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U+21DA LEFTWARDS TRIPLE ARROW ⇚ U+21DB RIGHTWARDS TRIPLE ARROW ⇛ U+21DC LEFTWARDS SQUIGGLE ARROW U+21DD RIGHTWARDS SQUIGGLE ARROW U+21DE UPWARDS ARROW WITH DOUBLE STROKE U+21DF DOWNWARDS ARROW WITH DOUBLE STROKE U+21E0 LEFTWARDS DASHED ARROW U+21E1 UPWARDS DASHED ARROW U+21E2 RIGHTWARDS DASHED ARROW U+21E3 DOWNWARDS DASHED ARROW U+21E4 LEFTWARDS ARROW TO BAR U+21E5 RIGHTWARDS ARROW TO BAR U+21E6 LEFTWARDS WHITE ARROW U+21E7 UPWARDS WHITE ARROW U+21E8 RIGHTWARDS WHITE ARROW U+21E9 DOWNWARDS WHITE ARROW U+21EA UPWARDS WHITE ARROW FROM BAR U+21EB UPWARDS WHITE ARROW ON PEDESTAL U+21EC UPWARDS WHITE ARROW ON PEDESTAL WITH HORIZONTAL BAR U+21ED UPWARDS WHITE ARROW ON PEDESTAL WITH VERTICAL BAR U+21EE UPWARDS WHITE DOUBLE ARROW U+21EF UPWARDS WHITE DOUBLE ARROW ON PEDESTAL U+21F0 RIGHTWARDS WHITE ARROW FROM WALL U+21F1 NORTH WEST ARROW TO CORNER U+21F2 SOUTH EAST ARROW TO CORNER U+21F3 UP DOWN WHITE ARROW U+21F4 RIGHT ARROW WITH SMALL CIRCLE U+21F5 DOWNWARDS ARROW LEFTWARDS OF UPWARDS ARROW U+21F6 THREE RIGHTWARDS ARROWS U+21F7 LEFTWARDS ARROW WITH VERTICAL STROKE U+21F8 RIGHTWARDS ARROW WITH VERTICAL STROKE U+21F9 LEFT RIGHT ARROW WITH VERTICAL STROKE U+21FA LEFTWARDS ARROW WITH DOUBLE VERTICAL STROKE U+21FB RIGHTWARDS ARROW WITH DOUBLE VERTICAL STROKE U+21FC LEFT RIGHT ARROW WITH DOUBLE VERTICAL STROKE U+21FD LEFTWARDS OPEN-HEADED ARROW U+21FE RIGHTWARDS OPEN-HEADED ARROW U+21FF LEFT RIGHT OPEN-HEADED ARROW U+2201 COMPLEMENT ∁ U+2202 PARTIAL DIFFERENTIAL ∂ DEL U+2204 THERE DOES NOT EXIST ̸ ∃ U+2206 INCREMENT ∆ ∏ U+220F N-ARY PRODUCT ∐ PRODUCT U+2210 N-ARY COPRODUCT ∑ COPRODUCT U+2211 N-ARY SUMMATION SUM U+2218 RING OPERATOR ◦ CIRC RING COMPOSE U+2219 BULLET OPERATOR • √ BULLET U+221A SQUARE ROOT SQRT U+221B CUBE ROOT CBRT U+221C FOURTH ROOT FOURTHROOT U+221D PROPORTIONAL TO ∝ PROPTO U+2223 DIVIDES | DIVIDES 366
U+2224 DOES NOT DIVIDE ∤ U+2225 PARALLEL TO ‖ PARALLEL U+2226 NOT PARALLEL TO ∦∫ NPARALLEL U+222B INTEGRAL U+222C DOUBLE INTEGRAL U+222D TRIPLE INTEGRAL U+222E CONTOUR INTEGRAL ∮ U+222F SURFACE INTEGRAL U+2230 VOLUME INTEGRAL U+2231 CLOCKWISE INTEGRAL U+2232 CLOCKWISE CONTOUR INTEGRAL U+2233 ANTICLOCKWISE CONTOUR INTEGRAL U+2234 THEREFORE ∴ U+2235 BECAUSE ∵ U+2236 RATIO U+2237 PROPORTION U+2239 EXCESS U+223A GEOMETRIC PROPORTION U+223B HOMOTHETIC U+223C TILDE OPERATOR ∼ U+223D REVERSED TILDE ∽ U+223E INVERTED LAZY S U+223F SINE WAVE U+2240 WREATH PRODUCT ≀ WREATH U+2241 NOT TILDE ≁ U+224B TRIPLE TILDE U+224F DIFFERENCE BETWEEN ≏ BUMPEQ . U+2250 APPROACHES THE LIMIT = DOTEQ U+2258 CORRESPONDS TO U+2259 ESTIMATES U+225A EQUIANGULAR TO U+225E MEASURED BY U+226C BETWEEN ≬ U+228C MULTISET U+229A CIRCLED RING OPERATOR ⊚ CIRCLEDRING U+229D CIRCLED DASH ⊖ U+22A2 RIGHT TACK ⊢ VDASH TURNSTILE U+22A3 LEFT TACK ⊣ DASHV U+22A6 ASSERTION ⊢ U+22A7 MODELS U+22A8 TRUE |= U+22A9 FORCES ⊩ U+22AA TRIPLE VERTICAL BAR RIGHT TURNSTILE U+22AB DOUBLE VERTICAL BAR DOUBLE RIGHT TURNSTILE U+22AC DOES NOT PROVE U+22AD NOT TRUE U+22AE DOES NOT FORCE U+22AF NEGATED DOUBLE VERTICAL BAR DOUBLE RIGHT TURNSTILE U+22B6 ORIGINAL OF U+22B7 IMAGE OF U+22B8 MULTIMAP ⊸ U+22B9 HERMITIAN CONJUGATE MATRIX 367
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The Fortress Language Specification
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4 Evaluation 36 4.1 Values . . . .
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12 Functions 85 12.1 Function Decla
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17.7 Coercions for Tuple and Arrow
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IV Fortress for Library Writers 214
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40.10The Trait Fortress.Core.Binary
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Chapter 1 Introduction The Fortress
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A note on the presented libraries i
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component HelloWorld export Executa
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foo:String → () baz: String → S
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fortress upgrade Gary with Ralph No
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halve(x : R64) : R64 = x/2 Predicta
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while x < 10 do print x x += 1 end
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It is also possible to convert a me
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factorial(n) = ∏ i←1:n i This f
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In both methods cons and append , t
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default distributions will provide
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Chapter 3 Programs A program consis
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Chapter 10), a literal (see Section
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(discussed in Section 13.13) to a s
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Chapter 5 Lexical Structure A Fortr
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U+0021 EXCLAMATION MARK ! U+0023 NU
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If a character (or any other syntac
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BIG SI unit absorbs abstract also a
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escape sequences are useful for ASC
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5.14.1 Multicharacter Enclosing Ope
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Note: the elegant way to write Avog
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• top-level variable declarations
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declares id to be a mutable variabl
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several new variables. This syntax
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Chapter 7 Names Names are used to r
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7.3 Qualified Names Fortress provid
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Fortress also allows coercion betwe
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• element types of a tuple type c
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TypeAlias ::= type Id [StaticParams
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‘}’. If such a clause contains
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Note that a trait declaration inher
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the parametric trait WrappedDiction
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Chapter 10 Objects An object is a v
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FldDef ::= FldMod ∗ Id [IsType] (
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Chapter 11 Static Parameters Trait,
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11.5 Operator and Identifier Parame
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Chapter 12 Functions Functions are
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swap(x :Object, y : Object) = (y, x
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A function declaration may be separ
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Chapter 13 Expressions Fortress is
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Negating the literal 0 produces a s
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the function is evaluated with the
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13.10 Assignments Syntax: Expr ::=
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treeSum(t : TreeLeaf) = 0 treeSum(t
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Several common mathematical operato
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The body of each iteration is run i
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to the value of the condition expre
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An atomic expression consists of th
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13.26 Try Expressions Syntax: Flow
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13.27.2 Static Expressions and Valu
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A = [1 2 3; 4 5 6; 7 8 9] then A 1,
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evaluates to the set {0, 4, 16} Map
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ignore(f x) is equivalent to: f x;
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trait CheckedException extends { Ex
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Chapter 15 Overloading and Multiple
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e called with. In other words, we e
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Chapter 16 Operators Operators are
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• “ordinary” operators: + −
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left context whitespace operator fi
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The rules below are designed to for
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16.8.3 Enclosing Operators When use
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16.8.9 Intersection, Union, and Set
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Chapter 17 Conversions and Coercion
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17.3 Coercion Invocations One may i
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arguments nor with variable number
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trait B extends A end trait C exten
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a new coercion in a subexpression
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for tokens. For readability, plural
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Chapter 19 Tests and Properties For
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19.5 Test Suites In order to allow
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Chapter 20 Type Inference Type infe
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3. If a i is a functional applicati
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21.2 Programming Discipline If Fort
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Here the call f(a, a) ends up setti
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2. Ancestors, dynamically ordered b
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The ways in which fortresses are ac
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Component equivalence is determined
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22.5 Type Inference for Components
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Fortress.IO Fortress.Crypto IronLin
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execute(componentName:String, args:
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3. If the replacement does not expo
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This method runs all test functions
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Fortress.IO Fortress.Crypto Fortres
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Part III Fortress APIs and Document
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23.1.3 hash(maxval: N64): N64 23.1.
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opr =(self,other: Boolean):Boolean
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24.1.20 getter hashCode(): N64 24.1
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False: BooleanInterval Uncertain: B
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24.2.11 opr ∧(self,other: Boolean
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24.2.19 possibly(self): Boolean 24.
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Chapter 25 Numbers 25.1 Rational Nu
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opr ⌈self ⌉: N realpart(self):
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25.1.11 opr (self,other: Q):Boolean
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25.1.26 floor(self): Z 25.1.27 opr
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Chapter 26 Negated Relational Opera
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26.1.26 opr ⊀T extends BinaryPred
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27.3 The Trait Fortress.Standard.Un
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Chapter 29 Dimensions and Units 29.
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unit secondOfAngle secondsOfAngle:
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Chapter 30 Tests 30.1 The Object Fo
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31.2 Convenience Types An optional
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Chapter 32 Parallelism and Locality
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y evaluating the two elements of th
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There is a DefaultDistribution whic
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v = spawn at a.region(i) do a i end
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expr ∑ type wrapper ∑ body R,
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32.8.3 Using Overloading to Adapt G
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Chapter 33 Overloaded Functional De
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coercions: T U ⇐⇒ T ♦ U ∧
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The More Specific Rule for Function
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An infix/multifix operator declarat
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may be so defined except ordinary p
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and here some of these computed dim
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unit name 1 name 2 name 3 : . . . u
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Chapter 36 Support for Domain-Speci
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Because syntax expanders are define
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Part V Fortress APIs and Documentat
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property ∀(a:T) ¬(a ∼ a) end A
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trait PartialOrderOperatorsT extend
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The HasMinimalElement trait specifi
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The trait Commutative requires the
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A value e is a left identity for a
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trait ApproximatelyHasInverses T ex
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end extends { ApproximateCommutativ
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A default definition is provided fo
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trait RationalQuantityunit U absorb
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(self,other:RationalQuantityU,ninf
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38.2.4 getter hashCode(): Z64 38.2.
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Chapter 39 Components and APIs We d
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Chapter 40 Memory Sequences and Bin
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updatenat k(j: IntegerStatick, v: T
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40.1.12 update(j:IndexInt, v: T): L
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40.3.2 opr nat m[r:RangeOfStaticSiz
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40.5 The Trait Fortress.Core.Binary
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40.5.5 bit(m:IndexInt): Bit The res
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40.6 The Trait Fortress.Core.Binary
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40.6.4 opr nat m[r:RangeOfStaticSiz
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40.6.16 opr ‖ nat m,bool bigEndia
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countLeadingZeroBits():IndexInt cou
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40.7.41 signedShift(j:IndexInt):T 4
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40.8.6 signedDivide(other: T,overfl
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littleEndian():BinaryEndianLinearEn
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40.9.16 opr ‖ nat m,nat radix,nat
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v: BinaryEndianWordb,bigEndianBytes
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40.10.9 opr nat m[r:RangeOfStaticSi
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40.10.24 splitnat b ′ () : Binary
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itAnd(selfStart: IndexInt,source:T,
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40.13.2 wrappingAdd(selfStart: Inde
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40.13.30 unsignedMax(selfStart: Ind
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40.13.55 gatherBits(selfStart:Index
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Appendix A Fortress Calculi A.1 Bas
Page 315 and 316: Values, evaluation contexts, redexe
Page 317 and 318: Expression typing: p; ∆; Γ ⊢ e
Page 319 and 320: α, β type variables f method name
Page 321 and 322: Function/method name lookup: Fname(
Page 323 and 324: One owner for all the visible metho
Page 325 and 326: Traits defining a method: defining
Page 327 and 328: Values, evaluation contexts, and re
Page 329 and 330: Valid function declarations: p ⊢
Page 331 and 332: Method definition lookup: visible p
Page 333 and 334: Values, intermediate expressions, e
Page 335 and 336: Field typing: p; ∆; Γ ⊢ vd ok
Page 337 and 338: Appendix B Overloaded Functional De
Page 339 and 340: Proof. We prove this for δ, but th
Page 341 and 342: Upgrade A predicate upg? takes two
Page 343 and 344: a is rendered as a foobar is render
Page 345 and 346: • If a portion is sub and another
Page 347 and 348: supercalifragilisticexpialidocious
Page 349 and 350: any alternative names, with undersc
Page 351 and 352: Appendix F Operator Precedence, Cha
Page 353 and 354: U+007C VERTICAL LINE | | U+2016 DOU
Page 355 and 356: The following two operators have th
Page 357 and 358: U+003D EQUALS SIGN = = EQ U+2243 AS
Page 359 and 360: U+22DD EQUAL TO OR GREATER-THAN U+2
Page 361 and 362: U+2289 NEITHER A SUPERSET OF NOR EQ
Page 363 and 364: F.4.8 Miscellaneous Relational Oper
Page 365: U+21A5 UPWARDS ARROW FROM BAR U+21A
Page 369 and 370: U+27F8 LONG LEFTWARDS DOUBLE ARROW
Page 371 and 372: U+2960 UPWARDS HARPOON WITH BARB LE
Page 373 and 374: U+29ED BLACK CIRCLE WITH DOWN ARROW
Page 375 and 376: Appendix G Concrete Syntax In this
Page 377 and 378: Extends ::= extends TraitTypes Excl
Page 379 and 380: StaticParam ::= Id [Extends] [absor
Page 381 and 382: Value ::= Literal | fn ValParam [Is
Page 383 and 384: Appendix H Generated Concrete Synta
Page 385 and 386: -> fn_decl -> object_decl -> var_de
Page 387 and 388: -> id extends_opt absorbs_opt -> "n
Page 389 and 390: -> w "excludes" w type_ref_list com
Page 391 and 392: -> obj_decl_body_elem br obj_decl_b
Page 393 and 394: -> op wr op_follows_op_w -> op wr e
Page 395 and 396: -> no_space_op_expr_result no_space
Page 397 and 398: id_opt -> -> w id with_opt -> -> w
Page 399 and 400: -> unpasting_elem rect_separator un
Page 401 and 402: -> "}" comprehension -> set_compreh
Page 403: -> do_expr -> for_expr -> spawn_exp
Page 406 and 407: -> type_ref -> "(" w type_ref w ","
Page 408 and 409: enc -> enc_literal op_or_enc -> op
Page 410 and 411: Bibliography [1] O. Agesen, L. Bak,
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