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Well-formed types: p; ∆ ⊢ τ ok [W-OBJ] [W-VAR] [W-TAPP] p; ∆ ⊢ Object ok α ∈ dom(∆) p; ∆ ⊢ α ok C −−−−−−−−−→ αextends N ∈ p p; ∆ ⊢ −→ τ ok p; ∆ ⊢ −→ τ
Appendix B Overloaded Functional Declarations As mentioned in Chapter 33, this appendix proves that the restrictions discussed in Chapter 33 guarantee no undefined nor ambiguous call at run time. B.1 Proof of Coercion Resolution for Functions This section proves that the restrictions discussed in the previous sections guarantee the static resolution of coercion (described in Section 17.5) is well defined for functions (the case for methods is analogous). Consider a static function call f (A) at some program point Z and its corresponding dynamic function call f (X). Let Σ be the set of parameter types of function declarations of f that are visible at Z and applicable to the static call f (A). Let Σ ′ be the set of parameter types of function declarations of f that are visible at Z and applicable with coercion to the static call f (A). Moreover, let σ ′ be the subset of Σ ′ for which no type in Σ ′ is more specific: σ ′ = {S ∈ Σ ′ | ¬∃S ′ ∈ Σ ′ : S ′ ⊳ S }. We prove the following: |Σ| = 0 and |Σ ′ | ≠ 0 imply |σ ′ | = 1. Informally, if no declaration is applicable to a static call but there is a declaration that is applicable with coercion then there exists a single most specific declaration that is applicable with coercion to the static call. Lemma 1. Given an acyclic, irreflexive binary relation R on a set S, and a finite nonempty subset A of S, the set {a ∈ A|¬∃a ′ ∈ A : (a ′ , a) ∈ R} is nonempty. Proof. Consider the relation R on S as a directed acyclic graph. Let A represent a subgraph. <strong>The</strong>n the Lemma amounts to proving that there exists a node in the graph represented by A with no edges pointing to it. This follows from the fact that A is finite and the graph is acyclic. Lemma 2. If |Σ ′ | ≥ 1 then |σ ′ | ≥ 1. Proof. Follows from Lemma 1 where S is the set of all types, A is Σ ′ , and the relation ⊳ is acyclic and irreflexive. Lemma 3. If |Σ| = 0 then |σ ′ | ≤ 1. 337
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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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Page 313 and 314: 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: Field typing: p; ∆; Γ ⊢ vd ok
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
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Page 361 and 362: U+2289 NEITHER A SUPERSET OF NOR EQ
Page 363 and 364: F.4.8 Miscellaneous Relational Oper
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Page 367 and 368: U+2224 DOES NOT DIVIDE ∤ U+2225 P
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
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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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