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The Fortress standard libraries also define other simple standard types such as Object, Exception, Boolean, and BooleanInterval as well as low-level binary data types such as LinearSequence, HeapSequence, and BinaryWord. See Parts III and V for discussions of the Fortress standard libraries. 8.8 Intersection and Union Types For every finite set of types, there is a type denoting a unique intersection of those types. The intersection of a set of types S is a subtype of every type T ∈ S and of the intersection of every subset of S. There is also a type denoting a unique union of those types. The union of a set of types S is a supertype of every type T ∈ S and of the union of every subset of S. Neither intersection types nor union types are first-class types; they are used solely for type inference (as described in Chapter 20) and they cannot be expressed directly in programs. The intersection of a set of types S is equal to a named type U when any subtype of every type T ∈ S and of the intersection of every subset of S is a subtype of U. Similarly, the union of a set of types S is equal to a named type U when any supertype of every type T ∈ S and of the union of every subset of S is a supertype of U. For example: trait S comprises {U, V } end trait T comprises {V, W } end trait U extends S excludes W end trait V extends {S, T } end trait W extends T end because of the comprises clauses of S and T and the excludes clause of U, any subtype of both S and T must be a subtype of V . Thus, V = S ∩ T . Intersection types (denoted by ∩) possess the following properties: • Commutativity: T ∩ U = U ∩ T . • Associativity: S ∩ (T ∩ U) = (S ∩ T) ∩ U. • Subsumption: If S ≼ T then S ∩ T = S. • Preservation of shared subtypes: If T ≼ S and T ≼ U then T ≼ S ∩ U. • Preservation of supertype: If S ≼ T then ∀U. S ∩ U ≼ T . • Distribution over union types: S ∩ (T ∪ U) = (S ∩ T) ∪ (S ∩ U). Union types (denoted by ∪) possess the following properties: • Commutativity: T ∪ U = U ∪ T . • Associativity: S ∪ (T ∪ U) = (S ∪ T) ∪ U. • Subsumption: If S ≼ T then S ∪ T = T . • Preservation of shared supertypes: If S ≼ T and U ≼ T then S ∪ U ≼ T . • Preservation of subtype: If T ≼ S then ∀U. T ≼ S ∪ U. • Distribution over intersection types: S ∪ (T ∩ U) = (S ∪ T) ∩ (S ∪ U). 8.9 Type Aliases Syntax: 68
TypeAlias ::= type Id [StaticParams] = TypeRef Fortress allows names to serve as aliases for more complex type instantiations. A type alias begins with the special reserved word type followed by the name of the alias type, followed by optional static parameters, followed by = , followed by the type it stands for. Parameterized type aliases are allowed but recursively defined type aliases are not. Here are some examples: type IntList = ListZ64 type BinOp = Float × Float → Float type SimpleFloatnat e,nat s = DetailedFloatUnity, e, s,false,false,false,false,true All uses of type aliases are expanded before type checking. Type aliases do not define new types nor nominal equivalence relations among types. 69
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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
Page 17 and 18: component HelloWorld export Executa
Page 19 and 20: foo:String → () baz: String → S
Page 21 and 22: fortress upgrade Gary with Ralph No
Page 23 and 24: halve(x : R64) : R64 = x/2 Predicta
Page 25 and 26: while x < 10 do print x x += 1 end
Page 27 and 28: It is also possible to convert a me
Page 29 and 30: factorial(n) = ∏ i←1:n i This f
Page 31 and 32: In both methods cons and append , t
Page 33 and 34: default distributions will provide
Page 35 and 36: Chapter 3 Programs A program consis
Page 37 and 38: Chapter 10), a literal (see Section
Page 39 and 40: (discussed in Section 13.13) to a s
Page 41 and 42: Chapter 5 Lexical Structure A Fortr
Page 43 and 44: U+0021 EXCLAMATION MARK ! U+0023 NU
Page 45 and 46: If a character (or any other syntac
Page 47 and 48: BIG SI unit absorbs abstract also a
Page 49 and 50: escape sequences are useful for ASC
Page 51 and 52: 5.14.1 Multicharacter Enclosing Ope
Page 53 and 54: Note: the elegant way to write Avog
Page 55 and 56: • top-level variable declarations
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Page 59 and 60: several new variables. This syntax
Page 61 and 62: Chapter 7 Names Names are used to r
Page 63 and 64: 7.3 Qualified Names Fortress provid
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Page 71 and 72: ‘}’. If such a clause contains
Page 73 and 74: Note that a trait declaration inher
Page 75 and 76: the parametric trait WrappedDiction
Page 77 and 78: Chapter 10 Objects An object is a v
Page 79 and 80: FldDef ::= FldMod ∗ Id [IsType] (
Page 81 and 82: Chapter 11 Static Parameters Trait,
Page 83 and 84: 11.5 Operator and Identifier Parame
Page 85 and 86: Chapter 12 Functions Functions are
Page 87 and 88: swap(x :Object, y : Object) = (y, x
Page 89 and 90: A function declaration may be separ
Page 91 and 92: Chapter 13 Expressions Fortress is
Page 93 and 94: Negating the literal 0 produces a s
Page 95 and 96: the function is evaluated with the
Page 97 and 98: 13.10 Assignments Syntax: Expr ::=
Page 99 and 100: treeSum(t : TreeLeaf) = 0 treeSum(t
Page 101 and 102: Several common mathematical operato
Page 103 and 104: The body of each iteration is run i
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Page 107 and 108: An atomic expression consists of th
Page 109 and 110: 13.26 Try Expressions Syntax: Flow
Page 111 and 112: 13.27.2 Static Expressions and Valu
Page 113 and 114: A = [1 2 3; 4 5 6; 7 8 9] then A 1,
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Page 117 and 118: 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
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Values, evaluation contexts, redexe
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Expression typing: p; ∆; Γ ⊢ e
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α, β type variables f method name
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Function/method name lookup: Fname(
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One owner for all the visible metho
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Traits defining a method: defining
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Values, evaluation contexts, and re
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Valid function declarations: p ⊢
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Method definition lookup: visible p
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Values, intermediate expressions, e
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Field typing: p; ∆; Γ ⊢ vd ok
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Appendix B Overloaded Functional De
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Proof. We prove this for δ, but th
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Upgrade A predicate upg? takes two
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a is rendered as a foobar is render
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• If a portion is sub and another
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supercalifragilisticexpialidocious
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any alternative names, with undersc
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Appendix F Operator Precedence, Cha
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U+007C VERTICAL LINE | | U+2016 DOU
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The following two operators have th
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U+003D EQUALS SIGN = = EQ U+2243 AS
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U+22DD EQUAL TO OR GREATER-THAN U+2
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U+2289 NEITHER A SUPERSET OF NOR EQ
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F.4.8 Miscellaneous Relational Oper
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U+21A5 UPWARDS ARROW FROM BAR U+21A
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U+2224 DOES NOT DIVIDE ∤ U+2225 P
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U+27F8 LONG LEFTWARDS DOUBLE ARROW
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U+2960 UPWARDS HARPOON WITH BARB LE
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U+29ED BLACK CIRCLE WITH DOWN ARROW
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Appendix G Concrete Syntax In this
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Extends ::= extends TraitTypes Excl
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StaticParam ::= Id [Extends] [absor
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Value ::= Literal | fn ValParam [Is
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Appendix H Generated Concrete Synta
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-> fn_decl -> object_decl -> var_de
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-> id extends_opt absorbs_opt -> "n
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-> w "excludes" w type_ref_list com
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-> obj_decl_body_elem br obj_decl_b
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-> op wr op_follows_op_w -> op wr e
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-> no_space_op_expr_result no_space
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id_opt -> -> w id with_opt -> -> w
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-> unpasting_elem rect_separator un
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-> "}" comprehension -> set_compreh
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-> do_expr -> for_expr -> spawn_exp
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-> type_ref -> "(" w type_ref w ","
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enc -> enc_literal op_or_enc -> op
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Bibliography [1] O. Agesen, L. Bak,
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