Neutral interop + governance layer for multi-agent systems. Any protocol in, any protocol out, with identity, budgets, and a tamper-evident audit trail. - shadowhunter-92/agentbridge
Then we can prove: \[ \dfrac { \dfrac { \dfrac { \dfrac { y : \gg(x : 1). \surd 0 \prov y : \gg(x : 1). \surd 0 }{ y :: \gg(x : 1). \surd 0, \star \prov y\tri 2 : \surd 0 }}{ y :: \gg(x : 1). \surd 0 \prov \mathbf{el}_\surd\ ( y\tri 2) : 0 }}{ y :: \gg(x : 1). \surd 0 \prov \babort(\mathbf{el}_\surd\ ( y\tri 2)) : \T }}{ y : \gg(x : 1). \surd 0 \prov {\triangle}(\babort(\mathbf{el}_\surd\ ( y\tri 2))) : \T } \] We can be careful and check that $\mathsf{fore}_2$ is really a type: \[ \dfrac { \dfrac { \dfrac{ \dfrac{} { C \div \surd \rtype, \star \prov C : \rtype } } { C \div \surd \rtype \prov \mathbf{el}_\surd C : \rtype } } { \cdots \prov \surd (\mathbf{el}_\surd C) : \rtype } \quad \dfrac{ \dfrac{ \dfrac{} { \cdots, \star \prov a' : \surd(\mathsf{fore}_1\ C\ t) } } { \cdots \prov \mathbf{el}_\surd\ a' : \mathsf{fore}_1\ C\ t } \quad \dfrac{} { \cdots \prov a : \mathsf{fore}_1\ C\ t } } { \cdots, a' : \surd (\mathbf{el}_\surd C) \prov \mathbf{el}_\surd\ a' \equiv a : \rtype } } { C : \surd \rtype, t : \T, a : \mathsf{fore}_1\ C\ t \prov (a' : \surd (\mathbf{el}_\surd C)) \x (\mathbf{el}_\surd\ a' \equiv a) : \rtype } \] Let's try to see if anything interesting happens when we do some round-trips. The computation of $\mathsf{back}$ of $\mathsf{fore}$ gives: \[\mathsf{back}\ (\mathsf{fore}_1\ C)\ (\mathsf{fore}_2\ C) \] \[= \mathbf{in}_\surd\ (\gg(a : \mathsf{fore}_1\ C\ \star).(\mathsf{fore}_2\ C\ \star\ a)) \] \[= \mathbf{in}_\surd\ (\gg(a : \mathbf{el}_\surd\ C).( (a' : \surd (\mathbf{el}_\surd C)) \x (\mathbf{el}_\surd\ a' \equiv a) ) \] so we'd want to show this is equal to $C$ at $\surd \rtype$. Although I can rely on univalence to tell me what equality of types means, I need to pin down what equality at $\surd$ means! Maybe there's some kind of extensionality; that all I need is equality under eliminations. Is this equality plausible semantically? If I hit with $\dash^*$ I get \[(\mathbf{in}_\surd\ (\gg(a : \mathbf{el}_\surd\ C).( (a' : \surd (\mathbf{el}_\surd C)) \x (\mathbf{el}_\surd\ a' \equiv a) ))^* \] \[= \pair{X^*}{ \sem X} \] for \[ X = \gg(a : \mathbf{el}_\surd\ C).( (a' : \surd (\mathbf{el}_\surd C)) \x (\mathbf{el}_\surd\ a' \equiv a) ) \] and I'd find (using some very sketchy reasoning at certain steps, but I think this is still vaguely plausible. I need to be more careful reasoning about universes and elements, I think) \[ \pair{X^*}{ \sem X} = \pair {(\mathbf{el}_\surd\ C)^*} {\lambda a^* : (\mathbf{el}_\surd\ C)^* . (a' : \surd (\mathbf{el}_\surd C)) \x (\mathbf{el}_\surd\ a' \equiv a)^* }\] \[ = \pair {C^*.1} {\lambda a^* : (C^*.1) . ((a' : \surd (\mathbf{el}_\surd C)) \x (\mathbf{el}_\surd\ a' \equiv a))^* }\] \[ = \pair {C^*.1} {\lambda a^* : (C^*.1) . ({a'}^* : (\surd (\mathbf{el}_\surd C))^*) \x (\mathbf{el}_\surd\ a' \equiv a)^* }\] \[ = \pair {C^*.1} {\lambda a^* : (C^*.1) . ({a'}^* : (\surd (\mathbf{el}_\surd C))^*) \x ({a'}^*.1 \equiv a^*) }\] \[ = \pair {C^*.1} {\lambda a^* : (C^*.1) . ({a'}^* : (y^* : (\mathbf{el}_\surd C)^*) \x \sem{\mathbf{el}_\surd C}(y^*)) \x ({a'}^*.1 \equiv a^*) }\] \[ = \pair {C^*.1} {\lambda a^* : (C^*.1) . ({a'}^* : (y^* : C^*.1) \x C^*.2\ y^*) \x ({a'}^*.1 \equiv a^*) }\] \[ = \pair {C^*.1} {\lambda a^* : (C^*.1) . ( C^*.2\ a^*) }\] \[ = \pair {C^*.1} {C^*.2 }\] \[ = C\] Let's try the other direction. Assume $A : \T \to \rtype$ and $B : (t : \T)(a : A\ t) \to \rtype$. We expect $\mathsf{fore}_1\ (\mathsf{back}\ A\ B)\ t = A\ t$ so we compute \[ \mathsf{fore}_1\ (\mathsf{back}\ A\ B)\ t = \mathbf{el}_\surd\ (\mathsf{back}\ A\ B) \] \[= \mathbf{el}_\surd\ (\mathbf{in}_\surd\ (\gg(a : A\ \star).(B\ \star\ a))) \] \[= \gg(a : A\ \star).(B\ \star\ a) \] At let's now compare semantics. We have a $t : \T$ present, so we only need to check the base, not the relation, for we can use $\sem \T(\_) = 0$ to abort. And indeed …
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Because apparently opening a brokerage app, checking Finviz, reading news, and then pretending I had a plan was too many steps, I made Simonbot. It’s a Discord bot for quick market lookups through /simon commands. It can check stock prices, daily change, recent stock news, top gainers/losers, intraday charts, and crypto prices with 24H change. Basically, it lets your Discord server say “wow, NVDA is up again” without anyone having to leave Discord. Truly revolutionary. Historians will study this. GitHub repo: https://github.com/Drewster6767/Simonbot Open source, still improving it, feedback/issues/roasts welcome. submitted by /u/Negative-Current809 [link] [Kommentare]