/usr/share/spark/checker/rules/ARITH.RUL is in spark 2012.0.deb-11.
This file is owned by root:root, with mode 0o644.
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% (C) Altran Praxis Limited
% -----------------------------------------------------------------------------
%
% The SPARK toolset is free software; you can redistribute it and/or modify it
% under terms of the GNU General Public License as published by the Free
% Software Foundation; either version 3, or (at your option) any later
% version. The SPARK toolset is distributed in the hope that it will be
% useful, but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General
% Public License for more details. You should have received a copy of the GNU
% General Public License distributed with the SPARK toolset; see file
% COPYING3. If not, go to http://www.gnu.org/licenses for a complete copy of
% the license.
%
% =============================================================================
%-------------------------------------------------------------------------------
% RULE FAMILIES CONTAINED HEREIN :-
%
% arith : simple arithmetic simplification rules
% assoc : associativity of + and *
% commut : commutativity of + and *
% distrib : distributivity of * over + and -
% minus : rules for unary and binary minus
% intdiv : rules for the integer div operator
%-------------------------------------------------------------------------------
% MODEL DECLARATIONS FOR THIS FILE :-
%
% rule_family arith:
% X * Y requires [ X:ir, Y:ir ] &
% X + Y requires [ X:ir, Y:ir ] &
% X - Y requires [ X:ir, Y:ir ] &
% X div Y requires [ X:i, Y:i ] &
% X / Y requires [ X:ir, Y:ir ].
%
% rule_family assoc:
% X + Y requires [ X:ir, Y:ir ] &
% X * Y requires [ X:ir, Y:ir ].
%
% rule_family commut:
% X + Y requires [ X:ir, Y:ir ] &
% X * Y requires [ X:ir, Y:ir ].
%
% rule_family distrib:
% X + Y requires [ X:ir, Y:ir ] &
% X - Y requires [ X:ir, Y:ir ] &
% X * Y requires [ X:ir, Y:ir ].
%
% rule_family minus:
% - X requires [ X:ir ] &
% X + Y requires [ X:ir, Y:ir ] &
% X - Y requires [ X:ir, Y:ir ] &
% X * Y requires [ X:ir, Y:ir ].
%
% rule_family intdiv:
% X div Y requires [ X:i, Y:i ] &
% X + Y requires [ X:i, Y:i ] &
% - X requires [ X:i ].
%-------------------------------------------------------------------------------
/*** SIMPLIFICATION OF ARITHMETIC EXPRESSIONS Rules ***/
arith(1): X*1 may_be_replaced_by X.
arith(2): 1*X may_be_replaced_by X.
arith(3): X+0 may_be_replaced_by X.
arith(4): 0+X may_be_replaced_by X.
arith(5): X - 0 may_be_replaced_by X.
arith(6): X*0 may_be_replaced_by 0.
arith(7): 0*X may_be_replaced_by 0.
arith(8): X div 1 may_be_replaced_by X.
arith(9): (X*N) div N may_be_replaced_by X if [ N<>0 ].
arith(10): (N*X) div N may_be_replaced_by X if [ N<>0 ].
arith(11): X/1 may_be_replaced_by X.
arith(12): (X/Y)*Y may_be_replaced_by X if [ Y<>0 ].
arith(13): Y*(X/Y) may_be_replaced_by X if [ Y<>0 ].
arith(14): (X*Y)/Y may_be_replaced_by X if [ Y<>0 ].
arith(15): (Y*X)/Y may_be_replaced_by X if [ Y<>0 ].
/*** ASSOCIATIVITY of + & * Rules ***/
assoc(1): (A+B)+C may_be_replaced_by A+(B+C).
assoc(2): A+(B+C) may_be_replaced_by (A+B)+C.
assoc(3): (A*B)*C may_be_replaced_by A*(B*C).
assoc(4): A*(B*C) may_be_replaced_by (A*B)*C.
/*** COMMUTATIVITY of + & * Rules ***/
commut(1): A+B may_be_replaced_by B+A.
commut(2): A*B may_be_replaced_by B*A.
/*** DISTRIBUTIVITY of * over + & - Rules ***/
distrib(1): A*(B+C) & A*B+A*C are_interchangeable.
distrib(2): (B+C)*A & A*B+A*C are_interchangeable.
distrib(3): A*(B-C) & A*B-A*C are_interchangeable.
distrib(4): (B-C)*A & A*B-A*C are_interchangeable.
/*** Rules for manipulation of unary and binary MINUS operators ***/
minus(1): X-X may_be_replaced_by 0.
minus(2): -(0) may_be_replaced_by 0.
minus(3): -(-X) may_be_replaced_by X.
minus(4): -(A+B) & -A+(-B) are_interchangeable.
minus(5): -(A+B) & -A-B are_interchangeable.
minus(6): -A+(-B) & -A-B are_interchangeable.
minus(7): A+(-B) & A-B are_interchangeable.
minus(8): A+(-B) & -(B-A) are_interchangeable.
minus(9): A-B & -(B-A) are_interchangeable.
minus(10): -A*B & A*(-B) are_interchangeable.
minus(11): -A*B & -(A*B) are_interchangeable.
minus(12): -(A*B) & A*(-B) are_interchangeable.
minus(13): -A*(-B) & A*B are_interchangeable.
/*** Some rules for INTEGER DIVISION ***/
intdiv(1): (A+B) div C & A div C+B div C are_interchangeable if [B=K*C, A*B>=0].
intdiv(2): (A+B) div C & A div C+B div C are_interchangeable if [B=C*K, A*B>=0].
intdiv(3): (A+B) div C & A div C+B div C are_interchangeable if [A=K*C, A*B>=0].
intdiv(4): (A+B) div C & A div C+B div C are_interchangeable if [A=C*K, A*B>=0].
intdiv(5): (A+B) div C & A div C+D are_interchangeable if
[goal(integer(B)), goal(integer(C)), goal(C\=0),
goal(D iss B div C), goal(B iss D*C), A*B>=0].
intdiv(6): (A+B) div C & D+B div C are_interchangeable if
[goal(integer(A)), goal(integer(C)), goal(C\=0),
goal(D iss A div C), goal(A iss D*C), A*B>=0].
intdiv(7): -A div B & A div (-B) are_interchangeable.
intdiv(8): -A div B & -(A div B) are_interchangeable.
intdiv(9): A div (-B) & -(A div B) are_interchangeable.
intdiv(10): -A div (-B) & A div B are_interchangeable.
intdiv(11): (A*B) div B may_be_replaced_by A if [B<>0].
intdiv(12): (A*B) div (C*B) may_be_replaced_by A div C if [B<>0].
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