Commit 48c7dfca by Ralf Jung

### add model for new terms

parent 06bb6d4d
 ... @@ -41,7 +41,7 @@ Below, $\melt$ ranges over $\monoid$ and $i$ ranges over $\set{1,2}$. ... @@ -41,7 +41,7 @@ Below, $\melt$ ranges over $\monoid$ and $i$ ranges over $\set{1,2}$. \term, \prop, \pred \bnfdef{}& \term, \prop, \pred \bnfdef{}& \var \mid \var \mid \sigfn(\term_1, \dots, \term_n) \mid \sigfn(\term_1, \dots, \term_n) \mid \textlog{abort}(\term) \mid \textlog{abort}\; \term \mid () \mid () \mid (\term, \term) \mid (\term, \term) \mid \pi_i\; \term \mid \pi_i\; \term \mid ... @@ -119,7 +119,7 @@ In writing $\vctx, x:\type$, we presuppose that $x$ is not already declared in $... @@ -119,7 +119,7 @@ In writing$\vctx, x:\type$, we presuppose that$x$is not already declared in$ %%% empty, unit, products, sums %%% empty, unit, products, sums \and \and \infer{\vctx \proves \wtt\term{0}} \infer{\vctx \proves \wtt\term{0}} {\vctx \proves \wtt{\textlog{abort}(\term)}\type} {\vctx \proves \wtt{\textlog{abort}\; \term}\type} \and \and \axiom{\vctx \proves \wtt{()}{1}} \axiom{\vctx \proves \wtt{()}{1}} \and \and ... ...
 ... @@ -9,11 +9,13 @@ The semantic domains are interpreted as follows: ... @@ -9,11 +9,13 @@ The semantic domains are interpreted as follows: \[ \[ \begin{array}[t]{@{}l@{\ }c@{\ }l@{}} \begin{array}[t]{@{}l@{\ }c@{\ }l@{}} \Sem{\Prop} &\eqdef& \UPred(\monoid) \\ \Sem{\Prop} &\eqdef& \UPred(\monoid) \\ \Sem{\textlog{M}} &\eqdef& \monoid \Sem{\textlog{M}} &\eqdef& \monoid \\ \Sem{0} &\eqdef& \Delta \emptyset \\ \Sem{1} &\eqdef& \Delta \{ () \} \end{array} \end{array} \qquad\qquad \qquad\qquad \begin{array}[t]{@{}l@{\ }c@{\ }l@{}} \begin{array}[t]{@{}l@{\ }c@{\ }l@{}} \Sem{1} &\eqdef& \Delta \{ () \} \\ \Sem{\type + \type'} &\eqdef& \Sem{\type} + \Sem{\type} \\ \Sem{\type \times \type'} &\eqdef& \Sem{\type} \times \Sem{\type} \\ \Sem{\type \times \type'} &\eqdef& \Sem{\type} \times \Sem{\type} \\ \Sem{\type \to \type'} &\eqdef& \Sem{\type} \nfn \Sem{\type} \\ \Sem{\type \to \type'} &\eqdef& \Sem{\type} \nfn \Sem{\type} \\ \end{array} \end{array} ... @@ -80,9 +82,15 @@ For every definition, we have to show all the side-conditions: The maps have to ... @@ -80,9 +82,15 @@ For every definition, we have to show all the side-conditions: The maps have to \Sem{\vctx \proves \MU \var:\type. \term : \type}_\gamma &\eqdef \Sem{\vctx \proves \MU \var:\type. \term : \type}_\gamma &\eqdef \mathit{fix}(\Lam \termB : \Sem{\type}. \Sem{\vctx, x : \type \proves \term : \type}_{\mapinsert \var \termB \gamma}) \\ \mathit{fix}(\Lam \termB : \Sem{\type}. \Sem{\vctx, x : \type \proves \term : \type}_{\mapinsert \var \termB \gamma}) \\ ~\\ ~\\ \Sem{\vctx \proves \textlog{abort}\;\term : \type}_\gamma &\eqdef \mathit{abort}_{\Sem\type}(\Sem{\vctx \proves \term:0}_\gamma) \\ \Sem{\vctx \proves () : 1}_\gamma &\eqdef () \\ \Sem{\vctx \proves () : 1}_\gamma &\eqdef () \\ \Sem{\vctx \proves (\term_1, \term_2) : \type_1 \times \type_2}_\gamma &\eqdef (\Sem{\vctx \proves \term_1 : \type_1}_\gamma, \Sem{\vctx \proves \term_2 : \type_2}_\gamma) \\ \Sem{\vctx \proves (\term_1, \term_2) : \type_1 \times \type_2}_\gamma &\eqdef (\Sem{\vctx \proves \term_1 : \type_1}_\gamma, \Sem{\vctx \proves \term_2 : \type_2}_\gamma) \\ \Sem{\vctx \proves \pi_i(\term) : \type_i}_\gamma &\eqdef \pi_i(\Sem{\vctx \proves \term : \type_1 \times \type_2}_\gamma) \\ \Sem{\vctx \proves \pi_i\; \term : \type_i}_\gamma &\eqdef \pi_i(\Sem{\vctx \proves \term : \type_1 \times \type_2}_\gamma) \\ \Sem{\vctx \proves \textlog{inj}_i\;\term : \type_1 + \type_2}_\gamma &\eqdef \mathit{inj}_i(\Sem{\vctx \proves \term : \type_i}_\gamma) \\ \Sem{\vctx \proves \textlog{match}\; \term \;\textlog{with}\; \Ret\textlog{inj}_1\; \var_1. \term_1 \mid \Ret\textlog{inj}_2\; \var_2. \term_2 \;\textlog{end} : \type }_\gamma &\eqdef \Sem{\vctx, \var_i:\type_i \proves \term_i : \type}_{\mapinsert{\var_i}\termB \gamma} \\ &\qquad \text{where $\Sem{\vctx \proves \term : \type_1 + \type_2}_\gamma = \mathit{inj}_i(\termB)$} \\ ~\\ ~\\ \Sem{ \melt : \textlog{M} }_\gamma &\eqdef \melt \\ \Sem{ \melt : \textlog{M} }_\gamma &\eqdef \melt \\ \Sem{\vctx \proves \mcore\term : \textlog{M}}_\gamma &\eqdef \mcore{\Sem{\vctx \proves \term : \textlog{M}}_\gamma} \\ \Sem{\vctx \proves \mcore\term : \textlog{M}}_\gamma &\eqdef \mcore{\Sem{\vctx \proves \term : \textlog{M}}_\gamma} \\ ... @@ -94,6 +102,7 @@ For every definition, we have to show all the side-conditions: The maps have to ... @@ -94,6 +102,7 @@ For every definition, we have to show all the side-conditions: The maps have to An environment $\vctx$ is interpreted as the set of An environment $\vctx$ is interpreted as the set of finite partial functions $\rho$, with $\dom(\rho) = \dom(\vctx)$ and finite partial functions $\rho$, with $\dom(\rho) = \dom(\vctx)$ and $\rho(x)\in\Sem{\vctx(x)}$. $\rho(x)\in\Sem{\vctx(x)}$. Above, $\mathit{fix}$ is the fixed-point on COFEs, and $\mathit{abort}_T$ is the unique function $\emptyset \to T$. \paragraph{Logical entailment.} \paragraph{Logical entailment.} We can now define \emph{semantic} logical entailment. We can now define \emph{semantic} logical entailment. ... ...
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