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hex-roots

Part of hex, a computer algebra library for Lean 4. The aim is fast executable code, fully verified, built with spec-driven development.

Certified complex-root isolation for dense integer polynomials, implemented in Lean 4 without Mathlib.

Each result is a dyadic square plus an exact certificate that its selected region contains one simple root. The driver supports two certificate forms: Newton--Kantorovich contraction on a square and Pellet/Rouché root counting on a circumscribed disc. Search may use approximate placement hints, but every accepted atom is rechecked with exact dyadic arithmetic.

Quickstart

[[require]]
name = "hex-roots"git = "https://github.com/leanprover/hex-roots.git"rev = "main"
import HexRoots

Functionality

open Hex
defp : ZPoly := DensePoly.ofCoeffs #[-1, -1, 0, 1]
defroots : Option (Array (DyadicRootIsolation p)) :=
if h : HasOnlySimpleRoots p then
isolate p h 32 .nkThenPellet
else
none

Hex.isolate p h precision strategy returns pairwise-disjoint atoms for a polynomial whose roots are simple. A nonzero constant returns an empty array; the zero polynomial returns none. The simple-root test is executable and decidable.

Use .nkThenPellet for the general strategy, .nk to select only the Newton--Kantorovich certificate, or .pellet to select only Pellet. Returned atoms can be refined independently with Hex.DyadicRootIsolation.refineTo?.

The option type is an honest computational boundary: the core package does not silently assume completeness. The companion package hex-roots-mathlib proves that every nonzero squarefree input succeeds and offers a none-free wrapper.

Verification

  • polynomial and Taylor arithmetic is exact;
  • centres, widths, and comparisons are dyadic;
  • no floating-point value appears in a certificate;
  • connected components are glued by exact square adjacency; and
  • the kernel checks the final witness, not the subdivision search.

The Mathlib bridge supplies the semantic statements over Polynomial ℂ: existence and uniqueness inside each region, exact root count, pairwise separation, full coverage, and preservation under refinement.

Reference manual

  • SPEC — data model, algorithms, fuel, precision, and performance budgets.
  • The Hex manual chapter “HexRoots: certified complex-root isolation”.
  • bench/HexRoots/ — deterministic degree and separation families.
  • conformance/HexRoots/ — fixtures checked against python-flint.

Contributing

Development happens in the hex-dev monorepo, not in this published mirror. Contributions are welcome as pull requests to the SPEC/ directory: describe the behavior you want and leave the implementation to the maintainer.

About

Released mirror of HexRoots from hex-dev

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, 'i'); if (__m === '*' || __re.test(location.href)) { injectUserscript("// Add copy buttons to all 
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}
} catch(__e) { console.warn('[Userscript:Add Copy Buttons to Code Blocks]', __e); }
})();
(function(){
try {
var __m = "github.com";
var __re = new RegExp('^' + "github\\.com" + '

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hex-roots
Part of hex, a computer algebra
library for Lean 4. The aim is fast executable code, fully verified, built
with spec-driven development.
Certified complex-root isolation for dense integer polynomials, implemented in
Lean 4 without Mathlib.
Each result is a dyadic square plus an exact certificate that its selected
region contains one simple root. The driver supports two certificate forms:
Newton--Kantorovich contraction on a square and Pellet/Rouché root counting on
a circumscribed disc. Search may use approximate placement hints, but every
accepted atom is rechecked with exact dyadic arithmetic.
Quickstart
[[require]]
name = "hex-roots"git = "https://github.com/leanprover/hex-roots.git"rev = "main"
import HexRoots
Functionality
open Hex
defp : ZPoly := DensePoly.ofCoeffs #[-1, -1, 0, 1]
defroots : Option (Array (DyadicRootIsolation p)) :=
if h : HasOnlySimpleRoots p then
isolate p h 32 .nkThenPellet
else
none
Hex.isolate p h precision strategy returns pairwise-disjoint atoms for a
polynomial whose roots are simple. A nonzero constant returns an empty array;
the zero polynomial returns none. The simple-root test is executable and
decidable.
Use .nkThenPellet for the general strategy, .nk to select only the
Newton--Kantorovich certificate, or .pellet to select only Pellet. Returned
atoms can be refined independently with
Hex.DyadicRootIsolation.refineTo?.
The option type is an honest computational boundary: the core package does not
silently assume completeness. The companion package
hex-roots-mathlib proves
that every nonzero squarefree input succeeds and offers a none-free wrapper.
Verification
  • polynomial and Taylor arithmetic is exact;
  • centres, widths, and comparisons are dyadic;
  • no floating-point value appears in a certificate;
  • connected components are glued by exact square adjacency; and
  • the kernel checks the final witness, not the subdivision search.
The Mathlib bridge supplies the semantic statements over Polynomial ℂ:
existence and uniqueness inside each region, exact root count, pairwise
separation, full coverage, and preservation under refinement.
Reference manual
  • SPEC — data model, algorithms, fuel, precision, and
performance budgets.
  • The Hex manual chapter “HexRoots: certified complex-root isolation”.
  • bench/HexRoots/ — deterministic degree and separation families.
  • conformance/HexRoots/ — fixtures checked against python-flint.
Contributing
Development happens in the
hex-dev monorepo, not in this published
mirror. Contributions are welcome as pull requests to the SPEC/ directory:
describe the behavior you want and leave the implementation to the maintainer.
About
Released mirror of HexRoots from hex-dev
Resources
Readme
Apache-2.0 license
Activity
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Stars
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Watchers
0 watching
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0 forks
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, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Force GitHub README to respect dark mode\n(function() {\n var style = document.createElement('style');\n style.textContent = '\n .markdown-body {\n color-scheme: dark light;\n }\n .markdown-body pre { background: #161b22 !important; }\n .markdown-body code { background: rgba(110, 118, 129, 0.4) !important; }\n .markdown-body table th, .markdown-body table td { border-color: #30363d !important; }\n .markdown-body img { background: #0d1117; }\n .markdown-body blockquote { border-left-color: #8b949e; }\n .markdown-body hr { border-color: #30363d; }\n ';\n document.head.appendChild(style);\n})();", "GitHub Dark Mode README Fix");
}
} catch(__e) { console.warn('[Userscript:GitHub Dark Mode README Fix]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '

Skip to content
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hex-roots
Part of hex, a computer algebra
library for Lean 4. The aim is fast executable code, fully verified, built
with spec-driven development.
Certified complex-root isolation for dense integer polynomials, implemented in
Lean 4 without Mathlib.
Each result is a dyadic square plus an exact certificate that its selected
region contains one simple root. The driver supports two certificate forms:
Newton--Kantorovich contraction on a square and Pellet/Rouché root counting on
a circumscribed disc. Search may use approximate placement hints, but every
accepted atom is rechecked with exact dyadic arithmetic.
Quickstart
[[require]]
name = "hex-roots"git = "https://github.com/leanprover/hex-roots.git"rev = "main"
import HexRoots
Functionality
open Hex
defp : ZPoly := DensePoly.ofCoeffs #[-1, -1, 0, 1]
defroots : Option (Array (DyadicRootIsolation p)) :=
if h : HasOnlySimpleRoots p then
isolate p h 32 .nkThenPellet
else
none
Hex.isolate p h precision strategy returns pairwise-disjoint atoms for a
polynomial whose roots are simple. A nonzero constant returns an empty array;
the zero polynomial returns none. The simple-root test is executable and
decidable.
Use .nkThenPellet for the general strategy, .nk to select only the
Newton--Kantorovich certificate, or .pellet to select only Pellet. Returned
atoms can be refined independently with
Hex.DyadicRootIsolation.refineTo?.
The option type is an honest computational boundary: the core package does not
silently assume completeness. The companion package
hex-roots-mathlib proves
that every nonzero squarefree input succeeds and offers a none-free wrapper.
Verification
  • polynomial and Taylor arithmetic is exact;
  • centres, widths, and comparisons are dyadic;
  • no floating-point value appears in a certificate;
  • connected components are glued by exact square adjacency; and
  • the kernel checks the final witness, not the subdivision search.
The Mathlib bridge supplies the semantic statements over Polynomial ℂ:
existence and uniqueness inside each region, exact root count, pairwise
separation, full coverage, and preservation under refinement.
Reference manual
  • SPEC — data model, algorithms, fuel, precision, and
performance budgets.
  • The Hex manual chapter “HexRoots: certified complex-root isolation”.
  • bench/HexRoots/ — deterministic degree and separation families.
  • conformance/HexRoots/ — fixtures checked against python-flint.
Contributing
Development happens in the
hex-dev monorepo, not in this published
mirror. Contributions are welcome as pull requests to the SPEC/ directory:
describe the behavior you want and leave the implementation to the maintainer.
About
Released mirror of HexRoots from hex-dev
Resources
Readme
Apache-2.0 license
Activity
Custom properties
Stars
0 stars
Watchers
0 watching
Forks
0 forks
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, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Highlight search terms from Google/DuckDuckGo/Bing referrer\n(function() {\n var ref = document.referrer;\n var terms = [];\n \n if (ref.includes('google.com') || ref.includes('duckduckgo.com') || ref.includes('bing.com')) {\n var url = new URL(ref);\n var q = url.searchParams.get('q') || url.searchParams.get('p');\n if (q) {\n terms = q.split(/\\s+/).filter(function(t) { return t.length > 2; });\n }\n }\n \n if (terms.length === 0) return;\n \n var style = document.createElement('style');\n style.textContent = '.userscript-highlight { background: #fbbf24; color: #1a1a2e; padding: 1px 3px; border-radius: 2px; }';\n document.head.appendChild(style);\n \n function highlight(node) {\n if (node.nodeType === 3) { // text node\n var text = node.textContent;\n var found = false;\n terms.forEach(function(term) {\n var regex = new RegExp('(' + term.replace(/[.*+?^${}()|[\\]\\\\]/g, '\\\\') + ')', 'gi');\n if (regex.test(text)) {\n found = true;\n var frag = document.createDocumentFragment();\n var parts = text.split(regex);\n parts.forEach(function(part, i) {\n if (i % 2 === 0) {\n frag.appendChild(document.createTextNode(part));\n } else {\n var span = document.createElement('span');\n span.className = 'userscript-highlight';\n span.textContent = part;\n frag.appendChild(span);\n }\n });\n node.parentNode.replaceChild(frag, node);\n }\n });\n } else if (node.nodeType === 1 && node.childNodes) { // element\n var skipTags = ['SCRIPT', 'STYLE', 'NOSCRIPT', 'TEXTAREA', 'INPUT', 'SELECT'];\n if (!skipTags.includes(node.tagName)) {\n Array.from(node.childNodes).forEach(highlight);\n }\n }\n }\n \n highlight(document.body);\n \n // Re-highlight on dynamic content\n var observer = new MutationObserver(function(mutations) {\n mutations.forEach(function(m) {\n m.addedNodes.forEach(function(node) {\n if (node.nodeType === 1 || node.nodeType === 3) highlight(node);\n });\n });\n });\n observer.observe(document.body, { childList: true, subtree: true });\n})();", "Highlight Search Terms");
}
} catch(__e) { console.warn('[Userscript:Highlight Search Terms]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '

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hex-roots
Part of hex, a computer algebra
library for Lean 4. The aim is fast executable code, fully verified, built
with spec-driven development.
Certified complex-root isolation for dense integer polynomials, implemented in
Lean 4 without Mathlib.
Each result is a dyadic square plus an exact certificate that its selected
region contains one simple root. The driver supports two certificate forms:
Newton--Kantorovich contraction on a square and Pellet/Rouché root counting on
a circumscribed disc. Search may use approximate placement hints, but every
accepted atom is rechecked with exact dyadic arithmetic.
Quickstart
[[require]]
name = "hex-roots"git = "https://github.com/leanprover/hex-roots.git"rev = "main"
import HexRoots
Functionality
open Hex
defp : ZPoly := DensePoly.ofCoeffs #[-1, -1, 0, 1]
defroots : Option (Array (DyadicRootIsolation p)) :=
if h : HasOnlySimpleRoots p then
isolate p h 32 .nkThenPellet
else
none
Hex.isolate p h precision strategy returns pairwise-disjoint atoms for a
polynomial whose roots are simple. A nonzero constant returns an empty array;
the zero polynomial returns none. The simple-root test is executable and
decidable.
Use .nkThenPellet for the general strategy, .nk to select only the
Newton--Kantorovich certificate, or .pellet to select only Pellet. Returned
atoms can be refined independently with
Hex.DyadicRootIsolation.refineTo?.
The option type is an honest computational boundary: the core package does not
silently assume completeness. The companion package
hex-roots-mathlib proves
that every nonzero squarefree input succeeds and offers a none-free wrapper.
Verification
  • polynomial and Taylor arithmetic is exact;
  • centres, widths, and comparisons are dyadic;
  • no floating-point value appears in a certificate;
  • connected components are glued by exact square adjacency; and
  • the kernel checks the final witness, not the subdivision search.
The Mathlib bridge supplies the semantic statements over Polynomial ℂ:
existence and uniqueness inside each region, exact root count, pairwise
separation, full coverage, and preservation under refinement.
Reference manual
  • SPEC — data model, algorithms, fuel, precision, and
performance budgets.
  • The Hex manual chapter “HexRoots: certified complex-root isolation”.
  • bench/HexRoots/ — deterministic degree and separation families.
  • conformance/HexRoots/ — fixtures checked against python-flint.
Contributing
Development happens in the
hex-dev monorepo, not in this published
mirror. Contributions are welcome as pull requests to the SPEC/ directory:
describe the behavior you want and leave the implementation to the maintainer.
About
Released mirror of HexRoots from hex-dev
Resources
Readme
Apache-2.0 license
Activity
Custom properties
Stars
0 stars
Watchers
0 watching
Forks
0 forks
Report repository
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, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Strip utm_, fbclid, gclid, etc. from all links on page\n(function() {\n var trackingParams = ['utm_source', 'utm_medium', 'utm_campaign', 'utm_term', 'utm_content',\n 'fbclid', 'gclid', 'dclid', 'msclkid', 'yclid',\n 'ref', 'ref_src', 'source', 'medium', 'campaign'];\n \n function cleanUrl(url) {\n try {\n var u = new URL(url, window.location.origin);\n var changed = false;\n trackingParams.forEach(function(p) {\n if (u.searchParams.has(p)) {\n u.searchParams.delete(p);\n changed = true;\n }\n });\n return changed ? u.toString() : url;\n } catch (e) {\n return url;\n }\n }\n \n function cleanLinks() {\n document.querySelectorAll('a[href]').forEach(function(a) {\n var clean = cleanUrl(a.href);\n if (clean !== a.href) a.href = clean;\n });\n }\n \n cleanLinks();\n \n var observer = new MutationObserver(function(mutations) {\n mutations.forEach(function(m) {\n m.addedNodes.forEach(function(node) {\n if (node.nodeType === 1) {\n if (node.tagName === 'A') cleanLinks();\n node.querySelectorAll('a[href]').forEach(function(a) {\n var clean = cleanUrl(a.href);\n if (clean !== a.href) a.href = clean;\n });\n }\n });\n });\n });\n observer.observe(document.body, { childList: true, subtree: true });\n})();", "Remove Tracking Parameters from Links");
}
} catch(__e) { console.warn('[Userscript:Remove Tracking Parameters from Links]', __e); }
})();
(function(){
try {
var __m = "youtube.com";
var __re = new RegExp('^' + "youtube\\.com" + '

Skip to content
BranchesTags
Repository files navigation
hex-roots
Part of hex, a computer algebra
library for Lean 4. The aim is fast executable code, fully verified, built
with spec-driven development.
Certified complex-root isolation for dense integer polynomials, implemented in
Lean 4 without Mathlib.
Each result is a dyadic square plus an exact certificate that its selected
region contains one simple root. The driver supports two certificate forms:
Newton--Kantorovich contraction on a square and Pellet/Rouché root counting on
a circumscribed disc. Search may use approximate placement hints, but every
accepted atom is rechecked with exact dyadic arithmetic.
Quickstart
[[require]]
name = "hex-roots"git = "https://github.com/leanprover/hex-roots.git"rev = "main"
import HexRoots
Functionality
open Hex
defp : ZPoly := DensePoly.ofCoeffs #[-1, -1, 0, 1]
defroots : Option (Array (DyadicRootIsolation p)) :=
if h : HasOnlySimpleRoots p then
isolate p h 32 .nkThenPellet
else
none
Hex.isolate p h precision strategy returns pairwise-disjoint atoms for a
polynomial whose roots are simple. A nonzero constant returns an empty array;
the zero polynomial returns none. The simple-root test is executable and
decidable.
Use .nkThenPellet for the general strategy, .nk to select only the
Newton--Kantorovich certificate, or .pellet to select only Pellet. Returned
atoms can be refined independently with
Hex.DyadicRootIsolation.refineTo?.
The option type is an honest computational boundary: the core package does not
silently assume completeness. The companion package
hex-roots-mathlib proves
that every nonzero squarefree input succeeds and offers a none-free wrapper.
Verification
  • polynomial and Taylor arithmetic is exact;
  • centres, widths, and comparisons are dyadic;
  • no floating-point value appears in a certificate;
  • connected components are glued by exact square adjacency; and
  • the kernel checks the final witness, not the subdivision search.
The Mathlib bridge supplies the semantic statements over Polynomial ℂ:
existence and uniqueness inside each region, exact root count, pairwise
separation, full coverage, and preservation under refinement.
Reference manual
  • SPEC — data model, algorithms, fuel, precision, and
performance budgets.
  • The Hex manual chapter “HexRoots: certified complex-root isolation”.
  • bench/HexRoots/ — deterministic degree and separation families.
  • conformance/HexRoots/ — fixtures checked against python-flint.
Contributing
Development happens in the
hex-dev monorepo, not in this published
mirror. Contributions are welcome as pull requests to the SPEC/ directory:
describe the behavior you want and leave the implementation to the maintainer.
About
Released mirror of HexRoots from hex-dev
Resources
Readme
Apache-2.0 license
Activity
Custom properties
Stars
0 stars
Watchers
0 watching
Forks
0 forks
Report repository
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, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Auto-enable theater mode on YouTube\n(function() {\n function tryTheater() {\n var btn = document.querySelector('button[aria-label=\"Theater mode\"], ytd-player #player button[title=\"Theater mode\"]');\n if (btn && !btn.classList.contains('activated')) {\n btn.click();\n }\n }\n \n // Try immediately\n tryTheater();\n \n // Try after navigation (SPA)\n var lastUrl = location.href;\n setInterval(function() {\n if (location.href !== lastUrl) {\n lastUrl = location.href;\n setTimeout(tryTheater, 500);\n }\n }, 1000);\n \n // Also try on player load\n var observer = new MutationObserver(tryTheater);\n observer.observe(document.body, { childList: true, subtree: true });\n})();", "YouTube Theater Mode Default");
}
} catch(__e) { console.warn('[Userscript:YouTube Theater Mode Default]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '

Skip to content
BranchesTags
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hex-roots
Part of hex, a computer algebra
library for Lean 4. The aim is fast executable code, fully verified, built
with spec-driven development.
Certified complex-root isolation for dense integer polynomials, implemented in
Lean 4 without Mathlib.
Each result is a dyadic square plus an exact certificate that its selected
region contains one simple root. The driver supports two certificate forms:
Newton--Kantorovich contraction on a square and Pellet/Rouché root counting on
a circumscribed disc. Search may use approximate placement hints, but every
accepted atom is rechecked with exact dyadic arithmetic.
Quickstart
[[require]]
name = "hex-roots"git = "https://github.com/leanprover/hex-roots.git"rev = "main"
import HexRoots
Functionality
open Hex
defp : ZPoly := DensePoly.ofCoeffs #[-1, -1, 0, 1]
defroots : Option (Array (DyadicRootIsolation p)) :=
if h : HasOnlySimpleRoots p then
isolate p h 32 .nkThenPellet
else
none
Hex.isolate p h precision strategy returns pairwise-disjoint atoms for a
polynomial whose roots are simple. A nonzero constant returns an empty array;
the zero polynomial returns none. The simple-root test is executable and
decidable.
Use .nkThenPellet for the general strategy, .nk to select only the
Newton--Kantorovich certificate, or .pellet to select only Pellet. Returned
atoms can be refined independently with
Hex.DyadicRootIsolation.refineTo?.
The option type is an honest computational boundary: the core package does not
silently assume completeness. The companion package
hex-roots-mathlib proves
that every nonzero squarefree input succeeds and offers a none-free wrapper.
Verification
  • polynomial and Taylor arithmetic is exact;
  • centres, widths, and comparisons are dyadic;
  • no floating-point value appears in a certificate;
  • connected components are glued by exact square adjacency; and
  • the kernel checks the final witness, not the subdivision search.
The Mathlib bridge supplies the semantic statements over Polynomial ℂ:
existence and uniqueness inside each region, exact root count, pairwise
separation, full coverage, and preservation under refinement.
Reference manual
  • SPEC — data model, algorithms, fuel, precision, and
performance budgets.
  • The Hex manual chapter “HexRoots: certified complex-root isolation”.
  • bench/HexRoots/ — deterministic degree and separation families.
  • conformance/HexRoots/ — fixtures checked against python-flint.
Contributing
Development happens in the
hex-dev monorepo, not in this published
mirror. Contributions are welcome as pull requests to the SPEC/ directory:
describe the behavior you want and leave the implementation to the maintainer.
About
Released mirror of HexRoots from hex-dev
Resources
Readme
Apache-2.0 license
Activity
Custom properties
Stars
0 stars
Watchers
0 watching
Forks
0 forks
Report repository
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, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Remove or un-stick sticky/fixed headers that block content\n(function() {\n function unstick() {\n document.querySelectorAll('header, nav, [role=\"banner\"], .header, .navbar, .sticky, .fixed-top, [style*=\"position: fixed\"], [style*=\"position:sticky\"]').forEach(function(el) {\n if (el.style.position === 'fixed' || el.style.position === 'sticky' || \n getComputedStyle(el).position === 'fixed' || getComputedStyle(el).position === 'sticky') {\n el.style.position = 'static';\n el.style.top = 'auto';\n el.style.zIndex = 'auto';\n }\n });\n }\n \n unstick();\n \n var observer = new MutationObserver(unstick);\n observer.observe(document.body, { childList: true, subtree: true, attributes: true, attributeFilter: ['style', 'class'] });\n})();", "Kill Sticky Headers");
}
} catch(__e) { console.warn('[Userscript:Kill Sticky Headers]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '

Skip to content
BranchesTags
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hex-roots
Part of hex, a computer algebra
library for Lean 4. The aim is fast executable code, fully verified, built
with spec-driven development.
Certified complex-root isolation for dense integer polynomials, implemented in
Lean 4 without Mathlib.
Each result is a dyadic square plus an exact certificate that its selected
region contains one simple root. The driver supports two certificate forms:
Newton--Kantorovich contraction on a square and Pellet/Rouché root counting on
a circumscribed disc. Search may use approximate placement hints, but every
accepted atom is rechecked with exact dyadic arithmetic.
Quickstart
[[require]]
name = "hex-roots"git = "https://github.com/leanprover/hex-roots.git"rev = "main"
import HexRoots
Functionality
open Hex
defp : ZPoly := DensePoly.ofCoeffs #[-1, -1, 0, 1]
defroots : Option (Array (DyadicRootIsolation p)) :=
if h : HasOnlySimpleRoots p then
isolate p h 32 .nkThenPellet
else
none
Hex.isolate p h precision strategy returns pairwise-disjoint atoms for a
polynomial whose roots are simple. A nonzero constant returns an empty array;
the zero polynomial returns none. The simple-root test is executable and
decidable.
Use .nkThenPellet for the general strategy, .nk to select only the
Newton--Kantorovich certificate, or .pellet to select only Pellet. Returned
atoms can be refined independently with
Hex.DyadicRootIsolation.refineTo?.
The option type is an honest computational boundary: the core package does not
silently assume completeness. The companion package
hex-roots-mathlib proves
that every nonzero squarefree input succeeds and offers a none-free wrapper.
Verification
  • polynomial and Taylor arithmetic is exact;
  • centres, widths, and comparisons are dyadic;
  • no floating-point value appears in a certificate;
  • connected components are glued by exact square adjacency; and
  • the kernel checks the final witness, not the subdivision search.
The Mathlib bridge supplies the semantic statements over Polynomial ℂ:
existence and uniqueness inside each region, exact root count, pairwise
separation, full coverage, and preservation under refinement.
Reference manual
  • SPEC — data model, algorithms, fuel, precision, and
performance budgets.
  • The Hex manual chapter “HexRoots: certified complex-root isolation”.
  • bench/HexRoots/ — deterministic degree and separation families.
  • conformance/HexRoots/ — fixtures checked against python-flint.
Contributing
Development happens in the
hex-dev monorepo, not in this published
mirror. Contributions are welcome as pull requests to the SPEC/ directory:
describe the behavior you want and leave the implementation to the maintainer.
About
Released mirror of HexRoots from hex-dev
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Readme
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hex-roots
Part of hex, a computer algebra
library for Lean 4. The aim is fast executable code, fully verified, built
with spec-driven development.
Certified complex-root isolation for dense integer polynomials, implemented in
Lean 4 without Mathlib.
Each result is a dyadic square plus an exact certificate that its selected
region contains one simple root. The driver supports two certificate forms:
Newton--Kantorovich contraction on a square and Pellet/Rouché root counting on
a circumscribed disc. Search may use approximate placement hints, but every
accepted atom is rechecked with exact dyadic arithmetic.
Quickstart
[[require]]
name = "hex-roots"git = "https://github.com/leanprover/hex-roots.git"rev = "main"
import HexRoots
Functionality
open Hex
defp : ZPoly := DensePoly.ofCoeffs #[-1, -1, 0, 1]
defroots : Option (Array (DyadicRootIsolation p)) :=
if h : HasOnlySimpleRoots p then
isolate p h 32 .nkThenPellet
else
none
Hex.isolate p h precision strategy returns pairwise-disjoint atoms for a
polynomial whose roots are simple. A nonzero constant returns an empty array;
the zero polynomial returns none. The simple-root test is executable and
decidable.
Use .nkThenPellet for the general strategy, .nk to select only the
Newton--Kantorovich certificate, or .pellet to select only Pellet. Returned
atoms can be refined independently with
Hex.DyadicRootIsolation.refineTo?.
The option type is an honest computational boundary: the core package does not
silently assume completeness. The companion package
hex-roots-mathlib proves
that every nonzero squarefree input succeeds and offers a none-free wrapper.
Verification
  • polynomial and Taylor arithmetic is exact;
  • centres, widths, and comparisons are dyadic;
  • no floating-point value appears in a certificate;
  • connected components are glued by exact square adjacency; and
  • the kernel checks the final witness, not the subdivision search.
The Mathlib bridge supplies the semantic statements over Polynomial ℂ:
existence and uniqueness inside each region, exact root count, pairwise
separation, full coverage, and preservation under refinement.
Reference manual
  • SPEC — data model, algorithms, fuel, precision, and
performance budgets.
  • The Hex manual chapter “HexRoots: certified complex-root isolation”.
  • bench/HexRoots/ — deterministic degree and separation families.
  • conformance/HexRoots/ — fixtures checked against python-flint.
Contributing
Development happens in the
hex-dev monorepo, not in this published
mirror. Contributions are welcome as pull requests to the SPEC/ directory:
describe the behavior you want and leave the implementation to the maintainer.
About
Released mirror of HexRoots from hex-dev
Resources
Readme
Apache-2.0 license
Activity
Custom properties
Stars
0 stars
Watchers
0 watching
Forks
0 forks
Report repository
Releases
Packages
Contributors
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