let i be Instruction of SCMPDS ; :: thesis: for s1, s2 being State of SCMPDS st DataPart s1 = DataPart s2 & InsCode i <> 3 holds
DataPart (Exec i,s1) = DataPart (Exec i,s2)
let s1, s2 be State of SCMPDS ; :: thesis: ( DataPart s1 = DataPart s2 & InsCode i <> 3 implies DataPart (Exec i,s1) = DataPart (Exec i,s2) )
assume that
A1: DataPart s1 = DataPart s2 and
A2: InsCode i <> 3 ; :: thesis: DataPart (Exec i,s1) = DataPart (Exec i,s2)

per cases ( InsCode i = 0 or InsCode i = 1 or InsCode i = 2 or InsCode i = 4 or InsCode i = 5 or InsCode i = 6 or InsCode i = 7 or InsCode i = 8 or InsCode i = 9 or InsCode i = 10 or InsCode i = 11 or InsCode i = 12 or InsCode i = 13 ) by A2, NAT_1:38, SCMPDS_2:15;

suppose InsCode i = 0 ; :: thesis: DataPart (Exec i,s1) = DataPart (Exec i,s2)

then A3: ex k1 being Integer st i = goto k1 by SCMPDS_2:35;

now

let a be Int_position ; :: thesis: (Exec i,s1) . a = (Exec i,s2) . a
thus (Exec i,s1) . a = s1 . a by A3, SCMPDS_2:66
.= s2 . a by A1, SCMPDS_4:23
.= (Exec i,s2) . a by A3, SCMPDS_2:66 ; :: thesis: verum

end;

hence DataPart (Exec i,s1) = DataPart (Exec i,s2) by SCMPDS_4:23; :: thesis: verum

end;

suppose InsCode i = 1 ; :: thesis: DataPart (Exec i,s1) = DataPart (Exec i,s2)

then consider a being Int_position such that
A4: i = return a by SCMPDS_2:36;

now

let b be Int_position ; :: thesis: (Exec i,s1) . b₁ = (Exec i,s2) . b₁

per cases ( a = b or a <> b ) ;

suppose A5: a = b ; :: thesis: (Exec i,s1) . b₁ = (Exec i,s2) . b₁

hence (Exec i,s1) . b = s1 . (DataLoc (s1 . a),RetSP ) by A4, SCMPDS_2:70
.= s2 . (DataLoc (s2 . a),RetSP ) by A1, SCMPDS_3:3
.= (Exec i,s2) . b by A4, A5, SCMPDS_2:70 ;
:: thesis: verum

end;

suppose A6: a <> b ; :: thesis: (Exec i,s1) . b₁ = (Exec i,s2) . b₁

hence (Exec i,s1) . b = s1 . b by A4, SCMPDS_2:70
.= s2 . b by A1, SCMPDS_4:23
.= (Exec i,s2) . b by A4, A6, SCMPDS_2:70 ;
:: thesis: verum

end;

end;

end;

hence DataPart (Exec i,s1) = DataPart (Exec i,s2) by SCMPDS_4:23; :: thesis: verum

end;

suppose InsCode i = 2 ; :: thesis: DataPart (Exec i,s1) = DataPart (Exec i,s2)

then consider a being Int_position , k1 being Integer such that
A7: i = a := k1 by SCMPDS_2:37;

now

let b be Int_position ; :: thesis: (Exec i,s1) . b₁ = (Exec i,s2) . b₁

per cases ( a = b or a <> b ) ;

suppose A8: a = b ; :: thesis: (Exec i,s1) . b₁ = (Exec i,s2) . b₁

hence (Exec i,s1) . b = k1 by A7, SCMPDS_2:57
.= (Exec i,s2) . b by A7, A8, SCMPDS_2:57 ;
:: thesis: verum

end;

suppose A9: a <> b ; :: thesis: (Exec i,s1) . b₁ = (Exec i,s2) . b₁

hence (Exec i,s1) . b = s1 . b by A7, SCMPDS_2:57
.= s2 . b by A1, SCMPDS_4:23
.= (Exec i,s2) . b by A7, A9, SCMPDS_2:57 ;
:: thesis: verum

end;

end;

end;

hence DataPart (Exec i,s1) = DataPart (Exec i,s2) by SCMPDS_4:23; :: thesis: verum

end;

suppose InsCode i = 4 ; :: thesis: DataPart (Exec i,s1) = DataPart (Exec i,s2)

then A10: ex a being Int_position ex k1, k2 being Integer st i = a,k1 <>0_goto k2 by SCMPDS_2:39;

now

let a be Int_position ; :: thesis: (Exec i,s1) . a = (Exec i,s2) . a
thus (Exec i,s1) . a = s1 . a by A10, SCMPDS_2:67
.= s2 . a by A1, SCMPDS_4:23
.= (Exec i,s2) . a by A10, SCMPDS_2:67 ; :: thesis: verum

end;

hence DataPart (Exec i,s1) = DataPart (Exec i,s2) by SCMPDS_4:23; :: thesis: verum

end;

suppose InsCode i = 5 ; :: thesis: DataPart (Exec i,s1) = DataPart (Exec i,s2)

then A11: ex a being Int_position ex k1, k2 being Integer st i = a,k1 <=0_goto k2 by SCMPDS_2:40;

now

let a be Int_position ; :: thesis: (Exec i,s1) . a = (Exec i,s2) . a
thus (Exec i,s1) . a = s1 . a by A11, SCMPDS_2:68
.= s2 . a by A1, SCMPDS_4:23
.= (Exec i,s2) . a by A11, SCMPDS_2:68 ; :: thesis: verum

end;

hence DataPart (Exec i,s1) = DataPart (Exec i,s2) by SCMPDS_4:23; :: thesis: verum

end;

suppose InsCode i = 6 ; :: thesis: DataPart (Exec i,s1) = DataPart (Exec i,s2)

then A12: ex a being Int_position ex k1, k2 being Integer st i = a,k1 >=0_goto k2 by SCMPDS_2:41;

now

let a be Int_position ; :: thesis: (Exec i,s1) . a = (Exec i,s2) . a
thus (Exec i,s1) . a = s1 . a by A12, SCMPDS_2:69
.= s2 . a by A1, SCMPDS_4:23
.= (Exec i,s2) . a by A12, SCMPDS_2:69 ; :: thesis: verum

end;

hence DataPart (Exec i,s1) = DataPart (Exec i,s2) by SCMPDS_4:23; :: thesis: verum

end;

suppose InsCode i = 7 ; :: thesis: DataPart (Exec i,s1) = DataPart (Exec i,s2)

then consider a being Int_position , k1, k2 being Integer such that
A13: i = a,k1 := k2 by SCMPDS_2:42;

now

let b be Int_position ; :: thesis: (Exec i,s1) . b₁ = (Exec i,s2) . b₁

per cases ( DataLoc (s1 . a),k1 = b or DataLoc (s1 . a),k1 <> b ) ;

suppose A14: DataLoc (s1 . a),k1 = b ; :: thesis: (Exec i,s1) . b₁ = (Exec i,s2) . b₁

then A15: DataLoc (s2 . a),k1 = b by A1, SCMPDS_4:23;
thus (Exec i,s1) . b = k2 by A13, A14, SCMPDS_2:58
.= (Exec i,s2) . b by A13, A15, SCMPDS_2:58 ; :: thesis: verum

end;

suppose A16: DataLoc (s1 . a),k1 <> b ; :: thesis: (Exec i,s1) . b₁ = (Exec i,s2) . b₁

then A17: DataLoc (s2 . a),k1 <> b by A1, SCMPDS_4:23;
thus (Exec i,s1) . b = s1 . b by A13, A16, SCMPDS_2:58
.= s2 . b by A1, SCMPDS_4:23
.= (Exec i,s2) . b by A13, A17, SCMPDS_2:58 ; :: thesis: verum

end;

end;

end;

hence DataPart (Exec i,s1) = DataPart (Exec i,s2) by SCMPDS_4:23; :: thesis: verum

end;

suppose InsCode i = 8 ; :: thesis: DataPart (Exec i,s1) = DataPart (Exec i,s2)

then consider a being Int_position , k1, k2 being Integer such that
A18: i = AddTo a,k1,k2 by SCMPDS_2:43;

now

let b be Int_position ; :: thesis: (Exec i,s1) . b₁ = (Exec i,s2) . b₁

per cases ( DataLoc (s1 . a),k1 = b or DataLoc (s1 . a),k1 <> b ) ;

suppose A19: DataLoc (s1 . a),k1 = b ; :: thesis: (Exec i,s1) . b₁ = (Exec i,s2) . b₁

then A20: DataLoc (s2 . a),k1 = b by A1, SCMPDS_4:23;
thus (Exec i,s1) . b = (s1 . (DataLoc (s1 . a),k1)) + k2 by A18, A19, SCMPDS_2:60
.= (s2 . (DataLoc (s2 . a),k1)) + k2 by A1, SCMPDS_3:3
.= (Exec i,s2) . b by A18, A20, SCMPDS_2:60 ; :: thesis: verum

end;

suppose A21: DataLoc (s1 . a),k1 <> b ; :: thesis: (Exec i,s1) . b₁ = (Exec i,s2) . b₁

then A22: DataLoc (s2 . a),k1 <> b by A1, SCMPDS_4:23;
thus (Exec i,s1) . b = s1 . b by A18, A21, SCMPDS_2:60
.= s2 . b by A1, SCMPDS_4:23
.= (Exec i,s2) . b by A18, A22, SCMPDS_2:60 ; :: thesis: verum

end;

end;

end;

hence DataPart (Exec i,s1) = DataPart (Exec i,s2) by SCMPDS_4:23; :: thesis: verum

end;

suppose InsCode i = 9 ; :: thesis: DataPart (Exec i,s1) = DataPart (Exec i,s2)

then consider a, b being Int_position , k1, k2 being Integer such that
A23: i = AddTo a,k1,b,k2 by SCMPDS_2:44;

now

let c be Int_position ; :: thesis: (Exec i,s1) . b₁ = (Exec i,s2) . b₁

per cases ( DataLoc (s1 . a),k1 = c or DataLoc (s1 . a),k1 <> c ) ;

suppose A24: DataLoc (s1 . a),k1 = c ; :: thesis: (Exec i,s1) . b₁ = (Exec i,s2) . b₁

then A25: DataLoc (s2 . a),k1 = c by A1, SCMPDS_4:23;
thus (Exec i,s1) . c = (s1 . (DataLoc (s1 . a),k1)) + (s1 . (DataLoc (s1 . b),k2)) by A23, A24, SCMPDS_2:61
.= (s2 . (DataLoc (s2 . a),k1)) + (s1 . (DataLoc (s1 . b),k2)) by A1, SCMPDS_3:3
.= (s2 . (DataLoc (s2 . a),k1)) + (s2 . (DataLoc (s2 . b),k2)) by A1, SCMPDS_3:3
.= (Exec i,s2) . c by A23, A25, SCMPDS_2:61 ; :: thesis: verum

end;

suppose A26: DataLoc (s1 . a),k1 <> c ; :: thesis: (Exec i,s1) . b₁ = (Exec i,s2) . b₁

then A27: DataLoc (s2 . a),k1 <> c by A1, SCMPDS_4:23;
thus (Exec i,s1) . c = s1 . c by A23, A26, SCMPDS_2:61
.= s2 . c by A1, SCMPDS_4:23
.= (Exec i,s2) . c by A23, A27, SCMPDS_2:61 ; :: thesis: verum

end;

end;

end;

hence DataPart (Exec i,s1) = DataPart (Exec i,s2) by SCMPDS_4:23; :: thesis: verum

end;

suppose InsCode i = 10 ; :: thesis: DataPart (Exec i,s1) = DataPart (Exec i,s2)

then consider a, b being Int_position , k1, k2 being Integer such that
A28: i = SubFrom a,k1,b,k2 by SCMPDS_2:45;

now

let c be Int_position ; :: thesis: (Exec i,s1) . b₁ = (Exec i,s2) . b₁

per cases ( DataLoc (s1 . a),k1 = c or DataLoc (s1 . a),k1 <> c ) ;

suppose A29: DataLoc (s1 . a),k1 = c ; :: thesis: (Exec i,s1) . b₁ = (Exec i,s2) . b₁

then A30: DataLoc (s2 . a),k1 = c by A1, SCMPDS_4:23;
thus (Exec i,s1) . c = (s1 . (DataLoc (s1 . a),k1)) - (s1 . (DataLoc (s1 . b),k2)) by A28, A29, SCMPDS_2:62
.= (s2 . (DataLoc (s2 . a),k1)) - (s1 . (DataLoc (s1 . b),k2)) by A1, SCMPDS_3:3
.= (s2 . (DataLoc (s2 . a),k1)) - (s2 . (DataLoc (s2 . b),k2)) by A1, SCMPDS_3:3
.= (Exec i,s2) . c by A28, A30, SCMPDS_2:62 ; :: thesis: verum

end;

suppose A31: DataLoc (s1 . a),k1 <> c ; :: thesis: (Exec i,s1) . b₁ = (Exec i,s2) . b₁

then A32: DataLoc (s2 . a),k1 <> c by A1, SCMPDS_4:23;
thus (Exec i,s1) . c = s1 . c by A28, A31, SCMPDS_2:62
.= s2 . c by A1, SCMPDS_4:23
.= (Exec i,s2) . c by A28, A32, SCMPDS_2:62 ; :: thesis: verum

end;

end;

end;

hence DataPart (Exec i,s1) = DataPart (Exec i,s2) by SCMPDS_4:23; :: thesis: verum

end;

suppose InsCode i = 11 ; :: thesis: DataPart (Exec i,s1) = DataPart (Exec i,s2)

then consider a, b being Int_position , k1, k2 being Integer such that
A33: i = MultBy a,k1,b,k2 by SCMPDS_2:46;

now

let c be Int_position ; :: thesis: (Exec i,s1) . b₁ = (Exec i,s2) . b₁

per cases ( DataLoc (s1 . a),k1 = c or DataLoc (s1 . a),k1 <> c ) ;

suppose A34: DataLoc (s1 . a),k1 = c ; :: thesis: (Exec i,s1) . b₁ = (Exec i,s2) . b₁

then A35: DataLoc (s2 . a),k1 = c by A1, SCMPDS_4:23;
thus (Exec i,s1) . c = (s1 . (DataLoc (s1 . a),k1)) * (s1 . (DataLoc (s1 . b),k2)) by A33, A34, SCMPDS_2:63
.= (s2 . (DataLoc (s2 . a),k1)) * (s1 . (DataLoc (s1 . b),k2)) by A1, SCMPDS_3:3
.= (s2 . (DataLoc (s2 . a),k1)) * (s2 . (DataLoc (s2 . b),k2)) by A1, SCMPDS_3:3
.= (Exec i,s2) . c by A33, A35, SCMPDS_2:63 ; :: thesis: verum

end;

suppose A36: DataLoc (s1 . a),k1 <> c ; :: thesis: (Exec i,s1) . b₁ = (Exec i,s2) . b₁

then A37: DataLoc (s2 . a),k1 <> c by A1, SCMPDS_4:23;
thus (Exec i,s1) . c = s1 . c by A33, A36, SCMPDS_2:63
.= s2 . c by A1, SCMPDS_4:23
.= (Exec i,s2) . c by A33, A37, SCMPDS_2:63 ; :: thesis: verum

end;

end;

end;

hence DataPart (Exec i,s1) = DataPart (Exec i,s2) by SCMPDS_4:23; :: thesis: verum

end;

suppose InsCode i = 12 ; :: thesis: DataPart (Exec i,s1) = DataPart (Exec i,s2)

then consider a, b being Int_position , k1, k2 being Integer such that
A38: i = Divide a,k1,b,k2 by SCMPDS_2:47;

now

let c be Int_position ; :: thesis: (Exec i,s1) . b₁ = (Exec i,s2) . b₁

per cases ( DataLoc (s1 . b),k2 = c or DataLoc (s1 . b),k2 <> c ) ;

suppose A39: DataLoc (s1 . b),k2 = c ; :: thesis: (Exec i,s1) . b₁ = (Exec i,s2) . b₁

then A40: DataLoc (s2 . b),k2 = c by A1, SCMPDS_4:23;
thus (Exec i,s1) . c = (s1 . (DataLoc (s1 . a),k1)) mod (s1 . (DataLoc (s1 . b),k2)) by A38, A39, SCMPDS_2:64
.= (s2 . (DataLoc (s2 . a),k1)) mod (s1 . (DataLoc (s1 . b),k2)) by A1, SCMPDS_3:3
.= (s2 . (DataLoc (s2 . a),k1)) mod (s2 . (DataLoc (s2 . b),k2)) by A1, SCMPDS_3:3
.= (Exec i,s2) . c by A38, A40, SCMPDS_2:64 ; :: thesis: verum

end;

suppose A41: DataLoc (s1 . b),k2 <> c ; :: thesis: (Exec i,s1) . b₁ = (Exec i,s2) . b₁

then A42: DataLoc (s2 . b),k2 <> c by A1, SCMPDS_4:23;

hereby :: thesis: verum

per cases ( DataLoc (s1 . a),k1 <> c or DataLoc (s1 . a),k1 = c ) ;

suppose A43: DataLoc (s1 . a),k1 <> c ; :: thesis: (Exec i,s1) . c = (Exec i,s2) . c

then A44: DataLoc (s2 . a),k1 <> c by A1, SCMPDS_4:23;
thus (Exec i,s1) . c = s1 . c by A38, A41, A43, SCMPDS_2:64
.= s2 . c by A1, SCMPDS_4:23
.= (Exec i,s2) . c by A38, A42, A44, SCMPDS_2:64 ; :: thesis: verum

end;

suppose A45: DataLoc (s1 . a),k1 = c ; :: thesis: (Exec i,s1) . c = (Exec i,s2) . c

then A46: DataLoc (s2 . a),k1 = c by A1, SCMPDS_4:23;
thus (Exec i,s1) . c = (s1 . (DataLoc (s1 . a),k1)) div (s1 . (DataLoc (s1 . b),k2)) by A38, A41, A45, SCMPDS_2:64
.= (s2 . (DataLoc (s2 . a),k1)) div (s1 . (DataLoc (s1 . b),k2)) by A1, SCMPDS_3:3
.= (s2 . (DataLoc (s2 . a),k1)) div (s2 . (DataLoc (s2 . b),k2)) by A1, SCMPDS_3:3
.= (Exec i,s2) . c by A38, A42, A46, SCMPDS_2:64 ; :: thesis: verum

end;

end;

end;

end;

end;

end;

hence DataPart (Exec i,s1) = DataPart (Exec i,s2) by SCMPDS_4:23; :: thesis: verum

end;

suppose InsCode i = 13 ; :: thesis: DataPart (Exec i,s1) = DataPart (Exec i,s2)

then consider a, b being Int_position , k1, k2 being Integer such that
A47: i = a,k1 := b,k2 by SCMPDS_2:48;

now

let c be Int_position ; :: thesis: (Exec i,s1) . b₁ = (Exec i,s2) . b₁

per cases ( DataLoc (s1 . a),k1 = c or DataLoc (s1 . a),k1 <> c ) ;

suppose A48: DataLoc (s1 . a),k1 = c ; :: thesis: (Exec i,s1) . b₁ = (Exec i,s2) . b₁

then A49: DataLoc (s2 . a),k1 = c by A1, SCMPDS_4:23;
thus (Exec i,s1) . c = s1 . (DataLoc (s1 . b),k2) by A47, A48, SCMPDS_2:59
.= s2 . (DataLoc (s2 . b),k2) by A1, SCMPDS_3:3
.= (Exec i,s2) . c by A47, A49, SCMPDS_2:59 ; :: thesis: verum

end;

suppose A50: DataLoc (s1 . a),k1 <> c ; :: thesis: (Exec i,s1) . b₁ = (Exec i,s2) . b₁

then A51: DataLoc (s2 . a),k1 <> c by A1, SCMPDS_4:23;
thus (Exec i,s1) . c = s1 . c by A47, A50, SCMPDS_2:59
.= s2 . c by A1, SCMPDS_4:23
.= (Exec i,s2) . c by A47, A51, SCMPDS_2:59 ; :: thesis: verum

end;

end;

end;

hence DataPart (Exec i,s1) = DataPart (Exec i,s2) by SCMPDS_4:23; :: thesis: verum

end;

end;