From 87d5edab00cf7c7774f85695d9d1c6fd5c11ac32 Mon Sep 17 00:00:00 2001 From: robdockins Date: Tue, 30 Jun 2020 10:58:25 -0700 Subject: [PATCH] Documentation updates (#779) * Add docstrings for all prelude functions and fix minor style issues. Fixes #771 * Update `CryptolPrims` documentation * Minor updates to the prelude * Update CHANGES * Updates to the cryptol book and CryptolPrims * Fix several additional docstrings * Specify and document properties of signed bitvector division. Fixes #677 * Fixup test * typos and style * Regenerate PDFs --- CHANGES.md | 30 +- docs/Cryptol.pdf | Bin 635706 -> 630950 bytes docs/CryptolPrims.md | 359 +++++++++++++----- docs/CryptolPrims.pdf | Bin 91235 -> 100258 bytes docs/ProgrammingCryptol.pdf | Bin 635706 -> 630950 bytes .../crashCourse/CrashCourse.tex | 31 +- .../highAssurance/HighAssurance.tex | 2 +- docs/ProgrammingCryptol/prims/Primitives.tex | 13 +- .../technicalities/TechAppendix.tex | 32 +- docs/Semantics.pdf | Bin 232117 -> 239978 bytes lib/Cryptol.cry | 153 +++++--- tests/issues/issue226.icry.stdout | 10 +- tests/regression/bvdiv.cry | 70 ++++ tests/regression/bvdiv.icry | 2 + tests/regression/bvdiv.icry.stdout | 27 ++ 15 files changed, 536 insertions(+), 193 deletions(-) create mode 100644 tests/regression/bvdiv.cry create mode 100644 tests/regression/bvdiv.icry create mode 100644 tests/regression/bvdiv.icry.stdout diff --git a/CHANGES.md b/CHANGES.md index daa61f77..41778b1e 100644 --- a/CHANGES.md +++ b/CHANGES.md @@ -24,8 +24,9 @@ There is also a new `Round` class for types that can sensibly be rounded to integers. This class has the methods `floor`, `ceiling`, - `trunc` and `round` for performing different kinds of - integer rounding. Currently `Rational` is the only meber of `Round`. + `trunc`, `roundToEven` and `roundAway` for performing different + kinds of integer rounding. Currently `Rational` is the only member + of `Round`. The type of `(^^)` is modified to be `{a, e} (Ring a, Integral e) => a -> e -> a`. This makes it clear @@ -35,6 +36,10 @@ Finally, the `lg2`, `(/$)` and `(%$)` methods of Arith have had their types specialized so operate only on bitvectors. +* Added an `Eq` class, and moved the equality operations + from `Cmp` into `Eq`. The `Z` type becomes a member of `Eq` + but not `Cmp`. + * Added a base `Rational` type. It is implemented as a pair of integers, quotiented in the usual way. As such, it reduces to the theory of integers and requires no new solver support (beyond @@ -45,6 +50,10 @@ * The `generate` function (and thus `x @ i= e` definitions) has had its type specialized so the index type is always `Integer`. +* The new typeclasses are arranged into a class hierarchy, and the + typechecker will use that information to infer superclass instances + from subclasses. + ## New features * Document the behavior of lifted selectors. @@ -57,10 +66,23 @@ * More detailed information about the status of various symbols in the output of the `:browse` command (issue #688). +* The `:safe` command will attempt to prove that a given Cryptol + term is safe; in other words, that it will not encounter a run-time + error for all inputs. Run-time errors arise from things like + division-by-zero, index-out-of-bounds situations and + explicit calls to `error` or `assert`. + +* The `:prove` and `:sat` commands now incorporate safety predicates + by default. In a `:sat` call, models will only be found that do not + cause run-time errors. For `:prove` calls, the safety conditions are + added as additional proof goals. The prior behavior + (which ignored safety conditions) can be restored using + `:set ignore-safety = on`. + ## Bug fixes -* Closed issues #346, #444, #614, #617, #660, #662, #663, #664, #667, #670, - #711 +* Closed issues #346, #444, #614, #617, #636, #660, #662, #663, #664, #667, #670, + #702, #711, #712, #716, #723, #725, #731 # 2.8.0 (September 4, 2019) diff --git a/docs/Cryptol.pdf b/docs/Cryptol.pdf index 648020b1e0b98bb364b1690a3f718f5c5b88e0d1..f4e88e79e4f280424c402da9cb60ccac5812b21c 100644 GIT binary patch delta 476234 zcmZs?Q~O0PmB zgy$#x1P2EvGeraww`PJ|)6+V~>1w{`+D%v630;^QD;^?ZbliW*y5H^T4(bV0E zLD}1Ku-4P|irFM}4HoK;EXX*~-lh!>8 z?RjowJyUBPGJBBLTjFn^U~l$YMX+y!X%a`WJ(M*(hJmx=*@q_S0Jd7rG|F_S@lt=H zU)yQj;JRwFIxOhDT=(!iFxDziPN_G`GM2{37POZ4<*bB|)BU$unUSgUMLHuLFna3i z9Lp`Y37{$SX$_W*zD{Wy=R3N6*MvvslF#T_ha95!!xW;V%I0(c0wG(% z8YGy#%K)86&o;tAq1@C3FUi0;nddHM(FW&Za_ri2B?a_JLPPhXMdf1|ceW|1I430M}H zL`6A?XTT6h0#Iz@us%&c6>1WCHwbc;{nljHJ;{6}5MsZ^n#||(h{xV@aTrUkVyLRy 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=> a -> a -> Bit - - min : {a} (Cmp a) => a -> a -> a - max : {a} (Cmp a) => a -> a -> a - - instance Cmp Bit - // No instance for functions. - instance (Cmp a, fin n) => Cmp [n]a - instance (Cmp a, Cmp b) => Cmp (a, b) - instance (Cmp a, Cmp b) => Cmp { x : a, y : b } - instance Cmp Integer - instance (fin n, n>=1) => Cmp (Z n) - instance Cmp Rational - -Signed Comparisons +Typeclass Heriarchy --------------------- - (<$) : {a} (SignedCmp a) => a -> a -> Bit - (>$) : {a} (SignedCmp a) => a -> a -> Bit - (<=$) : {a} (SignedCmp a) => a -> a -> Bit - (>=$) : {a} (SignedCmp a) => a -> a -> Bit + Zero + / \ + Logic Ring Eq + / \ / \ + Integral Field Cmp SignedCmp + \ / + Round + + +This diagram describes how the various typeclasses in +Cryptol are related. A type which is an instance of a +subclass is also always a member of all of its superclasses. +For example, any type which is a member of `Field` is also +a member of `Ring` and `Zero`. + + +Literals +----------------------- + type Literal : # -> * -> Prop + + number : {val, rep} Literal val rep => rep + length : {n, a, b} (fin n, Literal n b) => [n]a -> b + + // '[a..b]' is syntactic sugar for 'fromTo`{first=a,last=b}' + fromTo : {first, last, a} + (fin last, last >= first, + Literal first a, Literal last a) => + [1 + (last - first)]a + + // '[a,b..c]' is syntactic sugar for 'fromThenTo`{first=a,next=b,last=c}' + fromThenTo : {first, next, last, a, len} + ( fin first, fin next, fin last + , Literal first a, Literal next a, Literal last a + , first != next + , lengthFromThenTo first next last == len) => + [len]a - // No instance for Bit - // No instance for functions. - instance (fin n, n >= 1) => SignedCmp [n] - instance (SignedCmp a, fin n) => SignedCmp [n]a - // (for [n]a, where a is other than Bit) - instance (SignedCmp a, SignedCmp b) => SignedCmp (a, b) - instance (SignedCmp a, SignedCmp b) => SignedCmp { x : a, y : b } Zero ---- + type Zero : * -> Prop + zero : {a} (Zero a) => a Every base and structured type in Cryptol is a member of class `Zero`. -Arithmetic ----------- - - (+) : {a} (Ring a) => a -> a -> a - (-) : {a} (Ring a) => a -> a -> a - (*) : {a} (Ring a) => a -> a -> a - negate : {a} (Ring a) => a -> a - - (/) : {a} (Integral a) => a -> a -> a - (%) : {a} (Integral a) => a -> a -> a - - (/.) : {a} (Field a) => a -> a -> a - recip : {a} (Field a) => a -> a - - floor : {a} (Round a) => a -> Integer - ceiling : {a} (Round a) => a -> Integer - trunc : {a} (Round a) => a -> Integer - round : {a} (Round a) => a -> Integer - - (^^) : {a, e} (Ring a, Integral e) => a -> e -> a - - (/$) : {n} (fin n, n >= 1) => [n] -> [n] -> [n] - (%$) : {n} (fin n, n >= 1) => [n] -> [n] -> [n] - lg2 : {n} (fin n) => [n] -> [n] - -The prefix notation `- x` is syntactic sugar for `negate x`. - - // No instance for `Bit`. - instance (fin n) => Ring ([n]Bit) - instance (Ring a) => Ring ([n]a) - instance (Ring b) => Ring (a -> b) - instance (Ring a, Ring b) => Ring (a, b) - instance (Ring a, Ring b) => Ring { x : a, y : b } - instance Ring Integer - instance (fin n, n>=1) => Ring (Z n) - instance Ring Rational - -Note that because there is no instance for `Ring Bit` -the top two instances do not actually overlap. - - instance Integral Integer - instance (fin n) => Integral ([n]Bit) - - instance Field Rational - - instance Round Rational - Boolean ------- + type Logic : * -> Prop + False : Bit True : Bit @@ -104,6 +61,8 @@ Boolean (||) : {a} (Logic a) => a -> a -> a (^) : {a} (Logic a) => a -> a -> a complement : {a} (Logic a) => a -> a + // The prefix notation '~ x' is syntactic + // sugar for 'complement x'. (==>) : Bit -> Bit -> Bit (/\) : Bit -> Bit -> Bit @@ -118,17 +77,174 @@ Boolean // No instance for `Logic (Z n)`. // No instance for `Logic Rational`. + +Arithmetic +---------- + + type Ring : * -> Prop + + fromInteger : {a} (Ring a) => Integer -> a + (+) : {a} (Ring a) => a -> a -> a + (-) : {a} (Ring a) => a -> a -> a + (*) : {a} (Ring a) => a -> a -> a + negate : {a} (Ring a) => a -> a + // The prefix notation `- x` is syntactic + // sugar for `negate x`. + + + type Integral : * -> Prop + + (/) : {a} (Integral a) => a -> a -> a + (%) : {a} (Integral a) => a -> a -> a + toInteger : {a} (Integral a) => a -> Integer + infFrom : {a} (Integral a) => a -> [inf]a + // '[x...]' is syntactic sugar for 'infFrom x' + infFromThen : {a} (Integral a) => a -> a -> [inf]a + // [x,y...]' is syntactic sugar for 'infFromThen x y' + + + type Field : * -> Prop + + (/.) : {a} (Field a) => a -> a -> a + recip : {a} (Field a) => a -> a + + + type Round : * -> Prop + + floor : {a} (Round a) => a -> Integer + ceiling : {a} (Round a) => a -> Integer + trunc : {a} (Round a) => a -> Integer + roundAway : {a} (Round a) => a -> Integer + roundToEven : {a} (Round a) => a -> Integer + + (^^) : {a, e} (Ring a, Integral e) => a -> e -> a + + // No instance for `Bit`. + instance (fin n) => Ring ([n]Bit) + instance (Ring a) => Ring ([n]a) + instance (Ring b) => Ring (a -> b) + instance (Ring a, Ring b) => Ring (a, b) + instance (Ring a, Ring b) => Ring { x : a, y : b } + instance Ring Integer + instance (fin n, n>=1) => Ring (Z n) + instance Ring Rational + +Note that because there is no instance for `Ring Bit` +the top two instances do not actually overlap. + + instance Integral Integer + instance (fin n) => Integral ([n]Bit) + + instance Field Rational + + instance Round Rational + + +Equality Comparisons +------------------------ + + type Eq : * -> Prop + + (==) : {a} (Eq a) => a -> a -> Bit + (!=) : {a} (Eq a) => a -> a -> Bit + (===) : {a,b} (Eq b) => (a -> b) -> (a -> b) -> a -> Bit + (!==) : {a,b} (Eq b) => (a -> b) -> (a -> b) -> a -> Bit + + instance Eq Bit + instance (Eq a, fin n) => Eq [n]a + instance (Eq a, Eq b) => Eq (a, b) + instance (Eq a, Eq b) => Eq { x : a, y : b } + instance Eq Integer + instance Eq Rational + instance (fin n, n>=1) => Eq (Z n) + // No instance for functions. + + +Comparisons and Ordering +------------------------ + + type Cmp : * -> Prop + + (<) : {a} (Cmp a) => a -> a -> Bit + (>) : {a} (Cmp a) => a -> a -> Bit + (<=) : {a} (Cmp a) => a -> a -> Bit + (>=) : {a} (Cmp a) => a -> a -> Bit + + min : {a} (Cmp a) => a -> a -> a + max : {a} (Cmp a) => a -> a -> a + + abs : {a} (Cmp a, Ring a) => a -> a + + instance Cmp Bit + instance (Cmp a, fin n) => Cmp [n]a + instance (Cmp a, Cmp b) => Cmp (a, b) + instance (Cmp a, Cmp b) => Cmp { x : a, y : b } + instance Cmp Integer + instance Cmp Rational + // No instance for functions. + + +Signed Comparisons +--------------------- + + type SignedCmp : * -> Prop + + (<$) : {a} (SignedCmp a) => a -> a -> Bit + (>$) : {a} (SignedCmp a) => a -> a -> Bit + (<=$) : {a} (SignedCmp a) => a -> a -> Bit + (>=$) : {a} (SignedCmp a) => a -> a -> Bit + + // No instance for Bit + instance (fin n, n >= 1) => SignedCmp [n] + instance (SignedCmp a, fin n) => SignedCmp [n]a + // (for [n]a, where a is other than Bit) + instance (SignedCmp a, SignedCmp b) => SignedCmp (a, b) + instance (SignedCmp a, SignedCmp b) => SignedCmp { x : a, y : b } + // No instance for functions. + + +Bitvectors +----------- + + (/$) : {n} (fin n, n >= 1) => [n] -> [n] -> [n] + (%$) : {n} (fin n, n >= 1) => [n] -> [n] -> [n] + + carry : {n} (fin n) => [n] -> [n] -> Bit + scarry : {n} (fin n, n >= 1) => [n] -> [n] -> Bit + sborrow : {n} (fin n, n >= 1) => [n] -> [n] -> Bit + + zext : {m, n} (fin m, m >= n) => [n] -> [m] + sext : {m, n} (fin m, m >= n, n >= 1) => [n] -> [m] + + lg2 : {n} (fin n) => [n] -> [n] + + // Arithmetic shift only for bitvectors + (>>$) : {n, ix} (fin n, n >= 1, Integral ix) => [n] -> ix -> [n] + + +Rationals +--------- + + ratio : Integer -> Integer -> Rational + + +Z(n) +----- + + fromZ : {n} (fin n, n >= 1) => Z n -> Integer + + Sequences --------- - join : {parts,each,a} (fin each) => [parts][each]a -> [parts * each]a - split : {parts,each,a} (fin each) => [parts * each]a -> [parts][each]a + join : {parts,each,a} (fin each) => [parts][each]a -> [parts * each]a + split : {parts,each,a} (fin each) => [parts * each]a -> [parts][each]a - (#) : {front,back,a} (fin front) => [front]a -> [back]a -> [front + back]a - splitAt : {front,back,a} (fin front) => [from + back] a -> ([front] a, [back] a) + (#) : {front,back,a} (fin front) => [front]a -> [back]a -> [front + back]a + splitAt : {front,back,a} (fin front) => [from + back] a -> ([front] a, [back] a) - reverse : {n,a} (fin n) => [n]a -> [n]a - transpose : {n,m,a} [n][m]a -> [m][n]a + reverse : {n,a} (fin n) => [n]a -> [n]a + transpose : {n,m,a} [n][m]a -> [m][n]a (@) : {n,a,ix} (Integral ix) => [n]a -> ix -> a (@@) : {n,k,ix,a} (Integral ix) => [n]a -> [k]ix -> [k]a @@ -145,6 +261,10 @@ Sequences tail : {a, b} [1 + a]b -> [a]b last : {a, b} [1 + a]b -> b + // Declarations of the form 'x @ i = e' are syntactic + // sugar for 'x = generate (\i -> e)'. + generate : {n, a} (fin n, n >= 1) => (Integer -> a) -> [n]a + groupBy : {each,parts,elem} (fin each) => [parts * each]elem -> [parts][each]elem Function `groupBy` is the same as `split` but with its type arguments @@ -153,23 +273,60 @@ in a different order. Shift And Rotate ---------------- - (<<) : {n,ix,a} (Integral ix, Zero a) => [n]a -> ix -> [n]a - (>>) : {n,ix,a} (Integral ix, Zero a) => [n]a -> ix -> [n]a - (<<<) : {n,ix,a} (fin n, Integral ix) => [n]a -> ix -> [n]a - (>>>) : {n,ix,a} (fin n, Integral ix) => [n]a -> ix -> [n]a + (<<) : {n,ix,a} (Integral ix, Zero a) => [n]a -> ix -> [n]a + (>>) : {n,ix,a} (Integral ix, Zero a) => [n]a -> ix -> [n]a + (<<<) : {n,ix,a} (fin n, Integral ix) => [n]a -> ix -> [n]a + (>>>) : {n,ix,a} (fin n, Integral ix) => [n]a -> ix -> [n]a + + +GF(2) polynomials +----------------- + + pmult : {u, v} (fin u, fin v) => [1 + u] -> [1 + v] -> [1 + u + v] + pdiv : {u, v} (fin u, fin v) => [u] -> [v] -> [u] + pmod : {u, v} (fin u, fin v) => [u] -> [1 + v] -> [v] - // Arithmetic shift only for bitvectors - (>>$) : {n,ix} (fin n, n >= 1, Integral ix) => [n] -> ix -> [n] Random Values ------------- - random : {a} => [256] -> a + random : {a} => [256] -> a + + +Errors and Assertions +--------------------- + + undefined : {a} a + error : {a,n} (fin n) => String n -> a + assert : {a,n} (fin n) => Bit -> String n -> a -> a + Debugging --------- - undefined : {a} a - error : {n a} [n][8] -> a - trace : {n, a, b} (fin n) => [n][8] -> a -> b -> b - traceVal : {n, a} (fin n) => [n][8] -> a -> a + trace : {n, a, b} (fin n) => String n -> a -> b -> b + traceVal : {n, a} (fin n) => String n -> a -> a + + +Utility operations +------------------ + + and : {n} (fin n) => [n]Bit -> Bit + or : {n} (fin n) => [n]Bit -> Bit + all : {n, a} (fin n) => (a -> Bit) -> [n]a -> Bit + any : {n, a} (fin n) => (a -> Bit) -> [n]a -> Bit + elem : {n, a} (fin n, Eq a) => a -> [n]a -> Bit + + map : {n, a, b} (a -> b) -> [n]a -> [n]b + foldl : {n, a, b} (fin n) => (a -> b -> a) -> a -> [n]b -> a + foldr : {n, a, b} (fin n) => (a -> b -> b) -> b -> [n]a -> b + scanl : {n, b, a} (b -> a -> b) -> b -> [n]a -> [n+1]b + scanr : {n, a, b} (fin n) => (a -> b -> b) -> b -> [n]a -> [n+1]b + sum : {n, a} (fin n, Ring a) => [n]a -> 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\end{Exercise} \begin{Answer}\ansref{ex:recfun:4:1} Using a fold, it is easy to write {\tt elem}: @@ -2922,7 +2923,7 @@ Type classes are a way of describing behaviors shared by multiple types. As an example, consider the type of the function {\tt ==}: \begin{Verbatim} Cryptol> :t (==) - (==) : {a} (Cmp a) => a -> a -> Bit + (==) : {a} (Eq a) => a -> a -> Bit \end{Verbatim} This type signature may be read as, ``equality is an operator that @@ -2930,7 +2931,7 @@ takes two objects of any single type that can be compared and returns a Bit.'' Cryptol defines a collection of basic type classes: \texttt{Logic}, -\texttt{Zero}, \texttt{Cmp}, \texttt{SignedCmp}, \texttt{Ring}, +\texttt{Zero}, \texttt{Eq}, \texttt{Cmp}, \texttt{SignedCmp}, \texttt{Ring}, \texttt{Integral}, \texttt{Field}, \texttt{Round}, and \texttt{Literal}. These appear in the type signature of operators and functions that require them. If a function you define calls, for @@ -2952,15 +2953,17 @@ All of the built-in types of Cryptol are instances of class \texttt{Zero}. \item -The \texttt{Cmp} typeclass includes the binary relation operators -\texttt{<}, \texttt{>}, \texttt{<=}, \texttt{>=}, \texttt{==}, and -\texttt{!=}, as well as the binary functions \texttt{min} and -\texttt{max}. Note that equality and other comparisons are bundled -into a single typeclass. Function types are not in class \texttt{Cmp} +The \texttt{Eq} typeclass includes the equality testing operations +\texttt{==} and \texttt{!=}. Function types are not in class \texttt{Eq} and cannot be compared with \texttt{==}, but Cryptol does provide a special pointwise comparison operator for functions, \texttt{(===) : \{a b\} (Cmp b) => (a -> b) -> (a -> b) -> a -> Bit}. +\item +The \texttt{Cmp} typeclass includes the binary relation operators +\texttt{<}, \texttt{>}, \texttt{<=}, \texttt{>=}, as well as the +binary functions \texttt{min} and \texttt{max}. + \item The \texttt{SignedCmp} typeclass includes the binary relation operators \texttt{<\$}, \texttt{>\$}, \texttt{<=\$}, and @@ -2996,9 +2999,9 @@ Currently, only type \texttt{Rational} is a member of this class. \item The \texttt{Round} typeclass contains types that can be rounded to integers. It includes \texttt{floor}, \texttt{ceiling}, - \texttt{trunc} and \texttt{round} operations, which are all different - ways of rounding values to integers. -Currently, only type \texttt{Rational} is a member of this class. + \texttt{trunc}, \texttt{roundAway} and \texttt{roundToEven} + operations, which are all different ways of rounding values to + integers. Currently, only type \texttt{Rational} is a member of this class. \item The \texttt{Literal} typeclass includes numeric literals. @@ -3008,7 +3011,7 @@ The \texttt{Literal} typeclass includes numeric literals. Without including an explicit type declaration, define a function that Cryptol infers has the following type: \begin{Verbatim} - cmpRing : {a,b} (Cmp a, Ring b) => a -> a -> b -> b + cmpRing : {a,b} (Eq a, Ring b) => a -> a -> b -> b \end{Verbatim} \end{Exercise} \begin{Answer}\ansref{ex:tvar:1} @@ -3018,7 +3021,7 @@ This code: \end{code} yields the inferred type: \begin{Verbatim} - cmpRing : {a, b} (Cmp a, Ring b) => a -> a -> b -> b + cmpRing : {a, b} (Eq a, Ring b) => a -> a -> b -> b \end{Verbatim} \end{Answer} diff --git a/docs/ProgrammingCryptol/highAssurance/HighAssurance.tex b/docs/ProgrammingCryptol/highAssurance/HighAssurance.tex index 342a7eb4..fbbc7117 100644 --- a/docs/ProgrammingCryptol/highAssurance/HighAssurance.tex +++ b/docs/ProgrammingCryptol/highAssurance/HighAssurance.tex @@ -213,7 +213,7 @@ a different value: a property we might expect to hold intuitively. Here is the type of {\tt flipNeverIdentity}: \begin{Verbatim} Cryptol> :t flipNeverIdentity - flipNeverIdentity : {a} (Cmp a, Logic a) => a -> Bit + flipNeverIdentity : {a} (Eq a, Logic a) => a -> Bit \end{Verbatim} So, the only requirement on \texttt{flipNeverIdentity} is that it receives some type that supports comparisons and logical operations. diff --git a/docs/ProgrammingCryptol/prims/Primitives.tex b/docs/ProgrammingCryptol/prims/Primitives.tex index f3f44097..5e4c1617 100644 --- a/docs/ProgrammingCryptol/prims/Primitives.tex +++ b/docs/ProgrammingCryptol/prims/Primitives.tex @@ -15,11 +15,11 @@ primsPlaceHolder=1; \end{Verbatim} \paragraph*{Comparisons} \begin{Verbatim} - ==, != : {a} (Cmp a) => a -> a -> Bit + ==, != : {a} (Eq a) => a -> a -> Bit <, >, <=, >= : {a} (Cmp a) => a -> a -> Bit <$, >$, <=$, >=$ : {a} (SignedCmp a) => a -> a -> Bit min, max : {a} (Cmp a) => a -> a -> a - ===, !== : {a, b} (Cmp b) => (a -> b) -> (a -> b) -> a -> Bit + ===, !== : {a, b} (Eq b) => (a -> b) -> (a -> b) -> a -> Bit \end{Verbatim} \paragraph*{Arithmetic} \begin{Verbatim} @@ -42,9 +42,10 @@ primsPlaceHolder=1; floor : {a} (Round a) => a -> Integer ceiling : {a} (Round a) => a -> Integer trunc : {a} (Round a) => a -> Integer - round : {a} (Round a) => a -> Integer + roundAway : {a} (Round a) => a -> Integer + roundToEven : {a} (Round a) => a -> Integer \end{Verbatim} -\paragraph*{Polynomial arithmetic} +\paragraph*{GF(2) polynomial arithmetic} \begin{Verbatim} pdiv : {u, v} (fin u, fin v) => [u] -> [v] -> [u] pmod : {u, v} (fin u, fin v) => [u] -> [1 + v] -> [v] @@ -97,6 +98,7 @@ primsPlaceHolder=1; all, any : {n, a} (fin n) => (a -> Bit) -> [n]a -> Bit curry : {a, b, c} ((a, b) -> c) -> a -> b -> c uncurry : {a, b, c} (a -> b -> c) -> (a, b) -> c + elem : {n, a} (fin n, Eq a) => a -> [n]a -> Bit \end{Verbatim} \paragraph*{Miscellaneous} \begin{Verbatim} @@ -105,8 +107,9 @@ primsPlaceHolder=1; \end{Verbatim} \paragraph*{Representing exceptions} \begin{Verbatim} - error : {a, len} (fin len) => [len][8] -> a undefined : {a} a + error : {a, n} (fin n) => String n -> a + assert : {a, n} (fin n) => Bit -> String n -> a -> a trace : {n, a, b} [n][8] -> a -> b -> b traceVal : {n, a} [n][8] -> a -> a \end{Verbatim} diff --git a/docs/ProgrammingCryptol/technicalities/TechAppendix.tex b/docs/ProgrammingCryptol/technicalities/TechAppendix.tex index f3a428f0..1004f85f 100644 --- a/docs/ProgrammingCryptol/technicalities/TechAppendix.tex +++ b/docs/ProgrammingCryptol/technicalities/TechAppendix.tex @@ -130,7 +130,7 @@ type of an infix operator (e.g., {\tt +}, {\tt ==}, etc.) you will need to surround the operator with {\tt ()}, like this: \begin{Verbatim} Cryptol> :t (+) - (+) : {a} (Arith a) => a -> a -> a + (+) : {a} (Ring a) => a -> a -> a \end{Verbatim} \paragraph*{Browsing definitions} @@ -158,14 +158,15 @@ have defined, along with their types. \hline \textbf{Option} & \textbf{Default value} & \textbf{Meaning} \\ \hline - \texttt{ascii} & \texttt{off} & print sequences of bytes as a string \\ + \texttt{ascii} & \texttt{off} & print sequences of bytes as a string \\ \texttt{base} & \texttt{16} & numeric base for printing words \\ - \texttt{debug} & \texttt{off} & whether to print verbose debugging information \\ + \texttt{ignore-safety} & \texttt{off} & ignore safety predicates when performing \texttt{:sat} or \texttt{:prove} checks \\ \texttt{infLength} & \texttt{5} & number of elements to show from an infinite sequence \\ \texttt{prover} & \texttt{z3} & which SMT solver to use for \texttt{:prove} \\ \texttt{satNum} & \texttt{1} & maximum number of solutions to show for \texttt{:sat} \\ + \texttt{show-examples} & \texttt{on} & whether to print counterexamples and models \\ + \texttt{smtfile} & \texttt{-} & file to write SMT problems when \texttt{prover = offline} \\ \texttt{tests} & \texttt{100} & number of tests to run for \texttt{:check} \\ - \texttt{warnDefaulting} & \texttt{on} & \todo[inline]{talk to Iavor} \\ \hline \end{tabular*} \label{tab:set_options} @@ -173,8 +174,9 @@ have defined, along with their types. \paragraph*{Environment options} A variety of environment options are set through the use of the \texttt{:set} command. These options may change over time and some -options may be available only on specific platforms. The current -options are summarized in~\autoref{tab:set_options}. +options may be available only on specific platforms. The most useful +configuration options are summarized in~\autoref{tab:set_options}. +Help information about all flags can found by typing {\tt :help :set flag}. \todo[inline]{Ensure index references exist for all commands.} @@ -220,7 +222,7 @@ At the Cryptol prompt you can load a module by name with the {\tt :module} command.\indCmdLoadModule \paragraph*{} -The next three commands all operate on \emph{properties}. All take +The next commands all operate on \emph{properties}. All take either one or zero arguments. If one argument is provided, then that property is the focus of the command; otherwise all properties in the current context are checked. All three commands are covered in detail @@ -231,16 +233,30 @@ The \texttt{:check} command performs random value testing on a property to increase one's confidence that the property is valid. See~\autoref{sec:quickcheck} for more detailed information. +\paragraph*{Checking a property through exhaustive testing} +The \texttt{:exhaust} command enumerates all the possible input values +of a property to test that it is correct for every possible value. +This is infeasible except for properties with quite small input +domains! + \paragraph*{Verifying a property through automated theorem proving} The \texttt{:prove} command uses an external SMT solver to attempt to automatically formally prove that a given property is valid. See~\autoref{sec:formal-proofs} for more detailed information. \paragraph*{Finding a satisfying assignment for a property} -The \texttt{:sat} command uses an external SAT solver to attempt to +The \texttt{:sat} command uses an external SMT solver to attempt to find a satisfying assignment to a property. See~\autoref{sec:sat} for more detailed information. +\paragraph*{Proving that a property is safe} +The \texttt{:safe} command uses an external SMT solver to attempt to +prove that the given term is safe for all inputs, which means it cannot +encounter a run-time error. Unlike the other commands in this section, +\texttt{:safe} \emph{must} be given an argument and can be called on +values that compute types other than \texttt{Bit}. 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z4D>@3Qt4l_cIiiot~?|Mhw$o=^wSl6RzDpiY4Gy#n-0dBhM6BdW{K&_3wv9cQ|rk|cR zpM9A|hOSQc(Q#Rc@zmN($rZnrmQrG0^06LzA=kH@MBIAWVu&slzxwSBar@>D6g`rE z!3{~@Dmb9=GS&dm|KFyB;WB+-)bD-k-CwoEy~fSJZZK*DVNa= 1) => Z n : * -/** `Rational` is the type of rational numbers. - * Rational numbers form a Field (and thus a Ring). - * - * The `ratio` operation may be used to directly create - * rational values from as a ratio of integers, or - * the `fromInteger` method and the field operations - * can be used. - */ +/** + * 'Rational' is the type of rational numbers. + * Rational numbers form a Field (and thus a Ring). + * + * The 'ratio' operation may be used to directly create + * rational values from as a ratio of integers, or + * the 'fromInteger' method and the field operations + * can be used. + */ primitive type Rational : * type Bool = Bit @@ -255,13 +257,20 @@ primitive complement : {a} (Logic a) => a -> a // The Ring class ------------------------------------------------------- -/** Value types that support ring addition and multiplication. */ +/** + * Value types that support ring addition and multiplication. + * + * Floating-point values are only approximately a ring, but + * nonetheless inhabit this class. + */ primitive type Ring : * -> Prop /** * Converts an unbounded integer to a value in a Ring. When converting * to the bitvector type [n], the value is reduced modulo 2^^n. Likewise, - * when converting to Z n, the value is reduced modulo n. + * when converting to Z n, the value is reduced modulo n. When converting + * to a floating-point value, the value is rounded to the nearest + * representable value. */ primitive fromInteger : {a} (Ring a) => Integer -> a @@ -302,10 +311,11 @@ primitive negate : {a} (Ring a) => a -> a // The Integral class ------------------------------------------------- -/** Value types that correspond to a segment of the - * integers. These types support integer division and - * modulus, indexing into sequences, and enumeration. - */ +/** + * Value types that correspond to a segment of the + * integers. These types support integer division and + * modulus, indexing into sequences, and enumeration. + */ primitive type Integral : * -> Prop /** @@ -332,8 +342,8 @@ primitive toInteger : {a} (Integral a) => a -> Integer * Compute the exponentiation of a value in a ring. * * For type [n], the exponent is treated as unsigned. * * It is an error to raise a value to a negative integer exponent. - * * Satisfies: `x ^^ 0 == fromInteger 1` - * * Satisfies: `x ^^ e == x * x ^^ (e-1)` when `e > 0`. + * * Satisfies: 'x ^^ 0 == fromInteger 1' + * * Satisfies: 'x ^^ e == x * x ^^ (e-1)' when 'e > 0'. */ primitive (^^) : {a, e} (Ring a, Integral e) => a -> e -> a @@ -352,21 +362,22 @@ primitive infFromThen : {a} (Integral a) => a -> a -> [inf]a // The Field class ------------------------------------------------- -/** Value types that correspond to a field; that is, a - * ring also posessing multiplicative inverses for - * non-zero elements. - * - * Floating-point values are only approximately a field, - * but nonetheless inhabit this class. - */ +/** + * Value types that correspond to a field; that is, + * a ring also posessing multiplicative inverses for + * non-zero elements. + * + * Floating-point values are only approximately a field, + * but nonetheless inhabit this class. + */ primitive type Field : * -> Prop /** - * Reciprocal - * - * Compute the multiplicative inverse of an element of a field. - * The reciprocal of zero is undefined. - */ + * Reciprocal + * + * Compute the multiplicative inverse of an element of a field. + * The reciprocal of 0 is undefined. + */ primitive recip : {a} (Field a) => a -> a /** @@ -581,11 +592,19 @@ a ==> b = if a then b else True /** * 2's complement signed division. Division rounds toward 0. + * Division by 0 is undefined. + * + * * Satisfies 'x == x %$ y + (x /$ y) * y' for 'y != 0'. */ primitive (/$) : {n} (fin n, n >= 1) => [n] -> [n] -> [n] /** * 2's complement signed remainder. Division rounds toward 0. + * Division by 0 is undefined. Satisfies the following for 'y != 0' + * + * * 'x %$ y == x - (x /$ y) * y'. + * * 'x >=$ 0 ==> x %$ y >=$ 0' + * * 'x <=$ 0 ==> x %$ y <=$ 0' */ primitive (%$) : {n} (fin n, n >= 1) => [n] -> [n] -> [n] @@ -640,7 +659,7 @@ primitive (>>$) : {n, ix} (fin n, n >= 1, Integral ix) => [n] -> ix -> [n] * Log base two. * * For words, computes the ceiling of log, base 2, of a number. - * We set `lg2 0 = 0` + * We set 'lg2 0 = 0' */ primitive lg2 : {n} (fin n) => [n] -> [n] @@ -649,7 +668,9 @@ primitive lg2 : {n} (fin n) => [n] -> [n] /** * Compute the ratio of two integers as a rational. - * ratio x y = (fromInteger x /. fromInteger y) : Rational + * Ratio is undefined if the denominator is 0. + * + * 'ratio x y = (fromInteger x /. fromInteger y) : Rational' */ primitive ratio : Integer -> Integer -> Rational @@ -704,17 +725,26 @@ primitive reverse : {n, a} (fin n) => [n]a -> [n]a primitive transpose : {rows, cols, a} [rows][cols]a -> [cols][rows]a +/** + * Select the first (left-most) 'front' elements of a sequence. + */ take : {front, back, a} (fin front) => [front + back]a -> [front]a take (x # _) = x +/** + * Select all the elements after (to the right of) the 'front' elements of a sequence. + */ drop : {front, back, a} (fin front) => [front + back]a -> [back]a drop ((_ : [front] _) # y) = y +/** + * Drop the first (left-most) element of a sequence. + */ tail : {n, a} [1 + n]a -> [n]a tail xs = drop`{1} xs /** - * Return the left-most element of a sequence. + * Return the first (left-most) element of a sequence. */ head : {n, a} [1 + n]a -> a head xs = xs @ 0 @@ -725,13 +755,18 @@ head xs = xs @ 0 last : {n, a} (fin n) => [1 + n]a -> a last xs = xs ! 0 +/** + * Same as 'split', but with a different type argument order. + * Take a sequence of elements and break it into 'parts' sequences + * of 'each' elements. + */ groupBy : {each, parts, a} (fin each) => [parts * each]a -> [parts][each]a groupBy = split`{parts=parts} /** * Left shift. The first argument is the sequence to shift, the second is the * number of positions to shift by. -*/ + */ primitive (<<) : {n, ix, a} (Integral ix, Zero a) => [n]a -> ix -> [n]a /** @@ -890,19 +925,27 @@ pmod x y = if y == 0 then 0/0 else last zs // Utility operations ----------------------------------------------------------------- -primitive error : {a, len} (fin len) => String len -> a +/** + * Raise a run-time error with the given message. + * This function can be called at any type. + */ +primitive error : {a, n} (fin n) => String n -> a +/** + * Raise a run-time error with a generic message. + * This function can be called at any type. + */ undefined : {a} a undefined = error "undefined" /** * Assert that the given condition holds, and raise an error - * with the given message if it does not. + * with the given message if it does not. If the condition + * holds, return the third argument unchanged. */ -assert : {a, len} (fin len) => Bit -> String len -> a -> a +assert : {a, n} (fin n) => Bit -> String n -> a -> a assert pred msg x = if pred then x else error msg - /** * Generates random values from a seed. When called with a function, currently * generates a function that always returns zero. @@ -920,7 +963,7 @@ primitive random : {a} [256] -> a * which are unspecified. Thus, the output produced by this * operation may be difficult to predict. */ -primitive trace : {n, a, b} (fin n) => [n][8] -> a -> b -> b +primitive trace : {n, a, b} (fin n) => String n -> a -> b -> b /** * Debugging function for tracing values. The first argument is a string, @@ -933,7 +976,7 @@ primitive trace : {n, a, b} (fin n) => [n][8] -> a -> b -> b * which are unspecified. Thus, the output produced by this * operation may be difficult to predict. */ -traceVal : {n, a} (fin n) => [n][8] -> a -> a +traceVal : {n, a} (fin n) => String n -> a -> a traceVal msg x = trace msg x x @@ -1016,7 +1059,7 @@ repeat x = [ x | _ <- zero : [n] ] /** * 'elem x xs' returns true if x is equal to a value in xs. */ -elem : {n, a} (fin n, Cmp a) => a -> [n]a -> Bit +elem : {n, a} (fin n, Eq a) => a -> [n]a -> Bit elem a xs = any (\x -> x == a) xs /** diff --git a/tests/issues/issue226.icry.stdout b/tests/issues/issue226.icry.stdout index eb6ababa..071caa11 100644 --- a/tests/issues/issue226.icry.stdout +++ b/tests/issues/issue226.icry.stdout @@ -117,15 +117,15 @@ Symbols all : {n, a} (fin n) => (a -> Bit) -> [n]a -> Bit and : {n} (fin n) => [n] -> Bit any : {n, a} (fin n) => (a -> Bit) -> [n]a -> Bit - assert : {a, len} (fin len) => Bit -> String len -> a -> a + assert : {a, n} (fin n) => Bit -> String n -> a -> a carry : {n} (fin n) => [n] -> [n] -> Bit ceiling : {a} (Round a) => a -> Integer complement : {a} (Logic a) => a -> a curry : {a, b, c} ((a, b) -> c) -> a -> b -> c demote : {val, rep} (Literal val rep) => rep drop : {front, back, a} (fin front) => [front + back]a -> [back]a - elem : {n, a} (fin n, Cmp a) => a -> [n]a -> Bit - error : {a, len} (fin len) => String len -> a + elem : {n, a} (fin n, Eq a) => a -> [n]a -> Bit + error : {a, n} (fin n) => String n -> a False : Bit floor : {a} (Round a) => a -> Integer foldl : {n, a, b} (fin n) => (a -> b -> a) -> a -> [n]b -> a @@ -186,8 +186,8 @@ Symbols tail : {n, a} [1 + n]a -> [n]a take : {front, back, a} (fin front) => [front + back]a -> [front]a toInteger : {a} (Integral a) => a -> Integer - trace : {n, a, b} (fin n) => [n][8] -> a -> b -> b - traceVal : {n, a} (fin n) => [n][8] -> a -> a + trace : {n, a, b} (fin n) => String n -> a -> b -> b + traceVal : {n, a} (fin n) => String n -> a -> a transpose : {rows, cols, a} [rows][cols]a -> [cols][rows]a trunc : {a} (Round a) => a -> Integer uncurry : {a, b, c} (a -> b -> c) -> (a, b) -> c diff --git a/tests/regression/bvdiv.cry b/tests/regression/bvdiv.cry new file mode 100644 index 00000000..717146d0 --- /dev/null +++ b/tests/regression/bvdiv.cry @@ -0,0 +1,70 @@ +// unsigned division properties + +bvdiv_euc : {n} (fin n) => [n] -> [n] -> Bit +bvdiv_euc x y = y != 0 ==> (x/y)*y + x%y == x + +bvdiv_char : {n} (fin n) => [n] -> [n] -> Bit +bvdiv_char x y = y != 0 ==> floor r == toInteger (x / y) + where + r = ratio (toInteger x) (toInteger y) + +bvdiv_mod : {n} (fin n) => [n] -> [n] -> Bit +bvdiv_mod x y = y != 0 ==> (0 <= m /\ m < y) + where + m = x%y + + +// some utility definitions + +sabs : {n} (fin n, n>=1) => [n] -> [n] +sabs x = if x <=$ 0 then negate x else x + +toSignedInteger : {n} (fin n, n >=1 ) => [n] -> Integer +toSignedInteger x = if x <=$ 0 then negate (toInteger (negate x)) else toInteger x + +minint : {n} (fin n, n >= 1) => [n] +minint = 0b1 # zero + +maxint : {n} (fin n, n >= 1) => [n] +maxint = ~minint + + +// signed division properties + +bvsdiv_euc : {n} (fin n, n>=1) => [n] -> [n] -> Bit +bvsdiv_euc x y = y != 0 ==> (x/$y)*y + x%$y == x + +bvsdiv_char : {n} (fin n, n>=1) => [n] -> [n] -> Bit +bvsdiv_char x y = y != 0 ==> overflow \/ trunc r == toSignedInteger (x /$ y) + where + overflow = x == minint /\ y == (-1) + r = ratio (toSignedInteger x) (toSignedInteger y) + +bvsdiv_mod : {n} (fin n, n>=1) => [n] -> [n] -> Bit +bvsdiv_mod x y = y != 0 ==> (0 <= m /\ m < sabs y) + where + m = sabs (x%$y) + +bvsdiv_mod_sign : {n} (fin n, n>=1) => [n] -> [n] -> Bit +bvsdiv_mod_sign x y = + y != 0 ==> + if x >=$ 0 then x%$y >=$ 0 else x%$y <=$ 0 + +bvsdiv_doc_properties : {n} (fin n, n>=1) => [n] -> [n] -> Bit +bvsdiv_doc_properties x y = + y != 0 ==> + (x %$ y == x - (x /$ y) * y) /\ + (x >=$ 0 ==> x%$y >=$ 0) /\ + (x <=$ 0 ==> x%$y <=$ 0) + +// property instances to check + +property d1 = bvdiv_euc`{8} +property c1 = bvdiv_char`{8} +property m1 = bvdiv_mod`{8} + +property d2 = bvsdiv_euc`{8} +property c2 = bvsdiv_char`{8} +property m2 = bvsdiv_mod`{8} +property m3 = bvsdiv_mod_sign`{8} +property sdiv = bvsdiv_doc_properties`{8} diff --git a/tests/regression/bvdiv.icry b/tests/regression/bvdiv.icry new file mode 100644 index 00000000..d9ba6dfb --- /dev/null +++ b/tests/regression/bvdiv.icry @@ -0,0 +1,2 @@ +:l bvdiv.cry +:exhaust diff --git a/tests/regression/bvdiv.icry.stdout b/tests/regression/bvdiv.icry.stdout new file mode 100644 index 00000000..ae5c6a35 --- /dev/null +++ b/tests/regression/bvdiv.icry.stdout @@ -0,0 +1,27 @@ +Loading module Cryptol +Loading module Cryptol +Loading module Main +property d1 Using exhaustive testing. +Testing... Passed 65536 tests. +Q.E.D. +property c1 Using exhaustive testing. +Testing... Passed 65536 tests. +Q.E.D. +property m1 Using exhaustive testing. +Testing... Passed 65536 tests. +Q.E.D. +property d2 Using exhaustive testing. +Testing... Passed 65536 tests. +Q.E.D. +property c2 Using exhaustive testing. +Testing... Passed 65536 tests. +Q.E.D. +property m2 Using exhaustive testing. +Testing... Passed 65536 tests. +Q.E.D. +property m3 Using exhaustive testing. +Testing... Passed 65536 tests. +Q.E.D. +property sdiv Using exhaustive testing. +Testing... Passed 65536 tests. +Q.E.D.

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