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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<base href="html/">
<link href="coqdoc.css" rel="stylesheet" type="text/css" />
<style>
#correspondence th, td {
padding: 10px;
border-bottom:1px solid black;
}
#correspondence table {
border-collapse: collapse;
}
</style>
<title>Overview of Coq Development</title>
</head>
<body>
<div id="page">
<div id="header">
</div>
<div id="main">
<div id="toc">
<h2> Correspondence Table </h2>
<div id="correspondence">
<table>
<tr>
<th>Definition/Lemma</th>
<th>Location in Dissertation</th>
<th>File(s)</th>
<th>Name in Formalization</th>
<th>Remarks</th>
</tr>
<tr>
<td> Indexed Valuation </td>
<td> Section 2.2 </td>
<td> <a href="discprob.idxval.ival.html">proba/theories/idxval/ival.v</a>,<br />
<a href="discprob.idxval.ival_dist.html">proba/theories/idxval/ival_dist.v</a>
</td>
<td> <a href="discprob.idxval.ival.html#ival">ival</a>,
<a href="discprob.idxval.ival_dist.html#ivdist">ivdist</a>
</td>
<td> We first define indexed valuations as a record type without the requirement that values sum to one, and then later define an "extended" version of the record in which there is an additional proof that values sum to 1. </td>
</tr>
<tr>
<td> Equivalence of Indexed Valuations </td>
<td> Section 2.2 </td>
<td> <a href="discprob.idxval.ival.html">proba/theories/idxval/ival.v</a><br /> </td>
<td> <a href="discprob.idxval.ival.html#eq_ival">eq_ival</a>,
<a href="discprob.idxval.ival_dist.html#eq_ivd">eq_ivd</a> </td>
<td></td>
</tr>
<tr>
<td> Probabilistice Choice between Indexed Valuations </td>
<td> Section 2.2 </td>
<td> <a href="discprob.idxval.ival_dist.html">proba/theories/idxval/ival_dist.v</a> </td>
<td> <a href="discprob.idxval.ival_dist.html#ivdplus">ivdplus</a> </td>
<td>In the formalization, this is defined in terms of simpler operations (<a href="discprob.idxval.ival.html#iplus">iplus</a> and <a href="discprob.idxval.ival.html#iscale">iscale</a>) as in the work of Varacca and Winskel, whereas the dissertation presents the "beta reduced" definition.</td>
</tr>
<tr>
<td> Non-Empty Sets of Indexed Valuations </td>
<td> Section 2.2 </td>
<td> <a href="discprob.idxval.pival.html">proba/theories/idxval/pival.v</a><br />
<a href="discprob.idxval.pival_dist.html">proba/theories/idxval/pival_dist.v</a><br /> </td>
<td> <a href="discprob.idxval.pival.html#pival">pival</a>,
<a href="discprob.idxval.pival_dist.html#pidist">pidist</a> </td>
<td>As with indexed valuations, we first define a version where every element of the set is merely an <a href="discprob.idxval.ival.html#ival">ival</a>, and another where their values must also sum to 1</td>
</tr>
<tr>
<td> Equivalence/Subset Ordering for Sets of Indexed Valuations </td>
<td> Section 2.2 </td>
<td> <a href="discprob.idxval.pival.html">proba/theories/idxval/pival.v</a><br />
<a href="discprob.idxval.pival_dist.html">proba/theories/idxval/pival_dist.v</a><br /> </td>
<td> <a href="discprob.idxval.pival.html#eq_pival">eq_pival</a>,
<a href="discprob.idxval.pival.html#le_pival">le_pival</a>,
<a href="discprob.idxval.pival_dist.html#eq_pidist">eq_pidist</a>,
<a href="discprob.idxval.pival_dist.html#le_pidist">le_pidist</a> </td>
<td></td>
</tr>
<tr>
<td> Equational Laws for Combined Monad </td>
<td> Section 2.2, Fig. 2.1 </td>
<td> <a href="discprob.idxval.pival_dist.html">proba/theories/idxval/pival_dist.v</a></td>
<td>
<a href="discprob.idxval.pival_dist.html#pidist_plus_comm">pidist_plus_comm</a>,
<a href="discprob.idxval.pival_dist.html#pidist_plus_choice1">pidist_plus_choice1</a>,
<a href="discprob.idxval.pival_dist.html#pidist_union_idemp">pidist_union_idemp</a>,
<a href="discprob.idxval.pival_dist.html#pidist_union_comm">pidist_union_comm</a>,
<a href="discprob.idxval.pival_dist.html#pidist_union_assoc">pidist_union_assoc</a>,
<a href="discprob.idxval.pival_dist.html#pidist_union_bind">pidist_union_bind</a>,
<a href="discprob.idxval.pival_dist.html#pidist_plus_bind">pidist_plus_bind</a>
</td>
<td> Various other laws are also described in this file, including the monad laws
(<a href="discprob.idxval.pival_dist.html#pidist_assoc">pidist_assoc</a>,
<a href="discprob.idxval.pival_dist.html#pidist_left_id">pidist_left_id</a>,
<a href="discprob.idxval.pival_dist.html#pidist_right_id">pidist_right_id</a>)
</td>
</tr>
<tr>
<td> Monadic encoding of approximate counters </td>
<td> Section 2.2, Fig. 2.2 </td>
<td> <a href="iris.tests.approx_counter.faa_approx_counter.html">theories/tests/approx_counter/faa_approx_counter.v</a></td>
<td>
<a href="iris.tests.approx_counter.faa_approx_counter.html#approx_incr">approx_incr</a>
<a href="iris.tests.approx_counter.faa_approx_counter.html#approx_n">approx_n</a>
</td>
<td> </td>
</tr>
<tr>
<td> Expected value of indexed valuation </td>
<td> Section 2.3 </td>
<td> <a href="discprob.idxval.extrema.html">proba/theories/idxval/extrema.v</a></td>
<td>
<a href="discprob.idxval.extrema.html#Ex_ival">Ex_ival</a>,
<a href="discprob.idxval.extrema.html#ex_Ex_ival">ex_Ex_ival</a>,
<a href="discprob.idxval.extrema.html#is_Ex_ival">is_Ex_ival</a>,
<a href="discprob.idxval.extrema.html#Ex_ivd">Ex_ivd</a>,
<a href="discprob.idxval.extrema.html#ex_Ex_ivd">ex_Ex_ivd</a>,
<a href="discprob.idxval.extrema.html#is_Ex_ivd">is_Ex_ivd</a>,
</td>
<td> Recall that expected values may not exist (because they are infinite series). We follow what Boldo et al. do in their Coquelicot analysis library for infinite series: Ex_ival f I is a real number, which is equal to the expected value, if the expected value exists. ex_Ex_ival f I is a proposition stating stating this existence property, and is_Ex_ival f I v says that the expected value exists and is equal to v. <br /><br /> We first define these for ival that do not have probabilities summing to 1, then the _ivd variants are the versions with the latter restriction.
</td>
</tr>
<tr>
<td> Minimal and Maximal Expected Values </td>
<td> Section 2.3 </td>
<td> <a href="discprob.idxval.extrema.html">proba/theories/idxval/extrema.v</a></td>
<td>
<a href="discprob.idxval.extrema.html#Ex_min">Ex_min</a>,
<a href="discprob.idxval.extrema.html#Ex_max">Ex_max</a>,
<a href="discprob.idxval.extrema.html#ex_Ex_ival">ex_Ex_extrema</a>,
</td>
<td> </td>
</tr>
<tr>
<td> Rules for calculating extrema</td>
<td> Section 2.3, Fig. 2.4 </td>
<td> <a href="discprob.idxval.extrema.html">proba/theories/idxval/extrema.v</a></td>
<td>
<a href="discprob.idxval.extrema.html#Ex_min_mret">Ex_min_mret</a>,
<a href="discprob.idxval.extrema.html#Ex_min_plus_const_r">Ex_min_plus_const_r</a>,
<a href="discprob.idxval.extrema.html#Ex_min_scale_const">Ex_min_scale_const</a>,
<a href="discprob.idxval.extrema.html#Ex_min_pidist_plus">Ex_min_pidist_plus</a>,
<a href="discprob.idxval.extrema.html#Ex_min_comp">Ex_min_comp</a>,
<a href="discprob.idxval.extrema.html#Ex_min_bind_const_le">Ex_min_bind_const_le</a>,
<a href="discprob.idxval.extrema.html#Ex_min_bind_const_ge">Ex_min_bind_const_ge</a>
</td>
<td>As stated in the text, the rules are given there without the side conditions about the existence/finiteness of extrema.</td>
</tr>
<tr>
<td> Expected value of monadic specification of approximate counter</td>
<td> Section 2.3, Example 2.3 </td>
<td> <a href="iris.tests.approx_counter.faa_approx_counter.html">theories/tests/approx_counter/faa_approx_counter.v</a></td>
<td>
<a href="iris.tests.approx_counter.faa_approx_counter.html#Ex_min_approx_incr">Ex_min_approx_incr</a>
<a href="iris.tests.approx_counter.faa_approx_counter.html#Ex_min_approx_n">Ex_min_approx_n</a>
<a href="iris.tests.approx_counter.faa_approx_counter.html#Ex_max_approx_incr">Ex_max_approx_incr</a>
<a href="iris.tests.approx_counter.faa_approx_counter.html#Ex_max_approx_n">Ex_max_approx_n</a>
</td>
<td> </td>
</tr>
<tr>
<td>Coarser equivalence relation on indexed valuations</td>
<td> Section 2.3, Fig. 5 </td>
<td>
<a href="discprob.idxval.irrel_equiv.html">proba/theories/idxval/irrel_equiv.v</a><br />
<a href="discprob.idxval.irrel_coupling_alt.html">proba/theories/idxval/irrel_coupling_alt.v</a>
</td>
<td>
<a href="discprob.idxval.irrel_equiv.html#irrel_ivd">irrel_ivd</a>,
<a href="discprob.idxval.irrel_coupling_alt.html#irrel_ivd_alt">irrel_ivd_alt</a>,
</td>
<td>For historical reasons, much of the development uses irrel_ivd, which is
specified inductively using the rules in Fig. 5. However,
<a href="discprob.idxval.irrel_coupling_alt.html#irrel_ivd_alt">irrel_ivd_alt</a> show that this is equivalent to the definition used in the text in terms of expected values of bounded functions.</td>
</tr>
<tr>
<td>Coarser subset ordering on sets of indexed valuations</td>
<td>Section 2.3</td>
<td>
<a href="discprob.idxval.irrel_equiv.html">proba/theories/idxval/irrel_equiv.v</a><br />
</td>
<td>
<a href="discprob.idxval.irrel_equiv.html#irrel_pidist">irrel_pidist</a>,
</td>
<td> It is stated in Coq in terms of Ex_min instead of Ex_max, but see just
below for the Ex_max version.</td>
</tr>
<tr>
<td>Rules for coarser subset ordering on sets of indexed valuations</td>
<td>Section 2.3, Fig. 6.</td>
<td>
<a href="discprob.idxval.irrel_equiv.html">proba/theories/idxval/irrel_equiv.v</a><br /></td>
<td>
<a href="discprob.idxval.irrel_equiv.html#irrel_pidist_refl">irrel_pidist_refl</a>,
<a href="discprob.idxval.irrel_equiv.html#irrel_pidist_trans">irrel_pidist_trans</a>,
<a href="discprob.idxval.irrel_equiv.html#irrel_pidist_proper">irrel_pidist_proper</a>,
<a href="discprob.idxval.irrel_equiv.html#irrel_pidist_bind">irrel_pidist_bind</a>,
<a href="discprob.idxval.irrel_equiv.html#irrel_pidist_choice">irrel_pidist_choice</a>,
<a href="discprob.idxval.irrel_equiv.html#irrel_pidist_irrel">irrel_pidist_irrel</a>,
</td>
<td></td>
</tr>
<tr>
<td>Lemma 2.4 </td>
<td>Section 2.3</td>
<td>
<a href="discprob.idxval.irrel_equiv.html">proba/theories/idxval/irrel_equiv.v</a><br /></td>
<td>
<a href="discprob.idxval.irrel_equiv.html#irrel_bounded_supp_fun">
irrel_bounded_supp_fun</a>
<a href="discprob.idxval.irrel_equiv.html#irrel_pidist_bounded_supp_Ex_max">
irrel_pidist_bounded_supp_Ex_max</a>
</td>
<td></td>
</tr>
<tr>
<td>Lemma 2.5 (Markov's Inequality) </td>
<td>Section 2.4.1</td>
<td>
<a href="discprob.idxval.markov.html">proba/theories/idxval/markov.v</a><br /></td>
<td>
<a href="discprob.idxval.markov.html#general_markov_inequality">
general_markov_inequality</a>
</td>
<td></td>
</tr>
<tr>
<td>Theorem 2.6 (Chehbyshev's Inequality) </td>
<td>Section 2.4.2</td>
<td>
<a href="discprob.idxval.markov.html">proba/theories/idxval/markov.v</a><br /></td>
<td>
<a href="discprob.idxval.markov.html#chebyshev_inequality">chebyshev_inequality</a>
</td>
<td></td>
</tr>
<tr>
<td>Lemma 2.7</td>
<td>Section 2.4.2</td>
<td>
<a href="discprob.idxval.markov.html">proba/theories/idxval/markov.v</a><br /></td>
<td>
<a href="discprob.idxval.markov.html#Ex_max_variance">Ex_max_invariance</a>
</td>
<td></td>
</tr>
<tr>
<td>Non-deterministic Couplings Definition </td>
<td>Section 2.5</td>
<td>
<a href="discprob.idxval.irrel_equiv.html">proba/theories/idxval/irrel_equiv.v</a><br />
<a href="discprob.idxval.irrel_coupling_alt.html">proba/theories/idxval/irrel_coupling_alt.v</a><br /></td>
<td>
<a href="discprob.idxval.irrel_equiv.html#irrel_couplingP">
irrel_couplingP</a>
<a href="discprob.idxval.irrel_coupling_alt.html#idist_pidist_couplingP'">
idist_pidist_couplingP'</a>
</td>
<td>Again, for historical reasons, the development primarily uses
<a href="discprob.idxval.irrel_equiv.html#irrel_couplingP">
irrel_couplingP</a>, which is what is called a "non-deterministic
coupling up to irrelevance" in the original submitted
appendix. However, the definition currently given in the text
corresponds
to <a href="discprob.idxval.irrel_equiv.html#idist_pidist_couplingP'">
idist_pidist_couplingP'</a> which seems more natural to us in
hindsight (because one does not need to first introduce a
different notion and then quotient to get the "up to irrelevance"
version).
<a href="discprob.idxval.irrel_coupling_alt.html#irrel_couplingP_to_alt">irrel_couplingP_to_alt</a>
and
<a href="discprob.idxval.irrel_coupling_alt.html#alt_to_irrel_couplingP">alt_to_irrel_couplingP</a> show that these definitions are equivalent. Note that these are stated as records, and for technical reasons it's sometimes easier to use a version that is a proposition, <a href="discprob.idxval.irrel_equiv.html#irrel_coupling_propP">irrel_coupling_propP</a>
</td>
</tr>
<tr>
<td>Rules for Non-deterministic Couplings</td>
<td>Section 2.5, Fig. 2.7</td>
<td>
<a href="discprob.idxval.irrel_equiv.html">proba/theories/idxval/irrel_equiv.v</a></td>
<td>
<a href="discprob.idxval.irrel_equiv.html#irrel_coupling_mret">
irrel_coupling_mret</a>,
<a href="discprob.idxval.irrel_equiv.html#irrel_coupling_proper">
irrel_coupling_proper</a>,
<a href="discprob.idxval.irrel_equiv.html#irrel_coupling_conseq">
irrel_coupling_conseq</a>,
<a href="discprob.idxval.irrel_equiv.html#irrel_coupling_plus">
irrel_coupling_plus</a>,
<a href="discprob.idxval.irrel_equiv.html#irrel_coupling_trivial">
irrel_coupling_trivial</a>
</td>
<td>
</td>
</tr>
<tr>
<td>Theorem 2.8</td>
<td>Section 2.5</td>
<td> <a href="discprob.idxval.irrel_equiv.html">proba/theories/idxval/irrel_equiv.v</a></td>
<td>
<a href="discprob.idxval.irrel_equiv.html#irrel_coupling_eq_ex_Ex_supp">
irrel_coupling_eq_ex_Ex_supp</a>,
<a href="discprob.idxval.irrel_equiv.html#irrel_coupling_eq_Ex_min_supp">
irrel_coupling_eq_Ex_min_supp</a>
<a href="discprob.idxval.irrel_equiv.html#irrel_coupling_eq_Ex_max_supp">
irrel_coupling_eq_Ex_max_supp</a>,
</td>
<td>
</td>
</tr>
<tr>
<td>Background on Iris</td>
<td>Ch. 3 + 4</td>
<td>(various)</td>
<td>(various)</td>
<td> As these chapters provide background on original Iris, we will not provide links. See the Iris project site and appendix. </td>
</tr>
<tr>
<td>Concurrent Thread Reduction</td>
<td>Section 5.1, Fig. 5.1</td>
<td> <a href="iris.program_logic.language.html">theories/program_logic/language.v</a><br />
<a href="iris.program_logic.prob_language.html">theories/program_logic/prob_language.v</a></td>
<td>
<a href="iris.program_logic.language.html#istep">istep</a>,
<a href="iris.program_logic.prob_language.html#trace_step">trace_step</a>,
</td>
<td>
</td>
</tr>
<tr>
<td>Definition of bounded termination under a scheduler</td>
<td>Section 5.1.1</td>
<td>
<a href="iris.program_logic.prob_language.html">theories/program_logic/prob_language.v</a>
</td>
<td>
<a href="iris.program_logic.prob_language.html#terminates">terminates</a>
</td>
<td>
</td>
</tr>
<tr>
<td>Probabilistic Thread Reduction as an Indexed Valuation</td>
<td>Section 5.1.2</td>
<td>
<a href="iris.program_logic.prob_language.html">theories/program_logic/prob_language.v</a></td>
<td>
<a href="iris.program_logic.prob_language.html#ivdist_trace_stepN_aux">ivdist_trace_stepN_aux</a>,
<a href="iris.program_logic.prob_language.html#ivdist_tpool_stepN">ivdist_tpool_stepN</a>,
</td>
<td> </td>
</tr>
<tr>
<td>Randomized ML-Language</td>
<td>Section 5.1.3</td>
<td> <a href="iris.heap_lang.lang.html">theories/heap_lang/lang.v</a></td>
<td>
<a href="iris.heap_lang.lang.html#heap_lang.head_step">head_step</a>,
<a href="iris.heap_lang.lang.html#heap_lang.head_step_prob">head_step_prob</a>,
<a href="iris.heap_lang.lang.html#heap_lang.head_step_prob_rel">head_step_prob_rel</a>
</td>
<td>
As in the dissertation, the semantics are presented as a plain reduction relation
(<a href="iris.heap_lang.lang.html#heap_lang.head_step">head_step</a>) and
a function which assigns probabilities to a reduction
(<a href="iris.heap_lang.lang.html#heap_lang.head_step_prob">head_step_prob</a>). However,
we also prove that this is equivalent to a version where the relation is
directly annotated with the probability
(<a href="iris.heap_lang.lang.html#heap_lang.head_step_prob_rel">head_step_prob_rel</a>).
(see the equivalence proofs <a href="iris.heap_lang.lang.html#heap_lang.head_step_prob_rel_to_fun">head_step_prob_rel_to_fun</a> and <a href="iris.heap_lang.lang.html#heap_lang.head_step_prob_to_rel">head_step_prob_to_rel</a>) <br /> <br />
The other main difference is that the dissertation just describes a single
primitive called "flip" for simulating Bernoulli random variables. To make
it easier to add other primitives to the language, we allow various probabilistic operations (<a href="iris.heap_lang.lang.html#heap_lang.prob_bin_op">prob_bin_op</a>) to be added (which are supposed to represent different distributions, possibly taking some parameters), and then the semantics of these operations are specified by giving an indexed valuation that they represent (<a href="iris.heap_lang.lang.html#heap_lang.prob_bin_op_eval">prob_bin_op_eval</a>).
</td>
</tr>
<tr>
<td>New Probabilistic Rules</td>
<td>Section 5.2.1</td>
<td>
<a href="iris.heap_lang.lifting.html">theories/heap_lang/lifting.v</a><br />
<a href="iris.program_logic.prob_lifting.html">theories/program_logic/prob_lifting.v</a>
</td>
<td>
<a href="iris.heap_lang.lifting.html#wp_flip">wp_flip</a>,
<a href="iris.heap_lang.lifting.html#wp_irrel_flip">wp_irrel_flip</a>,
<a href="iris.program_logic.prob_lifting.html#ownProb_anti_irrel'">ownProb_anti_irrel'</a>
</td>
<td>
</td>
</tr>
<tr>
<td>Probabilistic Lifting Lemma</td>
<td>Section 5.2.2</td>
<td>
<a href="iris.program_logic.prob_lifting.html">theories/program_logic/prob_lifting.v</a>
</td>
<td>
<a href="iris.program_logic.prob_lifting.html#wp_lift_step_couple">wp_lift_step_couple</a>,
<a href="iris.program_logic.prob_lifting.html#wp_lift_step_nocouple">wp_lift_step_nocouple</a>,
<a href="iris.program_logic.prob_lifting.html#ownProb_anti_irrel'">ownProb_anti_irrel'</a>
</td>
<td>
The main difference is that the latest variants of Iris define a variant of the weakest precondition, in which it is annotated with an additonal "flag" s, which can take the value Stuck or NotStuck. If s is NotStuck then the weakest precondition behaves as in the dissertation: namely, for the weakest precondition to hold, the program must not get stuck. However, if s is Stuck, then we allow the program to get stuck during execution. Throughout, I only ever use the NotStuck version, but almost all of the rules generalize straightforwardly to the case where s is Stuck, so I support this.
</td>
</tr>
<tr>
<td>Theorem 5.1</td>
<td>Section 5.3</td>
<td>
<a href="iris.program_logic.prob_adequacy.html">theories/program_logic/prob_adequacy.v</a><br />
</td>
<td>
<a href="iris.program_logic.prob_adequacy.html#wp_prob_adequacy'">wp_prob_adequacy'</a>
</td>
<td>
</td>
</tr>
<tr>
<td>Corollary 5.2</td>
<td>Section 5.3</td>
<td>
<a href="iris.program_logic.prob_adequacy.html">theories/program_logic/prob_adequacy.v</a><br />
</td>
<td>
<a href="iris.program_logic.prob_adequacy.html#wp_prob_adequacy_ex_Ex">wp_prob_adequacy_ex_Ex</a>,
<a href="iris.program_logic.prob_adequacy.html#wp_prob_adequacy_Ex_min">wp_prob_adequacy_Ex_min</a>,
<a href="iris.program_logic.prob_adequacy.html#wp_prob_adequacy_Ex_max">wp_prob_adequacy_Ex_max</a>
</td>
<td>
</td>
</tr>
<tr>
<td>Theorem 5.3</td>
<td>Section 5.3</td>
<td>
<a href="iris.program_logic.prob_adequacy.html">theories/program_logic/prob_adequacy.v</a><br />
</td>
<td>
<a href="iris.program_logic.prob_adequacy.html#wp_prob_safety_adequacy">wp_prob_safety_adequacy</a>,
</td>
<td>
</td>
</tr>
<tr>
<td>Defintion of weakest precondition</td>
<td>Section 5.3</td>
<td>
<a href="iris.program_logic.weakestpre.html">theories/program_logic/weakestpre.v</a><br />
<a href="iris.program_logic.prob_lifting.html">theories/program_logic/prob_lifting.v</a><br />
</td>
<td>
<a href="iris.program_logic.weakestpre.html#wp_pre">wp_pre</a>,
<a href="iris.program_logic.weakestpre.html#wp_def">wp_def</a>,
<a href="iris.program_logic.prob_lifting.html#probG_irisG">probG_irisG</a>,
</td>
<td>
The definition in wp_pre makes reference to fields in a record, which are then instantiated with those of probG_irisG; the version shown in the dissertation is the "beta-reduced" form of this definition.
wp_def takes the fixed point of the guarded definition.
</td>
</tr>
<tr>
<td>Theorem 5.4</td>
<td>Section 5.3</td>
<td>
<a href="iris.program_logic.prob_adequacy.html">theories/program_logic/prob_adequacy.v</a><br />
</td>
<td>
<a href="iris.program_logic.prob_adequacy.html#wp_coupling">wp_coupling</a>,
</td>
<td>
</td>
</tr>
<tr>
<td>Lemma 5.5 (Instantiation of adequacy for ML-like language)</td>
<td>Section 5.3</td>
<td>
<a href="iris.heap_lang.adequacy.html">theories/heap_lang/adequacy.v</a><br />
</td>
<td>
<a href="iris.heap_lang.adequacy.html#heap_prob_adequacy">heap_prob_adequacy</a>,
<a href="iris.heap_lang.adequacy.html#heap_prob_adequacy_Ex_max">heap_prob_adequacy_Ex_max</a>,
<a href="iris.heap_lang.adequacy.html#heap_prob_adequacy_Ex_min">heap_prob_adequacy_Ex_min</a>,
</td>
<td>
Lemma 5.5 is an auxiliary lemma that is used in the insantiation proves of these lemmas; in the formalization it is not broken out as a separate lemma.
</td>
</tr>
<tr>
<td>Approximate counter algorithm</td>
<td>Section 6.1, Fig 6.1</td>
<td>
<a href="iris.tests.approx_counter.faa_approx_counter.html">theories/tests/approx_counter/faa_approx_counter.v</a>
</td>
<td>
<a href="iris.tests.approx_counter.faa_approx_counter.html#newcounter">newcounter</a>,
<a href="iris.tests.approx_counter.faa_approx_counter.html#incr">incr</a>,
<a href="iris.tests.approx_counter.faa_approx_counter.html#read">read</a>
</td>
<td>
</td>
</tr>
<tr>
<td>Approximate counter rules</td>
<td>Section 6.1, Fig. 6.2</td>
<td>
<a href="iris.tests.approx_counter.faa_approx_counter.html">theories/tests/approx_counter/faa_approx_counter.v</a>
</td>
<td>
<a href="iris.tests.approx_counter.faa_approx_counter.html#newcounter_spec'">newcounter_spec'</a>,
<a href="iris.tests.approx_counter.faa_approx_counter.html#Acounter_sep">Acounter_sep</a>,
<a href="iris.tests.approx_counter.faa_approx_counter.html#Acounter_join">Acounter_join</a>,
<a href="iris.tests.approx_counter.faa_approx_counter.html#incr_spec'">incr_spec'</a>,
<a href="iris.tests.approx_counter.faa_approx_counter.html#read_spec'">read_spec'</a>
</td>
<td>
</td>
</tr>
<tr>
<td>Example client for Approximate Counter</td>
<td>Section 6.1, Fig. 6.3</td>
<td>
<a href="iris.tests.approx_counter.faa_approx_counter.html">theories/tests/approx_counter/faa_approx_counter.v</a>
</td>
<td>
<a href="iris.tests.approx_counter.faa_approx_counter.html#list_client">list_client</a>
<a href="iris.tests.approx_counter.faa_approx_counter.html#list_client_Ex_ival">list_client_Ex_ival</a>
</td>
<td>
</td>
</tr>
<tr>
<td>Variations of Approximate Counter Example</td>
<td>Section 6.1.4</td>
<td>
<a href="iris.tests.approx_counter.faa_approx_counter_nomax.html">theories/tests/approx_counter/faa_approx_counter_nomax.v</a>
<a href="iris.tests.approx_counter.faa_approx_counter_arb.html">theories/tests/approx_counter/faa_approx_counter_arb.v</a>
</td>
<td>
(various)
</td>
<td>
<a href="iris.tests.approx_counter.faa_approx_counter_nomax.html#generic_client_chebyshev">generic_client_chebyshev</a> gives
the bound using Chebyshev's theorem.
</td>
</tr>
<tr>
<td>Skiplist Code</td>
<td>Section 6.2</td>
<td>
<a href="iris.tests.skiplist.code.html">theories/tests/skiplist/code.v</a>
</td>
<td>
(entire file)
</td>
<td>
</td>
</tr>
<tr>
<td>Skiplist Monadic Model</td>
<td>Section 6.2.1</td>
<td>
<a href="iris.tests.skiplist.idxval.html">theories/tests/skiplist/idxval.v</a>
</td>
<td>
<a href="iris.tests.skiplist.idxval.html#skiplist">skiplist</a>
</td>
<td>
Note that it is more complicated looking than the version in the dissertation because what is written in the paper is not structurally recursive, so we have to include additional information to convince Coq that it is terminating.
</td>
</tr>
<tr>
<td>Expected value for Skiplist Monadic Model</td>
<td>Section 6.2.1</td>
<td>
<a href="iris.tests.skiplist.idxval_expected.html">theories/tests/skiplist/idxval_expected.v</a>
</td>
<td>
<a href="iris.tests.skiplist.idxval_expected.html#Skiplist_Expected.skiplist_Ex_max">skiplist_Ex_max</a>
</td>
<td>
</td>
</tr>
<tr>
<td>Rules for Skiplist</td>
<td>Section 6.2.2, Fig. 6.7</td>
<td>
<a href="iris.tests.skiplist.spec_bundled.html">theories/tests/skiplist/spec_bundled.v</a>
</td>
<td>
<a href="iris.tests.skiplist.spec_bundled.html#Skiplist_Spec.new_spec">new_spec</a>,
<a href="iris.tests.skiplist.spec_bundled.html#Skiplist_Spec.add_spec">add_spec</a>,
<a href="iris.tests.skiplist.spec_bundled.html#Skiplist_Spec.mem_full_spec">mem_full_spec</a>,
<a href="iris.tests.skiplist.spec_bundled.html#Skiplist_Spec.skip_prop_sep">skip_prop_sep</a>,
<a href="iris.tests.skiplist.spec_bundled.html#Skiplist_Spec.skip_prop_join">skip_prop_join</a>
<a href="iris.tests.skiplist.spec_bundled.html#Skiplist_Spec.skip_prob_spec'">skip_prob_spec'</a>
</td>
<td>
</td>
</tr>
<tr>
<td>Example client for Skiplist</td>
<td>Section 6.2.2, mentioned but not shown</td>
<td>
<a href="iris.tests.skiplist.client.html">theories/tests/skiplist/client.v</a>
</td>
<td>
<a href="iris.tests.skiplist.client.html#skip_list_client">skip_list_client</a>,
<a href="iris.tests.skiplist.client.html#list_client_Ex_ival">list_client_Ex_ival</a>,
</td>
<td>
</td>
</tr>
</table>
</div>
</div>
</div>
</div>
</body>
</html>