,f)))) . s = IFGT (((. x) . s),((. (i,A,f)) . s),0,1) by Def31;
A8: ( (. x) . s <= i implies IFGT (((. x) . s),i,0,1) = 1 ) by XXREAL_0:def_11;
 (. x) . s = s . ((^ x) . s) by Def19;
hence  ( ( s . x <= i implies (f . (s,(x leq i))) . b = 1 ) & ( s . x > i implies (f . (s,(x leq i))) . b = 0 ) & ( s . x >= i implies (f . (s,(x geq i))) . b = 1 ) & ( s . x < i implies (f . (s,(x geq i))) . b = 0 ) & ( for z being Element of X st z <> b holds
( (f . (s,(x leq i))) . z = s . z & (f . (s,(x geq i))) . z = s . z ) ) ) by A1, A2, A8, A6, A4, A3, A7, A5, Th24, FUNCOP_1:7; ::_thesis: verum
end;

theorem Th35: :: AOFA_I00:35
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y being Variable of g holds
( ( s . x <= s . y implies (g . (s,(x leq y))) . b = 1 ) & ( s . x > s . y implies (g . (s,(x leq y))) . b = 0 ) & ( for z being Element of X st z <> b holds
(g . (s,(x leq y))) . z = s . z ) )
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y being Variable of g holds
( ( s . x <= s . y implies (g . (s,(x leq y))) . b = 1 ) & ( s . x > s . y implies (g . (s,(x leq y))) . b = 0 ) & ( for z being Element of X st z <> b holds
(g . (s,(x leq y))) . z = s . z ) )
let X be non empty countable set ; ::_thesis: for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y being Variable of g holds
( ( s . x <= s . y implies (g . (s,(x leq y))) . b = 1 ) & ( s . x > s . y implies (g . (s,(x leq y))) . b = 0 ) & ( for z being Element of X st z <> b holds
(g . (s,(x leq y))) . z = s . z ) )
let s be Element of Funcs (X,INT); ::_thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y being Variable of g holds
( ( s . x <= s . y implies (g . (s,(x leq y))) . b = 1 ) & ( s . x > s . y implies (g . (s,(x leq y))) . b = 0 ) & ( for z being Element of X st z <> b holds
(g . (s,(x leq y))) . z = s . z ) )
let b be Element of X; ::_thesis: for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y being Variable of g holds
( ( s . x <= s . y implies (g . (s,(x leq y))) . b = 1 ) & ( s . x > s . y implies (g . (s,(x leq y))) . b = 0 ) & ( for z being Element of X st z <> b holds
(g . (s,(x leq y))) . z = s . z ) )
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); ::_thesis: for x, y being Variable of f holds
( ( s . x <= s . y implies (f . (s,(x leq y))) . b = 1 ) & ( s . x > s . y implies (f . (s,(x leq y))) . b = 0 ) & ( for z being Element of X st z <> b holds
(f . (s,(x leq y))) . z = s . z ) )
reconsider b9 = b as Variable of f by Def2;
let x, y be Variable of f; ::_thesis: ( ( s . x <= s . y implies (f . (s,(x leq y))) . b = 1 ) & ( s . x > s . y implies (f . (s,(x leq y))) . b = 0 ) & ( for z being Element of X st z <> b holds
(f . (s,(x leq y))) . z = s . z ) )
set v = ^ b9;
A1: (^ b9) . s = b by FUNCOP_1:7;
A2: ( (. x) . s <= (. y) . s implies IFGT (((. x) . s),((. y) . s),0,1) = 1 ) by XXREAL_0:def_11;
 (. x) . s = s . ((^ x) . s) by Def19;
then A3: s . x = (. x) . s by FUNCOP_1:7;
A4: ( (. x) . s > (. y) . s implies IFGT (((. x) . s),((. y) . s),0,1) = 0 ) by XXREAL_0:def_11;
A5: (leq ((. x),(. y))) . s = IFGT (((. x) . s),((. y) . s),0,1) by Def31;
 (. y) . s = s . ((^ y) . s) by Def19;
hence  ( ( s . x <= s . y implies (f . (s,(x leq y))) . b = 1 ) & ( s . x > s . y implies (f . (s,(x leq y))) . b = 0 ) & ( for z being Element of X st z <> b holds
(f . (s,(x leq y))) . z = s . z ) ) by A3, A1, A2, A4, A5, Th24, FUNCOP_1:7; ::_thesis: verum
end;

theorem :: AOFA_I00:36
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g
for i being integer number holds
( ( s . x <= i implies g . (s,(x leq i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x leq i)) in (Funcs (X,INT)) \ (b,0) implies s . x <= i ) & ( s . x >= i implies g . (s,(x geq i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x geq i)) in (Funcs (X,INT)) \ (b,0) implies s . x >= i ) )
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g
for i being integer number holds
( ( s . x <= i implies g . (s,(x leq i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x leq i)) in (Funcs (X,INT)) \ (b,0) implies s . x <= i ) & ( s . x >= i implies g . (s,(x geq i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x geq i)) in (Funcs (X,INT)) \ (b,0) implies s . x >= i ) )
let X be non empty countable set ; ::_thesis: for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g
for i being integer number holds
( ( s . x <= i implies g . (s,(x leq i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x leq i)) in (Funcs (X,INT)) \ (b,0) implies s . x <= i ) & ( s . x >= i implies g . (s,(x geq i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x geq i)) in (Funcs (X,INT)) \ (b,0) implies s . x >= i ) )
let s be Element of Funcs (X,INT); ::_thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g
for i being integer number holds
( ( s . x <= i implies g . (s,(x leq i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x leq i)) in (Funcs (X,INT)) \ (b,0) implies s . x <= i ) & ( s . x >= i implies g . (s,(x geq i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x geq i)) in (Funcs (X,INT)) \ (b,0) implies s . x >= i ) )
let b be Element of X; ::_thesis: for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g
for i being integer number holds
( ( s . x <= i implies g . (s,(x leq i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x leq i)) in (Funcs (X,INT)) \ (b,0) implies s . x <= i ) & ( s . x >= i implies g . (s,(x geq i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x geq i)) in (Funcs (X,INT)) \ (b,0) implies s . x >= i ) )
let g be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); ::_thesis: for x being Variable of g
for i being integer number holds
( ( s . x <= i implies g . (s,(x leq i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x leq i)) in (Funcs (X,INT)) \ (b,0) implies s . x <= i ) & ( s . x >= i implies g . (s,(x geq i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x geq i)) in (Funcs (X,INT)) \ (b,0) implies s . x >= i ) )
let x be Variable of g; ::_thesis: for i being integer number holds
( ( s . x <= i implies g . (s,(x leq i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x leq i)) in (Funcs (X,INT)) \ (b,0) implies s . x <= i ) & ( s . x >= i implies g . (s,(x geq i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x geq i)) in (Funcs (X,INT)) \ (b,0) implies s . x >= i ) )
let i be integer number ; ::_thesis: ( ( s . x <= i implies g . (s,(x leq i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x leq i)) in (Funcs (X,INT)) \ (b,0) implies s . x <= i ) & ( s . x >= i implies g . (s,(x geq i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x geq i)) in (Funcs (X,INT)) \ (b,0) implies s . x >= i ) )
 ( g . (s,(x leq i)) in (Funcs (X,INT)) \ (b,0) iff (g . (s,(x leq i))) . b <> 0 ) by Th2;
hence  ( s . x <= i iff g . (s,(x leq i)) in (Funcs (X,INT)) \ (b,0) ) by Th34; ::_thesis: ( s . x >= i iff g . (s,(x geq i)) in (Funcs (X,INT)) \ (b,0) )
 ( g . (s,(x geq i)) in (Funcs (X,INT)) \ (b,0) iff (g . (s,(x geq i))) . b <> 0 ) by Th2;
hence  ( s . x >= i iff g . (s,(x geq i)) in (Funcs (X,INT)) \ (b,0) ) by Th34; ::_thesis: verum
end;

theorem :: AOFA_I00:37
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y being Variable of g holds
( ( s . x <= s . y implies g . (s,(x leq y)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x leq y)) in (Funcs (X,INT)) \ (b,0) implies s . x <= s . y ) & ( s . x >= s . y implies g . (s,(x geq y)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x geq y)) in (Funcs (X,INT)) \ (b,0) implies s . x >= s . y ) )
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y being Variable of g holds
( ( s . x <= s . y implies g . (s,(x leq y)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x leq y)) in (Funcs (X,INT)) \ (b,0) implies s . x <= s . y ) & ( s . x >= s . y implies g . (s,(x geq y)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x geq y)) in (Funcs (X,INT)) \ (b,0) implies s . x >= s . y ) )
let X be non empty countable set ; ::_thesis: for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y being Variable of g holds
( ( s . x <= s . y implies g . (s,(x leq y)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x leq y)) in (Funcs (X,INT)) \ (b,0) implies s . x <= s . y ) & ( s . x >= s . y implies g . (s,(x geq y)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x geq y)) in (Funcs (X,INT)) \ (b,0) implies s . x >= s . y ) )
let s be Element of Funcs (X,INT); ::_thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y being Variable of g holds
( ( s . x <= s . y implies g . (s,(x leq y)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x leq y)) in (Funcs (X,INT)) \ (b,0) implies s . x <= s . y ) & ( s . x >= s . y implies g . (s,(x geq y)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x geq y)) in (Funcs (X,INT)) \ (b,0) implies s . x >= s . y ) )
let b be Element of X; ::_thesis: for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y being Variable of g holds
( ( s . x <= s . y implies g . (s,(x leq y)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x leq y)) in (Funcs (X,INT)) \ (b,0) implies s . x <= s . y ) & ( s . x >= s . y implies g . (s,(x geq y)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x geq y)) in (Funcs (X,INT)) \ (b,0) implies s . x >= s . y ) )
let g be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); ::_thesis: for x, y being Variable of g holds
( ( s . x <= s . y implies g . (s,(x leq y)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x leq y)) in (Funcs (X,INT)) \ (b,0) implies s . x <= s . y ) & ( s . x >= s . y implies g . (s,(x geq y)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x geq y)) in (Funcs (X,INT)) \ (b,0) implies s . x >= s . y ) )
let x, y be Variable of g; ::_thesis: ( ( s . x <= s . y implies g . (s,(x leq y)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x leq y)) in (Funcs (X,INT)) \ (b,0) implies s . x <= s . y ) & ( s . x >= s . y implies g . (s,(x geq y)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x geq y)) in (Funcs (X,INT)) \ (b,0) implies s . x >= s . y ) )
 ( g . (s,(x leq y)) in (Funcs (X,INT)) \ (b,0) iff (g . (s,(x leq y))) . b <> 0 ) by Th2;
hence  ( s . x <= s . y iff g . (s,(x leq y)) in (Funcs (X,INT)) \ (b,0) ) by Th35; ::_thesis: ( s . x >= s . y iff g . (s,(x geq y)) in (Funcs (X,INT)) \ (b,0) )
 ( g . (s,(x geq y)) in (Funcs (X,INT)) \ (b,0) iff (g . (s,(x geq y))) . b <> 0 ) by Th2;
hence  ( s . x >= s . y iff g . (s,(x geq y)) in (Funcs (X,INT)) \ (b,0) ) by Th35; ::_thesis: verum
end;

theorem Th38: :: AOFA_I00:38
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g
for i being integer number holds
( ( s . x > i implies (g . (s,(x gt i))) . b = 1 ) & ( s . x <= i implies (g . (s,(x gt i))) . b = 0 ) & ( s . x < i implies (g . (s,(x lt i))) . b = 1 ) & ( s . x >= i implies (g . (s,(x lt i))) . b = 0 ) & ( for z being Element of X st z <> b holds
( (g . (s,(x gt i))) . z = s . z & (g . (s,(x lt i))) . z = s . z ) ) )
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g
for i being integer number holds
( ( s . x > i implies (g . (s,(x gt i))) . b = 1 ) & ( s . x <= i implies (g . (s,(x gt i))) . b = 0 ) & ( s . x < i implies (g . (s,(x lt i))) . b = 1 ) & ( s . x >= i implies (g . (s,(x lt i))) . b = 0 ) & ( for z being Element of X st z <> b holds
( (g . (s,(x gt i))) . z = s . z & (g . (s,(x lt i))) . z = s . z ) ) )
let X be non empty countable set ; ::_thesis: for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g
for i being integer number holds
( ( s . x > i implies (g . (s,(x gt i))) . b = 1 ) & ( s . x <= i implies (g . (s,(x gt i))) . b = 0 ) & ( s . x < i implies (g . (s,(x lt i))) . b = 1 ) & ( s . x >= i implies (g . (s,(x lt i))) . b = 0 ) & ( for z being Element of X st z <> b holds
( (g . (s,(x gt i))) . z = s . z & (g . (s,(x lt i))) . z = s . z ) ) )
let s be Element of Funcs (X,INT); ::_thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g
for i being integer number holds
( ( s . x > i implies (g . (s,(x gt i))) . b = 1 ) & ( s . x <= i implies (g . (s,(x gt i))) . b = 0 ) & ( s . x < i implies (g . (s,(x lt i))) . b = 1 ) & ( s . x >= i implies (g . (s,(x lt i))) . b = 0 ) & ( for z being Element of X st z <> b holds
( (g . (s,(x gt i))) . z = s . z & (g . (s,(x lt i))) . z = s . z ) ) )
let b be Element of X; ::_thesis: for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g
for i being integer number holds
( ( s . x > i implies (g . (s,(x gt i))) . b = 1 ) & ( s . x <= i implies (g . (s,(x gt i))) . b = 0 ) & ( s . x < i implies (g . (s,(x lt i))) . b = 1 ) & ( s . x >= i implies (g . (s,(x lt i))) . b = 0 ) & ( for z being Element of X st z <> b holds
( (g . (s,(x gt i))) . z = s . z & (g . (s,(x lt i))) . z = s . z ) ) )
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); ::_thesis: for x being Variable of f
for i being integer number holds
( ( s . x > i implies (f . (s,(x gt i))) . b = 1 ) & ( s . x <= i implies (f . (s,(x gt i))) . b = 0 ) & ( s . x < i implies (f . (s,(x lt i))) . b = 1 ) & ( s . x >= i implies (f . (s,(x lt i))) . b = 0 ) & ( for z being Element of X st z <> b holds
( (f . (s,(x gt i))) . z = s . z & (f . (s,(x lt i))) . z = s . z ) ) )
reconsider b9 = b as Variable of f by Def2;
let x be Variable of f; ::_thesis: for i being integer number holds
( ( s . x > i implies (f . (s,(x gt i))) . b = 1 ) & ( s . x <= i implies (f . (s,(x gt i))) . b = 0 ) & ( s . x < i implies (f . (s,(x lt i))) . b = 1 ) & ( s . x >= i implies (f . (s,(x lt i))) . b = 0 ) & ( for z being Element of X st z <> b holds
( (f . (s,(x gt i))) . z = s . z & (f . (s,(x lt i))) . z = s . z ) ) )
let i be integer number ; ::_thesis: ( ( s . x > i implies (f . (s,(x gt i))) . b = 1 ) & ( s . x <= i implies (f . (s,(x gt i))) . b = 0 ) & ( s . x < i implies (f . (s,(x lt i))) . b = 1 ) & ( s . x >= i implies (f . (s,(x lt i))) . b = 0 ) & ( for z being Element of X st z <> b holds
( (f . (s,(x gt i))) . z = s . z & (f . (s,(x lt i))) . z = s . z ) ) )
set v = ^ b9;
A1: (. (i,A,f)) . s = i by FUNCOP_1:7;
A2: (^ b9) . s = b by FUNCOP_1:7;
A3: ( (. x) . s >= i implies IFGT (i,((. x) . s),1,0) = 0 ) by XXREAL_0:def_11;
A4: ( (. x) . s < i implies IFGT (i,((. x) . s),1,0) = 1 ) by XXREAL_0:def_11;
A5: (gt ((. (i,A,f)),(. x))) . s = IFGT (((. (i,A,f)) . s),((. x) . s),1,0) by Def32;
A6: ( (. x) . s <= i implies IFGT (((. x) . s),i,1,0) = 0 ) by XXREAL_0:def_11;
A7: (gt ((. x),(. (i,A,f)))) . s = IFGT (((. x) . s),((. (i,A,f)) . s),1,0) by Def32;
A8: ( (. x) . s > i implies IFGT (((. x) . s),i,1,0) = 1 ) by XXREAL_0:def_11;
 (. x) . s = s . ((^ x) . s) by Def19;
hence  ( ( s . x > i implies (f . (s,(x gt i))) . b = 1 ) & ( s . x <= i implies (f . (s,(x gt i))) . b = 0 ) & ( s . x < i implies (f . (s,(x lt i))) . b = 1 ) & ( s . x >= i implies (f . (s,(x lt i))) . b = 0 ) & ( for z being Element of X st z <> b holds
( (f . (s,(x gt i))) . z = s . z & (f . (s,(x lt i))) . z = s . z ) ) ) by A1, A2, A8, A6, A4, A3, A7, A5, Th24, FUNCOP_1:7; ::_thesis: verum
end;

theorem Th39: :: AOFA_I00:39
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y being Variable of g holds
( ( s . x > s . y implies (g . (s,(x gt y))) . b = 1 ) & ( s . x <= s . y implies (g . (s,(x gt y))) . b = 0 ) & ( s . x < s . y implies (g . (s,(x lt y))) . b = 1 ) & ( s . x >= s . y implies (g . (s,(x lt y))) . b = 0 ) & ( for z being Element of X st z <> b holds
( (g . (s,(x gt y))) . z = s . z & (g . (s,(x lt y))) . z = s . z ) ) )
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y being Variable of g holds
( ( s . x > s . y implies (g . (s,(x gt y))) . b = 1 ) & ( s . x <= s . y implies (g . (s,(x gt y))) . b = 0 ) & ( s . x < s . y implies (g . (s,(x lt y))) . b = 1 ) & ( s . x >= s . y implies (g . (s,(x lt y))) . b = 0 ) & ( for z being Element of X st z <> b holds
( (g . (s,(x gt y))) . z = s . z & (g . (s,(x lt y))) . z = s . z ) ) )
let X be non empty countable set ; ::_thesis: for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y being Variable of g holds
( ( s . x > s . y implies (g . (s,(x gt y))) . b = 1 ) & ( s . x <= s . y implies (g . (s,(x gt y))) . b = 0 ) & ( s . x < s . y implies (g . (s,(x lt y))) . b = 1 ) & ( s . x >= s . y implies (g . (s,(x lt y))) . b = 0 ) & ( for z being Element of X st z <> b holds
( (g . (s,(x gt y))) . z = s . z & (g . (s,(x lt y))) . z = s . z ) ) )
let s be Element of Funcs (X,INT); ::_thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y being Variable of g holds
( ( s . x > s . y implies (g . (s,(x gt y))) . b = 1 ) & ( s . x <= s . y implies (g . (s,(x gt y))) . b = 0 ) & ( s . x < s . y implies (g . (s,(x lt y))) . b = 1 ) & ( s . x >= s . y implies (g . (s,(x lt y))) . b = 0 ) & ( for z being Element of X st z <> b holds
( (g . (s,(x gt y))) . z = s . z & (g . (s,(x lt y))) . z = s . z ) ) )
let b be Element of X; ::_thesis: for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y being Variable of g holds
( ( s . x > s . y implies (g . (s,(x gt y))) . b = 1 ) & ( s . x <= s . y implies (g . (s,(x gt y))) . b = 0 ) & ( s . x < s . y implies (g . (s,(x lt y))) . b = 1 ) & ( s . x >= s . y implies (g . (s,(x lt y))) . b = 0 ) & ( for z being Element of X st z <> b holds
( (g . (s,(x gt y))) . z = s . z & (g . (s,(x lt y))) . z = s . z ) ) )
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); ::_thesis: for x, y being Variable of f holds
( ( s . x > s . y implies (f . (s,(x gt y))) . b = 1 ) & ( s . x <= s . y implies (f . (s,(x gt y))) . b = 0 ) & ( s . x < s . y implies (f . (s,(x lt y))) . b = 1 ) & ( s . x >= s . y implies (f . (s,(x lt y))) . b = 0 ) & ( for z being Element of X st z <> b holds
( (f . (s,(x gt y))) . z = s . z & (f . (s,(x lt y))) . z = s . z ) ) )
reconsider b9 = b as Variable of f by Def2;
let x, y be Variable of f; ::_thesis: ( ( s . x > s . y implies (f . (s,(x gt y))) . b = 1 ) & ( s . x <= s . y implies (f . (s,(x gt y))) . b = 0 ) & ( s . x < s . y implies (f . (s,(x lt y))) . b = 1 ) & ( s . x >= s . y implies (f . (s,(x lt y))) . b = 0 ) & ( for z being Element of X st z <> b holds
( (f . (s,(x gt y))) . z = s . z & (f . (s,(x lt y))) . z = s . z ) ) )
set v = ^ b9;
A1: (^ b9) . s = b by FUNCOP_1:7;
A2: ( (. x) . s > (. y) . s implies IFGT (((. x) . s),((. y) . s),1,0) = 1 ) by XXREAL_0:def_11;
A3: (gt ((. x),(. y))) . s = IFGT (((. x) . s),((. y) . s),1,0) by Def32;
A4: ( (. x) . s < (. y) . s implies IFGT (((. y) . s),((. x) . s),1,0) = 1 ) by XXREAL_0:def_11;
 (. x) . s = s . ((^ x) . s) by Def19;
then A5: s . x = (. x) . s by FUNCOP_1:7;
A6: ( (. x) . s <= (. y) . s implies IFGT (((. x) . s),((. y) . s),1,0) = 0 ) by XXREAL_0:def_11;
A7: (gt ((. y),(. x))) . s = IFGT (((. y) . s),((. x) . s),1,0) by Def32;
A8: ( (. x) . s >= (. y) . s implies IFGT (((. y) . s),((. x) . s),1,0) = 0 ) by XXREAL_0:def_11;
 (. y) . s = s . ((^ y) . s) by Def19;
hence  ( ( s . x > s . y implies (f . (s,(x gt y))) . b = 1 ) & ( s . x <= s . y implies (f . (s,(x gt y))) . b = 0 ) & ( s . x < s . y implies (f . (s,(x lt y))) . b = 1 ) & ( s . x >= s . y implies (f . (s,(x lt y))) . b = 0 ) & ( for z being Element of X st z <> b holds
( (f . (s,(x gt y))) . z = s . z & (f . (s,(x lt y))) . z = s . z ) ) ) by A5, A1, A2, A6, A4, A8, A3, A7, Th24, FUNCOP_1:7; ::_thesis: verum
end;

theorem Th40: :: AOFA_I00:40
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g
for i being integer number holds
( ( s . x > i implies g . (s,(x gt i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x gt i)) in (Funcs (X,INT)) \ (b,0) implies s . x > i ) & ( s . x < i implies g . (s,(x lt i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x lt i)) in (Funcs (X,INT)) \ (b,0) implies s . x < i ) )
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g
for i being integer number holds
( ( s . x > i implies g . (s,(x gt i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x gt i)) in (Funcs (X,INT)) \ (b,0) implies s . x > i ) & ( s . x < i implies g . (s,(x lt i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x lt i)) in (Funcs (X,INT)) \ (b,0) implies s . x < i ) )
let X be non empty countable set ; ::_thesis: for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g
for i being integer number holds
( ( s . x > i implies g . (s,(x gt i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x gt i)) in (Funcs (X,INT)) \ (b,0) implies s . x > i ) & ( s . x < i implies g . (s,(x lt i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x lt i)) in (Funcs (X,INT)) \ (b,0) implies s . x < i ) )
let s be Element of Funcs (X,INT); ::_thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g
for i being integer number holds
( ( s . x > i implies g . (s,(x gt i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x gt i)) in (Funcs (X,INT)) \ (b,0) implies s . x > i ) & ( s . x < i implies g . (s,(x lt i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x lt i)) in (Funcs (X,INT)) \ (b,0) implies s . x < i ) )
let b be Element of X; ::_thesis: for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g
for i being integer number holds
( ( s . x > i implies g . (s,(x gt i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x gt i)) in (Funcs (X,INT)) \ (b,0) implies s . x > i ) & ( s . x < i implies g . (s,(x lt i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x lt i)) in (Funcs (X,INT)) \ (b,0) implies s . x < i ) )
let g be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); ::_thesis: for x being Variable of g
for i being integer number holds
( ( s . x > i implies g . (s,(x gt i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x gt i)) in (Funcs (X,INT)) \ (b,0) implies s . x > i ) & ( s . x < i implies g . (s,(x lt i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x lt i)) in (Funcs (X,INT)) \ (b,0) implies s . x < i ) )
let x be Variable of g; ::_thesis: for i being integer number holds
( ( s . x > i implies g . (s,(x gt i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x gt i)) in (Funcs (X,INT)) \ (b,0) implies s . x > i ) & ( s . x < i implies g . (s,(x lt i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x lt i)) in (Funcs (X,INT)) \ (b,0) implies s . x < i ) )
let i be integer number ; ::_thesis: ( ( s . x > i implies g . (s,(x gt i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x gt i)) in (Funcs (X,INT)) \ (b,0) implies s . x > i ) & ( s . x < i implies g . (s,(x lt i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x lt i)) in (Funcs (X,INT)) \ (b,0) implies s . x < i ) )
 ( g . (s,(x gt i)) in (Funcs (X,INT)) \ (b,0) iff (g . (s,(x gt i))) . b <> 0 ) by Th2;
hence  ( s . x > i iff g . (s,(x gt i)) in (Funcs (X,INT)) \ (b,0) ) by Th38; ::_thesis: ( s . x < i iff g . (s,(x lt i)) in (Funcs (X,INT)) \ (b,0) )
 ( g . (s,(x lt i)) in (Funcs (X,INT)) \ (b,0) iff (g . (s,(x lt i))) . b <> 0 ) by Th2;
hence  ( s . x < i iff g . (s,(x lt i)) in (Funcs (X,INT)) \ (b,0) ) by Th38; ::_thesis: verum
end;

theorem :: AOFA_I00:41
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y being Variable of g holds
( ( s . x > s . y implies g . (s,(x gt y)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x gt y)) in (Funcs (X,INT)) \ (b,0) implies s . x > s . y ) & ( s . x < s . y implies g . (s,(x lt y)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x lt y)) in (Funcs (X,INT)) \ (b,0) implies s . x < s . y ) )
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y being Variable of g holds
( ( s . x > s . y implies g . (s,(x gt y)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x gt y)) in (Funcs (X,INT)) \ (b,0) implies s . x > s . y ) & ( s . x < s . y implies g . (s,(x lt y)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x lt y)) in (Funcs (X,INT)) \ (b,0) implies s . x < s . y ) )
let X be non empty countable set ; ::_thesis: for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y being Variable of g holds
( ( s . x > s . y implies g . (s,(x gt y)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x gt y)) in (Funcs (X,INT)) \ (b,0) implies s . x > s . y ) & ( s . x < s . y implies g . (s,(x lt y)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x lt y)) in (Funcs (X,INT)) \ (b,0) implies s . x < s . y ) )
let s be Element of Funcs (X,INT); ::_thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y being Variable of g holds
( ( s . x > s . y implies g . (s,(x gt y)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x gt y)) in (Funcs (X,INT)) \ (b,0) implies s . x > s . y ) & ( s . x < s . y implies g . (s,(x lt y)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x lt y)) in (Funcs (X,INT)) \ (b,0) implies s . x < s . y ) )
let b be Element of X; ::_thesis: for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y being Variable of g holds
( ( s . x > s . y implies g . (s,(x gt y)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x gt y)) in (Funcs (X,INT)) \ (b,0) implies s . x > s . y ) & ( s . x < s . y implies g . (s,(x lt y)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x lt y)) in (Funcs (X,INT)) \ (b,0) implies s . x < s . y ) )
let g be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); ::_thesis: for x, y being Variable of g holds
( ( s . x > s . y implies g . (s,(x gt y)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x gt y)) in (Funcs (X,INT)) \ (b,0) implies s . x > s . y ) & ( s . x < s . y implies g . (s,(x lt y)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x lt y)) in (Funcs (X,INT)) \ (b,0) implies s . x < s . y ) )
let x, y be Variable of g; ::_thesis: ( ( s . x > s . y implies g . (s,(x gt y)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x gt y)) in (Funcs (X,INT)) \ (b,0) implies s . x > s . y ) & ( s . x < s . y implies g . (s,(x lt y)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x lt y)) in (Funcs (X,INT)) \ (b,0) implies s . x < s . y ) )
 ( g . (s,(x gt y)) in (Funcs (X,INT)) \ (b,0) iff (g . (s,(x gt y))) . b <> 0 ) by Th2;
hence  ( s . x > s . y iff g . (s,(x gt y)) in (Funcs (X,INT)) \ (b,0) ) by Th39; ::_thesis: ( s . x < s . y iff g . (s,(x lt y)) in (Funcs (X,INT)) \ (b,0) )
 ( g . (s,(x lt y)) in (Funcs (X,INT)) \ (b,0) iff (g . (s,(x lt y))) . b <> 0 ) by Th2;
hence  ( s . x < s . y iff g . (s,(x lt y)) in (Funcs (X,INT)) \ (b,0) ) by Th39; ::_thesis: verum
end;

theorem :: AOFA_I00:42
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for i being integer number
for x being Variable of f holds
( (f . (s,(x %= i))) . x = (s . x) mod i & ( for z being Element of X st z <> x holds
(f . (s,(x %= i))) . z = s . z ) )
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for i being integer number
for x being Variable of f holds
( (f . (s,(x %= i))) . x = (s . x) mod i & ( for z being Element of X st z <> x holds
(f . (s,(x %= i))) . z = s . z ) )
let X be non empty countable set ; ::_thesis: for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for i being integer number
for x being Variable of f holds
( (f . (s,(x %= i))) . x = (s . x) mod i & ( for z being Element of X st z <> x holds
(f . (s,(x %= i))) . z = s . z ) )
let s be Element of Funcs (X,INT); ::_thesis: for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for i being integer number
for x being Variable of f holds
( (f . (s,(x %= i))) . x = (s . x) mod i & ( for z being Element of X st z <> x holds
(f . (s,(x %= i))) . z = s . z ) )
let T be Subset of (Funcs (X,INT)); ::_thesis: for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for i being integer number
for x being Variable of f holds
( (f . (s,(x %= i))) . x = (s . x) mod i & ( for z being Element of X st z <> x holds
(f . (s,(x %= i))) . z = s . z ) )
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T; ::_thesis: for i being integer number
for x being Variable of f holds
( (f . (s,(x %= i))) . x = (s . x) mod i & ( for z being Element of X st z <> x holds
(f . (s,(x %= i))) . z = s . z ) )
let i be integer number ; ::_thesis: for x being Variable of f holds
( (f . (s,(x %= i))) . x = (s . x) mod i & ( for z being Element of X st z <> x holds
(f . (s,(x %= i))) . z = s . z ) )
let x be Variable of f; ::_thesis: ( (f . (s,(x %= i))) . x = (s . x) mod i & ( for z being Element of X st z <> x holds
(f . (s,(x %= i))) . z = s . z ) )
A1: (. (i,A,f)) . s = i by FUNCOP_1:7;
A2: (^ x) . s = x by FUNCOP_1:7;
A3: (. (^ x)) . s = s . ((^ x) . s) by Def19;
 ((. x) mod (. (i,A,f))) . s = ((. x) . s) mod ((. (i,A,f)) . s) by Def30;
hence  ( (f . (s,(x %= i))) . x = (s . x) mod i & ( for z being Element of X st z <> x holds
(f . (s,(x %= i))) . z = s . z ) ) by A1, A2, A3, Th24; ::_thesis: verum
end;

theorem Th43: :: AOFA_I00:43
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x %= t))) . x = (s . x) mod (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x %= t))) . z = s . z ) )
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x %= t))) . x = (s . x) mod (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x %= t))) . z = s . z ) )
let X be non empty countable set ; ::_thesis: for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x %= t))) . x = (s . x) mod (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x %= t))) . z = s . z ) )
let s be Element of Funcs (X,INT); ::_thesis: for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x %= t))) . x = (s . x) mod (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x %= t))) . z = s . z ) )
let T be Subset of (Funcs (X,INT)); ::_thesis: for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x %= t))) . x = (s . x) mod (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x %= t))) . z = s . z ) )
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T; ::_thesis: for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x %= t))) . x = (s . x) mod (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x %= t))) . z = s . z ) )
let x be Variable of f; ::_thesis: for t being INT-Expression of A,f holds
( (f . (s,(x %= t))) . x = (s . x) mod (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x %= t))) . z = s . z ) )
let t be INT-Expression of A,f; ::_thesis: ( (f . (s,(x %= t))) . x = (s . x) mod (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x %= t))) . z = s . z ) )
A1: (^ x) . s = x by FUNCOP_1:7;
A2: (. x) . s = s . x by Th22;
 ((. x) mod t) . s = ((. x) . s) mod (t . s) by Def30;
hence  ( (f . (s,(x %= t))) . x = (s . x) mod (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x %= t))) . z = s . z ) ) by A1, A2, Th24; ::_thesis: verum
end;

theorem Th44: :: AOFA_I00:44
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x, y being Variable of f holds
( (f . (s,(x %= y))) . x = (s . x) mod (s . y) & ( for z being Element of X st z <> x holds
(f . (s,(x %= y))) . z = s . z ) )
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x, y being Variable of f holds
( (f . (s,(x %= y))) . x = (s . x) mod (s . y) & ( for z being Element of X st z <> x holds
(f . (s,(x %= y))) . z = s . z ) )
let X be non empty countable set ; ::_thesis: for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x, y being Variable of f holds
( (f . (s,(x %= y))) . x = (s . x) mod (s . y) & ( for z being Element of X st z <> x holds
(f . (s,(x %= y))) . z = s . z ) )
let s be Element of Funcs (X,INT); ::_thesis: for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x, y being Variable of f holds
( (f . (s,(x %= y))) . x = (s . x) mod (s . y) & ( for z being Element of X st z <> x holds
(f . (s,(x %= y))) . z = s . z ) )
let T be Subset of (Funcs (X,INT)); ::_thesis: for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x, y being Variable of f holds
( (f . (s,(x %= y))) . x = (s . x) mod (s . y) & ( for z being Element of X st z <> x holds
(f . (s,(x %= y))) . z = s . z ) )
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T; ::_thesis: for x, y being Variable of f holds
( (f . (s,(x %= y))) . x = (s . x) mod (s . y) & ( for z being Element of X st z <> x holds
(f . (s,(x %= y))) . z = s . z ) )
let x, y be Variable of f; ::_thesis: ( (f . (s,(x %= y))) . x = (s . x) mod (s . y) & ( for z being Element of X st z <> x holds
(f . (s,(x %= y))) . z = s . z ) )
A1: x %= y = x %= (. y) ;
 (. y) . s = s . y by Th22;
hence  ( (f . (s,(x %= y))) . x = (s . x) mod (s . y) & ( for z being Element of X st z <> x holds
(f . (s,(x %= y))) . z = s . z ) ) by A1, Th43; ::_thesis: verum
end;

theorem Th45: :: AOFA_I00:45
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for i being integer number
for x being Variable of f holds
( (f . (s,(x /= i))) . x = (s . x) div i & ( for z being Element of X st z <> x holds
(f . (s,(x /= i))) . z = s . z ) )
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for i being integer number
for x being Variable of f holds
( (f . (s,(x /= i))) . x = (s . x) div i & ( for z being Element of X st z <> x holds
(f . (s,(x /= i))) . z = s . z ) )
let X be non empty countable set ; ::_thesis: for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for i being integer number
for x being Variable of f holds
( (f . (s,(x /= i))) . x = (s . x) div i & ( for z being Element of X st z <> x holds
(f . (s,(x /= i))) . z = s . z ) )
let s be Element of Funcs (X,INT); ::_thesis: for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for i being integer number
for x being Variable of f holds
( (f . (s,(x /= i))) . x = (s . x) div i & ( for z being Element of X st z <> x holds
(f . (s,(x /= i))) . z = s . z ) )
let T be Subset of (Funcs (X,INT)); ::_thesis: for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for i being integer number
for x being Variable of f holds
( (f . (s,(x /= i))) . x = (s . x) div i & ( for z being Element of X st z <> x holds
(f . (s,(x /= i))) . z = s . z ) )
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T; ::_thesis: for i being integer number
for x being Variable of f holds
( (f . (s,(x /= i))) . x = (s . x) div i & ( for z being Element of X st z <> x holds
(f . (s,(x /= i))) . z = s . z ) )
let i be integer number ; ::_thesis: for x being Variable of f holds
( (f . (s,(x /= i))) . x = (s . x) div i & ( for z being Element of X st z <> x holds
(f . (s,(x /= i))) . z = s . z ) )
let x be Variable of f; ::_thesis: ( (f . (s,(x /= i))) . x = (s . x) div i & ( for z being Element of X st z <> x holds
(f . (s,(x /= i))) . z = s . z ) )
A1: (. (i,A,f)) . s = i by FUNCOP_1:7;
A2: (^ x) . s = x by FUNCOP_1:7;
A3: (. (^ x)) . s = s . ((^ x) . s) by Def19;
 ((. x) div (. (i,A,f))) . s = ((. x) . s) div ((. (i,A,f)) . s) by Def29;
hence  ( (f . (s,(x /= i))) . x = (s . x) div i & ( for z being Element of X st z <> x holds
(f . (s,(x /= i))) . z = s . z ) ) by A1, A2, A3, Th24; ::_thesis: verum
end;

theorem Th46: :: AOFA_I00:46
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x /= t))) . x = (s . x) div (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x /= t))) . z = s . z ) )
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x /= t))) . x = (s . x) div (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x /= t))) . z = s . z ) )
let X be non empty countable set ; ::_thesis: for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x /= t))) . x = (s . x) div (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x /= t))) . z = s . z ) )
let s be Element of Funcs (X,INT); ::_thesis: for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x /= t))) . x = (s . x) div (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x /= t))) . z = s . z ) )
let T be Subset of (Funcs (X,INT)); ::_thesis: for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x /= t))) . x = (s . x) div (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x /= t))) . z = s . z ) )
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T; ::_thesis: for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x /= t))) . x = (s . x) div (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x /= t))) . z = s . z ) )
let x be Variable of f; ::_thesis: for t being INT-Expression of A,f holds
( (f . (s,(x /= t))) . x = (s . x) div (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x /= t))) . z = s . z ) )
let t be INT-Expression of A,f; ::_thesis: ( (f . (s,(x /= t))) . x = (s . x) div (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x /= t))) . z = s . z ) )
A1: (^ x) . s = x by FUNCOP_1:7;
A2: (. x) . s = s . x by Th22;
 ((. x) div t) . s = ((. x) . s) div (t . s) by Def29;
hence  ( (f . (s,(x /= t))) . x = (s . x) div (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x /= t))) . z = s . z ) ) by A1, A2, Th24; ::_thesis: verum
end;

theorem :: AOFA_I00:47
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x, y being Variable of f holds
( (f . (s,(x /= y))) . x = (s . x) div (s . y) & ( for z being Element of X st z <> x holds
(f . (s,(x /= y))) . z = s . z ) )
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x, y being Variable of f holds
( (f . (s,(x /= y))) . x = (s . x) div (s . y) & ( for z being Element of X st z <> x holds
(f . (s,(x /= y))) . z = s . z ) )
let X be non empty countable set ; ::_thesis: for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x, y being Variable of f holds
( (f . (s,(x /= y))) . x = (s . x) div (s . y) & ( for z being Element of X st z <> x holds
(f . (s,(x /= y))) . z = s . z ) )
let s be Element of Funcs (X,INT); ::_thesis: for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x, y being Variable of f holds
( (f . (s,(x /= y))) . x = (s . x) div (s . y) & ( for z being Element of X st z <> x holds
(f . (s,(x /= y))) . z = s . z ) )
let T be Subset of (Funcs (X,INT)); ::_thesis: for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x, y being Variable of f holds
( (f . (s,(x /= y))) . x = (s . x) div (s . y) & ( for z being Element of X st z <> x holds
(f . (s,(x /= y))) . z = s . z ) )
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T; ::_thesis: for x, y being Variable of f holds
( (f . (s,(x /= y))) . x = (s . x) div (s . y) & ( for z being Element of X st z <> x holds
(f . (s,(x /= y))) . z = s . z ) )
let x, y be Variable of f; ::_thesis: ( (f . (s,(x /= y))) . x = (s . x) div (s . y) & ( for z being Element of X st z <> x holds
(f . (s,(x /= y))) . z = s . z ) )
A1: x /= y = x /= (. y) ;
 (. y) . s = s . y by Th22;
hence  ( (f . (s,(x /= y))) . x = (s . x) div (s . y) & ( for z being Element of X st z <> x holds
(f . (s,(x /= y))) . z = s . z ) ) by A1, Th46; ::_thesis: verum
end;

theorem Th48: :: AOFA_I00:48
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for t being INT-Expression of A,g holds
( (g . (s,(t is_odd))) . b = (t . s) mod 2 & (g . (s,(t is_even))) . b = ((t . s) + 1) mod 2 & ( for z being Element of X st z <> b holds
( (g . (s,(t is_odd))) . z = s . z & (g . (s,(t is_even))) . z = s . z ) ) )
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for t being INT-Expression of A,g holds
( (g . (s,(t is_odd))) . b = (t . s) mod 2 & (g . (s,(t is_even))) . b = ((t . s) + 1) mod 2 & ( for z being Element of X st z <> b holds
( (g . (s,(t is_odd))) . z = s . z & (g . (s,(t is_even))) . z = s . z ) ) )
let X be non empty countable set ; ::_thesis: for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for t being INT-Expression of A,g holds
( (g . (s,(t is_odd))) . b = (t . s) mod 2 & (g . (s,(t is_even))) . b = ((t . s) + 1) mod 2 & ( for z being Element of X st z <> b holds
( (g . (s,(t is_odd))) . z = s . z & (g . (s,(t is_even))) . z = s . z ) ) )
let s be Element of Funcs (X,INT); ::_thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for t being INT-Expression of A,g holds
( (g . (s,(t is_odd))) . b = (t . s) mod 2 & (g . (s,(t is_even))) . b = ((t . s) + 1) mod 2 & ( for z being Element of X st z <> b holds
( (g . (s,(t is_odd))) . z = s . z & (g . (s,(t is_even))) . z = s . z ) ) )
let b be Element of X; ::_thesis: for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for t being INT-Expression of A,g holds
( (g . (s,(t is_odd))) . b = (t . s) mod 2 & (g . (s,(t is_even))) . b = ((t . s) + 1) mod 2 & ( for z being Element of X st z <> b holds
( (g . (s,(t is_odd))) . z = s . z & (g . (s,(t is_even))) . z = s . z ) ) )
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); ::_thesis: for t being INT-Expression of A,f holds
( (f . (s,(t is_odd))) . b = (t . s) mod 2 & (f . (s,(t is_even))) . b = ((t . s) + 1) mod 2 & ( for z being Element of X st z <> b holds
( (f . (s,(t is_odd))) . z = s . z & (f . (s,(t is_even))) . z = s . z ) ) )
let t be INT-Expression of A,f; ::_thesis: ( (f . (s,(t is_odd))) . b = (t . s) mod 2 & (f . (s,(t is_even))) . b = ((t . s) + 1) mod 2 & ( for z being Element of X st z <> b holds
( (f . (s,(t is_odd))) . z = s . z & (f . (s,(t is_even))) . z = s . z ) ) )
reconsider y = b as Variable of f by Def2;
A1: t is_odd = y := (t mod (. (2,A,f))) ;
 dom (t + 1) = Funcs (X,INT) by FUNCT_2:def_1;
then A2: (t + 1) . s = 1 + (t . s) by VALUED_1:def_2;
A3: (. (2,A,f)) . s = 2 by FUNCOP_1:7;
then A4: ((t + 1) mod (. (2,A,f))) . s = ((t + 1) . s) mod 2 by Def30;
 (t mod (. (2,A,f))) . s = (t . s) mod 2 by A3, Def30;
hence  ( (f . (s,(t is_odd))) . b = (t . s) mod 2 & (f . (s,(t is_even))) . b = ((t . s) + 1) mod 2 & ( for z being Element of X st z <> b holds
( (f . (s,(t is_odd))) . z = s . z & (f . (s,(t is_even))) . z = s . z ) ) ) by A1, A2, A4, Th26; ::_thesis: verum
end;

theorem Th49: :: AOFA_I00:49
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g holds
( (g . (s,(x is_odd))) . b = (s . x) mod 2 & (g . (s,(x is_even))) . b = ((s . x) + 1) mod 2 & ( for z being Element of X st z <> b holds
(g . (s,(x is_odd))) . z = s . z ) )
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g holds
( (g . (s,(x is_odd))) . b = (s . x) mod 2 & (g . (s,(x is_even))) . b = ((s . x) + 1) mod 2 & ( for z being Element of X st z <> b holds
(g . (s,(x is_odd))) . z = s . z ) )
let X be non empty countable set ; ::_thesis: for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g holds
( (g . (s,(x is_odd))) . b = (s . x) mod 2 & (g . (s,(x is_even))) . b = ((s . x) + 1) mod 2 & ( for z being Element of X st z <> b holds
(g . (s,(x is_odd))) . z = s . z ) )
let s be Element of Funcs (X,INT); ::_thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g holds
( (g . (s,(x is_odd))) . b = (s . x) mod 2 & (g . (s,(x is_even))) . b = ((s . x) + 1) mod 2 & ( for z being Element of X st z <> b holds
(g . (s,(x is_odd))) . z = s . z ) )
let b be Element of X; ::_thesis: for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g holds
( (g . (s,(x is_odd))) . b = (s . x) mod 2 & (g . (s,(x is_even))) . b = ((s . x) + 1) mod 2 & ( for z being Element of X st z <> b holds
(g . (s,(x is_odd))) . z = s . z ) )
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); ::_thesis: for x being Variable of f holds
( (f . (s,(x is_odd))) . b = (s . x) mod 2 & (f . (s,(x is_even))) . b = ((s . x) + 1) mod 2 & ( for z being Element of X st z <> b holds
(f . (s,(x is_odd))) . z = s . z ) )
let x be Variable of f; ::_thesis: ( (f . (s,(x is_odd))) . b = (s . x) mod 2 & (f . (s,(x is_even))) . b = ((s . x) + 1) mod 2 & ( for z being Element of X st z <> b holds
(f . (s,(x is_odd))) . z = s . z ) )
 (. x) . s = s . x by Th22;
hence  ( (f . (s,(x is_odd))) . b = (s . x) mod 2 & (f . (s,(x is_even))) . b = ((s . x) + 1) mod 2 & ( for z being Element of X st z <> b holds
(f . (s,(x is_odd))) . z = s . z ) ) by Th48; ::_thesis: verum
end;

theorem Th50: :: AOFA_I00:50
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for t being INT-Expression of A,g holds
( ( t . s is odd implies g . (s,(t is_odd)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(t is_odd)) in (Funcs (X,INT)) \ (b,0) implies t . s is odd ) & ( t . s is even implies g . (s,(t is_even)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(t is_even)) in (Funcs (X,INT)) \ (b,0) implies t . s is even ) )
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for t being INT-Expression of A,g holds
( ( t . s is odd implies g . (s,(t is_odd)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(t is_odd)) in (Funcs (X,INT)) \ (b,0) implies t . s is odd ) & ( t . s is even implies g . (s,(t is_even)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(t is_even)) in (Funcs (X,INT)) \ (b,0) implies t . s is even ) )
let X be non empty countable set ; ::_thesis: for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for t being INT-Expression of A,g holds
( ( t . s is odd implies g . (s,(t is_odd)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(t is_odd)) in (Funcs (X,INT)) \ (b,0) implies t . s is odd ) & ( t . s is even implies g . (s,(t is_even)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(t is_even)) in (Funcs (X,INT)) \ (b,0) implies t . s is even ) )
let s be Element of Funcs (X,INT); ::_thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for t being INT-Expression of A,g holds
( ( t . s is odd implies g . (s,(t is_odd)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(t is_odd)) in (Funcs (X,INT)) \ (b,0) implies t . s is odd ) & ( t . s is even implies g . (s,(t is_even)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(t is_even)) in (Funcs (X,INT)) \ (b,0) implies t . s is even ) )
let b be Element of X; ::_thesis: for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for t being INT-Expression of A,g holds
( ( t . s is odd implies g . (s,(t is_odd)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(t is_odd)) in (Funcs (X,INT)) \ (b,0) implies t . s is odd ) & ( t . s is even implies g . (s,(t is_even)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(t is_even)) in (Funcs (X,INT)) \ (b,0) implies t . s is even ) )
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); ::_thesis: for t being INT-Expression of A,f holds
( ( t . s is odd implies f . (s,(t is_odd)) in (Funcs (X,INT)) \ (b,0) ) & ( f . (s,(t is_odd)) in (Funcs (X,INT)) \ (b,0) implies t . s is odd ) & ( t . s is even implies f . (s,(t is_even)) in (Funcs (X,INT)) \ (b,0) ) & ( f . (s,(t is_even)) in (Funcs (X,INT)) \ (b,0) implies t . s is even ) )
let t be INT-Expression of A,f; ::_thesis: ( ( t . s is odd implies f . (s,(t is_odd)) in (Funcs (X,INT)) \ (b,0) ) & ( f . (s,(t is_odd)) in (Funcs (X,INT)) \ (b,0) implies t . s is odd ) & ( t . s is even implies f . (s,(t is_even)) in (Funcs (X,INT)) \ (b,0) ) & ( f . (s,(t is_even)) in (Funcs (X,INT)) \ (b,0) implies t . s is even ) )
A1: ( (t . s) mod 2 = 0 or (t . s) mod 2 = 1 ) by PRE_FF:6;
A2: t . s = (((t . s) div 2) * 2) + ((t . s) mod 2) by INT_1:59;
 (f . (s,(t is_odd))) . b = (t . s) mod 2 by Th48;
hence  ( t . s is odd iff f . (s,(t is_odd)) in (Funcs (X,INT)) \ (b,0) ) by A1, A2, Th2; ::_thesis: ( t . s is even iff f . (s,(t is_even)) in (Funcs (X,INT)) \ (b,0) )
A3: ( ((t . s) + 1) mod 2 = 0 or ((t . s) + 1) mod 2 = 1 ) by PRE_FF:6;
A4: (t . s) + 1 = ((((t . s) + 1) div 2) * 2) + (((t . s) + 1) mod 2) by INT_1:59;
 (f . (s,(t is_even))) . b = ((t . s) + 1) mod 2 by Th48;
hence  ( t . s is even iff f . (s,(t is_even)) in (Funcs (X,INT)) \ (b,0) ) by A3, A4, Th2; ::_thesis: verum
end;

theorem :: AOFA_I00:51
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g holds
( ( s . x is odd implies g . (s,(x is_odd)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x is_odd)) in (Funcs (X,INT)) \ (b,0) implies s . x is odd ) & ( s . x is even implies g . (s,(x is_even)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x is_even)) in (Funcs (X,INT)) \ (b,0) implies s . x is even ) )
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g holds
( ( s . x is odd implies g . (s,(x is_odd)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x is_odd)) in (Funcs (X,INT)) \ (b,0) implies s . x is odd ) & ( s . x is even implies g . (s,(x is_even)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x is_even)) in (Funcs (X,INT)) \ (b,0) implies s . x is even ) )
let X be non empty countable set ; ::_thesis: for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g holds
( ( s . x is odd implies g . (s,(x is_odd)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x is_odd)) in (Funcs (X,INT)) \ (b,0) implies s . x is odd ) & ( s . x is even implies g . (s,(x is_even)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x is_even)) in (Funcs (X,INT)) \ (b,0) implies s . x is even ) )
let s be Element of Funcs (X,INT); ::_thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g holds
( ( s . x is odd implies g . (s,(x is_odd)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x is_odd)) in (Funcs (X,INT)) \ (b,0) implies s . x is odd ) & ( s . x is even implies g . (s,(x is_even)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x is_even)) in (Funcs (X,INT)) \ (b,0) implies s . x is even ) )
let b be Element of X; ::_thesis: for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x being Variable of g holds
( ( s . x is odd implies g . (s,(x is_odd)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x is_odd)) in (Funcs (X,INT)) \ (b,0) implies s . x is odd ) & ( s . x is even implies g . (s,(x is_even)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x is_even)) in (Funcs (X,INT)) \ (b,0) implies s . x is even ) )
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); ::_thesis: for x being Variable of f holds
( ( s . x is odd implies f . (s,(x is_odd)) in (Funcs (X,INT)) \ (b,0) ) & ( f . (s,(x is_odd)) in (Funcs (X,INT)) \ (b,0) implies s . x is odd ) & ( s . x is even implies f . (s,(x is_even)) in (Funcs (X,INT)) \ (b,0) ) & ( f . (s,(x is_even)) in (Funcs (X,INT)) \ (b,0) implies s . x is even ) )
let x be Variable of f; ::_thesis: ( ( s . x is odd implies f . (s,(x is_odd)) in (Funcs (X,INT)) \ (b,0) ) & ( f . (s,(x is_odd)) in (Funcs (X,INT)) \ (b,0) implies s . x is odd ) & ( s . x is even implies f . (s,(x is_even)) in (Funcs (X,INT)) \ (b,0) ) & ( f . (s,(x is_even)) in (Funcs (X,INT)) \ (b,0) implies s . x is even ) )
 (. x) . s = s . x by Th22;
hence  ( ( s . x is odd implies f . (s,(x is_odd)) in (Funcs (X,INT)) \ (b,0) ) & ( f . (s,(x is_odd)) in (Funcs (X,INT)) \ (b,0) implies s . x is odd ) & ( s . x is even implies f . (s,(x is_even)) in (Funcs (X,INT)) \ (b,0) ) & ( f . (s,(x is_even)) in (Funcs (X,INT)) \ (b,0) implies s . x is even ) ) by Th50; ::_thesis: verum
end;

scheme :: AOFA_I00:sch 1
ForToIteration{ F1() ->  Euclidean preIfWhileAlgebra, F2() ->  non empty countable set , F3() ->  Element of F2(), F4() ->  Element of F1(), F5() ->  Element of F1(), F6() ->  Euclidean ExecutionFunction of F1(), Funcs (F2(),INT),(Funcs (F2(),INT)) \ (F3(),0), F7() ->  Variable of F6(), F8() ->  Variable of F6(), F9() ->  Element of Funcs (F2(),INT), F10() ->  INT-Expression of F1(),F6(), P1[ set ] } :
 ( P1[F6() . (F9(),F5())] & ( F10() . F9() <= F9() . F8() implies (F6() . (F9(),F5())) . F7() = (F9() . F8()) + 1 ) & ( F10() . F9() > F9() . F8() implies (F6() . (F9(),F5())) . F7() = F10() . F9() ) & (F6() . (F9(),F5())) . F8() = F9() . F8() )
provided
A1: F5() = for-do ((F7() := F10()),(F7() leq F8()),(F7() += 1),F4()) and
A2: P1[F6() . (F9(),(F7() := F10()))] and
A3: for s being Element of Funcs (F2(),INT) st P1[s] holds
( P1[F6() . (s,(F4() \; (F7() += 1)))] & P1[F6() . (s,(F7() leq F8()))] ) and
A4: for s being Element of Funcs (F2(),INT) st P1[s] holds
( (F6() . (s,F4())) . F7() = s . F7() & (F6() . (s,F4())) . F8() = s . F8() ) and
A5: ( F8() <> F7() & F8() <> F3() & F7() <> F3() )
proof
set C = F7() leq F8();
set S = Funcs (F2(),INT);
reconsider s1 = F6() . (F9(),(F7() := F10())) as Element of Funcs (F2(),INT) ;
reconsider s2 = F6() . (s1,(F7() leq F8())) as Element of Funcs (F2(),INT) ;
A6: P1[s2] by A2, A3;
defpred S1[ Element of Funcs (F2(),INT)] means ( P1[$1] & ( F10() . F9() <= F9() . F8() implies ($1 . F7()) - 1 <= F9() . F8() ) & $1 . F8() = F9() . F8() );
A7: s1 . F7() = F10() . F9() by Th26;
then A8: s2 . F7() = F10() . F9() by A5, Th35;
defpred S2[ Element of Funcs (F2(),INT)] means $1 . F7() <= $1 . F8();
set T = (Funcs (F2(),INT)) \ (F3(),0);
A9: F6() complies_with_while_wrt (Funcs (F2(),INT)) \ (F3(),0) by AOFA_000:def_32;
set II = F4() \; (F7() += 1);
A10: F6() . (F9(),F5()) = F6() . (s1,(while ((F7() leq F8()),(F4() \; (F7() += 1))))) by A1, AOFA_000:def_29;
 (F10() . F9()) - 1 < F10() . F9() by XREAL_1:44;
then A11: S1[s1] by A2, A5, A7, Th26, XXREAL_0:2;
A12: s1 . F8() = F9() . F8() by A5, Th26;
then A13: s2 . F8() = F9() . F8() by A5, Th35;
percases ( F10() . F9() <= F9() . F8() or F10() . F9() > F9() . F8() ) ;
supposeA14: F10() . F9() <= F9() . F8() ; ::_thesis: ( P1[F6() . (F9(),F5())] & ( F10() . F9() <= F9() . F8() implies (F6() . (F9(),F5())) . F7() = (F9() . F8()) + 1 ) & ( F10() . F9() > F9() . F8() implies (F6() . (F9(),F5())) . F7() = F10() . F9() ) & (F6() . (F9(),F5())) . F8() = F9() . F8() )
deffunc H1( Element of Funcs (F2(),INT)) ->  Element of NAT = In (((($1 . F8()) + 1) - ($1 . F7())),NAT);
defpred S3[ Element of Funcs (F2(),INT)] means ( P1[$1] & S2[$1] );
A15: for s being Element of Funcs (F2(),INT) st S1[s] & s in (Funcs (F2(),INT)) \ (F3(),0) & S2[s] holds
S1[F6() . (s,(F4() \; (F7() += 1)))]
proof
let s be Element of Funcs (F2(),INT); ::_thesis: ( S1[s] & s in (Funcs (F2(),INT)) \ (F3(),0) & S2[s] implies S1[F6() . (s,(F4() \; (F7() += 1)))] )
assume that
A16: S1[s] and
 s in (Funcs (F2(),INT)) \ (F3(),0) and
A17: S2[s] ; ::_thesis: S1[F6() . (s,(F4() \; (F7() += 1)))]
thus  P1[F6() . (s,(F4() \; (F7() += 1)))] by A3, A16; ::_thesis: ( ( F10() . F9() <= F9() . F8() implies ((F6() . (s,(F4() \; (F7() += 1)))) . F7()) - 1 <= F9() . F8() ) & (F6() . (s,(F4() \; (F7() += 1)))) . F8() = F9() . F8() )
reconsider s9 = F6() . (s,F4()) as Element of Funcs (F2(),INT) ;
reconsider s99 = F6() . (s9,(F7() += 1)) as Element of Funcs (F2(),INT) ;
A18: s99 = F6() . (s,(F4() \; (F7() += 1))) by AOFA_000:def_29;
A19: s99 . F7() = (s9 . F7()) + 1 by Th28;
 s9 . F8() = s . F8() by A4, A16;
hence  ( ( F10() . F9() <= F9() . F8() implies ((F6() . (s,(F4() \; (F7() += 1)))) . F7()) - 1 <= F9() . F8() ) & (F6() . (s,(F4() \; (F7() += 1)))) . F8() = F9() . F8() ) by A4, A5, A16, A17, A19, A18, Th28; ::_thesis: verum
end;
A20: for s being Element of Funcs (F2(),INT) st S3[s] holds
( ( S3[F6() . (s,((F4() \; (F7() += 1)) \; (F7() leq F8())))] implies F6() . (s,((F4() \; (F7() += 1)) \; (F7() leq F8()))) in (Funcs (F2(),INT)) \ (F3(),0) ) & ( F6() . (s,((F4() \; (F7() += 1)) \; (F7() leq F8()))) in (Funcs (F2(),INT)) \ (F3(),0) implies S3[F6() . (s,((F4() \; (F7() += 1)) \; (F7() leq F8())))] ) & H1(F6() . (s,((F4() \; (F7() += 1)) \; (F7() leq F8())))) < H1(s) )
proof
let s be Element of Funcs (F2(),INT); ::_thesis: ( S3[s] implies ( ( S3[F6() . (s,((F4() \; (F7() += 1)) \; (F7() leq F8())))] implies F6() . (s,((F4() \; (F7() += 1)) \; (F7() leq F8()))) in (Funcs (F2(),INT)) \ (F3(),0) ) & ( F6() . (s,((F4() \; (F7() += 1)) \; (F7() leq F8()))) in (Funcs (F2(),INT)) \ (F3(),0) implies S3[F6() . (s,((F4() \; (F7() += 1)) \; (F7() leq F8())))] ) & H1(F6() . (s,((F4() \; (F7() += 1)) \; (F7() leq F8())))) < H1(s) ) )
reconsider s9 = F6() . (s,F4()) as Element of Funcs (F2(),INT) ;
reconsider s99 = F6() . (s9,(F7() += 1)) as Element of Funcs (F2(),INT) ;
reconsider s999 = F6() . (s99,(F7() leq F8())) as Element of Funcs (F2(),INT) ;
A21: F6() . (s,((F4() \; (F7() += 1)) \; (F7() leq F8()))) = F6() . ((F6() . (s,(F4() \; (F7() += 1)))),(F7() leq F8())) by AOFA_000:def_29;
A22: s99 . F8() = s9 . F8() by A5, Th28;
A23: ( s99 . F7() <= s99 . F8() implies s999 . F3() = 1 ) by Th35;
A24: ( s99 . F7() > s99 . F8() implies s999 . F3() = 0 ) by Th35;
assume A25: S3[s] ; ::_thesis: ( ( S3[F6() . (s,((F4() \; (F7() += 1)) \; (F7() leq F8())))] implies F6() . (s,((F4() \; (F7() += 1)) \; (F7() leq F8()))) in (Funcs (F2(),INT)) \ (F3(),0) ) & ( F6() . (s,((F4() \; (F7() += 1)) \; (F7() leq F8()))) in (Funcs (F2(),INT)) \ (F3(),0) implies S3[F6() . (s,((F4() \; (F7() += 1)) \; (F7() leq F8())))] ) & H1(F6() . (s,((F4() \; (F7() += 1)) \; (F7() leq F8())))) < H1(s) )
then reconsider j = (s . F8()) - (s . F7()) as Element of NAT by INT_1:5;
A26: s999 . F7() = s99 . F7() by A5, Th35;
A27: s99 = F6() . (s,(F4() \; (F7() += 1))) by AOFA_000:def_29;
then  P1[s99] by A3, A25;
hence  ( S3[F6() . (s,((F4() \; (F7() += 1)) \; (F7() leq F8())))] iff F6() . (s,((F4() \; (F7() += 1)) \; (F7() leq F8()))) in (Funcs (F2(),INT)) \ (F3(),0) ) by A3, A5, A21, A27, A26, A24, A23, Th2, Th35; ::_thesis: H1(F6() . (s,((F4() \; (F7() += 1)) \; (F7() leq F8())))) < H1(s)
A28: s99 . F7() = (s9 . F7()) + 1 by Th28;
A29: s9 . F8() = s . F8() by A4, A25;
A30: s9 . F7() = s . F7() by A4, A25;
 s999 . F8() = s99 . F8() by A5, Th35;
then  H1(s999) = j by A26, A30, A29, A28, A22, FUNCT_7:def_1;
then  H1(s999) + 1 = H1(s) by FUNCT_7:def_1;
hence  H1(F6() . (s,((F4() \; (F7() += 1)) \; (F7() leq F8())))) < H1(s) by A21, A27, NAT_1:13; ::_thesis: verum
end;
A31: for s being Element of Funcs (F2(),INT) st S1[s] holds
( S1[F6() . (s,(F7() leq F8()))] & ( F6() . (s,(F7() leq F8())) in (Funcs (F2(),INT)) \ (F3(),0) implies S2[F6() . (s,(F7() leq F8()))] ) & ( S2[F6() . (s,(F7() leq F8()))] implies F6() . (s,(F7() leq F8())) in (Funcs (F2(),INT)) \ (F3(),0) ) )
proof
let s be Element of Funcs (F2(),INT); ::_thesis: ( S1[s] implies ( S1[F6() . (s,(F7() leq F8()))] & ( F6() . (s,(F7() leq F8())) in (Funcs (F2(),INT)) \ (F3(),0) implies S2[F6() . (s,(F7() leq F8()))] ) & ( S2[F6() . (s,(F7() leq F8()))] implies F6() . (s,(F7() leq F8())) in (Funcs (F2(),INT)) \ (F3(),0) ) ) )
A32: ( s . F7() > s . F8() implies (F6() . (s,(F7() leq F8()))) . F3() = 0 ) by Th35;
assume A33: S1[s] ; ::_thesis: ( S1[F6() . (s,(F7() leq F8()))] & ( F6() . (s,(F7() leq F8())) in (Funcs (F2(),INT)) \ (F3(),0) implies S2[F6() . (s,(F7() leq F8()))] ) & ( S2[F6() . (s,(F7() leq F8()))] implies F6() . (s,(F7() leq F8())) in (Funcs (F2(),INT)) \ (F3(),0) ) )
hence  P1[F6() . (s,(F7() leq F8()))] by A3; ::_thesis: ( ( F10() . F9() <= F9() . F8() implies ((F6() . (s,(F7() leq F8()))) . F7()) - 1 <= F9() . F8() ) & (F6() . (s,(F7() leq F8()))) . F8() = F9() . F8() & ( F6() . (s,(F7() leq F8())) in (Funcs (F2(),INT)) \ (F3(),0) implies S2[F6() . (s,(F7() leq F8()))] ) & ( S2[F6() . (s,(F7() leq F8()))] implies F6() . (s,(F7() leq F8())) in (Funcs (F2(),INT)) \ (F3(),0) ) )
A34: ( s . F7() <= s . F8() implies (F6() . (s,(F7() leq F8()))) . F3() = 1 ) by Th35;
 (F6() . (s,(F7() leq F8()))) . F7() = s . F7() by A5, Th35;
hence  ( ( F10() . F9() <= F9() . F8() implies ((F6() . (s,(F7() leq F8()))) . F7()) - 1 <= F9() . F8() ) & (F6() . (s,(F7() leq F8()))) . F8() = F9() . F8() & ( F6() . (s,(F7() leq F8())) in (Funcs (F2(),INT)) \ (F3(),0) implies S2[F6() . (s,(F7() leq F8()))] ) & ( S2[F6() . (s,(F7() leq F8()))] implies F6() . (s,(F7() leq F8())) in (Funcs (F2(),INT)) \ (F3(),0) ) ) by A5, A33, A32, A34, Th2, Th35; ::_thesis: verum
end;
 s2 . F3() = 1 by A7, A12, A14, Th35;
then A35: ( F6() . (s1,(F7() leq F8())) in (Funcs (F2(),INT)) \ (F3(),0) iff S3[F6() . (s1,(F7() leq F8()))] ) by A2, A3, A5, A12, A8, A14, Th35;
A36: F6() iteration_terminates_for (F4() \; (F7() += 1)) \; (F7() leq F8()),F6() . (s1,(F7() leq F8())) from AOFA_000:sch_3(A35, A20);
A37: ( S1[F6() . (s1,(while ((F7() leq F8()),(F4() \; (F7() += 1)))))] & not S2[F6() . (s1,(while ((F7() leq F8()),(F4() \; (F7() += 1)))))] ) from AOFA_000:sch_5(A11, A36, A15, A31);
then  (F6() . (F9(),F5())) . F7() >= (F9() . F8()) + 1 by A10, INT_1:7;
then  ((F6() . (F9(),F5())) . F7()) - 1 >= ((F9() . F8()) + 1) - 1 by XREAL_1:13;
then  ((F6() . (F9(),F5())) . F7()) - 1 = F9() . F8() by A10, A14, A37, XXREAL_0:1;
hence  ( P1[F6() . (F9(),F5())] & ( F10() . F9() <= F9() . F8() implies (F6() . (F9(),F5())) . F7() = (F9() . F8()) + 1 ) & ( F10() . F9() > F9() . F8() implies (F6() . (F9(),F5())) . F7() = F10() . F9() ) & (F6() . (F9(),F5())) . F8() = F9() . F8() ) by A1, A14, A37, AOFA_000:def_29; ::_thesis: verum
end;
supposeA38: F10() . F9() > F9() . F8() ; ::_thesis: ( P1[F6() . (F9(),F5())] & ( F10() . F9() <= F9() . F8() implies (F6() . (F9(),F5())) . F7() = (F9() . F8()) + 1 ) & ( F10() . F9() > F9() . F8() implies (F6() . (F9(),F5())) . F7() = F10() . F9() ) & (F6() . (F9(),F5())) . F8() = F9() . F8() )
then  s2 . F3() = 0 by A7, A12, Th35;
then  s2 nin (Funcs (F2(),INT)) \ (F3(),0) by Th2;
hence  ( P1[F6() . (F9(),F5())] & ( F10() . F9() <= F9() . F8() implies (F6() . (F9(),F5())) . F7() = (F9() . F8()) + 1 ) & ( F10() . F9() > F9() . F8() implies (F6() . (F9(),F5())) . F7() = F10() . F9() ) & (F6() . (F9(),F5())) . F8() = F9() . F8() ) by A9, A8, A13, A10, A6, A38, AOFA_000:def_31; ::_thesis: verum
end;
end;
end;

scheme :: AOFA_I00:sch 2
ForDowntoIteration{ F1() ->  Euclidean preIfWhileAlgebra, F2() ->  non empty countable set , F3() ->  Element of F2(), F4() ->  Element of F1(), F5() ->  Element of F1(), F6() ->  Euclidean ExecutionFunction of F1(), Funcs (F2(),INT),(Funcs (F2(),INT)) \ (F3(),0), F7() ->  Variable of F6(), F8() ->  Variable of F6(), F9() ->  Element of Funcs (F2(),INT), F10() ->  INT-Expression of F1(),F6(), P1[ set ] } :
 ( P1[F6() . (F9(),F5())] & ( F10() . F9() >= F9() . F8() implies (F6() . (F9(),F5())) . F7() = (F9() . F8()) - 1 ) & ( F10() . F9() < F9() . F8() implies (F6() . (F9(),F5())) . F7() = F10() . F9() ) & (F6() . (F9(),F5())) . F8() = F9() . F8() )
provided
A1: F5() = for-do ((F7() := F10()),((. F8()) leq (. F7())),(F7() += (- 1)),F4()) and
A2: P1[F6() . (F9(),(F7() := F10()))] and
A3: for s being Element of Funcs (F2(),INT) st P1[s] holds
( P1[F6() . (s,(F4() \; (F7() += (- 1))))] & P1[F6() . (s,(F8() leq F7()))] ) and
A4: for s being Element of Funcs (F2(),INT) st P1[s] holds
( (F6() . (s,F4())) . F7() = s . F7() & (F6() . (s,F4())) . F8() = s . F8() ) and
A5: ( F8() <> F7() & F8() <> F3() & F7() <> F3() )
proof
set S = Funcs (F2(),INT);
reconsider s1 = F6() . (F9(),(F7() := F10())) as Element of Funcs (F2(),INT) ;
set C = F8() leq F7();
reconsider s2 = F6() . (s1,(F8() leq F7())) as Element of Funcs (F2(),INT) ;
A6: P1[s2] by A2, A3;
defpred S1[ Element of Funcs (F2(),INT)] means ( P1[$1] & ( F10() . F9() >= F9() . F8() implies ($1 . F7()) + 1 >= F9() . F8() ) & $1 . F8() = F9() . F8() );
A7: s1 . F7() = F10() . F9() by Th26;
then A8: S1[s1] by A2, A5, Th26, XREAL_1:39;
A9: s2 . F7() = F10() . F9() by A5, A7, Th35;
defpred S2[ Element of Funcs (F2(),INT)] means $1 . F7() >= $1 . F8();
set T = (Funcs (F2(),INT)) \ (F3(),0);
A10: F6() complies_with_while_wrt (Funcs (F2(),INT)) \ (F3(),0) by AOFA_000:def_32;
set II = F4() \; (F7() += (- 1));
A11: F6() . (F9(),F5()) = F6() . (s1,(while ((F8() leq F7()),(F4() \; (F7() += (- 1)))))) by A1, AOFA_000:def_29;
A12: s1 . F8() = F9() . F8() by A5, Th26;
then A13: s2 . F8() = F9() . F8() by A5, Th35;
percases ( F10() . F9() >= F9() . F8() or F10() . F9() < F9() . F8() ) ;
supposeA14: F10() . F9() >= F9() . F8() ; ::_thesis: ( P1[F6() . (F9(),F5())] & ( F10() . F9() >= F9() . F8() implies (F6() . (F9(),F5())) . F7() = (F9() . F8()) - 1 ) & ( F10() . F9() < F9() . F8() implies (F6() . (F9(),F5())) . F7() = F10() . F9() ) & (F6() . (F9(),F5())) . F8() = F9() . F8() )
deffunc H1( Element of Funcs (F2(),INT)) ->  Element of NAT = In (((($1 . F7()) + 1) - ($1 . F8())),NAT);
defpred S3[ Element of Funcs (F2(),INT)] means ( P1[$1] & S2[$1] );
A15: for s being Element of Funcs (F2(),INT) st S1[s] & s in (Funcs (F2(),INT)) \ (F3(),0) & S2[s] holds
S1[F6() . (s,(F4() \; (F7() += (- 1))))]
proof
let s be Element of Funcs (F2(),INT); ::_thesis: ( S1[s] & s in (Funcs (F2(),INT)) \ (F3(),0) & S2[s] implies S1[F6() . (s,(F4() \; (F7() += (- 1))))] )
assume that
A16: S1[s] and
 s in (Funcs (F2(),INT)) \ (F3(),0) and
A17: S2[s] ; ::_thesis: S1[F6() . (s,(F4() \; (F7() += (- 1))))]
thus  P1[F6() . (s,(F4() \; (F7() += (- 1))))] by A3, A16; ::_thesis: ( ( F10() . F9() >= F9() . F8() implies ((F6() . (s,(F4() \; (F7() += (- 1))))) . F7()) + 1 >= F9() . F8() ) & (F6() . (s,(F4() \; (F7() += (- 1))))) . F8() = F9() . F8() )
reconsider s99 = F6() . (s,F4()) as Element of Funcs (F2(),INT) ;
reconsider s999 = F6() . (s99,(F7() += (- 1))) as Element of Funcs (F2(),INT) ;
A18: s999 = F6() . (s,(F4() \; (F7() += (- 1)))) by AOFA_000:def_29;
A19: s999 . F7() = (s99 . F7()) - 1 by Th28;
 s99 . F8() = s . F8() by A4, A16;
hence  ( ( F10() . F9() >= F9() . F8() implies ((F6() . (s,(F4() \; (F7() += (- 1))))) . F7()) + 1 >= F9() . F8() ) & (F6() . (s,(F4() \; (F7() += (- 1))))) . F8() = F9() . F8() ) by A4, A5, A16, A17, A19, A18, Th28; ::_thesis: verum
end;
A20: for s being Element of Funcs (F2(),INT) st S3[s] holds
( ( S3[F6() . (s,((F4() \; (F7() += (- 1))) \; (F8() leq F7())))] implies F6() . (s,((F4() \; (F7() += (- 1))) \; (F8() leq F7()))) in (Funcs (F2(),INT)) \ (F3(),0) ) & ( F6() . (s,((F4() \; (F7() += (- 1))) \; (F8() leq F7()))) in (Funcs (F2(),INT)) \ (F3(),0) implies S3[F6() . (s,((F4() \; (F7() += (- 1))) \; (F8() leq F7())))] ) & H1(F6() . (s,((F4() \; (F7() += (- 1))) \; (F8() leq F7())))) < H1(s) )
proof
let s be Element of Funcs (F2(),INT); ::_thesis: ( S3[s] implies ( ( S3[F6() . (s,((F4() \; (F7() += (- 1))) \; (F8() leq F7())))] implies F6() . (s,((F4() \; (F7() += (- 1))) \; (F8() leq F7()))) in (Funcs (F2(),INT)) \ (F3(),0) ) & ( F6() . (s,((F4() \; (F7() += (- 1))) \; (F8() leq F7()))) in (Funcs (F2(),INT)) \ (F3(),0) implies S3[F6() . (s,((F4() \; (F7() += (- 1))) \; (F8() leq F7())))] ) & H1(F6() . (s,((F4() \; (F7() += (- 1))) \; (F8() leq F7())))) < H1(s) ) )
reconsider s9 = F6() . (s,F4()) as Element of Funcs (F2(),INT) ;
reconsider s99 = F6() . (s9,(F7() += (- 1))) as Element of Funcs (F2(),INT) ;
reconsider s999 = F6() . (s99,(F8() leq F7())) as Element of Funcs (F2(),INT) ;
A21: F6() . (s,((F4() \; (F7() += (- 1))) \; (F8() leq F7()))) = F6() . ((F6() . (s,(F4() \; (F7() += (- 1))))),(F8() leq F7())) by AOFA_000:def_29;
A22: s99 . F8() = s9 . F8() by A5, Th28;
A23: ( s99 . F7() >= s99 . F8() implies s999 . F3() = 1 ) by Th35;
A24: ( s99 . F7() < s99 . F8() implies s999 . F3() = 0 ) by Th35;
assume A25: S3[s] ; ::_thesis: ( ( S3[F6() . (s,((F4() \; (F7() += (- 1))) \; (F8() leq F7())))] implies F6() . (s,((F4() \; (F7() += (- 1))) \; (F8() leq F7()))) in (Funcs (F2(),INT)) \ (F3(),0) ) & ( F6() . (s,((F4() \; (F7() += (- 1))) \; (F8() leq F7()))) in (Funcs (F2(),INT)) \ (F3(),0) implies S3[F6() . (s,((F4() \; (F7() += (- 1))) \; (F8() leq F7())))] ) & H1(F6() . (s,((F4() \; (F7() += (- 1))) \; (F8() leq F7())))) < H1(s) )
then reconsider j = (s . F7()) - (s . F8()) as Element of NAT by INT_1:5;
A26: s999 . F7() = s99 . F7() by A5, Th35;
A27: s99 = F6() . (s,(F4() \; (F7() += (- 1)))) by AOFA_000:def_29;
then  P1[s99] by A3, A25;
hence  ( S3[F6() . (s,((F4() \; (F7() += (- 1))) \; (F8() leq F7())))] iff F6() . (s,((F4() \; (F7() += (- 1))) \; (F8() leq F7()))) in (Funcs (F2(),INT)) \ (F3(),0) ) by A3, A5, A21, A27, A26, A24, A23, Th2, Th35; ::_thesis: H1(F6() . (s,((F4() \; (F7() += (- 1))) \; (F8() leq F7())))) < H1(s)
A28: s99 . F7() = (s9 . F7()) - 1 by Th28;
A29: s9 . F8() = s . F8() by A4, A25;
A30: s9 . F7() = s . F7() by A4, A25;
 s999 . F8() = s99 . F8() by A5, Th35;
then  H1(s999) = j by A26, A30, A29, A28, A22, FUNCT_7:def_1;
then  H1(s999) + 1 = H1(s) by FUNCT_7:def_1;
hence  H1(F6() . (s,((F4() \; (F7() += (- 1))) \; (F8() leq F7())))) < H1(s) by A21, A27, NAT_1:13; ::_thesis: verum
end;
A31: for s being Element of Funcs (F2(),INT) st S1[s] holds
( S1[F6() . (s,(F8() leq F7()))] & ( F6() . (s,(F8() leq F7())) in (Funcs (F2(),INT)) \ (F3(),0) implies S2[F6() . (s,(F8() leq F7()))] ) & ( S2[F6() . (s,(F8() leq F7()))] implies F6() . (s,(F8() leq F7())) in (Funcs (F2(),INT)) \ (F3(),0) ) )
proof
let s be Element of Funcs (F2(),INT); ::_thesis: ( S1[s] implies ( S1[F6() . (s,(F8() leq F7()))] & ( F6() . (s,(F8() leq F7())) in (Funcs (F2(),INT)) \ (F3(),0) implies S2[F6() . (s,(F8() leq F7()))] ) & ( S2[F6() . (s,(F8() leq F7()))] implies F6() . (s,(F8() leq F7())) in (Funcs (F2(),INT)) \ (F3(),0) ) ) )
A32: ( s . F7() < s . F8() implies (F6() . (s,(F8() leq F7()))) . F3() = 0 ) by Th35;
assume A33: S1[s] ; ::_thesis: ( S1[F6() . (s,(F8() leq F7()))] & ( F6() . (s,(F8() leq F7())) in (Funcs (F2(),INT)) \ (F3(),0) implies S2[F6() . (s,(F8() leq F7()))] ) & ( S2[F6() . (s,(F8() leq F7()))] implies F6() . (s,(F8() leq F7())) in (Funcs (F2(),INT)) \ (F3(),0) ) )
hence  P1[F6() . (s,(F8() leq F7()))] by A3; ::_thesis: ( ( F10() . F9() >= F9() . F8() implies ((F6() . (s,(F8() leq F7()))) . F7()) + 1 >= F9() . F8() ) & (F6() . (s,(F8() leq F7()))) . F8() = F9() . F8() & ( F6() . (s,(F8() leq F7())) in (Funcs (F2(),INT)) \ (F3(),0) implies S2[F6() . (s,(F8() leq F7()))] ) & ( S2[F6() . (s,(F8() leq F7()))] implies F6() . (s,(F8() leq F7())) in (Funcs (F2(),INT)) \ (F3(),0) ) )
A34: ( s . F7() >= s . F8() implies (F6() . (s,(F8() leq F7()))) . F3() = 1 ) by Th35;
 (F6() . (s,(F8() leq F7()))) . F7() = s . F7() by A5, Th35;
hence  ( ( F10() . F9() >= F9() . F8() implies ((F6() . (s,(F8() leq F7()))) . F7()) + 1 >= F9() . F8() ) & (F6() . (s,(F8() leq F7()))) . F8() = F9() . F8() & ( F6() . (s,(F8() leq F7())) in (Funcs (F2(),INT)) \ (F3(),0) implies S2[F6() . (s,(F8() leq F7()))] ) & ( S2[F6() . (s,(F8() leq F7()))] implies F6() . (s,(F8() leq F7())) in (Funcs (F2(),INT)) \ (F3(),0) ) ) by A5, A33, A32, A34, Th2, Th35; ::_thesis: verum
end;
 s2 . F3() = 1 by A7, A12, A14, Th35;
then A35: ( F6() . (s1,(F8() leq F7())) in (Funcs (F2(),INT)) \ (F3(),0) iff S3[F6() . (s1,(F8() leq F7()))] ) by A2, A3, A5, A12, A9, A14, Th35;
A36: F6() iteration_terminates_for (F4() \; (F7() += (- 1))) \; (F8() leq F7()),F6() . (s1,(F8() leq F7())) from AOFA_000:sch_3(A35, A20);
A37: ( S1[F6() . (s1,(while ((F8() leq F7()),(F4() \; (F7() += (- 1))))))] & not S2[F6() . (s1,(while ((F8() leq F7()),(F4() \; (F7() += (- 1))))))] ) from AOFA_000:sch_5(A8, A36, A15, A31);
then  ((F6() . (F9(),F5())) . F7()) + 1 <= ((F9() . F8()) + 1) - 1 by A11, INT_1:7;
then  ((F6() . (F9(),F5())) . F7()) + 1 = F9() . F8() by A11, A14, A37, XXREAL_0:1;
hence  ( P1[F6() . (F9(),F5())] & ( F10() . F9() >= F9() . F8() implies (F6() . (F9(),F5())) . F7() = (F9() . F8()) - 1 ) & ( F10() . F9() < F9() . F8() implies (F6() . (F9(),F5())) . F7() = F10() . F9() ) & (F6() . (F9(),F5())) . F8() = F9() . F8() ) by A1, A14, A37, AOFA_000:def_29; ::_thesis: verum
end;
supposeA38: F10() . F9() < F9() . F8() ; ::_thesis: ( P1[F6() . (F9(),F5())] & ( F10() . F9() >= F9() . F8() implies (F6() . (F9(),F5())) . F7() = (F9() . F8()) - 1 ) & ( F10() . F9() < F9() . F8() implies (F6() . (F9(),F5())) . F7() = F10() . F9() ) & (F6() . (F9(),F5())) . F8() = F9() . F8() )
then  s2 . F3() = 0 by A7, A12, Th35;
then  s2 nin (Funcs (F2(),INT)) \ (F3(),0) by Th2;
hence  ( P1[F6() . (F9(),F5())] & ( F10() . F9() >= F9() . F8() implies (F6() . (F9(),F5())) . F7() = (F9() . F8()) - 1 ) & ( F10() . F9() < F9() . F8() implies (F6() . (F9(),F5())) . F7() = F10() . F9() ) & (F6() . (F9(),F5())) . F8() = F9() . F8() ) by A10, A9, A13, A11, A6, A38, AOFA_000:def_31; ::_thesis: verum
end;
end;
end;

begin


theorem Th52: :: AOFA_I00:52
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for I being Element of A
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) holds
g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n))
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for I being Element of A
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) holds
g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n))
let X be non empty countable set ; ::_thesis: for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for I being Element of A
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) holds
g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n))
let s be Element of Funcs (X,INT); ::_thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for I being Element of A
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) holds
g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n))
let b be Element of X; ::_thesis: for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for I being Element of A
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) holds
g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n))
let g be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); ::_thesis: for I being Element of A
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) holds
g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n))
let I be Element of A; ::_thesis: for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) holds
g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n))
let i, n be Variable of g; ::_thesis: ( ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) implies g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n)) )
given d being Function such that A1: d . b = 0 and
A2: d . n = 1 and
A3: d . i = 2 ; ::_thesis: ( ex s being Element of Funcs (X,INT) st
( (g . (s,I)) . n = s . n implies not (g . (s,I)) . i = s . i ) or g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n)) )
set J = i += 1;
set C = i leq n;
set S = Funcs (X,INT);
set h = g;
assume A4: for s being Element of Funcs (X,INT) holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ; ::_thesis: g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n))
deffunc H1( Element of Funcs (X,INT)) ->  Element of NAT = In (((($1 . n) + 1) - ($1 . i)),NAT);
defpred S1[ Element of Funcs (X,INT)] means $1 . i <= $1 . n;
set T = (Funcs (X,INT)) \ (b,0);
A5: i <> n by A2, A3;
A6: for s being Element of Funcs (X,INT) st S1[s] holds
( ( S1[g . (s,((I \; (i += 1)) \; (i leq n)))] implies g . (s,((I \; (i += 1)) \; (i leq n))) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,((I \; (i += 1)) \; (i leq n))) in (Funcs (X,INT)) \ (b,0) implies S1[g . (s,((I \; (i += 1)) \; (i leq n)))] ) & H1(g . (s,((I \; (i += 1)) \; (i leq n)))) < H1(s) )
proof
let s be Element of Funcs (X,INT); ::_thesis: ( S1[s] implies ( ( S1[g . (s,((I \; (i += 1)) \; (i leq n)))] implies g . (s,((I \; (i += 1)) \; (i leq n))) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,((I \; (i += 1)) \; (i leq n))) in (Funcs (X,INT)) \ (b,0) implies S1[g . (s,((I \; (i += 1)) \; (i leq n)))] ) & H1(g . (s,((I \; (i += 1)) \; (i leq n)))) < H1(s) ) )
set s1 = g . (s,I);
set q = g . (s,(I \; (i += 1)));
set q1 = g . ((g . (s,(I \; (i += 1)))),(i leq n));
A7: g . (s,(I \; (i += 1))) = g . ((g . (s,I)),(i += 1)) by AOFA_000:def_29;
 (g . (s,I)) . i = s . i by A4;
then  (g . (s,(I \; (i += 1)))) . i = (s . i) + 1 by A7, Th28;
then A8: (g . ((g . (s,(I \; (i += 1)))),(i leq n))) . i = (s . i) + 1 by A1, A3, Th35;
A9: ( (g . (s,(I \; (i += 1)))) . i > (g . (s,(I \; (i += 1)))) . n implies (g . ((g . (s,(I \; (i += 1)))),(i leq n))) . b = 0 ) by Th35;
assume  S1[s] ; ::_thesis: ( ( S1[g . (s,((I \; (i += 1)) \; (i leq n)))] implies g . (s,((I \; (i += 1)) \; (i leq n))) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,((I \; (i += 1)) \; (i leq n))) in (Funcs (X,INT)) \ (b,0) implies S1[g . (s,((I \; (i += 1)) \; (i leq n)))] ) & H1(g . (s,((I \; (i += 1)) \; (i leq n)))) < H1(s) )
then reconsider ni = (s . n) - (s . i) as Element of NAT by INT_1:3, XREAL_1:48;
A10: g . ((g . (s,(I \; (i += 1)))),(i leq n)) = g . (s,((I \; (i += 1)) \; (i leq n))) by AOFA_000:def_29;
A11: ( (g . (s,(I \; (i += 1)))) . i <= (g . (s,(I \; (i += 1)))) . n implies (g . ((g . (s,(I \; (i += 1)))),(i leq n))) . b = 1 ) by Th35;
 (g . ((g . (s,(I \; (i += 1)))),(i leq n))) . i = (g . (s,(I \; (i += 1)))) . i by A1, A3, Th35;
hence  ( S1[g . (s,((I \; (i += 1)) \; (i leq n)))] iff g . (s,((I \; (i += 1)) \; (i leq n))) in (Funcs (X,INT)) \ (b,0) ) by A1, A2, A9, A11, A10, Th2, Th35; ::_thesis: H1(g . (s,((I \; (i += 1)) \; (i leq n)))) < H1(s)
A12: H1(s) = ni + 1 by FUNCT_7:def_1;
 (g . (s,I)) . n = s . n by A4;
then  (g . (s,(I \; (i += 1)))) . n = s . n by A5, A7, Th28;
then  (g . ((g . (s,(I \; (i += 1)))),(i leq n))) . n = s . n by A1, A2, Th35;
then  H1(g . ((g . (s,(I \; (i += 1)))),(i leq n))) = ni by A8, FUNCT_7:def_1;
hence  H1(g . (s,((I \; (i += 1)) \; (i leq n)))) < H1(s) by A10, A12, NAT_1:13; ::_thesis: verum
end;
reconsider s1 = g . (s,(i leq n)) as Element of Funcs (X,INT) ;
A13: ( s . i > s . n implies s1 . b = 0 ) by Th35;
A14: ( s . i <= s . n implies s1 . b = 1 ) by Th35;
 s1 . i = s . i by A1, A3, Th35;
then A15: ( s1 in (Funcs (X,INT)) \ (b,0) iff S1[s1] ) by A1, A2, A13, A14, Th2, Th35;
 g iteration_terminates_for (I \; (i += 1)) \; (i leq n),s1 from AOFA_000:sch_3(A15, A6);
hence  g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n)) ; ::_thesis: verum
end;

theorem Th53: :: AOFA_I00:53
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for P being set
for I being Element of A
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i & g . (s,I) in P & g . (s,(i leq n)) in P & g . (s,(i += 1)) in P ) ) & s in P holds
g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n))
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for P being set
for I being Element of A
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i & g . (s,I) in P & g . (s,(i leq n)) in P & g . (s,(i += 1)) in P ) ) & s in P holds
g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n))
let X be non empty countable set ; ::_thesis: for s being Element of Funcs (X,INT)
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for P being set
for I being Element of A
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i & g . (s,I) in P & g . (s,(i leq n)) in P & g . (s,(i += 1)) in P ) ) & s in P holds
g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n))
let s be Element of Funcs (X,INT); ::_thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for P being set
for I being Element of A
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i & g . (s,I) in P & g . (s,(i leq n)) in P & g . (s,(i += 1)) in P ) ) & s in P holds
g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n))
let b be Element of X; ::_thesis: for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for P being set
for I being Element of A
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i & g . (s,I) in P & g . (s,(i leq n)) in P & g . (s,(i += 1)) in P ) ) & s in P holds
g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n))
let g be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); ::_thesis: for P being set
for I being Element of A
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i & g . (s,I) in P & g . (s,(i leq n)) in P & g . (s,(i += 1)) in P ) ) & s in P holds
g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n))
let P be set ; ::_thesis: for I being Element of A
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i & g . (s,I) in P & g . (s,(i leq n)) in P & g . (s,(i += 1)) in P ) ) & s in P holds
g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n))
let I be Element of A; ::_thesis: for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i & g . (s,I) in P & g . (s,(i leq n)) in P & g . (s,(i += 1)) in P ) ) & s in P holds
g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n))
let i, n be Variable of g; ::_thesis: ( ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i & g . (s,I) in P & g . (s,(i leq n)) in P & g . (s,(i += 1)) in P ) ) & s in P implies g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n)) )
given d being Function such that A1: d . b = 0 and
A2: d . n = 1 and
A3: d . i = 2 ; ::_thesis: ( ex s being Element of Funcs (X,INT) st
( s in P & not ( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i & g . (s,I) in P & g . (s,(i leq n)) in P & g . (s,(i += 1)) in P ) ) or not s in P or g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n)) )
set J = i += 1;
set C = i leq n;
set S = Funcs (X,INT);
set h = g;
assume that
A4: for s being Element of Funcs (X,INT) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i & g . (s,I) in P & g . (s,(i leq n)) in P & g . (s,(i += 1)) in P ) and
A5: s in P ; ::_thesis: g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n))
defpred S1[ Element of Funcs (X,INT)] means $1 in P;
reconsider s1 = g . (s,(i leq n)) as Element of Funcs (X,INT) ;
A6: S1[s1] by A4, A5;
deffunc H1( Element of Funcs (X,INT)) ->  Element of NAT = In (((($1 . n) + 1) - ($1 . i)),NAT);
defpred S2[ Element of Funcs (X,INT)] means $1 . i <= $1 . n;
set T = (Funcs (X,INT)) \ (b,0);
A7: i <> n by A2, A3;
A8: for s being Element of Funcs (X,INT) st S1[s] & s in (Funcs (X,INT)) \ (b,0) & S2[s] holds
( S1[g . (s,((I \; (i += 1)) \; (i leq n)))] & ( S2[g . (s,((I \; (i += 1)) \; (i leq n)))] implies g . (s,((I \; (i += 1)) \; (i leq n))) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,((I \; (i += 1)) \; (i leq n))) in (Funcs (X,INT)) \ (b,0) implies S2[g . (s,((I \; (i += 1)) \; (i leq n)))] ) & H1(g . (s,((I \; (i += 1)) \; (i leq n)))) < H1(s) )
proof
let s be Element of Funcs (X,INT); ::_thesis: ( S1[s] & s in (Funcs (X,INT)) \ (b,0) & S2[s] implies ( S1[g . (s,((I \; (i += 1)) \; (i leq n)))] & ( S2[g . (s,((I \; (i += 1)) \; (i leq n)))] implies g . (s,((I \; (i += 1)) \; (i leq n))) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,((I \; (i += 1)) \; (i leq n))) in (Funcs (X,INT)) \ (b,0) implies S2[g . (s,((I \; (i += 1)) \; (i leq n)))] ) & H1(g . (s,((I \; (i += 1)) \; (i leq n)))) < H1(s) ) )
assume that
A9: S1[s] and
 s in (Funcs (X,INT)) \ (b,0) and
A10: S2[s] ; ::_thesis: ( S1[g . (s,((I \; (i += 1)) \; (i leq n)))] & ( S2[g . (s,((I \; (i += 1)) \; (i leq n)))] implies g . (s,((I \; (i += 1)) \; (i leq n))) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,((I \; (i += 1)) \; (i leq n))) in (Funcs (X,INT)) \ (b,0) implies S2[g . (s,((I \; (i += 1)) \; (i leq n)))] ) & H1(g . (s,((I \; (i += 1)) \; (i leq n)))) < H1(s) )
set q = g . (s,(I \; (i += 1)));
set s1 = g . (s,I);
set q1 = g . ((g . (s,(I \; (i += 1)))),(i leq n));
A11: g . (s,(I \; (i += 1))) = g . ((g . (s,I)),(i += 1)) by AOFA_000:def_29;
 (g . (s,I)) . i = s . i by A4, A9;
then  (g . (s,(I \; (i += 1)))) . i = (s . i) + 1 by A11, Th28;
then A12: (g . ((g . (s,(I \; (i += 1)))),(i leq n))) . i = (s . i) + 1 by A1, A3, Th35;
reconsider ni = (s . n) - (s . i) as Element of NAT by A10, INT_1:3, XREAL_1:48;
A13: H1(s) = ni + 1 by FUNCT_7:def_1;
A14: ( (g . (s,(I \; (i += 1)))) . i <= (g . (s,(I \; (i += 1)))) . n implies (g . ((g . (s,(I \; (i += 1)))),(i leq n))) . b = 1 ) by Th35;
A15: g . ((g . (s,(I \; (i += 1)))),(i leq n)) = g . (s,((I \; (i += 1)) \; (i leq n))) by AOFA_000:def_29;
 S1[g . (s,I)] by A4, A9;
then  S1[g . (s,(I \; (i += 1)))] by A4, A11;
hence  S1[g . (s,((I \; (i += 1)) \; (i leq n)))] by A4, A15; ::_thesis: ( ( S2[g . (s,((I \; (i += 1)) \; (i leq n)))] implies g . (s,((I \; (i += 1)) \; (i leq n))) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,((I \; (i += 1)) \; (i leq n))) in (Funcs (X,INT)) \ (b,0) implies S2[g . (s,((I \; (i += 1)) \; (i leq n)))] ) & H1(g . (s,((I \; (i += 1)) \; (i leq n)))) < H1(s) )
A16: ( (g . (s,(I \; (i += 1)))) . i > (g . (s,(I \; (i += 1)))) . n implies (g . ((g . (s,(I \; (i += 1)))),(i leq n))) . b = 0 ) by Th35;
 (g . ((g . (s,(I \; (i += 1)))),(i leq n))) . i = (g . (s,(I \; (i += 1)))) . i by A1, A3, Th35;
hence  ( S2[g . (s,((I \; (i += 1)) \; (i leq n)))] iff g . (s,((I \; (i += 1)) \; (i leq n))) in (Funcs (X,INT)) \ (b,0) ) by A1, A2, A16, A14, A15, Th2, Th35; ::_thesis: H1(g . (s,((I \; (i += 1)) \; (i leq n)))) < H1(s)
 (g . (s,I)) . n = s . n by A4, A9;
then  (g . (s,(I \; (i += 1)))) . n = s . n by A7, A11, Th28;
then  (g . ((g . (s,(I \; (i += 1)))),(i leq n))) . n = s . n by A1, A2, Th35;
then  H1(g . ((g . (s,(I \; (i += 1)))),(i leq n))) = ni by A12, FUNCT_7:def_1;
hence  H1(g . (s,((I \; (i += 1)) \; (i leq n)))) < H1(s) by A15, A13, NAT_1:13; ::_thesis: verum
end;
A17: ( s . i > s . n implies s1 . b = 0 ) by Th35;
A18: ( s . i <= s . n implies s1 . b = 1 ) by Th35;
 s1 . i = s . i by A1, A3, Th35;
then A19: ( s1 in (Funcs (X,INT)) \ (b,0) iff S2[s1] ) by A1, A2, A17, A18, Th2, Th35;
 g iteration_terminates_for (I \; (i += 1)) \; (i leq n),s1 from AOFA_000:sch_4(A6, A19, A8);
hence  g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n)) ; ::_thesis: verum
end;

theorem Th54: :: AOFA_I00:54
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for t being INT-Expression of A,f
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for I being Element of A st I is_terminating_wrt g holds
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) holds
for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for t being INT-Expression of A,f
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for I being Element of A st I is_terminating_wrt g holds
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) holds
for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g
let X be non empty countable set ; ::_thesis: for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for t being INT-Expression of A,f
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for I being Element of A st I is_terminating_wrt g holds
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) holds
for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g
let T be Subset of (Funcs (X,INT)); ::_thesis: for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for t being INT-Expression of A,f
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for I being Element of A st I is_terminating_wrt g holds
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) holds
for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T; ::_thesis: for t being INT-Expression of A,f
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for I being Element of A st I is_terminating_wrt g holds
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) holds
for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g
let t be INT-Expression of A,f; ::_thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for I being Element of A st I is_terminating_wrt g holds
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) holds
for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g
let b be Element of X; ::_thesis: for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for I being Element of A st I is_terminating_wrt g holds
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) holds
for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g
let g be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); ::_thesis: for I being Element of A st I is_terminating_wrt g holds
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) holds
for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g
set S = Funcs (X,INT);
set T = (Funcs (X,INT)) \ (b,0);
let I be Element of A; ::_thesis: ( I is_terminating_wrt g implies for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) holds
for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g )
assume A1: I is_terminating_wrt g ; ::_thesis: for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) holds
for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g
let i, n be Variable of g; ::_thesis: ( ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) implies for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g )
 i += 1 is_terminating_wrt g by AOFA_000:104;
then A2: I \; (i += 1) is_terminating_wrt g by A1, AOFA_000:110;
set Q = while ((i leq n),(I \; (i += 1)));
given d being Function such that A3: d . b = 0 and
A4: d . n = 1 and
A5: d . i = 2 ; ::_thesis: ( ex s being Element of Funcs (X,INT) st
( (g . (s,I)) . n = s . n implies not (g . (s,I)) . i = s . i ) or for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g )
assume A6: for s being Element of Funcs (X,INT) holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ; ::_thesis: for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g
let s be Element of Funcs (X,INT); :: according to AOFA_000:def_37 ::_thesis: [s,(for-do ((i := t),(i leq n),(i += 1),I))] in TerminatingPrograms (A,(Funcs (X,INT)),((Funcs (X,INT)) \ (b,0)),g)
A7: [s,(i := t)] in TerminatingPrograms (A,(Funcs (X,INT)),((Funcs (X,INT)) \ (b,0)),g) by AOFA_000:def_36;
A8: i leq n is_terminating_wrt g by AOFA_000:104;
 g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . ((g . (s,(i := t))),(i leq n)) by A3, A4, A5, A6, Th52;
then  [(g . (s,(i := t))),(while ((i leq n),(I \; (i += 1))))] in TerminatingPrograms (A,(Funcs (X,INT)),((Funcs (X,INT)) \ (b,0)),g) by A8, A2, AOFA_000:114;
hence  [s,(for-do ((i := t),(i leq n),(i += 1),I))] in TerminatingPrograms (A,(Funcs (X,INT)),((Funcs (X,INT)) \ (b,0)),g) by A7, AOFA_000:def_35; ::_thesis: verum
end;

theorem :: AOFA_I00:55
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for t being INT-Expression of A,f
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for P being set
for I being Element of A st I is_terminating_wrt g,P holds
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) & P is_invariant_wrt i := t,g & P is_invariant_wrt I,g & P is_invariant_wrt i leq n,g & P is_invariant_wrt i += 1,g holds
for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g,P
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for t being INT-Expression of A,f
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for P being set
for I being Element of A st I is_terminating_wrt g,P holds
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) & P is_invariant_wrt i := t,g & P is_invariant_wrt I,g & P is_invariant_wrt i leq n,g & P is_invariant_wrt i += 1,g holds
for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g,P
let X be non empty countable set ; ::_thesis: for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for t being INT-Expression of A,f
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for P being set
for I being Element of A st I is_terminating_wrt g,P holds
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) & P is_invariant_wrt i := t,g & P is_invariant_wrt I,g & P is_invariant_wrt i leq n,g & P is_invariant_wrt i += 1,g holds
for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g,P
let T be Subset of (Funcs (X,INT)); ::_thesis: for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for t being INT-Expression of A,f
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for P being set
for I being Element of A st I is_terminating_wrt g,P holds
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) & P is_invariant_wrt i := t,g & P is_invariant_wrt I,g & P is_invariant_wrt i leq n,g & P is_invariant_wrt i += 1,g holds
for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g,P
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T; ::_thesis: for t being INT-Expression of A,f
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for P being set
for I being Element of A st I is_terminating_wrt g,P holds
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) & P is_invariant_wrt i := t,g & P is_invariant_wrt I,g & P is_invariant_wrt i leq n,g & P is_invariant_wrt i += 1,g holds
for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g,P
let t be INT-Expression of A,f; ::_thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for P being set
for I being Element of A st I is_terminating_wrt g,P holds
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) & P is_invariant_wrt i := t,g & P is_invariant_wrt I,g & P is_invariant_wrt i leq n,g & P is_invariant_wrt i += 1,g holds
for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g,P
let b be Element of X; ::_thesis: for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for P being set
for I being Element of A st I is_terminating_wrt g,P holds
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) & P is_invariant_wrt i := t,g & P is_invariant_wrt I,g & P is_invariant_wrt i leq n,g & P is_invariant_wrt i += 1,g holds
for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g,P
let g be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); ::_thesis: for P being set
for I being Element of A st I is_terminating_wrt g,P holds
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) & P is_invariant_wrt i := t,g & P is_invariant_wrt I,g & P is_invariant_wrt i leq n,g & P is_invariant_wrt i += 1,g holds
for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g,P
set Z = Funcs (X,INT);
set T = (Funcs (X,INT)) \ (b,0);
let S be set ; ::_thesis: for I being Element of A st I is_terminating_wrt g,S holds
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) st s in S holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) & S is_invariant_wrt i := t,g & S is_invariant_wrt I,g & S is_invariant_wrt i leq n,g & S is_invariant_wrt i += 1,g holds
for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g,S
let I be Element of A; ::_thesis: ( I is_terminating_wrt g,S implies for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) st s in S holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) & S is_invariant_wrt i := t,g & S is_invariant_wrt I,g & S is_invariant_wrt i leq n,g & S is_invariant_wrt i += 1,g holds
for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g,S )
assume A1: I is_terminating_wrt g,S ; ::_thesis: for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) st s in S holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) & S is_invariant_wrt i := t,g & S is_invariant_wrt I,g & S is_invariant_wrt i leq n,g & S is_invariant_wrt i += 1,g holds
for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g,S
let i, n be Variable of g; ::_thesis: ( ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) st s in S holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) & S is_invariant_wrt i := t,g & S is_invariant_wrt I,g & S is_invariant_wrt i leq n,g & S is_invariant_wrt i += 1,g implies for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g,S )
given d being Function such that A2: d . b = 0 and
A3: d . n = 1 and
A4: d . i = 2 ; ::_thesis: ( ex s being Element of Funcs (X,INT) st
( s in S & not ( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) or not S is_invariant_wrt i := t,g or not S is_invariant_wrt I,g or not S is_invariant_wrt i leq n,g or not S is_invariant_wrt i += 1,g or for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g,S )
set C = i leq n;
set J = I \; (i += 1);
assume that
A5: for s being Element of Funcs (X,INT) st s in S holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) and
A6: S is_invariant_wrt i := t,g and
A7: S is_invariant_wrt I,g and
A8: S is_invariant_wrt i leq n,g and
A9: S is_invariant_wrt i += 1,g ; ::_thesis: for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g,S
let s be Element of Funcs (X,INT); :: according to AOFA_000:def_38 ::_thesis: ( not s in S or [s,(for-do ((i := t),(i leq n),(i += 1),I))] in TerminatingPrograms (A,(Funcs (X,INT)),((Funcs (X,INT)) \ (b,0)),g) )
assume  s in S ; ::_thesis: [s,(for-do ((i := t),(i leq n),(i += 1),I))] in TerminatingPrograms (A,(Funcs (X,INT)),((Funcs (X,INT)) \ (b,0)),g)
then A10: g . (s,(i := t)) in S by A6, AOFA_000:def_39;
 for s being Element of Funcs (X,INT) st s in S holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i & g . (s,I) in S & g . (s,(i leq n)) in S & g . (s,(i += 1)) in S ) by A5, A7, A8, A9, AOFA_000:def_39;
then A11: g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . ((g . (s,(i := t))),(i leq n)) by A2, A3, A4, A10, Th53;
set Q = while ((i leq n),(I \; (i += 1)));
A12: i leq n is_terminating_wrt g by AOFA_000:104;
A13: [s,(i := t)] in TerminatingPrograms (A,(Funcs (X,INT)),((Funcs (X,INT)) \ (b,0)),g) by AOFA_000:def_36;
 S is_invariant_wrt I \; (i += 1),g by A7, A9, AOFA_000:109;
then A14: for s being Element of Funcs (X,INT) st s in S & g . ((g . (s,(I \; (i += 1)))),(i leq n)) in (Funcs (X,INT)) \ (b,0) holds
g . (s,(I \; (i += 1))) in S by AOFA_000:def_39;
 i += 1 is_terminating_wrt g,S by AOFA_000:107;
then  [(g . (s,(i := t))),(while ((i leq n),(I \; (i += 1))))] in TerminatingPrograms (A,(Funcs (X,INT)),((Funcs (X,INT)) \ (b,0)),g) by A1, A7, A8, A10, A11, A12, A14, AOFA_000:111, AOFA_000:116;
hence  [s,(for-do ((i := t),(i leq n),(i += 1),I))] in TerminatingPrograms (A,(Funcs (X,INT)),((Funcs (X,INT)) \ (b,0)),g) by A13, AOFA_000:def_35; ::_thesis: verum
end;

begin

definition
let X be non empty countable set ;
let A be Euclidean preIfWhileAlgebra;
let T be Subset of (Funcs (X,INT));
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T;
let s be Element of Funcs (X,INT);
let I be Element of A;
:: original: .
redefine funcf . (s,I) ->  Element of Funcs (X,INT);
coherence
 f . (s,I) is Element of Funcs (X,INT)
proof
 f . (s,I) is Element of Funcs (X,INT) ;
hence  f . (s,I) is Element of Funcs (X,INT) ; ::_thesis: verum
end;
end;

theorem :: AOFA_I00:56
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for n, s, i being Variable of g st ex d being Function st
( d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) holds
(s := 1) \; (for-do ((i := 2),(i leq n),(i += 1),(s *= i))) is_terminating_wrt g
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for n, s, i being Variable of g st ex d being Function st
( d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) holds
(s := 1) \; (for-do ((i := 2),(i leq n),(i += 1),(s *= i))) is_terminating_wrt g
let X be non empty countable set ; ::_thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for n, s, i being Variable of g st ex d being Function st
( d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) holds
(s := 1) \; (for-do ((i := 2),(i leq n),(i += 1),(s *= i))) is_terminating_wrt g
let b be Element of X; ::_thesis: for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for n, s, i being Variable of g st ex d being Function st
( d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) holds
(s := 1) \; (for-do ((i := 2),(i leq n),(i += 1),(s *= i))) is_terminating_wrt g
let g be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); ::_thesis: for n, s, i being Variable of g st ex d being Function st
( d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) holds
(s := 1) \; (for-do ((i := 2),(i leq n),(i += 1),(s *= i))) is_terminating_wrt g
set S = Funcs (X,INT);
set T = (Funcs (X,INT)) \ (b,0);
let n, s, i be Variable of g; ::_thesis: ( ex d being Function st
( d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) implies (s := 1) \; (for-do ((i := 2),(i leq n),(i += 1),(s *= i))) is_terminating_wrt g )
given d being Function such that A1: d . n = 1 and
A2: d . s = 2 and
A3: d . i = 3 and
A4: d . b = 4 ; ::_thesis: (s := 1) \; (for-do ((i := 2),(i leq n),(i += 1),(s *= i))) is_terminating_wrt g
A5: s <> n by A1, A2;
 s <> i by A2, A3;
then A6: for q being Element of Funcs (X,INT) holds
( (g . (q,(s *= i))) . n = q . n & (g . (q,(s *= i))) . i = q . i ) by A5, Th33;
A7: n <> b by A1, A4;
A8: b <> i by A3, A4;
let q be Element of Funcs (X,INT); :: according to AOFA_000:def_37 ::_thesis: [q,((s := 1) \; (for-do ((i := 2),(i leq n),(i += 1),(s *= i))))] in TerminatingPrograms (A,(Funcs (X,INT)),((Funcs (X,INT)) \ (b,0)),g)
set t = . (2,A,g);
 i <> n by A1, A3;
then  ex d9 being Function st
( d9 . b = 0 & d9 . n = 1 & d9 . i = 2 ) by A8, A7, Th1;
then  for-do ((i := (. (2,A,g))),(i leq n),(i += 1),(s *= i)) is_terminating_wrt g by A6, Th54, AOFA_000:104;
then A9: [(g . (q,(s := 1))),(for-do ((i := (. (2,A,g))),(i leq n),(i += 1),(s *= i)))] in TerminatingPrograms (A,(Funcs (X,INT)),((Funcs (X,INT)) \ (b,0)),g) by AOFA_000:def_37;
 [q,(s := 1)] in TerminatingPrograms (A,(Funcs (X,INT)),((Funcs (X,INT)) \ (b,0)),g) by AOFA_000:def_36;
hence  [q,((s := 1) \; (for-do ((i := 2),(i leq n),(i += 1),(s *= i))))] in TerminatingPrograms (A,(Funcs (X,INT)),((Funcs (X,INT)) \ (b,0)),g) by A9, AOFA_000:def_35; ::_thesis: verum
end;

theorem :: AOFA_I00:57
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for n, s, i being Variable of g st ex d being Function st
( d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) holds
for q being Element of Funcs (X,INT)
for N being Nat st N = q . n holds
(g . (q,((s := 1) \; (for-do ((i := 2),(i leq n),(i += 1),(s *= i)))))) . s = N !
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for n, s, i being Variable of g st ex d being Function st
( d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) holds
for q being Element of Funcs (X,INT)
for N being Nat st N = q . n holds
(g . (q,((s := 1) \; (for-do ((i := 2),(i leq n),(i += 1),(s *= i)))))) . s = N !
let X be non empty countable set ; ::_thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for n, s, i being Variable of g st ex d being Function st
( d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) holds
for q being Element of Funcs (X,INT)
for N being Nat st N = q . n holds
(g . (q,((s := 1) \; (for-do ((i := 2),(i leq n),(i += 1),(s *= i)))))) . s = N !
let b be Element of X; ::_thesis: for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for n, s, i being Variable of g st ex d being Function st
( d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) holds
for q being Element of Funcs (X,INT)
for N being Nat st N = q . n holds
(g . (q,((s := 1) \; (for-do ((i := 2),(i leq n),(i += 1),(s *= i)))))) . s = N !
let g be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); ::_thesis: for n, s, i being Variable of g st ex d being Function st
( d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) holds
for q being Element of Funcs (X,INT)
for N being Nat st N = q . n holds
(g . (q,((s := 1) \; (for-do ((i := 2),(i leq n),(i += 1),(s *= i)))))) . s = N !
set f = g;
let n, s, i be Variable of g; ::_thesis: ( ex d being Function st
( d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) implies for q being Element of Funcs (X,INT)
for N being Nat st N = q . n holds
(g . (q,((s := 1) \; (for-do ((i := 2),(i leq n),(i += 1),(s *= i)))))) . s = N ! )
given d being Function such that A1: d . n = 1 and
A2: d . s = 2 and
A3: d . i = 3 and
A4: d . b = 4 ; ::_thesis: for q being Element of Funcs (X,INT)
for N being Nat st N = q . n holds
(g . (q,((s := 1) \; (for-do ((i := 2),(i leq n),(i += 1),(s *= i)))))) . s = N !
A5: ( n <> i & n <> b & i <> b ) by A1, A3, A4;
set S = Funcs (X,INT);
let q be Element of Funcs (X,INT); ::_thesis: for N being Nat st N = q . n holds
(g . (q,((s := 1) \; (for-do ((i := 2),(i leq n),(i += 1),(s *= i)))))) . s = N !
reconsider q1 = g . (q,(s := 1)) as Element of Funcs (X,INT) ;
defpred S1[ Element of Funcs (X,INT)] means  ex K being Nat st
( K = ($1 . i) - 1 & $1 . s = K ! );
reconsider a = . (2,A,g) as INT-Expression of A,g ;
reconsider q2 = g . (q1,(i := 2)) as Element of Funcs (X,INT) ;
A6: q2 . i = 2 by Th25;
A7: s <> i by A2, A3;
then  q2 . s = q1 . s by Th25;
then A8: S1[g . (q1,(i := a))] by A6, Th25, NEWTON:13;
set I = s *= i;
A9: a . q1 = 2 by FUNCOP_1:7;
A10: s <> b by A2, A4;
A11: for q being Element of Funcs (X,INT) st S1[q] holds
( S1[g . (q,((s *= i) \; (i += 1)))] & S1[g . (q,(i leq n))] )
proof
let q be Element of Funcs (X,INT); ::_thesis: ( S1[q] implies ( S1[g . (q,((s *= i) \; (i += 1)))] & S1[g . (q,(i leq n))] ) )
given Ki being Nat such that A12: Ki = (q . i) - 1 and
A13: q . s = Ki ! ; ::_thesis: ( S1[g . (q,((s *= i) \; (i += 1)))] & S1[g . (q,(i leq n))] )
reconsider q3 = g . (q,(s *= i)) as Element of Funcs (X,INT) ;
reconsider q4 = g . (q3,(i += 1)) as Element of Funcs (X,INT) ;
A14: q3 . s = (q . s) * (q . i) by Th33;
 q4 . s = q3 . s by A7, Th28;
then A15: q4 . s = (Ki + 1) ! by A12, A13, A14, NEWTON:15;
A16: q4 = g . (q,((s *= i) \; (i += 1))) by AOFA_000:def_29;
 q4 . i = (q3 . i) + 1 by Th28;
then  Ki + 1 = (q4 . i) - 1 by A7, A12, Th33;
hence  S1[g . (q,((s *= i) \; (i += 1)))] by A16, A15; ::_thesis: S1[g . (q,(i leq n))]
reconsider q9 = g . (q,(i leq n)) as Element of Funcs (X,INT) ;
A17: q9 . i = q . i by A5, Th35;
 q9 . s = q . s by A10, Th35;
hence  S1[g . (q,(i leq n))] by A12, A13, A17; ::_thesis: verum
end;
reconsider F = for-do ((i := a),(i leq n),(i += 1),(s *= i)) as Element of A ;
let N be Nat; ::_thesis: ( N = q . n implies (g . (q,((s := 1) \; (for-do ((i := 2),(i leq n),(i += 1),(s *= i)))))) . s = N ! )
assume A18: N = q . n ; ::_thesis: (g . (q,((s := 1) \; (for-do ((i := 2),(i leq n),(i += 1),(s *= i)))))) . s = N !
A19: F = for-do ((i := a),(i leq n),(i += 1),(s *= i)) ;
A20: s <> n by A1, A2;
then A21: q1 . n = q . n by Th25;
A22: for q being Element of Funcs (X,INT) st S1[q] holds
( (g . (q,(s *= i))) . i = q . i & (g . (q,(s *= i))) . n = q . n ) by A20, A7, Th32;
A23: ( S1[g . (q1,F)] & ( a . q1 <= q1 . n implies (g . (q1,F)) . i = (q1 . n) + 1 ) & ( a . q1 > q1 . n implies (g . (q1,F)) . i = a . q1 ) & (g . (q1,F)) . n = q1 . n ) from AOFA_I00:sch_1(A19, A8, A11, A22, A5);
thus (g . (q,((s := 1) \; (for-do ((i := 2),(i leq n),(i += 1),(s *= i)))))) . s = (g . (q1,F)) . s by AOFA_000:def_29
.= N ! by A18, A21, A23, A9, NAT_1:26, NEWTON:12, NEWTON:13 ; ::_thesis: verum
end;

theorem :: AOFA_I00:58
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, n, s, i being Variable of g st ex d being Function st
( d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) holds
(s := 1) \; (for-do ((i := 1),(i leq n),(i += 1),(s *= x))) is_terminating_wrt g
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, n, s, i being Variable of g st ex d being Function st
( d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) holds
(s := 1) \; (for-do ((i := 1),(i leq n),(i += 1),(s *= x))) is_terminating_wrt g
let X be non empty countable set ; ::_thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, n, s, i being Variable of g st ex d being Function st
( d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) holds
(s := 1) \; (for-do ((i := 1),(i leq n),(i += 1),(s *= x))) is_terminating_wrt g
let b be Element of X; ::_thesis: for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, n, s, i being Variable of g st ex d being Function st
( d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) holds
(s := 1) \; (for-do ((i := 1),(i leq n),(i += 1),(s *= x))) is_terminating_wrt g
let g be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); ::_thesis: for x, n, s, i being Variable of g st ex d being Function st
( d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) holds
(s := 1) \; (for-do ((i := 1),(i leq n),(i += 1),(s *= x))) is_terminating_wrt g
set S = Funcs (X,INT);
set T = (Funcs (X,INT)) \ (b,0);
let x, n, s, i be Variable of g; ::_thesis: ( ex d being Function st
( d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) implies (s := 1) \; (for-do ((i := 1),(i leq n),(i += 1),(s *= x))) is_terminating_wrt g )
given d being Function such that A1: d . n = 1 and
A2: d . s = 2 and
A3: d . i = 3 and
A4: d . b = 4 ; ::_thesis: (s := 1) \; (for-do ((i := 1),(i leq n),(i += 1),(s *= x))) is_terminating_wrt g
A5: s <> n by A1, A2;
 s <> i by A2, A3;
then A6: for q being Element of Funcs (X,INT) holds
( (g . (q,(s *= x))) . n = q . n & (g . (q,(s *= x))) . i = q . i ) by A5, Th33;
A7: n <> b by A1, A4;
A8: b <> i by A3, A4;
let q be Element of Funcs (X,INT); :: according to AOFA_000:def_37 ::_thesis: [q,((s := 1) \; (for-do ((i := 1),(i leq n),(i += 1),(s *= x))))] in TerminatingPrograms (A,(Funcs (X,INT)),((Funcs (X,INT)) \ (b,0)),g)
set t = . (1,A,g);
 i <> n by A1, A3;
then  ex d9 being Function st
( d9 . b = 0 & d9 . n = 1 & d9 . i = 2 ) by A8, A7, Th1;
then  for-do ((i := (. (1,A,g))),(i leq n),(i += 1),(s *= x)) is_terminating_wrt g by A6, Th54, AOFA_000:104;
then A9: [(g . (q,(s := 1))),(for-do ((i := (. (1,A,g))),(i leq n),(i += 1),(s *= x)))] in TerminatingPrograms (A,(Funcs (X,INT)),((Funcs (X,INT)) \ (b,0)),g) by AOFA_000:def_37;
 [q,(s := 1)] in TerminatingPrograms (A,(Funcs (X,INT)),((Funcs (X,INT)) \ (b,0)),g) by AOFA_000:def_36;
hence  [q,((s := 1) \; (for-do ((i := 1),(i leq n),(i += 1),(s *= x))))] in TerminatingPrograms (A,(Funcs (X,INT)),((Funcs (X,INT)) \ (b,0)),g) by A9, AOFA_000:def_35; ::_thesis: verum
end;

theorem :: AOFA_I00:59
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, n, s, i being Variable of g st ex d being Function st
( d . x = 0 & d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) holds
for q being Element of Funcs (X,INT)
for N being Nat st N = q . n holds
(g . (q,((s := 1) \; (for-do ((i := 1),(i leq n),(i += 1),(s *= x)))))) . s = (q . x) |^ N
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, n, s, i being Variable of g st ex d being Function st
( d . x = 0 & d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) holds
for q being Element of Funcs (X,INT)
for N being Nat st N = q . n holds
(g . (q,((s := 1) \; (for-do ((i := 1),(i leq n),(i += 1),(s *= x)))))) . s = (q . x) |^ N
let X be non empty countable set ; ::_thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, n, s, i being Variable of g st ex d being Function st
( d . x = 0 & d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) holds
for q being Element of Funcs (X,INT)
for N being Nat st N = q . n holds
(g . (q,((s := 1) \; (for-do ((i := 1),(i leq n),(i += 1),(s *= x)))))) . s = (q . x) |^ N
let b be Element of X; ::_thesis: for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, n, s, i being Variable of g st ex d being Function st
( d . x = 0 & d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) holds
for q being Element of Funcs (X,INT)
for N being Nat st N = q . n holds
(g . (q,((s := 1) \; (for-do ((i := 1),(i leq n),(i += 1),(s *= x)))))) . s = (q . x) |^ N
let g be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); ::_thesis: for x, n, s, i being Variable of g st ex d being Function st
( d . x = 0 & d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) holds
for q being Element of Funcs (X,INT)
for N being Nat st N = q . n holds
(g . (q,((s := 1) \; (for-do ((i := 1),(i leq n),(i += 1),(s *= x)))))) . s = (q . x) |^ N
set f = g;
let x, n, s, i be Variable of g; ::_thesis: ( ex d being Function st
( d . x = 0 & d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) implies for q being Element of Funcs (X,INT)
for N being Nat st N = q . n holds
(g . (q,((s := 1) \; (for-do ((i := 1),(i leq n),(i += 1),(s *= x)))))) . s = (q . x) |^ N )
given d being Function such that A1: d . x = 0 and
A2: d . n = 1 and
A3: d . s = 2 and
A4: d . i = 3 and
A5: d . b = 4 ; ::_thesis: for q being Element of Funcs (X,INT)
for N being Nat st N = q . n holds
(g . (q,((s := 1) \; (for-do ((i := 1),(i leq n),(i += 1),(s *= x)))))) . s = (q . x) |^ N
A6: ( n <> i & n <> b & i <> b ) by A2, A4, A5;
set S = Funcs (X,INT);
let q be Element of Funcs (X,INT); ::_thesis: for N being Nat st N = q . n holds
(g . (q,((s := 1) \; (for-do ((i := 1),(i leq n),(i += 1),(s *= x)))))) . s = (q . x) |^ N
reconsider q1 = g . (q,(s := 1)) as Element of Funcs (X,INT) ;
reconsider q2 = g . (q1,(i := 1)) as Element of Funcs (X,INT) ;
A7: s <> i by A3, A4;
then A8: q2 . s = q1 . s by Th25;
defpred S1[ Element of Funcs (X,INT)] means  ex K being Nat st
( K = ($1 . i) - 1 & $1 . s = (q . x) |^ K & $1 . x = q . x );
set I = s *= x;
set q0 = q;
A9: (q . x) |^ 0 = 1 by NEWTON:4;
A10: s <> n by A2, A3;
then A11: for q being Element of Funcs (X,INT) st S1[q] holds
( (g . (q,(s *= x))) . i = q . i & (g . (q,(s *= x))) . n = q . n ) by A7, Th32;
A12: s <> b by A3, A5;
A13: for q being Element of Funcs (X,INT) st S1[q] holds
( S1[g . (q,((s *= x) \; (i += 1)))] & S1[g . (q,(i leq n))] )
proof
let q be Element of Funcs (X,INT); ::_thesis: ( S1[q] implies ( S1[g . (q,((s *= x) \; (i += 1)))] & S1[g . (q,(i leq n))] ) )
given Ki being Nat such that A14: Ki = (q . i) - 1 and
A15: q . s = (q . x) |^ Ki and
A16: q . x = q . x ; ::_thesis: ( S1[g . (q,((s *= x) \; (i += 1)))] & S1[g . (q,(i leq n))] )
reconsider q3 = g . (q,(s *= x)) as Element of Funcs (X,INT) ;
reconsider q4 = g . (q3,(i += 1)) as Element of Funcs (X,INT) ;
A17: q3 . s = (q . s) * (q . x) by Th33;
 q4 . s = q3 . s by A7, Th28;
then A18: q4 . s = (q . x) |^ (Ki + 1) by A15, A16, A17, NEWTON:6;
A19: q4 = g . (q,((s *= x) \; (i += 1))) by AOFA_000:def_29;
A20: q3 . x = q . x by A1, A3, Th33;
 q4 . i = (q3 . i) + 1 by Th28;
then  Ki + 1 = (q4 . i) - 1 by A7, A14, Th33;
hence  S1[g . (q,((s *= x) \; (i += 1)))] by A1, A4, A16, A19, A20, A18, Th28; ::_thesis: S1[g . (q,(i leq n))]
reconsider q9 = g . (q,(i leq n)) as Element of Funcs (X,INT) ;
A21: q9 . s = q . s by A12, Th35;
A22: q9 . i = q . i by A6, Th35;
 q9 . x = q . x by A1, A5, Th35;
hence  S1[g . (q,(i leq n))] by A14, A15, A16, A21, A22; ::_thesis: verum
end;
reconsider a = . (1,A,g) as INT-Expression of A,g ;
reconsider F = for-do ((i := a),(i leq n),(i += 1),(s *= x)) as Element of A ;
A23: F = for-do ((i := a),(i leq n),(i += 1),(s *= x)) ;
A24: q2 . i = 1 by Th25;
A25: q2 . x = q1 . x by A1, A4, Th25;
A26: q1 . s = 1 by Th25;
 q1 . x = q . x by A1, A3, Th25;
then A27: S1[g . (q1,(i := a))] by A26, A8, A24, A25, A9;
A28: ( S1[g . (q1,F)] & ( a . q1 <= q1 . n implies (g . (q1,F)) . i = (q1 . n) + 1 ) & ( a . q1 > q1 . n implies (g . (q1,F)) . i = a . q1 ) & (g . (q1,F)) . n = q1 . n ) from AOFA_I00:sch_1(A23, A27, A13, A11, A6);
A29: ((q . n) + 1) - 1 = q . n ;
A30: q1 . n = q . n by A10, Th25;
let N be Nat; ::_thesis: ( N = q . n implies (g . (q,((s := 1) \; (for-do ((i := 1),(i leq n),(i += 1),(s *= x)))))) . s = (q . x) |^ N )
assume A31: N = q . n ; ::_thesis: (g . (q,((s := 1) \; (for-do ((i := 1),(i leq n),(i += 1),(s *= x)))))) . s = (q . x) |^ N
A32: ( N = 0 or N >= 1 ) by NAT_1:25;
thus (g . (q,((s := 1) \; (for-do ((i := 1),(i leq n),(i += 1),(s *= x)))))) . s = (g . (q1,F)) . s by AOFA_000:def_29
.= (q . x) |^ N by A31, A30, A29, A28, A32, FUNCOP_1:7 ; ::_thesis: verum
end;

theorem :: AOFA_I00:60
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for n, x, y, z, i being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . x = 2 & d . y = 3 & d . z = 4 & d . i = 5 ) holds
((x := 0) \; (y := 1)) \; (for-do ((i := 1),(i leq n),(i += 1),(((z := x) \; (x := y)) \; (y += z)))) is_terminating_wrt g
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for n, x, y, z, i being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . x = 2 & d . y = 3 & d . z = 4 & d . i = 5 ) holds
((x := 0) \; (y := 1)) \; (for-do ((i := 1),(i leq n),(i += 1),(((z := x) \; (x := y)) \; (y += z)))) is_terminating_wrt g
let X be non empty countable set ; ::_thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for n, x, y, z, i being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . x = 2 & d . y = 3 & d . z = 4 & d . i = 5 ) holds
((x := 0) \; (y := 1)) \; (for-do ((i := 1),(i leq n),(i += 1),(((z := x) \; (x := y)) \; (y += z)))) is_terminating_wrt g
let b be Element of X; ::_thesis: for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for n, x, y, z, i being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . x = 2 & d . y = 3 & d . z = 4 & d . i = 5 ) holds
((x := 0) \; (y := 1)) \; (for-do ((i := 1),(i leq n),(i += 1),(((z := x) \; (x := y)) \; (y += z)))) is_terminating_wrt g
let g be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); ::_thesis: for n, x, y, z, i being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . x = 2 & d . y = 3 & d . z = 4 & d . i = 5 ) holds
((x := 0) \; (y := 1)) \; (for-do ((i := 1),(i leq n),(i += 1),(((z := x) \; (x := y)) \; (y += z)))) is_terminating_wrt g
set S = Funcs (X,INT);
set T = (Funcs (X,INT)) \ (b,0);
let n, x, y, z, i be Variable of g; ::_thesis: ( ex d being Function st
( d . b = 0 & d . n = 1 & d . x = 2 & d . y = 3 & d . z = 4 & d . i = 5 ) implies ((x := 0) \; (y := 1)) \; (for-do ((i := 1),(i leq n),(i += 1),(((z := x) \; (x := y)) \; (y += z)))) is_terminating_wrt g )
given d being Function such that A1: d . b = 0 and
A2: d . n = 1 and
A3: d . x = 2 and
A4: d . y = 3 and
A5: d . z = 4 and
A6: d . i = 5 ; ::_thesis: ((x := 0) \; (y := 1)) \; (for-do ((i := 1),(i leq n),(i += 1),(((z := x) \; (x := y)) \; (y += z)))) is_terminating_wrt g
A7: i <> y by A4, A6;
A8: n <> z by A2, A5;
A9: n <> y by A2, A4;
A10: n <> x by A2, A3;
A11: i <> z by A5, A6;
set I = ((z := x) \; (x := y)) \; (y += z);
set I1 = z := x;
set I2 = x := y;
set I3 = y += z;
A12: i <> x by A3, A6;
A13: for q being Element of Funcs (X,INT) holds
( (g . (q,(((z := x) \; (x := y)) \; (y += z)))) . n = q . n & (g . (q,(((z := x) \; (x := y)) \; (y += z)))) . i = q . i )
proof
let q be Element of Funcs (X,INT); ::_thesis: ( (g . (q,(((z := x) \; (x := y)) \; (y += z)))) . n = q . n & (g . (q,(((z := x) \; (x := y)) \; (y += z)))) . i = q . i )
thus (g . (q,(((z := x) \; (x := y)) \; (y += z)))) . n = (g . ((g . (q,((z := x) \; (x := y)))),(y += z))) . n by AOFA_000:def_29
.= (g . (q,((z := x) \; (x := y)))) . n by A9, Th30
.= (g . ((g . (q,(z := x))),(x := y))) . n by AOFA_000:def_29
.= (g . (q,(z := x))) . n by A10, Th27
.= q . n by A8, Th27 ; ::_thesis: (g . (q,(((z := x) \; (x := y)) \; (y += z)))) . i = q . i
thus (g . (q,(((z := x) \; (x := y)) \; (y += z)))) . i = (g . ((g . (q,((z := x) \; (x := y)))),(y += z))) . i by AOFA_000:def_29
.= (g . (q,((z := x) \; (x := y)))) . i by A7, Th30
.= (g . ((g . (q,(z := x))),(x := y))) . i by AOFA_000:def_29
.= (g . (q,(z := x))) . i by A12, Th27
.= q . i by A11, Th27 ; ::_thesis: verum
end;
let s be Element of Funcs (X,INT); :: according to AOFA_000:def_37 ::_thesis: [s,(((x := 0) \; (y := 1)) \; (for-do ((i := 1),(i leq n),(i += 1),(((z := x) \; (x := y)) \; (y += z)))))] in TerminatingPrograms (A,(Funcs (X,INT)),((Funcs (X,INT)) \ (b,0)),g)
set s2 = g . (s,((x := 0) \; (y := 1)));
 i <> n by A2, A6;
then  ex d9 being Function st
( d9 . b = 0 & d9 . n = 1 & d9 . i = 2 ) by A1, A2, A6, Th1;
then  for-do ((i := 1),(i leq n),(i += 1),(((z := x) \; (x := y)) \; (y += z))) is_terminating_wrt g by A13, Th54, AOFA_000:104;
then A14: [(g . (s,((x := 0) \; (y := 1)))),(for-do ((i := 1),(i leq n),(i += 1),(((z := x) \; (x := y)) \; (y += z))))] in TerminatingPrograms (A,(Funcs (X,INT)),((Funcs (X,INT)) \ (b,0)),g) by AOFA_000:def_37;
 [s,((x := 0) \; (y := 1))] in TerminatingPrograms (A,(Funcs (X,INT)),((Funcs (X,INT)) \ (b,0)),g) by AOFA_000:def_36;
hence  [s,(((x := 0) \; (y := 1)) \; (for-do ((i := 1),(i leq n),(i += 1),(((z := x) \; (x := y)) \; (y += z)))))] in TerminatingPrograms (A,(Funcs (X,INT)),((Funcs (X,INT)) \ (b,0)),g) by A14, AOFA_000:def_35; ::_thesis: verum
end;

theorem :: AOFA_I00:61
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for n, x, y, z, i being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . x = 2 & d . y = 3 & d . z = 4 & d . i = 5 ) holds
for s being Element of Funcs (X,INT)
for N being Element of NAT st N = s . n holds
(g . (s,(((x := 0) \; (y := 1)) \; (for-do ((i := 1),(i leq n),(i += 1),(((z := x) \; (x := y)) \; (y += z))))))) . x = Fib N
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for n, x, y, z, i being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . x = 2 & d . y = 3 & d . z = 4 & d . i = 5 ) holds
for s being Element of Funcs (X,INT)
for N being Element of NAT st N = s . n holds
(g . (s,(((x := 0) \; (y := 1)) \; (for-do ((i := 1),(i leq n),(i += 1),(((z := x) \; (x := y)) \; (y += z))))))) . x = Fib N
let X be non empty countable set ; ::_thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for n, x, y, z, i being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . x = 2 & d . y = 3 & d . z = 4 & d . i = 5 ) holds
for s being Element of Funcs (X,INT)
for N being Element of NAT st N = s . n holds
(g . (s,(((x := 0) \; (y := 1)) \; (for-do ((i := 1),(i leq n),(i += 1),(((z := x) \; (x := y)) \; (y += z))))))) . x = Fib N
let b be Element of X; ::_thesis: for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for n, x, y, z, i being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . x = 2 & d . y = 3 & d . z = 4 & d . i = 5 ) holds
for s being Element of Funcs (X,INT)
for N being Element of NAT st N = s . n holds
(g . (s,(((x := 0) \; (y := 1)) \; (for-do ((i := 1),(i leq n),(i += 1),(((z := x) \; (x := y)) \; (y += z))))))) . x = Fib N
let g be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); ::_thesis: for n, x, y, z, i being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . x = 2 & d . y = 3 & d . z = 4 & d . i = 5 ) holds
for s being Element of Funcs (X,INT)
for N being Element of NAT st N = s . n holds
(g . (s,(((x := 0) \; (y := 1)) \; (for-do ((i := 1),(i leq n),(i += 1),(((z := x) \; (x := y)) \; (y += z))))))) . x = Fib N
A1: 0 + 1 = 1 ;
set S = Funcs (X,INT);
let n, x, y, z, i be Variable of g; ::_thesis: ( ex d being Function st
( d . b = 0 & d . n = 1 & d . x = 2 & d . y = 3 & d . z = 4 & d . i = 5 ) implies for s being Element of Funcs (X,INT)
for N being Element of NAT st N = s . n holds
(g . (s,(((x := 0) \; (y := 1)) \; (for-do ((i := 1),(i leq n),(i += 1),(((z := x) \; (x := y)) \; (y += z))))))) . x = Fib N )
given d being Function such that A2: d . b = 0 and
A3: d . n = 1 and
A4: d . x = 2 and
A5: d . y = 3 and
A6: d . z = 4 and
A7: d . i = 5 ; ::_thesis: for s being Element of Funcs (X,INT)
for N being Element of NAT st N = s . n holds
(g . (s,(((x := 0) \; (y := 1)) \; (for-do ((i := 1),(i leq n),(i += 1),(((z := x) \; (x := y)) \; (y += z))))))) . x = Fib N
A8: n <> x by A3, A4;
A9: y <> z by A5, A6;
A10: x <> z by A4, A6;
A11: i <> z by A6, A7;
A12: n <> y by A3, A5;
let s be Element of Funcs (X,INT); ::_thesis: for N being Element of NAT st N = s . n holds
(g . (s,(((x := 0) \; (y := 1)) \; (for-do ((i := 1),(i leq n),(i += 1),(((z := x) \; (x := y)) \; (y += z))))))) . x = Fib N
reconsider s1 = g . (s,(x := 0)) as Element of Funcs (X,INT) ;
reconsider s2 = g . (s1,(y := 1)) as Element of Funcs (X,INT) ;
reconsider s4 = g . (s2,(i := 1)) as Element of Funcs (X,INT) ;
A13: s4 . i = 1 by Th25;
A14: i <> y by A5, A7;
then A15: s4 . y = s2 . y by Th25;
set I = ((z := x) \; (x := y)) \; (y += z);
A16: s2 . y = 1 by Th25;
A17: i <> x by A4, A7;
then A18: s4 . x = s2 . x by Th25;
defpred S1[ Element of Funcs (X,INT)] means  ex N being Element of NAT st
( N = ($1 . i) - 1 & $1 . x = Fib N & $1 . y = Fib (N + 1) );
reconsider a = . (1,A,g) as INT-Expression of A,g ;
A19: s1 . x = 0 by Th25;
A20: x <> y by A4, A5;
then  s2 . x = s1 . x by Th25;
then A21: S1[g . (s2,(i := a))] by A19, A16, A15, A18, A13, A1, PRE_FF:1;
A22: n <> z by A3, A6;
A23: now__::_thesis:_for_s_being_Element_of_Funcs_(X,INT)_holds_
(_(g_._(s,(((z_:=_x)_\;_(x_:=_y))_\;_(y_+=_z))))_._i_=_s_._i_&_(g_._(s,(((z_:=_x)_\;_(x_:=_y))_\;_(y_+=_z))))_._n_=_s_._n_&_(g_._(s,(((z_:=_x)_\;_(x_:=_y))_\;_(y_+=_z))))_._x_=_s_._y_&_(g_._(s,(((z_:=_x)_\;_(x_:=_y))_\;_(y_+=_z))))_._y_=_(s_._x)_+_(s_._y)_)
let s be Element of Funcs (X,INT); ::_thesis: ( (g . (s,(((z := x) \; (x := y)) \; (y += z)))) . i = s . i & (g . (s,(((z := x) \; (x := y)) \; (y += z)))) . n = s . n & (g . (s,(((z := x) \; (x := y)) \; (y += z)))) . x = s . y & (g . (s,(((z := x) \; (x := y)) \; (y += z)))) . y = (s . x) + (s . y) )
reconsider s1 = g . (s,(z := x)) as Element of Funcs (X,INT) ;
reconsider s2 = g . (s1,(x := y)) as Element of Funcs (X,INT) ;
reconsider s3 = g . (s2,(y += z)) as Element of Funcs (X,INT) ;
A24: g . (s,(((z := x) \; (x := y)) \; (y += z))) = g . ((g . (s,((z := x) \; (x := y)))),(y += z)) by AOFA_000:def_29
.= s3 by AOFA_000:def_29 ;
A25: s1 . z = s . x by Th27;
A26: s2 . n = s1 . n by A8, Th27;
A27: s1 . i = s . i by A11, Th27;
A28: s2 . z = s1 . z by A10, Th27;
A29: s2 . y = s1 . y by A20, Th27;
A30: s2 . i = s1 . i by A17, Th27;
A31: s1 . y = s . y by A9, Th27;
A32: s2 . x = s1 . y by Th27;
 s1 . n = s . n by A22, Th27;
hence  ( (g . (s,(((z := x) \; (x := y)) \; (y += z)))) . i = s . i & (g . (s,(((z := x) \; (x := y)) \; (y += z)))) . n = s . n & (g . (s,(((z := x) \; (x := y)) \; (y += z)))) . x = s . y & (g . (s,(((z := x) \; (x := y)) \; (y += z)))) . y = (s . x) + (s . y) ) by A12, A14, A20, A31, A25, A27, A26, A32, A29, A28, A30, A24, Th30; ::_thesis: verum
end;
A33: for s being Element of Funcs (X,INT) st S1[s] holds
( S1[g . (s,((((z := x) \; (x := y)) \; (y += z)) \; (i += 1)))] & S1[g . (s,(i leq n))] )
proof
let s be Element of Funcs (X,INT); ::_thesis: ( S1[s] implies ( S1[g . (s,((((z := x) \; (x := y)) \; (y += z)) \; (i += 1)))] & S1[g . (s,(i leq n))] ) )
given N being Element of NAT  such that A34: N = (s . i) - 1 and
A35: s . x = Fib N and
A36: s . y = Fib (N + 1) ; ::_thesis: ( S1[g . (s,((((z := x) \; (x := y)) \; (y += z)) \; (i += 1)))] & S1[g . (s,(i leq n))] )
reconsider s1 = g . (s,(((z := x) \; (x := y)) \; (y += z))) as Element of Funcs (X,INT) ;
reconsider s2 = g . (s1,(i += 1)) as Element of Funcs (X,INT) ;
A37: s1 . x = s . y by A23;
A38: s2 . i = (s1 . i) + 1 by Th28;
A39: s1 . y = (s . x) + (s . y) by A23;
A40: s2 . y = s1 . y by A14, Th28;
thus  S1[g . (s,((((z := x) \; (x := y)) \; (y += z)) \; (i += 1)))] ::_thesis: S1[g . (s,(i leq n))]
proof
take  N + 1 ; ::_thesis: ( N + 1 = ((g . (s,((((z := x) \; (x := y)) \; (y += z)) \; (i += 1)))) . i) - 1 & (g . (s,((((z := x) \; (x := y)) \; (y += z)) \; (i += 1)))) . x = Fib (N + 1) & (g . (s,((((z := x) \; (x := y)) \; (y += z)) \; (i += 1)))) . y = Fib ((N + 1) + 1) )
 g . (s,((((z := x) \; (x := y)) \; (y += z)) \; (i += 1))) = s2 by AOFA_000:def_29;
hence  ( N + 1 = ((g . (s,((((z := x) \; (x := y)) \; (y += z)) \; (i += 1)))) . i) - 1 & (g . (s,((((z := x) \; (x := y)) \; (y += z)) \; (i += 1)))) . x = Fib (N + 1) & (g . (s,((((z := x) \; (x := y)) \; (y += z)) \; (i += 1)))) . y = Fib ((N + 1) + 1) ) by A17, A23, A34, A35, A36, A37, A39, A40, A38, Th28, PRE_FF:1; ::_thesis: verum
end;
take  N ; ::_thesis: ( N = ((g . (s,(i leq n))) . i) - 1 & (g . (s,(i leq n))) . x = Fib N & (g . (s,(i leq n))) . y = Fib (N + 1) )
thus  ( N = ((g . (s,(i leq n))) . i) - 1 & (g . (s,(i leq n))) . x = Fib N & (g . (s,(i leq n))) . y = Fib (N + 1) ) by A2, A4, A5, A7, A34, A35, A36, Th35; ::_thesis: verum
end;
reconsider F = for-do ((i := a),(i leq n),(i += 1),(((z := x) \; (x := y)) \; (y += z))) as Element of A ;
reconsider s3 = g . (s2,(for-do ((i := 1),(i leq n),(i += 1),(((z := x) \; (x := y)) \; (y += z))))) as Element of Funcs (X,INT) ;
let N be Element of NAT ; ::_thesis: ( N = s . n implies (g . (s,(((x := 0) \; (y := 1)) \; (for-do ((i := 1),(i leq n),(i += 1),(((z := x) \; (x := y)) \; (y += z))))))) . x = Fib N )
assume A41: N = s . n ; ::_thesis: (g . (s,(((x := 0) \; (y := 1)) \; (for-do ((i := 1),(i leq n),(i += 1),(((z := x) \; (x := y)) \; (y += z))))))) . x = Fib N
A42: F = for-do ((i := a),(i leq n),(i += 1),(((z := x) \; (x := y)) \; (y += z))) ;
 g . (s,((x := 0) \; (y := 1))) = s2 by AOFA_000:def_29;
then A43: g . (s,(((x := 0) \; (y := 1)) \; F)) = s3 by AOFA_000:def_29;
A44: ( ( 0 <= N & N <= 0 ) or N >= 0 + 1 ) by NAT_1:13;
A45: ( n <> i & n <> b & i <> b ) by A2, A3, A7;
A46: a . s2 = 1 by FUNCOP_1:7;
A47: for s being Element of Funcs (X,INT) st S1[s] holds
( (g . (s,(((z := x) \; (x := y)) \; (y += z)))) . i = s . i & (g . (s,(((z := x) \; (x := y)) \; (y += z)))) . n = s . n ) by A23;
A48: ( S1[g . (s2,F)] & ( a . s2 <= s2 . n implies (g . (s2,F)) . i = (s2 . n) + 1 ) & ( a . s2 > s2 . n implies (g . (s2,F)) . i = a . s2 ) & (g . (s2,F)) . n = s2 . n ) from AOFA_I00:sch_1(A42, A21, A33, A47, A45);
 s2 . n = s1 . n by A12, Th25;
hence  (g . (s,(((x := 0) \; (y := 1)) \; (for-do ((i := 1),(i leq n),(i += 1),(((z := x) \; (x := y)) \; (y += z))))))) . x = Fib N by A41, A43, A8, A48, A44, A46, Th25; ::_thesis: verum
end;

Lm1:  for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
for s being Element of Funcs (X,INT) holds
( (g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))) . y = (s . x) mod (s . y) & ( for n, m being Element of NAT holds
not ( m = s . y & ( s in (Funcs (X,INT)) \ (b,0) implies m > 0 ) & ( m > 0 implies s in (Funcs (X,INT)) \ (b,0) ) & not g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0),s ) ) )
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
for s being Element of Funcs (X,INT) holds
( (g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))) . y = (s . x) mod (s . y) & ( for n, m being Element of NAT holds
not ( m = s . y & ( s in (Funcs (X,INT)) \ (b,0) implies m > 0 ) & ( m > 0 implies s in (Funcs (X,INT)) \ (b,0) ) & not g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0),s ) ) )
let X be non empty countable set ; ::_thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
for s being Element of Funcs (X,INT) holds
( (g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))) . y = (s . x) mod (s . y) & ( for n, m being Element of NAT holds
not ( m = s . y & ( s in (Funcs (X,INT)) \ (b,0) implies m > 0 ) & ( m > 0 implies s in (Funcs (X,INT)) \ (b,0) ) & not g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0),s ) ) )
let b be Element of X; ::_thesis: for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
for s being Element of Funcs (X,INT) holds
( (g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))) . y = (s . x) mod (s . y) & ( for n, m being Element of NAT holds
not ( m = s . y & ( s in (Funcs (X,INT)) \ (b,0) implies m > 0 ) & ( m > 0 implies s in (Funcs (X,INT)) \ (b,0) ) & not g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0),s ) ) )
let g be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); ::_thesis: for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
for s being Element of Funcs (X,INT) holds
( (g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))) . y = (s . x) mod (s . y) & ( for n, m being Element of NAT holds
not ( m = s . y & ( s in (Funcs (X,INT)) \ (b,0) implies m > 0 ) & ( m > 0 implies s in (Funcs (X,INT)) \ (b,0) ) & not g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0),s ) ) )
set h = g;
set S = Funcs (X,INT);
set T = (Funcs (X,INT)) \ (b,0);
let x, y, z be Variable of g; ::_thesis: ( ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) implies for s being Element of Funcs (X,INT) holds
( (g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))) . y = (s . x) mod (s . y) & ( for n, m being Element of NAT holds
not ( m = s . y & ( s in (Funcs (X,INT)) \ (b,0) implies m > 0 ) & ( m > 0 implies s in (Funcs (X,INT)) \ (b,0) ) & not g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0),s ) ) ) )
given d being Function such that A1: d . b = 0 and
A2: d . x = 1 and
A3: d . y = 2 and
A4: d . z = 3 ; ::_thesis: for s being Element of Funcs (X,INT) holds
( (g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))) . y = (s . x) mod (s . y) & ( for n, m being Element of NAT holds
not ( m = s . y & ( s in (Funcs (X,INT)) \ (b,0) implies m > 0 ) & ( m > 0 implies s in (Funcs (X,INT)) \ (b,0) ) & not g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0),s ) ) )
A5: z <> x by A2, A4;
let s be Element of Funcs (X,INT); ::_thesis: ( (g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))) . y = (s . x) mod (s . y) & ( for n, m being Element of NAT holds
not ( m = s . y & ( s in (Funcs (X,INT)) \ (b,0) implies m > 0 ) & ( m > 0 implies s in (Funcs (X,INT)) \ (b,0) ) & not g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0),s ) ) )
set I = (((z := x) \; (z %= y)) \; (x := y)) \; (y := z);
A6: y <> z by A3, A4;
A7: x <> y by A2, A3;
A8: now__::_thesis:_for_s_being_Element_of_Funcs_(X,INT)_holds_
(_(g_._(s,((((z_:=_x)_\;_(z_%=_y))_\;_(x_:=_y))_\;_(y_:=_z))))_._x_=_s_._y_&_(g_._(s,((((z_:=_x)_\;_(z_%=_y))_\;_(x_:=_y))_\;_(y_:=_z))))_._y_=_(s_._x)_mod_(s_._y)_)
let s be Element of Funcs (X,INT); ::_thesis: ( (g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))) . y = (s . x) mod (s . y) )
reconsider s1 = g . (s,(z := x)) as Element of Funcs (X,INT) ;
reconsider s2 = g . (s1,(z %= y)) as Element of Funcs (X,INT) ;
reconsider s3 = g . (s2,(x := y)) as Element of Funcs (X,INT) ;
reconsider s4 = g . (s3,(y := z)) as Element of Funcs (X,INT) ;
A9: g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) = g . ((g . (s,(((z := x) \; (z %= y)) \; (x := y)))),(y := z)) by AOFA_000:def_29
.= g . ((g . ((g . (s,((z := x) \; (z %= y)))),(x := y))),(y := z)) by AOFA_000:def_29
.= s4 by AOFA_000:def_29 ;
A10: s1 . z = s . x by Th27;
A11: s2 . y = s1 . y by A6, Th44;
A12: s2 . z = (s1 . z) mod (s1 . y) by Th44;
A13: s3 . z = s2 . z by A5, Th27;
A14: s3 . x = s2 . y by Th27;
 s1 . y = s . y by A6, Th27;
hence  ( (g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))) . y = (s . x) mod (s . y) ) by A7, A9, A10, A11, A12, A14, A13, Th27; ::_thesis: verum
end;
hence  ( (g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))) . y = (s . x) mod (s . y) ) ; ::_thesis: for n, m being Element of NAT holds
not ( m = s . y & ( s in (Funcs (X,INT)) \ (b,0) implies m > 0 ) & ( m > 0 implies s in (Funcs (X,INT)) \ (b,0) ) & not g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0),s )
deffunc H1( Element of Funcs (X,INT)) ->  Element of NAT = In (($1 . y),NAT);
defpred S1[ Element of Funcs (X,INT)] means $1 . y > 0 ;
set C = y gt 0;
A15: for s being Element of Funcs (X,INT) st S1[s] holds
( ( S1[g . (s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0)))] implies g . (s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0))) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0))) in (Funcs (X,INT)) \ (b,0) implies S1[g . (s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0)))] ) & H1(g . (s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0)))) < H1(s) )
proof
let s be Element of Funcs (X,INT); ::_thesis: ( S1[s] implies ( ( S1[g . (s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0)))] implies g . (s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0))) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0))) in (Funcs (X,INT)) \ (b,0) implies S1[g . (s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0)))] ) & H1(g . (s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0)))) < H1(s) ) )
assume A16: s . y > 0 ; ::_thesis: ( ( S1[g . (s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0)))] implies g . (s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0))) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0))) in (Funcs (X,INT)) \ (b,0) implies S1[g . (s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0)))] ) & H1(g . (s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0)))) < H1(s) )
reconsider s9 = g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) as Element of Funcs (X,INT) ;
A17: s9 . y = (s . x) mod (s . y) by A8;
then A18: 0 <= s9 . y by A16, NEWTON:64;
reconsider s99 = g . (s9,(y gt 0)) as Element of Funcs (X,INT) ;
A19: g . (s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0))) = s99 by AOFA_000:def_29;
A20: ( s9 . y <= 0 implies s99 . b = 0 ) by Th38;
 ( s9 . y > 0 implies s99 . b = 1 ) by Th38;
hence  ( S1[g . (s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0)))] iff g . (s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0))) in (Funcs (X,INT)) \ (b,0) ) by A19, A20, Th2, Th38; ::_thesis: H1(g . (s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0)))) < H1(s)
 s99 . y = s9 . y by A1, A3, Th38;
then A21: H1(s99) = s9 . y by A18, FUNCT_7:def_1, INT_1:3;
A22: s9 . y < s . y by A16, A17, NEWTON:65;
then  H1(s) = s . y by A18, FUNCT_7:def_1, INT_1:3;
hence  H1(g . (s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0)))) < H1(s) by A22, A21, AOFA_000:def_29; ::_thesis: verum
end;
let n, m be Element of NAT ; ::_thesis: not ( m = s . y & ( s in (Funcs (X,INT)) \ (b,0) implies m > 0 ) & ( m > 0 implies s in (Funcs (X,INT)) \ (b,0) ) & not g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0),s )
assume A23: m = s . y ; ::_thesis: ( ( s in (Funcs (X,INT)) \ (b,0) & not m > 0 ) or ( m > 0 & not s in (Funcs (X,INT)) \ (b,0) ) or g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0),s )
assume  ( s in (Funcs (X,INT)) \ (b,0) iff m > 0 ) ; ::_thesis: g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0),s
then A24: ( s in (Funcs (X,INT)) \ (b,0) iff S1[s] ) by A23;
 g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0),s from AOFA_000:sch_3(A24, A15);
hence  g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0),s ; ::_thesis: verum
end;

theorem :: AOFA_I00:62
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
while ((y gt 0),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) is_terminating_wrt g, {  s where s is Element of Funcs (X,INT) : ( s . x > s . y & s . y >= 0 )  }
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
while ((y gt 0),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) is_terminating_wrt g, {  s where s is Element of Funcs (X,INT) : ( s . x > s . y & s . y >= 0 )  }
let X be non empty countable set ; ::_thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
while ((y gt 0),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) is_terminating_wrt g, {  s where s is Element of Funcs (X,INT) : ( s . x > s . y & s . y >= 0 )  }
let b be Element of X; ::_thesis: for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
while ((y gt 0),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) is_terminating_wrt g, {  s where s is Element of Funcs (X,INT) : ( s . x > s . y & s . y >= 0 )  }
let g be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); ::_thesis: for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
while ((y gt 0),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) is_terminating_wrt g, {  s where s is Element of Funcs (X,INT) : ( s . x > s . y & s . y >= 0 )  }
set S = Funcs (X,INT);
set T = (Funcs (X,INT)) \ (b,0);
let x, y, z be Variable of g; ::_thesis: ( ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) implies while ((y gt 0),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) is_terminating_wrt g, {  s where s is Element of Funcs (X,INT) : ( s . x > s . y & s . y >= 0 )  }  )
set P =  {  s where s is Element of Funcs (X,INT) : ( s . x > s . y & s . y >= 0 )  }  ;
given d being Function such that A1: d . b = 0 and
A2: d . x = 1 and
A3: d . y = 2 and
A4: d . z = 3 ; ::_thesis: while ((y gt 0),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) is_terminating_wrt g, {  s where s is Element of Funcs (X,INT) : ( s . x > s . y & s . y >= 0 )  }
set C = y gt 0;
set I = (((z := x) \; (z %= y)) \; (x := y)) \; (y := z);
A5:  {  s where s is Element of Funcs (X,INT) : ( s . x > s . y & s . y >= 0 )  }  is_invariant_wrt y gt 0,g
proof
let s be Element of Funcs (X,INT); :: according to AOFA_000:def_39 ::_thesis: ( not s in  {  s where s is Element of Funcs (X,INT) : ( s . x > s . y & s . y >= 0 )  }  or g . (s,(y gt 0)) in  {  s where s is Element of Funcs (X,INT) : ( s . x > s . y & s . y >= 0 )  }  )
set s1 = g . (s,(y gt 0));
assume  s in  {  s where s is Element of Funcs (X,INT) : ( s . x > s . y & s . y >= 0 )  }  ; ::_thesis: g . (s,(y gt 0)) in  {  s where s is Element of Funcs (X,INT) : ( s . x > s . y & s . y >= 0 )  }
then A6: ex s9 being Element of Funcs (X,INT) st
( s9 = s & s9 . x > s9 . y & s9 . y >= 0 ) ;
A7: (g . (s,(y gt 0))) . y = s . y by A1, A3, Th38;
 (g . (s,(y gt 0))) . x = s . x by A1, A2, Th38;
hence  g . (s,(y gt 0)) in  {  s where s is Element of Funcs (X,INT) : ( s . x > s . y & s . y >= 0 )  }  by A6, A7; ::_thesis: verum
end;
A8: now__::_thesis:_for_s_being_Element_of_Funcs_(X,INT)_st_s_in__{__s_where_s_is_Element_of_Funcs_(X,INT)_:_(_s_._x_>_s_._y_&_s_._y_>=_0_)__}__&_g_._((g_._(s,((((z_:=_x)_\;_(z_%=_y))_\;_(x_:=_y))_\;_(y_:=_z)))),(y_gt_0))_in_(Funcs_(X,INT))_\_(b,0)_holds_
g_._(s,((((z_:=_x)_\;_(z_%=_y))_\;_(x_:=_y))_\;_(y_:=_z)))_in__{__s_where_s_is_Element_of_Funcs_(X,INT)_:_(_s_._x_>_s_._y_&_s_._y_>=_0_)__}_
let s be Element of Funcs (X,INT); ::_thesis: ( s in  {  s where s is Element of Funcs (X,INT) : ( s . x > s . y & s . y >= 0 )  }  & g . ((g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))),(y gt 0)) in (Funcs (X,INT)) \ (b,0) implies g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) in  {  s where s is Element of Funcs (X,INT) : ( s . x > s . y & s . y >= 0 )  }  )
assume  s in  {  s where s is Element of Funcs (X,INT) : ( s . x > s . y & s . y >= 0 )  }  ; ::_thesis: ( g . ((g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))),(y gt 0)) in (Funcs (X,INT)) \ (b,0) implies g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) in  {  s where s is Element of Funcs (X,INT) : ( s . x > s . y & s . y >= 0 )  }  )
then  ex s9 being Element of Funcs (X,INT) st
( s9 = s & s9 . x > s9 . y & s9 . y >= 0 ) ;
then reconsider m = s . y as Element of NAT by INT_1:3;
assume  g . ((g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))),(y gt 0)) in (Funcs (X,INT)) \ (b,0) ; ::_thesis: g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) in  {  s where s is Element of Funcs (X,INT) : ( s . x > s . y & s . y >= 0 )  }
then  (g . ((g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))),(y gt 0))) . b <> 0 by Th2;
then A9: (g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))) . y > 0 by Th38;
A10: (g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))) . x = s . y by A1, A2, A3, A4, Lm1;
A11: (g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))) . y = (s . x) mod (s . y) by A1, A2, A3, A4, Lm1;
then  m <> 0 by A9, INT_1:def_10;
then  (g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))) . x > (g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))) . y by A11, A10, NEWTON:65;
hence  g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) in  {  s where s is Element of Funcs (X,INT) : ( s . x > s . y & s . y >= 0 )  }  by A9; ::_thesis: verum
end;
A12: now__::_thesis:_for_s_being_Element_of_Funcs_(X,INT)_st_g_._(s,(y_gt_0))_in__{__s_where_s_is_Element_of_Funcs_(X,INT)_:_(_s_._x_>_s_._y_&_s_._y_>=_0_)__}__holds_
g_iteration_terminates_for_((((z_:=_x)_\;_(z_%=_y))_\;_(x_:=_y))_\;_(y_:=_z))_\;_(y_gt_0),g_._(s,(y_gt_0))
let s be Element of Funcs (X,INT); ::_thesis: ( g . (s,(y gt 0)) in  {  s where s is Element of Funcs (X,INT) : ( s . x > s . y & s . y >= 0 )  }  implies g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0),g . (s,(y gt 0)) )
set s1 = g . (s,(y gt 0));
A13: ( s . y <= 0 implies (g . (s,(y gt 0))) . b = 0 ) by Th38;
assume  g . (s,(y gt 0)) in  {  s where s is Element of Funcs (X,INT) : ( s . x > s . y & s . y >= 0 )  }  ; ::_thesis: g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0),g . (s,(y gt 0))
then  ex s9 being Element of Funcs (X,INT) st
( s9 = g . (s,(y gt 0)) & s9 . x > s9 . y & s9 . y >= 0 ) ;
then reconsider m = (g . (s,(y gt 0))) . y as Element of NAT by INT_1:3;
 ( s . y > 0 implies (g . (s,(y gt 0))) . b = 1 ) by Th38;
then  ( g . (s,(y gt 0)) in (Funcs (X,INT)) \ (b,0) iff m > 0 ) by A13, Th2, Th38;
hence  g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0),g . (s,(y gt 0)) by A1, A2, A3, A4, Lm1; ::_thesis: verum
end;
 y gt 0 is_terminating_wrt g by AOFA_000:104;
hence  while ((y gt 0),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) is_terminating_wrt g, {  s where s is Element of Funcs (X,INT) : ( s . x > s . y & s . y >= 0 )  }  by A5, A8, A12, AOFA_000:107, AOFA_000:118; ::_thesis: verum
end;

theorem :: AOFA_I00:63
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
for s being Element of Funcs (X,INT)
for n, m being Element of NAT st n = s . x & m = s . y & n > m holds
(g . (s,(while ((y gt 0),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))))) . x = n gcd m
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
for s being Element of Funcs (X,INT)
for n, m being Element of NAT st n = s . x & m = s . y & n > m holds
(g . (s,(while ((y gt 0),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))))) . x = n gcd m
let X be non empty countable set ; ::_thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
for s being Element of Funcs (X,INT)
for n, m being Element of NAT st n = s . x & m = s . y & n > m holds
(g . (s,(while ((y gt 0),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))))) . x = n gcd m
let b be Element of X; ::_thesis: for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
for s being Element of Funcs (X,INT)
for n, m being Element of NAT st n = s . x & m = s . y & n > m holds
(g . (s,(while ((y gt 0),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))))) . x = n gcd m
let g be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); ::_thesis: for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
for s being Element of Funcs (X,INT)
for n, m being Element of NAT st n = s . x & m = s . y & n > m holds
(g . (s,(while ((y gt 0),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))))) . x = n gcd m
set h = g;
set S = Funcs (X,INT);
set T = (Funcs (X,INT)) \ (b,0);
let x, y, z be Variable of g; ::_thesis: ( ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) implies for s being Element of Funcs (X,INT)
for n, m being Element of NAT st n = s . x & m = s . y & n > m holds
(g . (s,(while ((y gt 0),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))))) . x = n gcd m )
given d being Function such that A1: d . b = 0 and
A2: d . x = 1 and
A3: d . y = 2 and
A4: d . z = 3 ; ::_thesis: for s being Element of Funcs (X,INT)
for n, m being Element of NAT st n = s . x & m = s . y & n > m holds
(g . (s,(while ((y gt 0),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))))) . x = n gcd m
set C = y gt 0;
set I = (((z := x) \; (z %= y)) \; (x := y)) \; (y := z);
let s be Element of Funcs (X,INT); ::_thesis: for n, m being Element of NAT st n = s . x & m = s . y & n > m holds
(g . (s,(while ((y gt 0),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))))) . x = n gcd m
reconsider fin = g . (s,(while ((y gt 0),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))))) as Element of Funcs (X,INT) ;
defpred S1[ Element of Funcs (X,INT)] means ( fin . x divides $1 . x & fin . x divides $1 . y );
A5: for s being Element of Funcs (X,INT) st S1[g . (s,(y gt 0))] holds
S1[s] by A1, A2, A3, Th38;
A6: for s being Element of Funcs (X,INT) st S1[g . ((g . (s,(y gt 0))),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))] & g . (s,(y gt 0)) in (Funcs (X,INT)) \ (b,0) holds
S1[g . (s,(y gt 0))]
proof
let s be Element of Funcs (X,INT); ::_thesis: ( S1[g . ((g . (s,(y gt 0))),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))] & g . (s,(y gt 0)) in (Funcs (X,INT)) \ (b,0) implies S1[g . (s,(y gt 0))] )
assume A7: S1[g . ((g . (s,(y gt 0))),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))] ; ::_thesis: ( not g . (s,(y gt 0)) in (Funcs (X,INT)) \ (b,0) or S1[g . (s,(y gt 0))] )
reconsider s1 = g . (s,(y gt 0)) as Element of Funcs (X,INT) ;
reconsider s2 = g . (s1,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) as Element of Funcs (X,INT) ;
A8: ( s . y <= 0 implies s1 . b = 0 ) by Th38;
A9: s1 . x = s . x by A1, A2, Th38;
A10: s2 . y = (s1 . x) mod (s1 . y) by A1, A2, A3, A4, Lm1;
A11: s2 . x = s1 . y by A1, A2, A3, A4, Lm1;
A12: s1 . y = s . y by A1, A3, Th38;
assume  g . (s,(y gt 0)) in (Funcs (X,INT)) \ (b,0) ; ::_thesis: S1[g . (s,(y gt 0))]
then  s . x = (((s . x) div (s . y)) * (s2 . x)) + ((s2 . y) * 1) by A9, A12, A8, A11, A10, Th2, NEWTON:66;
hence  S1[g . (s,(y gt 0))] by A1, A2, A3, A4, A7, A9, Lm1, WSIERP_1:5; ::_thesis: verum
end;
reconsider s1 = g . (s,(y gt 0)) as Element of Funcs (X,INT) ;
A13: s1 . y = s . y by A1, A3, Th38;
A14: ( s . y <= 0 implies s1 . b = 0 ) by Th38;
let n, m be Element of NAT ; ::_thesis: ( n = s . x & m = s . y & n > m implies (g . (s,(while ((y gt 0),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))))) . x = n gcd m )
defpred S2[ Element of Funcs (X,INT)] means ( n gcd m divides $1 . x & n gcd m divides $1 . y & $1 . x > $1 . y & $1 . y >= 0 );
defpred S3[ Element of Funcs (X,INT)] means $1 . y > 0 ;
assume that
A15: n = s . x and
A16: m = s . y and
A17: n > m ; ::_thesis: (g . (s,(while ((y gt 0),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))))) . x = n gcd m
 ( s . y > 0 implies s1 . b = 1 ) by Th38;
then  ( s1 in (Funcs (X,INT)) \ (b,0) iff m > 0 ) by A16, A14, Th2;
then A18: g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0),g . (s,(y gt 0)) by A1, A2, A3, A4, A16, A13, Lm1;
A19: for s being Element of Funcs (X,INT) st S2[s] & s in (Funcs (X,INT)) \ (b,0) & S3[s] holds
S2[g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))]
proof
let s be Element of Funcs (X,INT); ::_thesis: ( S2[s] & s in (Funcs (X,INT)) \ (b,0) & S3[s] implies S2[g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))] )
reconsider s99 = g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) as Element of Funcs (X,INT) ;
assume A20: S2[s] ; ::_thesis: ( not s in (Funcs (X,INT)) \ (b,0) or not S3[s] or S2[g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))] )
then reconsider n9 = s . x, m9 = s . y as Element of NAT by INT_1:3;
assume that
 s in (Funcs (X,INT)) \ (b,0) and
A21: S3[s] ; ::_thesis: S2[g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))]
A22: s99 . x = s . y by A1, A2, A3, A4, Lm1;
A23: s99 . y = (s . x) mod (s . y) by A1, A2, A3, A4, Lm1;
 n gcd m divides n9 mod m9 by A20, NAT_D:11;
hence  S2[g . (s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))] by A20, A21, A22, A23, NEWTON:65; ::_thesis: verum
end;
A24: for s being Element of Funcs (X,INT) st S2[s] holds
( S2[g . (s,(y gt 0))] & ( g . (s,(y gt 0)) in (Funcs (X,INT)) \ (b,0) implies S3[g . (s,(y gt 0))] ) & ( S3[g . (s,(y gt 0))] implies g . (s,(y gt 0)) in (Funcs (X,INT)) \ (b,0) ) )
proof
let s be Element of Funcs (X,INT); ::_thesis: ( S2[s] implies ( S2[g . (s,(y gt 0))] & ( g . (s,(y gt 0)) in (Funcs (X,INT)) \ (b,0) implies S3[g . (s,(y gt 0))] ) & ( S3[g . (s,(y gt 0))] implies g . (s,(y gt 0)) in (Funcs (X,INT)) \ (b,0) ) ) )
assume A25: S2[s] ; ::_thesis: ( S2[g . (s,(y gt 0))] & ( g . (s,(y gt 0)) in (Funcs (X,INT)) \ (b,0) implies S3[g . (s,(y gt 0))] ) & ( S3[g . (s,(y gt 0))] implies g . (s,(y gt 0)) in (Funcs (X,INT)) \ (b,0) ) )
reconsider s9 = g . (s,(y gt 0)) as Element of Funcs (X,INT) ;
 s9 . y = s . y by A1, A3, Th38;
hence  S2[g . (s,(y gt 0))] by A1, A2, A25, Th38; ::_thesis: ( g . (s,(y gt 0)) in (Funcs (X,INT)) \ (b,0) iff S3[g . (s,(y gt 0))] )
A26: ( s . y <= 0 implies s9 . b = 0 ) by Th38;
 ( s . y > 0 implies s9 . b = 1 ) by Th38;
hence  ( g . (s,(y gt 0)) in (Funcs (X,INT)) \ (b,0) iff S3[g . (s,(y gt 0))] ) by A26, Th2, Th38; ::_thesis: verum
end;
A27: S2[s] by A15, A16, A17, NAT_D:def_5;
A28: ( S2[g . (s,(while ((y gt 0),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))))] & not S3[g . (s,(while ((y gt 0),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))))] ) from AOFA_000:sch_5(A27, A18, A19, A24);
then  fin . y = 0 ;
then A29: S1[g . (s,(while ((y gt 0),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))))] by INT_2:12;
 S1[s] from AOFA_000:sch_6(A29, A18, A6, A5);
then  fin . x divides n gcd m by A15, A16, INT_2:22;
then  ( fin . x = n gcd m or fin . x = - (n gcd m) ) by A28, INT_2:11;
hence  (g . (s,(while ((y gt 0),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))))) . x = n gcd m by A28; ::_thesis: verum
end;

Lm2:  for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
for s being Element of Funcs (X,INT) holds
( (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . y = abs ((s . x) - (s . y)) & ( for n, m being Element of NAT st n = s . x & m = s . y & ( s in (Funcs (X,INT)) \ (b,0) implies m > 0 ) & ( m > 0 implies s in (Funcs (X,INT)) \ (b,0) ) holds
g iteration_terminates_for ((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0),s ) )
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
for s being Element of Funcs (X,INT) holds
( (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . y = abs ((s . x) - (s . y)) & ( for n, m being Element of NAT st n = s . x & m = s . y & ( s in (Funcs (X,INT)) \ (b,0) implies m > 0 ) & ( m > 0 implies s in (Funcs (X,INT)) \ (b,0) ) holds
g iteration_terminates_for ((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0),s ) )
let X be non empty countable set ; ::_thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
for s being Element of Funcs (X,INT) holds
( (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . y = abs ((s . x) - (s . y)) & ( for n, m being Element of NAT st n = s . x & m = s . y & ( s in (Funcs (X,INT)) \ (b,0) implies m > 0 ) & ( m > 0 implies s in (Funcs (X,INT)) \ (b,0) ) holds
g iteration_terminates_for ((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0),s ) )
let b be Element of X; ::_thesis: for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
for s being Element of Funcs (X,INT) holds
( (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . y = abs ((s . x) - (s . y)) & ( for n, m being Element of NAT st n = s . x & m = s . y & ( s in (Funcs (X,INT)) \ (b,0) implies m > 0 ) & ( m > 0 implies s in (Funcs (X,INT)) \ (b,0) ) holds
g iteration_terminates_for ((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0),s ) )
let g be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); ::_thesis: for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
for s being Element of Funcs (X,INT) holds
( (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . y = abs ((s . x) - (s . y)) & ( for n, m being Element of NAT st n = s . x & m = s . y & ( s in (Funcs (X,INT)) \ (b,0) implies m > 0 ) & ( m > 0 implies s in (Funcs (X,INT)) \ (b,0) ) holds
g iteration_terminates_for ((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0),s ) )
set S = Funcs (X,INT);
set T = (Funcs (X,INT)) \ (b,0);
let x, y, z be Variable of g; ::_thesis: ( ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) implies for s being Element of Funcs (X,INT) holds
( (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . y = abs ((s . x) - (s . y)) & ( for n, m being Element of NAT st n = s . x & m = s . y & ( s in (Funcs (X,INT)) \ (b,0) implies m > 0 ) & ( m > 0 implies s in (Funcs (X,INT)) \ (b,0) ) holds
g iteration_terminates_for ((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0),s ) ) )
given d being Function such that A1: d . b = 0 and
A2: d . x = 1 and
A3: d . y = 2 and
A4: d . z = 3 ; ::_thesis: for s being Element of Funcs (X,INT) holds
( (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . y = abs ((s . x) - (s . y)) & ( for n, m being Element of NAT st n = s . x & m = s . y & ( s in (Funcs (X,INT)) \ (b,0) implies m > 0 ) & ( m > 0 implies s in (Funcs (X,INT)) \ (b,0) ) holds
g iteration_terminates_for ((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0),s ) )
A5: y <> z by A3, A4;
let s be Element of Funcs (X,INT); ::_thesis: ( (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . y = abs ((s . x) - (s . y)) & ( for n, m being Element of NAT st n = s . x & m = s . y & ( s in (Funcs (X,INT)) \ (b,0) implies m > 0 ) & ( m > 0 implies s in (Funcs (X,INT)) \ (b,0) ) holds
g iteration_terminates_for ((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0),s ) )
set J = if-then ((z lt 0),(z *= (- 1)));
A6: g complies_with_if_wrt (Funcs (X,INT)) \ (b,0) by AOFA_000:def_32;
A7: z <> x by A2, A4;
set I = (((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z);
A8: x <> y by A2, A3;
A9: now__::_thesis:_for_s_being_Element_of_Funcs_(X,INT)_holds_
(_(g_._(s,((((z_:=_((._x)_-_(._y)))_\;_(if-then_((z_lt_0),(z_*=_(-_1)))))_\;_(x_:=_y))_\;_(y_:=_z))))_._x_=_s_._y_&_(g_._(s,((((z_:=_((._x)_-_(._y)))_\;_(if-then_((z_lt_0),(z_*=_(-_1)))))_\;_(x_:=_y))_\;_(y_:=_z))))_._y_=_abs_((s_._x)_-_(s_._y))_)
let s be Element of Funcs (X,INT); ::_thesis: ( (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . y = abs ((s . x) - (s . y)) )
set s1 = g . (s,(z := ((. x) - (. y))));
set s2 = g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0));
set q = g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1)))));
set qz = g . ((g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))),(z *= (- 1)));
A10: ((g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . z) * (- 1) = - ((g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . z) ;
set s3 = g . ((g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1)))))),(x := y));
set s4 = g . ((g . ((g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1)))))),(x := y))),(y := z));
A11: g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z))) = g . ((g . (s,(((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)))),(y := z)) by AOFA_000:def_29
.= g . ((g . ((g . (s,((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))))),(x := y))),(y := z)) by AOFA_000:def_29
.= g . ((g . ((g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1)))))),(x := y))),(y := z)) by AOFA_000:def_29 ;
 ( (g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . b = 1 implies g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0)) in (Funcs (X,INT)) \ (b,0) ) ;
then A12: ( (g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . b = 1 implies g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1))))) = g . ((g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))),(z *= (- 1))) ) by A6, AOFA_000:def_30;
A13: (. x) . s = s . x by Th22;
A14: (g . ((g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))),(z *= (- 1)))) . y = (g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . y by A5, Th31;
A15: (. y) . s = s . y by Th22;
 ((. x) - (. y)) . s = ((. x) . s) - ((. y) . s) by Def11;
then A16: (g . (s,(z := ((. x) - (. y))))) . z = (s . x) - (s . y) by A13, A15, Th26;
A17: ( (g . (s,(z := ((. x) - (. y))))) . z < 0 implies (g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . b = 1 ) by Th38;
A18: (g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . z = (g . (s,(z := ((. x) - (. y))))) . z by A1, A4, Th38;
A19: (g . ((g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))),(z *= (- 1)))) . z = ((g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . z) * (- 1) by Th31;
A20: (g . ((g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1)))))),(x := y))) . z = (g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1)))))) . z by A7, Th27;
A21: (g . ((g . ((g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1)))))),(x := y))),(y := z))) . y = (g . ((g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1)))))),(x := y))) . z by Th27;
 ( (g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . b = 0 implies g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0)) nin (Funcs (X,INT)) \ (b,0) ) by Th2;
then A22: ( (g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . b = 0 implies g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1))))) = g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0)) ) by A6, AOFA_000:80;
A23: ( (g . (s,(z := ((. x) - (. y))))) . z >= 0 implies (g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . b = 0 ) by Th38;
A24: (g . (s,(z := ((. x) - (. y))))) . y = s . y by A5, Th26;
A25: (g . ((g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1)))))),(x := y))) . x = (g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1)))))) . y by Th27;
 (g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . y = (g . (s,(z := ((. x) - (. y))))) . y by A1, A3, Th38;
hence  ( (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . y = abs ((s . x) - (s . y)) ) by A8, A22, A12, A18, A17, A23, A14, A19, A10, A11, A24, A16, A25, A20, A21, Th27, ABSVALUE:def_1; ::_thesis: verum
end;
hence  ( (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . y = abs ((s . x) - (s . y)) ) ; ::_thesis: for n, m being Element of NAT st n = s . x & m = s . y & ( s in (Funcs (X,INT)) \ (b,0) implies m > 0 ) & ( m > 0 implies s in (Funcs (X,INT)) \ (b,0) ) holds
g iteration_terminates_for ((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0),s
deffunc H1( Element of Funcs (X,INT)) ->  Element of NAT = IFEQ (($1 . y),0,0,(IFEQ (($1 . x),0,2,(IFEQ (($1 . x),($1 . y),1,(In ((max ((2 * ($1 . x)),((2 * ($1 . y)) + 1))),NAT)))))));
defpred S1[ Element of Funcs (X,INT)] means ( $1 . x >= 0 & $1 . y > 0 );
set C = y gt 0;
A26: for s being Element of Funcs (X,INT) st S1[s] holds
( ( S1[g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))] implies g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0))) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0))) in (Funcs (X,INT)) \ (b,0) implies S1[g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))] ) & H1(g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))) < H1(s) )
proof
let s be Element of Funcs (X,INT); ::_thesis: ( S1[s] implies ( ( S1[g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))] implies g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0))) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0))) in (Funcs (X,INT)) \ (b,0) implies S1[g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))] ) & H1(g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))) < H1(s) ) )
assume that
A27: s . x >= 0 and
A28: s . y > 0 ; ::_thesis: ( ( S1[g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))] implies g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0))) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0))) in (Funcs (X,INT)) \ (b,0) implies S1[g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))] ) & H1(g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))) < H1(s) )
reconsider s9 = g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z))) as Element of Funcs (X,INT) ;
reconsider s99 = g . (s9,(y gt 0)) as Element of Funcs (X,INT) ;
A29: s9 . y = abs ((s . x) - (s . y)) by A9;
then reconsider nx = s . x, ny = s . y, nn = s99 . y as Element of NAT by A1, A3, A27, A28, Th38, INT_1:3;
A30: g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0))) = s99 by AOFA_000:def_29;
A31: s99 . x = s9 . x by A1, A2, Th38;
A32: ( s9 . y <= 0 implies s99 . b = 0 ) by Th38;
A33: ( s9 . y > 0 implies s99 . b = 1 ) by Th38;
hence  ( S1[g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))] iff g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0))) in (Funcs (X,INT)) \ (b,0) ) by A9, A28, A30, A31, A32, Th2, Th38; ::_thesis: H1(g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))) < H1(s)
A34: s9 . x = s . y by A9;
A35: H1(s99) = IFEQ (nn,0,0,(IFEQ (ny,0,2,(IFEQ (ny,nn,1,(max ((2 * ny),((2 * nn) + 1)))))))) by A31, A34, FUNCT_7:def_1;
 (2 * ny) + 1 > 2 * ny by NAT_1:13;
then A36: max ((2 * nx),((2 * ny) + 1)) > 2 * ny by XXREAL_0:30;
A37: s99 . y = s9 . y by A1, A3, Th38;
A38: H1(s) = IFEQ (ny,0,0,(IFEQ (nx,0,2,(IFEQ (nx,ny,1,(max ((2 * nx),((2 * ny) + 1)))))))) by FUNCT_7:def_1
.= IFEQ (nx,0,2,(IFEQ (nx,ny,1,(max ((2 * nx),((2 * ny) + 1)))))) by A28, FUNCOP_1:def_8 ;
percases ( nx = ny or nx > ny or ( nx < ny & nx > 0 ) or nx = 0 ) by XXREAL_0:1;
supposeA39: nx = ny ; ::_thesis: H1(g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))) < H1(s)
then A40: IFEQ (nx,ny,1,(max ((2 * nx),((2 * ny) + 1)))) = 1 by FUNCOP_1:def_8;
A41: nn = 0 by A37, A29, A39, ABSVALUE:2;
 H1(s) = IFEQ (nx,ny,1,(max ((2 * nx),((2 * ny) + 1)))) by A28, A38, A39, FUNCOP_1:def_8;
hence  H1(g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))) < H1(s) by A30, A40, A41, FUNCOP_1:def_8; ::_thesis: verum
end;
supposeA42: nx > ny ; ::_thesis: H1(g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))) < H1(s)
then  IFEQ (nx,ny,1,(max ((2 * nx),((2 * ny) + 1)))) = max ((2 * nx),((2 * ny) + 1)) by FUNCOP_1:def_8;
then A43: H1(s) = max ((2 * nx),((2 * ny) + 1)) by A38, A42, FUNCOP_1:def_8;
A44: nx - ny > 0 by A42, XREAL_1:50;
then A45: H1(s99) = IFEQ (ny,0,2,(IFEQ (ny,nn,1,(max ((2 * ny),((2 * nn) + 1)))))) by A37, A33, A32, A29, A35, ABSVALUE:def_1, FUNCOP_1:def_8
.= IFEQ (ny,nn,1,(max ((2 * ny),((2 * nn) + 1)))) by A28, FUNCOP_1:def_8 ;
then A46: ( ny = nn implies H1(s99) = 1 ) by FUNCOP_1:def_8;
 nn = nx - ny by A37, A29, A44, ABSVALUE:def_1;
then  nn < nx by A28, XREAL_1:44;
then  nn + 1 <= nx by NAT_1:13;
then A47: 2 * (nn + 1) <= 2 * nx by XREAL_1:64;
A48: 1 <= (2 * nn) + 1 by NAT_1:11;
A49: ( ny <> nn implies H1(s99) = max ((2 * ny),((2 * nn) + 1)) ) by A45, FUNCOP_1:def_8;
 (2 * nn) + 2 = ((2 * nn) + 1) + 1 ;
then  (2 * nn) + 1 < 2 * nx by A47, NAT_1:13;
then  (2 * nn) + 1 < H1(s) by A43, XXREAL_0:30;
hence  H1(g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))) < H1(s) by A30, A36, A43, A46, A49, A48, XXREAL_0:2, XXREAL_0:29; ::_thesis: verum
end;
supposeA50: ( nx < ny & nx > 0 ) ; ::_thesis: H1(g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))) < H1(s)
A51: - (nx - ny) = ny - nx ;
A52: nx - ny < 0 by A50, XREAL_1:49;
then A53: nn = - (nx - ny) by A37, A29, ABSVALUE:def_1;
then A54: nn < ny by A50, A51, XREAL_1:44;
 2 * nn < 2 * ny by A50, A53, A51, XREAL_1:44, XREAL_1:68;
then  (2 * nn) + 1 < (2 * ny) + 1 by XREAL_1:6;
then A55: (2 * nn) + 1 < max ((2 * nx),((2 * ny) + 1)) by XXREAL_0:30;
 nn > 0 by A37, A29, A52, A51, ABSVALUE:def_1;
then A56: H1(s99) = IFEQ (ny,0,2,(IFEQ (ny,nn,1,(max ((2 * ny),((2 * nn) + 1)))))) by A35, FUNCOP_1:def_8
.= IFEQ (ny,nn,1,(max ((2 * ny),((2 * nn) + 1)))) by A28, FUNCOP_1:def_8
.= max ((2 * ny),((2 * nn) + 1)) by A54, FUNCOP_1:def_8 ;
H1(s) = IFEQ (nx,ny,1,(max ((2 * nx),((2 * ny) + 1)))) by A38, A50, FUNCOP_1:def_8
.= max ((2 * nx),((2 * ny) + 1)) by A50, FUNCOP_1:def_8 ;
hence  H1(g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))) < H1(s) by A30, A36, A55, A56, XXREAL_0:29; ::_thesis: verum
end;
supposeA57: nx = 0 ; ::_thesis: H1(g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))) < H1(s)
then A58: H1(s) = 2 by A38, FUNCOP_1:def_8;
A59: nn = - (nx - ny) by A28, A37, A29, A57, ABSVALUE:def_1
.= ny by A57 ;
then H1(s99) = IFEQ (ny,0,2,(IFEQ (ny,nn,1,(max ((2 * ny),((2 * nn) + 1)))))) by A28, A35, FUNCOP_1:def_8
.= IFEQ (ny,nn,1,(max ((2 * ny),((2 * nn) + 1)))) by A28, FUNCOP_1:def_8
.= 1 by A59, FUNCOP_1:def_8 ;
hence  H1(g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))) < H1(s) by A30, A58; ::_thesis: verum
end;
end;
end;
let n, m be Element of NAT ; ::_thesis: ( n = s . x & m = s . y & ( s in (Funcs (X,INT)) \ (b,0) implies m > 0 ) & ( m > 0 implies s in (Funcs (X,INT)) \ (b,0) ) implies g iteration_terminates_for ((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0),s )
assume that
A60: n = s . x and
A61: m = s . y ; ::_thesis: ( ( s in (Funcs (X,INT)) \ (b,0) & not m > 0 ) or ( m > 0 & not s in (Funcs (X,INT)) \ (b,0) ) or g iteration_terminates_for ((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0),s )
assume  ( s in (Funcs (X,INT)) \ (b,0) iff m > 0 ) ; ::_thesis: g iteration_terminates_for ((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0),s
then A62: ( s in (Funcs (X,INT)) \ (b,0) iff S1[s] ) by A60, A61;
 g iteration_terminates_for ((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0),s from AOFA_000:sch_3(A62, A26);
hence  g iteration_terminates_for ((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0),s ; ::_thesis: verum
end;

theorem :: AOFA_I00:64
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
while ((y gt 0),((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z))) is_terminating_wrt g, {  s where s is Element of Funcs (X,INT) : ( s . x >= 0 & s . y >= 0 )  }
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
while ((y gt 0),((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z))) is_terminating_wrt g, {  s where s is Element of Funcs (X,INT) : ( s . x >= 0 & s . y >= 0 )  }
let X be non empty countable set ; ::_thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
while ((y gt 0),((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z))) is_terminating_wrt g, {  s where s is Element of Funcs (X,INT) : ( s . x >= 0 & s . y >= 0 )  }
let b be Element of X; ::_thesis: for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
while ((y gt 0),((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z))) is_terminating_wrt g, {  s where s is Element of Funcs (X,INT) : ( s . x >= 0 & s . y >= 0 )  }
let g be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); ::_thesis: for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
while ((y gt 0),((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z))) is_terminating_wrt g, {  s where s is Element of Funcs (X,INT) : ( s . x >= 0 & s . y >= 0 )  }
set S = Funcs (X,INT);
set T = (Funcs (X,INT)) \ (b,0);
let x, y, z be Variable of g; ::_thesis: ( ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) implies while ((y gt 0),((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z))) is_terminating_wrt g, {  s where s is Element of Funcs (X,INT) : ( s . x >= 0 & s . y >= 0 )  }  )
set P =  {  s where s is Element of Funcs (X,INT) : ( s . x >= 0 & s . y >= 0 )  }  ;
given d being Function such that A1: d . b = 0 and
A2: d . x = 1 and
A3: d . y = 2 and
A4: d . z = 3 ; ::_thesis: while ((y gt 0),((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z))) is_terminating_wrt g, {  s where s is Element of Funcs (X,INT) : ( s . x >= 0 & s . y >= 0 )  }
set C = y gt 0;
set I = (((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z);
A5: now__::_thesis:_for_s_being_Element_of_Funcs_(X,INT)_st_s_in__{__s_where_s_is_Element_of_Funcs_(X,INT)_:_(_s_._x_>=_0_&_s_._y_>=_0_)__}__&_g_._((g_._(s,((((z_:=_((._x)_-_(._y)))_\;_(if-then_((z_lt_0),(z_*=_(-_1)))))_\;_(x_:=_y))_\;_(y_:=_z)))),(y_gt_0))_in_(Funcs_(X,INT))_\_(b,0)_holds_
g_._(s,((((z_:=_((._x)_-_(._y)))_\;_(if-then_((z_lt_0),(z_*=_(-_1)))))_\;_(x_:=_y))_\;_(y_:=_z)))_in__{__s_where_s_is_Element_of_Funcs_(X,INT)_:_(_s_._x_>=_0_&_s_._y_>=_0_)__}_
let s be Element of Funcs (X,INT); ::_thesis: ( s in  {  s where s is Element of Funcs (X,INT) : ( s . x >= 0 & s . y >= 0 )  }  & g . ((g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))),(y gt 0)) in (Funcs (X,INT)) \ (b,0) implies g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z))) in  {  s where s is Element of Funcs (X,INT) : ( s . x >= 0 & s . y >= 0 )  }  )
assume  s in  {  s where s is Element of Funcs (X,INT) : ( s . x >= 0 & s . y >= 0 )  }  ; ::_thesis: ( g . ((g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))),(y gt 0)) in (Funcs (X,INT)) \ (b,0) implies g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z))) in  {  s where s is Element of Funcs (X,INT) : ( s . x >= 0 & s . y >= 0 )  }  )
then  ex s9 being Element of Funcs (X,INT) st
( s9 = s & s9 . x >= 0 & s9 . y >= 0 ) ;
then reconsider m = s . y as Element of NAT by INT_1:3;
assume  g . ((g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))),(y gt 0)) in (Funcs (X,INT)) \ (b,0) ; ::_thesis: g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z))) in  {  s where s is Element of Funcs (X,INT) : ( s . x >= 0 & s . y >= 0 )  }
A6: (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . x = m by A1, A2, A3, A4, Lm2;
 (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . y = abs ((s . x) - (s . y)) by A1, A2, A3, A4, Lm2;
hence  g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z))) in  {  s where s is Element of Funcs (X,INT) : ( s . x >= 0 & s . y >= 0 )  }  by A6; ::_thesis: verum
end;
A7: now__::_thesis:_for_s_being_Element_of_Funcs_(X,INT)_st_g_._(s,(y_gt_0))_in__{__s_where_s_is_Element_of_Funcs_(X,INT)_:_(_s_._x_>=_0_&_s_._y_>=_0_)__}__holds_
g_iteration_terminates_for_((((z_:=_((._x)_-_(._y)))_\;_(if-then_((z_lt_0),(z_*=_(-_1)))))_\;_(x_:=_y))_\;_(y_:=_z))_\;_(y_gt_0),g_._(s,(y_gt_0))
let s be Element of Funcs (X,INT); ::_thesis: ( g . (s,(y gt 0)) in  {  s where s is Element of Funcs (X,INT) : ( s . x >= 0 & s . y >= 0 )  }  implies g iteration_terminates_for ((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0),g . (s,(y gt 0)) )
set s1 = g . (s,(y gt 0));
A8: ( s . y <= 0 implies (g . (s,(y gt 0))) . b = 0 ) by Th38;
assume  g . (s,(y gt 0)) in  {  s where s is Element of Funcs (X,INT) : ( s . x >= 0 & s . y >= 0 )  }  ; ::_thesis: g iteration_terminates_for ((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0),g . (s,(y gt 0))
then  ex s9 being Element of Funcs (X,INT) st
( s9 = g . (s,(y gt 0)) & s9 . x >= 0 & s9 . y >= 0 ) ;
then reconsider n = (g . (s,(y gt 0))) . x, m = (g . (s,(y gt 0))) . y as Element of NAT by INT_1:3;
A9: n = n ;
 ( s . y > 0 implies (g . (s,(y gt 0))) . b = 1 ) by Th38;
then  ( g . (s,(y gt 0)) in (Funcs (X,INT)) \ (b,0) iff m > 0 ) by A8, Th2, Th38;
hence  g iteration_terminates_for ((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0),g . (s,(y gt 0)) by A1, A2, A3, A4, A9, Lm2; ::_thesis: verum
end;
A10:  {  s where s is Element of Funcs (X,INT) : ( s . x >= 0 & s . y >= 0 )  }  is_invariant_wrt y gt 0,g
proof
let s be Element of Funcs (X,INT); :: according to AOFA_000:def_39 ::_thesis: ( not s in  {  s where s is Element of Funcs (X,INT) : ( s . x >= 0 & s . y >= 0 )  }  or g . (s,(y gt 0)) in  {  s where s is Element of Funcs (X,INT) : ( s . x >= 0 & s . y >= 0 )  }  )
set s1 = g . (s,(y gt 0));
assume  s in  {  s where s is Element of Funcs (X,INT) : ( s . x >= 0 & s . y >= 0 )  }  ; ::_thesis: g . (s,(y gt 0)) in  {  s where s is Element of Funcs (X,INT) : ( s . x >= 0 & s . y >= 0 )  }
then A11: ex s9 being Element of Funcs (X,INT) st
( s9 = s & s9 . x >= 0 & s9 . y >= 0 ) ;
A12: (g . (s,(y gt 0))) . y = s . y by A1, A3, Th38;
 (g . (s,(y gt 0))) . x = s . x by A1, A2, Th38;
hence  g . (s,(y gt 0)) in  {  s where s is Element of Funcs (X,INT) : ( s . x >= 0 & s . y >= 0 )  }  by A11, A12; ::_thesis: verum
end;
 y gt 0 is_terminating_wrt g by AOFA_000:104;
hence  while ((y gt 0),((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z))) is_terminating_wrt g, {  s where s is Element of Funcs (X,INT) : ( s . x >= 0 & s . y >= 0 )  }  by A10, A5, A7, AOFA_000:107, AOFA_000:118; ::_thesis: verum
end;

theorem :: AOFA_I00:65
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
for s being Element of Funcs (X,INT)
for n, m being Element of NAT st n = s . x & m = s . y & n > 0 holds
(g . (s,(while ((y gt 0),((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))))) . x = n gcd m
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
for s being Element of Funcs (X,INT)
for n, m being Element of NAT st n = s . x & m = s . y & n > 0 holds
(g . (s,(while ((y gt 0),((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))))) . x = n gcd m
let X be non empty countable set ; ::_thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
for s being Element of Funcs (X,INT)
for n, m being Element of NAT st n = s . x & m = s . y & n > 0 holds
(g . (s,(while ((y gt 0),((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))))) . x = n gcd m
let b be Element of X; ::_thesis: for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
for s being Element of Funcs (X,INT)
for n, m being Element of NAT st n = s . x & m = s . y & n > 0 holds
(g . (s,(while ((y gt 0),((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))))) . x = n gcd m
let g be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); ::_thesis: for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
for s being Element of Funcs (X,INT)
for n, m being Element of NAT st n = s . x & m = s . y & n > 0 holds
(g . (s,(while ((y gt 0),((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))))) . x = n gcd m
set h = g;
set S = Funcs (X,INT);
set T = (Funcs (X,INT)) \ (b,0);
A1: g complies_with_if_wrt (Funcs (X,INT)) \ (b,0) by AOFA_000:def_32;
let x, y, z be Variable of g; ::_thesis: ( ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) implies for s being Element of Funcs (X,INT)
for n, m being Element of NAT st n = s . x & m = s . y & n > 0 holds
(g . (s,(while ((y gt 0),((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))))) . x = n gcd m )
given d being Function such that A2: d . b = 0 and
A3: d . x = 1 and
A4: d . y = 2 and
A5: d . z = 3 ; ::_thesis: for s being Element of Funcs (X,INT)
for n, m being Element of NAT st n = s . x & m = s . y & n > 0 holds
(g . (s,(while ((y gt 0),((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))))) . x = n gcd m
set C = y gt 0;
let s be Element of Funcs (X,INT); ::_thesis: for n, m being Element of NAT st n = s . x & m = s . y & n > 0 holds
(g . (s,(while ((y gt 0),((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))))) . x = n gcd m
A6: y <> z by A4, A5;
reconsider s1 = g . (s,(y gt 0)) as Element of Funcs (X,INT) ;
A7: s1 . x = s . x by A2, A3, Th38;
A8: s1 . y = s . y by A2, A4, Th38;
A9: ( s . y <= 0 implies s1 . b = 0 ) by Th38;
defpred S1[ Element of Funcs (X,INT)] means ( $1 . x > 0 & $1 . y > 0 );
let n, m be Element of NAT ; ::_thesis: ( n = s . x & m = s . y & n > 0 implies (g . (s,(while ((y gt 0),((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))))) . x = n gcd m )
defpred S2[ Element of Funcs (X,INT)] means ( n gcd m divides $1 . x & n gcd m divides $1 . y & $1 . x > 0 & $1 . y >= 0 & ( for c being Nat st c divides $1 . x & c divides $1 . y holds
c divides n gcd m ) );
set J = if-then ((z lt 0),(z *= (- 1)));
set I = (((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z);
assume that
A10: n = s . x and
A11: m = s . y and
A12: n > 0 ; ::_thesis: (g . (s,(while ((y gt 0),((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))))) . x = n gcd m
 ( s . y > 0 implies s1 . b = 1 ) by Th38;
then  ( s1 in (Funcs (X,INT)) \ (b,0) iff S1[s1] ) by A10, A12, A9, Th2, Th38;
then A13: g iteration_terminates_for ((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0),g . (s,(y gt 0)) by A2, A3, A4, A5, A10, A11, A12, A7, A8, Lm2;
A14: z <> x by A3, A5;
A15: x <> y by A3, A4;
A16: now__::_thesis:_for_s_being_Element_of_Funcs_(X,INT)_holds_
(_(g_._(s,((((z_:=_((._x)_-_(._y)))_\;_(if-then_((z_lt_0),(z_*=_(-_1)))))_\;_(x_:=_y))_\;_(y_:=_z))))_._x_=_s_._y_&_(g_._(s,((((z_:=_((._x)_-_(._y)))_\;_(if-then_((z_lt_0),(z_*=_(-_1)))))_\;_(x_:=_y))_\;_(y_:=_z))))_._y_=_abs_((s_._x)_-_(s_._y))_)
let s be Element of Funcs (X,INT); ::_thesis: ( (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . y = abs ((s . x) - (s . y)) )
set s1 = g . (s,(z := ((. x) - (. y))));
set s2 = g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0));
set q = g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1)))));
set qz = g . ((g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))),(z *= (- 1)));
A17: ((g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . z) * (- 1) = - ((g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . z) ;
set s3 = g . ((g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1)))))),(x := y));
set s4 = g . ((g . ((g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1)))))),(x := y))),(y := z));
A18: g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z))) = g . ((g . (s,(((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)))),(y := z)) by AOFA_000:def_29
.= g . ((g . ((g . (s,((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))))),(x := y))),(y := z)) by AOFA_000:def_29
.= g . ((g . ((g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1)))))),(x := y))),(y := z)) by AOFA_000:def_29 ;
 ( (g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . b = 1 implies g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0)) in (Funcs (X,INT)) \ (b,0) ) ;
then A19: ( (g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . b = 1 implies g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1))))) = g . ((g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))),(z *= (- 1))) ) by A1, AOFA_000:def_30;
A20: (. x) . s = s . x by Th22;
A21: (g . ((g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))),(z *= (- 1)))) . y = (g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . y by A6, Th31;
A22: (. y) . s = s . y by Th22;
 ((. x) - (. y)) . s = ((. x) . s) - ((. y) . s) by Def11;
then A23: (g . (s,(z := ((. x) - (. y))))) . z = (s . x) - (s . y) by A20, A22, Th26;
A24: ( (g . (s,(z := ((. x) - (. y))))) . z < 0 implies (g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . b = 1 ) by Th38;
A25: (g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . z = (g . (s,(z := ((. x) - (. y))))) . z by A2, A5, Th38;
A26: (g . ((g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))),(z *= (- 1)))) . z = ((g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . z) * (- 1) by Th31;
A27: (g . ((g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1)))))),(x := y))) . z = (g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1)))))) . z by A14, Th27;
A28: (g . ((g . ((g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1)))))),(x := y))),(y := z))) . y = (g . ((g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1)))))),(x := y))) . z by Th27;
 ( (g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . b = 0 implies g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0)) nin (Funcs (X,INT)) \ (b,0) ) by Th2;
then A29: ( (g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . b = 0 implies g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1))))) = g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0)) ) by A1, AOFA_000:80;
A30: ( (g . (s,(z := ((. x) - (. y))))) . z >= 0 implies (g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . b = 0 ) by Th38;
A31: (g . (s,(z := ((. x) - (. y))))) . y = s . y by A6, Th26;
A32: (g . ((g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1)))))),(x := y))) . x = (g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1)))))) . y by Th27;
 (g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . y = (g . (s,(z := ((. x) - (. y))))) . y by A2, A4, Th38;
hence  ( (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . y = abs ((s . x) - (s . y)) ) by A15, A29, A19, A25, A24, A30, A21, A26, A17, A18, A31, A23, A32, A27, A28, Th27, ABSVALUE:def_1; ::_thesis: verum
end;
A33: for s being Element of Funcs (X,INT) st S2[s] & s in (Funcs (X,INT)) \ (b,0) & S1[s] holds
S2[g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))]
proof
let s be Element of Funcs (X,INT); ::_thesis: ( S2[s] & s in (Funcs (X,INT)) \ (b,0) & S1[s] implies S2[g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))] )
reconsider s99 = g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z))) as Element of Funcs (X,INT) ;
A34: abs (n gcd m) = n gcd m by ABSVALUE:def_1;
A35: s99 . y = abs ((s . x) - (s . y)) by A16;
assume A36: S2[s] ; ::_thesis: ( not s in (Funcs (X,INT)) \ (b,0) or not S1[s] or S2[g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))] )
then reconsider n9 = s . x, m9 = s . y as Element of NAT by INT_1:3;
assume that
 s in (Funcs (X,INT)) \ (b,0) and
A37: S1[s] ; ::_thesis: S2[g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))]
 n gcd m divides n9 - m9 by A36, PREPOWER:94;
hence  ( n gcd m divides (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . x & n gcd m divides (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . y & (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . x > 0 & (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . y >= 0 ) by A16, A36, A37, A35, A34, INT_2:16; ::_thesis: for c being Nat st c divides (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . x & c divides (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . y holds
c divides n gcd m
let c be Nat; ::_thesis: ( c divides (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . x & c divides (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . y implies c divides n gcd m )
reconsider c9 = c as Element of NAT by ORDINAL1:def_12;
assume that
A38: c divides (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . x and
A39: c divides (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . y ; ::_thesis: c divides n gcd m
A40: abs c = c by ABSVALUE:def_1;
A41: s99 . x = s . y by A16;
 c9 divides abs (n9 - m9) by A16, A39;
then A42: c divides n9 - m9 by A40, INT_2:16;
 c divides m9 by A16, A38;
then  c divides (n9 - m9) + m9 by A42, WSIERP_1:4;
hence  c divides n gcd m by A36, A41, A38; ::_thesis: verum
end;
A43: for s being Element of Funcs (X,INT) st S2[s] holds
( S2[g . (s,(y gt 0))] & ( g . (s,(y gt 0)) in (Funcs (X,INT)) \ (b,0) implies S1[g . (s,(y gt 0))] ) & ( S1[g . (s,(y gt 0))] implies g . (s,(y gt 0)) in (Funcs (X,INT)) \ (b,0) ) )
proof
let s be Element of Funcs (X,INT); ::_thesis: ( S2[s] implies ( S2[g . (s,(y gt 0))] & ( g . (s,(y gt 0)) in (Funcs (X,INT)) \ (b,0) implies S1[g . (s,(y gt 0))] ) & ( S1[g . (s,(y gt 0))] implies g . (s,(y gt 0)) in (Funcs (X,INT)) \ (b,0) ) ) )
assume A44: S2[s] ; ::_thesis: ( S2[g . (s,(y gt 0))] & ( g . (s,(y gt 0)) in (Funcs (X,INT)) \ (b,0) implies S1[g . (s,(y gt 0))] ) & ( S1[g . (s,(y gt 0))] implies g . (s,(y gt 0)) in (Funcs (X,INT)) \ (b,0) ) )
reconsider s9 = g . (s,(y gt 0)) as Element of Funcs (X,INT) ;
A45: s9 . y = s . y by A2, A4, Th38;
 s9 . x = s . x by A2, A3, Th38;
hence  S2[g . (s,(y gt 0))] by A44, A45; ::_thesis: ( g . (s,(y gt 0)) in (Funcs (X,INT)) \ (b,0) iff S1[g . (s,(y gt 0))] )
A46: ( s . y <= 0 implies s9 . b = 0 ) by Th38;
 ( s . y > 0 implies s9 . b = 1 ) by Th38;
hence  ( g . (s,(y gt 0)) in (Funcs (X,INT)) \ (b,0) iff S1[g . (s,(y gt 0))] ) by A44, A46, Th2, Th38; ::_thesis: verum
end;
reconsider fin = g . (s,(while ((y gt 0),((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z))))) as Element of Funcs (X,INT) ;
A47: S2[s] by A10, A11, A12, NAT_D:def_5;
A48: ( S2[g . (s,(while ((y gt 0),((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))))] & not S1[g . (s,(while ((y gt 0),((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))))] ) from AOFA_000:sch_5(A47, A13, A33, A43);
then reconsider fn = fin . x as Element of NAT by INT_1:3;
A49: fn divides 0 by NAT_D:6;
 fin . y = 0 by A48;
then  fn divides n gcd m by A48, A49;
hence  (g . (s,(while ((y gt 0),((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))))) . x = n gcd m by A48, NAT_D:5; ::_thesis: verum
end;

theorem :: AOFA_I00:66
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y, m being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . m = 3 ) holds
(y := 1) \; (while ((m gt 0),(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))) is_terminating_wrt g, {  s where s is Element of Funcs (X,INT) : s . m >= 0  }
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y, m being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . m = 3 ) holds
(y := 1) \; (while ((m gt 0),(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))) is_terminating_wrt g, {  s where s is Element of Funcs (X,INT) : s . m >= 0  }
let X be non empty countable set ; ::_thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y, m being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . m = 3 ) holds
(y := 1) \; (while ((m gt 0),(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))) is_terminating_wrt g, {  s where s is Element of Funcs (X,INT) : s . m >= 0  }
let b be Element of X; ::_thesis: for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y, m being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . m = 3 ) holds
(y := 1) \; (while ((m gt 0),(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))) is_terminating_wrt g, {  s where s is Element of Funcs (X,INT) : s . m >= 0  }
let g be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); ::_thesis: for x, y, m being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . m = 3 ) holds
(y := 1) \; (while ((m gt 0),(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))) is_terminating_wrt g, {  s where s is Element of Funcs (X,INT) : s . m >= 0  }
set S = Funcs (X,INT);
set T = (Funcs (X,INT)) \ (b,0);
let x, y, m be Variable of g; ::_thesis: ( ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . m = 3 ) implies (y := 1) \; (while ((m gt 0),(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))) is_terminating_wrt g, {  s where s is Element of Funcs (X,INT) : s . m >= 0  }  )
set P =  {  s where s is Element of Funcs (X,INT) : s . m >= 0  }  ;
given d being Function such that A1: d . b = 0 and
A2: d . x = 1 and
A3: d . y = 2 and
A4: d . m = 3 ; ::_thesis: (y := 1) \; (while ((m gt 0),(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))) is_terminating_wrt g, {  s where s is Element of Funcs (X,INT) : s . m >= 0  }
set C = m gt 0;
A5: y := 1 is_terminating_wrt g, {  s where s is Element of Funcs (X,INT) : s . m >= 0  }  by AOFA_000:107;
deffunc H1( Element of Funcs (X,INT)) ->  Element of NAT = In (($1 . m),NAT);
defpred S1[ Element of Funcs (X,INT)] means $1 . m > 0 ;
set I = if-then ((m is_odd),(y *= x));
set J = ((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x);
A6: g complies_with_if_wrt (Funcs (X,INT)) \ (b,0) by AOFA_000:def_32;
A7:  {  s where s is Element of Funcs (X,INT) : s . m >= 0  }  is_invariant_wrt m gt 0,g
proof
let s be Element of Funcs (X,INT); :: according to AOFA_000:def_39 ::_thesis: ( not s in  {  s where s is Element of Funcs (X,INT) : s . m >= 0  }  or g . (s,(m gt 0)) in  {  s where s is Element of Funcs (X,INT) : s . m >= 0  }  )
assume  s in  {  s where s is Element of Funcs (X,INT) : s . m >= 0  }  ; ::_thesis: g . (s,(m gt 0)) in  {  s where s is Element of Funcs (X,INT) : s . m >= 0  }
then A8: ex s9 being Element of Funcs (X,INT) st
( s = s9 & s9 . m >= 0 ) ;
 (g . (s,(m gt 0))) . m = s . m by A1, A4, Th38;
hence  g . (s,(m gt 0)) in  {  s where s is Element of Funcs (X,INT) : s . m >= 0  }  by A8; ::_thesis: verum
end;
A9: for s being Element of Funcs (X,INT) st s in  {  s where s is Element of Funcs (X,INT) : s . m >= 0  }  & g . ((g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))),(m gt 0)) in (Funcs (X,INT)) \ (b,0) holds
g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x))) in  {  s where s is Element of Funcs (X,INT) : s . m >= 0  }
proof
let s be Element of Funcs (X,INT); ::_thesis: ( s in  {  s where s is Element of Funcs (X,INT) : s . m >= 0  }  & g . ((g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))),(m gt 0)) in (Funcs (X,INT)) \ (b,0) implies g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x))) in  {  s where s is Element of Funcs (X,INT) : s . m >= 0  }  )
assume  s in  {  s where s is Element of Funcs (X,INT) : s . m >= 0  }  ; ::_thesis: ( not g . ((g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))),(m gt 0)) in (Funcs (X,INT)) \ (b,0) or g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x))) in  {  s where s is Element of Funcs (X,INT) : s . m >= 0  }  )
assume  g . ((g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))),(m gt 0)) in (Funcs (X,INT)) \ (b,0) ; ::_thesis: g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x))) in  {  s where s is Element of Funcs (X,INT) : s . m >= 0  }
then  (g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))) . m > 0 by Th40;
hence  g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x))) in  {  s where s is Element of Funcs (X,INT) : s . m >= 0  }  ; ::_thesis: verum
end;
A10: m <> y by A3, A4;
A11:  {  s where s is Element of Funcs (X,INT) : s . m >= 0  }  is_invariant_wrt y := 1,g
proof
let s be Element of Funcs (X,INT); :: according to AOFA_000:def_39 ::_thesis: ( not s in  {  s where s is Element of Funcs (X,INT) : s . m >= 0  }  or g . (s,(y := 1)) in  {  s where s is Element of Funcs (X,INT) : s . m >= 0  }  )
assume  s in  {  s where s is Element of Funcs (X,INT) : s . m >= 0  }  ; ::_thesis: g . (s,(y := 1)) in  {  s where s is Element of Funcs (X,INT) : s . m >= 0  }
then A12: ex s9 being Element of Funcs (X,INT) st
( s = s9 & s9 . m >= 0 ) ;
 (g . (s,(y := 1))) . m = s . m by A10, Th25;
hence  g . (s,(y := 1)) in  {  s where s is Element of Funcs (X,INT) : s . m >= 0  }  by A12; ::_thesis: verum
end;
A13: m <> x by A2, A4;
A14: for s being Element of Funcs (X,INT) st g . (s,(m gt 0)) in  {  s where s is Element of Funcs (X,INT) : s . m >= 0  }  holds
g iteration_terminates_for (((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0),g . (s,(m gt 0))
proof
A15: for s being Element of Funcs (X,INT) st S1[s] holds
( ( S1[g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0)))] implies g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0))) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0))) in (Funcs (X,INT)) \ (b,0) implies S1[g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0)))] ) & H1(g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0)))) < H1(s) )
proof
let s be Element of Funcs (X,INT); ::_thesis: ( S1[s] implies ( ( S1[g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0)))] implies g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0))) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0))) in (Funcs (X,INT)) \ (b,0) implies S1[g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0)))] ) & H1(g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0)))) < H1(s) ) )
assume A16: s . m > 0 ; ::_thesis: ( ( S1[g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0)))] implies g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0))) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0))) in (Funcs (X,INT)) \ (b,0) implies S1[g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0)))] ) & H1(g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0)))) < H1(s) )
A17: H1(s) = s . m by A16, FUNCT_7:def_1, INT_1:3;
set q1 = g . (s,(if-then ((m is_odd),(y *= x))));
set q0 = g . (s,(m is_odd));
set sJ = g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)));
set sC = g . ((g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))),(m gt 0));
A18: g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0))) = g . ((g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))),(m gt 0)) by AOFA_000:def_29;
A19: ( (g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))) . m <= 0 implies (g . ((g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))),(m gt 0))) . b = 0 ) by Th38;
 ( (g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))) . m > 0 implies (g . ((g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))),(m gt 0))) . b = 1 ) by Th38;
hence  ( S1[g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0)))] iff g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0))) in (Funcs (X,INT)) \ (b,0) ) by A19, A18, Th2, Th38; ::_thesis: H1(g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0)))) < H1(s)
set q2 = g . ((g . (s,(if-then ((m is_odd),(y *= x))))),(m /= 2));
set q3 = g . ((g . ((g . (s,(if-then ((m is_odd),(y *= x))))),(m /= 2))),(x *= x));
A20: ( g . (s,(if-then ((m is_odd),(y *= x)))) = g . ((g . (s,(m is_odd))),(y *= x)) or g . (s,(if-then ((m is_odd),(y *= x)))) = g . ((g . (s,(m is_odd))),(EmptyIns A)) ) by A6, AOFA_000:def_30;
 g . ((g . (s,(if-then ((m is_odd),(y *= x))))),(m /= 2)) = g . (s,((if-then ((m is_odd),(y *= x))) \; (m /= 2))) by AOFA_000:def_29;
then  g . ((g . ((g . (s,(if-then ((m is_odd),(y *= x))))),(m /= 2))),(x *= x)) = g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x))) by AOFA_000:def_29;
then A21: (g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))) . m = (g . ((g . (s,(if-then ((m is_odd),(y *= x))))),(m /= 2))) . m by A13, Th33
.= ((g . (s,(if-then ((m is_odd),(y *= x))))) . m) div 2 by Th45
.= ((g . (s,(m is_odd))) . m) div 2 by A10, A20, Th33, AOFA_000:def_28
.= (s . m) div 2 by A1, A4, Th49 ;
A22: (g . ((g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))),(m gt 0))) . m = (g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))) . m by A1, A4, Th38;
then  (g . ((g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))),(m gt 0))) . m in NAT by A16, A21, INT_1:3, INT_1:61;
then  H1(g . ((g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))),(m gt 0))) = (g . ((g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))),(m gt 0))) . m by FUNCT_7:def_1;
hence  H1(g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0)))) < H1(s) by A16, A22, A18, A21, A17, INT_1:56; ::_thesis: verum
end;
let s0 be Element of Funcs (X,INT); ::_thesis: ( g . (s0,(m gt 0)) in  {  s where s is Element of Funcs (X,INT) : s . m >= 0  }  implies g iteration_terminates_for (((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0),g . (s0,(m gt 0)) )
assume  g . (s0,(m gt 0)) in  {  s where s is Element of Funcs (X,INT) : s . m >= 0  }  ; ::_thesis: g iteration_terminates_for (((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0),g . (s0,(m gt 0))
set s1 = g . (s0,(m gt 0));
A23: ( s0 . m <= 0 implies (g . (s0,(m gt 0))) . b = 0 ) by Th38;
 ( s0 . m > 0 implies (g . (s0,(m gt 0))) . b = 1 ) by Th38;
then A24: ( g . (s0,(m gt 0)) in (Funcs (X,INT)) \ (b,0) iff S1[g . (s0,(m gt 0))] ) by A23, Th2, Th38;
thus  g iteration_terminates_for (((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0),g . (s0,(m gt 0)) from AOFA_000:sch_3(A24, A15); ::_thesis: verum
end;
 ((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x) is_terminating_wrt g, {  s where s is Element of Funcs (X,INT) : s . m >= 0  }  by AOFA_000:107;
then  while ((m gt 0),(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x))) is_terminating_wrt g, {  s where s is Element of Funcs (X,INT) : s . m >= 0  }  by A7, A9, A14, AOFA_000:104, AOFA_000:118;
hence  (y := 1) \; (while ((m gt 0),(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))) is_terminating_wrt g, {  s where s is Element of Funcs (X,INT) : s . m >= 0  }  by A5, A11, AOFA_000:111; ::_thesis: verum
end;

theorem :: AOFA_I00:67
for A being Euclidean preIfWhileAlgebra
for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y, m being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . m = 3 ) holds
for s being Element of Funcs (X,INT)
for n being Nat st n = s . m holds
(g . (s,((y := 1) \; (while ((m gt 0),(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x))))))) . y = (s . x) |^ n
proof
let A be Euclidean preIfWhileAlgebra; ::_thesis: for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y, m being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . m = 3 ) holds
for s being Element of Funcs (X,INT)
for n being Nat st n = s . m holds
(g . (s,((y := 1) \; (while ((m gt 0),(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x))))))) . y = (s . x) |^ n
let X be non empty countable set ; ::_thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y, m being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . m = 3 ) holds
for s being Element of Funcs (X,INT)
for n being Nat st n = s . m holds
(g . (s,((y := 1) \; (while ((m gt 0),(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x))))))) . y = (s . x) |^ n
let b be Element of X; ::_thesis: for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for x, y, m being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . m = 3 ) holds
for s being Element of Funcs (X,INT)
for n being Nat st n = s . m holds
(g . (s,((y := 1) \; (while ((m gt 0),(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x))))))) . y = (s . x) |^ n
let g be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); ::_thesis: for x, y, m being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . m = 3 ) holds
for s being Element of Funcs (X,INT)
for n being Nat st n = s . m holds
(g . (s,((y := 1) \; (while ((m gt 0),(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x))))))) . y = (s . x) |^ n
set S = Funcs (X,INT);
set T = (Funcs (X,INT)) \ (b,0);
A1: g complies_with_if_wrt (Funcs (X,INT)) \ (b,0) by AOFA_000:def_32;
let x, y, m be Variable of g; ::_thesis: ( ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . m = 3 ) implies for s being Element of Funcs (X,INT)
for n being Nat st n = s . m holds
(g . (s,((y := 1) \; (while ((m gt 0),(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x))))))) . y = (s . x) |^ n )
given d being Function such that A2: d . b = 0 and
A3: d . x = 1 and
A4: d . y = 2 and
A5: d . m = 3 ; ::_thesis: for s being Element of Funcs (X,INT)
for n being Nat st n = s . m holds
(g . (s,((y := 1) \; (while ((m gt 0),(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x))))))) . y = (s . x) |^ n
defpred S1[ Element of Funcs (X,INT)] means $1 . m > 0 ;
set C = m gt 0;
let s be Element of Funcs (X,INT); ::_thesis: for n being Nat st n = s . m holds
(g . (s,((y := 1) \; (while ((m gt 0),(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x))))))) . y = (s . x) |^ n
let n be Nat; ::_thesis: ( n = s . m implies (g . (s,((y := 1) \; (while ((m gt 0),(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x))))))) . y = (s . x) |^ n )
defpred S2[ Element of Funcs (X,INT)] means ( (s . x) |^ n = ($1 . y) * (($1 . x) to_power ($1 . m)) & $1 . m >= 0 );
deffunc H1( Element of Funcs (X,INT)) ->  Element of NAT = In (($1 . m),NAT);
set I = if-then ((m is_odd),(y *= x));
set J = ((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x);
set s0 = g . (s,(y := 1));
A6: m <> y by A4, A5;
then A7: (g . (s,(y := 1))) . m = s . m by Th25;
A8: for s being Element of Funcs (X,INT) st S2[s] holds
( S2[g . (s,(m gt 0))] & ( g . (s,(m gt 0)) in (Funcs (X,INT)) \ (b,0) implies S1[g . (s,(m gt 0))] ) & ( S1[g . (s,(m gt 0))] implies g . (s,(m gt 0)) in (Funcs (X,INT)) \ (b,0) ) )
proof
let s be Element of Funcs (X,INT); ::_thesis: ( S2[s] implies ( S2[g . (s,(m gt 0))] & ( g . (s,(m gt 0)) in (Funcs (X,INT)) \ (b,0) implies S1[g . (s,(m gt 0))] ) & ( S1[g . (s,(m gt 0))] implies g . (s,(m gt 0)) in (Funcs (X,INT)) \ (b,0) ) ) )
assume A9: S2[s] ; ::_thesis: ( S2[g . (s,(m gt 0))] & ( g . (s,(m gt 0)) in (Funcs (X,INT)) \ (b,0) implies S1[g . (s,(m gt 0))] ) & ( S1[g . (s,(m gt 0))] implies g . (s,(m gt 0)) in (Funcs (X,INT)) \ (b,0) ) )
set s1 = g . (s,(m gt 0));
A10: (g . (s,(m gt 0))) . x = s . x by A2, A3, Th38;
 (g . (s,(m gt 0))) . m = s . m by A2, A5, Th38;
hence  S2[g . (s,(m gt 0))] by A2, A4, A9, A10, Th38; ::_thesis: ( g . (s,(m gt 0)) in (Funcs (X,INT)) \ (b,0) iff S1[g . (s,(m gt 0))] )
A11: ( s . m <= 0 implies (g . (s,(m gt 0))) . b = 0 ) by Th38;
 ( s . m > 0 implies (g . (s,(m gt 0))) . b = 1 ) by Th38;
hence  ( g . (s,(m gt 0)) in (Funcs (X,INT)) \ (b,0) iff S1[g . (s,(m gt 0))] ) by A11, Th2, Th38; ::_thesis: verum
end;
A12: (g . (s,(y := 1))) . y = 1 by Th25;
set fs = g . ((g . (s,(y := 1))),(while ((m gt 0),(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))));
set s1 = g . ((g . (s,(y := 1))),(m gt 0));
assume A13: n = s . m ; ::_thesis: (g . (s,((y := 1) \; (while ((m gt 0),(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x))))))) . y = (s . x) |^ n
A14: ((g . ((g . (s,(y := 1))),(while ((m gt 0),(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))))) . x) to_power 0 = 1 by POWER:24;
A15: m <> x by A3, A5;
A16: for s being Element of Funcs (X,INT) st S1[s] holds
( ( S1[g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0)))] implies g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0))) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0))) in (Funcs (X,INT)) \ (b,0) implies S1[g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0)))] ) & H1(g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0)))) < H1(s) )
proof
let s be Element of Funcs (X,INT); ::_thesis: ( S1[s] implies ( ( S1[g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0)))] implies g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0))) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0))) in (Funcs (X,INT)) \ (b,0) implies S1[g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0)))] ) & H1(g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0)))) < H1(s) ) )
assume A17: s . m > 0 ; ::_thesis: ( ( S1[g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0)))] implies g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0))) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0))) in (Funcs (X,INT)) \ (b,0) implies S1[g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0)))] ) & H1(g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0)))) < H1(s) )
A18: H1(s) = s . m by A17, FUNCT_7:def_1, INT_1:3;
set q1 = g . (s,(if-then ((m is_odd),(y *= x))));
set q0 = g . (s,(m is_odd));
set sJ = g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)));
set sC = g . ((g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))),(m gt 0));
A19: g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0))) = g . ((g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))),(m gt 0)) by AOFA_000:def_29;
A20: ( (g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))) . m <= 0 implies (g . ((g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))),(m gt 0))) . b = 0 ) by Th38;
 ( (g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))) . m > 0 implies (g . ((g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))),(m gt 0))) . b = 1 ) by Th38;
hence  ( S1[g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0)))] iff g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0))) in (Funcs (X,INT)) \ (b,0) ) by A20, A19, Th2, Th38; ::_thesis: H1(g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0)))) < H1(s)
set q2 = g . ((g . (s,(if-then ((m is_odd),(y *= x))))),(m /= 2));
set q3 = g . ((g . ((g . (s,(if-then ((m is_odd),(y *= x))))),(m /= 2))),(x *= x));
A21: ( g . (s,(if-then ((m is_odd),(y *= x)))) = g . ((g . (s,(m is_odd))),(y *= x)) or g . (s,(if-then ((m is_odd),(y *= x)))) = g . ((g . (s,(m is_odd))),(EmptyIns A)) ) by A1, AOFA_000:def_30;
 g . ((g . (s,(if-then ((m is_odd),(y *= x))))),(m /= 2)) = g . (s,((if-then ((m is_odd),(y *= x))) \; (m /= 2))) by AOFA_000:def_29;
then  g . ((g . ((g . (s,(if-then ((m is_odd),(y *= x))))),(m /= 2))),(x *= x)) = g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x))) by AOFA_000:def_29;
then A22: (g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))) . m = (g . ((g . (s,(if-then ((m is_odd),(y *= x))))),(m /= 2))) . m by A15, Th33
.= ((g . (s,(if-then ((m is_odd),(y *= x))))) . m) div 2 by Th45
.= ((g . (s,(m is_odd))) . m) div 2 by A6, A21, Th33, AOFA_000:def_28
.= (s . m) div 2 by A2, A5, Th49 ;
A23: (g . ((g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))),(m gt 0))) . m = (g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))) . m by A2, A5, Th38;
then  (g . ((g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))),(m gt 0))) . m in NAT by A17, A22, INT_1:3, INT_1:61;
then  H1(g . ((g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))),(m gt 0))) = (g . ((g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))),(m gt 0))) . m by FUNCT_7:def_1;
hence  H1(g . (s,((((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0)))) < H1(s) by A17, A23, A19, A22, A18, INT_1:56; ::_thesis: verum
end;
set q = s;
A24: x <> y by A3, A4;
A25: for s being Element of Funcs (X,INT) st S2[s] & s in (Funcs (X,INT)) \ (b,0) & S1[s] holds
S2[g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))]
proof
let s be Element of Funcs (X,INT); ::_thesis: ( S2[s] & s in (Funcs (X,INT)) \ (b,0) & S1[s] implies S2[g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))] )
assume that
A26: S2[s] and
 s in (Funcs (X,INT)) \ (b,0) and
 S1[s] ; ::_thesis: S2[g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))]
reconsider sm = s . m as Element of NAT by A26, INT_1:3;
 s . m = (((s . m) div 2) * 2) + ((s . m) mod 2) by NEWTON:66;
then A27: (s . x) |^ n = (s . y) * (((s . x) to_power ((sm div 2) * 2)) * ((s . x) to_power (sm mod 2))) by A26, FIB_NUM2:5
.= ((s . y) * ((s . x) to_power (sm mod 2))) * ((s . x) to_power ((sm div 2) * 2))
.= ((s . y) * ((s . x) to_power (sm mod 2))) * (((s . x) to_power 2) to_power (sm div 2)) by NEWTON:9
.= ((s . y) * ((s . x) to_power (sm mod 2))) * (((s . x) * (s . x)) to_power (sm div 2)) by NEWTON:81 ;
set q1 = g . (s,(if-then ((m is_odd),(y *= x))));
set q0 = g . (s,(m is_odd));
set sJ = g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)));
set q2 = g . ((g . (s,(if-then ((m is_odd),(y *= x))))),(m /= 2));
set q3 = g . ((g . ((g . (s,(if-then ((m is_odd),(y *= x))))),(m /= 2))),(x *= x));
A28: ( g . (s,(if-then ((m is_odd),(y *= x)))) = g . ((g . (s,(m is_odd))),(y *= x)) or g . (s,(if-then ((m is_odd),(y *= x)))) = g . ((g . (s,(m is_odd))),(EmptyIns A)) ) by A1, AOFA_000:def_30;
A29: (g . ((g . (s,(if-then ((m is_odd),(y *= x))))),(m /= 2))) . x = (g . (s,(if-then ((m is_odd),(y *= x))))) . x by A15, Th45
.= (g . (s,(m is_odd))) . x by A24, A28, Th33, AOFA_000:def_28 ;
A30: (g . ((g . (s,(if-then ((m is_odd),(y *= x))))),(m /= 2))) . y = (g . (s,(if-then ((m is_odd),(y *= x))))) . y by A6, Th45;
A31: (g . (s,(m is_odd))) . y = s . y by A2, A4, Th49;
A32: (g . (s,(m is_odd))) . x = s . x by A2, A3, Th49;
 g . ((g . (s,(if-then ((m is_odd),(y *= x))))),(m /= 2)) = g . (s,((if-then ((m is_odd),(y *= x))) \; (m /= 2))) by AOFA_000:def_29;
then A33: g . ((g . ((g . (s,(if-then ((m is_odd),(y *= x))))),(m /= 2))),(x *= x)) = g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x))) by AOFA_000:def_29;
then A34: (g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))) . y = (g . ((g . (s,(if-then ((m is_odd),(y *= x))))),(m /= 2))) . y by A24, Th33;
A35: sm div 2 = (s . m) div 2 ;
A36: now__::_thesis:_(s_._y)_*_((s_._x)_to_power_(sm_mod_2))_=_(g_._(s,(((if-then_((m_is_odd),(y_*=_x)))_\;_(m_/=_2))_\;_(x_*=_x))))_._y
A37: (g . (s,(m is_odd))) . b = (s . m) mod 2 by Th49;
percases ( (g . (s,(m is_odd))) . b = 0 or (g . (s,(m is_odd))) . b = 1 ) by A35, A37, NAT_D:12;
supposeA38: (g . (s,(m is_odd))) . b = 0 ; ::_thesis: (s . y) * ((s . x) to_power (sm mod 2)) = (g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))) . y
then  g . (s,(m is_odd)) nin (Funcs (X,INT)) \ (b,0) by Th2;
then  g . (s,(if-then ((m is_odd),(y *= x)))) = g . ((g . (s,(m is_odd))),(EmptyIns A)) by A1, AOFA_000:def_30;
then A39: (g . (s,(if-then ((m is_odd),(y *= x))))) . y = (g . (s,(m is_odd))) . y by AOFA_000:def_28;
A40: (s . y) * 1 = s . y ;
 (s . x) to_power 0 = 1 by POWER:24;
hence  (s . y) * ((s . x) to_power (sm mod 2)) = (g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))) . y by A34, A30, A31, A38, A39, A40, Th49; ::_thesis: verum
end;
supposeA41: (g . (s,(m is_odd))) . b = 1 ; ::_thesis: (s . y) * ((s . x) to_power (sm mod 2)) = (g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))) . y
then  g . (s,(m is_odd)) in (Funcs (X,INT)) \ (b,0) ;
then  g . (s,(if-then ((m is_odd),(y *= x)))) = g . ((g . (s,(m is_odd))),(y *= x)) by A1, AOFA_000:def_30;
then A42: (g . (s,(if-then ((m is_odd),(y *= x))))) . y = ((g . (s,(m is_odd))) . y) * ((g . (s,(m is_odd))) . x) by Th33;
 (s . x) to_power 1 = s . x by POWER:25;
hence  (s . y) * ((s . x) to_power (sm mod 2)) = (g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))) . y by A32, A34, A30, A31, A41, A42, Th49; ::_thesis: verum
end;
end;
end;
(g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))) . m = (g . ((g . (s,(if-then ((m is_odd),(y *= x))))),(m /= 2))) . m by A15, A33, Th33
.= ((g . (s,(if-then ((m is_odd),(y *= x))))) . m) div 2 by Th45
.= ((g . (s,(m is_odd))) . m) div 2 by A6, A28, Th33, AOFA_000:def_28
.= (s . m) div 2 by A2, A5, Th49 ;
hence  S2[g . (s,(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))] by A33, A29, A32, A36, A27, Th33; ::_thesis: verum
end;
A43: ( (g . (s,(y := 1))) . m <= 0 implies (g . ((g . (s,(y := 1))),(m gt 0))) . b = 0 ) by Th38;
 ( (g . (s,(y := 1))) . m > 0 implies (g . ((g . (s,(y := 1))),(m gt 0))) . b = 1 ) by Th38;
then A44: ( g . ((g . (s,(y := 1))),(m gt 0)) in (Funcs (X,INT)) \ (b,0) iff S1[g . ((g . (s,(y := 1))),(m gt 0))] ) by A43, Th2, Th38;
A45: g iteration_terminates_for (((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)) \; (m gt 0),g . ((g . (s,(y := 1))),(m gt 0)) from AOFA_000:sch_3(A44, A16);
 (g . (s,(y := 1))) . x = s . x by A24, Th25;
then A46: S2[g . (s,(y := 1))] by A13, A7, A12, POWER:41;
A47: ( S2[g . ((g . (s,(y := 1))),(while ((m gt 0),(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))))] & not S1[g . ((g . (s,(y := 1))),(while ((m gt 0),(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))))] ) from AOFA_000:sch_5(A46, A45, A25, A8);
then  (g . ((g . (s,(y := 1))),(while ((m gt 0),(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))))) . m = 0 ;
hence  (g . (s,((y := 1) \; (while ((m gt 0),(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x))))))) . y = (s . x) |^ n by A47, A14, AOFA_000:def_29; ::_thesis: verum
end;