diff --git a/404.html b/404.html index c1813c03..119413d0 100644 --- a/404.html +++ b/404.html @@ -3,7 +3,7 @@ -Page Not Found | Caesar +Page Not Found | Caesar @@ -12,12 +12,13 @@ + - - - + + + -
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\ No newline at end of file diff --git a/about/index.html b/about/index.html index 1f0a874c..9af32980 100644 --- a/about/index.html +++ b/about/index.html @@ -3,7 +3,7 @@ -About Caesar | Caesar +About Caesar | Caesar @@ -12,12 +12,13 @@ + - - - + + + -
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About Caesar

Caesar is a deductive verification infrastructure specifically designed for probabilistic programs. That means it is a tool to provide formal guarantees for programs that incorporate probabilistic behaviours, such as drawing random numbers or making random choices.

Probabilistic programs can be found in randomized algorithms, communication protocols, machine learning models, and many other domains. When designing or analyzing such programs, many questions can be formulated with respect to some kind of expected value, such as the probability of reaching a certain state, the expected number of steps until a certain event occurs, or the expected value of some variable at the end of the program. We support the use of different proof rules, from areas such as martingale analysis or domain theory, thus enabling reasoning about programs with infinite state spaces and unbounded loops.

Caesar aims to be a verification infrastructure that combines many different techniques to formally reason about such expected values. We want to provide a tool that automates the verification process as much as possible, while still allowing the user to guide the verification process and provide additional information when needed. This differs from probabilistic model checkers such as Storm or PRISM, which are more "push-button" approaches that require less user interaction but struggle with e.g. infinite state spaces.

Caesar is an open-source project from RWTH Aachen University (MOVES group), Saarland University (QUAVE group), Denmark Technical University (SSE section), and University College London (PPLV group). The source code is available on GitHub. The name "Caesar" comes from "veni, vidi, vc", where we let "vc" stand for "verification conditions".

Main Features

  • Verification of expected values in probabilistic programs
  • Support for infinite state spaces, unbounded loops, and recursion
  • Support for different proof rules, such as from martingale analysis or domain theory
  • Integration with SMT solvers and probabilistic model checkers
  • Program slicing for error localization and hints
  • Integration with Visual Studio Code

Key Components

The Quantitative Intermediate Verification Language HeyVL

The architecture of Caesar is inspired by modern program verifiers that use an intermediate verification language (IVL) to encode a program, its specification, and proof rules into a common representation. Our quantitative IVL is called HeyVL. HeyVL generalizes classical IVLs to the quantitative setting through HeyLo, our new quantitative logic developed for Caesar. The names come from Heyting algebras, which underlie the logic.

In Caesar, HeyVL also has high-level constructs such as loops or procedure calls, but these are all encoded into a smaller core language. This core language and many encodings are described in our OOPSLA 2023 publication "A Deductive Verification Infrastructure for Probabilistic Programs".

Backends: SMT Solving and Probabilistic Model Checking

Caesar uses the Z3 SMT solver to reason about HeyVL programs. This allows Caesar to use symbolic reasoning to prove properties of probabilistic programs, enabling reasoning about infinite state spaces and unbounded loops. It is also possible to use other SMT solvers (by emitting SMT-LIB).

In addition, Caesar has a model checking backend that emits files in the JANI format, a JSON-based interchange format for probabilistic models. This is possible for an "executable" subset of HeyVL programs that can be translated into JANI models. The result can be fed into probabilistic model checkers such as Storm.

Slicing for Error Localization and Hints

Caesar includes a program slicing engine called Brutus (get it?) that reduces HeyVL programs with respect to certain properties. Slicing is Caesar's theoretical and practical foundation for error localization and hints. For example, by removing assertions from the program, we can find out the remaining assertions which must cause a verification failure.

Visual Studio Code Integration

We value usability and user-friendliness highly and want to make Caesar as easy to use as possible. It is our belief that UX is a key factor in making formal verification succeed in practice. This is why we put a lot of effort into integrating Caesar with Visual Studio Code with our Caesar for VSCode extension.

Disclaimer: (Un)Related Work

Caesar should not be confused with the set of tools in the CADP toolbox by INRIA, which includes tools like the CAESAR or CAESAR.ADT compilers, or the OPEN/CAESAR software architecture.

+
Skip to main content

About Caesar

Caesar is a deductive verification infrastructure specifically designed for probabilistic programs. That means it is a tool to provide formal guarantees for programs that incorporate probabilistic behaviours, such as drawing random numbers or making random choices.

Probabilistic programs can be found in randomized algorithms, communication protocols, machine learning models, and many other domains. When designing or analyzing such programs, many questions can be formulated with respect to some kind of expected value, such as the probability of reaching a certain state, the expected number of steps until a certain event occurs, or the expected value of some variable at the end of the program. We support the use of different proof rules, from areas such as martingale analysis or domain theory, thus enabling reasoning about programs with infinite state spaces and unbounded loops.

Caesar aims to be a verification infrastructure that combines many different techniques to formally reason about such expected values. We want to provide a tool that automates the verification process as much as possible, while still allowing the user to guide the verification process and provide additional information when needed. This differs from probabilistic model checkers such as Storm or PRISM, which are more "push-button" approaches that require less user interaction but struggle with e.g. infinite state spaces.

Caesar is an open-source project from RWTH Aachen University (MOVES group), Saarland University (QUAVE group), Denmark Technical University (SSE section), and University College London (PPLV group). The source code is available on GitHub. The name "Caesar" comes from "veni, vidi, vc", where we let "vc" stand for "verification conditions".

Main Features

  • Verification of expected values in probabilistic programs
  • Support for infinite state spaces, unbounded loops, and recursion
  • Support for different proof rules, such as from martingale analysis or domain theory
  • Integration with SMT solvers and probabilistic model checkers
  • Program slicing for error localization and hints
  • Integration with Visual Studio Code

Key Components

The Quantitative Intermediate Verification Language HeyVL

The architecture of Caesar is inspired by modern program verifiers that use an intermediate verification language (IVL) to encode a program, its specification, and proof rules into a common representation. Our quantitative IVL is called HeyVL. HeyVL generalizes classical IVLs to the quantitative setting through HeyLo, our new quantitative logic developed for Caesar. The names come from Heyting algebras, which underlie the logic.

In Caesar, HeyVL also has high-level constructs such as loops or procedure calls, but these are all encoded into a smaller core language. This core language and many encodings are described in our OOPSLA 2023 publication "A Deductive Verification Infrastructure for Probabilistic Programs".

Backends: SMT Solving and Probabilistic Model Checking

Caesar uses the Z3 SMT solver to reason about HeyVL programs. This allows Caesar to use symbolic reasoning to prove properties of probabilistic programs, enabling reasoning about infinite state spaces and unbounded loops. It is also possible to use other SMT solvers (by emitting SMT-LIB).

In addition, Caesar has a model checking backend that emits files in the JANI format, a JSON-based interchange format for probabilistic models. This is possible for an "executable" subset of HeyVL programs that can be translated into JANI models. The result can be fed into probabilistic model checkers such as Storm.

Slicing for Error Localization and Hints

Caesar includes a program slicing engine called Brutus (get it?) that reduces HeyVL programs with respect to certain properties. Slicing is Caesar's theoretical and practical foundation for error localization and hints. For example, by removing assertions from the program, we can find out the remaining assertions which must cause a verification failure.

Visual Studio Code Integration

We value usability and user-friendliness highly and want to make Caesar as easy to use as possible. It is our belief that UX is a key factor in making formal verification succeed in practice. This is why we put a lot of effort into integrating Caesar with Visual Studio Code with our Caesar for VSCode extension.

Disclaimer: (Un)Related Work

Caesar should not be confused with the set of tools in the CADP toolbox by INRIA, which includes tools like the CAESAR or CAESAR.ADT compilers, or the OPEN/CAESAR software architecture.

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button,.codeBlockContainer_Ckt0{background:var(--prism-background-color);color:var(--prism-color)}.authorListItem_n3yI{list-style-type:none;margin-bottom:2rem}.authorCol_Hf19{max-width:inherit!important}.imageOnlyAuthorRow_pa_O{display:flex;flex-flow:row wrap}.imageOnlyAuthorCol_G86a{margin-left:.3rem;margin-right:.3rem}.features_t9lD{align-items:center;display:flex;padding:2rem 0;width:100%}.featureSvg_GfXr{height:100px;width:200px}[data-theme=dark] .invertDark_jaEU{filter:invert(100%)}.codeBlockContainer_Ckt0{border-radius:var(--ifm-code-border-radius);box-shadow:var(--ifm-global-shadow-lw);margin-bottom:var(--ifm-leading)}.codeBlockContent_biex{border-radius:inherit;direction:ltr;position:relative}.codeBlockTitle_Ktv7{border-bottom:1px solid var(--ifm-color-emphasis-300);border-top-left-radius:inherit;border-top-right-radius:inherit;font-size:var(--ifm-code-font-size);font-weight:500;padding:.75rem 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configuration");return f.isRefined(this.disjunctiveFacetsRefinements,e,t)?this.setQueryParameters({disjunctiveFacetsRefinements:f.removeRefinement(this.disjunctiveFacetsRefinements,e,t)}):this},removeTagRefinement:function(e){if(!this.isTagRefined(e))return this;var t={tagRefinements:this.tagRefinements.filter((function(t){return t!==e}))};return this.setQueryParameters(t)},toggleRefinement:function(e,t){return this.toggleFacetRefinement(e,t)},toggleFacetRefinement:function(e,t){if(this.isHierarchicalFacet(e))return this.toggleHierarchicalFacetRefinement(e,t);if(this.isConjunctiveFacet(e))return this.toggleConjunctiveFacetRefinement(e,t);if(this.isDisjunctiveFacet(e))return this.toggleDisjunctiveFacetRefinement(e,t);throw new Error("Cannot refine the undeclared facet "+e+"; it should be added to the helper options facets, disjunctiveFacets or hierarchicalFacets")},toggleConjunctiveFacetRefinement:function(e,t){if(!this.isConjunctiveFacet(e))throw new Error(e+" is not defined in the facets attribute of the helper configuration");return this.setQueryParameters({facetsRefinements:f.toggleRefinement(this.facetsRefinements,e,t)})},toggleExcludeFacetRefinement:function(e,t){if(!this.isConjunctiveFacet(e))throw new Error(e+" is not defined in the facets attribute of the helper configuration");return this.setQueryParameters({facetsExcludes:f.toggleRefinement(this.facetsExcludes,e,t)})},toggleDisjunctiveFacetRefinement:function(e,t){if(!this.isDisjunctiveFacet(e))throw new Error(e+" is not defined in the disjunctiveFacets attribute of the helper configuration");return this.setQueryParameters({disjunctiveFacetsRefinements:f.toggleRefinement(this.disjunctiveFacetsRefinements,e,t)})},toggleHierarchicalFacetRefinement:function(e,t){if(!this.isHierarchicalFacet(e))throw new Error(e+" is not defined in the hierarchicalFacets attribute of the helper configuration");var r=this._getHierarchicalFacetSeparator(this.getHierarchicalFacetByName(e)),i={};return void 0!==this.hierarchicalFacetsRefinements[e]&&this.hierarchicalFacetsRefinements[e].length>0&&(this.hierarchicalFacetsRefinements[e][0]===t||0===this.hierarchicalFacetsRefinements[e][0].indexOf(t+r))?-1===t.indexOf(r)?i[e]=[]:i[e]=[t.slice(0,t.lastIndexOf(r))]:i[e]=[t],this.setQueryParameters({hierarchicalFacetsRefinements:n(i,this.hierarchicalFacetsRefinements)})},addHierarchicalFacetRefinement:function(e,t){if(this.isHierarchicalFacetRefined(e))throw new Error(e+" is already refined.");if(!this.isHierarchicalFacet(e))throw new Error(e+" is not defined in the hierarchicalFacets attribute of the helper configuration.");var r={};return r[e]=[t],this.setQueryParameters({hierarchicalFacetsRefinements:n(r,this.hierarchicalFacetsRefinements)})},removeHierarchicalFacetRefinement:function(e){if(!this.isHierarchicalFacetRefined(e))return this;var t={};return t[e]=[],this.setQueryParameters({hierarchicalFacetsRefinements:n(t,this.hierarchicalFacetsRefinements)})},toggleTagRefinement:function(e){return this.isTagRefined(e)?this.removeTagRefinement(e):this.addTagRefinement(e)},isDisjunctiveFacet:function(e){return this.disjunctiveFacets.indexOf(e)>-1},isHierarchicalFacet:function(e){return void 0!==this.getHierarchicalFacetByName(e)},isConjunctiveFacet:function(e){return this.facets.indexOf(e)>-1},isFacetRefined:function(e,t){return!!this.isConjunctiveFacet(e)&&f.isRefined(this.facetsRefinements,e,t)},isExcludeRefined:function(e,t){return!!this.isConjunctiveFacet(e)&&f.isRefined(this.facetsExcludes,e,t)},isDisjunctiveFacetRefined:function(e,t){return!!this.isDisjunctiveFacet(e)&&f.isRefined(this.disjunctiveFacetsRefinements,e,t)},isHierarchicalFacetRefined:function(e,t){if(!this.isHierarchicalFacet(e))return!1;var r=this.getHierarchicalRefinement(e);return t?-1!==r.indexOf(t):r.length>0},isNumericRefined:function(e,t,r){if(void 0===r&&void 0===t)return Boolean(this.numericRefinements[e]);var n=this.numericRefinements[e]&&void 0!==this.numericRefinements[e][t];if(void 0===r||!n)return n;var s,a,c=u(r),o=void 0!==(s=this.numericRefinements[e][t],a=c,i(s,(function(e){return l(e,a)})));return n&&o},isTagRefined:function(e){return-1!==this.tagRefinements.indexOf(e)},getRefinedDisjunctiveFacets:function(){var e=this,t=s(Object.keys(this.numericRefinements).filter((function(t){return Object.keys(e.numericRefinements[t]).length>0})),this.disjunctiveFacets);return Object.keys(this.disjunctiveFacetsRefinements).filter((function(t){return e.disjunctiveFacetsRefinements[t].length>0})).concat(t).concat(this.getRefinedHierarchicalFacets()).sort()},getRefinedHierarchicalFacets:function(){var e=this;return s(this.hierarchicalFacets.map((function(e){return e.name})),Object.keys(this.hierarchicalFacetsRefinements).filter((function(t){return e.hierarchicalFacetsRefinements[t].length>0}))).sort()},getUnrefinedDisjunctiveFacets:function(){var e=this.getRefinedDisjunctiveFacets();return this.disjunctiveFacets.filter((function(t){return-1===e.indexOf(t)}))},managedParameters:["index","facets","disjunctiveFacets","facetsRefinements","hierarchicalFacets","facetsExcludes","disjunctiveFacetsRefinements","numericRefinements","tagRefinements","hierarchicalFacetsRefinements"],getQueryParams:function(){var e=this.managedParameters,t={},r=this;return Object.keys(this).forEach((function(n){var i=r[n];-1===e.indexOf(n)&&void 0!==i&&(t[n]=i)})),t},setQueryParameter:function(e,t){if(this[e]===t)return this;var r={};return r[e]=t,this.setQueryParameters(r)},setQueryParameters:function(e){if(!e)return this;var t=m.validate(this,e);if(t)throw t;var r=this,n=m._parseNumbers(e),i=Object.keys(this).reduce((function(e,t){return e[t]=r[t],e}),{}),s=Object.keys(n).reduce((function(e,t){var r=void 0!==e[t],i=void 0!==n[t];return r&&!i?o(e,[t]):(i&&(e[t]=n[t]),e)}),i);return new this.constructor(s)},resetPage:function(){return void 0===this.page?this:this.setPage(0)},_getHierarchicalFacetSortBy:function(e){return e.sortBy||["isRefined:desc","name:asc"]},_getHierarchicalFacetSeparator:function(e){return e.separator||" > "},_getHierarchicalRootPath:function(e){return e.rootPath||null},_getHierarchicalShowParentLevel:function(e){return"boolean"!=typeof e.showParentLevel||e.showParentLevel},getHierarchicalFacetByName:function(e){return i(this.hierarchicalFacets,(function(t){return t.name===e}))},getHierarchicalFacetBreadcrumb:function(e){if(!this.isHierarchicalFacet(e))return[];var t=this.getHierarchicalRefinement(e)[0];if(!t)return[];var r=this._getHierarchicalFacetSeparator(this.getHierarchicalFacetByName(e));return t.split(r).map((function(e){return e.trim()}))},toString:function(){return JSON.stringify(this,null,2)}},e.exports=m},7762:(e,t,r)=>{"use strict";e.exports=function(e){return function(t,r){var n=e.hierarchicalFacets[r],u=e.hierarchicalFacetsRefinements[n.name]&&e.hierarchicalFacetsRefinements[n.name][0]||"",h=e._getHierarchicalFacetSeparator(n),f=e._getHierarchicalRootPath(n),l=e._getHierarchicalShowParentLevel(n),m=s(e._getHierarchicalFacetSortBy(n)),d=t.every((function(e){return e.exhaustive})),p=function(e,t,r,n,s){return function(u,h,f){var l=u;if(f>0){var m=0;for(l=u;m{"use strict";var n=r(7004),i=r(7285),s=r(2914),a=r(2902),c=r(1072),o=r(742),u=r(1600),h=s.escapeFacetValue,f=s.unescapeFacetValue,l=r(7762);function m(e){var t={};return e.forEach((function(e,r){t[e]=r})),t}function d(e,t,r){t&&t[r]&&(e.stats=t[r])}function p(e,t,r){var s=t[0]||{};this._rawResults=t;var o=this;Object.keys(s).forEach((function(e){o[e]=s[e]}));var u=i(r,{persistHierarchicalRootCount:!1});Object.keys(u).forEach((function(e){o[e]=u[e]})),this.processingTimeMS=t.reduce((function(e,t){return void 0===t.processingTimeMS?e:e+t.processingTimeMS}),0),this.disjunctiveFacets=[],this.hierarchicalFacets=e.hierarchicalFacets.map((function(){return[]})),this.facets=[];var h=e.getRefinedDisjunctiveFacets(),p=m(e.facets),g=m(e.disjunctiveFacets),v=1,y=s.facets||{};Object.keys(y).forEach((function(t){var r,n,i=y[t],u=(r=e.hierarchicalFacets,n=t,a(r,(function(e){return(e.attributes||[]).indexOf(n)>-1})));if(u){var h=u.attributes.indexOf(t),f=c(e.hierarchicalFacets,(function(e){return e.name===u.name}));o.hierarchicalFacets[f][h]={attribute:t,data:i,exhaustive:s.exhaustiveFacetsCount}}else{var l,m=-1!==e.disjunctiveFacets.indexOf(t),v=-1!==e.facets.indexOf(t);m&&(l=g[t],o.disjunctiveFacets[l]={name:t,data:i,exhaustive:s.exhaustiveFacetsCount},d(o.disjunctiveFacets[l],s.facets_stats,t)),v&&(l=p[t],o.facets[l]={name:t,data:i,exhaustive:s.exhaustiveFacetsCount},d(o.facets[l],s.facets_stats,t))}})),this.hierarchicalFacets=n(this.hierarchicalFacets),h.forEach((function(r){var n=t[v],a=n&&n.facets?n.facets:{},u=e.getHierarchicalFacetByName(r);Object.keys(a).forEach((function(t){var r,h=a[t];if(u){r=c(e.hierarchicalFacets,(function(e){return e.name===u.name}));var l=c(o.hierarchicalFacets[r],(function(e){return e.attribute===t}));if(-1===l)return;o.hierarchicalFacets[r][l].data=o.persistHierarchicalRootCount?i(o.hierarchicalFacets[r][l].data,h):i(h,o.hierarchicalFacets[r][l].data)}else{r=g[t];var m=s.facets&&s.facets[t]||{};o.disjunctiveFacets[r]={name:t,data:i(m,h),exhaustive:n.exhaustiveFacetsCount},d(o.disjunctiveFacets[r],n.facets_stats,t),e.disjunctiveFacetsRefinements[t]&&e.disjunctiveFacetsRefinements[t].forEach((function(n){!o.disjunctiveFacets[r].data[n]&&e.disjunctiveFacetsRefinements[t].indexOf(f(n))>-1&&(o.disjunctiveFacets[r].data[n]=0)}))}})),v++})),e.getRefinedHierarchicalFacets().forEach((function(r){var n=e.getHierarchicalFacetByName(r),s=e._getHierarchicalFacetSeparator(n),a=e.getHierarchicalRefinement(r);0===a.length||a[0].split(s).length<2||t.slice(v).forEach((function(t){var r=t&&t.facets?t.facets:{};Object.keys(r).forEach((function(t){var u=r[t],h=c(e.hierarchicalFacets,(function(e){return e.name===n.name})),f=c(o.hierarchicalFacets[h],(function(e){return e.attribute===t}));if(-1!==f){var l={};if(a.length>0&&!o.persistHierarchicalRootCount){var m=a[0].split(s)[0];l[m]=o.hierarchicalFacets[h][f].data[m]}o.hierarchicalFacets[h][f].data=i(l,u,o.hierarchicalFacets[h][f].data)}})),v++}))})),Object.keys(e.facetsExcludes).forEach((function(t){var r=e.facetsExcludes[t],n=p[t];o.facets[n]={name:t,data:y[t],exhaustive:s.exhaustiveFacetsCount},r.forEach((function(e){o.facets[n]=o.facets[n]||{name:t},o.facets[n].data=o.facets[n].data||{},o.facets[n].data[e]=0}))})),this.hierarchicalFacets=this.hierarchicalFacets.map(l(e)),this.facets=n(this.facets),this.disjunctiveFacets=n(this.disjunctiveFacets),this._state=e}function g(e,t){function r(e){return e.name===t}if(e._state.isConjunctiveFacet(t)){var n=a(e.facets,r);return n?Object.keys(n.data).map((function(r){var i=h(r);return{name:r,escapedValue:i,count:n.data[r],isRefined:e._state.isFacetRefined(t,i),isExcluded:e._state.isExcludeRefined(t,r)}})):[]}if(e._state.isDisjunctiveFacet(t)){var i=a(e.disjunctiveFacets,r);return i?Object.keys(i.data).map((function(r){var n=h(r);return{name:r,escapedValue:n,count:i.data[r],isRefined:e._state.isDisjunctiveFacetRefined(t,n)}})):[]}if(e._state.isHierarchicalFacet(t)){var s=a(e.hierarchicalFacets,r);if(!s)return s;var c=e._state.getHierarchicalFacetByName(t),o=e._state._getHierarchicalFacetSeparator(c),u=f(e._state.getHierarchicalRefinement(t)[0]||"");0===u.indexOf(c.rootPath)&&(u=u.replace(c.rootPath+o,""));var l=u.split(o);return l.unshift(t),v(s,l,0),s}}function v(e,t,r){e.isRefined=e.name===(t[r]&&t[r].trim()),e.data&&e.data.forEach((function(e){v(e,t,r+1)}))}function y(e,t,r,n){if(n=n||0,Array.isArray(t))return e(t,r[n]);if(!t.data||0===t.data.length)return t;var s=t.data.map((function(t){return y(e,t,r,n+1)})),a=e(s,r[n]);return i({data:a},t)}function R(e,t){var r=a(e,(function(e){return e.name===t}));return r&&r.stats}function F(e,t,r,n,i){var s=a(i,(function(e){return e.name===r})),c=s&&s.data&&s.data[n]?s.data[n]:0,o=s&&s.exhaustive||!1;return{type:t,attributeName:r,name:n,count:c,exhaustive:o}}p.prototype.getFacetByName=function(e){function t(t){return t.name===e}return a(this.facets,t)||a(this.disjunctiveFacets,t)||a(this.hierarchicalFacets,t)},p.DEFAULT_SORT=["isRefined:desc","count:desc","name:asc"],p.prototype.getFacetValues=function(e,t){var r=g(this,e);if(r){var n,s=i(t,{sortBy:p.DEFAULT_SORT,facetOrdering:!(t&&t.sortBy)}),a=this;if(Array.isArray(r))n=[e];else n=a._state.getHierarchicalFacetByName(r.name).attributes;return y((function(e,t){if(s.facetOrdering){var r=function(e,t){return e.renderingContent&&e.renderingContent.facetOrdering&&e.renderingContent.facetOrdering.values&&e.renderingContent.facetOrdering.values[t]}(a,t);if(r)return function(e,t){var r=[],n=[],i=t.hide||[],s=(t.order||[]).reduce((function(e,t,r){return e[t]=r,e}),{});e.forEach((function(e){var t=e.path||e.name,a=i.indexOf(t)>-1;a||void 0===s[t]?a||n.push(e):r[s[t]]=e})),r=r.filter((function(e){return e}));var a,c=t.sortRemainingBy;return"hidden"===c?r:(a="alpha"===c?[["path","name"],["asc","asc"]]:[["count"],["desc"]],r.concat(u(n,a[0],a[1])))}(e,r)}if(Array.isArray(s.sortBy)){var n=o(s.sortBy,p.DEFAULT_SORT);return u(e,n[0],n[1])}if("function"==typeof s.sortBy)return function(e,t){return t.sort(e)}(s.sortBy,e);throw new Error("options.sortBy is optional but if defined it must be either an array of string (predicates) or a sorting function")}),r,n)}},p.prototype.getFacetStats=function(e){return this._state.isConjunctiveFacet(e)?R(this.facets,e):this._state.isDisjunctiveFacet(e)?R(this.disjunctiveFacets,e):void 0},p.prototype.getRefinements=function(){var e=this._state,t=this,r=[];return Object.keys(e.facetsRefinements).forEach((function(n){e.facetsRefinements[n].forEach((function(i){r.push(F(e,"facet",n,i,t.facets))}))})),Object.keys(e.facetsExcludes).forEach((function(n){e.facetsExcludes[n].forEach((function(i){r.push(F(e,"exclude",n,i,t.facets))}))})),Object.keys(e.disjunctiveFacetsRefinements).forEach((function(n){e.disjunctiveFacetsRefinements[n].forEach((function(i){r.push(F(e,"disjunctive",n,i,t.disjunctiveFacets))}))})),Object.keys(e.hierarchicalFacetsRefinements).forEach((function(n){e.hierarchicalFacetsRefinements[n].forEach((function(i){r.push(function(e,t,r,n){var i=e.getHierarchicalFacetByName(t),s=e._getHierarchicalFacetSeparator(i),c=r.split(s),o=a(n,(function(e){return e.name===t})),u=c.reduce((function(e,t){var r=e&&a(e.data,(function(e){return e.name===t}));return void 0!==r?r:e}),o),h=u&&u.count||0,f=u&&u.exhaustive||!1,l=u&&u.path||"";return{type:"hierarchical",attributeName:t,name:l,count:h,exhaustive:f}}(e,n,i,t.hierarchicalFacets))}))})),Object.keys(e.numericRefinements).forEach((function(t){var n=e.numericRefinements[t];Object.keys(n).forEach((function(e){n[e].forEach((function(n){r.push({type:"numeric",attributeName:t,name:n,numericValue:n,operator:e})}))}))})),e.tagRefinements.forEach((function(e){r.push({type:"tag",attributeName:"_tags",name:e})})),r},e.exports=p},6924:(e,t,r)=>{"use strict";var n=r(2733),i=r(1817),s=r(2914).escapeFacetValue,a=r(1641),c=r(5901),o=r(9628),u=r(3416),h=r(8110),f=r(4824),l=r(9827),m=r(6872),d=r(1974),p=r(2164),g=r(3235);function v(e,t,r,n){"function"==typeof e.addAlgoliaAgent&&e.addAlgoliaAgent("JS Helper ("+g+")"),this.setClient(e);var i=r||{};i.index=t,this.state=m.make(i),this.recommendState=new h({params:i.recommendState}),this.lastResults=null,this.lastRecommendResults=null,this._queryId=0,this._recommendQueryId=0,this._lastQueryIdReceived=-1,this._lastRecommendQueryIdReceived=-1,this.derivedHelpers=[],this._currentNbQueries=0,this._currentNbRecommendQueries=0,this._searchResultsOptions=n,this._recommendCache={}}function y(e){if(e<0)throw new Error("Page requested below 0.");return this._change({state:this.state.setPage(e),isPageReset:!1}),this}function R(){return this.state.page}a(v,n),v.prototype.search=function(){return this._search({onlyWithDerivedHelpers:!1}),this},v.prototype.searchOnlyWithDerivedHelpers=function(){return this._search({onlyWithDerivedHelpers:!0}),this},v.prototype.recommend=function(){return this._recommend(),this},v.prototype.getQuery=function(){var e=this.state;return l._getHitsSearchParams(e)},v.prototype.searchOnce=function(e,t){var r=e?this.state.setQueryParameters(e):this.state,n=l._getQueries(r.index,r),i=this;if(this._currentNbQueries++,this.emit("searchOnce",{state:r}),!t)return this.client.search(n).then((function(e){return i._currentNbQueries--,0===i._currentNbQueries&&i.emit("searchQueueEmpty"),{content:new d(r,e.results),state:r,_originalResponse:e}}),(function(e){throw i._currentNbQueries--,0===i._currentNbQueries&&i.emit("searchQueueEmpty"),e}));this.client.search(n).then((function(e){i._currentNbQueries--,0===i._currentNbQueries&&i.emit("searchQueueEmpty"),t(null,new d(r,e.results),r)})).catch((function(e){i._currentNbQueries--,0===i._currentNbQueries&&i.emit("searchQueueEmpty"),t(e,null,r)}))},v.prototype.findAnswers=function(e){console.warn("[algoliasearch-helper] answers is no longer supported");var t=this.state,r=this.derivedHelpers[0];if(!r)return Promise.resolve([]);var n=r.getModifiedState(t),i=c({attributesForPrediction:e.attributesForPrediction,nbHits:e.nbHits},{params:u(l._getHitsSearchParams(n),["attributesToSnippet","hitsPerPage","restrictSearchableAttributes","snippetEllipsisText"])}),s="search for answers was called, but this client does not have a function client.initIndex(index).findAnswers";if("function"!=typeof this.client.initIndex)throw new Error(s);var a=this.client.initIndex(n.index);if("function"!=typeof a.findAnswers)throw new Error(s);return a.findAnswers(n.query,e.queryLanguages,i)},v.prototype.searchForFacetValues=function(e,t,r,n){var i="function"==typeof this.client.searchForFacetValues&&"function"!=typeof this.client.searchForFacets,a="function"==typeof this.client.initIndex;if(!i&&!a&&"function"!=typeof this.client.search)throw new Error("search for facet values (searchable) was called, but this client does not have a function client.searchForFacetValues or client.initIndex(index).searchForFacetValues");var c=this.state.setQueryParameters(n||{}),o=c.isDisjunctiveFacet(e),u=l.getSearchForFacetQuery(e,t,r,c);this._currentNbQueries++;var h,f=this;return i?h=this.client.searchForFacetValues([{indexName:c.index,params:u}]):a?h=this.client.initIndex(c.index).searchForFacetValues(u):(delete u.facetName,h=this.client.search([{type:"facet",facet:e,indexName:c.index,params:u}]).then((function(e){return e.results[0]}))),this.emit("searchForFacetValues",{state:c,facet:e,query:t}),h.then((function(t){return f._currentNbQueries--,0===f._currentNbQueries&&f.emit("searchQueueEmpty"),(t=Array.isArray(t)?t[0]:t).facetHits.forEach((function(t){t.escapedValue=s(t.value),t.isRefined=o?c.isDisjunctiveFacetRefined(e,t.escapedValue):c.isFacetRefined(e,t.escapedValue)})),t}),(function(e){throw f._currentNbQueries--,0===f._currentNbQueries&&f.emit("searchQueueEmpty"),e}))},v.prototype.setQuery=function(e){return this._change({state:this.state.resetPage().setQuery(e),isPageReset:!0}),this},v.prototype.clearRefinements=function(e){return this._change({state:this.state.resetPage().clearRefinements(e),isPageReset:!0}),this},v.prototype.clearTags=function(){return this._change({state:this.state.resetPage().clearTags(),isPageReset:!0}),this},v.prototype.addDisjunctiveFacetRefinement=function(e,t){return this._change({state:this.state.resetPage().addDisjunctiveFacetRefinement(e,t),isPageReset:!0}),this},v.prototype.addDisjunctiveRefine=function(){return this.addDisjunctiveFacetRefinement.apply(this,arguments)},v.prototype.addHierarchicalFacetRefinement=function(e,t){return this._change({state:this.state.resetPage().addHierarchicalFacetRefinement(e,t),isPageReset:!0}),this},v.prototype.addNumericRefinement=function(e,t,r){return this._change({state:this.state.resetPage().addNumericRefinement(e,t,r),isPageReset:!0}),this},v.prototype.addFacetRefinement=function(e,t){return this._change({state:this.state.resetPage().addFacetRefinement(e,t),isPageReset:!0}),this},v.prototype.addRefine=function(){return this.addFacetRefinement.apply(this,arguments)},v.prototype.addFacetExclusion=function(e,t){return this._change({state:this.state.resetPage().addExcludeRefinement(e,t),isPageReset:!0}),this},v.prototype.addExclude=function(){return this.addFacetExclusion.apply(this,arguments)},v.prototype.addTag=function(e){return this._change({state:this.state.resetPage().addTagRefinement(e),isPageReset:!0}),this},v.prototype.addFrequentlyBoughtTogether=function(e){return this._recommendChange({state:this.recommendState.addFrequentlyBoughtTogether(e)}),this},v.prototype.addRelatedProducts=function(e){return this._recommendChange({state:this.recommendState.addRelatedProducts(e)}),this},v.prototype.addTrendingItems=function(e){return this._recommendChange({state:this.recommendState.addTrendingItems(e)}),this},v.prototype.addTrendingFacets=function(e){return this._recommendChange({state:this.recommendState.addTrendingFacets(e)}),this},v.prototype.addLookingSimilar=function(e){return this._recommendChange({state:this.recommendState.addLookingSimilar(e)}),this},v.prototype.removeNumericRefinement=function(e,t,r){return this._change({state:this.state.resetPage().removeNumericRefinement(e,t,r),isPageReset:!0}),this},v.prototype.removeDisjunctiveFacetRefinement=function(e,t){return this._change({state:this.state.resetPage().removeDisjunctiveFacetRefinement(e,t),isPageReset:!0}),this},v.prototype.removeDisjunctiveRefine=function(){return this.removeDisjunctiveFacetRefinement.apply(this,arguments)},v.prototype.removeHierarchicalFacetRefinement=function(e){return this._change({state:this.state.resetPage().removeHierarchicalFacetRefinement(e),isPageReset:!0}),this},v.prototype.removeFacetRefinement=function(e,t){return this._change({state:this.state.resetPage().removeFacetRefinement(e,t),isPageReset:!0}),this},v.prototype.removeRefine=function(){return this.removeFacetRefinement.apply(this,arguments)},v.prototype.removeFacetExclusion=function(e,t){return this._change({state:this.state.resetPage().removeExcludeRefinement(e,t),isPageReset:!0}),this},v.prototype.removeExclude=function(){return this.removeFacetExclusion.apply(this,arguments)},v.prototype.removeTag=function(e){return this._change({state:this.state.resetPage().removeTagRefinement(e),isPageReset:!0}),this},v.prototype.removeFrequentlyBoughtTogether=function(e){return this._recommendChange({state:this.recommendState.removeParams(e)}),this},v.prototype.removeRelatedProducts=function(e){return this._recommendChange({state:this.recommendState.removeParams(e)}),this},v.prototype.removeTrendingItems=function(e){return this._recommendChange({state:this.recommendState.removeParams(e)}),this},v.prototype.removeTrendingFacets=function(e){return this._recommendChange({state:this.recommendState.removeParams(e)}),this},v.prototype.removeLookingSimilar=function(e){return this._recommendChange({state:this.recommendState.removeParams(e)}),this},v.prototype.toggleFacetExclusion=function(e,t){return this._change({state:this.state.resetPage().toggleExcludeFacetRefinement(e,t),isPageReset:!0}),this},v.prototype.toggleExclude=function(){return this.toggleFacetExclusion.apply(this,arguments)},v.prototype.toggleRefinement=function(e,t){return this.toggleFacetRefinement(e,t)},v.prototype.toggleFacetRefinement=function(e,t){return this._change({state:this.state.resetPage().toggleFacetRefinement(e,t),isPageReset:!0}),this},v.prototype.toggleRefine=function(){return this.toggleFacetRefinement.apply(this,arguments)},v.prototype.toggleTag=function(e){return this._change({state:this.state.resetPage().toggleTagRefinement(e),isPageReset:!0}),this},v.prototype.nextPage=function(){var e=this.state.page||0;return this.setPage(e+1)},v.prototype.previousPage=function(){var e=this.state.page||0;return this.setPage(e-1)},v.prototype.setCurrentPage=y,v.prototype.setPage=y,v.prototype.setIndex=function(e){return this._change({state:this.state.resetPage().setIndex(e),isPageReset:!0}),this},v.prototype.setQueryParameter=function(e,t){return this._change({state:this.state.resetPage().setQueryParameter(e,t),isPageReset:!0}),this},v.prototype.setState=function(e){return this._change({state:m.make(e),isPageReset:!1}),this},v.prototype.overrideStateWithoutTriggeringChangeEvent=function(e){return this.state=new m(e),this},v.prototype.hasRefinements=function(e){return!!o(this.state.getNumericRefinements(e))||(this.state.isConjunctiveFacet(e)?this.state.isFacetRefined(e):this.state.isDisjunctiveFacet(e)?this.state.isDisjunctiveFacetRefined(e):!!this.state.isHierarchicalFacet(e)&&this.state.isHierarchicalFacetRefined(e))},v.prototype.isExcluded=function(e,t){return this.state.isExcludeRefined(e,t)},v.prototype.isDisjunctiveRefined=function(e,t){return this.state.isDisjunctiveFacetRefined(e,t)},v.prototype.hasTag=function(e){return this.state.isTagRefined(e)},v.prototype.isTagRefined=function(){return this.hasTagRefinements.apply(this,arguments)},v.prototype.getIndex=function(){return this.state.index},v.prototype.getCurrentPage=R,v.prototype.getPage=R,v.prototype.getTags=function(){return this.state.tagRefinements},v.prototype.getRefinements=function(e){var t=[];if(this.state.isConjunctiveFacet(e))this.state.getConjunctiveRefinements(e).forEach((function(e){t.push({value:e,type:"conjunctive"})})),this.state.getExcludeRefinements(e).forEach((function(e){t.push({value:e,type:"exclude"})}));else if(this.state.isDisjunctiveFacet(e)){this.state.getDisjunctiveRefinements(e).forEach((function(e){t.push({value:e,type:"disjunctive"})}))}var r=this.state.getNumericRefinements(e);return Object.keys(r).forEach((function(e){var n=r[e];t.push({value:n,operator:e,type:"numeric"})})),t},v.prototype.getNumericRefinement=function(e,t){return this.state.getNumericRefinement(e,t)},v.prototype.getHierarchicalFacetBreadcrumb=function(e){return this.state.getHierarchicalFacetBreadcrumb(e)},v.prototype._search=function(e){var t=this.state,r=[],n=[];e.onlyWithDerivedHelpers||(n=l._getQueries(t.index,t),r.push({state:t,queriesCount:n.length,helper:this}),this.emit("search",{state:t,results:this.lastResults}));var i=this.derivedHelpers.map((function(e){var n=e.getModifiedState(t),i=n.index?l._getQueries(n.index,n):[];return r.push({state:n,queriesCount:i.length,helper:e}),e.emit("search",{state:n,results:e.lastResults}),i})),s=Array.prototype.concat.apply(n,i),a=this._queryId++;if(this._currentNbQueries++,!s.length)return 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t=this.graphManager.getLayout();this.displacementX=t.coolingFactor*(this.springForceX+this.repulsionForceX+this.gravitationForceX)/this.noOfChildren,this.displacementY=t.coolingFactor*(this.springForceY+this.repulsionForceY+this.gravitationForceY)/this.noOfChildren,Math.abs(this.displacementX)>t.coolingFactor*t.maxNodeDisplacement&&(this.displacementX=t.coolingFactor*t.maxNodeDisplacement*r.sign(this.displacementX)),Math.abs(this.displacementY)>t.coolingFactor*t.maxNodeDisplacement&&(this.displacementY=t.coolingFactor*t.maxNodeDisplacement*r.sign(this.displacementY)),null==this.child||0==this.child.getNodes().length?this.moveBy(this.displacementX,this.displacementY):this.propogateDisplacementToChildren(this.displacementX,this.displacementY),t.totalDisplacement+=Math.abs(this.displacementX)+Math.abs(this.displacementY),this.springForceX=0,this.springForceY=0,this.repulsionForceX=0,this.repulsionForceY=0,this.gravitationForceX=0,this.gravitationForceY=0,this.displacementX=0,this.displacementY=0},o.prototype.propogateDisplacementToChildren=function(t,e){for(var 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this.initSpringEmbedder(),this.runSpringEmbedder(),!0},_.prototype.tick=function(){if(this.totalIterations++,this.totalIterations===this.maxIterations&&!this.isTreeGrowing&&!this.isGrowthFinished){if(!(this.prunedNodesAll.length>0))return!0;this.isTreeGrowing=!0}if(this.totalIterations%l.CONVERGENCE_CHECK_PERIOD==0&&!this.isTreeGrowing&&!this.isGrowthFinished){if(this.isConverged()){if(!(this.prunedNodesAll.length>0))return!0;this.isTreeGrowing=!0}this.coolingCycle++,0==this.layoutQuality?this.coolingAdjuster=this.coolingCycle:1==this.layoutQuality&&(this.coolingAdjuster=this.coolingCycle/3),this.coolingFactor=Math.max(this.initialCoolingFactor-Math.pow(this.coolingCycle,Math.log(100*(this.initialCoolingFactor-this.finalTemperature))/Math.log(this.maxCoolingCycle))/100*this.coolingAdjuster,this.finalTemperature),this.animationPeriod=Math.ceil(this.initialAnimationPeriod*Math.sqrt(this.coolingFactor))}if(this.isTreeGrowing){if(this.growTreeIterations%10==0)if(this.prunedNodesAll.length>0){this.graphManager.updateBounds(),this.updateGrid(),this.growTree(this.prunedNodesAll),this.graphManager.resetAllNodesToApplyGravitation();var t=new Set(this.getAllNodes()),e=this.nodesWithGravity.filter((function(e){return t.has(e)}));this.graphManager.setAllNodesToApplyGravitation(e),this.graphManager.updateBounds(),this.updateGrid(),this.coolingFactor=l.DEFAULT_COOLING_FACTOR_INCREMENTAL}else this.isTreeGrowing=!1,this.isGrowthFinished=!0;this.growTreeIterations++}if(this.isGrowthFinished){if(this.isConverged())return!0;this.afterGrowthIterations%10==0&&(this.graphManager.updateBounds(),this.updateGrid()),this.coolingFactor=l.DEFAULT_COOLING_FACTOR_INCREMENTAL*((100-this.afterGrowthIterations)/100),this.afterGrowthIterations++}var i=!this.isTreeGrowing&&!this.isGrowthFinished,n=this.growTreeIterations%10==1&&this.isTreeGrowing||this.afterGrowthIterations%10==1&&this.isGrowthFinished;return this.totalDisplacement=0,this.graphManager.updateBounds(),this.calcSpringForces(),this.calcRepulsionForces(i,n),this.calcGravitationalForces(),this.moveNodes(),this.animate(),!1},_.prototype.getPositionsData=function(){for(var t=this.graphManager.getAllNodes(),e={},i=0;i1)for(a=0;an&&(n=Math.floor(s.y)),o=Math.floor(s.x+h.DEFAULT_COMPONENT_SEPERATION)}this.transform(new u(c.WORLD_CENTER_X-s.x/2,c.WORLD_CENTER_Y-s.y/2))},_.radialLayout=function(t,e,i){var n=Math.max(this.maxDiagonalInTree(t),h.DEFAULT_RADIAL_SEPARATION);_.branchRadialLayout(e,null,0,359,0,n);var r=y.calculateBounds(t),o=new E;o.setDeviceOrgX(r.getMinX()),o.setDeviceOrgY(r.getMinY()),o.setWorldOrgX(i.x),o.setWorldOrgY(i.y);for(var s=0;s1;){var E=y[0];y.splice(0,1);var m=c.indexOf(E);m>=0&&c.splice(m,1),p--,g--}u=null!=e?(c.indexOf(y[0])+1)%p:0;for(var v=Math.abs(n-i)/g,N=u;d!=g;N=++N%p){var A=c[N].getOtherEnd(t);if(A!=e){var L=(i+d*v)%360,T=(L+v)%360;_.branchRadialLayout(A,t,L,T,r+o,o),d++}}},_.maxDiagonalInTree=function(t){for(var e=p.MIN_VALUE,i=0;ie&&(e=n)}return e},_.prototype.calcRepulsionRange=function(){return 2*(this.level+1)*this.idealEdgeLength},_.prototype.groupZeroDegreeMembers=function(){var t=this,e={};this.memberGroups={},this.idToDummyNode={};for(var i=[],n=this.graphManager.getAllNodes(),r=0;r1){var n="DummyCompound_"+i;t.memberGroups[n]=e[i];var r=e[i][0].getParent(),o=new s(t.graphManager);o.id=n,o.paddingLeft=r.paddingLeft||0,o.paddingRight=r.paddingRight||0,o.paddingBottom=r.paddingBottom||0,o.paddingTop=r.paddingTop||0,t.idToDummyNode[n]=o;var a=t.getGraphManager().add(t.newGraph(),o),h=r.getChild();h.add(o);for(var l=0;l=0;t--){var e=this.compoundOrder[t],i=e.id,n=e.paddingLeft,r=e.paddingTop;this.adjustLocations(this.tiledMemberPack[i],e.rect.x,e.rect.y,n,r)}},_.prototype.repopulateZeroDegreeMembers=function(){var t=this,e=this.tiledZeroDegreePack;Object.keys(e).forEach((function(i){var n=t.idToDummyNode[i],r=n.paddingLeft,o=n.paddingTop;t.adjustLocations(e[i],n.rect.x,n.rect.y,r,o)}))},_.prototype.getToBeTiled=function(t){var e=t.id;if(null!=this.toBeTiled[e])return this.toBeTiled[e];var i=t.getChild();if(null==i)return this.toBeTiled[e]=!1,!1;for(var n=i.getNodes(),r=0;r0)return this.toBeTiled[e]=!1,!1;if(null!=o.getChild()){if(!this.getToBeTiled(o))return this.toBeTiled[e]=!1,!1}else this.toBeTiled[o.id]=!1}return this.toBeTiled[e]=!0,!0},_.prototype.getNodeDegree=function(t){t.id;for(var e=t.getEdges(),i=0,n=0;nh&&(h=c.rect.height)}i+=h+t.verticalPadding}},_.prototype.tileCompoundMembers=function(t,e){var i=this;this.tiledMemberPack=[],Object.keys(t).forEach((function(n){var r=e[n];i.tiledMemberPack[n]=i.tileNodes(t[n],r.paddingLeft+r.paddingRight),r.rect.width=i.tiledMemberPack[n].width,r.rect.height=i.tiledMemberPack[n].height}))},_.prototype.tileNodes=function(t,e){var i={rows:[],rowWidth:[],rowHeight:[],width:0,height:e,verticalPadding:h.TILING_PADDING_VERTICAL,horizontalPadding:h.TILING_PADDING_HORIZONTAL};t.sort((function(t,e){return t.rect.width*t.rect.height>e.rect.width*e.rect.height?-1:t.rect.width*t.rect.height0&&(o+=t.horizontalPadding),t.rowWidth[i]=o,t.width0&&(s+=t.verticalPadding);var a=0;s>t.rowHeight[i]&&(a=t.rowHeight[i],t.rowHeight[i]=s,a=t.rowHeight[i]-a),t.height+=a,t.rows[i].push(e)},_.prototype.getShortestRowIndex=function(t){for(var e=-1,i=Number.MAX_VALUE,n=0;ni&&(e=n,i=t.rowWidth[n]);return e},_.prototype.canAddHorizontal=function(t,e,i){var n=this.getShortestRowIndex(t);if(n<0)return!0;var r=t.rowWidth[n];if(r+t.horizontalPadding+e<=t.width)return!0;var o,s,a=0;return t.rowHeight[n]0&&(a=i+t.verticalPadding-t.rowHeight[n]),o=t.width-r>=e+t.horizontalPadding?(t.height+a)/(r+e+t.horizontalPadding):(t.height+a)/t.width,a=i+t.verticalPadding,(s=t.widtho&&e!=i){n.splice(-1,1),t.rows[i].push(r),t.rowWidth[e]=t.rowWidth[e]-o,t.rowWidth[i]=t.rowWidth[i]+o,t.width=t.rowWidth[instance.getLongestRowIndex(t)];for(var s=Number.MIN_VALUE,a=0;as&&(s=n[a].height);e>0&&(s+=t.verticalPadding);var h=t.rowHeight[e]+t.rowHeight[i];t.rowHeight[e]=s,t.rowHeight[i]0)for(var c=r;c<=o;c++)h[0]+=this.grid[c][s-1].length+this.grid[c][s].length-1;if(o0)for(c=s;c<=a;c++)h[3]+=this.grid[r-1][c].length+this.grid[r][c].length-1;for(var g,u,d=p.MAX_VALUE,f=0;f0&&(s=i.getGraphManager().add(i.newGraph(),o),this.processChildrenList(s,g,i))}},g.prototype.stop=function(){return this.stopped=!0,this};var d=function(t){t("layout","cose-bilkent",g)};"undefined"!=typeof cytoscape&&d(cytoscape),t.exports=d}])},t.exports=n(i(7799))},3143:function(t){var e;e=function(){return function(t){var e={};function i(n){if(e[n])return e[n].exports;var r=e[n]={i:n,l:!1,exports:{}};return t[n].call(r.exports,r,r.exports,i),r.l=!0,r.exports}return i.m=t,i.c=e,i.i=function(t){return t},i.d=function(t,e,n){i.o(t,e)||Object.defineProperty(t,e,{configurable:!1,enumerable:!0,get:n})},i.n=function(t){var e=t&&t.__esModule?function(){return t.default}:function(){return t};return i.d(e,"a",e),e},i.o=function(t,e){return Object.prototype.hasOwnProperty.call(t,e)},i.p="",i(i.s=26)}([function(t,e,i){"use strict";function n(){}n.QUALITY=1,n.DEFAULT_CREATE_BENDS_AS_NEEDED=!1,n.DEFAULT_INCREMENTAL=!1,n.DEFAULT_ANIMATION_ON_LAYOUT=!0,n.DEFAULT_ANIMATION_DURING_LAYOUT=!1,n.DEFAULT_ANIMATION_PERIOD=50,n.DEFAULT_UNIFORM_LEAF_NODE_SIZES=!1,n.DEFAULT_GRAPH_MARGIN=15,n.NODE_DIMENSIONS_INCLUDE_LABELS=!1,n.SIMPLE_NODE_SIZE=40,n.SIMPLE_NODE_HALF_SIZE=n.SIMPLE_NODE_SIZE/2,n.EMPTY_COMPOUND_NODE_SIZE=40,n.MIN_EDGE_LENGTH=1,n.WORLD_BOUNDARY=1e6,n.INITIAL_WORLD_BOUNDARY=n.WORLD_BOUNDARY/1e3,n.WORLD_CENTER_X=1200,n.WORLD_CENTER_Y=900,t.exports=n},function(t,e,i){"use strict";var n=i(2),r=i(8),o=i(9);function s(t,e,i){n.call(this,i),this.isOverlapingSourceAndTarget=!1,this.vGraphObject=i,this.bendpoints=[],this.source=t,this.target=e}for(var a in s.prototype=Object.create(n.prototype),n)s[a]=n[a];s.prototype.getSource=function(){return this.source},s.prototype.getTarget=function(){return this.target},s.prototype.isInterGraph=function(){return this.isInterGraph},s.prototype.getLength=function(){return this.length},s.prototype.isOverlapingSourceAndTarget=function(){return this.isOverlapingSourceAndTarget},s.prototype.getBendpoints=function(){return this.bendpoints},s.prototype.getLca=function(){return this.lca},s.prototype.getSourceInLca=function(){return this.sourceInLca},s.prototype.getTargetInLca=function(){return this.targetInLca},s.prototype.getOtherEnd=function(t){if(this.source===t)return this.target;if(this.target===t)return this.source;throw"Node is not incident with this edge"},s.prototype.getOtherEndInGraph=function(t,e){for(var i=this.getOtherEnd(t),n=e.getGraphManager().getRoot();;){if(i.getOwner()==e)return i;if(i.getOwner()==n)break;i=i.getOwner().getParent()}return null},s.prototype.updateLength=function(){var t=new Array(4);this.isOverlapingSourceAndTarget=r.getIntersection(this.target.getRect(),this.source.getRect(),t),this.isOverlapingSourceAndTarget||(this.lengthX=t[0]-t[2],this.lengthY=t[1]-t[3],Math.abs(this.lengthX)<1&&(this.lengthX=o.sign(this.lengthX)),Math.abs(this.lengthY)<1&&(this.lengthY=o.sign(this.lengthY)),this.length=Math.sqrt(this.lengthX*this.lengthX+this.lengthY*this.lengthY))},s.prototype.updateLengthSimple=function(){this.lengthX=this.target.getCenterX()-this.source.getCenterX(),this.lengthY=this.target.getCenterY()-this.source.getCenterY(),Math.abs(this.lengthX)<1&&(this.lengthX=o.sign(this.lengthX)),Math.abs(this.lengthY)<1&&(this.lengthY=o.sign(this.lengthY)),this.length=Math.sqrt(this.lengthX*this.lengthX+this.lengthY*this.lengthY)},t.exports=s},function(t,e,i){"use strict";t.exports=function(t){this.vGraphObject=t}},function(t,e,i){"use strict";var n=i(2),r=i(10),o=i(13),s=i(0),a=i(16),h=i(4);function l(t,e,i,s){null==i&&null==s&&(s=e),n.call(this,s),null!=t.graphManager&&(t=t.graphManager),this.estimatedSize=r.MIN_VALUE,this.inclusionTreeDepth=r.MAX_VALUE,this.vGraphObject=s,this.edges=[],this.graphManager=t,this.rect=null!=i&&null!=e?new o(e.x,e.y,i.width,i.height):new o}for(var c in l.prototype=Object.create(n.prototype),n)l[c]=n[c];l.prototype.getEdges=function(){return this.edges},l.prototype.getChild=function(){return this.child},l.prototype.getOwner=function(){return this.owner},l.prototype.getWidth=function(){return this.rect.width},l.prototype.setWidth=function(t){this.rect.width=t},l.prototype.getHeight=function(){return this.rect.height},l.prototype.setHeight=function(t){this.rect.height=t},l.prototype.getCenterX=function(){return this.rect.x+this.rect.width/2},l.prototype.getCenterY=function(){return this.rect.y+this.rect.height/2},l.prototype.getCenter=function(){return new h(this.rect.x+this.rect.width/2,this.rect.y+this.rect.height/2)},l.prototype.getLocation=function(){return new h(this.rect.x,this.rect.y)},l.prototype.getRect=function(){return this.rect},l.prototype.getDiagonal=function(){return Math.sqrt(this.rect.width*this.rect.width+this.rect.height*this.rect.height)},l.prototype.getHalfTheDiagonal=function(){return Math.sqrt(this.rect.height*this.rect.height+this.rect.width*this.rect.width)/2},l.prototype.setRect=function(t,e){this.rect.x=t.x,this.rect.y=t.y,this.rect.width=e.width,this.rect.height=e.height},l.prototype.setCenter=function(t,e){this.rect.x=t-this.rect.width/2,this.rect.y=e-this.rect.height/2},l.prototype.setLocation=function(t,e){this.rect.x=t,this.rect.y=e},l.prototype.moveBy=function(t,e){this.rect.x+=t,this.rect.y+=e},l.prototype.getEdgeListToNode=function(t){var e=[],i=this;return i.edges.forEach((function(n){if(n.target==t){if(n.source!=i)throw"Incorrect edge source!";e.push(n)}})),e},l.prototype.getEdgesBetween=function(t){var e=[],i=this;return i.edges.forEach((function(n){if(n.source!=i&&n.target!=i)throw"Incorrect edge source and/or target";n.target!=t&&n.source!=t||e.push(n)})),e},l.prototype.getNeighborsList=function(){var t=new Set,e=this;return e.edges.forEach((function(i){if(i.source==e)t.add(i.target);else{if(i.target!=e)throw"Incorrect incidency!";t.add(i.source)}})),t},l.prototype.withChildren=function(){var t=new Set;if(t.add(this),null!=this.child)for(var e=this.child.getNodes(),i=0;ie&&(this.rect.x-=(this.labelWidth-e)/2,this.setWidth(this.labelWidth)),this.labelHeight>i&&("center"==this.labelPos?this.rect.y-=(this.labelHeight-i)/2:"top"==this.labelPos&&(this.rect.y-=this.labelHeight-i),this.setHeight(this.labelHeight))}}},l.prototype.getInclusionTreeDepth=function(){if(this.inclusionTreeDepth==r.MAX_VALUE)throw"assert failed";return this.inclusionTreeDepth},l.prototype.transform=function(t){var e=this.rect.x;e>s.WORLD_BOUNDARY?e=s.WORLD_BOUNDARY:e<-s.WORLD_BOUNDARY&&(e=-s.WORLD_BOUNDARY);var i=this.rect.y;i>s.WORLD_BOUNDARY?i=s.WORLD_BOUNDARY:i<-s.WORLD_BOUNDARY&&(i=-s.WORLD_BOUNDARY);var n=new h(e,i),r=t.inverseTransformPoint(n);this.setLocation(r.x,r.y)},l.prototype.getLeft=function(){return this.rect.x},l.prototype.getRight=function(){return this.rect.x+this.rect.width},l.prototype.getTop=function(){return this.rect.y},l.prototype.getBottom=function(){return this.rect.y+this.rect.height},l.prototype.getParent=function(){return null==this.owner?null:this.owner.getParent()},t.exports=l},function(t,e,i){"use strict";function n(t,e){null==t&&null==e?(this.x=0,this.y=0):(this.x=t,this.y=e)}n.prototype.getX=function(){return this.x},n.prototype.getY=function(){return this.y},n.prototype.setX=function(t){this.x=t},n.prototype.setY=function(t){this.y=t},n.prototype.getDifference=function(t){return new DimensionD(this.x-t.x,this.y-t.y)},n.prototype.getCopy=function(){return new n(this.x,this.y)},n.prototype.translate=function(t){return this.x+=t.width,this.y+=t.height,this},t.exports=n},function(t,e,i){"use strict";var n=i(2),r=i(10),o=i(0),s=i(6),a=i(3),h=i(1),l=i(13),c=i(12),g=i(11);function u(t,e,i){n.call(this,i),this.estimatedSize=r.MIN_VALUE,this.margin=o.DEFAULT_GRAPH_MARGIN,this.edges=[],this.nodes=[],this.isConnected=!1,this.parent=t,null!=e&&e instanceof s?this.graphManager=e:null!=e&&e instanceof Layout&&(this.graphManager=e.graphManager)}for(var d in u.prototype=Object.create(n.prototype),n)u[d]=n[d];u.prototype.getNodes=function(){return this.nodes},u.prototype.getEdges=function(){return this.edges},u.prototype.getGraphManager=function(){return this.graphManager},u.prototype.getParent=function(){return this.parent},u.prototype.getLeft=function(){return this.left},u.prototype.getRight=function(){return this.right},u.prototype.getTop=function(){return this.top},u.prototype.getBottom=function(){return this.bottom},u.prototype.isConnected=function(){return this.isConnected},u.prototype.add=function(t,e,i){if(null==e&&null==i){var n=t;if(null==this.graphManager)throw"Graph has no graph mgr!";if(this.getNodes().indexOf(n)>-1)throw"Node already in graph!";return n.owner=this,this.getNodes().push(n),n}var r=t;if(!(this.getNodes().indexOf(e)>-1&&this.getNodes().indexOf(i)>-1))throw"Source or target not in graph!";if(e.owner!=i.owner||e.owner!=this)throw"Both owners must be this graph!";return e.owner!=i.owner?null:(r.source=e,r.target=i,r.isInterGraph=!1,this.getEdges().push(r),e.edges.push(r),i!=e&&i.edges.push(r),r)},u.prototype.remove=function(t){var e=t;if(t instanceof a){if(null==e)throw"Node is null!";if(null==e.owner||e.owner!=this)throw"Owner graph is invalid!";if(null==this.graphManager)throw"Owner graph manager is invalid!";for(var i=e.edges.slice(),n=i.length,r=0;r-1&&c>-1))throw"Source and/or target doesn't know this edge!";if(o.source.edges.splice(l,1),o.target!=o.source&&o.target.edges.splice(c,1),-1==(s=o.source.owner.getEdges().indexOf(o)))throw"Not in owner's edge list!";o.source.owner.getEdges().splice(s,1)}},u.prototype.updateLeftTop=function(){for(var t,e,i,n=r.MAX_VALUE,o=r.MAX_VALUE,s=this.getNodes(),a=s.length,h=0;h(t=l.getTop())&&(n=t),o>(e=l.getLeft())&&(o=e)}return n==r.MAX_VALUE?null:(i=null!=s[0].getParent().paddingLeft?s[0].getParent().paddingLeft:this.margin,this.left=o-i,this.top=n-i,new c(this.left,this.top))},u.prototype.updateBounds=function(t){for(var e,i,n,o,s,a=r.MAX_VALUE,h=-r.MAX_VALUE,c=r.MAX_VALUE,g=-r.MAX_VALUE,u=this.nodes,d=u.length,p=0;p(e=f.getLeft())&&(a=e),h<(i=f.getRight())&&(h=i),c>(n=f.getTop())&&(c=n),g<(o=f.getBottom())&&(g=o)}var y=new l(a,c,h-a,g-c);a==r.MAX_VALUE&&(this.left=this.parent.getLeft(),this.right=this.parent.getRight(),this.top=this.parent.getTop(),this.bottom=this.parent.getBottom()),s=null!=u[0].getParent().paddingLeft?u[0].getParent().paddingLeft:this.margin,this.left=y.x-s,this.right=y.x+y.width+s,this.top=y.y-s,this.bottom=y.y+y.height+s},u.calculateBounds=function(t){for(var e,i,n,o,s=r.MAX_VALUE,a=-r.MAX_VALUE,h=r.MAX_VALUE,c=-r.MAX_VALUE,g=t.length,u=0;u(e=d.getLeft())&&(s=e),a<(i=d.getRight())&&(a=i),h>(n=d.getTop())&&(h=n),c<(o=d.getBottom())&&(c=o)}return new l(s,h,a-s,c-h)},u.prototype.getInclusionTreeDepth=function(){return this==this.graphManager.getRoot()?1:this.parent.getInclusionTreeDepth()},u.prototype.getEstimatedSize=function(){if(this.estimatedSize==r.MIN_VALUE)throw"assert failed";return this.estimatedSize},u.prototype.calcEstimatedSize=function(){for(var t=0,e=this.nodes,i=e.length,n=0;n=this.nodes.length){var h=0;r.forEach((function(e){e.owner==t&&h++})),h==this.nodes.length&&(this.isConnected=!0)}}else this.isConnected=!0},t.exports=u},function(t,e,i){"use strict";var n,r=i(1);function o(t){n=i(5),this.layout=t,this.graphs=[],this.edges=[]}o.prototype.addRoot=function(){var t=this.layout.newGraph(),e=this.layout.newNode(null),i=this.add(t,e);return this.setRootGraph(i),this.rootGraph},o.prototype.add=function(t,e,i,n,r){if(null==i&&null==n&&null==r){if(null==t)throw"Graph is null!";if(null==e)throw"Parent node is null!";if(this.graphs.indexOf(t)>-1)throw"Graph already in this graph mgr!";if(this.graphs.push(t),null!=t.parent)throw"Already has a parent!";if(null!=e.child)throw"Already has a child!";return t.parent=e,e.child=t,t}r=i,i=t;var o=(n=e).getOwner(),s=r.getOwner();if(null==o||o.getGraphManager()!=this)throw"Source not in this graph mgr!";if(null==s||s.getGraphManager()!=this)throw"Target not in this graph mgr!";if(o==s)return i.isInterGraph=!1,o.add(i,n,r);if(i.isInterGraph=!0,i.source=n,i.target=r,this.edges.indexOf(i)>-1)throw"Edge already in inter-graph edge list!";if(this.edges.push(i),null==i.source||null==i.target)throw"Edge source and/or target is null!";if(-1!=i.source.edges.indexOf(i)||-1!=i.target.edges.indexOf(i))throw"Edge already in source and/or target incidency list!";return i.source.edges.push(i),i.target.edges.push(i),i},o.prototype.remove=function(t){if(t instanceof n){var e=t;if(e.getGraphManager()!=this)throw"Graph not in this graph mgr";if(e!=this.rootGraph&&(null==e.parent||e.parent.graphManager!=this))throw"Invalid parent node!";for(var i,o=[],s=(o=o.concat(e.getEdges())).length,a=0;a=e.getRight()?i[0]+=Math.min(e.getX()-t.getX(),t.getRight()-e.getRight()):e.getX()<=t.getX()&&e.getRight()>=t.getRight()&&(i[0]+=Math.min(t.getX()-e.getX(),e.getRight()-t.getRight())),t.getY()<=e.getY()&&t.getBottom()>=e.getBottom()?i[1]+=Math.min(e.getY()-t.getY(),t.getBottom()-e.getBottom()):e.getY()<=t.getY()&&e.getBottom()>=t.getBottom()&&(i[1]+=Math.min(t.getY()-e.getY(),e.getBottom()-t.getBottom()));var o=Math.abs((e.getCenterY()-t.getCenterY())/(e.getCenterX()-t.getCenterX()));e.getCenterY()===t.getCenterY()&&e.getCenterX()===t.getCenterX()&&(o=1);var s=o*i[0],a=i[1]/o;i[0]s)return i[0]=n,i[1]=h,i[2]=o,i[3]=m,!1;if(ro)return i[0]=a,i[1]=r,i[2]=E,i[3]=s,!1;if(no?(i[0]=c,i[1]=g,L=!0):(i[0]=l,i[1]=h,L=!0):O===I&&(n>o?(i[0]=a,i[1]=h,L=!0):(i[0]=u,i[1]=g,L=!0)),-D===I?o>n?(i[2]=_,i[3]=m,T=!0):(i[2]=E,i[3]=y,T=!0):D===I&&(o>n?(i[2]=f,i[3]=y,T=!0):(i[2]=v,i[3]=m,T=!0)),L&&T)return!1;if(n>o?r>s?(w=this.getCardinalDirection(O,I,4),R=this.getCardinalDirection(D,I,2)):(w=this.getCardinalDirection(-O,I,3),R=this.getCardinalDirection(-D,I,1)):r>s?(w=this.getCardinalDirection(-O,I,1),R=this.getCardinalDirection(-D,I,3)):(w=this.getCardinalDirection(O,I,2),R=this.getCardinalDirection(D,I,4)),!L)switch(w){case 1:M=h,C=n+-p/I,i[0]=C,i[1]=M;break;case 2:C=u,M=r+d*I,i[0]=C,i[1]=M;break;case 3:M=g,C=n+p/I,i[0]=C,i[1]=M;break;case 4:C=c,M=r+-d*I,i[0]=C,i[1]=M}if(!T)switch(R){case 1:G=y,x=o+-A/I,i[2]=x,i[3]=G;break;case 2:x=v,G=s+N*I,i[2]=x,i[3]=G;break;case 3:G=m,x=o+A/I,i[2]=x,i[3]=G;break;case 4:x=_,G=s+-N*I,i[2]=x,i[3]=G}}return!1},r.getCardinalDirection=function(t,e,i){return t>e?i:1+i%4},r.getIntersection=function(t,e,i,r){if(null==r)return this.getIntersection2(t,e,i);var o,s,a,h,l,c,g,u=t.x,d=t.y,p=e.x,f=e.y,y=i.x,E=i.y,_=r.x,m=r.y;return 0==(g=(o=f-d)*(h=y-_)-(s=m-E)*(a=u-p))?null:new n((a*(c=_*E-y*m)-h*(l=p*d-u*f))/g,(s*l-o*c)/g)},r.angleOfVector=function(t,e,i,n){var r=void 0;return t!==i?(r=Math.atan((n-e)/(i-t)),i0?1:t<0?-1:0},n.floor=function(t){return t<0?Math.ceil(t):Math.floor(t)},n.ceil=function(t){return t<0?Math.floor(t):Math.ceil(t)},t.exports=n},function(t,e,i){"use strict";function n(){}n.MAX_VALUE=2147483647,n.MIN_VALUE=-2147483648,t.exports=n},function(t,e,i){"use strict";var n=function(){function t(t,e){for(var i=0;i0&&e;){for(a.push(l[0]);a.length>0&&e;){var c=a[0];a.splice(0,1),s.add(c);var g=c.getEdges();for(o=0;o-1&&l.splice(f,1)}s=new Set,h=new Map}else t=[]}return t},u.prototype.createDummyNodesForBendpoints=function(t){for(var e=[],i=t.source,n=this.graphManager.calcLowestCommonAncestor(t.source,t.target),r=0;r0){for(var r=this.edgeToDummyNodes.get(i),o=0;o=0&&e.splice(g,1),c.getNeighborsList().forEach((function(t){if(i.indexOf(t)<0){var e=n.get(t)-1;1==e&&h.push(t),n.set(t,e)}}))}i=i.concat(h),1!=e.length&&2!=e.length||(r=!0,o=e[0])}return o},u.prototype.setGraphManager=function(t){this.graphManager=t},t.exports=u},function(t,e,i){"use strict";function n(){}n.seed=1,n.x=0,n.nextDouble=function(){return n.x=1e4*Math.sin(n.seed++),n.x-Math.floor(n.x)},t.exports=n},function(t,e,i){"use strict";var n=i(4);function r(t,e){this.lworldOrgX=0,this.lworldOrgY=0,this.ldeviceOrgX=0,this.ldeviceOrgY=0,this.lworldExtX=1,this.lworldExtY=1,this.ldeviceExtX=1,this.ldeviceExtY=1}r.prototype.getWorldOrgX=function(){return this.lworldOrgX},r.prototype.setWorldOrgX=function(t){this.lworldOrgX=t},r.prototype.getWorldOrgY=function(){return this.lworldOrgY},r.prototype.setWorldOrgY=function(t){this.lworldOrgY=t},r.prototype.getWorldExtX=function(){return this.lworldExtX},r.prototype.setWorldExtX=function(t){this.lworldExtX=t},r.prototype.getWorldExtY=function(){return this.lworldExtY},r.prototype.setWorldExtY=function(t){this.lworldExtY=t},r.prototype.getDeviceOrgX=function(){return this.ldeviceOrgX},r.prototype.setDeviceOrgX=function(t){this.ldeviceOrgX=t},r.prototype.getDeviceOrgY=function(){return this.ldeviceOrgY},r.prototype.setDeviceOrgY=function(t){this.ldeviceOrgY=t},r.prototype.getDeviceExtX=function(){return this.ldeviceExtX},r.prototype.setDeviceExtX=function(t){this.ldeviceExtX=t},r.prototype.getDeviceExtY=function(){return this.ldeviceExtY},r.prototype.setDeviceExtY=function(t){this.ldeviceExtY=t},r.prototype.transformX=function(t){var e=0,i=this.lworldExtX;return 0!=i&&(e=this.ldeviceOrgX+(t-this.lworldOrgX)*this.ldeviceExtX/i),e},r.prototype.transformY=function(t){var e=0,i=this.lworldExtY;return 0!=i&&(e=this.ldeviceOrgY+(t-this.lworldOrgY)*this.ldeviceExtY/i),e},r.prototype.inverseTransformX=function(t){var e=0,i=this.ldeviceExtX;return 0!=i&&(e=this.lworldOrgX+(t-this.ldeviceOrgX)*this.lworldExtX/i),e},r.prototype.inverseTransformY=function(t){var e=0,i=this.ldeviceExtY;return 0!=i&&(e=this.lworldOrgY+(t-this.ldeviceOrgY)*this.lworldExtY/i),e},r.prototype.inverseTransformPoint=function(t){return new n(this.inverseTransformX(t.x),this.inverseTransformY(t.y))},t.exports=r},function(t,e,i){"use strict";var n=i(15),r=i(7),o=i(0),s=i(8),a=i(9);function h(){n.call(this),this.useSmartIdealEdgeLengthCalculation=r.DEFAULT_USE_SMART_IDEAL_EDGE_LENGTH_CALCULATION,this.idealEdgeLength=r.DEFAULT_EDGE_LENGTH,this.springConstant=r.DEFAULT_SPRING_STRENGTH,this.repulsionConstant=r.DEFAULT_REPULSION_STRENGTH,this.gravityConstant=r.DEFAULT_GRAVITY_STRENGTH,this.compoundGravityConstant=r.DEFAULT_COMPOUND_GRAVITY_STRENGTH,this.gravityRangeFactor=r.DEFAULT_GRAVITY_RANGE_FACTOR,this.compoundGravityRangeFactor=r.DEFAULT_COMPOUND_GRAVITY_RANGE_FACTOR,this.displacementThresholdPerNode=3*r.DEFAULT_EDGE_LENGTH/100,this.coolingFactor=r.DEFAULT_COOLING_FACTOR_INCREMENTAL,this.initialCoolingFactor=r.DEFAULT_COOLING_FACTOR_INCREMENTAL,this.totalDisplacement=0,this.oldTotalDisplacement=0,this.maxIterations=r.MAX_ITERATIONS}for(var l in 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t,e,i,n,s,a,h=this.getGraphManager().getAllEdges(),l=0;lr.ADAPTATION_LOWER_NODE_LIMIT&&(this.coolingFactor=Math.max(this.coolingFactor*r.COOLING_ADAPTATION_FACTOR,this.coolingFactor-(t-r.ADAPTATION_LOWER_NODE_LIMIT)/(r.ADAPTATION_UPPER_NODE_LIMIT-r.ADAPTATION_LOWER_NODE_LIMIT)*this.coolingFactor*(1-r.COOLING_ADAPTATION_FACTOR))),this.maxNodeDisplacement=r.MAX_NODE_DISPLACEMENT_INCREMENTAL):(t>r.ADAPTATION_LOWER_NODE_LIMIT?this.coolingFactor=Math.max(r.COOLING_ADAPTATION_FACTOR,1-(t-r.ADAPTATION_LOWER_NODE_LIMIT)/(r.ADAPTATION_UPPER_NODE_LIMIT-r.ADAPTATION_LOWER_NODE_LIMIT)*(1-r.COOLING_ADAPTATION_FACTOR)):this.coolingFactor=1,this.initialCoolingFactor=this.coolingFactor,this.maxNodeDisplacement=r.MAX_NODE_DISPLACEMENT),this.maxIterations=Math.max(5*this.getAllNodes().length,this.maxIterations),this.totalDisplacementThreshold=this.displacementThresholdPerNode*this.getAllNodes().length,this.repulsionRange=this.calcRepulsionRange()},h.prototype.calcSpringForces=function(){for(var t,e=this.getAllEdges(),i=0;i0&&void 0!==arguments[0])||arguments[0],a=arguments.length>1&&void 0!==arguments[1]&&arguments[1],h=this.getAllNodes();if(this.useFRGridVariant)for(this.totalIterations%r.GRID_CALCULATION_CHECK_PERIOD==1&&s&&this.updateGrid(),o=new Set,t=0;t(h=e.getEstimatedSize()*this.gravityRangeFactor)||a>h)&&(t.gravitationForceX=-this.gravityConstant*r,t.gravitationForceY=-this.gravityConstant*o):(s>(h=e.getEstimatedSize()*this.compoundGravityRangeFactor)||a>h)&&(t.gravitationForceX=-this.gravityConstant*r*this.compoundGravityConstant,t.gravitationForceY=-this.gravityConstant*o*this.compoundGravityConstant)},h.prototype.isConverged=function(){var t,e=!1;return this.totalIterations>this.maxIterations/3&&(e=Math.abs(this.totalDisplacement-this.oldTotalDisplacement)<2),t=this.totalDisplacement=a.length||l>=a[0].length))for(var c=0;ct}}]),t}();t.exports=o},function(t,e,i){"use strict";var n=function(){function t(t,e){for(var i=0;i2&&void 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i.hash=e,i}this.trace(t)}),"parseError"),parse:(0,s.K2)((function(t){var e=this,i=[0],n=[],r=[null],o=[],a=this.table,h="",l=0,c=0,g=0,u=o.slice.call(arguments,1),d=Object.create(this.lexer),p={yy:{}};for(var f in this.yy)Object.prototype.hasOwnProperty.call(this.yy,f)&&(p.yy[f]=this.yy[f]);d.setInput(t,p.yy),p.yy.lexer=d,p.yy.parser=this,void 0===d.yylloc&&(d.yylloc={});var y=d.yylloc;o.push(y);var E=d.options&&d.options.ranges;function _(){var t;return"number"!=typeof(t=n.pop()||d.lex()||1)&&(t instanceof Array&&(t=(n=t).pop()),t=e.symbols_[t]||t),t}"function"==typeof p.yy.parseError?this.parseError=p.yy.parseError:this.parseError=Object.getPrototypeOf(this).parseError,(0,s.K2)((function(t){i.length=i.length-2*t,r.length=r.length-t,o.length=o.length-t}),"popStack"),(0,s.K2)(_,"lex");for(var m,v,N,A,L,T,O,D,I,w={};;){if(N=i[i.length-1],this.defaultActions[N]?A=this.defaultActions[N]:(null==m&&(m=_()),A=a[N]&&a[N][m]),void 0===A||!A.length||!A[0]){var R="";for(T in 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t=this.graphManager.getLayout();this.displacementX=t.coolingFactor*(this.springForceX+this.repulsionForceX+this.gravitationForceX)/this.noOfChildren,this.displacementY=t.coolingFactor*(this.springForceY+this.repulsionForceY+this.gravitationForceY)/this.noOfChildren,Math.abs(this.displacementX)>t.coolingFactor*t.maxNodeDisplacement&&(this.displacementX=t.coolingFactor*t.maxNodeDisplacement*r.sign(this.displacementX)),Math.abs(this.displacementY)>t.coolingFactor*t.maxNodeDisplacement&&(this.displacementY=t.coolingFactor*t.maxNodeDisplacement*r.sign(this.displacementY)),null==this.child||0==this.child.getNodes().length?this.moveBy(this.displacementX,this.displacementY):this.propogateDisplacementToChildren(this.displacementX,this.displacementY),t.totalDisplacement+=Math.abs(this.displacementX)+Math.abs(this.displacementY),this.springForceX=0,this.springForceY=0,this.repulsionForceX=0,this.repulsionForceY=0,this.gravitationForceX=0,this.gravitationForceY=0,this.displacementX=0,this.displacementY=0},o.prototype.propogateDisplacementToChildren=function(t,e){for(var i,n=this.getChild().getNodes(),r=0;r0)this.positionNodesRadially(t);else{this.reduceTrees(),this.graphManager.resetAllNodesToApplyGravitation();var e=new Set(this.getAllNodes()),i=this.nodesWithGravity.filter((function(t){return e.has(t)}));this.graphManager.setAllNodesToApplyGravitation(i),this.positionNodesRandomly()}}return this.initSpringEmbedder(),this.runSpringEmbedder(),!0},_.prototype.tick=function(){if(this.totalIterations++,this.totalIterations===this.maxIterations&&!this.isTreeGrowing&&!this.isGrowthFinished){if(!(this.prunedNodesAll.length>0))return!0;this.isTreeGrowing=!0}if(this.totalIterations%l.CONVERGENCE_CHECK_PERIOD==0&&!this.isTreeGrowing&&!this.isGrowthFinished){if(this.isConverged()){if(!(this.prunedNodesAll.length>0))return!0;this.isTreeGrowing=!0}this.coolingCycle++,0==this.layoutQuality?this.coolingAdjuster=this.coolingCycle:1==this.layoutQuality&&(this.coolingAdjuster=this.coolingCycle/3),this.coolingFactor=Math.max(this.initialCoolingFactor-Math.pow(this.coolingCycle,Math.log(100*(this.initialCoolingFactor-this.finalTemperature))/Math.log(this.maxCoolingCycle))/100*this.coolingAdjuster,this.finalTemperature),this.animationPeriod=Math.ceil(this.initialAnimationPeriod*Math.sqrt(this.coolingFactor))}if(this.isTreeGrowing){if(this.growTreeIterations%10==0)if(this.prunedNodesAll.length>0){this.graphManager.updateBounds(),this.updateGrid(),this.growTree(this.prunedNodesAll),this.graphManager.resetAllNodesToApplyGravitation();var t=new Set(this.getAllNodes()),e=this.nodesWithGravity.filter((function(e){return t.has(e)}));this.graphManager.setAllNodesToApplyGravitation(e),this.graphManager.updateBounds(),this.updateGrid(),this.coolingFactor=l.DEFAULT_COOLING_FACTOR_INCREMENTAL}else this.isTreeGrowing=!1,this.isGrowthFinished=!0;this.growTreeIterations++}if(this.isGrowthFinished){if(this.isConverged())return!0;this.afterGrowthIterations%10==0&&(this.graphManager.updateBounds(),this.updateGrid()),this.coolingFactor=l.DEFAULT_COOLING_FACTOR_INCREMENTAL*((100-this.afterGrowthIterations)/100),this.afterGrowthIterations++}var i=!this.isTreeGrowing&&!this.isGrowthFinished,n=this.growTreeIterations%10==1&&this.isTreeGrowing||this.afterGrowthIterations%10==1&&this.isGrowthFinished;return this.totalDisplacement=0,this.graphManager.updateBounds(),this.calcSpringForces(),this.calcRepulsionForces(i,n),this.calcGravitationalForces(),this.moveNodes(),this.animate(),!1},_.prototype.getPositionsData=function(){for(var t=this.graphManager.getAllNodes(),e={},i=0;i1)for(a=0;an&&(n=Math.floor(s.y)),o=Math.floor(s.x+h.DEFAULT_COMPONENT_SEPERATION)}this.transform(new u(c.WORLD_CENTER_X-s.x/2,c.WORLD_CENTER_Y-s.y/2))},_.radialLayout=function(t,e,i){var n=Math.max(this.maxDiagonalInTree(t),h.DEFAULT_RADIAL_SEPARATION);_.branchRadialLayout(e,null,0,359,0,n);var r=y.calculateBounds(t),o=new E;o.setDeviceOrgX(r.getMinX()),o.setDeviceOrgY(r.getMinY()),o.setWorldOrgX(i.x),o.setWorldOrgY(i.y);for(var s=0;s1;){var E=y[0];y.splice(0,1);var m=c.indexOf(E);m>=0&&c.splice(m,1),p--,g--}u=null!=e?(c.indexOf(y[0])+1)%p:0;for(var v=Math.abs(n-i)/g,N=u;d!=g;N=++N%p){var A=c[N].getOtherEnd(t);if(A!=e){var L=(i+d*v)%360,T=(L+v)%360;_.branchRadialLayout(A,t,L,T,r+o,o),d++}}},_.maxDiagonalInTree=function(t){for(var e=p.MIN_VALUE,i=0;ie&&(e=n)}return e},_.prototype.calcRepulsionRange=function(){return 2*(this.level+1)*this.idealEdgeLength},_.prototype.groupZeroDegreeMembers=function(){var t=this,e={};this.memberGroups={},this.idToDummyNode={};for(var i=[],n=this.graphManager.getAllNodes(),r=0;r1){var n="DummyCompound_"+i;t.memberGroups[n]=e[i];var r=e[i][0].getParent(),o=new s(t.graphManager);o.id=n,o.paddingLeft=r.paddingLeft||0,o.paddingRight=r.paddingRight||0,o.paddingBottom=r.paddingBottom||0,o.paddingTop=r.paddingTop||0,t.idToDummyNode[n]=o;var a=t.getGraphManager().add(t.newGraph(),o),h=r.getChild();h.add(o);for(var l=0;l=0;t--){var e=this.compoundOrder[t],i=e.id,n=e.paddingLeft,r=e.paddingTop;this.adjustLocations(this.tiledMemberPack[i],e.rect.x,e.rect.y,n,r)}},_.prototype.repopulateZeroDegreeMembers=function(){var t=this,e=this.tiledZeroDegreePack;Object.keys(e).forEach((function(i){var n=t.idToDummyNode[i],r=n.paddingLeft,o=n.paddingTop;t.adjustLocations(e[i],n.rect.x,n.rect.y,r,o)}))},_.prototype.getToBeTiled=function(t){var e=t.id;if(null!=this.toBeTiled[e])return this.toBeTiled[e];var i=t.getChild();if(null==i)return this.toBeTiled[e]=!1,!1;for(var n=i.getNodes(),r=0;r0)return this.toBeTiled[e]=!1,!1;if(null!=o.getChild()){if(!this.getToBeTiled(o))return this.toBeTiled[e]=!1,!1}else this.toBeTiled[o.id]=!1}return this.toBeTiled[e]=!0,!0},_.prototype.getNodeDegree=function(t){t.id;for(var e=t.getEdges(),i=0,n=0;nh&&(h=c.rect.height)}i+=h+t.verticalPadding}},_.prototype.tileCompoundMembers=function(t,e){var i=this;this.tiledMemberPack=[],Object.keys(t).forEach((function(n){var r=e[n];i.tiledMemberPack[n]=i.tileNodes(t[n],r.paddingLeft+r.paddingRight),r.rect.width=i.tiledMemberPack[n].width,r.rect.height=i.tiledMemberPack[n].height}))},_.prototype.tileNodes=function(t,e){var i={rows:[],rowWidth:[],rowHeight:[],width:0,height:e,verticalPadding:h.TILING_PADDING_VERTICAL,horizontalPadding:h.TILING_PADDING_HORIZONTAL};t.sort((function(t,e){return t.rect.width*t.rect.height>e.rect.width*e.rect.height?-1:t.rect.width*t.rect.height0&&(o+=t.horizontalPadding),t.rowWidth[i]=o,t.width0&&(s+=t.verticalPadding);var a=0;s>t.rowHeight[i]&&(a=t.rowHeight[i],t.rowHeight[i]=s,a=t.rowHeight[i]-a),t.height+=a,t.rows[i].push(e)},_.prototype.getShortestRowIndex=function(t){for(var e=-1,i=Number.MAX_VALUE,n=0;ni&&(e=n,i=t.rowWidth[n]);return e},_.prototype.canAddHorizontal=function(t,e,i){var n=this.getShortestRowIndex(t);if(n<0)return!0;var r=t.rowWidth[n];if(r+t.horizontalPadding+e<=t.width)return!0;var o,s,a=0;return t.rowHeight[n]0&&(a=i+t.verticalPadding-t.rowHeight[n]),o=t.width-r>=e+t.horizontalPadding?(t.height+a)/(r+e+t.horizontalPadding):(t.height+a)/t.width,a=i+t.verticalPadding,(s=t.widtho&&e!=i){n.splice(-1,1),t.rows[i].push(r),t.rowWidth[e]=t.rowWidth[e]-o,t.rowWidth[i]=t.rowWidth[i]+o,t.width=t.rowWidth[instance.getLongestRowIndex(t)];for(var s=Number.MIN_VALUE,a=0;as&&(s=n[a].height);e>0&&(s+=t.verticalPadding);var h=t.rowHeight[e]+t.rowHeight[i];t.rowHeight[e]=s,t.rowHeight[i]0)for(var c=r;c<=o;c++)h[0]+=this.grid[c][s-1].length+this.grid[c][s].length-1;if(o0)for(c=s;c<=a;c++)h[3]+=this.grid[r-1][c].length+this.grid[r][c].length-1;for(var g,u,d=p.MAX_VALUE,f=0;f0&&(s=i.getGraphManager().add(i.newGraph(),o),this.processChildrenList(s,g,i))}},g.prototype.stop=function(){return this.stopped=!0,this};var d=function(t){t("layout","cose-bilkent",g)};"undefined"!=typeof cytoscape&&d(cytoscape),t.exports=d}])},t.exports=n(i(7799))},3143:function(t){var e;e=function(){return function(t){var e={};function i(n){if(e[n])return e[n].exports;var r=e[n]={i:n,l:!1,exports:{}};return t[n].call(r.exports,r,r.exports,i),r.l=!0,r.exports}return i.m=t,i.c=e,i.i=function(t){return t},i.d=function(t,e,n){i.o(t,e)||Object.defineProperty(t,e,{configurable:!1,enumerable:!0,get:n})},i.n=function(t){var e=t&&t.__esModule?function(){return t.default}:function(){return t};return i.d(e,"a",e),e},i.o=function(t,e){return Object.prototype.hasOwnProperty.call(t,e)},i.p="",i(i.s=26)}([function(t,e,i){"use strict";function n(){}n.QUALITY=1,n.DEFAULT_CREATE_BENDS_AS_NEEDED=!1,n.DEFAULT_INCREMENTAL=!1,n.DEFAULT_ANIMATION_ON_LAYOUT=!0,n.DEFAULT_ANIMATION_DURING_LAYOUT=!1,n.DEFAULT_ANIMATION_PERIOD=50,n.DEFAULT_UNIFORM_LEAF_NODE_SIZES=!1,n.DEFAULT_GRAPH_MARGIN=15,n.NODE_DIMENSIONS_INCLUDE_LABELS=!1,n.SIMPLE_NODE_SIZE=40,n.SIMPLE_NODE_HALF_SIZE=n.SIMPLE_NODE_SIZE/2,n.EMPTY_COMPOUND_NODE_SIZE=40,n.MIN_EDGE_LENGTH=1,n.WORLD_BOUNDARY=1e6,n.INITIAL_WORLD_BOUNDARY=n.WORLD_BOUNDARY/1e3,n.WORLD_CENTER_X=1200,n.WORLD_CENTER_Y=900,t.exports=n},function(t,e,i){"use strict";var n=i(2),r=i(8),o=i(9);function s(t,e,i){n.call(this,i),this.isOverlapingSourceAndTarget=!1,this.vGraphObject=i,this.bendpoints=[],this.source=t,this.target=e}for(var a in s.prototype=Object.create(n.prototype),n)s[a]=n[a];s.prototype.getSource=function(){return this.source},s.prototype.getTarget=function(){return this.target},s.prototype.isInterGraph=function(){return this.isInterGraph},s.prototype.getLength=function(){return this.length},s.prototype.isOverlapingSourceAndTarget=function(){return this.isOverlapingSourceAndTarget},s.prototype.getBendpoints=function(){return this.bendpoints},s.prototype.getLca=function(){return this.lca},s.prototype.getSourceInLca=function(){return this.sourceInLca},s.prototype.getTargetInLca=function(){return this.targetInLca},s.prototype.getOtherEnd=function(t){if(this.source===t)return this.target;if(this.target===t)return this.source;throw"Node is not incident with this edge"},s.prototype.getOtherEndInGraph=function(t,e){for(var i=this.getOtherEnd(t),n=e.getGraphManager().getRoot();;){if(i.getOwner()==e)return i;if(i.getOwner()==n)break;i=i.getOwner().getParent()}return null},s.prototype.updateLength=function(){var t=new Array(4);this.isOverlapingSourceAndTarget=r.getIntersection(this.target.getRect(),this.source.getRect(),t),this.isOverlapingSourceAndTarget||(this.lengthX=t[0]-t[2],this.lengthY=t[1]-t[3],Math.abs(this.lengthX)<1&&(this.lengthX=o.sign(this.lengthX)),Math.abs(this.lengthY)<1&&(this.lengthY=o.sign(this.lengthY)),this.length=Math.sqrt(this.lengthX*this.lengthX+this.lengthY*this.lengthY))},s.prototype.updateLengthSimple=function(){this.lengthX=this.target.getCenterX()-this.source.getCenterX(),this.lengthY=this.target.getCenterY()-this.source.getCenterY(),Math.abs(this.lengthX)<1&&(this.lengthX=o.sign(this.lengthX)),Math.abs(this.lengthY)<1&&(this.lengthY=o.sign(this.lengthY)),this.length=Math.sqrt(this.lengthX*this.lengthX+this.lengthY*this.lengthY)},t.exports=s},function(t,e,i){"use strict";t.exports=function(t){this.vGraphObject=t}},function(t,e,i){"use strict";var n=i(2),r=i(10),o=i(13),s=i(0),a=i(16),h=i(4);function l(t,e,i,s){null==i&&null==s&&(s=e),n.call(this,s),null!=t.graphManager&&(t=t.graphManager),this.estimatedSize=r.MIN_VALUE,this.inclusionTreeDepth=r.MAX_VALUE,this.vGraphObject=s,this.edges=[],this.graphManager=t,this.rect=null!=i&&null!=e?new o(e.x,e.y,i.width,i.height):new o}for(var c in l.prototype=Object.create(n.prototype),n)l[c]=n[c];l.prototype.getEdges=function(){return this.edges},l.prototype.getChild=function(){return this.child},l.prototype.getOwner=function(){return this.owner},l.prototype.getWidth=function(){return this.rect.width},l.prototype.setWidth=function(t){this.rect.width=t},l.prototype.getHeight=function(){return this.rect.height},l.prototype.setHeight=function(t){this.rect.height=t},l.prototype.getCenterX=function(){return this.rect.x+this.rect.width/2},l.prototype.getCenterY=function(){return this.rect.y+this.rect.height/2},l.prototype.getCenter=function(){return new h(this.rect.x+this.rect.width/2,this.rect.y+this.rect.height/2)},l.prototype.getLocation=function(){return new h(this.rect.x,this.rect.y)},l.prototype.getRect=function(){return this.rect},l.prototype.getDiagonal=function(){return Math.sqrt(this.rect.width*this.rect.width+this.rect.height*this.rect.height)},l.prototype.getHalfTheDiagonal=function(){return Math.sqrt(this.rect.height*this.rect.height+this.rect.width*this.rect.width)/2},l.prototype.setRect=function(t,e){this.rect.x=t.x,this.rect.y=t.y,this.rect.width=e.width,this.rect.height=e.height},l.prototype.setCenter=function(t,e){this.rect.x=t-this.rect.width/2,this.rect.y=e-this.rect.height/2},l.prototype.setLocation=function(t,e){this.rect.x=t,this.rect.y=e},l.prototype.moveBy=function(t,e){this.rect.x+=t,this.rect.y+=e},l.prototype.getEdgeListToNode=function(t){var e=[],i=this;return i.edges.forEach((function(n){if(n.target==t){if(n.source!=i)throw"Incorrect edge source!";e.push(n)}})),e},l.prototype.getEdgesBetween=function(t){var e=[],i=this;return i.edges.forEach((function(n){if(n.source!=i&&n.target!=i)throw"Incorrect edge source and/or target";n.target!=t&&n.source!=t||e.push(n)})),e},l.prototype.getNeighborsList=function(){var t=new Set,e=this;return e.edges.forEach((function(i){if(i.source==e)t.add(i.target);else{if(i.target!=e)throw"Incorrect incidency!";t.add(i.source)}})),t},l.prototype.withChildren=function(){var t=new Set;if(t.add(this),null!=this.child)for(var e=this.child.getNodes(),i=0;ie&&(this.rect.x-=(this.labelWidth-e)/2,this.setWidth(this.labelWidth)),this.labelHeight>i&&("center"==this.labelPos?this.rect.y-=(this.labelHeight-i)/2:"top"==this.labelPos&&(this.rect.y-=this.labelHeight-i),this.setHeight(this.labelHeight))}}},l.prototype.getInclusionTreeDepth=function(){if(this.inclusionTreeDepth==r.MAX_VALUE)throw"assert failed";return this.inclusionTreeDepth},l.prototype.transform=function(t){var e=this.rect.x;e>s.WORLD_BOUNDARY?e=s.WORLD_BOUNDARY:e<-s.WORLD_BOUNDARY&&(e=-s.WORLD_BOUNDARY);var i=this.rect.y;i>s.WORLD_BOUNDARY?i=s.WORLD_BOUNDARY:i<-s.WORLD_BOUNDARY&&(i=-s.WORLD_BOUNDARY);var n=new h(e,i),r=t.inverseTransformPoint(n);this.setLocation(r.x,r.y)},l.prototype.getLeft=function(){return this.rect.x},l.prototype.getRight=function(){return this.rect.x+this.rect.width},l.prototype.getTop=function(){return this.rect.y},l.prototype.getBottom=function(){return this.rect.y+this.rect.height},l.prototype.getParent=function(){return null==this.owner?null:this.owner.getParent()},t.exports=l},function(t,e,i){"use strict";function n(t,e){null==t&&null==e?(this.x=0,this.y=0):(this.x=t,this.y=e)}n.prototype.getX=function(){return this.x},n.prototype.getY=function(){return this.y},n.prototype.setX=function(t){this.x=t},n.prototype.setY=function(t){this.y=t},n.prototype.getDifference=function(t){return new DimensionD(this.x-t.x,this.y-t.y)},n.prototype.getCopy=function(){return new n(this.x,this.y)},n.prototype.translate=function(t){return this.x+=t.width,this.y+=t.height,this},t.exports=n},function(t,e,i){"use strict";var n=i(2),r=i(10),o=i(0),s=i(6),a=i(3),h=i(1),l=i(13),c=i(12),g=i(11);function u(t,e,i){n.call(this,i),this.estimatedSize=r.MIN_VALUE,this.margin=o.DEFAULT_GRAPH_MARGIN,this.edges=[],this.nodes=[],this.isConnected=!1,this.parent=t,null!=e&&e instanceof s?this.graphManager=e:null!=e&&e instanceof Layout&&(this.graphManager=e.graphManager)}for(var d in u.prototype=Object.create(n.prototype),n)u[d]=n[d];u.prototype.getNodes=function(){return this.nodes},u.prototype.getEdges=function(){return this.edges},u.prototype.getGraphManager=function(){return this.graphManager},u.prototype.getParent=function(){return this.parent},u.prototype.getLeft=function(){return this.left},u.prototype.getRight=function(){return this.right},u.prototype.getTop=function(){return this.top},u.prototype.getBottom=function(){return this.bottom},u.prototype.isConnected=function(){return this.isConnected},u.prototype.add=function(t,e,i){if(null==e&&null==i){var n=t;if(null==this.graphManager)throw"Graph has no graph mgr!";if(this.getNodes().indexOf(n)>-1)throw"Node already in graph!";return n.owner=this,this.getNodes().push(n),n}var r=t;if(!(this.getNodes().indexOf(e)>-1&&this.getNodes().indexOf(i)>-1))throw"Source or target not in graph!";if(e.owner!=i.owner||e.owner!=this)throw"Both owners must be this graph!";return e.owner!=i.owner?null:(r.source=e,r.target=i,r.isInterGraph=!1,this.getEdges().push(r),e.edges.push(r),i!=e&&i.edges.push(r),r)},u.prototype.remove=function(t){var e=t;if(t instanceof a){if(null==e)throw"Node is null!";if(null==e.owner||e.owner!=this)throw"Owner graph is invalid!";if(null==this.graphManager)throw"Owner graph manager is invalid!";for(var i=e.edges.slice(),n=i.length,r=0;r-1&&c>-1))throw"Source and/or target doesn't know this edge!";if(o.source.edges.splice(l,1),o.target!=o.source&&o.target.edges.splice(c,1),-1==(s=o.source.owner.getEdges().indexOf(o)))throw"Not in owner's edge list!";o.source.owner.getEdges().splice(s,1)}},u.prototype.updateLeftTop=function(){for(var t,e,i,n=r.MAX_VALUE,o=r.MAX_VALUE,s=this.getNodes(),a=s.length,h=0;h(t=l.getTop())&&(n=t),o>(e=l.getLeft())&&(o=e)}return n==r.MAX_VALUE?null:(i=null!=s[0].getParent().paddingLeft?s[0].getParent().paddingLeft:this.margin,this.left=o-i,this.top=n-i,new c(this.left,this.top))},u.prototype.updateBounds=function(t){for(var e,i,n,o,s,a=r.MAX_VALUE,h=-r.MAX_VALUE,c=r.MAX_VALUE,g=-r.MAX_VALUE,u=this.nodes,d=u.length,p=0;p(e=f.getLeft())&&(a=e),h<(i=f.getRight())&&(h=i),c>(n=f.getTop())&&(c=n),g<(o=f.getBottom())&&(g=o)}var y=new l(a,c,h-a,g-c);a==r.MAX_VALUE&&(this.left=this.parent.getLeft(),this.right=this.parent.getRight(),this.top=this.parent.getTop(),this.bottom=this.parent.getBottom()),s=null!=u[0].getParent().paddingLeft?u[0].getParent().paddingLeft:this.margin,this.left=y.x-s,this.right=y.x+y.width+s,this.top=y.y-s,this.bottom=y.y+y.height+s},u.calculateBounds=function(t){for(var e,i,n,o,s=r.MAX_VALUE,a=-r.MAX_VALUE,h=r.MAX_VALUE,c=-r.MAX_VALUE,g=t.length,u=0;u(e=d.getLeft())&&(s=e),a<(i=d.getRight())&&(a=i),h>(n=d.getTop())&&(h=n),c<(o=d.getBottom())&&(c=o)}return new l(s,h,a-s,c-h)},u.prototype.getInclusionTreeDepth=function(){return this==this.graphManager.getRoot()?1:this.parent.getInclusionTreeDepth()},u.prototype.getEstimatedSize=function(){if(this.estimatedSize==r.MIN_VALUE)throw"assert failed";return this.estimatedSize},u.prototype.calcEstimatedSize=function(){for(var t=0,e=this.nodes,i=e.length,n=0;n=this.nodes.length){var h=0;r.forEach((function(e){e.owner==t&&h++})),h==this.nodes.length&&(this.isConnected=!0)}}else this.isConnected=!0},t.exports=u},function(t,e,i){"use strict";var n,r=i(1);function o(t){n=i(5),this.layout=t,this.graphs=[],this.edges=[]}o.prototype.addRoot=function(){var t=this.layout.newGraph(),e=this.layout.newNode(null),i=this.add(t,e);return this.setRootGraph(i),this.rootGraph},o.prototype.add=function(t,e,i,n,r){if(null==i&&null==n&&null==r){if(null==t)throw"Graph is null!";if(null==e)throw"Parent node is null!";if(this.graphs.indexOf(t)>-1)throw"Graph already in this graph mgr!";if(this.graphs.push(t),null!=t.parent)throw"Already has a parent!";if(null!=e.child)throw"Already has a child!";return t.parent=e,e.child=t,t}r=i,i=t;var o=(n=e).getOwner(),s=r.getOwner();if(null==o||o.getGraphManager()!=this)throw"Source not in this graph mgr!";if(null==s||s.getGraphManager()!=this)throw"Target not in this graph mgr!";if(o==s)return i.isInterGraph=!1,o.add(i,n,r);if(i.isInterGraph=!0,i.source=n,i.target=r,this.edges.indexOf(i)>-1)throw"Edge already in inter-graph edge list!";if(this.edges.push(i),null==i.source||null==i.target)throw"Edge source and/or target is null!";if(-1!=i.source.edges.indexOf(i)||-1!=i.target.edges.indexOf(i))throw"Edge already in source and/or target incidency list!";return i.source.edges.push(i),i.target.edges.push(i),i},o.prototype.remove=function(t){if(t instanceof n){var e=t;if(e.getGraphManager()!=this)throw"Graph not in this graph mgr";if(e!=this.rootGraph&&(null==e.parent||e.parent.graphManager!=this))throw"Invalid parent node!";for(var 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t>e?i:1+i%4},r.getIntersection=function(t,e,i,r){if(null==r)return this.getIntersection2(t,e,i);var o,s,a,h,l,c,g,u=t.x,d=t.y,p=e.x,f=e.y,y=i.x,E=i.y,_=r.x,m=r.y;return 0==(g=(o=f-d)*(h=y-_)-(s=m-E)*(a=u-p))?null:new n((a*(c=_*E-y*m)-h*(l=p*d-u*f))/g,(s*l-o*c)/g)},r.angleOfVector=function(t,e,i,n){var r=void 0;return t!==i?(r=Math.atan((n-e)/(i-t)),i0?1:t<0?-1:0},n.floor=function(t){return t<0?Math.ceil(t):Math.floor(t)},n.ceil=function(t){return t<0?Math.floor(t):Math.ceil(t)},t.exports=n},function(t,e,i){"use strict";function n(){}n.MAX_VALUE=2147483647,n.MIN_VALUE=-2147483648,t.exports=n},function(t,e,i){"use strict";var n=function(){function t(t,e){for(var i=0;i0&&e;){for(a.push(l[0]);a.length>0&&e;){var c=a[0];a.splice(0,1),s.add(c);var g=c.getEdges();for(o=0;o-1&&l.splice(f,1)}s=new Set,h=new Map}else t=[]}return t},u.prototype.createDummyNodesForBendpoints=function(t){for(var e=[],i=t.source,n=this.graphManager.calcLowestCommonAncestor(t.source,t.target),r=0;r0){for(var 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sans-serif',this.fontSize="16px",this.labelBackground="#181818",this.textColor="#ccc",this.THEME_COLOR_LIMIT=12,this.nodeBkg="calculated",this.nodeBorder="calculated",this.clusterBkg="calculated",this.clusterBorder="calculated",this.defaultLinkColor="calculated",this.titleColor="#F9FFFE",this.edgeLabelBackground="calculated",this.actorBorder="calculated",this.actorBkg="calculated",this.actorTextColor="calculated",this.actorLineColor="calculated",this.signalColor="calculated",this.signalTextColor="calculated",this.labelBoxBkgColor="calculated",this.labelBoxBorderColor="calculated",this.labelTextColor="calculated",this.loopTextColor="calculated",this.noteBorderColor="calculated",this.noteBkgColor="#fff5ad",this.noteTextColor="calculated",this.activationBorderColor="calculated",this.activationBkgColor="calculated",this.sequenceNumberColor="black",this.sectionBkgColor=(0,u.A)("#EAE8D9",30),this.altSectionBkgColor="calculated",this.sectionBkgColor2="#EAE8D9",this.excludeBkgColor=(0,u.A)(this.sectionBkgColor,10),this.taskBorderColor=(0,s.A)(255,255,255,70),this.taskBkgColor="calculated",this.taskTextColor="calculated",this.taskTextLightColor="calculated",this.taskTextOutsideColor="calculated",this.taskTextClickableColor="#003163",this.activeTaskBorderColor=(0,s.A)(255,255,255,50),this.activeTaskBkgColor="#81B1DB",this.gridColor="calculated",this.doneTaskBkgColor="calculated",this.doneTaskBorderColor="grey",this.critBorderColor="#E83737",this.critBkgColor="#E83737",this.taskTextDarkColor="calculated",this.todayLineColor="#DB5757",this.personBorder=this.primaryBorderColor,this.personBkg=this.mainBkg,this.archEdgeColor="calculated",this.archEdgeArrowColor="calculated",this.archEdgeWidth="3",this.archGroupBorderColor=this.primaryBorderColor,this.archGroupBorderWidth="2px",this.labelColor="calculated",this.errorBkgColor="#a44141",this.errorTextColor="#ddd"}updateColors(){this.secondBkg=(0,d.A)(this.mainBkg,16),this.lineColor=this.mainContrastColor,this.arrowheadColor=this.mainContrastColor,this.nodeBkg=this.mainBkg,this.nodeBorder=this.border1,this.clusterBkg=this.secondBkg,this.clusterBorder=this.border2,this.defaultLinkColor=this.lineColor,this.edgeLabelBackground=(0,d.A)(this.labelBackground,25),this.actorBorder=this.border1,this.actorBkg=this.mainBkg,this.actorTextColor=this.mainContrastColor,this.actorLineColor=this.actorBorder,this.signalColor=this.mainContrastColor,this.signalTextColor=this.mainContrastColor,this.labelBoxBkgColor=this.actorBkg,this.labelBoxBorderColor=this.actorBorder,this.labelTextColor=this.mainContrastColor,this.loopTextColor=this.mainContrastColor,this.noteBorderColor=this.secondaryBorderColor,this.noteBkgColor=this.secondBkg,this.noteTextColor=this.secondaryTextColor,this.activationBorderColor=this.border1,this.activationBkgColor=this.secondBkg,this.altSectionBkgColor=this.background,this.taskBkgColor=(0,d.A)(this.mainBkg,23),this.taskTextColor=this.darkTextColor,this.taskTextLightColor=this.mainContrastColor,this.taskTextOutsideColor=this.taskTextLightColor,this.gridColor=this.mainContrastColor,this.doneTaskBkgColor=this.mainContrastColor,this.taskTextDarkColor=this.darkTextColor,this.archEdgeColor=this.lineColor,this.archEdgeArrowColor=this.lineColor,this.transitionColor=this.transitionColor||this.lineColor,this.transitionLabelColor=this.transitionLabelColor||this.textColor,this.stateLabelColor=this.stateLabelColor||this.stateBkg||this.primaryTextColor,this.stateBkg=this.stateBkg||this.mainBkg,this.labelBackgroundColor=this.labelBackgroundColor||this.stateBkg,this.compositeBackground=this.compositeBackground||this.background||this.tertiaryColor,this.altBackground=this.altBackground||"#555",this.compositeTitleBackground=this.compositeTitleBackground||this.mainBkg,this.compositeBorder=this.compositeBorder||this.nodeBorder,this.innerEndBackground=this.primaryBorderColor,this.specialStateColor="#f4f4f4",this.errorBkgColor=this.errorBkgColor||this.tertiaryColor,this.errorTextColor=this.errorTextColor||this.tertiaryTextColor,this.fillType0=this.primaryColor,this.fillType1=this.secondaryColor,this.fillType2=o(this.primaryColor,{h:64}),this.fillType3=o(this.secondaryColor,{h:64}),this.fillType4=o(this.primaryColor,{h:-64}),this.fillType5=o(this.secondaryColor,{h:-64}),this.fillType6=o(this.primaryColor,{h:128}),this.fillType7=o(this.secondaryColor,{h:128}),this.cScale1=this.cScale1||"#0b0000",this.cScale2=this.cScale2||"#4d1037",this.cScale3=this.cScale3||"#3f5258",this.cScale4=this.cScale4||"#4f2f1b",this.cScale5=this.cScale5||"#6e0a0a",this.cScale6=this.cScale6||"#3b0048",this.cScale7=this.cScale7||"#995a01",this.cScale8=this.cScale8||"#154706",this.cScale9=this.cScale9||"#161722",this.cScale10=this.cScale10||"#00296f",this.cScale11=this.cScale11||"#01629c",this.cScale12=this.cScale12||"#010029",this.cScale0=this.cScale0||this.primaryColor,this.cScale1=this.cScale1||this.secondaryColor,this.cScale2=this.cScale2||this.tertiaryColor,this.cScale3=this.cScale3||o(this.primaryColor,{h:30}),this.cScale4=this.cScale4||o(this.primaryColor,{h:60}),this.cScale5=this.cScale5||o(this.primaryColor,{h:90}),this.cScale6=this.cScale6||o(this.primaryColor,{h:120}),this.cScale7=this.cScale7||o(this.primaryColor,{h:150}),this.cScale8=this.cScale8||o(this.primaryColor,{h:210}),this.cScale9=this.cScale9||o(this.primaryColor,{h:270}),this.cScale10=this.cScale10||o(this.primaryColor,{h:300}),this.cScale11=this.cScale11||o(this.primaryColor,{h:330});for(let t=0;t{this[e]=t[e]})),this.updateColors(),e.forEach((e=>{this[e]=t[e]}))}},P=y((t=>{const e=new R;return e.calculate(t),e}),"getThemeVariables"),z=class{static{y(this,"Theme")}constructor(){this.background="#f4f4f4",this.primaryColor="#ECECFF",this.secondaryColor=o(this.primaryColor,{h:120}),this.secondaryColor="#ffffde",this.tertiaryColor=o(this.primaryColor,{h:-160}),this.primaryBorderColor=O(this.primaryColor,this.darkMode),this.secondaryBorderColor=O(this.secondaryColor,this.darkMode),this.tertiaryBorderColor=O(this.tertiaryColor,this.darkMode),this.primaryTextColor=c(this.primaryColor),this.secondaryTextColor=c(this.secondaryColor),this.tertiaryTextColor=c(this.tertiaryColor),this.lineColor=c(this.background),this.textColor=c(this.background),this.background="white",this.mainBkg="#ECECFF",this.secondBkg="#ffffde",this.lineColor="#333333",this.border1="#9370DB",this.border2="#aaaa33",this.arrowheadColor="#333333",this.fontFamily='"trebuchet ms", verdana, arial, sans-serif',this.fontSize="16px",this.labelBackground="rgba(232,232,232, 0.8)",this.textColor="#333",this.THEME_COLOR_LIMIT=12,this.nodeBkg="calculated",this.nodeBorder="calculated",this.clusterBkg="calculated",this.clusterBorder="calculated",this.defaultLinkColor="calculated",this.titleColor="calculated",this.edgeLabelBackground="calculated",this.actorBorder="calculated",this.actorBkg="calculated",this.actorTextColor="black",this.actorLineColor="calculated",this.signalColor="calculated",this.signalTextColor="calculated",this.labelBoxBkgColor="calculated",this.labelBoxBorderColor="calculated",this.labelTextColor="calculated",this.loopTextColor="calculated",this.noteBorderColor="calculated",this.noteBkgColor="#fff5ad",this.noteTextColor="calculated",this.activationBorderColor="#666",this.activationBkgColor="#f4f4f4",this.sequenceNumberColor="white",this.sectionBkgColor="calculated",this.altSectionBkgColor="calculated",this.sectionBkgColor2="calculated",this.excludeBkgColor="#eeeeee",this.taskBorderColor="calculated",this.taskBkgColor="calculated",this.taskTextLightColor="calculated",this.taskTextColor=this.taskTextLightColor,this.taskTextDarkColor="calculated",this.taskTextOutsideColor=this.taskTextDarkColor,this.taskTextClickableColor="calculated",this.activeTaskBorderColor="calculated",this.activeTaskBkgColor="calculated",this.gridColor="calculated",this.doneTaskBkgColor="calculated",this.doneTaskBorderColor="calculated",this.critBorderColor="calculated",this.critBkgColor="calculated",this.todayLineColor="calculated",this.sectionBkgColor=(0,s.A)(102,102,255,.49),this.altSectionBkgColor="white",this.sectionBkgColor2="#fff400",this.taskBorderColor="#534fbc",this.taskBkgColor="#8a90dd",this.taskTextLightColor="white",this.taskTextColor="calculated",this.taskTextDarkColor="black",this.taskTextOutsideColor="calculated",this.taskTextClickableColor="#003163",this.activeTaskBorderColor="#534fbc",this.activeTaskBkgColor="#bfc7ff",this.gridColor="lightgrey",this.doneTaskBkgColor="lightgrey",this.doneTaskBorderColor="grey",this.critBorderColor="#ff8888",this.critBkgColor="red",this.todayLineColor="red",this.personBorder=this.primaryBorderColor,this.personBkg=this.mainBkg,this.archEdgeColor="calculated",this.archEdgeArrowColor="calculated",this.archEdgeWidth="3",this.archGroupBorderColor=this.primaryBorderColor,this.archGroupBorderWidth="2px",this.labelColor="black",this.errorBkgColor="#552222",this.errorTextColor="#552222",this.updateColors()}updateColors(){this.cScale0=this.cScale0||this.primaryColor,this.cScale1=this.cScale1||this.secondaryColor,this.cScale2=this.cScale2||this.tertiaryColor,this.cScale3=this.cScale3||o(this.primaryColor,{h:30}),this.cScale4=this.cScale4||o(this.primaryColor,{h:60}),this.cScale5=this.cScale5||o(this.primaryColor,{h:90}),this.cScale6=this.cScale6||o(this.primaryColor,{h:120}),this.cScale7=this.cScale7||o(this.primaryColor,{h:150}),this.cScale8=this.cScale8||o(this.primaryColor,{h:210}),this.cScale9=this.cScale9||o(this.primaryColor,{h:270}),this.cScale10=this.cScale10||o(this.primaryColor,{h:300}),this.cScale11=this.cScale11||o(this.primaryColor,{h:330}),this.cScalePeer1=this.cScalePeer1||(0,u.A)(this.secondaryColor,45),this.cScalePeer2=this.cScalePeer2||(0,u.A)(this.tertiaryColor,40);for(let t=0;t{this[e]=t[e]})),this.updateColors(),e.forEach((e=>{this[e]=t[e]}))}},K=y((t=>{const e=new z;return e.calculate(t),e}),"getThemeVariables"),j=class{static{y(this,"Theme")}constructor(){this.background="#f4f4f4",this.primaryColor="#cde498",this.secondaryColor="#cdffb2",this.background="white",this.mainBkg="#cde498",this.secondBkg="#cdffb2",this.lineColor="green",this.border1="#13540c",this.border2="#6eaa49",this.arrowheadColor="green",this.fontFamily='"trebuchet ms", verdana, arial, sans-serif',this.fontSize="16px",this.tertiaryColor=(0,d.A)("#cde498",10),this.primaryBorderColor=O(this.primaryColor,this.darkMode),this.secondaryBorderColor=O(this.secondaryColor,this.darkMode),this.tertiaryBorderColor=O(this.tertiaryColor,this.darkMode),this.primaryTextColor=c(this.primaryColor),this.secondaryTextColor=c(this.secondaryColor),this.tertiaryTextColor=c(this.primaryColor),this.lineColor=c(this.background),this.textColor=c(this.background),this.THEME_COLOR_LIMIT=12,this.nodeBkg="calculated",this.nodeBorder="calculated",this.clusterBkg="calculated",this.clusterBorder="calculated",this.defaultLinkColor="calculated",this.titleColor="#333",this.edgeLabelBackground="#e8e8e8",this.actorBorder="calculated",this.actorBkg="calculated",this.actorTextColor="black",this.actorLineColor="calculated",this.signalColor="#333",this.signalTextColor="#333",this.labelBoxBkgColor="calculated",this.labelBoxBorderColor="#326932",this.labelTextColor="calculated",this.loopTextColor="calculated",this.noteBorderColor="calculated",this.noteBkgColor="#fff5ad",this.noteTextColor="calculated",this.activationBorderColor="#666",this.activationBkgColor="#f4f4f4",this.sequenceNumberColor="white",this.sectionBkgColor="#6eaa49",this.altSectionBkgColor="white",this.sectionBkgColor2="#6eaa49",this.excludeBkgColor="#eeeeee",this.taskBorderColor="calculated",this.taskBkgColor="#487e3a",this.taskTextLightColor="white",this.taskTextColor="calculated",this.taskTextDarkColor="black",this.taskTextOutsideColor="calculated",this.taskTextClickableColor="#003163",this.activeTaskBorderColor="calculated",this.activeTaskBkgColor="calculated",this.gridColor="lightgrey",this.doneTaskBkgColor="lightgrey",this.doneTaskBorderColor="grey",this.critBorderColor="#ff8888",this.critBkgColor="red",this.todayLineColor="red",this.personBorder=this.primaryBorderColor,this.personBkg=this.mainBkg,this.archEdgeColor="calculated",this.archEdgeArrowColor="calculated",this.archEdgeWidth="3",this.archGroupBorderColor=this.primaryBorderColor,this.archGroupBorderWidth="2px",this.labelColor="black",this.errorBkgColor="#552222",this.errorTextColor="#552222"}updateColors(){this.actorBorder=(0,u.A)(this.mainBkg,20),this.actorBkg=this.mainBkg,this.labelBoxBkgColor=this.actorBkg,this.labelTextColor=this.actorTextColor,this.loopTextColor=this.actorTextColor,this.noteBorderColor=this.border2,this.noteTextColor=this.actorTextColor,this.actorLineColor=this.actorBorder,this.cScale0=this.cScale0||this.primaryColor,this.cScale1=this.cScale1||this.secondaryColor,this.cScale2=this.cScale2||this.tertiaryColor,this.cScale3=this.cScale3||o(this.primaryColor,{h:30}),this.cScale4=this.cScale4||o(this.primaryColor,{h:60}),this.cScale5=this.cScale5||o(this.primaryColor,{h:90}),this.cScale6=this.cScale6||o(this.primaryColor,{h:120}),this.cScale7=this.cScale7||o(this.primaryColor,{h:150}),this.cScale8=this.cScale8||o(this.primaryColor,{h:210}),this.cScale9=this.cScale9||o(this.primaryColor,{h:270}),this.cScale10=this.cScale10||o(this.primaryColor,{h:300}),this.cScale11=this.cScale11||o(this.primaryColor,{h:330}),this.cScalePeer1=this.cScalePeer1||(0,u.A)(this.secondaryColor,45),this.cScalePeer2=this.cScalePeer2||(0,u.A)(this.tertiaryColor,40);for(let t=0;t{this[e]=t[e]})),this.updateColors(),e.forEach((e=>{this[e]=t[e]}))}},q=y((t=>{const e=new j;return e.calculate(t),e}),"getThemeVariables"),W=class{static{y(this,"Theme")}constructor(){this.primaryColor="#eee",this.contrast="#707070",this.secondaryColor=(0,d.A)(this.contrast,55),this.background="#ffffff",this.tertiaryColor=o(this.primaryColor,{h:-160}),this.primaryBorderColor=O(this.primaryColor,this.darkMode),this.secondaryBorderColor=O(this.secondaryColor,this.darkMode),this.tertiaryBorderColor=O(this.tertiaryColor,this.darkMode),this.primaryTextColor=c(this.primaryColor),this.secondaryTextColor=c(this.secondaryColor),this.tertiaryTextColor=c(this.tertiaryColor),this.lineColor=c(this.background),this.textColor=c(this.background),this.mainBkg="#eee",this.secondBkg="calculated",this.lineColor="#666",this.border1="#999",this.border2="calculated",this.note="#ffa",this.text="#333",this.critical="#d42",this.done="#bbb",this.arrowheadColor="#333333",this.fontFamily='"trebuchet ms", verdana, arial, sans-serif',this.fontSize="16px",this.THEME_COLOR_LIMIT=12,this.nodeBkg="calculated",this.nodeBorder="calculated",this.clusterBkg="calculated",this.clusterBorder="calculated",this.defaultLinkColor="calculated",this.titleColor="calculated",this.edgeLabelBackground="white",this.actorBorder="calculated",this.actorBkg="calculated",this.actorTextColor="calculated",this.actorLineColor=this.actorBorder,this.signalColor="calculated",this.signalTextColor="calculated",this.labelBoxBkgColor="calculated",this.labelBoxBorderColor="calculated",this.labelTextColor="calculated",this.loopTextColor="calculated",this.noteBorderColor="calculated",this.noteBkgColor="calculated",this.noteTextColor="calculated",this.activationBorderColor="#666",this.activationBkgColor="#f4f4f4",this.sequenceNumberColor="white",this.sectionBkgColor="calculated",this.altSectionBkgColor="white",this.sectionBkgColor2="calculated",this.excludeBkgColor="#eeeeee",this.taskBorderColor="calculated",this.taskBkgColor="calculated",this.taskTextLightColor="white",this.taskTextColor="calculated",this.taskTextDarkColor="calculated",this.taskTextOutsideColor="calculated",this.taskTextClickableColor="#003163",this.activeTaskBorderColor="calculated",this.activeTaskBkgColor="calculated",this.gridColor="calculated",this.doneTaskBkgColor="calculated",this.doneTaskBorderColor="calculated",this.critBkgColor="calculated",this.critBorderColor="calculated",this.todayLineColor="calculated",this.personBorder=this.primaryBorderColor,this.personBkg=this.mainBkg,this.archEdgeColor="calculated",this.archEdgeArrowColor="calculated",this.archEdgeWidth="3",this.archGroupBorderColor=this.primaryBorderColor,this.archGroupBorderWidth="2px",this.labelColor="black",this.errorBkgColor="#552222",this.errorTextColor="#552222"}updateColors(){this.secondBkg=(0,d.A)(this.contrast,55),this.border2=this.contrast,this.actorBorder=(0,d.A)(this.border1,23),this.actorBkg=this.mainBkg,this.actorTextColor=this.text,this.actorLineColor=this.actorBorder,this.signalColor=this.text,this.signalTextColor=this.text,this.labelBoxBkgColor=this.actorBkg,this.labelBoxBorderColor=this.actorBorder,this.labelTextColor=this.text,this.loopTextColor=this.text,this.noteBorderColor="#999",this.noteBkgColor="#666",this.noteTextColor="#fff",this.cScale0=this.cScale0||"#555",this.cScale1=this.cScale1||"#F4F4F4",this.cScale2=this.cScale2||"#555",this.cScale3=this.cScale3||"#BBB",this.cScale4=this.cScale4||"#777",this.cScale5=this.cScale5||"#999",this.cScale6=this.cScale6||"#DDD",this.cScale7=this.cScale7||"#FFF",this.cScale8=this.cScale8||"#DDD",this.cScale9=this.cScale9||"#BBB",this.cScale10=this.cScale10||"#999",this.cScale11=this.cScale11||"#777";for(let t=0;t{this[e]=t[e]})),this.updateColors(),e.forEach((e=>{this[e]=t[e]}))}},U={base:{getThemeVariables:I},dark:{getThemeVariables:P},default:{getThemeVariables:K},forest:{getThemeVariables:q},neutral:{getThemeVariables:y((t=>{const e=new W;return e.calculate(t),e}),"getThemeVariables")}},H={flowchart:{useMaxWidth:!0,titleTopMargin:25,subGraphTitleMargin:{top:0,bottom:0},diagramPadding:8,htmlLabels:!0,nodeSpacing:50,rankSpacing:50,curve:"basis",padding:15,defaultRenderer:"dagre-wrapper",wrappingWidth:200},sequence:{useMaxWidth:!0,hideUnusedParticipants:!1,activationWidth:10,diagramMarginX:50,diagramMarginY:10,actorMargin:50,width:150,height:65,boxMargin:10,boxTextMargin:5,noteMargin:10,messageMargin:35,messageAlign:"center",mirrorActors:!0,forceMenus:!1,bottomMarginAdj:1,rightAngles:!1,showSequenceNumbers:!1,actorFontSize:14,actorFontFamily:'"Open Sans", sans-serif',actorFontWeight:400,noteFontSize:14,noteFontFamily:'"trebuchet ms", verdana, arial, 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if(r=this.tokenizer.tag(t))t=t.substring(r.raw.length),n=e[e.length-1],n&&"text"===r.type&&"text"===n.type?(n.raw+=r.raw,n.text+=r.text):e.push(r);else if(r=this.tokenizer.link(t))t=t.substring(r.raw.length),e.push(r);else if(r=this.tokenizer.reflink(t,this.tokens.links))t=t.substring(r.raw.length),n=e[e.length-1],n&&"text"===r.type&&"text"===n.type?(n.raw+=r.raw,n.text+=r.text):e.push(r);else if(r=this.tokenizer.emStrong(t,l,s))t=t.substring(r.raw.length),e.push(r);else if(r=this.tokenizer.codespan(t))t=t.substring(r.raw.length),e.push(r);else if(r=this.tokenizer.br(t))t=t.substring(r.raw.length),e.push(r);else if(r=this.tokenizer.del(t))t=t.substring(r.raw.length),e.push(r);else if(r=this.tokenizer.autolink(t))t=t.substring(r.raw.length),e.push(r);else if(this.state.inLink||!(r=this.tokenizer.url(t))){if(i=t,this.options.extensions&&this.options.extensions.startInline){let e=1/0;const r=t.slice(1);let n;this.options.extensions.startInline.forEach((t=>{n=t.call({lexer:this},r),"number"==typeof n&&n>=0&&(e=Math.min(e,n))})),e<1/0&&e>=0&&(i=t.substring(0,e+1))}if(r=this.tokenizer.inlineText(i))t=t.substring(r.raw.length),"_"!==r.raw.slice(-1)&&(s=r.raw.slice(-1)),o=!0,n=e[e.length-1],n&&"text"===n.type?(n.raw+=r.raw,n.text+=r.text):e.push(r);else if(t){const e="Infinite loop on byte: "+t.charCodeAt(0);if(this.options.silent){console.error(e);break}throw new Error(e)}}else t=t.substring(r.raw.length),e.push(r);return e}}class lt{options;parser;constructor(t){this.options=t||s}space(t){return""}code({text:t,lang:e,escaped:r}){const n=(e||"").match(/^\S*/)?.[0],i=t.replace(/\n$/,"")+"\n";return n?'
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After downloading and storing it in ",(0,d.jsx)("code",{children:"examples/lossy_list.heyvl"})," you can try Caesar on it:"]}),(0,d.jsx)(c.A,{language:"bash",children:"caesar verify examples/lossy_list.heyvl"}),(0,d.jsxs)("p",{children:["Caesar will print: ",(0,d.jsx)("code",{children:"examples/lossy_list.heyvl: Verified."})]})]})]})]})})}const u={heroBanner:"heroBanner_qdFl",heroBannerRow:"heroBannerRow_F26w",heroTitle:"heroTitle_qg2I",heroLogo:"heroLogo_U6bI",quickButtons:"quickButtons_fNHp",heroImageWrapper:"heroImageWrapper_Vsqa"};function m(){const{siteConfig:e}=(0,a.A)();return(0,d.jsx)("header",{className:n("hero hero--dark",u.heroBanner),children:(0,d.jsx)("div",{className:"container",children:(0,d.jsxs)("div",{className:`row ${u.heroBannerRow}`,children:[(0,d.jsxs)("div",{className:"col col--7",children:[(0,d.jsxs)("h1",{className:n("hero__title",u.heroTitle),children:[(0,d.jsx)("img",{src:"/img/laurel.svg",className:u.heroLogo,alt:""}),(0,d.jsx)("span",{children:e.title}),(0,d.jsx)("img",{src:"/img/laurel.svg",className:u.heroLogo,style:{transform:"scale(-1, 1)"},alt:""})]}),(0,d.jsxs)("p",{className:"hero__subtitle",children:[(0,d.jsx)("span",{style:{display:"inline-block"},children:"A Deductive Verifier\xa0"}),(0,d.jsx)("span",{style:{display:"inline-block"},children:"for Probabilistic Programs"})]}),(0,d.jsxs)("div",{className:u.quickButtons,children:[(0,d.jsx)(r.A,{className:"button button--primary",to:"/docs/getting-started",children:"Get Started \u2192"}),(0,d.jsx)(r.A,{className:"button button--primary",to:"https://marketplace.visualstudio.com/items?itemName=rwth-moves.caesar",children:"VSCode Extension"}),(0,d.jsx)(r.A,{className:"button button--primary",to:"/docs/",children:"Docs"})]})]}),(0,d.jsx)("div",{className:`col col--5 ${u.heroImageWrapper}`,children:(0,d.jsx)(r.A,{to:"/docs/",className:"margin--md shadow--md",children:(0,d.jsx)("img",{src:"img/architecture-oopsla23.svg",alt:"Architecture diagram for Caesar"})})})]})})})}function g(){const{siteConfig:e}=(0,a.A)();return(0,d.jsxs)(o.A,{title:"Caesar Verification Infrastructure",description:`${e.tagline}`,children:[(0,d.jsx)(m,{}),(0,d.jsx)("main",{children:(0,d.jsx)(x,{})})]})}}}]); \ No newline at end of file +"use strict";(self.webpackChunkcaesar_website=self.webpackChunkcaesar_website||[]).push([[2634],{2083:(e,s,i)=>{i.r(s),i.d(s,{default:()=>g});i(6540);function t(e){var s,i,n="";if("string"==typeof e||"number"==typeof e)n+=e;else if("object"==typeof e)if(Array.isArray(e))for(s=0;s(0,d.jsx)(p,{...e},s)))}),(0,d.jsx)("hr",{style:{marginTop:"2.5em",marginBottom:"2.5em"}}),(0,d.jsx)("div",{className:"row",children:(0,d.jsxs)("div",{className:"col col--12",children:[(0,d.jsx)("h2",{children:"Features of HeyVL \u2014 Lossy List Traversal Example"}),(0,d.jsx)("p",{children:"Let's look at a program that traverses a list but has a chance of crashing during the traversal. 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+ 0.4 \\not\\geq \\mathbb{E}(\\mathtt{x})"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.7063em",verticalAlign:"-0.0951em"}}),(0,l.jsx)(e.span,{className:"mord",children:(0,l.jsx)(e.span,{className:"mord mathtt",children:"init_x"})}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mbin",children:"+"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,l.jsx)(e.span,{className:"mord",children:"0.4"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:(0,l.jsx)(e.span,{className:"mord 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mathtt",children:"x"}),(0,l.jsx)(e.span,{className:"mclose",children:")"})]})]})]}),"."]})]})]})]}),"\n",(0,l.jsx)(e.h2,{id:"writing-specifications",children:"Writing Specifications"}),"\n",(0,l.jsxs)(e.p,{children:["The specifications in HeyVL follow the ",(0,l.jsx)(e.em,{children:"probabilistic predicate transformer"})," paradigm.\nCaesar transforms a ",(0,l.jsx)(e.em,{children:"post"})," backwards through the program to compute its expected value.\nThe result is then checked against the ",(0,l.jsx)(e.em,{children:"pre"}),", for each possible initial state."]}),"\n",(0,l.jsxs)(e.p,{children:["There is an inductive definition on the ",(0,l.jsx)(e.a,{href:"/docs/heyvl/statements",children:"statements documentation page"})," for the ",(0,l.jsx)(e.em,{children:"verification pre-expectation transformer"})," ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{mathvariant:"normal",children:"v"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"p"})]}),(0,l.jsx)(e.mo,{mathvariant:"normal",lspace:"0em",rspace:"0em",children:"\u27e6"}),(0,l.jsx)(e.mrow,{}),(0,l.jsx)(e.mi,{children:"S"}),(0,l.jsx)(e.mo,{mathvariant:"normal",lspace:"0em",rspace:"0em",children:"\u27e7"}),(0,l.jsx)(e.mo,{stretchy:"false",children:"("}),(0,l.jsx)(e.mi,{children:"\u03c6"}),(0,l.jsx)(e.mo,{stretchy:"false",children:")"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\mathrm{vp}\\llbracket{}S\\rrbracket(\\varphi)"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mord",children:(0,l.jsx)(e.span,{className:"mord mathrm",children:"vp"})}),(0,l.jsxs)(e.span,{className:"mopen",children:[(0,l.jsx)(e.span,{className:"mopen",children:"["}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"-0.1778em"}}),(0,l.jsx)(e.span,{className:"mopen",children:"["})]}),(0,l.jsx)(e.span,{className:"mord"}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.05764em"},children:"S"}),(0,l.jsxs)(e.span,{className:"mclose",children:[(0,l.jsx)(e.span,{className:"mclose",children:"]"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"-0.1778em"}}),(0,l.jsx)(e.span,{className:"mclose",children:"]"})]}),(0,l.jsx)(e.span,{className:"mopen",children:"("}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"\u03c6"}),(0,l.jsx)(e.span,{className:"mclose",children:")"})]})})]})," that Caesar uses to compute the expected value of the post ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{children:"\u03c6"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\varphi"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"\u03c6"})]})})]})," after running statements ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{children:"S"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"S"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.05764em"},children:"S"})]})})]}),".\nHowever, for now it is sufficient to think of expected values as they intuitively are."]}),"\n",(0,l.jsxs)(e.h3,{id:"the-specification-type-eureal-for-quantitative-reasoning",children:["The Specification Type: ",(0,l.jsx)(e.code,{children:"EUReal"})," for Quantitative Reasoning"]}),"\n",(0,l.jsxs)(e.p,{children:["In HeyVL, the ",(0,l.jsx)(e.code,{children:"pre"})," and ",(0,l.jsx)(e.code,{children:"post"})," expressions have type ",(0,l.jsx)(e.a,{href:"/docs/stdlib/numbers#eureal",children:(0,l.jsx)(e.code,{children:"EUReal"})}),", i.e. non-negative real numbers or infinity.\nWe have ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mtext,{mathvariant:"monospace",children:"EUReal"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mo,{stretchy:"false",children:"{"}),(0,l.jsx)(e.mtext,{children:"\u2009"}),(0,l.jsx)(e.mi,{children:"x"}),(0,l.jsx)(e.mo,{children:"\u2208"}),(0,l.jsx)(e.mi,{mathvariant:"double-struck",children:"R"}),(0,l.jsx)(e.mo,{children:"\u2223"}),(0,l.jsx)(e.mi,{children:"x"}),(0,l.jsx)(e.mo,{children:"\u2265"}),(0,l.jsx)(e.mn,{children:"0"}),(0,l.jsx)(e.mtext,{children:"\u2009"}),(0,l.jsx)(e.mo,{stretchy:"false",children:"}"}),(0,l.jsx)(e.mo,{children:"\u222a"}),(0,l.jsx)(e.mo,{stretchy:"false",children:"{"}),(0,l.jsx)(e.mtext,{children:"\u2009"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u221e"}),(0,l.jsx)(e.mtext,{children:"\u2009"}),(0,l.jsx)(e.mo,{stretchy:"false",children:"}"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\texttt{EUReal} = \\set{ x \\in \\mathbb{R} \\mid x \\geq 0 } \\cup \\set{ \\infty }"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6111em"}}),(0,l.jsx)(e.span,{className:"mord text",children:(0,l.jsx)(e.span,{className:"mord texttt",children:"EUReal"})}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mopen",children:"{"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"x"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"\u2208"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mord mathbb",children:"R"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"\u2223"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.7719em",verticalAlign:"-0.136em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"x"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"\u2265"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mord",children:"0"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mclose",children:"}"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mbin",children:"\u222a"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mopen",children:"{"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mord",children:"\u221e"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mclose",children:"}"})]})]})]}),"."]}),"\n",(0,l.jsxs)(e.p,{children:["This allows us to reason about various kinds expected values of probabilistic programs.\nHowever, all of these expected values are limited to non-negative values.\nThis is seldom a practical limitation.",(0,l.jsx)(e.sup,{children:(0,l.jsx)(e.a,{href:"#user-content-fn-on-negative-numbers",id:"user-content-fnref-on-negative-numbers","data-footnote-ref":!0,"aria-describedby":"footnote-label",children:"1"})})]}),"\n",(0,l.jsxs)(e.p,{children:["HeyVL supports the syntax of the ",(0,l.jsxs)(e.a,{href:"https://doi.org/10.1145/3434320",children:[(0,l.jsx)(e.em,{children:"relatively complete assertion language"})," by Batz et al"]}),".\nThis means we can express practically all expected values of interest in our syntax.\nFor more information, read the ",(0,l.jsx)(e.a,{href:"/docs/heyvl/expressions#relative-completeness",children:"Caesar documentation on relative completeness"}),"."]}),"\n",(0,l.jsx)(e.h3,{id:"embedding-boolean-specifications",children:"Embedding Boolean Specifications"}),"\n",(0,l.jsxs)(e.p,{children:["The ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{mathvariant:"normal",children:"v"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"p"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\mathrm{vp}"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,l.jsx)(e.span,{className:"mord",children:(0,l.jsx)(e.span,{className:"mord mathrm",children:"vp"})})]})})]})," semantics of HeyVL is a generalization of classical (Boolean) predicate transformer reasonining as found in verifiers such as ",(0,l.jsx)(e.a,{href:"https://dafny.org/",children:"Dafny"}),".\nThe Boolean reasoning can be fully embedded into our quantitative setting."]}),"\n",(0,l.jsxs)(e.p,{children:["The ",(0,l.jsx)(e.a,{href:"/docs/heyvl/expressions",children:(0,l.jsx)(e.em,{children:"embed expression"})})," ",(0,l.jsx)(e.code,{children:"?(b)"})," takes a Boolean expression ",(0,l.jsx)(e.code,{children:"b"})," and maps it to ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u221e"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\infty"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(e.span,{className:"mord",children:"\u221e"})]})})]})," if ",(0,l.jsx)(e.code,{children:"b"})," evaluates to true in the current state.\nOtherwise, ",(0,l.jsx)(e.code,{children:"?(b)"})," evaluates to ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mn,{children:"0"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"0"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,l.jsx)(e.span,{className:"mord",children:"0"})]})})]}),"."]}),"\n",(0,l.jsxs)(e.p,{children:["The following procedure named ",(0,l.jsx)(e.code,{children:"forty_two"})," accepts a single parameter, ",(0,l.jsx)(e.code,{children:"x"})," of type ",(0,l.jsx)(e.code,{children:"UInt"})," and returns a value ",(0,l.jsx)(e.code,{children:"y"})," of type ",(0,l.jsx)(e.code,{children:"UInt"}),".\nThe procedure has a body with just a single assignment statement that increments ",(0,l.jsx)(e.code,{children:"x"})," by 1 and saves the result in ",(0,l.jsx)(e.code,{children:"y"}),"."]}),"\n",(0,l.jsx)(e.pre,{children:(0,l.jsx)(e.code,{className:"language-heyvl",children:"proc forty_two(x: UInt) -> (y: UInt)\n pre ?(x == 41)\n post ?(y == 42)\n{\n y = x + 1\n}\n"})}),"\n",(0,l.jsxs)(e.p,{children:["The ",(0,l.jsx)(e.code,{children:"proc"})," verifies that ",(0,l.jsx)(e.code,{children:"y == 42"})," holds after all executions (",(0,l.jsx)(e.code,{children:"post"}),") when we started in a state that satisfied ",(0,l.jsx)(e.code,{children:"x == 41"})," (",(0,l.jsx)(e.code,{children:"pre"}),")."]}),"\n",(0,l.jsxs)(e.p,{children:["Why do the semantics work out so neatly?\nLet's examine the semantics manually to see how it works.\nRecall that the ",(0,l.jsx)(e.a,{href:"/docs/heyvl/statements#semantics",children:"verification pre-expectation transformer"})," ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{mathvariant:"normal",children:"v"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"p"})]}),(0,l.jsx)(e.mo,{mathvariant:"normal",lspace:"0em",rspace:"0em",children:"\u27e6"}),(0,l.jsx)(e.mi,{children:"S"}),(0,l.jsx)(e.mo,{mathvariant:"normal",lspace:"0em",rspace:"0em",children:"\u27e7"}),(0,l.jsx)(e.mo,{stretchy:"false",children:"("}),(0,l.jsx)(e.mi,{children:"\u03c6"}),(0,l.jsx)(e.mo,{stretchy:"false",children:")"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\mathrm{vp}\\llbracket{S}\\rrbracket(\\varphi)"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mord",children:(0,l.jsx)(e.span,{className:"mord 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",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"x"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mn,{children:"41"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"x = 41"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(e.span,{className:"mord 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42"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,l.jsx)(e.span,{className:"mord",children:"42"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,l.jsx)(e.span,{className:"mord",children:"42"})]})]})]}),", e.g. the initial state ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"x"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mn,{children:"0"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"x = 0"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(e.span,{className:"mord 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",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"y"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mn,{children:"42"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"y = 42"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,l.jsx)(e.span,{className:"mord 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41"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"x"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,l.jsx)(e.span,{className:"mord",children:"41"})]})]})]}),'?".']}),"\n",(0,l.jsxs)(a,{children:[(0,l.jsxs)("summary",{children:["How To: Obtaining the above formula via ",(0,l.jsx)("code",{children:"--print-theorem --no-slice-error"}),"."]}),(0,l.jsxs)(e.p,{children:["Using the ",(0,l.jsx)("code",{children:"--print-theorem"})," command-line flag, you can print the theorem Caesar tries to prove about your (co)procedures.\nThe result 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",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u221e"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\infty"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(e.span,{className:"mord",children:"\u221e"})]})})]})," otherwise.",(0,l.jsx)(e.sup,{children:(0,l.jsx)(e.a,{href:"#user-content-fn-bang-question-operator",id:"user-content-fnref-bang-question-operator","data-footnote-ref":!0,"aria-describedby":"footnote-label",children:"2"})})]}),"\n",(0,l.jsx)(e.pre,{children:(0,l.jsx)(e.code,{className:"language-heyvl",children:"coproc forty_two_upper2(x: UInt) -> (y: UInt)\n pre !?(x == 41)\n post !?(y == 42)\n{\n y = 42\n}\n"})}),"\n",(0,l.jsxs)(e.p,{children:["In this example, the ",(0,l.jsx)(e.code,{children:"pre"})," works as expected.\nIt tells the verifier to verify only from initial states that satisfy ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"x"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mn,{children:"41"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"x = 41"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(e.span,{className:"mord 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42"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"y"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,l.jsx)(e.span,{className:"mord",children:"42"})]})]})]})," \u2014 the set of initial states that reach ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"y"}),(0,l.jsx)(e.mo,{mathvariant:"normal",children:"\u2260"}),(0,l.jsx)(e.mn,{children:"42"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"y \\neq 42"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"y"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsxs)(e.span,{className:"mrel",children:[(0,l.jsx)(e.span,{className:"mrel",children:(0,l.jsx)(e.span,{className:"mord 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",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"x"}),(0,l.jsx)(e.mo,{mathvariant:"normal",children:"\u2260"}),(0,l.jsx)(e.mn,{children:"41"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"x \\neq 41"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"x"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsxs)(e.span,{className:"mrel",children:[(0,l.jsx)(e.span,{className:"mrel",children:(0,l.jsx)(e.span,{className:"mord 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annotations are logically combined.",(0,l.jsx)(e.sup,{children:(0,l.jsx)(e.a,{href:"#user-content-fn-spec-combination",id:"user-content-fnref-spec-combination","data-footnote-ref":!0,"aria-describedby":"footnote-label",children:"3"})}),"\n",(0,l.jsx)(e.code,{children:"\u2293"})," is the minimum operator and ",(0,l.jsx)(e.code,{children:"\u2294"})," maximum operator (see ",(0,l.jsx)(e.a,{href:"/docs/heyvl/expressions",children:"expressions documentation"}),")."]}),"\n",(0,l.jsxs)(e.table,{children:[(0,l.jsx)(e.thead,{children:(0,l.jsxs)(e.tr,{children:[(0,l.jsx)(e.th,{}),(0,l.jsx)(e.th,{children:(0,l.jsx)(e.code,{children:"pre A pre B"})}),(0,l.jsx)(e.th,{children:(0,l.jsx)(e.code,{children:"post C post D"})})]})}),(0,l.jsxs)(e.tbody,{children:[(0,l.jsxs)(e.tr,{children:[(0,l.jsx)(e.td,{children:(0,l.jsx)(e.code,{children:"proc"})}),(0,l.jsx)(e.td,{children:(0,l.jsx)(e.code,{children:"pre (A \u2293 B)"})}),(0,l.jsx)(e.td,{children:(0,l.jsx)(e.code,{children:"post (C \u2293 D)"})})]}),(0,l.jsxs)(e.tr,{children:[(0,l.jsx)(e.td,{children:(0,l.jsx)(e.code,{children:"coproc"})}),(0,l.jsx)(e.td,{children:(0,l.jsx)(e.code,{children:"pre (A \u2294 B)"})}),(0,l.jsx)(e.td,{children:(0,l.jsx)(e.code,{children:"post (C \u2294 D)"})})]})]})]}),"\n",(0,l.jsxs)(e.p,{children:["In ",(0,l.jsx)(e.code,{children:"proc"}),"s, we combine ",(0,l.jsx)(e.code,{children:"pre"}),"s and ",(0,l.jsx)(e.code,{children:"post"}),"s with the minimum ",(0,l.jsx)(e.code,{children:"\u2293"}),".\nIn ",(0,l.jsx)(e.code,{children:"coproc"}),"s, the combination is dual and uses the maximum ",(0,l.jsx)(e.code,{children:"\u2294"}),"."]}),"\n",(0,l.jsx)(e.p,{children:"This generalizes the Boolean setting neatly:"}),"\n",(0,l.jsxs)(e.table,{children:[(0,l.jsx)(e.thead,{children:(0,l.jsxs)(e.tr,{children:[(0,l.jsx)(e.th,{}),(0,l.jsx)(e.th,{children:(0,l.jsx)(e.code,{children:"pre ?(A) pre ?(B)"})}),(0,l.jsx)(e.th,{children:(0,l.jsx)(e.code,{children:"post ?(C) post ?(D)"})})]})}),(0,l.jsxs)(e.tbody,{children:[(0,l.jsxs)(e.tr,{children:[(0,l.jsx)(e.td,{children:(0,l.jsx)(e.code,{children:"proc"})}),(0,l.jsx)(e.td,{children:(0,l.jsx)(e.code,{children:"pre ?(A && B)"})}),(0,l.jsx)(e.td,{children:(0,l.jsx)(e.code,{children:"post ?(C && D)"})})]}),(0,l.jsxs)(e.tr,{children:[(0,l.jsx)(e.td,{children:(0,l.jsx)(e.code,{children:"coproc"})}),(0,l.jsx)(e.td,{children:(0,l.jsx)(e.code,{children:"pre ?(A || B)"})}),(0,l.jsx)(e.td,{children:(0,l.jsx)(e.code,{children:"post ?(C || D)"})})]})]})]}),"\n",(0,l.jsxs)(e.p,{children:["If ",(0,l.jsxs)(e.a,{href:"#usually-you-want-coembed",children:["we use ",(0,l.jsx)(e.code,{children:"!?(b)"})," in ",(0,l.jsx)(e.code,{children:"coproc"}),"s"]}),", then we obtain that ",(0,l.jsx)(e.code,{children:"pre !?(A) pre !?(B)"})," is equivalent to ",(0,l.jsx)(e.code,{children:"pre !?(A && B)"})," as one might expect.\nSame for ",(0,l.jsx)(e.code,{children:"post"})," annotations."]}),"\n",(0,l.jsxs)(e.p,{children:["The specification is optional; if it's not provided, Caesar will add a default specification: ",(0,l.jsx)(e.code,{children:"pre ?(true)"})," and ",(0,l.jsx)(e.code,{children:"post ?(true)"})," for procedures and ",(0,l.jsx)(e.code,{children:"pre ?(false)"})," and ",(0,l.jsx)(e.code,{children:"post ?(false)"})," for coprocedures."]}),"\n",(0,l.jsxs)(e.p,{children:["The defaults correspond to the empty conjunction (",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mo,{children:"\u22c0"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u2205"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{mathvariant:"normal",children:"t"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"r"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"u"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"e"})]})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\bigwedge \\emptyset = \\mathrm{true}"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mop op-symbol small-op",style:{position:"relative",top:"0em"},children:"\u22c0"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mord",children:"\u2205"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6151em"}}),(0,l.jsx)(e.span,{className:"mord",children:(0,l.jsx)(e.span,{className:"mord mathrm",children:"true"})})]})]})]}),") and empty disjunction (",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mo,{children:"\u22c1"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u2205"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{mathvariant:"normal",children:"f"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"a"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"l"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"s"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"e"})]})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\bigvee \\emptyset = \\mathrm{false}"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mop op-symbol small-op",style:{position:"relative",top:"0em"},children:"\u22c1"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mord",children:"\u2205"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6944em"}}),(0,l.jsx)(e.span,{className:"mord",children:(0,l.jsx)(e.span,{className:"mord mathrm",children:"false"})})]})]})]}),").\nThe quantitative setting behaves the same, we have ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"inf"}),(0,l.jsx)(e.mo,{children:"\u2061"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u2205"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u221e"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\inf \\emptyset = \\infty"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.8056em",verticalAlign:"-0.0556em"}}),(0,l.jsxs)(e.span,{className:"mop",children:["in",(0,l.jsx)(e.span,{style:{marginRight:"0.07778em"},children:"f"})]}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mord",children:"\u2205"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(e.span,{className:"mord",children:"\u221e"})]})]})]})," and ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"sup"}),(0,l.jsx)(e.mo,{children:"\u2061"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u2205"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mn,{children:"0"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\sup \\emptyset = 0"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.9444em",verticalAlign:"-0.1944em"}}),(0,l.jsx)(e.span,{className:"mop",children:"sup"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mord",children:"\u2205"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,l.jsx)(e.span,{className:"mord",children:"0"})]})]})]}),"."]}),"\n",(0,l.jsx)(e.h3,{id:"procs-without-body",children:"Procedures Without a Body"}),"\n",(0,l.jsxs)(e.p,{children:["Procedures and coprocedures can be written without a corresponding body.\nThis allows us to ",(0,l.jsx)(e.em,{children:(0,l.jsx)(e.a,{href:"https://www.cs.ox.ac.uk/publications/books/PfS/",children:"program from specifications"})}),", i.e. write and verify programs incrementally.\nWe can start with a specification and step-by-step fill in executable code (or just don't if we don't feel like it)."]}),"\n",(0,l.jsxs)(e.p,{children:["Since ",(0,l.jsx)(e.a,{href:"#calling-procedures",children:"procedure calls only need the callee specification"}),", we can just write a procedure specification and use it in other places.\nThere does not need to be procedure body of the callee at all."]}),"\n",(0,l.jsxs)(e.p,{children:["This can be used to introduce unsoundness even without using ",(0,l.jsxs)(e.a,{href:"/docs/heyvl/statements#assert-and-assume",children:[(0,l.jsx)(e.code,{children:"assume"})," statements"]}),".\nCalling the following procedure is essentially the same as writing ",(0,l.jsx)(e.code,{children:"assume ?(false)"}),"."]}),"\n",(0,l.jsx)(e.pre,{children:(0,l.jsx)(e.code,{className:"language-heyvl",children:"proc unsound() -> ()\n pre ?(true)\n post ?(false)\n"})}),"\n",(0,l.jsx)(e.h2,{id:"calling-procedures",children:"Calling Procedures"}),"\n",(0,l.jsxs)(e.p,{children:["We can call ",(0,l.jsx)(e.code,{children:"proc"}),"s from ",(0,l.jsx)(e.code,{children:"proc"}),"s and ",(0,l.jsx)(e.code,{children:"coproc"}),"s from ",(0,l.jsx)(e.code,{children:"coproc"}),"s.\nMixing both is unsound and will result in an error.\nIn the following, we will talk about procedures, but everything applies to coprocedures as well."]}),"\n",(0,l.jsxs)(e.p,{children:["Procedure calls make use of the procedure's specification ",(0,l.jsx)(e.em,{children:"only"})," and do not inspect the procedure body.\nSomewhat informally, we could say that ",(0,l.jsx)(e.em,{children:"assuming"})," the callee procedure verifies, the procedure call can be replaced by the procedure's body and the program will still verify.\nThis enables modular reasoning: one can verify a big program and we can re-use already-verified parts of it in other parts."]}),"\n",(0,l.jsx)(e.h3,{id:"example-a-spare-engine",children:"Example: A Spare Engine"}),"\n",(0,l.jsxs)(e.p,{children:["The following example models running an engine with a spare engine available.\nIn ",(0,l.jsx)(e.code,{children:"runPrimaryOrSpare"}),", the primary engine works with probability of 90%.\nIf it fails, we have another spare available that is guaranteed to work with the same probability."]}),"\n",(0,l.jsx)(e.pre,{children:(0,l.jsx)(e.code,{className:"language-heyvl",children:"proc spare() -> (working: Bool)\n pre 0.9\n post [working]\n{\n working = flip(0.95)\n}\n\nproc runPrimaryOrSpare() -> (working: Bool)\n pre 0.99\n post [working]\n{\n working = flip(0.9)\n if !working {\n working = spare()\n } else {}\n}\n"})}),"\n",(0,l.jsxs)(e.p,{children:["The specification of ",(0,l.jsx)(e.code,{children:"spare"})," says ",(0,l.jsx)(e.code,{children:"working"})," is true with probability of at least ",(0,l.jsx)(e.code,{children:"0.9"})," (and in fact the coin flip says it is ",(0,l.jsx)(e.code,{children:"0.95"}),").\nThus, we can verify a success probability of ",(0,l.jsx)(e.code,{children:"runOrSpare"})," of at least ",(0,l.jsx)(e.code,{children:"0.99"}),".\nHowever, we cannot verify a lower bound of ",(0,l.jsx)(e.code,{children:"0.995"})," because the specification of ",(0,l.jsx)(e.code,{children:"spare"})," does not guarantee it (even though it is true here)."]}),"\n",(0,l.jsx)(e.h3,{id:"multiple-or-no-return-values",children:"Multiple Or No Return Values"}),"\n",(0,l.jsxs)(e.p,{children:["If the callee has multiple return values, multiple variables may be assigned to by comma-separating them.\nFor example, if ",(0,l.jsx)(e.code,{children:"spare"})," returned two variables, we could obtain the results via ",(0,l.jsx)(e.code,{children:"working1, working2 = spare()"}),"."]}),"\n",(0,l.jsxs)(e.p,{children:["If the callee has no return values, we just call it without the assignment.\nFor example, we simply write ",(0,l.jsx)(e.code,{children:"spare()"})," as a statement to call it without assignment."]}),"\n",(0,l.jsx)(e.h3,{id:"assert-assume-understanding-of-procedure-calls",children:"Assert-Assume Understanding of Procedure Calls"}),"\n",(0,l.jsxs)(e.p,{children:["Internally, procedure calls are translated to HeyVL verification statements (see ",(0,l.jsx)(e.a,{href:"/docs/heyvl/statements",children:"statements documentation"}),").\nWe can understand how procedure calls work by understanding this encoding."]}),"\n",(0,l.jsxs)(e.p,{children:["During the verification of ",(0,l.jsx)(e.code,{children:"runPrimaryOrSpare"}),", we replace the call to ",(0,l.jsx)(e.code,{children:"spare"})," by a sequence of ",(0,l.jsx)(e.code,{children:"assert"}),", ",(0,l.jsx)(e.code,{children:"havoc,"})," ",(0,l.jsx)(e.code,{children:"validate"}),", and ",(0,l.jsx)(e.code,{children:"assume"})," statements."]}),"\n",(0,l.jsx)(e.pre,{children:(0,l.jsx)(e.code,{className:"language-heyvl",children:"proc runPrimaryOrSpare() -> (working: Bool)\n pre 0.99\n post [working]\n{\n working = flip(0.9)\n if !working {\n assert 0.95\n havoc success\n validate\n assume [success]\n } else {}\n}\n"})}),"\n",(0,l.jsxs)(a,{children:[(0,l.jsxs)("summary",{children:["How To: Obtaining intermediate encodings via ",(0,l.jsx)("code",{children:"--print-core"}),"."]}),(0,l.jsxs)(e.p,{children:["To obtain the intermediate encodings from Caesar, we can use the ",(0,l.jsx)("code",{children:"--print-core"})," command-line flag.\nThis will print the fully desugared HeyVL code for each procedure to standard output.\nFor example:"]}),(0,l.jsx)(e.pre,{children:(0,l.jsx)(e.code,{className:"language-bash",children:"caesar verify --print-core example.heyvl\n"})})]}),"\n",(0,l.jsx)(e.p,{children:"We can now read the encoding as follows:"}),"\n",(0,l.jsxs)(e.ol,{children:["\n",(0,l.jsxs)(e.li,{children:["Before the call, ",(0,l.jsx)(e.code,{children:"assert"})," the ",(0,l.jsx)(e.code,{children:"pre"})," ",(0,l.jsx)(e.code,{children:"spare()"}),". This means that the expected value (going backwards) is at most ",(0,l.jsx)(e.code,{children:"0.95"}),"."]}),"\n",(0,l.jsxs)(e.li,{children:["We then forget the current value of ",(0,l.jsx)(e.code,{children:"success"})," via ",(0,l.jsx)(e.code,{children:"havoc"}),". This models that ",(0,l.jsx)(e.code,{children:"spare"})," can have arbitrary effects on return values."]}),"\n",(0,l.jsxs)(e.li,{children:["We ",(0,l.jsx)(e.code,{children:"validate"})," (the following assumption must be absolutely true)."]}),"\n",(0,l.jsxs)(e.li,{children:["We can only ",(0,l.jsx)(e.code,{children:"assume"})," the ",(0,l.jsx)(e.code,{children:"post"})," of ",(0,l.jsx)(e.code,{children:"spare"}),", i.e. that ",(0,l.jsx)(e.code,{children:"success"})," is true."]}),"\n"]}),"\n",(0,l.jsx)(e.p,{children:"This generalizes the Boolean setting:"}),"\n",(0,l.jsxs)(e.ol,{children:["\n",(0,l.jsxs)(e.li,{children:["Before the call, the ",(0,l.jsx)(e.code,{children:"pre"})," must hold."]}),"\n",(0,l.jsx)(e.li,{children:"Forget the current value of modified variables."}),"\n",(0,l.jsxs)(e.li,{children:["(The ",(0,l.jsx)(e.code,{children:"validate"})," does not have any effect in the Boolean setting.)"]}),"\n",(0,l.jsxs)(e.li,{children:["Assume that the ",(0,l.jsx)(e.code,{children:"post"})," holds after the call."]}),"\n"]}),"\n",(0,l.jsxs)(e.p,{children:["A more formal treatment of the encoding and semantics of procedure calls can be found in our ",(0,l.jsx)(e.a,{href:"/docs/publications#oopsla-23",children:"OOPSLA '23 paper"}),"."]}),"\n","\n",(0,l.jsxs)(e.section,{"data-footnotes":!0,className:"footnotes",children:[(0,l.jsx)(e.h2,{className:"sr-only",id:"footnote-label",children:"Footnotes"}),"\n",(0,l.jsxs)(e.ol,{children:["\n",(0,l.jsxs)(e.li,{id:"user-content-fn-on-negative-numbers",children:["\n",(0,l.jsxs)(e.p,{children:["Many verification tasks that require reasoning with negative numbers can be embedded in this framework. First, note that we can still have negative numbers in our program states, we just have to ensure that the ",(0,l.jsx)(e.code,{children:"post"})," is non-negative. ",(0,l.jsxs)(e.a,{href:"https://dl.acm.org/doi/pdf/10.1145/3434320#page=24",children:["Chapter 11 of the paper ",(0,l.jsx)(e.em,{children:'"Relatively Complete Verification for Probabilistic Programs"'})]})," by Batz et al. might be of interest for further reading. ",(0,l.jsx)(e.a,{href:"#user-content-fnref-on-negative-numbers","data-footnote-backref":"","aria-label":"Back to reference 1",className:"data-footnote-backref",children:"\u21a9"})]}),"\n"]}),"\n",(0,l.jsxs)(e.li,{id:"user-content-fn-bang-question-operator",children:["\n",(0,l.jsxs)(e.p,{children:[(0,l.jsx)(e.code,{children:"!?(b)"})," is not a special kind of operator, it is simply the ",(0,l.jsx)(e.code,{children:"!"})," operator applied to ",(0,l.jsx)(e.code,{children:"?(b)"}),", i.e. ",(0,l.jsx)(e.code,{children:"!(?(b))"}),".\nThe negation operator ",(0,l.jsx)(e.code,{children:"!"})," is defined on ",(0,l.jsx)(e.code,{children:"EUReal"})," by ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mo,{stretchy:"false",children:"!"}),(0,l.jsx)(e.mn,{children:"0"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u221e"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"!0 = 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",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{children:"\u03c6"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\varphi"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"\u03c6"})]})})]})," after running statements ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{children:"S"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"S"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.05764em"},children:"S"})]})})]}),".\nHowever, for now it is sufficient to think of expected values as they intuitively are."]}),"\n",(0,l.jsxs)(e.h3,{id:"the-specification-type-eureal-for-quantitative-reasoning",children:["The Specification Type: ",(0,l.jsx)(e.code,{children:"EUReal"})," for Quantitative Reasoning"]}),"\n",(0,l.jsxs)(e.p,{children:["In HeyVL, the ",(0,l.jsx)(e.code,{children:"pre"})," and ",(0,l.jsx)(e.code,{children:"post"})," expressions have type ",(0,l.jsx)(e.a,{href:"/docs/stdlib/numbers#eureal",children:(0,l.jsx)(e.code,{children:"EUReal"})}),", i.e. non-negative real numbers or infinity.\nWe have ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mtext,{mathvariant:"monospace",children:"EUReal"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mo,{stretchy:"false",children:"{"}),(0,l.jsx)(e.mtext,{children:"\u2009"}),(0,l.jsx)(e.mi,{children:"x"}),(0,l.jsx)(e.mo,{children:"\u2208"}),(0,l.jsx)(e.mi,{mathvariant:"double-struck",children:"R"}),(0,l.jsx)(e.mo,{children:"\u2223"}),(0,l.jsx)(e.mi,{children:"x"}),(0,l.jsx)(e.mo,{children:"\u2265"}),(0,l.jsx)(e.mn,{children:"0"}),(0,l.jsx)(e.mtext,{children:"\u2009"}),(0,l.jsx)(e.mo,{stretchy:"false",children:"}"}),(0,l.jsx)(e.mo,{children:"\u222a"}),(0,l.jsx)(e.mo,{stretchy:"false",children:"{"}),(0,l.jsx)(e.mtext,{children:"\u2009"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u221e"}),(0,l.jsx)(e.mtext,{children:"\u2009"}),(0,l.jsx)(e.mo,{stretchy:"false",children:"}"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\texttt{EUReal} = \\set{ x \\in \\mathbb{R} \\mid x \\geq 0 } \\cup \\set{ \\infty }"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6111em"}}),(0,l.jsx)(e.span,{className:"mord text",children:(0,l.jsx)(e.span,{className:"mord texttt",children:"EUReal"})}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mopen",children:"{"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"x"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"\u2208"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mord mathbb",children:"R"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"\u2223"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.7719em",verticalAlign:"-0.136em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"x"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"\u2265"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mord",children:"0"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mclose",children:"}"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mbin",children:"\u222a"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mopen",children:"{"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mord",children:"\u221e"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mclose",children:"}"})]})]})]}),"."]}),"\n",(0,l.jsxs)(e.p,{children:["This allows us to reason about various kinds expected values of probabilistic programs.\nHowever, all of these expected values are limited to non-negative values.\nThis is seldom a practical limitation.",(0,l.jsx)(e.sup,{children:(0,l.jsx)(e.a,{href:"#user-content-fn-on-negative-numbers",id:"user-content-fnref-on-negative-numbers","data-footnote-ref":!0,"aria-describedby":"footnote-label",children:"1"})})]}),"\n",(0,l.jsxs)(e.p,{children:["HeyVL supports the syntax of the ",(0,l.jsxs)(e.a,{href:"https://doi.org/10.1145/3434320",children:[(0,l.jsx)(e.em,{children:"relatively complete assertion language"})," by Batz et al"]}),".\nThis means we can express practically all expected values of interest in our syntax.\nFor more information, read the ",(0,l.jsx)(e.a,{href:"/docs/heyvl/expressions#relative-completeness",children:"Caesar documentation on relative completeness"}),"."]}),"\n",(0,l.jsx)(e.h3,{id:"embedding-boolean-specifications",children:"Embedding Boolean Specifications"}),"\n",(0,l.jsxs)(e.p,{children:["The ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{mathvariant:"normal",children:"v"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"p"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\mathrm{vp}"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,l.jsx)(e.span,{className:"mord",children:(0,l.jsx)(e.span,{className:"mord mathrm",children:"vp"})})]})})]})," semantics of HeyVL is a generalization of classical (Boolean) predicate transformer reasonining as found in verifiers such as ",(0,l.jsx)(e.a,{href:"https://dafny.org/",children:"Dafny"}),".\nThe Boolean reasoning can be fully embedded into our quantitative setting."]}),"\n",(0,l.jsxs)(e.p,{children:["The ",(0,l.jsx)(e.a,{href:"/docs/heyvl/expressions",children:(0,l.jsx)(e.em,{children:"embed expression"})})," ",(0,l.jsx)(e.code,{children:"?(b)"})," takes a Boolean expression ",(0,l.jsx)(e.code,{children:"b"})," and maps it to ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u221e"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\infty"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(e.span,{className:"mord",children:"\u221e"})]})})]})," if ",(0,l.jsx)(e.code,{children:"b"})," evaluates to true in the current state.\nOtherwise, ",(0,l.jsx)(e.code,{children:"?(b)"})," evaluates to ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mn,{children:"0"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"0"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,l.jsx)(e.span,{className:"mord",children:"0"})]})})]}),"."]}),"\n",(0,l.jsxs)(e.p,{children:["The following procedure named ",(0,l.jsx)(e.code,{children:"forty_two"})," accepts a single parameter, ",(0,l.jsx)(e.code,{children:"x"})," of type ",(0,l.jsx)(e.code,{children:"UInt"})," and returns a value ",(0,l.jsx)(e.code,{children:"y"})," of type ",(0,l.jsx)(e.code,{children:"UInt"}),".\nThe procedure has a body with just a single assignment statement that increments ",(0,l.jsx)(e.code,{children:"x"})," by 1 and saves the result in ",(0,l.jsx)(e.code,{children:"y"}),"."]}),"\n",(0,l.jsx)(e.pre,{children:(0,l.jsx)(e.code,{className:"language-heyvl",children:"proc forty_two(x: UInt) -> (y: UInt)\n pre ?(x == 41)\n post ?(y == 42)\n{\n y = x + 1\n}\n"})}),"\n",(0,l.jsxs)(e.p,{children:["The ",(0,l.jsx)(e.code,{children:"proc"})," verifies that ",(0,l.jsx)(e.code,{children:"y == 42"})," holds after all executions (",(0,l.jsx)(e.code,{children:"post"}),") when we started in a state that satisfied ",(0,l.jsx)(e.code,{children:"x == 41"})," (",(0,l.jsx)(e.code,{children:"pre"}),")."]}),"\n",(0,l.jsxs)(e.p,{children:["Why do the semantics work out so neatly?\nLet's examine the semantics manually to see how it works.\nRecall that the ",(0,l.jsx)(e.a,{href:"/docs/heyvl/statements#semantics",children:"verification pre-expectation transformer"})," ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{mathvariant:"normal",children:"v"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"p"})]}),(0,l.jsx)(e.mo,{mathvariant:"normal",lspace:"0em",rspace:"0em",children:"\u27e6"}),(0,l.jsx)(e.mi,{children:"S"}),(0,l.jsx)(e.mo,{mathvariant:"normal",lspace:"0em",rspace:"0em",children:"\u27e7"}),(0,l.jsx)(e.mo,{stretchy:"false",children:"("}),(0,l.jsx)(e.mi,{children:"\u03c6"}),(0,l.jsx)(e.mo,{stretchy:"false",children:")"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\mathrm{vp}\\llbracket{S}\\rrbracket(\\varphi)"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mord",children:(0,l.jsx)(e.span,{className:"mord mathrm",children:"vp"})}),(0,l.jsxs)(e.span,{className:"mopen",children:[(0,l.jsx)(e.span,{className:"mopen",children:"["}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"-0.1778em"}}),(0,l.jsx)(e.span,{className:"mopen",children:"["})]}),(0,l.jsx)(e.span,{className:"mord",children:(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.05764em"},children:"S"})}),(0,l.jsxs)(e.span,{className:"mclose",children:[(0,l.jsx)(e.span,{className:"mclose",children:"]"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"-0.1778em"}}),(0,l.jsx)(e.span,{className:"mclose",children:"]"})]}),(0,l.jsx)(e.span,{className:"mopen",children:"("}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"\u03c6"}),(0,l.jsx)(e.span,{className:"mclose",children:")"})]})})]})," in effect computes the expected value ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{mathvariant:"double-struck",children:"E"}),(0,l.jsx)(e.mo,{stretchy:"false",children:"("}),(0,l.jsx)(e.mi,{children:"\u03c6"}),(0,l.jsx)(e.mo,{stretchy:"false",children:")"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\mathbb{E}(\\varphi)"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mord mathbb",children:"E"}),(0,l.jsx)(e.span,{className:"mopen",children:"("}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"\u03c6"}),(0,l.jsx)(e.span,{className:"mclose",children:")"})]})})]})," of ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{children:"\u03c6"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\varphi"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"\u03c6"})]})})]})," after running statement 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41"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"x"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,l.jsx)(e.span,{className:"mord",children:"41"})]})]})]})," holds in the beginning, then we know that 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(",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{mathvariant:"normal",children:"t"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"r"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"u"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"e"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\mathrm{true}"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6151em"}}),(0,l.jsx)(e.span,{className:"mord",children:(0,l.jsx)(e.span,{className:"mord mathrm",children:"true"})})]})})]}),") about the final state."]}),"\n",(0,l.jsxs)(e.p,{children:["In short: we can use embed expressions ",(0,l.jsx)(e.code,{children:"?(b)"})," to write Boolean assumptions 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",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"x"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mn,{children:"41"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"x = 41"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(e.span,{className:"mord 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42"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,l.jsx)(e.span,{className:"mord",children:"42"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,l.jsx)(e.span,{className:"mord",children:"42"})]})]})]}),", e.g. the initial state ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"x"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mn,{children:"0"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"x = 0"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(e.span,{className:"mord 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",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"y"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mn,{children:"42"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"y = 42"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,l.jsx)(e.span,{className:"mord 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41"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"x"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,l.jsx)(e.span,{className:"mord",children:"41"})]})]})]}),'?".']}),"\n",(0,l.jsxs)(a,{children:[(0,l.jsxs)("summary",{children:["How To: Obtaining the above formula via ",(0,l.jsx)("code",{children:"--print-theorem --no-slice-error"}),"."]}),(0,l.jsxs)(e.p,{children:["Using the ",(0,l.jsx)("code",{children:"--print-theorem"})," command-line flag, you can print the theorem Caesar tries to prove about your (co)procedures.\nThe result 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",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u221e"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\infty"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(e.span,{className:"mord",children:"\u221e"})]})})]})," otherwise.",(0,l.jsx)(e.sup,{children:(0,l.jsx)(e.a,{href:"#user-content-fn-bang-question-operator",id:"user-content-fnref-bang-question-operator","data-footnote-ref":!0,"aria-describedby":"footnote-label",children:"2"})})]}),"\n",(0,l.jsx)(e.pre,{children:(0,l.jsx)(e.code,{className:"language-heyvl",children:"coproc forty_two_upper2(x: UInt) -> (y: UInt)\n pre !?(x == 41)\n post !?(y == 42)\n{\n y = 42\n}\n"})}),"\n",(0,l.jsxs)(e.p,{children:["In this example, the ",(0,l.jsx)(e.code,{children:"pre"})," works as expected.\nIt tells the verifier to verify only from initial states that satisfy ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"x"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mn,{children:"41"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"x = 41"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(e.span,{className:"mord 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42"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"y"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,l.jsx)(e.span,{className:"mord",children:"42"})]})]})]})," \u2014 the set of initial states that reach ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"y"}),(0,l.jsx)(e.mo,{mathvariant:"normal",children:"\u2260"}),(0,l.jsx)(e.mn,{children:"42"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"y \\neq 42"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"y"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsxs)(e.span,{className:"mrel",children:[(0,l.jsx)(e.span,{className:"mrel",children:(0,l.jsx)(e.span,{className:"mord 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",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"x"}),(0,l.jsx)(e.mo,{mathvariant:"normal",children:"\u2260"}),(0,l.jsx)(e.mn,{children:"41"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"x \\neq 41"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"x"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsxs)(e.span,{className:"mrel",children:[(0,l.jsx)(e.span,{className:"mrel",children:(0,l.jsx)(e.span,{className:"mord 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annotations are logically combined.",(0,l.jsx)(e.sup,{children:(0,l.jsx)(e.a,{href:"#user-content-fn-spec-combination",id:"user-content-fnref-spec-combination","data-footnote-ref":!0,"aria-describedby":"footnote-label",children:"3"})}),"\n",(0,l.jsx)(e.code,{children:"\u2293"})," is the minimum operator and ",(0,l.jsx)(e.code,{children:"\u2294"})," maximum operator (see ",(0,l.jsx)(e.a,{href:"/docs/heyvl/expressions",children:"expressions documentation"}),")."]}),"\n",(0,l.jsxs)(e.table,{children:[(0,l.jsx)(e.thead,{children:(0,l.jsxs)(e.tr,{children:[(0,l.jsx)(e.th,{}),(0,l.jsx)(e.th,{children:(0,l.jsx)(e.code,{children:"pre A pre B"})}),(0,l.jsx)(e.th,{children:(0,l.jsx)(e.code,{children:"post C post D"})})]})}),(0,l.jsxs)(e.tbody,{children:[(0,l.jsxs)(e.tr,{children:[(0,l.jsx)(e.td,{children:(0,l.jsx)(e.code,{children:"proc"})}),(0,l.jsx)(e.td,{children:(0,l.jsx)(e.code,{children:"pre (A \u2293 B)"})}),(0,l.jsx)(e.td,{children:(0,l.jsx)(e.code,{children:"post (C \u2293 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specification is optional; if it's not provided, Caesar will add a default specification: ",(0,l.jsx)(e.code,{children:"pre ?(true)"})," and ",(0,l.jsx)(e.code,{children:"post ?(true)"})," for procedures and ",(0,l.jsx)(e.code,{children:"pre ?(false)"})," and ",(0,l.jsx)(e.code,{children:"post ?(false)"})," for coprocedures."]}),"\n",(0,l.jsxs)(e.p,{children:["The defaults correspond to the empty conjunction (",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mo,{children:"\u22c0"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u2205"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{mathvariant:"normal",children:"t"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"r"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"u"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"e"})]})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\bigwedge \\emptyset = \\mathrm{true}"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mop op-symbol 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(",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mo,{children:"\u22c1"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u2205"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{mathvariant:"normal",children:"f"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"a"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"l"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"s"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"e"})]})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\bigvee \\emptyset = \\mathrm{false}"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mop op-symbol small-op",style:{position:"relative",top:"0em"},children:"\u22c1"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mord",children:"\u2205"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6944em"}}),(0,l.jsx)(e.span,{className:"mord",children:(0,l.jsx)(e.span,{className:"mord mathrm",children:"false"})})]})]})]}),").\nThe quantitative setting behaves the same, we have ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"inf"}),(0,l.jsx)(e.mo,{children:"\u2061"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u2205"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u221e"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\inf \\emptyset = \\infty"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.8056em",verticalAlign:"-0.0556em"}}),(0,l.jsxs)(e.span,{className:"mop",children:["in",(0,l.jsx)(e.span,{style:{marginRight:"0.07778em"},children:"f"})]}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mord",children:"\u2205"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(e.span,{className:"mord",children:"\u221e"})]})]})]})," and ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"sup"}),(0,l.jsx)(e.mo,{children:"\u2061"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u2205"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mn,{children:"0"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\sup \\emptyset = 0"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.9444em",verticalAlign:"-0.1944em"}}),(0,l.jsx)(e.span,{className:"mop",children:"sup"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mord",children:"\u2205"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,l.jsx)(e.span,{className:"mord",children:"0"})]})]})]}),"."]}),"\n",(0,l.jsx)(e.h3,{id:"procs-without-body",children:"Procedures Without a Body"}),"\n",(0,l.jsxs)(e.p,{children:["Procedures and coprocedures can be written without a corresponding body.\nThis allows us to ",(0,l.jsx)(e.em,{children:(0,l.jsx)(e.a,{href:"https://www.cs.ox.ac.uk/publications/books/PfS/",children:"program from specifications"})}),", i.e. write and verify programs incrementally.\nWe can start with a specification and step-by-step fill in executable code (or just don't if we don't feel like it)."]}),"\n",(0,l.jsxs)(e.p,{children:["Since ",(0,l.jsx)(e.a,{href:"#calling-procedures",children:"procedure calls only need the callee specification"}),", we can just write a procedure specification and use it in other places.\nThere does not need to be procedure body of the callee at all."]}),"\n",(0,l.jsxs)(e.p,{children:["This can be used to introduce unsoundness even without using ",(0,l.jsxs)(e.a,{href:"/docs/heyvl/statements#assert-and-assume",children:[(0,l.jsx)(e.code,{children:"assume"})," statements"]}),".\nCalling the following procedure is essentially the same as writing ",(0,l.jsx)(e.code,{children:"assume ?(false)"}),"."]}),"\n",(0,l.jsx)(e.pre,{children:(0,l.jsx)(e.code,{className:"language-heyvl",children:"proc unsound() -> ()\n pre ?(true)\n post ?(false)\n"})}),"\n",(0,l.jsx)(e.h2,{id:"calling-procedures",children:"Calling Procedures"}),"\n",(0,l.jsxs)(e.p,{children:["We can call ",(0,l.jsx)(e.code,{children:"proc"}),"s from ",(0,l.jsx)(e.code,{children:"proc"}),"s and ",(0,l.jsx)(e.code,{children:"coproc"}),"s from ",(0,l.jsx)(e.code,{children:"coproc"}),"s.\nMixing both is unsound and will result in an error.\nIn the following, we will talk about procedures, but everything applies to coprocedures as well."]}),"\n",(0,l.jsxs)(e.p,{children:["Procedure calls make use of the procedure's specification ",(0,l.jsx)(e.em,{children:"only"})," and do not inspect the procedure body.\nSomewhat informally, we could say that ",(0,l.jsx)(e.em,{children:"assuming"})," the callee procedure verifies, the procedure call can be replaced by the procedure's body and the program will still verify.\nThis enables modular reasoning: one can verify a big program and we can re-use already-verified parts of it in other parts."]}),"\n",(0,l.jsx)(e.h3,{id:"example-a-spare-engine",children:"Example: A Spare Engine"}),"\n",(0,l.jsxs)(e.p,{children:["The following example models running an engine with a spare engine available.\nIn ",(0,l.jsx)(e.code,{children:"runPrimaryOrSpare"}),", the primary engine works with probability of 90%.\nIf it fails, we have another spare available that is guaranteed to work with the same probability."]}),"\n",(0,l.jsx)(e.pre,{children:(0,l.jsx)(e.code,{className:"language-heyvl",children:"proc spare() -> (working: Bool)\n pre 0.9\n post [working]\n{\n working = flip(0.95)\n}\n\nproc runPrimaryOrSpare() -> (working: Bool)\n pre 0.99\n post [working]\n{\n working = flip(0.9)\n if !working {\n working = spare()\n } else {}\n}\n"})}),"\n",(0,l.jsxs)(e.p,{children:["The specification of ",(0,l.jsx)(e.code,{children:"spare"})," says ",(0,l.jsx)(e.code,{children:"working"})," is true with probability of at least ",(0,l.jsx)(e.code,{children:"0.9"})," (and in fact the coin flip says it is ",(0,l.jsx)(e.code,{children:"0.95"}),").\nThus, we can verify a success probability of ",(0,l.jsx)(e.code,{children:"runOrSpare"})," of at least ",(0,l.jsx)(e.code,{children:"0.99"}),".\nHowever, we cannot verify a lower bound of ",(0,l.jsx)(e.code,{children:"0.995"})," because the specification of ",(0,l.jsx)(e.code,{children:"spare"})," does not guarantee it (even though it is true here)."]}),"\n",(0,l.jsx)(e.h3,{id:"multiple-or-no-return-values",children:"Multiple Or No Return Values"}),"\n",(0,l.jsxs)(e.p,{children:["If the callee has multiple return values, multiple variables may be assigned to by comma-separating them.\nFor example, if ",(0,l.jsx)(e.code,{children:"spare"})," returned two variables, we could obtain the results via ",(0,l.jsx)(e.code,{children:"working1, working2 = spare()"}),"."]}),"\n",(0,l.jsxs)(e.p,{children:["If the callee has no return values, we just call it without the assignment.\nFor example, we simply write ",(0,l.jsx)(e.code,{children:"spare()"})," as a statement to call it without assignment."]}),"\n",(0,l.jsx)(e.h3,{id:"assert-assume-understanding-of-procedure-calls",children:"Assert-Assume Understanding of Procedure Calls"}),"\n",(0,l.jsxs)(e.p,{children:["Internally, procedure calls are translated to HeyVL verification statements (see ",(0,l.jsx)(e.a,{href:"/docs/heyvl/statements",children:"statements documentation"}),").\nWe can understand how procedure calls work by understanding this encoding."]}),"\n",(0,l.jsxs)(e.p,{children:["During the verification of ",(0,l.jsx)(e.code,{children:"runPrimaryOrSpare"}),", we replace the call to ",(0,l.jsx)(e.code,{children:"spare"})," by a sequence of ",(0,l.jsx)(e.code,{children:"assert"}),", ",(0,l.jsx)(e.code,{children:"havoc,"})," ",(0,l.jsx)(e.code,{children:"validate"}),", and ",(0,l.jsx)(e.code,{children:"assume"})," statements."]}),"\n",(0,l.jsx)(e.pre,{children:(0,l.jsx)(e.code,{className:"language-heyvl",children:"proc runPrimaryOrSpare() -> (working: Bool)\n pre 0.99\n post [working]\n{\n working = flip(0.9)\n if !working {\n assert 0.95\n havoc success\n validate\n assume [success]\n } else {}\n}\n"})}),"\n",(0,l.jsxs)(a,{children:[(0,l.jsxs)("summary",{children:["How To: Obtaining intermediate encodings via ",(0,l.jsx)("code",{children:"--print-core"}),"."]}),(0,l.jsxs)(e.p,{children:["To obtain the intermediate encodings from Caesar, we can use the ",(0,l.jsx)("code",{children:"--print-core"})," command-line flag.\nThis will print the fully desugared HeyVL code for each procedure to standard output.\nFor example:"]}),(0,l.jsx)(e.pre,{children:(0,l.jsx)(e.code,{className:"language-bash",children:"caesar verify --print-core example.heyvl\n"})})]}),"\n",(0,l.jsx)(e.p,{children:"We can now read the encoding as follows:"}),"\n",(0,l.jsxs)(e.ol,{children:["\n",(0,l.jsxs)(e.li,{children:["Before the call, ",(0,l.jsx)(e.code,{children:"assert"})," the ",(0,l.jsx)(e.code,{children:"pre"})," ",(0,l.jsx)(e.code,{children:"spare()"}),". This means that the expected value (going backwards) is at most ",(0,l.jsx)(e.code,{children:"0.95"}),"."]}),"\n",(0,l.jsxs)(e.li,{children:["We then forget the current value of ",(0,l.jsx)(e.code,{children:"success"})," via ",(0,l.jsx)(e.code,{children:"havoc"}),". This models that ",(0,l.jsx)(e.code,{children:"spare"})," can have arbitrary effects on return values."]}),"\n",(0,l.jsxs)(e.li,{children:["We ",(0,l.jsx)(e.code,{children:"validate"})," (the following assumption must be absolutely true)."]}),"\n",(0,l.jsxs)(e.li,{children:["We can only ",(0,l.jsx)(e.code,{children:"assume"})," the ",(0,l.jsx)(e.code,{children:"post"})," of ",(0,l.jsx)(e.code,{children:"spare"}),", i.e. that ",(0,l.jsx)(e.code,{children:"success"})," is true."]}),"\n"]}),"\n",(0,l.jsx)(e.p,{children:"This generalizes the Boolean setting:"}),"\n",(0,l.jsxs)(e.ol,{children:["\n",(0,l.jsxs)(e.li,{children:["Before the call, the ",(0,l.jsx)(e.code,{children:"pre"})," must hold."]}),"\n",(0,l.jsx)(e.li,{children:"Forget the current value of modified variables."}),"\n",(0,l.jsxs)(e.li,{children:["(The ",(0,l.jsx)(e.code,{children:"validate"})," does not have any effect in the Boolean setting.)"]}),"\n",(0,l.jsxs)(e.li,{children:["Assume that the ",(0,l.jsx)(e.code,{children:"post"})," holds after the call."]}),"\n"]}),"\n",(0,l.jsxs)(e.p,{children:["A more formal treatment of the encoding and semantics of procedure calls can be found in our ",(0,l.jsx)(e.a,{href:"/docs/publications#oopsla-23",children:"OOPSLA '23 paper"}),"."]}),"\n","\n",(0,l.jsxs)(e.section,{"data-footnotes":!0,className:"footnotes",children:[(0,l.jsx)(e.h2,{className:"sr-only",id:"footnote-label",children:"Footnotes"}),"\n",(0,l.jsxs)(e.ol,{children:["\n",(0,l.jsxs)(e.li,{id:"user-content-fn-on-negative-numbers",children:["\n",(0,l.jsxs)(e.p,{children:["Many verification tasks that require reasoning with negative numbers can be embedded in this framework. First, note that we can still have negative numbers in our program states, we just have to ensure that the ",(0,l.jsx)(e.code,{children:"post"})," is non-negative. ",(0,l.jsxs)(e.a,{href:"https://dl.acm.org/doi/pdf/10.1145/3434320#page=24",children:["Chapter 11 of the paper ",(0,l.jsx)(e.em,{children:'"Relatively Complete Verification for Probabilistic Programs"'})]})," by Batz et al. might be of interest for further reading. ",(0,l.jsx)(e.a,{href:"#user-content-fnref-on-negative-numbers","data-footnote-backref":"","aria-label":"Back to reference 1",className:"data-footnote-backref",children:"\u21a9"})]}),"\n"]}),"\n",(0,l.jsxs)(e.li,{id:"user-content-fn-bang-question-operator",children:["\n",(0,l.jsxs)(e.p,{children:[(0,l.jsx)(e.code,{children:"!?(b)"})," is not a special kind of operator, it is simply the ",(0,l.jsx)(e.code,{children:"!"})," operator applied to ",(0,l.jsx)(e.code,{children:"?(b)"}),", i.e. ",(0,l.jsx)(e.code,{children:"!(?(b))"}),".\nThe negation operator ",(0,l.jsx)(e.code,{children:"!"})," is defined on ",(0,l.jsx)(e.code,{children:"EUReal"})," by ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mo,{stretchy:"false",children:"!"}),(0,l.jsx)(e.mn,{children:"0"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u221e"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"!0 = \\infty"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6944em"}}),(0,l.jsx)(e.span,{className:"mclose",children:"!"}),(0,l.jsx)(e.span,{className:"mord",children:"0"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(e.span,{className:"mord",children:"\u221e"})]})]})]})," and ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mo,{stretchy:"false",children:"!"}),(0,l.jsx)(e.mi,{children:"x"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mn,{children:"0"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"!x = 0"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6944em"}}),(0,l.jsx)(e.span,{className:"mclose",children:"!"}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"x"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,l.jsx)(e.span,{className:"mord",children:"0"})]})]})]})," for all ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"x"}),(0,l.jsx)(e.mo,{mathvariant:"normal",children:"\u2260"}),(0,l.jsx)(e.mn,{children:"0"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"x \\neq 0"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"x"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsxs)(e.span,{className:"mrel",children:[(0,l.jsx)(e.span,{className:"mrel",children:(0,l.jsx)(e.span,{className:"mord vbox",children:(0,l.jsx)(e.span,{className:"thinbox",children:(0,l.jsxs)(e.span,{className:"rlap",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,l.jsx)(e.span,{className:"inner",children:(0,l.jsx)(e.span,{className:"mord",children:(0,l.jsx)(e.span,{className:"mrel",children:"\ue020"})})}),(0,l.jsx)(e.span,{className:"fix"})]})})})}),(0,l.jsx)(e.span,{className:"mrel",children:"="})]}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,l.jsx)(e.span,{className:"mord",children:"0"})]})]})]}),".\nThus, ",(0,l.jsx)(e.code,{children:"!?(b)"})," is logically equivalent to ",(0,l.jsx)(e.code,{children:"?(!b)"}),".\nIn the ",(0,l.jsx)(e.a,{href:"/docs/publications#oopsla-23",children:"OOPSLA '23 paper"}),", we denoted ",(0,l.jsx)(e.code,{children:"!?(b)"})," by ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{mathvariant:"normal",children:"c"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"o"}),(0,l.jsx)(e.mo,{stretchy:"false",children:"?"})]}),(0,l.jsx)(e.mo,{stretchy:"false",children:"("}),(0,l.jsx)(e.mi,{children:"b"}),(0,l.jsx)(e.mo,{stretchy:"false",children:")"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\mathrm{co?}(b)"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord 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