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extends { BinaryOperatorT, ⊙, UnaryOperatorT, ∼, EquivalenceRelationT, = } where { T coerces ComplementBound⊙ } property ∀(a:T) (a ⊙ (∼ a)) :=: ComplementBound⊙ end A set of values with a binary operator ⊙ has complements if and only if there is a specific value e such that for every value a there is another value a ′ such that the result of applying ⊙ to a and a ′ (in either order) equals the specified value e . This value may be obtained by coercing the object named ComplementBound⊙ to type T . The unary operator ∼ returns the complement of its argument with respect to the operator ⊙ . Note that the trait HasComplements is similar to the trait HasInverses, but HasComplements does not require that that specified element be an identity of the binary operator. trait DeMorganT extends DeMorganT, , , ∼,opr ,opr ,opr ∼ extends { BinaryOperatorT, , BinaryOperatorT, , UnaryOperatorT, ∼, EquivalenceRelationT, = } property ∀(a:T, b: T) (∼ (a b)) :=: ((∼ a) (∼ b)) end This trait expresses De Morgan’s law for two binary operators and and a unary operator ∼ : the expressions ∼ (a b) and (∼ a) (∼ b) produce equal results. trait BooleanAlgebraT extends BooleanAlgebraT, , , ∼, ∨,opr ,opr ,opr ∼,opr ∨ extends { CommutativeT, , AssociativeT, , CommutativeT, , AssociativeT, , IdempotentBinaryOperatorT, , IdempotentBinaryOperatorT, , HasIdentityT, , HasIdentityT, , HasComplementsT, , ∼, HasComplementsT, , ∼, DistributiveT, , , DistributiveT, , , DeMorganT, , , ∼, DeMorganT, , , ∼, RingT, ∨,IDENTITY, } property ∀(a:T) (∼ (∼ a)) :=: a opr ∨(self,other: T):T = (self (∼ other)) ((∼ self) other) property castT(Identity∨) = castT(Identity) property castT(ComplementBound) = castT(Identity) property castT(ComplementBound) = castT(Identity) end A boolean algebra is an algebraic structure consisting of a set of values, three binary operators , , and ∨ , and a unary operator ∼ , such that the operators obey a number of specific properties: • is commutative, associative, and idempotent, and has a unique identity • is commutative, associative, and idempotent, and has a unique identity • has complements with respect to ∼ • has complements with respect to ∼ • and distribute over each other • De Morgan’s law applies to , , and ∼ , and also to , , and ∼ • The values form a ring with ∨ as the addition operator, IDENTITY as the additive inverse operator, and as the multiplication operator 260
A default definition is provided for the ∨ operator in terms of , , and ∼ . The type Boolean is the most familiar example of a boolean algebra. The power set of a set (that is, the set of all subsets of the set) also forms a boolean algebra with operators ∩ , ∪ , set complement, and symmetric set difference; the empty set is the identity for ∪ , and the original set is the identity for ∩ . 261
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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.
Page 209 and 210: unit secondOfAngle secondsOfAngle:
Page 211 and 212: Chapter 30 Tests 30.1 The Object Fo
Page 213 and 214: 31.2 Convenience Types An optional
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Page 217 and 218: y evaluating the two elements of th
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Page 227 and 228: Chapter 33 Overloaded Functional De
Page 229 and 230: coercions: T U ⇐⇒ T ♦ U ∧
Page 231 and 232: The More Specific Rule for Function
Page 233 and 234: An infix/multifix operator declarat
Page 235 and 236: may be so defined except ordinary p
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Page 241 and 242: Chapter 36 Support for Domain-Speci
Page 243 and 244: Because syntax expanders are define
Page 245 and 246: Part V Fortress APIs and Documentat
Page 247 and 248: property ∀(a:T) ¬(a ∼ a) end A
Page 249 and 250: trait PartialOrderOperatorsT extend
Page 251 and 252: The HasMinimalElement trait specifi
Page 253 and 254: The trait Commutative requires the
Page 255 and 256: A value e is a left identity for a
Page 257 and 258: trait ApproximatelyHasInverses T ex
Page 259: end extends { ApproximateCommutativ
Page 263 and 264: trait RationalQuantityunit U absorb
Page 265 and 266: (self,other:RationalQuantityU,ninf
Page 267 and 268: 38.2.4 getter hashCode(): Z64 38.2.
Page 269 and 270: Chapter 39 Components and APIs We d
Page 271 and 272: Chapter 40 Memory Sequences and Bin
Page 273 and 274: updatenat k(j: IntegerStatick, v: T
Page 275 and 276: 40.1.12 update(j:IndexInt, v: T): L
Page 277 and 278: 40.3.2 opr nat m[r:RangeOfStaticSiz
Page 279 and 280: 40.5 The Trait Fortress.Core.Binary
Page 281 and 282: 40.5.5 bit(m:IndexInt): Bit The res
Page 283 and 284: 40.6 The Trait Fortress.Core.Binary
Page 285 and 286: 40.6.4 opr nat m[r:RangeOfStaticSiz
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Page 289 and 290: countLeadingZeroBits():IndexInt cou
Page 291 and 292: 40.7.41 signedShift(j:IndexInt):T 4
Page 293 and 294: 40.8.6 signedDivide(other: T,overfl
Page 295 and 296: littleEndian():BinaryEndianLinearEn
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Page 301 and 302: 40.10.9 opr nat m[r:RangeOfStaticSi
Page 303 and 304: 40.10.24 splitnat b ′ () : Binary
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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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