base-logic.tex 12.4 KB
Newer Older
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
\section{Base Logic}
\label{sec:base-logic}

The base logic is parameterized by an arbitrary CMRA $\monoid$ having a unit.
This defines the structure of resources that can be owned.

As usual for higher-order logics, you can furthermore pick a \emph{signature} $\Sig = (\SigType, \SigFn, \SigAx)$ to add more types, symbols and axioms to the language.
You have to make sure that $\SigType$ includes the base types:
\[
	\SigType \supseteq \{ \textlog{M}, \Prop \}
\]
Elements of $\SigType$ are ranged over by $\sigtype$.

Each function symbol in $\SigFn$ has an associated \emph{arity} comprising a natural number $n$ and an ordered list of $n+1$ types $\type$ (the grammar of $\type$ is defined below, and depends only on $\SigType$).
We write
\[
	\sigfn : \type_1, \dots, \type_n \to \type_{n+1} \in \SigFn
\]
to express that $\sigfn$ is a function symbol with the indicated arity.

Furthermore, $\SigAx$ is a set of \emph{axioms}, that is, terms $\term$ of type $\Prop$.
Again, the grammar of terms and their typing rules are defined below, and depends only on $\SigType$ and $\SigFn$, not on $\SigAx$.
Elements of $\SigAx$ are ranged over by $\sigax$.

\subsection{Grammar}\label{sec:grammar}

\paragraph{Syntax.}
Iris syntax is built up from a signature $\Sig$ and a countably infinite set $\textdom{Var}$ of variables (ranged over by metavariables $x$, $y$, $z$).
Below, $\melt$ ranges over $\monoid$ and $i$ ranges over $\set{1,2}$.

\begin{align*}
  \type \bnfdef{}&
      \sigtype \mid
      1 \mid
      \type \times \type \mid
      \type \to \type
\\[0.4em]
  \term, \prop, \pred \bnfdef{}&
      \var \mid
      \sigfn(\term_1, \dots, \term_n) \mid
      () \mid
      (\term, \term) \mid
      \pi_i\; \term \mid
      \Lam \var:\type.\term \mid
      \term(\term)  \mid
      \melt \mid
      \mcore\term \mid
      \term \mtimes \term \mid
\\&
    \FALSE \mid
    \TRUE \mid
    \term =_\type \term \mid
    \prop \Ra \prop \mid
    \prop \land \prop \mid
    \prop \lor \prop \mid
    \prop * \prop \mid
    \prop \wand \prop \mid
\\&
    \MU \var:\type. \term  \mid
    \Exists \var:\type. \prop \mid
    \All \var:\type. \prop \mid
\\&
Ralf Jung's avatar
Ralf Jung committed
63
    \ownM{\term} \mid \mval(\term) \mid
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
    \always\prop \mid
    {\later\prop} \mid
    \upd \prop\mid
\end{align*}
Recursive predicates must be \emph{guarded}: in $\MU \var. \term$, the variable $\var$ can only appear under the later $\later$ modality.

Note that the modalities $\upd$, $\always$ and $\later$ bind more tightly than $*$, $\wand$, $\land$, $\lor$, and $\Ra$.


\paragraph{Variable conventions.}
We assume that, if a term occurs multiple times in a rule, its free variables are exactly those binders which are available at every occurrence.


\subsection{Types}\label{sec:types}

Iris terms are simply-typed.
The judgment $\vctx \proves \wtt{\term}{\type}$ expresses that, in variable context $\vctx$, the term $\term$ has type $\type$.

A variable context, $\vctx = x_1:\type_1, \dots, x_n:\type_n$, declares a list of variables and their types.
In writing $\vctx, x:\type$, we presuppose that $x$ is not already declared in $\vctx$.

\judgment[Well-typed terms]{\vctx \proves_\Sig \wtt{\term}{\type}}
\begin{mathparpagebreakable}
%%% variables and function symbols
	\axiom{x : \type \proves \wtt{x}{\type}}
\and
	\infer{\vctx \proves \wtt{\term}{\type}}
		{\vctx, x:\type' \proves \wtt{\term}{\type}}
\and
	\infer{\vctx, x:\type', y:\type' \proves \wtt{\term}{\type}}
		{\vctx, x:\type' \proves \wtt{\term[x/y]}{\type}}
\and
	\infer{\vctx_1, x:\type', y:\type'', \vctx_2 \proves \wtt{\term}{\type}}
		{\vctx_1, x:\type'', y:\type', \vctx_2 \proves \wtt{\term[y/x,x/y]}{\type}}
\and
	\infer{
		\vctx \proves \wtt{\term_1}{\type_1} \and
		\cdots \and
		\vctx \proves \wtt{\term_n}{\type_n} \and
		\sigfn : \type_1, \dots, \type_n \to \type_{n+1} \in \SigFn
	}{
		\vctx \proves \wtt {\sigfn(\term_1, \dots, \term_n)} {\type_{n+1}}
	}
%%% products
\and
	\axiom{\vctx \proves \wtt{()}{1}}
\and
	\infer{\vctx \proves \wtt{\term}{\type_1} \and \vctx \proves \wtt{\termB}{\type_2}}
		{\vctx \proves \wtt{(\term,\termB)}{\type_1 \times \type_2}}
\and
	\infer{\vctx \proves \wtt{\term}{\type_1 \times \type_2} \and i \in \{1, 2\}}
		{\vctx \proves \wtt{\pi_i\,\term}{\type_i}}
%%% functions
\and
	\infer{\vctx, x:\type \proves \wtt{\term}{\type'}}
		{\vctx \proves \wtt{\Lam x. \term}{\type \to \type'}}
\and
	\infer
	{\vctx \proves \wtt{\term}{\type \to \type'} \and \wtt{\termB}{\type}}
	{\vctx \proves \wtt{\term(\termB)}{\type'}}
%%% monoids
\and
        \infer{}{\vctx \proves \wtt\munit{\textlog{M}}}
\and
	\infer{\vctx \proves \wtt\melt{\textlog{M}}}{\vctx \proves \wtt{\mcore\melt}{\textlog{M}}}
\and
	\infer{\vctx \proves \wtt{\melt}{\textlog{M}} \and \vctx \proves \wtt{\meltB}{\textlog{M}}}
		{\vctx \proves \wtt{\melt \mtimes \meltB}{\textlog{M}}}
%%% props and predicates
\\
	\axiom{\vctx \proves \wtt{\FALSE}{\Prop}}
\and
	\axiom{\vctx \proves \wtt{\TRUE}{\Prop}}
\and
	\infer{\vctx \proves \wtt{\term}{\type} \and \vctx \proves \wtt{\termB}{\type}}
		{\vctx \proves \wtt{\term =_\type \termB}{\Prop}}
\and
	\infer{\vctx \proves \wtt{\prop}{\Prop} \and \vctx \proves \wtt{\propB}{\Prop}}
		{\vctx \proves \wtt{\prop \Ra \propB}{\Prop}}
\and
	\infer{\vctx \proves \wtt{\prop}{\Prop} \and \vctx \proves \wtt{\propB}{\Prop}}
		{\vctx \proves \wtt{\prop \land \propB}{\Prop}}
\and
	\infer{\vctx \proves \wtt{\prop}{\Prop} \and \vctx \proves \wtt{\propB}{\Prop}}
		{\vctx \proves \wtt{\prop \lor \propB}{\Prop}}
\and
	\infer{\vctx \proves \wtt{\prop}{\Prop} \and \vctx \proves \wtt{\propB}{\Prop}}
		{\vctx \proves \wtt{\prop * \propB}{\Prop}}
\and
	\infer{\vctx \proves \wtt{\prop}{\Prop} \and \vctx \proves \wtt{\propB}{\Prop}}
		{\vctx \proves \wtt{\prop \wand \propB}{\Prop}}
\and
	\infer{
		\vctx, \var:\type \proves \wtt{\term}{\type} \and
		\text{$\var$ is guarded in $\term$}
	}{
		\vctx \proves \wtt{\MU \var:\type. \term}{\type}
	}
\and
	\infer{\vctx, x:\type \proves \wtt{\prop}{\Prop}}
		{\vctx \proves \wtt{\Exists x:\type. \prop}{\Prop}}
\and
	\infer{\vctx, x:\type \proves \wtt{\prop}{\Prop}}
		{\vctx \proves \wtt{\All x:\type. \prop}{\Prop}}
\and
	\infer{\vctx \proves \wtt{\melt}{\textlog{M}}}
Ralf Jung's avatar
Ralf Jung committed
170
		{\vctx \proves \wtt{\ownM{\melt}}{\Prop}}
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
\and
	\infer{\vctx \proves \wtt{\melt}{\type} \and \text{$\type$ is a CMRA}}
		{\vctx \proves \wtt{\mval(\melt)}{\Prop}}
\and
	\infer{\vctx \proves \wtt{\prop}{\Prop}}
		{\vctx \proves \wtt{\always\prop}{\Prop}}
\and
	\infer{\vctx \proves \wtt{\prop}{\Prop}}
		{\vctx \proves \wtt{\later\prop}{\Prop}}
\and
	\infer{
		\vctx \proves \wtt{\prop}{\Prop}
	}{
		\vctx \proves \wtt{\upd \prop}{\Prop}
	}
\end{mathparpagebreakable}

\subsection{Proof rules}
\label{sec:proof-rules}

The judgment $\vctx \mid \pfctx \proves \prop$ says that with free variables $\vctx$, proposition $\prop$ holds whenever all assumptions $\pfctx$ hold.
We implicitly assume that an arbitrary variable context, $\vctx$, is added to every constituent of the rules.
Furthermore, an arbitrary \emph{boxed} assertion context $\always\pfctx$ may be added to every constituent.
Axioms $\vctx \mid \prop \provesIff \propB$ indicate that both $\vctx \mid \prop \proves \propB$ and $\vctx \mid \propB \proves \prop$ can be derived.

\judgment{\vctx \mid \pfctx \proves \prop}
\paragraph{Laws of intuitionistic higher-order logic with equality.}
This is entirely standard.
\begin{mathparpagebreakable}
\infer[Asm]
  {\prop \in \pfctx}
  {\pfctx \proves \prop}
\and
\infer[Eq]
  {\pfctx \proves \prop \\ \pfctx \proves \term =_\type \term'}
  {\pfctx \proves \prop[\term'/\term]}
\and
\infer[Refl]
  {}
  {\pfctx \proves \term =_\type \term}
\and
\infer[$\bot$E]
  {\pfctx \proves \FALSE}
  {\pfctx \proves \prop}
\and
\infer[$\top$I]
  {}
  {\pfctx \proves \TRUE}
\and
\infer[$\wedge$I]
  {\pfctx \proves \prop \\ \pfctx \proves \propB}
  {\pfctx \proves \prop \wedge \propB}
\and
\infer[$\wedge$EL]
  {\pfctx \proves \prop \wedge \propB}
  {\pfctx \proves \prop}
\and
\infer[$\wedge$ER]
  {\pfctx \proves \prop \wedge \propB}
  {\pfctx \proves \propB}
\and
\infer[$\vee$IL]
  {\pfctx \proves \prop }
  {\pfctx \proves \prop \vee \propB}
\and
\infer[$\vee$IR]
  {\pfctx \proves \propB}
  {\pfctx \proves \prop \vee \propB}
\and
\infer[$\vee$E]
  {\pfctx \proves \prop \vee \propB \\
   \pfctx, \prop \proves \propC \\
   \pfctx, \propB \proves \propC}
  {\pfctx \proves \propC}
\and
\infer[$\Ra$I]
  {\pfctx, \prop \proves \propB}
  {\pfctx \proves \prop \Ra \propB}
\and
\infer[$\Ra$E]
  {\pfctx \proves \prop \Ra \propB \\ \pfctx \proves \prop}
  {\pfctx \proves \propB}
\and
\infer[$\forall$I]
  { \vctx,\var : \type\mid\pfctx \proves \prop}
  {\vctx\mid\pfctx \proves \forall \var: \type.\; \prop}
\and
\infer[$\forall$E]
  {\vctx\mid\pfctx \proves \forall \var :\type.\; \prop \\
   \vctx \proves \wtt\term\type}
  {\vctx\mid\pfctx \proves \prop[\term/\var]}
\and
\infer[$\exists$I]
  {\vctx\mid\pfctx \proves \prop[\term/\var] \\
   \vctx \proves \wtt\term\type}
  {\vctx\mid\pfctx \proves \exists \var: \type. \prop}
\and
\infer[$\exists$E]
  {\vctx\mid\pfctx \proves \exists \var: \type.\; \prop \\
   \vctx,\var : \type\mid\pfctx , \prop \proves \propB}
  {\vctx\mid\pfctx \proves \propB}
% \and
% \infer[$\lambda$]
%   {}
%   {\pfctx \proves (\Lam\var: \type. \prop)(\term) =_{\type\to\type'} \prop[\term/\var]}
% \and
% \infer[$\mu$]
%   {}
%   {\pfctx \proves \mu\var: \type. \prop =_{\type} \prop[\mu\var: \type. \prop/\var]}
\end{mathparpagebreakable}
Furthermore, we have the usual $\eta$ and $\beta$ laws for projections, $\lambda$ and $\mu$.


\paragraph{Laws of (affine) bunched implications.}
\begin{mathpar}
\begin{array}{rMcMl}
  \TRUE * \prop &\provesIff& \prop \\
  \prop * \propB &\provesIff& \propB * \prop \\
  (\prop * \propB) * \propC &\provesIff& \prop * (\propB * \propC)
\end{array}
\and
\infer[$*$-mono]
  {\prop_1 \proves \propB_1 \and
   \prop_2 \proves \propB_2}
  {\prop_1 * \prop_2 \proves \propB_1 * \propB_2}
\and
\inferB[$\wand$I-E]
  {\prop * \propB \proves \propC}
  {\prop \proves \propB \wand \propC}
\end{mathpar}

302
\paragraph{Laws for the always modality.}
303
\begin{mathpar}
304
305
306
\infer[$\always$-mono]
  {\prop \proves \propB}
  {\always{\prop} \proves \always{\propB}}
307
\and
308
309
310
311
312
\begin{array}[c]{rMcMl}
  \always{\prop} &\proves& \prop \\
  \always{(\prop \land \propB)} &\proves& \always{(\prop * \propB)} \\
  \always{\prop} \land \propB &\proves& \always{\prop} * \propB
\end{array}
313
\and
314
315
316
317
\begin{array}[c]{rMcMl}
  \always{\prop} &\proves& \always\always\prop \\
  \All x. \always{\prop} &\proves& \always{\All x. \prop} \\
  \always{\Exists x. \prop} &\proves& \Exists x. \always{\prop}
318
319
320
\end{array}
\end{mathpar}

321

322
323
324
\paragraph{Laws for the later modality.}
\begin{mathpar}
\infer[$\later$-mono]
325
326
  {\prop \proves \propB}
  {\later\prop \proves \later{\propB}}
327
328
329
330
331
332
\and
\infer[L{\"o}b]
  {}
  {(\later\prop\Ra\prop) \proves \prop}
\and
\begin{array}[c]{rMcMl}
333
334
335
  \All x. \later\prop &\proves& \later{\All x.\prop} \\
  \later\Exists x. \prop &\proves& \later\FALSE \lor {\Exists x.\later\prop}  \\
  \later\prop &\proves& \later\FALSE \lor (\later\FALSE \Ra \prop) \\
336
337
338
\end{array}
\and
\begin{array}[c]{rMcMl}
339
340
  \later{(\prop * \propB)} &\provesIff& \later\prop * \later\propB \\
  \always{\later\prop} &\provesIff& \later\always{\prop} \\
341
342
343
\end{array}
\end{mathpar}

344
345

\paragraph{Laws for ghosts and validity.}
346
\begin{mathpar}
347
\begin{array}{rMcMl}
Ralf Jung's avatar
Ralf Jung committed
348
349
350
351
\ownM{\melt} * \ownM{\meltB} &\provesIff&  \ownM{\melt \mtimes \meltB} \\
\ownM\melt &\proves& \always{\ownM{\mcore\melt}} \\
\TRUE &\proves&  \ownM{\munit} \\
\later\ownM\melt &\proves& \Exists\meltB. \ownM\meltB \land \later(\melt = \meltB)
352
353
\end{array}
\and
354
355
356
\infer[valid-intro]
{\melt \in \mval}
{\TRUE \vdash \mval(\melt)}
357
\and
358
359
360
\infer[valid-elim]
{\melt \notin \mval_0}
{\mval(\melt) \proves \FALSE}
361
\and
362
\begin{array}{rMcMl}
Ralf Jung's avatar
Ralf Jung committed
363
\ownM{\melt} &\proves& \mval(\melt) \\
364
365
366
\mval(\melt \mtimes \meltB) &\proves& \mval(\melt) \\
\mval(\melt) &\proves& \always\mval(\melt)
\end{array}
367
368
\end{mathpar}

369

370
371
\paragraph{Laws for the update modality.}
\begin{mathpar}
372
373
374
375
\infer[upd-mono]
{\prop \proves \propB}
{\upd\prop \proves \upd\propB}

376
377
378
379
380
381
382
383
384
385
386
387
\infer[upd-intro]
{}{\prop \proves \upd \prop}

\infer[upd-trans]
{}
{\upd \upd \prop \proves \upd \prop}

\infer[upd-frame]
{}{\propB * \upd\prop \proves \upd (\propB * \prop)}

\inferH{upd-update}
{\melt \mupd \meltsB}
Ralf Jung's avatar
Ralf Jung committed
388
{\ownM\melt \proves \upd \Exists\meltB\in\meltsB. \ownM\meltB}
389
390
391
392
\end{mathpar}

\subsection{Soundness}

393
394
395
396
397
The soundness statement of the logic reads as follows: For any $n$, we have
\begin{align*}
  \lnot(\TRUE \vdash (\upd\later)^n \FALSE)
\end{align*}
where $(\upd\later)^n$ is short for $\upd\later$ being nested $n$ times.
398
399
400
401
402
403


%%% Local Variables:
%%% mode: latex
%%% TeX-master: "iris"
%%% End: