Verifikation reaktiver Systeme - Universität Kaiserslautern

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val SHIFT_def = Define ‘SHIFT(inp:num->num, out:num->num) =
!t. out t = (inp t) DIV 2‘;
val SUB_def = Define ‘SUB(in1:num->num,in2:num->num,out:num->num) =
!t. out t = (in1 t - in2 t)‘;
val LESS_def = Define ‘LESS(in1:num->num,in2:num->num,out:num->bool) =
!t. out t = (in1 t < in2 t)‘;
val LSB_def = Define ‘LSB(inp:num->num,out:num->bool) =
!t. out t = ((inp t) MOD 2 = 1)‘;
Then, the composition is described by:
val BGCD_IMP_def = Define
‘BGCD_IMP(initX:num, initY:num,
regX:num->num, regY:num->num, regK:num->num) =
? inX, inY, regXshift, regYshift, oddX, oddY,
regXcor, regYcor, evenX, evenY, evenXY,
oddXY, oddXYy, oddXYx, subX, subY, subXY, subYX.
REG(inX,regX,initX) /\ REG(inY,regY,initY) /\
SHIFT(regX,regXshift) /\ SHIFT(regY,regYshift) /\
LSB(regX,oddX) /\ LSB(regY,oddY) /\
MUX(oddX,regX,regXshift,regXcor) /\
MUX(oddY,regY,regYshift,regYcor) /\
NOT(oddX,evenX) /\ NOT(oddY,evenY) /\
AND(evenX,evenY,evenXY)/\
COUNTER(evenXY,regK)/\
AND(oddX, oddY, oddXY) /\
LESS(regXcor,regYcor,oddXYx) /\ LESS(regYcor,regXcor,oddXYy) /\
AND(oddXY,oddXYx,subX) /\ AND(oddXY,oddXYy,subY) /\
SUB(regXcor,regYcor,subXY) /\ SUB(regYcor,regXcor,subYX) /\
MUX(subX,regXcor,subXY,inX) /\ MUX(subY,regYcor,subYX,inY)‘;
Now, it is possible to prove the invariant and the correctness of the component.
The proof for this can be found in A.1. Considering this example, the
benefit from the theorem proving method becomes obvious: The number of possible
states and the complexity of the components is big. But this does not affect
the theorem proving approach, which abstracts from the size of the registers and
make it even possible to prove the correctness for arbitrary sizes.
3.3 Rational Numbers
This section shows are more comprehensive example introducing the type of
rational numbers Q. This theory can be used to verify hardware that is based on
rational arithmetic operations. Applications of this can be found in areas that
need precise calculations of basic arithmetics, e.g. linear geometry primitives
used in safety-critical environments.
The rational numbers are defined in two steps. First, fractions consisting of a
pair of integers are introduced. Subsequently, rational numbers are constructed
as equivalence classes of them using the quotient theory.