FORP(1) FORP(1) NAME forp - formula prover SYNOPSIS forp [ -m ] [ file ] DESCRIPTION Forp is a tool for proving formulae involving finite- precision arithmetic. Given a formula it will attempt to find a counterexample; if it can't find one the formula has been proven correct. Forp is invoked on an input file with the syntax as defined below. If no input file is provided, standard input is used instead. The -m flag instructs forp to produce a table of all counterexamples rather than report just one. Note that counterexamples may report bits as ?, meaning that either value will lead to a counterexample. The input file consists of statements terminated by semi- colons and comments using C syntax (using // or /* */ syn- tax). Valid statements are Variable definitions, roughly: type var ; Expressions (including assignments): expr ; Assertions: obviously expr ; Assumptions: assume expr ; Assertions are formulae to be proved. If multiple asser- tions are given, they are effectively "and"-ed together. Each input file must have at least one assertion to be valid. Assumptions are formulae that are assumed, i.e. counterexamples that would violate assumptions are never considered. Exercise care with them, as contradictory assumptions will lead to any formula being true (the logician's principle of explosion). Variables can be defined with C notation, but the only types supported are bit and 1D arrays of bit (corresponding to machine integers of the specified size). Signed integers are indicated with the keyword signed. Like int in C, the bit keyword can be omitted in the presence of signed. For example, bit a, b[4], c[8]; signed bit d[3]; signed e[16]; is a set of valid declarations. Page 1 Plan 9 (printed 4/22/21) FORP(1) FORP(1) Unlike a programming language, it is perfectly legitimate to use a variable before it is assigned value; this means the variable is an "input" variable. Forp tries to find assign- ments for all input variables that render the assertions invalid. Expressions can be formed just as in C, however when used in an expression, all variables are automatically promoted to an infinite size signed type. The valid operators are listed below, in decreasing precedence. Note that logical operations treat all non-zero values as 1, whereas bitwise operators operate on all bits independently. [] Array indexing. The syntax is var[a:b], with a denoting the MSB and b denoting the LSB. Omiting :b addresses a single bit. The result is always treated as unsigned. !, ~, +, - (Unary operators) Logical and bitwise "not", unary plus (no-op), arithmetic negation. Because of promotion, ~ and - operate beyond the width of variables. *, /, % Multiplication, division, modulo. Division and modulo add an assumption that the divisor is non-zero. +, - Addition, subtraction. <<, >> Left shift, arithmetic right shift. Because of promotion, this is effectively a logical right shift on unsigned variables. <, <=, >, >= Less than, less than or equal to, greater than, greather than or equal to. ==, != Equal to, not equal to. & Bitwise "and". ^ Bitwise "xor". | Bitwise "or". && Logical "and" || Logical "or". <=>, => Logical equivalence and logical implication (equivalent to (a != 0) == (b != 0) and !a || b, respectively). Page 2 Plan 9 (printed 4/22/21) FORP(1) FORP(1) ?: Ternary operator (a?b:c equals b if a is true and c otherwise). = Assignment. One subtle point concerning assignments is that they forci- bly override any previous values, i.e. expressions use the value of the latest assignments preceding them. Note that the values reported as the counterexample are always the values given by the last assignment. EXAMPLES We know that, mathematically, a + b ≥ a if b ≥ 0 (which is always true for an unsigned number). We can ask forp to prove this using bit a[32], b[32]; obviously a + b >= a; Forp will report "Proved", since it cannot find a counterex- ample for which this is not true. In C, on the other hand, we know that this is not necessarily true. The reason is that, depending on the types involved, results are trun- cated. We can emulate this by writing bit a[32], b[32], c[32]; c = a + b; obviously c >= a; Given this, forp will now report it as incorrect by provid- ing a counterexample, for example a = 10000000000000000000000000000000 b = 10000000000000000000000000000000 c = 00000000000000000000000000000000 Can we use c < a to check for overflow? We can ask forp to confirm this using bit a[32], b[32], c[32]; c = a + b; obviously c < a <=> c != a+b; Here the statement to be proved is "c is less than a if and only if c does not equal the mathematical sum a + b (i.e. overflow has occured)". SOURCE /sys/src/cmd/forp SEE ALSO spin(1) Page 3 Plan 9 (printed 4/22/21) FORP(1) FORP(1) BUGS Any proof is only as good as the assumptions made, in par- ticular care has to be taken with respect to truncation of intermediate results. Array indices must be constants. Left shifting can produce a huge number of intermediate bits. Forp will try to identify the minimum needed number but it may be a good idea to help it by assigning the result of a left shift to a variable. HISTORY Forp first appeared in 9front in March, 2018. Page 4 Plan 9 (printed 4/22/21)