adt-row-sub.scrbl (4101B)
1 #lang scribble/manual 2 3 @require["util.rkt" 4 "adt-utils.rkt" 5 (for-label (only-meta-in 0 typed/racket))] 6 @(use-mathjax) 7 8 @title[#:style (with-html5 manual-doc-style) 9 #:version (version-text)]{Subtyping (with ρ)} 10 11 Variants which do not involve a row type are nothing more than a syntactic 12 restriction on a union of types. The variant only allows constructors to be 13 present in the union, as specified by the type validity rule 14 @refrule[@textsc{TE-Variant}]. 15 16 @$inferrule[- 17 @=:[@variantτ[@repeated{τ}] @un[@repeated{τ}]] 18 @${@textsc{S-Variant-Union}}] 19 20 From @refrule[@textsc{S-Variant-Union}], we can derive that the empty variant 21 is equivalent to the bottom type. 22 23 @$p[@$infertree[(@refrule[@textsc{S-Variant-Union}] 24 @refrule[@textsc{S-Bot}] 25 ⇒ 26 @refrule[@textsc{S-Eq-Transitive}]) 27 ⇒ 28 @=:[@variantτ[] ⊥] 29 @${@textsc{S-Variant-Empty}}] 30 @$infertree[(((@${@=:[τ τ′]} ⇒ @refrule[@${@textsc{S-Eq}_1}]) 31 (@${@=:[τ′ τ″]} ⇒ @refrule[@${@textsc{S-Eq}_1}]) 32 ⇒ 33 @refrule[@textsc{S-Eq-Transitive}]) 34 ((@${@=:[τ′ τ″]} ⇒ @refrule[@${@textsc{S-Eq}_2}]) 35 (@${@=:[τ τ′]} ⇒ @refrule[@${@textsc{S-Eq}_2}]) 36 ⇒ 37 @refrule[@textsc{S-Eq-Transitive}]) 38 ⇒ 39 @refrule[@${@textsc{S-Eq}_3}]) 40 ⇒ 41 @=:[τ τ″] 42 @${@textsc{S-Eq-Transitive}}]] 43 44 45 @$p[@$inferrule[@${∃ i . @<:[τ @${σᵢ}]} 46 @${@<:[τ @variantτ[@repeated{σᵢ}]]} 47 @${@textsc{S-VariantSuper}}] 48 49 @$inferrule[@${@repeated[@<:[τᵢ @${σ}]]} 50 @${@<:[@variantτ[@repeated{τᵢ}] σ]} 51 @${@textsc{S-VariantSub}}]] 52 53 54 55 56 57 58 59 @;{@$inferrule[- 60 @<:[@ctorτ[@κof[τ]] @ctorTop[τ]] 61 @${@textsc{S-CtorTop}}]} 62 63 @$inferrule[ 64 @${@<:[τ τ′]} 65 @<:[@ctorτ[@κof[τ]] @ctorτ[@κof[τ′]]] 66 @${@textsc{S-Ctor}}] 67 68 @$inferrule[ 69 @${@repeated{@<:[τᵢ τ′ᵢ]}} 70 @<:[@recordτ[@repeated{@${@|ɐ|ᵢ : τᵢ}}] @recordτ[@repeated{@${@|ɐ|ᵢ : τ'ᵢ}}]] 71 @${@textsc{S-Record}}] 72 73 @$inferrule[ 74 @${@repeated{@<:[τⱼ τ′ⱼ]}} 75 @<:[@recordτ[@repeated{@${@|ɐ|ⱼ : τⱼ}}] 76 @recordτ[@|ρf| @repeatset{-@|ɐ|ᵢ} @repeatset{+@|ɐ|ⱼ:τ'ⱼ}]] 77 @${@textsc{S-RecordF}}] 78 79 @$inferrule[@${@repeated{@<:[τₖ τ′ₖ]}} 80 @<:[@recordτ[@|ρf| 81 @repeatset{-@|ɐ|ᵢ} 82 @repeatset{-@|ɐ|ⱼ} 83 @repeatset{+@|ɐ|ₖ:τₖ} 84 @repeatset{+@|ɐ|ₗ:τₗ}] 85 @recordτ[@|ρf| 86 @repeatset{-@|ɐ|ᵢ} 87 @repeatset{+@|ɐ|ₖ:τ'ₖ}]] 88 #:wide #t 89 @${@textsc{S-FRecordF}}] 90 91 @$inferrule[@${@<:[@${τ[@repeated{@|ρf|ᵢ ↦ @|ρf|′ᵢ}]} σ]} 92 @${@<:[@∀f[(@repeated{@|ρf|ᵢ}) τ] 93 @∀f[(@repeated{@|ρf|′ᵢ}) σ]]} 94 @${@textsc{S-PolyF-}α@textsc{-Equiv}}] 95 96 @$inferrule[@${@<:[@${τ[@repeated{@|ρc|ᵢ ↦ @|ρc|′ᵢ}]} σ]} 97 @${@<:[@∀c[(@repeated{@|ρc|ᵢ}) τ] 98 @∀c[(@repeated{@|ρc|′ᵢ}) σ]]} 99 @${@textsc{S-PolyC-}α@textsc{-Equiv}}] 100 101 @; TODO: can these rules be used to combine a record type and a record*? 102 @; predicate to make sure some fields are absent, without checking anything 103 @; for the other fields? 104 105 @; TODO: instantiation rules. ∀ rules. Etc. 106 107 @;{ 108 Permutation of the fields of a record type produces an equivalent type: 109 110 @$${ 111 @$inferrule[ 112 @${\phantom{x}} 113 @<:[@recordτ[@repeated{@${@|ɐ|ᵢ : τᵢ}} @repeated{@${@|ɐ|ⱼ : τⱼ}}] 114 @recordτ[@repeated{@${@|ɐ|ⱼ : τⱼ}} @repeated{@${@|ɐ|ᵢ : τᵢ}}]] 115 @${@textsc{S-Record-Permuation}} 116 ] 117 } 118 } 119 120 @todo{propagate our types up/down unions like the primitive ones (I think 121 Typed Racket does not do this everywhere it “should”).}