reflect_(Either a b) t =
  case t of ...

NO — we need more complicated model and meta-language

Danvy: shift/reset

Soundness: If $\Gamma \vdash p:A$, then in any model for any world, if $w \Vdash \Gamma$, then $w \Vdash A$

Completeness: if in any model, in any world $w \Vdash \Gamma$ implies $w \Vdash A$, then there exists a term $p$ such that $\Gamma \vdash p:A$

We can construct an Universal Model for proving completeness

$K$ (possible worlds) will be a set of different contexts $\Gamma$

$\Gamma \leq \Gamma'$ if $\Gamma \subseteq \Gamma'$

Forcing: $\Gamma \Vdash A$ — a set of normal form derivations $\Gamma \vdash^{nf} A$, where $A$ — closed and atomic

Then, for example, by definition

$\Gamma \Vdash A \to B$ when for all $\Gamma' \geq \Gamma$ $\Gamma' \vdash^{nf} A$ implies $\Gamma' \vdash^{nf} B$ — a function in meta-language

Forcing:

$$\forall C\, \forall \Gamma' \geq \Gamma \left( \forall \Gamma'' \geq \Gamma'\, \Gamma'' \Vdash_s A \to \Gamma'' \vdash^{nf} C \right) \to \Gamma' \vdash^{nf} C$$

In Haskell:

"unit": $\eta: w \Vdash_s A \to w \Vdash A$, $\eta(a) = \lambda k.\, k a$

"bind": $* : (\forall w' \geq w\, w' \Vdash_s A \to w' \Vdash B) \to w \Vdash A \to w \Vdash B$,

    $\phi*a = \lambda k.\, a\ (\lambda b.\, \phi\ b\ k)$

Forcing ($w \Vdash A$) is kinda "Cont" monad storing strong forcing ($w \Vdash_s A$)

"run": $\mu : w \Vdash X \to w \Vdash_s X$ for atomic $X$, by definition of $\Vdash$

nbe(t (Left 3)):

nbe(t (Right ())):

Strong forcing:

$\Gamma; \Delta \Vdash_s \text{box}\ A$ is a pair of $\Gamma; \emptyset \vdash t$ and $\Gamma; \Delta \Vdash A$

eval(box t) = $\eta( <\text{graft}\ t, \text{eval}\ t>)$

eval(let box u = t in e) =

$$\lambda k.\, (\text{eval}\ t) (<t, s> \to (\text{eval}\ b) k)$$

reify([] A)(t) = $t (<t, s> \to \text{box}\ t)$

reflect([] A)(t) =

$$\lambda k.\, \text{let box}\ u = t\ \text{in}\ k <u, \text{reflect}_A u>$$

nbe (exp 3)

Full normalization: nbe(let box u = exp 3 in u)