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By Jeremy Gibbons (editor)

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Pn Γ{σx → (λN x1 . . e, τ )}, x, σ, p1 . . C xi , γ xi )], C xi , γ, ε E-A PP 4 Γ, e, τ, ε ∆{γy → (λN x1 . . xk xk+1 . . f, µ)}, y, γ, q1 . . qk ∆⊕[σx → (λN xk+1 . . f, µ[x1 /q1 . . xk /qk ])], y, γ, q1 . . qk p1 . . e, τ )}, x, σ, p1 . . pm Θ, w, ξ, ri Γ⊕[pi → (lf i , τi [xi /pi ] FV(lf i ))], e, σ[xi /pi ], ri ∆, w, γ, si Γ, letrec xi = lf i in e, σ, ri ∆, w, γ, si Γ, e, σ, ε pi ∈ POINTERS \ Dom(Γ) ∆, Ck yj , γ, ε ∆, ek , σ[xkj /γyj ], qi Θ, w, ξ, ri Γ, case e of Ci xij → ei , σ, qi Θ, w, ξ, ri E-A PP 5 E-L ETREC E-C ASE Figure 7.

Qk , UPD(p, p1 . . pm ) : Si stg =⇒ enter, ∆⊕[p → (λN xk+1 . . f, µ[x1 /q1 . . xk /qk ])], y, γ, (q1 . . qk p1 . . pm ), Si enter, Γ{σx → (λN x1 . . e, τ )}, x, σ, p1 . . pn , Si stg =⇒ eval, Γ, e, τ [x1 /p1 . . xm /pm ], pm+1 . . 5 (15) S-L ETREC (3) SE -C ASE (4) SA-C ASE (6) Figure 10. The STG machine the semantics with environments we allow a variable to be an exceeding index if it is in the domain of the associated environment (as in Definition 4). Hence, our approach is not “locally nameless”, as we work on free de Bruijn indices.

Pn where n < m, P ROPOSITION 9 (completeness). For a closed expression e, the following hold: E ∆{γy → λN x1 . . xk xk+1 . . f µ}, y, γ, q1 . . qk UPD(p, p1 . . pn ) : Si where k < n 1. If (∅ : e ↓ ∆ : p pi ), there exist ∆• , x and σ such that stg ∗ eval, ∅, e, ε, ε, ε =⇒ ∆• , x, σ, pi and σx = p. 2. If (∅ : e ↓ ∆ : C pi ), there exist ∆• , xi and σ such that: stg ∗ eval, ∅, e, ε, ε, ε =⇒ ∆• , C xi , σ, ε and σxi = pi . dcps =⇒ E ∆⊕[p → λN xk+1 . . f µ[x1 /q1 . . xk /qk ]], y, γ, q1 .

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