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.announcementBar_mb4j{display:none}.skipToContent_fXgn{background-color:var(--ifm-background-surface-color);color:var(--ifm-color-emphasis-900);left:100%;padding:calc(var(--ifm-global-spacing)/2) var(--ifm-global-spacing);position:fixed;top:1rem;z-index:calc(var(--ifm-z-index-fixed) + 1)}.skipToContent_fXgn:focus{box-shadow:var(--ifm-global-shadow-md);left:1rem}.closeButton_CVFx{line-height:0;padding:0}.content_knG7{font-size:85%;padding:5px 0;text-align:center}.content_knG7 a{color:inherit;text-decoration:underline}.announcementBar_mb4j{align-items:center;background-color:var(--ifm-color-white);border-bottom:1px solid var(--ifm-color-emphasis-100);color:var(--ifm-color-black);display:flex;height:var(--docusaurus-announcement-bar-height)}.announcementBarPlaceholder_vyr4{flex:0 0 10px}.announcementBarClose_gvF7{align-self:stretch;flex:0 0 30px}.toggle_vylO{height:2rem;width:2rem}.toggleButton_gllP{align-items:center;border-radius:50%;display:flex;height:100%;justify-content:center;transition:background var(--ifm-transition-fast);width:100%}.toggleButton_gllP:hover{background:var(--ifm-color-emphasis-200)}.toggleButtonDisabled_aARS{cursor:not-allowed}.darkNavbarColorModeToggle_X3D1:hover{background:var(--ifm-color-gray-800)}[data-theme=dark] .themedImage--dark_i4oU,[data-theme=light] .themedImage--light_HNdA,html:not([data-theme]) .themedComponent--light_NU7w{display:initial}.iconExternalLink_nPIU{margin-left:.3rem}.iconLanguage_nlXk{margin-right:5px;vertical-align:text-bottom}.navbarHideable_m1mJ{transition:transform var(--ifm-transition-fast) ease}.navbarHidden_jGov{transform:translate3d(0,calc(-100% - 2px),0)}.errorBoundaryError_a6uf{color:red;white-space:pre-wrap}.footerLogoLink_BH7S{opacity:.5;transition:opacity var(--ifm-transition-fast) var(--ifm-transition-timing-default)}.footerLogoLink_BH7S:hover,.hash-link:focus,:hover>.hash-link{opacity:1}.mainWrapper_z2l0{display:flex;flex:1 0 auto;flex-direction:column}.docusaurus-mt-lg{margin-top:3rem}#__docusaurus{display:flex;flex-direction:column;min-height:100%}.sidebar_re4s{overflow-y:auto;position:sticky;top:calc(var(--ifm-navbar-height) + 2rem)}.sidebarItemTitle_pO2u{font-size:var(--ifm-h3-font-size)}.container_mt6G,.sidebarItemList_Yudw{font-size:.9rem}.sidebarItem__DBe{margin-top:.7rem}.sidebarItemLink_mo7H{color:var(--ifm-font-color-base);display:block}.sidebarItemLinkActive_I1ZP{color:var(--ifm-color-primary)!important}.cardContainer_fWXF{--ifm-link-color:var(--ifm-color-emphasis-800);--ifm-link-hover-color:var(--ifm-color-emphasis-700);--ifm-link-hover-decoration:none;border:1px solid var(--ifm-color-emphasis-200);box-shadow:0 1.5px 3px 0 #00000026;transition:all var(--ifm-transition-fast) ease;transition-property:border,box-shadow}.cardContainer_fWXF:hover{border-color:var(--ifm-color-primary);box-shadow:0 3px 6px 0 #0003}.cardTitle_rnsV{font-size:1.2rem}.cardDescription_PWke{font-size:.8rem}.backToTopButton_sjWU{background-color:var(--ifm-color-emphasis-200);border-radius:50%;bottom:1.3rem;box-shadow:var(--ifm-global-shadow-lw);height:3rem;opacity:0;position:fixed;right:1.3rem;transform:scale(0);transition:all var(--ifm-transition-fast) var(--ifm-transition-timing-default);visibility:hidden;width:3rem;z-index:calc(var(--ifm-z-index-fixed) - 1)}.buttonGroup__atx button,.codeBlockContainer_Ckt0{background:var(--prism-background-color);color:var(--prism-color)}.backToTopButton_sjWU:after{background-color:var(--ifm-color-emphasis-1000);content:" ";display:inline-block;height:100%;-webkit-mask:var(--ifm-menu-link-sublist-icon) 50%/2rem 2rem no-repeat;mask:var(--ifm-menu-link-sublist-icon) 50%/2rem 2rem no-repeat;width:100%}.backToTopButtonShow_xfvO{opacity:1;transform:scale(1);visibility:visible}[data-theme=dark]:root{--docusaurus-collapse-button-bg:#ffffff0d;--docusaurus-collapse-button-bg-hover:#ffffff1a}.collapseSidebarButton_PEFL{display:none;margin:0}.docMainContainer_gTbr,.docPage__5DB{display:flex;width:100%}.docPage__5DB{flex:1 0}.docsWrapper_BCFX{display:flex;flex:1 0 auto}.authorCol_Hf19{flex-grow:1!important;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 var(--ifm-pre-padding)}.codeBlock_bY9V{--ifm-pre-background:var(--prism-background-color);margin:0;padding:0}.codeBlockTitle_Ktv7+.codeBlockContent_biex .codeBlock_bY9V{border-top-left-radius:0;border-top-right-radius:0}.codeBlockLines_e6Vv{float:left;font:inherit;min-width:100%;padding:var(--ifm-pre-padding)}.codeBlockLinesWithNumbering_o6Pm{display:table;padding:var(--ifm-pre-padding) 0}.buttonGroup__atx{column-gap:.2rem;display:flex;position:absolute;right:calc(var(--ifm-pre-padding)/2);top:calc(var(--ifm-pre-padding)/2)}.buttonGroup__atx button{align-items:center;border:1px solid var(--ifm-color-emphasis-300);border-radius:var(--ifm-global-radius);display:flex;line-height:0;opacity:0;padding:.4rem;transition:opacity var(--ifm-transition-fast) ease-in-out}.buttonGroup__atx button:focus-visible,.buttonGroup__atx button:hover{opacity:1!important}.theme-code-block:hover .buttonGroup__atx button{opacity:.4}:where(:root){--docusaurus-highlighted-code-line-bg:#484d5b}:where([data-theme=dark]){--docusaurus-highlighted-code-line-bg:#646464}.theme-code-block-highlighted-line{background-color:var(--docusaurus-highlighted-code-line-bg);display:block;margin:0 calc(var(--ifm-pre-padding)*-1);padding:0 var(--ifm-pre-padding)}.codeLine_lJS_{counter-increment:a;display:table-row}.codeLineNumber_Tfdd{background:var(--ifm-pre-background);display:table-cell;left:0;overflow-wrap:normal;padding:0 var(--ifm-pre-padding);position:sticky;text-align:right;width:1%}.codeLineNumber_Tfdd:before{content:counter(a);opacity:.4}.codeLineContent_feaV{padding-right:var(--ifm-pre-padding)}.iconEdit_Z9Sw{margin-right:.3em;vertical-align:sub}.theme-code-block:hover 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1rem;position:relative}.tagWithCount_h2kH:after,.tagWithCount_h2kH:before{border:1px solid var(--docusaurus-tag-list-border);content:"";position:absolute;top:50%;transition:inherit}.tagWithCount_h2kH:before{border-bottom:0;border-right:0;height:1.18rem;right:100%;transform:translate(50%,-50%) rotate(-45deg);width:1.18rem}.tagWithCount_h2kH:after{border-radius:50%;height:.5rem;left:0;transform:translateY(-50%);width:.5rem}.tagWithCount_h2kH span{background:var(--ifm-color-secondary);border-radius:var(--ifm-global-radius);color:var(--ifm-color-black);font-size:.7rem;line-height:1.2;margin-left:.3rem;padding:.1rem .4rem}.tag_Nnez{display:inline-block;margin:.5rem .5rem 0 1rem}.wordWrapButtonIcon_Bwma{height:1.2rem;width:1.2rem}.tableOfContentsInline_prmo ul{font-size:medium;list-style-type:disc;padding-top:0}.heroBanner_qdFl{overflow:hidden;padding:4rem 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var(--docusaurus-details-decoration-color);border-style:solid;border-width:var(--docusaurus-details-summary-arrow-size);content:"";left:0;position:absolute;top:.45rem;transform:rotate(0);transform-origin:calc(var(--docusaurus-details-summary-arrow-size)/2) 50%;transition:var(--docusaurus-details-transition)}.collapsibleContent_i85q{border-top:1px solid var(--docusaurus-details-decoration-color);margin-top:1rem;padding-top:1rem}.details_b_Ee{--docusaurus-details-decoration-color:var(--ifm-alert-border-color);--docusaurus-details-transition:transform var(--ifm-transition-fast) ease;border:1px solid var(--ifm-alert-border-color);margin:0 0 var(--ifm-spacing-vertical)}.anchorWithStickyNavbar_LWe7{scroll-margin-top:calc(var(--ifm-navbar-height) + .5rem)}.anchorWithHideOnScrollNavbar_WYt5{scroll-margin-top:.5rem}.hash-link{opacity:0;padding-left:.5rem;transition:opacity 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(min-width:997px){.collapseSidebarButton_PEFL,.expandButton_m80_{background-color:var(--docusaurus-collapse-button-bg)}:root{--docusaurus-announcement-bar-height:30px}.announcementBarClose_gvF7,.announcementBarPlaceholder_vyr4{flex-basis:50px}.searchBox_ZlJk{padding:var(--ifm-navbar-item-padding-vertical) var(--ifm-navbar-item-padding-horizontal)}.collapseSidebarButton_PEFL{border:1px solid var(--ifm-toc-border-color);border-radius:0;bottom:0;display:block!important;height:40px;position:sticky}.collapseSidebarButtonIcon_kv0_{margin-top:4px;transform:rotate(180deg)}.expandButtonIcon_BlDH,[dir=rtl] .collapseSidebarButtonIcon_kv0_{transform:rotate(0)}.collapseSidebarButton_PEFL:focus,.collapseSidebarButton_PEFL:hover,.expandButton_m80_:focus,.expandButton_m80_:hover{background-color:var(--docusaurus-collapse-button-bg-hover)}.menuHtmlItem_M9Kj{padding:var(--ifm-menu-link-padding-vertical) var(--ifm-menu-link-padding-horizontal)}.menu_SIkG{flex-grow:1;padding:.5rem}@supports 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(max-width:996px){.col{--ifm-col-width:100%;flex-basis:var(--ifm-col-width);margin-left:0}.footer{--ifm-footer-padding-horizontal:0}.colorModeToggle_DEke,.footer__link-separator,.navbar__item,.sidebar_re4s,.tableOfContents_bqdL{display:none}.footer__col{margin-bottom:calc(var(--ifm-spacing-vertical)*3)}.footer__link-item{display:block}.hero{padding-left:0;padding-right:0}.navbar>.container,.navbar>.container-fluid{padding:0}.navbar__toggle{display:inherit}.navbar__search-input{width:9rem}.pills--block,.tabs--block{flex-direction:column}.searchBox_ZlJk{position:absolute;right:var(--ifm-navbar-padding-horizontal)}.docItemContainer_F8PC{padding:0 .3rem}}@media (max-width:576px){.markdown h1:first-child{--ifm-h1-font-size:2rem}.markdown>h2{--ifm-h2-font-size:1.5rem}.markdown>h3{--ifm-h3-font-size:1.25rem}.title_f1Hy{font-size:2rem}}@media (hover:hover){.backToTopButton_sjWU:hover{background-color:var(--ifm-color-emphasis-300)}}@media 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We model a 50% chance of crashing by a coin flip (",n.createElement("code",null,"flip(0.5)"),") leading to ",n.createElement("code",null,"assert [false]"),"."),n.createElement(m.Z,{language:"heyvl"},"proc lossy_list(init_l: List) -> (l: List)\n pre [len(init_l) == 1] * 0.5 // quantitative specification!\n post [true]\n{\n l = init_l\n @invariant(exp(0.5, len(l)))\n while len(l) > 0 {\n var prob_choice: Bool = flip(0.5) // coin flip\n if prob_choice {\n l = pop(l) // next list element\n } else {\n assert [false] // crash\n }\n }\n}\n")),n.createElement("div",{className:"col col--6"},n.createElement("h3",null,"\ud83d\udcd0 Reading The Spec"),n.createElement("p",null,"Let's focus on the ",n.createElement("i",null,"quantitative specification")," of ",n.createElement("code",null,"lossy_list"),":"),n.createElement(m.Z,{language:"heyvl"},"pre [len(init_l) == 1] * 0.5\npost [true]"),n.createElement("p",null,"The ",n.createElement("code",null,"post")," says that we are looking at the expected value of ",n.createElement("code",null,"[true]")," (i.e. 1) in the final states of the program. 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It says that if the length of the list is 1, then the lower bound is ",n.createElement("code",null,"0.5")," and otherwise ",n.createElement("code",null,"0"),"."),n.createElement("p",null,"To verify the spec, the ",n.createElement("code",null,"while")," loop has an ",n.createElement("code",null,"@invariant")," annotation with a ",n.createElement(r.Z,{to:"/docs/proof-rules/induction"},"probabilistic invariant"),"."))),n.createElement("div",{className:"row"},n.createElement("div",{className:"col col--6"},n.createElement("h3",null,"\ud83c\udf0d Axiomatizing Exponentials and Lists"),n.createElement("p",null,"Here is a strength of deductive verification: users can axomatize additional functions and data types that can be used for verification! 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a.languages.reason.function,function(e){e.languages.sass=e.languages.extend("css",{comment:{pattern:/^([ \t]*)\/[\/*].*(?:(?:\r?\n|\r)\1[ \t].+)*/m,lookbehind:!0,greedy:!0}}),e.languages.insertBefore("sass","atrule",{"atrule-line":{pattern:/^(?:[ \t]*)[@+=].+/m,greedy:!0,inside:{atrule:/(?:@[\w-]+|[+=])/}}}),delete e.languages.sass.atrule;var t=/\$[-\w]+|#\{\$[-\w]+\}/,n=[/[+*\/%]|[=!]=|<=?|>=?|\b(?:and|not|or)\b/,{pattern:/(\s)-(?=\s)/,lookbehind:!0}];e.languages.insertBefore("sass","property",{"variable-line":{pattern:/^[ \t]*\$.+/m,greedy:!0,inside:{punctuation:/:/,variable:t,operator:n}},"property-line":{pattern:/^[ \t]*(?:[^:\s]+ *:.*|:[^:\s].*)/m,greedy:!0,inside:{property:[/[^:\s]+(?=\s*:)/,{pattern:/(:)[^:\s]+/,lookbehind:!0}],punctuation:/:/,variable:t,operator:n,important:e.languages.sass.important}}}),delete e.languages.sass.property,delete e.languages.sass.important,e.languages.insertBefore("sass","punctuation",{selector:{pattern:/^([ 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t={pattern:/(\b\d+)(?:%|[a-z]+)/,lookbehind:!0},n={pattern:/(^|[^\w.-])-?(?:\d+(?:\.\d+)?|\.\d+)/,lookbehind:!0},r={comment:{pattern:/(^|[^\\])(?:\/\*[\s\S]*?\*\/|\/\/.*)/,lookbehind:!0},url:{pattern:/\burl\((["']?).*?\1\)/i,greedy:!0},string:{pattern:/("|')(?:(?!\1)[^\\\r\n]|\\(?:\r\n|[\s\S]))*\1/,greedy:!0},interpolation:null,func:null,important:/\B!(?:important|optional)\b/i,keyword:{pattern:/(^|\s+)(?:(?:else|for|if|return|unless)(?=\s|$)|@[\w-]+)/,lookbehind:!0},hexcode:/#[\da-f]{3,6}/i,color:[/\b(?:AliceBlue|AntiqueWhite|Aqua|Aquamarine|Azure|Beige|Bisque|Black|BlanchedAlmond|Blue|BlueViolet|Brown|BurlyWood|CadetBlue|Chartreuse|Chocolate|Coral|CornflowerBlue|Cornsilk|Crimson|Cyan|DarkBlue|DarkCyan|DarkGoldenRod|DarkGr[ae]y|DarkGreen|DarkKhaki|DarkMagenta|DarkOliveGreen|DarkOrange|DarkOrchid|DarkRed|DarkSalmon|DarkSeaGreen|DarkSlateBlue|DarkSlateGr[ae]y|DarkTurquoise|DarkViolet|DeepPink|DeepSkyBlue|DimGr[ae]y|DodgerBlue|FireBrick|FloralWhite|ForestGreen|Fuchsia|Gainsboro|GhostWhite|Gold|GoldenRod|Gr[ae]y|Green|GreenYellow|HoneyDew|HotPink|IndianRed|Indigo|Ivory|Khaki|Lavender|LavenderBlush|LawnGreen|LemonChiffon|LightBlue|LightCoral|LightCyan|LightGoldenRodYellow|LightGr[ae]y|LightGreen|LightPink|LightSalmon|LightSeaGreen|LightSkyBlue|LightSlateGr[ae]y|LightSteelBlue|LightYellow|Lime|LimeGreen|Linen|Magenta|Maroon|MediumAquaMarine|MediumBlue|MediumOrchid|MediumPurple|MediumSeaGreen|MediumSlateBlue|MediumSpringGreen|MediumTurquoise|MediumVioletRed|MidnightBlue|MintCream|MistyRose|Moccasin|NavajoWhite|Navy|OldLace|Olive|OliveDrab|Orange|OrangeRed|Orchid|PaleGoldenRod|PaleGreen|PaleTurquoise|PaleVioletRed|PapayaWhip|PeachPuff|Peru|Pink|Plum|PowderBlue|Purple|Red|RosyBrown|RoyalBlue|SaddleBrown|Salmon|SandyBrown|SeaGreen|SeaShell|Sienna|Silver|SkyBlue|SlateBlue|SlateGr[ae]y|Snow|SpringGreen|SteelBlue|Tan|Teal|Thistle|Tomato|Transparent|Turquoise|Violet|Wheat|White|WhiteSmoke|Yellow|YellowGreen)\b/i,{pattern:/\b(?:hsl|rgb)\(\s*\d{1,3}\s*,\s*\d{1,3}%?\s*,\s*\d{1,3}%?\s*\)\B|\b(?:hsl|rgb)a\(\s*\d{1,3}\s*,\s*\d{1,3}%?\s*,\s*\d{1,3}%?\s*,\s*(?:0|0?\.\d+|1)\s*\)\B/i,inside:{unit:t,number:n,function:/[\w-]+(?=\()/,punctuation:/[(),]/}}],entity:/\\[\da-f]{1,8}/i,unit:t,boolean:/\b(?:false|true)\b/,operator:[/~|[+!\/%<>?=]=?|[-:]=|\*[*=]?|\.{2,3}|&&|\|\||\B-\B|\b(?:and|in|is(?: 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