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model.rkt
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model.rkt
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#lang racket
(require redex)
(define-language
layouts
[l ::=
; unions
l+
; structs
l×
; atoms
la]
[l+ ::= (+ l l)]
[l× ::= (× l l)]
[la ::=
; zst
ε
; void
∅
lb]
[lb ::=
; uninit bytes
(u (name len natural))
; init bytes with valid range
(side-condition
(i (name len natural)
(name min natural)
(name max natural))
(< -1
(term min)
(term max)
(expt 255 (term len))))]
)
; add two numbers
(define-metafunction layouts
⊕ : natural natural -> natural
[(⊕ natural_0 natural_1)
,(+ (term natural_0) (term natural_1))])
; saturating sub
(define-metafunction layouts
⊖ : natural natural -> natural
[(⊖ natural_0 natural_1)
,(min 0 (- (term natural_0) (term natural_1)))])
; integer divide
(define-metafunction layouts
⊘ : natural natural -> natural
[(⊘ natural_0 natural_1)
,(quotient (term natural_0) (term natural_1))])
; max
(define-metafunction layouts
⊔ : natural natural -> natural
[(⊔ natural_0 natural_1)
,(max (term natural_0) (term natural_1))])
; min
(define-metafunction layouts
⊓ : natural natural -> natural
[(⊓ natural_0 natural_1)
,(min (term natural_0) (term natural_1))])
; exponentiation
(define-metafunction layouts
^ : natural natural -> natural
[(^ natural_0 natural_1)
,(expt (term natural_0) (term natural_1))])
; len of an `i` or `u` byte sequence
(define-metafunction layouts
📏 : lb -> natural_len
[(📏 (u natural_len))
natural_len]
[(📏 (i natural_len natural_min natural_max))
natural_len])
; neq
(define-metafunction layouts
≠ : natural natural -> boolean
[(≠ natural natural) #f]
[(≠ natural_0 natural_1) #t])
; eq
(define-metafunction layouts
= : natural natural -> boolean
[(= natural natural) #t]
[(= natural_0 natural_1) #f])
; lte
(define-metafunction layouts
≤ : natural natural -> boolean
[(≤ natural_0 natural_1)
,(<= (term natural_0) (term natural_1))])
; gte
(define-metafunction layouts
≥ : natural natural -> boolean
[(≥ natural_0 natural_1)
,(>= (term natural_0) (term natural_1))])
; snip bytes
; todo: test this! I just eyeballed it.
(define-metafunction layouts
✂ : lb natural -> (lb lb)
[(✂ (u natural_len) natural_snip)
((u (⊓ natural_len natural_snip))
(u (⊖ natural_len natural_snip)))]
; this is the routine for little-endian targets
[(✂ (i natural_len natural_min natural_max) natural_snip)
((i natural_snip
(⊓ natural_len natural_snip)
(⊓ natural_min (^ 2 (⊓ natural_len natural_snip)))
(⊓ natural_max (^ 2 (⊓ natural_len natural_snip))))
(i (⊖ natural_len natural_snip)
(⊘ (⊔ natural_min (^ 2 (⊖ natural_len natural_snip))))
(⊘ natural_max (^ 2 (⊖ natural_len natural_snip)))))])
; append
(define-metafunction layouts
⧺ : l l -> l
; (⧺ × l)
[(⧺ (× l_0 l_1) l_2)
(× l_0 (⧺ l_1 l_2))]
; (⧺ + l)
[(⧺ (+ l_0 l_1) l_2)
(+ (⧺ l_0 l_2)
(⧺ l_1 l_2))]
; (⧺ a l)
[(⧺ la l)
(× la l)]
)
; simplify a term
(define-judgment-form layouts
#:mode (~ I O)
#:contract (~ l l)
; reassociate
[(~ (× l_0 l_1)
(⧺ l_0 l_1))
reassociate]
; left-distribute
[(~ (× l_m (+ l_a0 l_a1))
(+ (× l_m l_a0) (× l_m l_a1)))
left-distribute]
; right-distribute
[(~ (× (+ l_a0 l_a1) l_m)
(+ (× l_a0 l_m) (× l_a1 l_m)))
right-distribute]
; eliminate ε from ×
[(~ (× l ε) l) unwrap-×-l]
[(~ (× ε l) l) unwrap-×-r]
; eliminate empty bytes
; u~ε
[(~ (u 0)
ε)
u~ε]
; i~ε
[(~ (i 0 _ _)
ε)
i~ε]
)
; is a layout transmutable to another layout?
(define-judgment-form layouts
#:mode (→ I I)
#:contract (→ l l)
; algebraic transformations of src and dst
[ (~ l_src l_src′) (→ l_src′ l_dst)
----------------------------------- src′
(→ l_src l_dst) ]
[ (~ l_dst l_dst′) (→ l_src l_dst′)
----------------------------------- dst′
(→ l_src l_dst) ]
;; +→∗
; 1. +→×
; 2. +→a
; 3. +→+
; every variant of the src must be transmutable to dst
[(→ l_src_0 l_dst) (→ l_src_1 l_dst)
----------------------------------- +→∗
(→ (+ l_src_0 l_src_1) l_dst) ]
;; ∗→+
; 3. +→+
; 4. a→+
; 5. ×→+
; src is transmutable to the left variant of dst
[ (→ l_src l_dst_0)
------------------------------ ∗→+-L
(→ l_src (+ l_dst_0 l_dst_1))]
; src is transmutable to the right variant of dst
[ (→ l_src l_dst_1)
------------------------------ ∗→+-R
(→ l_src (+ l_dst_0 l_dst_1))]
;; 6. ×→×
; the cases where the first operand of ×
; isn't an atom are handled by ~
[ (⇝ la_src la_dst la_src′ la_dst′)
(→ (× la_src′ l_src) (× la_dst′ l_dst))
----------------------------------------- ×→×
(→ (× la_src l_src) (× la_dst l_dst)) ]
;; 7. ×→a
[(⇝ la_src la_dst la_src′ la_dst′)
(→ (× la_src′ l_src) la_dst′)
--------------------------------- ×→a
(→ (× la_src l_src) la_dst) ]
;; 8. a→×
[(⇝ la_src la_dst la_src′ la_dst′)
(→ la_src′ (× la_dst′ l_dst))
--------------------------------- a→×
(→ la_src (× la_dst l_dst)) ]
;; 9. a→a
[(⇝ la_src la_dst la_src′ la_dst′)
(→ la_src′ la_dst′)
--------------------------------- a→a
(→ la_src la_dst) ]
)
; nom
; ⇝ consumes a src atom and dst atom,
; chomps as much as possible off src and dst
; and produces their remainders
(define-judgment-form layouts
#:mode (⇝ I I O O)
#:contract (⇝ la la la la)
; ∅ is transmutable into everything
; fine since ∅ isn't constructible
[(⇝ ∅ l
ε l)]
; b→b; with length-equalizing snip
[(side-condition
(≠ (📏 lb_src)
(📏 lb_dst)))
(where (lb_src′ lb_src″)
(✂ lb_src))
(where (lb_dst′ lb_dst″)
(✂ lb_dst))
(→ lb_src′ lb_dst′)
----------------------------------
(⇝ lb_src lb_dst lb_src″ lb_dst″)]
; b→b; without length-equalizing snip
[(side-condition
(= (📏 lb_src)
(📏 lb_dst)))
(→ lb_src lb_dst)
----------------------
(⇝ lb_src lb_dst ε ε)]
)
(define-judgment-form layouts
#:mode (⊆ I I)
#:contract (⊆ lb lb)
[(⊆ (u natural_len)
(u natural_len))]
[(⊆ (i natural_len
natural_min
natural_max)
(u natural_len))]
[(side-condition
(≥ natural_src_min natural_dst_min))
(side-condition
(≤ natural_src_max natural_dst_max))
-------------------------------------
(⊆ (i natural_len
natural_src_min
natural_src_max)
(i natural_len
natural_dst_min
natural_dst_max))])
; shorthand for test cases
(define-metafunction layouts
u8 : (side-condition (name value natural)
(<= 0 (term value) 255)) -> l⋅
[(u8 natural)
(init 1 natural ,(+ (term natural) 1))])