Triángulo de Pascal by V. A. Uspenski

By V. A. Uspenski

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Abstract machines Both Fun languages are different than STG, thus their STG-like machines differ from the original STG machine. The most striking difference is the lack of enter, eval and return instructions which are STG-tuned incarnations of the eval (E) and apply (A) instructions arising naturally from the D-CPS transformation. [4] Alberto de la Encina and Ricardo Pe˜na. Formally deriving an STG machine. In Miller [9], pages 102–112. [5] Alberto de la Encina and Ricardo Pe˜na. Proving the correctness of the STG machine.

F µ[x1 /q1 . . xk /qk ]], y, γ, q1 . . qk p1 . . pn , Si . 2 Soundness and completeness Introduction of the STG instructions So far we have used two kinds of “instructions:” eval (E) and apply (A), where E intuitively means that we are currently evaluating an expression, and A means that we have just finished evaluating an expression and we need an element from the stack of continuations to go on. P ROPOSITION 10 (soundness). For a closed expression e, the following hold: stg ∗ 1. If eval, ∅, e, ε, ε, ε =⇒ ∆• , x, σ, pi then there exists ∆ such that (∅ : e ↓ ∆ : (σx) pi ).

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 . . qk p1 . . pn , Si . 2 Soundness and completeness Introduction of the STG instructions So far we have used two kinds of “instructions:” eval (E) and apply (A), where E intuitively means that we are currently evaluating an expression, and A means that we have just finished evaluating an expression and we need an element from the stack of continuations to go on.

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