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2
DynaCertBench / original /dynacertbench_executable.json
Zhipeng Cao
Publish 105 DynaCertBench problems from pinned public GitHub release
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{
"schema_version": "dynacertbench.executable",
"description": "Question-only executable DynaCertBench ordered by active LaTeX systeminstance blocks.",
"source_dataset": "dynacertbench_by_problem.json",
"paper_snapshot_id": "paper",
"record_count": 105,
"excluded_count": 0,
"records": [
{
"record_id": "DCB-LIT-P1-001",
"problem": {
"problem_id": "dcb_lit_p1_001",
"task_type": "stability",
"name": "DynaCertBench System 001 · DCB-LIT-P1-001 · Exercise~4.30, \\cite{khalil2002nonlinear}",
"dynamics": "dx1/dt = -((1)/(2))*tan(((pi*x1)/(2))+((pi)/(4)))+x2+((1)/(2)), dx2/dt = x1-((1)/(2))*tan(((pi*x2)/(2))+((pi)/(4)))+((1)/(2))",
"source": "paper System 1; canonical record DCB-LIT-P1-001",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P1-001.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.",
"The source epsilon inner radius is provenance only; formal N includes the equilibrium."
],
"equilibrium": [
0,
0
],
"stability_kind": "local_asymptotic",
"neighborhood": {
"type": "constraints",
"constraints": [
{
"lhs": "x1**2 + x2**2",
"relation": "le",
"rhs": "(0.25)**2"
}
],
"bounds": [
[
"-0.25",
"0.25"
],
[
"-0.25",
"0.25"
]
]
},
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[
"-0.25",
"0.25"
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"-0.25",
"0.25"
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},
"candidate": null,
"paper_system_number": 1,
"paper_system_label": "System 1",
"local_name": "S001__DCB-LIT-P1-001"
},
{
"record_id": "DCB-LIT-P1-002",
"problem": {
"problem_id": "dcb_lit_p1_002",
"task_type": "stability",
"name": "DynaCertBench System 002 · DCB-LIT-P1-002 · Table~1 row~1 (NonPoly0), \\cite{abate2021fossil}",
"dynamics": "dx1/dt = -x1+x1*x2, dx2/dt = -x2",
"source": "paper System 2; canonical record DCB-LIT-P1-002",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P1-002.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.",
"The source epsilon inner radius is provenance only; formal N includes the equilibrium."
],
"equilibrium": [
0,
0
],
"stability_kind": "local_asymptotic",
"neighborhood": {
"type": "constraints",
"constraints": [
{
"lhs": "x1**2 + x2**2",
"relation": "le",
"rhs": "(1)**2"
}
],
"bounds": [
[
"-1",
"1"
],
[
"-1",
"1"
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]
},
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"-1",
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[
"-1",
"1"
]
]
},
"candidate": null,
"paper_system_number": 2,
"paper_system_label": "System 2",
"local_name": "S002__DCB-LIT-P1-002"
},
{
"record_id": "DCB-LIT-P1-003",
"problem": {
"problem_id": "dcb_lit_p1_003",
"task_type": "stability",
"name": "DynaCertBench System 003 · DCB-LIT-P1-003 · Table~1 row~2 (NonPoly1), \\cite{abate2021fossil}",
"dynamics": "dx1/dt = -x1+2*x1**2*x2, dx2/dt = -x2",
"source": "paper System 3; canonical record DCB-LIT-P1-003",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P1-003.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"equilibrium": [
0,
0
],
"stability_kind": "local_asymptotic",
"neighborhood": {
"type": "constraints",
"constraints": [
{
"lhs": "x1**2 + x2**2",
"relation": "le",
"rhs": "(10)**2"
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{
"lhs": "x1",
"relation": "ge",
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{
"lhs": "x2",
"relation": "ge",
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"0",
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},
"candidate": null,
"paper_system_number": 3,
"paper_system_label": "System 3",
"local_name": "S003__DCB-LIT-P1-003"
},
{
"record_id": "DCB-LIT-P1-004",
"problem": {
"problem_id": "dcb_lit_p1_004",
"task_type": "stability",
"name": "DynaCertBench System 004 · DCB-LIT-P1-004 · Table~1 row~3 (NonPoly2), \\cite{abate2021fossil}",
"dynamics": "dx1/dt = -x1, dx2/dt = -2*x2+0.1*x1*x2**2+x3, dx3/dt = -x3-1.5*x2",
"source": "paper System 4; canonical record DCB-LIT-P1-004",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P1-004.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"equilibrium": [
0,
0,
0
],
"stability_kind": "local_asymptotic",
"neighborhood": {
"type": "constraints",
"constraints": [
{
"lhs": "x1**2 + x2**2 + x3**2",
"relation": "le",
"rhs": "(10)**2"
},
{
"lhs": "x1",
"relation": "ge",
"rhs": "0"
},
{
"lhs": "x2",
"relation": "ge",
"rhs": "0"
},
{
"lhs": "x3",
"relation": "ge",
"rhs": "0"
}
],
"bounds": [
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"0",
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"0",
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[
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"0",
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},
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"paper_system_number": 4,
"paper_system_label": "System 4",
"local_name": "S004__DCB-LIT-P1-004"
},
{
"record_id": "DCB-LIT-P1-005",
"problem": {
"problem_id": "dcb_lit_p1_005",
"task_type": "stability",
"name": "DynaCertBench System 005 · DCB-LIT-P1-005 · Table~1 row~4 (NonPoly3), \\cite{abate2021fossil}",
"dynamics": "dx1/dt = -3*x1-0.1*x1*x2**3, dx2/dt = -x2+x3, dx3/dt = -x3",
"source": "paper System 5; canonical record DCB-LIT-P1-005",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P1-005.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"equilibrium": [
0,
0,
0
],
"stability_kind": "local_asymptotic",
"neighborhood": {
"type": "constraints",
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{
"lhs": "x1**2 + x2**2 + x3**2",
"relation": "le",
"rhs": "(10)**2"
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{
"lhs": "x1",
"relation": "ge",
"rhs": "0"
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{
"lhs": "x2",
"relation": "ge",
"rhs": "0"
},
{
"lhs": "x3",
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"rhs": "0"
}
],
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"0",
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[
"0",
"10"
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]
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"0",
"10"
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"0",
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"0",
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"candidate": null,
"paper_system_number": 5,
"paper_system_label": "System 5",
"local_name": "S005__DCB-LIT-P1-005"
},
{
"record_id": "DCB-LIT-P1-006",
"problem": {
"problem_id": "dcb_lit_p1_006",
"task_type": "stability",
"name": "DynaCertBench System 006 · DCB-LIT-P1-006 · Table~1 row~5 (Poly1), \\cite{abate2021fossil}",
"dynamics": "dx1/dt = -x1**3-x1*x3**2, dx2/dt = -x2-x1**2*x2, dx3/dt = -x3+3*x1**2*x3-3*x3",
"source": "paper System 6; canonical record DCB-LIT-P1-006",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P1-006.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.",
"The source epsilon inner radius is provenance only; formal N includes the equilibrium."
],
"equilibrium": [
0,
0,
0
],
"stability_kind": "local_asymptotic",
"neighborhood": {
"type": "constraints",
"constraints": [
{
"lhs": "x1**2 + x2**2 + x3**2",
"relation": "le",
"rhs": "(10)**2"
}
],
"bounds": [
[
"-10",
"10"
],
[
"-10",
"10"
],
[
"-10",
"10"
]
]
},
"analysis_bounds": [
[
"-10",
"10"
],
[
"-10",
"10"
],
[
"-10",
"10"
]
]
},
"candidate": null,
"paper_system_number": 6,
"paper_system_label": "System 6",
"local_name": "S006__DCB-LIT-P1-006"
},
{
"record_id": "DCB-LIT-P1-007",
"problem": {
"problem_id": "dcb_lit_p1_007",
"task_type": "stability",
"name": "DynaCertBench System 007 · DCB-LIT-P1-007 · Table~1 row~6 (Poly2), \\cite{abate2021fossil}",
"dynamics": "dx1/dt = -x1**3+x2, dx2/dt = -x1-x2",
"source": "paper System 7; canonical record DCB-LIT-P1-007",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P1-007.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.",
"The source epsilon inner radius is provenance only; formal N includes the equilibrium."
],
"equilibrium": [
0,
0
],
"stability_kind": "local_asymptotic",
"neighborhood": {
"type": "constraints",
"constraints": [
{
"lhs": "x1**2 + x2**2",
"relation": "le",
"rhs": "(10)**2"
}
],
"bounds": [
[
"-10",
"10"
],
[
"-10",
"10"
]
]
},
"analysis_bounds": [
[
"-10",
"10"
],
[
"-10",
"10"
]
]
},
"candidate": null,
"paper_system_number": 7,
"paper_system_label": "System 7",
"local_name": "S007__DCB-LIT-P1-007"
},
{
"record_id": "DCB-LIT-P1-008",
"problem": {
"problem_id": "dcb_lit_p1_008",
"task_type": "stability",
"name": "DynaCertBench System 008 · DCB-LIT-P1-008 · Table~1 row~7 (Poly3), \\cite{abate2021fossil}",
"dynamics": "dx1/dt = -x1**3-x2**2, dx2/dt = x1*x2-x2**3",
"source": "paper System 8; canonical record DCB-LIT-P1-008",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P1-008.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.",
"The source epsilon inner radius is provenance only; formal N includes the equilibrium."
],
"equilibrium": [
0,
0
],
"stability_kind": "local_asymptotic",
"neighborhood": {
"type": "constraints",
"constraints": [
{
"lhs": "x1**2 + x2**2",
"relation": "le",
"rhs": "(10)**2"
}
],
"bounds": [
[
"-10",
"10"
],
[
"-10",
"10"
]
]
},
"analysis_bounds": [
[
"-10",
"10"
],
[
"-10",
"10"
]
]
},
"candidate": null,
"paper_system_number": 8,
"paper_system_label": "System 8",
"local_name": "S008__DCB-LIT-P1-008"
},
{
"record_id": "DCB-LIT-P1-009",
"problem": {
"problem_id": "dcb_lit_p1_009",
"task_type": "stability",
"name": "DynaCertBench System 009 · DCB-LIT-P1-009 · Table~1 row~8 (Poly4), \\cite{abate2021fossil}",
"dynamics": "dx1/dt = -x1-1.5*x1**2*x2**3, dx2/dt = -x2**3+0.5*x1**3*x2**2",
"source": "paper System 9; canonical record DCB-LIT-P1-009",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P1-009.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.",
"The source epsilon inner radius is provenance only; formal N includes the equilibrium."
],
"equilibrium": [
0,
0
],
"stability_kind": "local_asymptotic",
"neighborhood": {
"type": "constraints",
"constraints": [
{
"lhs": "x1**2 + x2**2",
"relation": "le",
"rhs": "(10)**2"
}
],
"bounds": [
[
"-10",
"10"
],
[
"-10",
"10"
]
]
},
"analysis_bounds": [
[
"-10",
"10"
],
[
"-10",
"10"
]
]
},
"candidate": null,
"paper_system_number": 9,
"paper_system_label": "System 9",
"local_name": "S009__DCB-LIT-P1-009"
},
{
"record_id": "DCB-LIT-P1-010",
"problem": {
"problem_id": "dcb_lit_p1_010",
"task_type": "stability",
"name": "DynaCertBench System 010 · DCB-LIT-P1-010 · Table~4 \\#2 (SineModel), \\cite{edwards2024fossil}",
"dynamics": "dx1/dt = x2+u1, dx2/dt = u2+((3/20)*(981/100)*(1/2)*sin(x1)-(1/10)*x2)/((3/20)*(1/2)**2)",
"source": "paper System 10; canonical record DCB-LIT-P1-010",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P1-010.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.",
"The source epsilon inner radius is provenance only; formal N includes the equilibrium."
],
"controller_parameters": "u1,u2",
"equilibrium": [
0,
0
],
"stability_kind": "local_asymptotic",
"neighborhood": {
"type": "constraints",
"constraints": [
{
"lhs": "x1**2 + x2**2",
"relation": "le",
"rhs": "(1)**2"
}
],
"bounds": [
[
"-1",
"1"
],
[
"-1",
"1"
]
]
},
"analysis_bounds": [
[
"-1",
"1"
],
[
"-1",
"1"
]
]
},
"candidate": null,
"paper_system_number": 10,
"paper_system_label": "System 10",
"local_name": "S010__DCB-LIT-P1-010"
},
{
"record_id": "DCB-LIT-P1-011",
"problem": {
"problem_id": "dcb_lit_p1_011",
"task_type": "stability",
"name": "DynaCertBench System 011 · DCB-LIT-P1-011 · Table~4 \\#14 (DTAhmadi2), \\cite{edwards2024fossil}",
"dynamics": "x1[k+1] = 0.687*x1+0.558*x2-0.0001*x1*x2, x2[k+1] = -0.292*x1+0.773*x2",
"source": "paper System 11; canonical record DCB-LIT-P1-011",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P1-011.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.",
"The source epsilon inner radius is provenance only; formal N includes the equilibrium."
],
"equilibrium": [
0,
0
],
"stability_kind": "local_asymptotic",
"neighborhood": {
"type": "constraints",
"constraints": [
{
"lhs": "x1**2 + x2**2",
"relation": "le",
"rhs": "(1)**2"
}
],
"bounds": [
[
"-1",
"1"
],
[
"-1",
"1"
]
]
},
"analysis_bounds": [
[
"-1",
"1"
],
[
"-1",
"1"
]
]
},
"candidate": null,
"paper_system_number": 11,
"paper_system_label": "System 11",
"local_name": "S011__DCB-LIT-P1-011"
},
{
"record_id": "DCB-LIT-P1-012",
"problem": {
"problem_id": "dcb_lit_p1_012",
"task_type": "stability",
"name": "DynaCertBench System 012 · DCB-LIT-P1-012 · Table~4 \\#15 (DTAhmadi), \\cite{edwards2024fossil}",
"dynamics": "x1[k+1] = 0.3*x1+u1, x2[k+1] = -x1+0.5*x2+((7)/(18))*x1**2+u2",
"source": "paper System 12; canonical record DCB-LIT-P1-012",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P1-012.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.",
"The source epsilon inner radius is provenance only; formal N includes the equilibrium."
],
"controller_parameters": "u1,u2",
"equilibrium": [
0,
0
],
"stability_kind": "local_asymptotic",
"neighborhood": {
"type": "constraints",
"constraints": [
{
"lhs": "x1**2 + x2**2",
"relation": "le",
"rhs": "(1)**2"
}
],
"bounds": [
[
"-1",
"1"
],
[
"-1",
"1"
]
]
},
"analysis_bounds": [
[
"-1",
"1"
],
[
"-1",
"1"
]
]
},
"candidate": null,
"paper_system_number": 12,
"paper_system_label": "System 12",
"local_name": "S012__DCB-LIT-P1-012"
},
{
"record_id": "DCB-LIT-P1-013",
"problem": {
"problem_id": "dcb_lit_p1_013",
"task_type": "stability",
"name": "DynaCertBench System 013 · DCB-LIT-P1-013 · Table~1 row~4, \\cite{zou2025analytical}",
"dynamics": "dx1/dt = -3*x1**3+3*x1*x3-9*x1, dx2/dt = -x1**3-5*x2+5*x3**3, dx3/dt = -9*x3**3",
"source": "paper System 13; canonical record DCB-LIT-P1-013",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P1-013.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.",
"The source epsilon inner hole is retained only as provenance; N is the origin-containing outer box."
],
"equilibrium": [
0,
0,
0
],
"stability_kind": "local_asymptotic",
"neighborhood": {
"type": "box",
"bounds": [
[
"-1",
"1"
],
[
"-1",
"1"
],
[
"-1",
"1"
]
]
},
"analysis_bounds": [
[
"-1",
"1"
],
[
"-1",
"1"
],
[
"-1",
"1"
]
]
},
"candidate": null,
"paper_system_number": 13,
"paper_system_label": "System 13",
"local_name": "S013__DCB-LIT-P1-013"
},
{
"record_id": "DCB-LIT-P1-014",
"problem": {
"problem_id": "dcb_lit_p1_014",
"task_type": "stability",
"name": "DynaCertBench System 014 · DCB-LIT-P1-014 · Table~1 row~5, \\cite{zou2025analytical}",
"dynamics": "dx1/dt = x2, dx2/dt = -sin(x1)*cos(x1)-x2-sin(x3)*cos(x3), dx3/dt = x2-x3",
"source": "paper System 14; canonical record DCB-LIT-P1-014",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P1-014.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.",
"The source epsilon inner hole is retained only as provenance; N is the origin-containing outer box."
],
"equilibrium": [
0,
0,
0
],
"stability_kind": "local_asymptotic",
"neighborhood": {
"type": "box",
"bounds": [
[
"-1.5",
"1.5"
],
[
"-1.5",
"1.5"
],
[
"-1.5",
"1.5"
]
]
},
"analysis_bounds": [
[
"-1.5",
"1.5"
],
[
"-1.5",
"1.5"
],
[
"-1.5",
"1.5"
]
]
},
"candidate": null,
"paper_system_number": 14,
"paper_system_label": "System 14",
"local_name": "S014__DCB-LIT-P1-014"
},
{
"record_id": "DCB-LIT-P1-015",
"problem": {
"problem_id": "dcb_lit_p1_015",
"task_type": "stability",
"name": "DynaCertBench System 015 · DCB-LIT-P1-015 · Table~1 row~6, \\cite{zou2025analytical}",
"dynamics": "dq/dt = w1-w2, dw1/dt = (1/2)*(-1-2*w1-sin(q)+cos(q)), dw2/dt = (1/2)*(-1-2*w2+sin(q)+cos(q))",
"source": "paper System 15; canonical record DCB-LIT-P1-015",
"assumptions": [
"Audited center-of-inertia reduction with delta1=q/2 and delta2=-q/2.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.",
"The source epsilon inner hole is retained only as provenance; N is the origin-containing outer box."
],
"equilibrium": [
0,
0,
0
],
"stability_kind": "local_asymptotic",
"neighborhood": {
"type": "box",
"bounds": [
[
"-1.5",
"1.5"
],
[
"-2",
"2"
],
[
"-2",
"2"
]
]
},
"analysis_bounds": [
[
"-1.5",
"1.5"
],
[
"-2",
"2"
],
[
"-2",
"2"
]
]
},
"candidate": null,
"paper_system_number": 15,
"paper_system_label": "System 15",
"local_name": "S015__DCB-LIT-P1-015"
},
{
"record_id": "DCB-LIT-P1-016",
"problem": {
"problem_id": "dcb_lit_p1_016",
"task_type": "stability",
"name": "DynaCertBench System 016 · DCB-LIT-P1-016 · Table~1 row~7, \\cite{zou2025analytical}",
"dynamics": "dx1/dt = -x1+0.5*x2-0.1*x5**2, dx2/dt = -0.5*x1-x2, dx3/dt = -x3+0.5*x4-0.1*x1**2, dx4/dt = -0.5*x3-x4, dx5/dt = -x5+0.5*x6, dx6/dt = -0.5*x5-x6+0.1*x2**2",
"source": "paper System 16; canonical record DCB-LIT-P1-016",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P1-016.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.",
"The source epsilon inner hole is retained only as provenance; N is the origin-containing outer box."
],
"equilibrium": [
0,
0,
0,
0,
0,
0
],
"stability_kind": "local_asymptotic",
"neighborhood": {
"type": "box",
"bounds": [
[
"-1",
"1"
],
[
"-1",
"1"
],
[
"-1",
"1"
],
[
"-1",
"1"
],
[
"-1",
"1"
],
[
"-1",
"1"
]
]
},
"analysis_bounds": [
[
"-1",
"1"
],
[
"-1",
"1"
],
[
"-1",
"1"
],
[
"-1",
"1"
],
[
"-1",
"1"
],
[
"-1",
"1"
]
]
},
"candidate": null,
"paper_system_number": 16,
"paper_system_label": "System 16",
"local_name": "S016__DCB-LIT-P1-016"
},
{
"record_id": "DCB-LIT-P1-017",
"problem": {
"problem_id": "dcb_lit_p1_017",
"task_type": "stability",
"name": "DynaCertBench System 017 · DCB-LIT-P1-017 · Table~1 row~8, \\cite{zou2025analytical}",
"dynamics": "dx1/dt = x2, dx2/dt = -((3)/(2))*x6*x4-2*x1-((5)/(2))*x2-((1)/(2))*x4*sin(x2)*cos(x4), dx3/dt = x4, dx4/dt = ((3)/(2))*x6*x2-2*x3-10*x4+((1)/(2))*x2*sin(x2)*cos(x4), dx5/dt = x6, dx6/dt = -5*x5-((4)/(5))*x6",
"source": "paper System 17; canonical record DCB-LIT-P1-017",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P1-017.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.",
"The source epsilon inner hole is retained only as provenance; N is the origin-containing outer box."
],
"equilibrium": [
0,
0,
0,
0,
0,
0
],
"stability_kind": "local_asymptotic",
"neighborhood": {
"type": "box",
"bounds": [
[
"-3",
"3"
],
[
"-3",
"3"
],
[
"-3",
"3"
],
[
"-3",
"3"
],
[
"-3",
"3"
],
[
"-3",
"3"
]
]
},
"analysis_bounds": [
[
"-3",
"3"
],
[
"-3",
"3"
],
[
"-3",
"3"
],
[
"-3",
"3"
],
[
"-3",
"3"
],
[
"-3",
"3"
]
]
},
"candidate": null,
"paper_system_number": 17,
"paper_system_label": "System 17",
"local_name": "S017__DCB-LIT-P1-017"
},
{
"record_id": "DCB-LIT-P1-018",
"problem": {
"problem_id": "dcb_lit_p1_018",
"task_type": "stability",
"name": "DynaCertBench System 018 · DCB-LIT-P1-018 · Table~1 row~9, \\cite{zou2025analytical}",
"dynamics": "dd1/dt = w1-(w1+w2+w3)/3, dd2/dt = w2-(w1+w2+w3)/3, dw1/dt = -w1-(1/2)*(sin(d1-d2)+sin(2*d1+d2)), dw2/dt = -w2-(1/2)*(sin(d2-d1)+sin(d1+2*d2)), dw3/dt = -w3+(1/2)*(sin(2*d1+d2)+sin(d1+2*d2))",
"source": "paper System 18; canonical record DCB-LIT-P1-018",
"assumptions": [
"Audited center-of-inertia reduction with delta3=-(delta1+delta2).",
"The retained common-frequency mode decays; angle derivatives use omega_i minus the mean frequency.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.",
"The source epsilon inner hole is retained only as provenance; N is the origin-containing outer constrained region."
],
"equilibrium": [
0,
0,
0,
0,
0
],
"stability_kind": "local_asymptotic",
"neighborhood": {
"type": "constraints",
"bounds": [
[
"-0.75",
"0.75"
],
[
"-0.75",
"0.75"
],
[
"-1.2",
"1.2"
],
[
"-1.2",
"1.2"
],
[
"-1.2",
"1.2"
]
],
"constraints": [
{
"lhs": "d1+d2",
"relation": "le",
"rhs": "0.75"
},
{
"lhs": "d1+d2",
"relation": "ge",
"rhs": "-0.75"
}
]
},
"analysis_bounds": [
[
"-0.75",
"0.75"
],
[
"-0.75",
"0.75"
],
[
"-1.2",
"1.2"
],
[
"-1.2",
"1.2"
],
[
"-1.2",
"1.2"
]
]
},
"candidate": null,
"paper_system_number": 18,
"paper_system_label": "System 18",
"local_name": "S018__DCB-LIT-P1-018"
},
{
"record_id": "DCB-LIT-P1-019",
"problem": {
"problem_id": "dcb_lit_p1_019",
"task_type": "stability",
"name": "DynaCertBench System 019 · DCB-LIT-P1-019 · Table~1 row~10, \\cite{zou2025analytical}",
"dynamics": "dx1/dt = -x1+0.5*x2-0.1*x5**2, dx2/dt = -0.5*x1-x2+0.1*x8, dx3/dt = -x3+0.5*x4-0.1*x1**2, dx4/dt = -0.5*x3-x4, dx5/dt = -x5+0.5*x6, dx6/dt = -0.5*x5-x6+0.1*x2**2, dx7/dt = x8, dx8/dt = -sin(x7)*cos(x7)-x8-sin(x9)*cos(x9)-0.1*x2, dx9/dt = x8-x9",
"source": "paper System 19; canonical record DCB-LIT-P1-019",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P1-019.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.",
"The source epsilon inner hole is retained only as provenance; N is the origin-containing outer box."
],
"equilibrium": [
0,
0,
0,
0,
0,
0,
0,
0,
0
],
"stability_kind": "local_asymptotic",
"neighborhood": {
"type": "box",
"bounds": [
[
"-1.5",
"1.5"
],
[
"-1.5",
"1.5"
],
[
"-1.5",
"1.5"
],
[
"-1.5",
"1.5"
],
[
"-1.5",
"1.5"
],
[
"-1.5",
"1.5"
],
[
"-1.5",
"1.5"
],
[
"-1.5",
"1.5"
],
[
"-1.5",
"1.5"
]
]
},
"analysis_bounds": [
[
"-1.5",
"1.5"
],
[
"-1.5",
"1.5"
],
[
"-1.5",
"1.5"
],
[
"-1.5",
"1.5"
],
[
"-1.5",
"1.5"
],
[
"-1.5",
"1.5"
],
[
"-1.5",
"1.5"
],
[
"-1.5",
"1.5"
],
[
"-1.5",
"1.5"
]
]
},
"candidate": null,
"paper_system_number": 19,
"paper_system_label": "System 19",
"local_name": "S019__DCB-LIT-P1-019"
},
{
"record_id": "DCB-LIT-P1-020",
"problem": {
"problem_id": "dcb_lit_p1_020",
"task_type": "stability",
"name": "DynaCertBench System 020 · DCB-LIT-P1-020 · Appendix Table~6, \\cite{zou2025analytical}",
"dynamics": "dx1/dt = -8*x1*x2**2-10*x2**4, dx2/dt = -8*x2**3+3*x2**2-8*x2, dx3/dt = -x3",
"source": "paper System 20; canonical record DCB-LIT-P1-020",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P1-020.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.",
"The source epsilon inner hole and epsilon-stability label are retained only as provenance; the formal task uses the origin-containing outer box and ordinary local Lyapunov stability."
],
"equilibrium": [
0,
0,
0
],
"stability_kind": "local_lyapunov",
"neighborhood": {
"type": "box",
"bounds": [
[
"-1",
"1"
],
[
"-1",
"1"
],
[
"-1",
"1"
]
]
},
"analysis_bounds": [
[
"-1",
"1"
],
[
"-1",
"1"
],
[
"-1",
"1"
]
]
},
"candidate": null,
"paper_system_number": 20,
"paper_system_label": "System 20",
"local_name": "S020__DCB-LIT-P1-020"
},
{
"record_id": "DCB-LIT-P1-021",
"problem": {
"problem_id": "dcb_lit_p1_021",
"task_type": "stability",
"name": "DynaCertBench System 021 · DCB-LIT-P1-021 · Main theorem, \\cite{liu2026globally}",
"dynamics": "dx1/dt = 4*x1**3-x1**2*x2-6*x1*x2**2-x2**3, dx2/dt = x1**3+4*x1**2*x2+x1*x2**2-6*x2**3",
"source": "paper System 21; canonical record DCB-LIT-P1-021",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P1-021.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"equilibrium": [
0,
0
],
"stability_kind": "global_asymptotic",
"neighborhood": {
"type": "unbounded"
},
"analysis_bounds": [
[
-10,
10
],
[
-10,
10
]
],
"formal_domain": "unbounded"
},
"candidate": null,
"paper_system_number": 21,
"paper_system_label": "System 21",
"local_name": "S021__DCB-LIT-P1-021"
},
{
"record_id": "DCB-LIT-P2-001",
"problem": {
"problem_id": "dcb_lit_p2_001",
"task_type": "roa",
"name": "DynaCertBench System 022 · DCB-LIT-P2-001 · Section~5.3 and Appendix~C.3.3, \\cite{zhou2022neural}",
"dynamics": "dx1/dt = x2, dx2/dt = ((3/20)*(981/100)*(1/2)*sin(x1)+(20*tanh(-(23286329/1000000)*x1-(105411/20000)*x2))-(1/10)*x2)/((3/20)*(1/2)**2)",
"source": "paper System 22; canonical record DCB-LIT-P2-001",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P2-001.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"equilibrium": [
0,
0
],
"stability_kind": "local_asymptotic",
"neighborhood": {
"type": "ball",
"center": [
"0",
"0"
],
"radius": "4"
},
"analysis_bounds": [
[
"-4",
"4"
],
[
"-4",
"4"
]
]
},
"candidate": null,
"paper_system_number": 22,
"paper_system_label": "System 22",
"local_name": "S022__DCB-LIT-P2-001"
},
{
"record_id": "DCB-LIT-P2-002",
"problem": {
"problem_id": "dcb_lit_p2_002",
"task_type": "roa",
"name": "DynaCertBench System 023 · DCB-LIT-P2-002 · Section~5.1 and Appendix~C.3.1, \\cite{zhou2022neural}",
"dynamics": "dx1/dt = -x2, dx2/dt = x1 + (x1**2 - 1)*x2",
"equilibrium": [
0,
0
],
"stability_kind": "local_asymptotic",
"neighborhood": {
"type": "ball",
"center": [
0,
0
],
"radius": "6/5"
},
"analysis_bounds": [
[
"-6/5",
"6/5"
],
[
"-6/5",
"6/5"
]
],
"source": "paper System 23; canonical record DCB-LIT-P2-002",
"assumptions": [
"The formal neighborhood is the closed ball x1^2+x2^2 <= (6/5)^2.",
"No separate initial set is supplied; the target is to certify Omega_c itself."
]
},
"candidate": null,
"paper_system_number": 23,
"paper_system_label": "System 23",
"local_name": "S023__DCB-LIT-P2-002"
},
{
"record_id": "DCB-LIT-P2-003",
"problem": {
"problem_id": "dcb_lit_p2_003",
"task_type": "roa",
"name": "DynaCertBench System 024 · DCB-LIT-P2-003 · Example~3.1, \\cite{liu2023towards}",
"dynamics": "dx/dt = -x + x**3",
"equilibrium": [
0
],
"stability_kind": "local_asymptotic",
"neighborhood": {
"type": "constraint",
"lhs": "x**2",
"relation": "lt",
"rhs": "1",
"bounds": [
[
-1,
1
]
]
},
"analysis_bounds": [
[
-1,
1
]
],
"source": "paper System 24; canonical record DCB-LIT-P2-003",
"assumptions": [
"The strict constraint x^2 < 1 represents the source neighborhood (-1,1).",
"No separate initial set is supplied; the target is to certify Omega_c itself."
]
},
"candidate": null,
"paper_system_number": 24,
"paper_system_label": "System 24",
"local_name": "S024__DCB-LIT-P2-003"
},
{
"record_id": "DCB-LIT-P2-004",
"problem": {
"problem_id": "dcb_lit_p2_004",
"task_type": "roa",
"name": "DynaCertBench System 025 · DCB-LIT-P2-004 · Example~6.2 and Table~II, \\cite{liu2023towards}",
"dynamics": "dx1/dt = x2, dx2/dt = -2*x1 + 1/3*x1**3 - x2",
"equilibrium": [
0,
0
],
"stability_kind": "local_asymptotic",
"neighborhood": {
"type": "box",
"bounds": [
[
-6,
6
],
[
-6,
6
]
]
},
"analysis_bounds": [
[
-6,
6
],
[
-6,
6
]
],
"source": "paper System 25; canonical record DCB-LIT-P2-004",
"assumptions": [
"No separate initial set is supplied; the target is to certify Omega_c itself."
]
},
"candidate": null,
"paper_system_number": 25,
"paper_system_label": "System 25",
"local_name": "S025__DCB-LIT-P2-004"
},
{
"record_id": "DCB-LIT-P2-005",
"problem": {
"problem_id": "dcb_lit_p2_005",
"task_type": "roa",
"name": "DynaCertBench System 026 · DCB-LIT-P2-005 · Table~4 \\#3 (LorenzSystem), \\cite{edwards2024fossil}",
"dynamics": "dx1/dt = -10*(x1-x2) + u1, dx2/dt = 28*x1 - x2 - x1*x3 + u2, dx3/dt = x1*x2 - 8/3*x3 + u3",
"controller_parameters": "u1,u2,u3",
"equilibrium": [
0,
0,
0
],
"stability_kind": "local_asymptotic",
"neighborhood": {
"type": "ball",
"center": [
0,
0,
0
],
"radius": 2
},
"initial_set": {
"type": "ball",
"center": [
0,
0,
0
],
"radius": "3/10"
},
"analysis_bounds": [
[
-2,
2
],
[
-2,
2
],
[
-2,
2
]
],
"source": "paper System 26; canonical record DCB-LIT-P2-005",
"assumptions": [
"The source leaves u1,u2,u3 unconstrained over the reals, so no artificial control box is imposed.",
"The supplied initial ball must be contained in the synthesized Omega_c."
]
},
"candidate": null,
"paper_system_number": 26,
"paper_system_label": "System 26",
"local_name": "S026__DCB-LIT-P2-005"
},
{
"record_id": "DCB-LIT-P2-006",
"paper_system_number": 27,
"paper_system_label": "System 27",
"local_name": "S027__DCB-LIT-P2-006",
"problem": {
"problem_id": "dcb_lit_p2_006",
"task_type": "roa",
"name": "DynaCertBench System 027 · DCB-LIT-P2-006 · Example~5.2 and Table~2, \\cite{liu2024tool}",
"dynamics": "dx1/dt = x2, dx2/dt = -1/2*x2-sin(x1+pi/3)+sin(pi/3)",
"source": "paper System 27; canonical record DCB-LIT-P2-006",
"assumptions": [
"Transcribed from paper-4-20-20260904 System 27.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.",
"The final sine offset is positive in PDF 4-20, making the origin an equilibrium."
],
"analysis_bounds": [
[
"-2",
"3"
],
[
"-3",
"3/2"
]
],
"equilibrium": [
0,
0
],
"stability_kind": "local_asymptotic",
"neighborhood": {
"type": "box",
"bounds": [
[
"-2",
"3"
],
[
"-3",
"3/2"
]
]
}
},
"candidate": null
},
{
"record_id": "DCB-LIT-P2-007",
"problem": {
"problem_id": "dcb_lit_p2_007",
"task_type": "roa",
"name": "DynaCertBench System 028 · DCB-LIT-P2-007 · Example~5.4 and Table~3, \\cite{liu2024tool}",
"dynamics": "dx1/dt = -x1 + 1/2*x2 - 1/10*x9**2, dx2/dt = -1/2*x1 - x2, dx3/dt = -x3 + 1/2*x4 - 1/10*x1**2, dx4/dt = -1/2*x3 - x4, dx5/dt = -x5 + 1/2*x6 + 1/10*x7**2, dx6/dt = -1/2*x5 - x6, dx7/dt = -x7 + 1/2*x8, dx8/dt = -1/2*x7 - x8, dx9/dt = -x9 + 1/2*x10, dx10/dt = -1/2*x9 - x10 + 1/10*x2**2",
"equilibrium": [
0,
0,
0,
0,
0,
0,
0,
0,
0,
0
],
"stability_kind": "local_asymptotic",
"neighborhood": {
"type": "box",
"bounds": [
[
-1,
1
],
[
-1,
1
],
[
-1,
1
],
[
-1,
1
],
[
-1,
1
],
[
-1,
1
],
[
-1,
1
],
[
-1,
1
],
[
-1,
1
],
[
-1,
1
]
]
},
"analysis_bounds": [
[
-1,
1
],
[
-1,
1
],
[
-1,
1
],
[
-1,
1
],
[
-1,
1
],
[
-1,
1
],
[
-1,
1
],
[
-1,
1
],
[
-1,
1
],
[
-1,
1
]
],
"source": "paper System 28; canonical record DCB-LIT-P2-007",
"assumptions": [
"No separate initial set is supplied; the target is to certify Omega_c itself."
]
},
"candidate": null,
"paper_system_number": 28,
"paper_system_label": "System 28",
"local_name": "S028__DCB-LIT-P2-007"
},
{
"record_id": "DCB-LIT-P2-008",
"problem": {
"problem_id": "dcb_lit_p2_008",
"task_type": "roa",
"name": "DynaCertBench System 029 · DCB-LIT-P2-008 · Section~5.1, with verification details in Appendix~E.2 (\\(n=1\\)), \\cite{meng2024physics}",
"dynamics": "dx/dt = -x - x**3",
"equilibrium": [
0
],
"stability_kind": "local_asymptotic",
"neighborhood": {
"type": "box",
"bounds": [
[
-1,
1
]
]
},
"analysis_bounds": [
[
-1,
1
]
],
"source": "paper System 29; canonical record DCB-LIT-P2-008",
"assumptions": [
"No separate initial set is supplied; the target is to certify Omega_c itself."
]
},
"candidate": null,
"paper_system_number": 29,
"paper_system_label": "System 29",
"local_name": "S029__DCB-LIT-P2-008"
},
{
"record_id": "DCB-LIT-P3-001",
"problem": {
"problem_id": "dcb_lit_p3_001",
"task_type": "invariant",
"name": "DynaCertBench System 030 · DCB-LIT-P3-001 · Appendix~A.1 (PVTOL), \\cite{wu2023neural}",
"dynamics": "x1[k+1] = x1+1/20*(x4*cos(x3)-x5*sin(x3)), x2[k+1] = x2+1/20*(x4*sin(x3)+x5*cos(x3)), x3[k+1] = x3+1/20*x6, x4[k+1] = x4+1/20*(x5*x6-49/5*sin(x3)), x5[k+1] = x5+1/20*(-x4*x6-49/5*cos(x3)+(u1+u2)/4), x6[k+1] = x6+1/20*((1/4)/(19/400)*(u1-u2))",
"source": "paper System 30; canonical record DCB-LIT-P3-001",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P3-001.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"controller_parameters": "u1,u2",
"control_bounds": [
[
0,
"196/5"
],
[
0,
"196/5"
]
],
"unsafe_set": {
"type": "box",
"bounds": [
[
"-(1)",
"1"
],
[
"-(1)",
"1"
],
[
"-(1)",
"1"
],
[
"-(1)",
"1"
],
[
"-(1)",
"1"
],
[
"-(1)",
"1"
]
],
"complement": true
},
"analysis_bounds": [
[
"-1",
"1"
],
[
"-1",
"1"
],
[
"-1",
"1"
],
[
"-1",
"1"
],
[
"-1",
"1"
],
[
"-1",
"1"
]
],
"safety_interpretation": "outside_X_is_unsafe"
},
"candidate": null,
"paper_system_number": 30,
"paper_system_label": "System 30",
"local_name": "S030__DCB-LIT-P3-001"
},
{
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"problem": {
"problem_id": "dcb_lit_p3_002",
"task_type": "invariant",
"name": "DynaCertBench System 031 · DCB-LIT-P3-002 · Example~1, \\cite{samanipour2026relu}",
"dynamics": "dx1/dt = x2, dx2/dt = sin(x1)-3*x1-3*x2",
"source": "paper System 31; canonical record DCB-LIT-P3-002",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P3-002.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.",
"The paper fixes u=-3*x1-3*x2; it is substituted into the closed-loop dynamics, not synthesized as a free input."
],
"unsafe_set": {
"type": "box",
"bounds": [
[
"-(pi)",
"pi"
],
[
"-(pi)",
"pi"
]
],
"complement": true
},
"analysis_bounds": [
[
"-4",
"4"
],
[
"-4",
"4"
]
],
"safety_interpretation": "outside_X_is_unsafe"
},
"candidate": null,
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"paper_system_label": "System 31",
"local_name": "S031__DCB-LIT-P3-002"
},
{
"record_id": "DCB-LIT-P3-003",
"problem": {
"problem_id": "dcb_lit_p3_003",
"task_type": "invariant",
"name": "DynaCertBench System 032 · DCB-LIT-P3-003 · Example~2, \\cite{samanipour2026relu}",
"dynamics": "dx1/dt = x1**2+x1*x2+x1, dx2/dt = x1*x2+x2**2+x2",
"source": "paper System 32; canonical record DCB-LIT-P3-003",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P3-003.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"unsafe_set": {
"type": "box",
"bounds": [
[
"-1",
"0"
],
[
"-1",
"0"
]
]
},
"analysis_bounds": [
[
"-2",
"2"
],
[
"-2",
"2"
]
],
"safety_interpretation": "conditional_in_X"
},
"candidate": null,
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"paper_system_label": "System 32",
"local_name": "S032__DCB-LIT-P3-003"
},
{
"record_id": "DCB-LIT-P3-004",
"problem": {
"problem_id": "dcb_lit_p3_004",
"task_type": "invariant",
"name": "DynaCertBench System 033 · DCB-LIT-P3-004 · Example~3, \\cite{samanipour2026relu}",
"dynamics": "dx1/dt = x2, dx2/dt = ((x1+2.4109*x2+34.3620*x3+10.7009*x4)+(1/10)*x4**2*sin(x3)-(981/1000)*sin(x3)*cos(x3))/(1+(1/10)*sin(x3)**2), dx3/dt = x4, dx4/dt = ((10791/1000)*sin(x3)-(x1+2.4109*x2+34.3620*x3+10.7009*x4)*cos(x3)-(1/10)*x4**2*sin(x3)*cos(x3))/(1+(1/10)*sin(x3)**2)",
"source": "paper System 33; canonical record DCB-LIT-P3-004",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P3-004.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.",
"The paper's fixed linear policy is substituted exactly into the closed-loop dynamics; it is not synthesized as a free input."
],
"unsafe_set": {
"type": "box",
"bounds": [
[
"-(1)",
"1"
],
[
"-(1)",
"1"
],
[
"-(1)",
"1"
],
[
"-(1)",
"1"
]
],
"complement": true
},
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[
"-1",
"1"
],
[
"-1",
"1"
],
[
"-1",
"1"
],
[
"-1",
"1"
]
],
"safety_interpretation": "outside_X_is_unsafe"
},
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"paper_system_number": 33,
"paper_system_label": "System 33",
"local_name": "S033__DCB-LIT-P3-004"
},
{
"record_id": "DCB-LIT-P3-005",
"problem": {
"problem_id": "dcb_lit_p3_005",
"task_type": "invariant",
"name": "DynaCertBench System 034 · DCB-LIT-P3-005 · Section~6.1.1, \\cite{vaisi2025extracting}",
"dynamics": "x1[k+1] = x1+0.01*x2, x2[k+1] = x2+0.01*(9.8*sin(x1)+u)",
"source": "paper System 34; canonical record DCB-LIT-P3-005",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P3-005.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"controller_parameters": "u",
"unsafe_set": {
"type": "box",
"bounds": [
[
"-(pi/4)",
"pi/4"
],
[
"-(pi/4)",
"pi/4"
]
],
"complement": true
},
"analysis_bounds": [
[
"-1",
"1"
],
[
"-1",
"1"
]
],
"safety_interpretation": "outside_X_is_unsafe",
"control_bounds": [
[
-10,
10
]
]
},
"candidate": null,
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"paper_system_label": "System 34",
"local_name": "S034__DCB-LIT-P3-005"
},
{
"record_id": "DCB-LIT-P3-006",
"problem": {
"problem_id": "dcb_lit_p3_006",
"task_type": "invariant",
"name": "DynaCertBench System 035 · DCB-LIT-P3-006 · Section~6.1.2, \\cite{vaisi2025extracting}",
"dynamics": "x1[k+1] = x2, x2[k+1] = 29.4*sin(x1)+300*cos(x1)*tan(x3)+15*u*cos(x1)*cos(x3)**2, x3[k+1] = u",
"source": "paper System 35; canonical record DCB-LIT-P3-006",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P3-006.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"controller_parameters": "u",
"unsafe_set": {
"type": "box",
"bounds": [
[
"-(11/5)",
"11/5"
],
[
"-(11/5)",
"11/5"
],
[
"-(11/5)",
"11/5"
]
],
"complement": true
},
"analysis_bounds": [
[
"-11/5",
"11/5"
],
[
"-11/5",
"11/5"
],
[
"-11/5",
"11/5"
]
],
"safety_interpretation": "outside_X_is_unsafe",
"control_bounds": [
[
-10,
10
]
]
},
"candidate": null,
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"paper_system_label": "System 35",
"local_name": "S035__DCB-LIT-P3-006"
},
{
"record_id": "DCB-LIT-P4-001",
"problem": {
"problem_id": "dcb_lit_p4_001",
"task_type": "safety",
"name": "DynaCertBench System 036 · DCB-LIT-P4-001 · Theorem~4, \\cite{sogokon2018vector}",
"dynamics": "dx1/dt = x2, dx2/dt = x1",
"source": "paper System 36; canonical record DCB-LIT-P4-001",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-001.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"initial_set": {
"type": "constraints",
"constraints": [
{
"lhs": "x1",
"relation": "le",
"rhs": "0"
},
{
"lhs": "x2",
"relation": "le",
"rhs": "0"
}
]
},
"unsafe_set": {
"type": "constraint",
"lhs": "x1",
"relation": "gt",
"rhs": "0"
},
"analysis_bounds": [
[
-3,
3
],
[
-3,
3
]
],
"formal_domain": "unbounded"
},
"candidate": null,
"paper_system_number": 36,
"paper_system_label": "System 36",
"local_name": "S036__DCB-LIT-P4-001"
},
{
"record_id": "DCB-LIT-P4-002",
"problem": {
"problem_id": "dcb_lit_p4_002",
"task_type": "safety",
"name": "DynaCertBench System 037 · DCB-LIT-P4-002 · Example~1, \\cite{sogokon2018vector}",
"dynamics": "dx1/dt = 2*x1+x2+2*x3-x1**2+x1*x3-x3**2, dx2/dt = -2+x1-x2-x2**2, dx3/dt = -2-x2-x3+x1**2-x1*x3",
"source": "paper System 37; canonical record DCB-LIT-P4-002",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-002.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"initial_set": {
"type": "constraints",
"constraints": [
{
"lhs": "x1",
"relation": "le",
"rhs": "0"
},
{
"lhs": "x2",
"relation": "le",
"rhs": "0"
},
{
"lhs": "x3",
"relation": "le",
"rhs": "0"
}
]
},
"unsafe_set": {
"type": "constraint",
"lhs": "x1+x2+x3",
"relation": "ge",
"rhs": "1"
},
"analysis_bounds": [
[
-3,
3
],
[
-3,
3
],
[
-3,
3
]
],
"formal_domain": "unbounded"
},
"candidate": null,
"paper_system_number": 37,
"paper_system_label": "System 37",
"local_name": "S037__DCB-LIT-P4-002"
},
{
"record_id": "DCB-LIT-P4-003",
"problem": {
"problem_id": "dcb_lit_p4_003",
"task_type": "safety",
"name": "DynaCertBench System 038 · DCB-LIT-P4-003 · Example~3.1, \\cite{zhao2020synthesizing}",
"dynamics": "dx1/dt = x2, dx2/dt = -x1+((1)/(3))*x1**3-x2",
"source": "paper System 38; canonical record DCB-LIT-P4-003",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-003.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"initial_set": {
"type": "ball",
"center": [
"3/2",
"0"
],
"radius": "1/2"
},
"unsafe_set": {
"type": "ball",
"center": [
"-1",
"-1"
],
"radius": "2/5"
},
"analysis_bounds": [
[
"-7/2",
"2"
],
[
"-2",
"1"
]
]
},
"candidate": null,
"paper_system_number": 38,
"paper_system_label": "System 38",
"local_name": "S038__DCB-LIT-P4-003"
},
{
"record_id": "DCB-LIT-P4-004",
"problem": {
"problem_id": "dcb_lit_p4_004",
"task_type": "safety",
"name": "DynaCertBench System 039 · DCB-LIT-P4-004 · Table~1 row~9 (Barr1), \\cite{abate2021fossil}",
"dynamics": "dx1/dt = x2+2*x1*x2, dx2/dt = -x1-x2**2+2*x1**2",
"source": "paper System 39; canonical record DCB-LIT-P4-004",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-004.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"initial_set": {
"type": "box",
"bounds": [
[
"0",
"1"
],
[
"1",
"2"
]
]
},
"unsafe_set": {
"type": "constraint",
"lhs": "x1+x2**2",
"relation": "le",
"rhs": "0",
"bounds": [
[
"-2",
"2"
],
[
"-2",
"2"
]
]
},
"analysis_bounds": [
[
"-2",
"2"
],
[
"-2",
"2"
]
]
},
"candidate": null,
"paper_system_number": 39,
"paper_system_label": "System 39",
"local_name": "S039__DCB-LIT-P4-004"
},
{
"record_id": "DCB-LIT-P4-005",
"problem": {
"problem_id": "dcb_lit_p4_005",
"task_type": "safety",
"name": "DynaCertBench System 040 · DCB-LIT-P4-005 · Table~1 row~12 (Barr4), \\cite{abate2021fossil}",
"dynamics": "dx1/dt = sin(x3), dx2/dt = cos(x3), dx3/dt = -sin(x3)+((3*(x1*sin(x3)+x2*cos(x3)))/(0.5+x1**2+x2**2))",
"source": "paper System 40; canonical record DCB-LIT-P4-005",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-005.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"initial_set": {
"type": "box",
"bounds": [
[
"-1/10",
"1/10"
],
[
"-2",
"-9/5"
],
[
"-13/25",
"13/25"
]
]
},
"unsafe_set": {
"type": "constraint",
"lhs": "x1**2+x2**2",
"relation": "le",
"rhs": "1/25",
"bounds": [
[
"-2",
"2"
],
[
"-2",
"2"
],
[
"-157/100",
"157/100"
]
]
},
"analysis_bounds": [
[
"-2",
"2"
],
[
"-2",
"2"
],
[
"-157/100",
"157/100"
]
]
},
"candidate": null,
"paper_system_number": 40,
"paper_system_label": "System 40",
"local_name": "S040__DCB-LIT-P4-005"
},
{
"record_id": "DCB-LIT-P4-006",
"problem": {
"problem_id": "dcb_lit_p4_006",
"task_type": "safety",
"name": "DynaCertBench System 041 · DCB-LIT-P4-006 · Table~1 row~17 (Hi-Ord8), \\cite{abate2021fossil}",
"dynamics": "dx0/dt = x1, dx1/dt = x2, dx2/dt = x3, dx3/dt = x4, dx4/dt = x5, dx5/dt = x6, dx6/dt = x7, dx7/dt = -(20*x7 + 170*x6 + 800*x5 + 2273*x4 + 3980*x3 + 4180*x2 + 2400*x1 + 576*x0)",
"initial_set": {
"type": "box",
"bounds": [
[
"9/10",
"11/10"
],
[
"9/10",
"11/10"
],
[
"9/10",
"11/10"
],
[
"9/10",
"11/10"
],
[
"9/10",
"11/10"
],
[
"9/10",
"11/10"
],
[
"9/10",
"11/10"
],
[
"9/10",
"11/10"
]
]
},
"unsafe_set": {
"type": "box",
"bounds": [
[
"-11/5",
"-9/5"
],
[
"-11/5",
"-9/5"
],
[
"-11/5",
"-9/5"
],
[
"-11/5",
"-9/5"
],
[
"-11/5",
"-9/5"
],
[
"-11/5",
"-9/5"
],
[
"-11/5",
"-9/5"
],
[
"-11/5",
"-9/5"
]
]
},
"analysis_bounds": [
[
"-11/5",
"11/5"
],
[
"-11/5",
"11/5"
],
[
"-11/5",
"11/5"
],
[
"-11/5",
"11/5"
],
[
"-11/5",
"11/5"
],
[
"-11/5",
"11/5"
],
[
"-11/5",
"11/5"
],
[
"-11/5",
"11/5"
]
],
"source": "paper System 41; canonical record DCB-LIT-P4-006",
"assumptions": [
"The working domain is X=[-11/5, 11/5]^8.",
"The initial set is [9/10, 11/10]^8.",
"The unsafe set is [-11/5, -9/5]^8."
]
},
"candidate": null,
"paper_system_number": 41,
"paper_system_label": "System 41",
"local_name": "S041__DCB-LIT-P4-006"
},
{
"record_id": "DCB-LIT-P4-007",
"problem": {
"problem_id": "dcb_lit_p4_007",
"task_type": "safety",
"name": "DynaCertBench System 042 · DCB-LIT-P4-007 · Table~1 (arch2[36], instance~1), \\cite{wu2024completeness}",
"dynamics": "dx1/dt = x1**2+x2**2-1, dx2/dt = 5*(x1*x2-1)",
"source": "paper System 42; canonical record DCB-LIT-P4-007",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-007.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"initial_set": {
"type": "ball",
"center": [
"0",
"2"
],
"radius": "3/2"
},
"unsafe_set": {
"type": "box",
"bounds": [
[
"-3",
"-1"
],
[
"-3",
"-2"
]
]
},
"analysis_bounds": [
[
-4,
4
],
[
-4,
4
]
],
"formal_domain": "unbounded"
},
"candidate": null,
"paper_system_number": 42,
"paper_system_label": "System 42",
"local_name": "S042__DCB-LIT-P4-007"
},
{
"record_id": "DCB-LIT-P4-008",
"problem": {
"problem_id": "dcb_lit_p4_008",
"task_type": "safety",
"name": "DynaCertBench System 043 · DCB-LIT-P4-008 · Table~1 (arch3[36], instance~1), \\cite{wu2024completeness}",
"dynamics": "dx1/dt = x1-x1**3+x2-x1*x2**2, dx2/dt = -x1+x2-x1**2*x2-x2**3",
"source": "paper System 43; canonical record DCB-LIT-P4-008",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-008.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"initial_set": {
"type": "ball",
"center": [
"0",
"0"
],
"radius": "1/5"
},
"unsafe_set": {
"type": "ball",
"center": [
"5/2",
"5/2"
],
"radius": "1/2"
},
"analysis_bounds": [
[
-1,
4
],
[
-1,
4
]
],
"formal_domain": "unbounded"
},
"candidate": null,
"paper_system_number": 43,
"paper_system_label": "System 43",
"local_name": "S043__DCB-LIT-P4-008"
},
{
"record_id": "DCB-LIT-P4-009",
"problem": {
"problem_id": "dcb_lit_p4_009",
"task_type": "safety",
"name": "DynaCertBench System 044 · DCB-LIT-P4-009 · Example~5 and Table~1, \\cite{liu2015abstraction}",
"dynamics": "dx1/dt = exp(-x1)+x2-1, dx2/dt = -sin(x1)**2",
"source": "paper System 44; canonical record DCB-LIT-P4-009",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-009.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"initial_set": {
"type": "ball",
"center": [
"-1/2",
"1/2"
],
"radius": "2/5"
},
"unsafe_set": {
"type": "ball",
"center": [
"7/10",
"-7/10"
],
"radius": "3/10"
},
"analysis_bounds": [
[
-2,
2
],
[
-2,
2
]
]
},
"candidate": null,
"paper_system_number": 44,
"paper_system_label": "System 44",
"local_name": "S044__DCB-LIT-P4-009"
},
{
"record_id": "DCB-LIT-P4-010",
"problem": {
"problem_id": "dcb_lit_p4_010",
"task_type": "safety",
"name": "DynaCertBench System 045 · DCB-LIT-P4-010 · Example~6 and Table~1, \\cite{liu2015abstraction}",
"dynamics": "dx1/dt = -(((1/5)*10*x1*x2)/(x1+x2+x3))-(1/125)*x1, dx2/dt = (((1/5)*10*x1*x2)/(x1+x2+x3))-((1/125)+(1/10))*x2, dx3/dt = 1/10*x2-19/20*x3",
"source": "paper System 45; canonical record DCB-LIT-P4-010",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-010.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"initial_set": {
"type": "box",
"bounds": [
[
"1997/200",
"1999/200"
],
[
"1/200",
"3/200"
],
[
"0",
"3/1000"
]
]
},
"unsafe_set": {
"type": "constraint",
"lhs": "x3",
"relation": "gt",
"rhs": "1"
},
"analysis_bounds": [
[
0,
"10013/1000"
],
[
0,
"10013/1000"
],
[
0,
"10013/1000"
]
]
},
"candidate": null,
"paper_system_number": 45,
"paper_system_label": "System 45",
"local_name": "S045__DCB-LIT-P4-010"
},
{
"record_id": "DCB-LIT-P4-011",
"problem": {
"problem_id": "dcb_lit_p4_011",
"task_type": "safety",
"name": "DynaCertBench System 046 · DCB-LIT-P4-011 · Table~1 (lotka[14], instance~1), \\cite{wu2024completeness}",
"dynamics": "dx1/dt = x1*(1-x3), dx2/dt = x2*(1-2*x3), dx3/dt = x3*(x1+x2-1)",
"source": "paper System 46; canonical record DCB-LIT-P4-011",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-011.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"initial_set": {
"type": "ball",
"center": [
"1",
"1",
"3/2"
],
"radius": "3/10"
},
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"-1",
"-1",
"-3/2"
],
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},
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-3,
3
],
[
-3,
3
],
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-3,
3
]
],
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},
{
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"name": "DynaCertBench System 047 · DCB-LIT-P4-012 · Table~4 \\#5 (ctrlbarr obstacle), \\cite{edwards2024fossil}",
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"Deterministically normalized from canonical record DCB-LIT-P4-012.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
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[
"4",
"6"
],
[
"4",
"6"
],
[
"-pi/2",
"pi/2"
]
]
},
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"-9",
"-7"
],
[
"-9",
"-6"
],
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"-pi/2",
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]
]
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-10,
10
],
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-10,
10
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-4,
4
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},
{
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"task_type": "safety",
"name": "DynaCertBench System 048 · DCB-LIT-P4-013 · Table~4 \\#17 (ctrlbarr 2roomtemp DT), \\cite{edwards2024fossil}",
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"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-013.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
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"type": "box",
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[
"17",
"18"
],
[
"17",
"18"
]
]
},
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"28",
"30"
],
[
"28",
"30"
]
]
},
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17,
30
],
[
17,
30
]
]
},
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},
{
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"task_type": "safety",
"name": "DynaCertBench System 049 · DCB-LIT-P4-014 · ~\\texttt{S001}, \\cite{taheri2025barrierbench}",
"dynamics": "dx1/dt = x2+u0, dx2/dt = -0.8*x1-0.15*x2+0.2*x1**(2)+u1, dx3/dt = -0.1*x3+0.1*x1+u2",
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"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-014.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
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"type": "ball",
"center": [
"0",
"0",
"0"
],
"radius": "0.4"
},
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"type": "ball",
"center": [
"0",
"0",
"0"
],
"radius": "3",
"complement": true
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"-4",
"4"
],
[
"-4",
"4"
],
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"-4",
"4"
]
],
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},
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},
{
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"problem": {
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"task_type": "safety",
"name": "DynaCertBench System 050 · DCB-LIT-P4-015 · ~\\texttt{S002}, \\cite{taheri2025barrierbench}",
"dynamics": "dx1/dt = x2+u0, dx2/dt = -0.95*sin(x1)-0.02*x2+0.01*x3+u1, dx3/dt = -0.1*x3+0.02*x1+u2, dx4/dt = -0.03*x4+0.01*x2+u3",
"source": "paper System 50; canonical record DCB-LIT-P4-015",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-015.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
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"type": "ball",
"center": [
"0",
"0",
"0",
"0"
],
"radius": "0.3"
},
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"center": [
"0",
"0",
"0",
"0"
],
"radius": "2.5",
"complement": true
},
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"-7/2",
"7/2"
],
[
"-7/2",
"7/2"
],
[
"-7/2",
"7/2"
],
[
"-7/2",
"7/2"
]
],
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},
{
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"problem": {
"problem_id": "dcb_lit_p4_016",
"task_type": "safety",
"name": "DynaCertBench System 051 · DCB-LIT-P4-016 · ~\\texttt{S004}, \\cite{taheri2025barrierbench}",
"dynamics": "dx1/dt = 10*(x2-x1)+u0, dx2/dt = 28*x1-x2-x1*x3+u1, dx3/dt = x1*x2-((8)/(3))*x3+u2, dx4/dt = -x2*x3+2.7*x4+u3",
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"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-016.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
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"type": "ball",
"center": [
"0",
"0",
"0",
"0"
],
"radius": "0.1"
},
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"0",
"0",
"0",
"0"
],
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},
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"-26",
"26"
],
[
"-26",
"26"
],
[
"-26",
"26"
],
[
"-26",
"26"
]
],
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},
{
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"problem": {
"problem_id": "dcb_lit_p4_017",
"task_type": "safety",
"name": "DynaCertBench System 052 · DCB-LIT-P4-017 · ~\\texttt{S008}, \\cite{taheri2025barrierbench}",
"dynamics": "dx1/dt = -2*x1+6.7*x2-x2*x3+u0, dx2/dt = x1+u1, dx3/dt = -x3+x2**(2)+u2",
"source": "paper System 52; canonical record DCB-LIT-P4-017",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-017.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
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"type": "ball",
"center": [
"0",
"0",
"0"
],
"radius": "0.2"
},
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"type": "ball",
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"0",
"0",
"0"
],
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"complement": true
},
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"-6",
"6"
],
[
"-6",
"6"
],
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"-6",
"6"
]
],
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},
{
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"problem": {
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"task_type": "safety",
"name": "DynaCertBench System 053 · DCB-LIT-P4-018 · ~\\texttt{S011}, \\cite{taheri2025barrierbench}",
"dynamics": "dx1/dt = x2+u0, dx2/dt = -0.3*x1-0.2*(1+cos(x1))*x2-0.1*x1**(3)+u1",
"source": "paper System 53; canonical record DCB-LIT-P4-018",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-018.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
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"type": "ball",
"center": [
"0",
"0"
],
"radius": "0.6"
},
"unsafe_set": {
"type": "ball",
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"2",
"2"
],
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},
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"-8/5",
"4"
],
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"-8/5",
"4"
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},
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"paper_system_label": "System 53",
"local_name": "S053__DCB-LIT-P4-018"
},
{
"record_id": "DCB-LIT-P4-019",
"problem": {
"problem_id": "dcb_lit_p4_019",
"task_type": "safety",
"name": "DynaCertBench System 054 · DCB-LIT-P4-019 · ~\\texttt{S019}, \\cite{taheri2025barrierbench}",
"dynamics": "dx1/dt = -x1+x2*x3, dx2/dt = -x2+x1*x3, dx3/dt = -x3-x1*x2",
"source": "paper System 54; canonical record DCB-LIT-P4-019",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-019.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
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"type": "ball",
"center": [
"0",
"0",
"0"
],
"radius": "0.2"
},
"unsafe_set": {
"type": "ball",
"center": [
"0",
"0",
"0"
],
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"complement": true
},
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"-2",
"2"
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[
"-2",
"2"
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"-2",
"2"
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"candidate": null,
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},
{
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"problem": {
"problem_id": "dcb_lit_p4_020",
"task_type": "safety",
"name": "DynaCertBench System 055 · DCB-LIT-P4-020 · ~\\texttt{S024}, \\cite{taheri2025barrierbench}",
"dynamics": "dx1/dt = x2+u0, dx2/dt = -x1-0.4*x2+0.15*sin(x3)+u1, dx3/dt = x4+u2, dx4/dt = -0.6*x3-0.3*x4+u3",
"source": "paper System 55; canonical record DCB-LIT-P4-020",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-020.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"controller_parameters": "u0,u1,u2,u3",
"initial_set": {
"type": "box",
"bounds": [
[
"0.5",
"0.8"
],
[
"0.3",
"0.6"
],
[
"0.5",
"0.8"
],
[
"0.3",
"0.6"
]
]
},
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"type": "union",
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{
"type": "box",
"bounds": [
[
"2.5",
"3.5"
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[
"2",
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],
[
"2.5",
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[
"2",
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]
},
{
"type": "box",
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[
"-3.5",
"-2.5"
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[
"-3",
"-2"
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[
"-3.5",
"-2.5"
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[
"-3",
"-2"
]
]
}
]
},
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[
"-1/2",
"9/5"
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[
"-7/10",
"8/5"
],
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"-1/2",
"9/5"
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"-7/10",
"8/5"
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"paper_system_label": "System 55",
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},
{
"record_id": "DCB-LIT-P4-021",
"problem": {
"problem_id": "dcb_lit_p4_021",
"task_type": "safety",
"name": "DynaCertBench System 056 · DCB-LIT-P4-021 · ~\\texttt{S028}, \\cite{taheri2025barrierbench}",
"dynamics": "dx1/dt = -0.4*x1+0.25*x2*x3-0.15*x1**(3), dx2/dt = -0.5*x2+0.3*x1*x3-0.2*x2**(3), dx3/dt = -0.6*x3+0.2*x1*x2-0.18*x1**(2)*x2",
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"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-021.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
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"type": "ball",
"center": [
"0",
"0",
"0"
],
"radius": "0.5"
},
"unsafe_set": {
"type": "ball",
"center": [
"0",
"0",
"0"
],
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},
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[
"-9/2",
"9/2"
],
[
"-9/2",
"9/2"
],
[
"-9/2",
"9/2"
]
],
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},
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"paper_system_label": "System 56",
"local_name": "S056__DCB-LIT-P4-021"
},
{
"record_id": "DCB-LIT-P4-022",
"problem": {
"problem_id": "dcb_lit_p4_022",
"task_type": "safety",
"name": "DynaCertBench System 057 · DCB-LIT-P4-022 · ~\\texttt{S031}, \\cite{taheri2025barrierbench}",
"dynamics": "dx1/dt = -0.3*x1+0.2*x2*x4-0.15*x3**(2), dx2/dt = -0.4*x2+0.25*x1*x3-0.18*x2*x4, dx3/dt = -0.5*x3+0.3*x1*x4+0.12*x2**(2), dx4/dt = -0.6*x4+0.2*x2*x3-0.1*x1**(2)*x4",
"source": "paper System 57; canonical record DCB-LIT-P4-022",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-022.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
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"type": "ball",
"center": [
"0",
"0",
"0",
"0"
],
"radius": "0.4"
},
"unsafe_set": {
"type": "ball",
"center": [
"0",
"0",
"0",
"0"
],
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},
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[
"-4",
"4"
],
[
"-4",
"4"
],
[
"-4",
"4"
],
[
"-4",
"4"
]
],
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},
"candidate": null,
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"paper_system_label": "System 57",
"local_name": "S057__DCB-LIT-P4-022"
},
{
"record_id": "DCB-LIT-P4-023",
"problem": {
"problem_id": "dcb_lit_p4_023",
"task_type": "safety",
"name": "DynaCertBench System 058 · DCB-LIT-P4-023 · ~\\texttt{S036}, \\cite{taheri2025barrierbench}",
"dynamics": "x1[k+1] = 0.85*x1[k]+u0[k], x2[k+1] = 0.8*x2[k]+0.15*x1[k]+u1[k]",
"source": "paper System 58; canonical record DCB-LIT-P4-023",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-023.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"controller_parameters": "u0,u1",
"initial_set": {
"type": "box",
"bounds": [
[
"1.5",
"2"
],
[
"1.2",
"1.8"
]
]
},
"unsafe_set": {
"type": "union",
"sets": [
{
"type": "box",
"bounds": [
[
"4.5",
"5.5"
],
[
"4",
"5"
]
]
},
{
"type": "box",
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[
"-5.5",
"-4.5"
],
[
"-5",
"-4"
]
]
}
]
},
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[
"1/2",
"3"
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[
"1/5",
"14/5"
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],
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"paper_system_label": "System 58",
"local_name": "S058__DCB-LIT-P4-023"
},
{
"record_id": "DCB-LIT-P4-024",
"problem": {
"problem_id": "dcb_lit_p4_024",
"task_type": "safety",
"name": "DynaCertBench System 059 · DCB-LIT-P4-024 · ~\\texttt{S037}, \\cite{taheri2025barrierbench}",
"dynamics": "dx1/dt = x2+u0, dx2/dt = -0.2*x1-0.1*(x1**(2)-1)*x2+u1",
"source": "paper System 59; canonical record DCB-LIT-P4-024",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-024.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"controller_parameters": "u0,u1",
"initial_set": {
"type": "ball",
"center": [
"0",
"0"
],
"radius": "0.5"
},
"unsafe_set": {
"type": "ball",
"center": [
"0",
"0"
],
"radius": "2.8",
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},
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[
"-19/5",
"19/5"
],
[
"-19/5",
"19/5"
]
],
"formal_domain": "unbounded"
},
"candidate": null,
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"paper_system_label": "System 59",
"local_name": "S059__DCB-LIT-P4-024"
},
{
"record_id": "DCB-LIT-P4-025",
"problem": {
"problem_id": "dcb_lit_p4_025",
"task_type": "safety",
"name": "DynaCertBench System 060 · DCB-LIT-P4-025 · ~\\texttt{S042}, \\cite{taheri2025barrierbench}",
"dynamics": "dx1/dt = x2+u0, dx2/dt = -0.6*x1-0.1*x2+0.05*x3+u1, dx3/dt = x4+u2, dx4/dt = -0.8*x3-0.15*x4+0.05*x1+u3",
"source": "paper System 60; canonical record DCB-LIT-P4-025",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-025.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"controller_parameters": "u0,u1,u2,u3",
"initial_set": {
"type": "ball",
"center": [
"0",
"0",
"0",
"0"
],
"radius": "0.6"
},
"unsafe_set": {
"type": "ball",
"center": [
"0",
"0",
"0",
"0"
],
"radius": "3.5",
"complement": true
},
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[
"-9/2",
"9/2"
],
[
"-9/2",
"9/2"
],
[
"-9/2",
"9/2"
],
[
"-9/2",
"9/2"
]
],
"formal_domain": "unbounded"
},
"candidate": null,
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"paper_system_label": "System 60",
"local_name": "S060__DCB-LIT-P4-025"
},
{
"record_id": "DCB-LIT-P4-026",
"problem": {
"problem_id": "dcb_lit_p4_026",
"task_type": "safety",
"name": "DynaCertBench System 061 · DCB-LIT-P4-026 · ~\\texttt{S048}, \\cite{taheri2025barrierbench}",
"dynamics": "dx1/dt = -0.5*x1+0.3*x2*sin(x3)-0.1*x1**(2)*x2, dx2/dt = -0.6*x2+0.5*x1*cos(x3)+0.15*x3**(2)*x1, dx3/dt = -0.7*x3-0.1*x3**(3)",
"source": "paper System 61; canonical record DCB-LIT-P4-026",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-026.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"initial_set": {
"type": "ball",
"center": [
"0",
"0",
"0"
],
"radius": "0.4"
},
"unsafe_set": {
"type": "ball",
"center": [
"0",
"0",
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],
"radius": "3.5",
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},
"analysis_bounds": [
[
"-9/2",
"9/2"
],
[
"-9/2",
"9/2"
],
[
"-9/2",
"9/2"
]
],
"formal_domain": "unbounded"
},
"candidate": null,
"paper_system_number": 61,
"paper_system_label": "System 61",
"local_name": "S061__DCB-LIT-P4-026"
},
{
"record_id": "DCB-LIT-P4-027",
"problem": {
"problem_id": "dcb_lit_p4_027",
"task_type": "safety",
"name": "DynaCertBench System 062 · DCB-LIT-P4-027 · ~\\texttt{S052}, \\cite{taheri2025barrierbench}",
"dynamics": "dx1/dt = x2+u0, dx2/dt = x3+u1, dx3/dt = -6*x3-2.92*x2-1.2*x1+x1**(2)+u2",
"source": "paper System 62; canonical record DCB-LIT-P4-027",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-027.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"controller_parameters": "u0,u1,u2",
"initial_set": {
"type": "ball",
"center": [
"0",
"0",
"0"
],
"radius": "0.1"
},
"unsafe_set": {
"type": "ball",
"center": [
"0",
"0",
"0"
],
"radius": "2",
"complement": true
},
"analysis_bounds": [
[
"-3",
"3"
],
[
"-3",
"3"
],
[
"-3",
"3"
]
],
"formal_domain": "unbounded"
},
"candidate": null,
"paper_system_number": 62,
"paper_system_label": "System 62",
"local_name": "S062__DCB-LIT-P4-027"
},
{
"record_id": "DCB-LIT-P4-028",
"problem": {
"problem_id": "dcb_lit_p4_028",
"task_type": "safety",
"name": "DynaCertBench System 063 · DCB-LIT-P4-028 · ~\\texttt{S057}, \\cite{taheri2025barrierbench}",
"dynamics": "dx1/dt = -x1+0.08*x2+x1**(2)*x2+u0, dx2/dt = 0.6-0.08*x2-x1**(2)*x2+u1",
"source": "paper System 63; canonical record DCB-LIT-P4-028",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-028.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"controller_parameters": "u0,u1",
"initial_set": {
"type": "box",
"bounds": [
[
"0.1",
"0.5"
],
[
"1",
"2"
]
]
},
"unsafe_set": {
"type": "ball",
"center": [
"0",
"0"
],
"radius": "5",
"complement": true
},
"analysis_bounds": [
[
"-6",
"6"
],
[
"-6",
"6"
]
],
"formal_domain": "unbounded"
},
"candidate": null,
"paper_system_number": 63,
"paper_system_label": "System 63",
"local_name": "S063__DCB-LIT-P4-028"
},
{
"record_id": "DCB-LIT-P4-029",
"problem": {
"problem_id": "dcb_lit_p4_029",
"task_type": "safety",
"name": "DynaCertBench System 064 · DCB-LIT-P4-029 · ~\\texttt{S058}, \\cite{taheri2025barrierbench}",
"dynamics": "dx1/dt = x2+u0, dx2/dt = -0.1*x1-0.05*x2+0.02*x1*x3+u1, dx3/dt = -0.2*x3+0.1*x1+u2",
"source": "paper System 64; canonical record DCB-LIT-P4-029",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-029.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"controller_parameters": "u0,u1,u2",
"initial_set": {
"type": "ball",
"center": [
"0",
"0",
"0"
],
"radius": "0.5"
},
"unsafe_set": {
"type": "ball",
"center": [
"0",
"0",
"0"
],
"radius": "3",
"complement": true
},
"analysis_bounds": [
[
"-4",
"4"
],
[
"-4",
"4"
],
[
"-4",
"4"
]
],
"formal_domain": "unbounded"
},
"candidate": null,
"paper_system_number": 64,
"paper_system_label": "System 64",
"local_name": "S064__DCB-LIT-P4-029"
},
{
"record_id": "DCB-LIT-P4-030",
"problem": {
"problem_id": "dcb_lit_p4_030",
"task_type": "safety",
"name": "DynaCertBench System 065 · DCB-LIT-P4-030 · ~\\texttt{S059}, \\cite{taheri2025barrierbench}",
"dynamics": "dx1/dt = -0.25*x1-x2**(2)-x3**(2)+0.75+u0, dx2/dt = -x2+x1*x3+u1, dx3/dt = -x3+x1*x2-0.1*x1+u2",
"source": "paper System 65; canonical record DCB-LIT-P4-030",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-030.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"controller_parameters": "u0,u1,u2",
"initial_set": {
"type": "ball",
"center": [
"0",
"0",
"0"
],
"radius": "0.2"
},
"unsafe_set": {
"type": "ball",
"center": [
"0",
"0",
"0"
],
"radius": "3",
"complement": true
},
"analysis_bounds": [
[
"-4",
"4"
],
[
"-4",
"4"
],
[
"-4",
"4"
]
],
"formal_domain": "unbounded"
},
"candidate": null,
"paper_system_number": 65,
"paper_system_label": "System 65",
"local_name": "S065__DCB-LIT-P4-030"
},
{
"record_id": "DCB-LIT-P4-031",
"problem": {
"problem_id": "dcb_lit_p4_031",
"task_type": "safety",
"name": "DynaCertBench System 066 · DCB-LIT-P4-031 · ~\\texttt{S060}, \\cite{taheri2025barrierbench}",
"dynamics": "dx1/dt = x1*(x1-0.1)*(1-x1)-0.3*x1*x2+u0, dx2/dt = -0.2*x2+0.1*x1*x2+u1",
"source": "paper System 66; canonical record DCB-LIT-P4-031",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-031.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"controller_parameters": "u0,u1",
"initial_set": {
"type": "box",
"bounds": [
[
"0.2",
"0.8"
],
[
"0.1",
"0.6"
]
]
},
"unsafe_set": {
"type": "box",
"bounds": [
[
"1.5",
"2"
],
[
"1",
"1.5"
]
]
},
"analysis_bounds": [
[
"-4/5",
"3"
],
[
"-9/10",
"5/2"
]
],
"formal_domain": "unbounded"
},
"candidate": null,
"paper_system_number": 66,
"paper_system_label": "System 66",
"local_name": "S066__DCB-LIT-P4-031"
},
{
"record_id": "DCB-LIT-P4-032",
"problem": {
"problem_id": "dcb_lit_p4_032",
"task_type": "safety",
"name": "DynaCertBench System 067 · DCB-LIT-P4-032 · ~\\texttt{S062}, \\cite{taheri2025barrierbench}",
"dynamics": "dx1/dt = x2+u0, dx2/dt = -x1+x2*x3+u1, dx3/dt = 1-x2**(2)+u2",
"source": "paper System 67; canonical record DCB-LIT-P4-032",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-032.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"controller_parameters": "u0,u1,u2",
"initial_set": {
"type": "ball",
"center": [
"0",
"0",
"0"
],
"radius": "0.2"
},
"unsafe_set": {
"type": "ball",
"center": [
"0",
"0",
"0"
],
"radius": "2.5",
"complement": true
},
"analysis_bounds": [
[
"-7/2",
"7/2"
],
[
"-7/2",
"7/2"
],
[
"-7/2",
"7/2"
]
],
"formal_domain": "unbounded"
},
"candidate": null,
"paper_system_number": 67,
"paper_system_label": "System 67",
"local_name": "S067__DCB-LIT-P4-032"
},
{
"record_id": "DCB-LIT-P4-033",
"problem": {
"problem_id": "dcb_lit_p4_033",
"task_type": "safety",
"name": "DynaCertBench System 068 · DCB-LIT-P4-033 · ~\\texttt{S076}, \\cite{taheri2025barrierbench}",
"dynamics": "dx1/dt = x3+x2*x1+u0, dx2/dt = -x2-x1**(2)+u1, dx3/dt = -x1-0.5*x3+u2",
"source": "paper System 68; canonical record DCB-LIT-P4-033",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-033.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"controller_parameters": "u0,u1,u2",
"initial_set": {
"type": "ball",
"center": [
"0",
"0",
"0"
],
"radius": "0.3"
},
"unsafe_set": {
"type": "ball",
"center": [
"0",
"0",
"0"
],
"radius": "3.5",
"complement": true
},
"analysis_bounds": [
[
"-9/2",
"9/2"
],
[
"-9/2",
"9/2"
],
[
"-9/2",
"9/2"
]
],
"formal_domain": "unbounded"
},
"candidate": null,
"paper_system_number": 68,
"paper_system_label": "System 68",
"local_name": "S068__DCB-LIT-P4-033"
},
{
"record_id": "DCB-LIT-P4-034",
"problem": {
"problem_id": "dcb_lit_p4_034",
"task_type": "safety",
"name": "DynaCertBench System 069 · DCB-LIT-P4-034 · ~\\texttt{S077}, \\cite{taheri2025barrierbench}",
"dynamics": "dx1/dt = x2+u0, dx2/dt = -0.8*x1-0.3*x2*(1+x1**(2))+0.1*x3+u1, dx3/dt = x4+u2, dx4/dt = -0.9*x3-0.25*x4*(1+x3**(2))+0.1*x1+u3",
"source": "paper System 69; canonical record DCB-LIT-P4-034",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-034.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"controller_parameters": "u0,u1,u2,u3",
"initial_set": {
"type": "ball",
"center": [
"0",
"0",
"0",
"0"
],
"radius": "0.35"
},
"unsafe_set": {
"type": "ball",
"center": [
"0",
"0",
"0",
"0"
],
"radius": "2.8",
"complement": true
},
"analysis_bounds": [
[
"-19/5",
"19/5"
],
[
"-19/5",
"19/5"
],
[
"-19/5",
"19/5"
],
[
"-19/5",
"19/5"
]
],
"formal_domain": "unbounded"
},
"candidate": null,
"paper_system_number": 69,
"paper_system_label": "System 69",
"local_name": "S069__DCB-LIT-P4-034"
},
{
"record_id": "DCB-LIT-P4-035",
"problem": {
"problem_id": "dcb_lit_p4_035",
"task_type": "safety",
"name": "DynaCertBench System 070 · DCB-LIT-P4-035 · ~\\texttt{S087}, \\cite{taheri2025barrierbench}",
"dynamics": "x1[k+1] = -1/100*x2, x2[k+1] = 1/100*x1",
"formal_domain": "unbounded",
"initial_set": {
"type": "box",
"bounds": [
[
"1/10",
"2/5"
],
[
"1/10",
"11/20"
]
]
},
"unsafe_set": {
"type": "box",
"bounds": [
[
"9/20",
"1/2"
],
[
"3/5",
1
]
]
},
"analysis_bounds": [
[
-2,
2
],
[
-2,
2
]
],
"source": "paper System 70; canonical record DCB-LIT-P4-035",
"assumptions": [
"The source declares X = R^2, so all formal conditions are global.",
"The displayed Euler map simplifies exactly to x1+ = -x2/100 and x2+ = x1/100.",
"The finite analysis box is only a FastCheck screening box."
]
},
"candidate": null,
"paper_system_number": 70,
"paper_system_label": "System 70",
"local_name": "S070__DCB-LIT-P4-035"
},
{
"record_id": "DCB-LIT-P4-036",
"problem": {
"problem_id": "dcb_lit_p4_036",
"task_type": "safety",
"name": "DynaCertBench System 071 · DCB-LIT-P4-036 · ~\\texttt{U000}, \\cite{taheri2025barrierbench}",
"dynamics": "dx1/dt = -0.5*x1+0.3*x2, dx2/dt = -0.4*x2+0.15*cos(x1)*sin(x3), dx3/dt = -0.6*x3+0.1*sin(x2)*cos(x4), dx4/dt = -0.5*x4",
"source": "paper System 71; canonical record DCB-LIT-P4-036",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-036.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"initial_set": {
"type": "ball",
"center": [
"0",
"0",
"0",
"0"
],
"radius": "0.3"
},
"unsafe_set": {
"type": "ball",
"center": [
"0",
"0",
"0",
"0"
],
"radius": "2.5",
"complement": true
},
"analysis_bounds": [
[
"-7/2",
"7/2"
],
[
"-7/2",
"7/2"
],
[
"-7/2",
"7/2"
],
[
"-7/2",
"7/2"
]
],
"formal_domain": "unbounded"
},
"candidate": null,
"paper_system_number": 71,
"paper_system_label": "System 71",
"local_name": "S071__DCB-LIT-P4-036"
},
{
"record_id": "DCB-LIT-P4-037",
"problem": {
"problem_id": "dcb_lit_p4_037",
"task_type": "safety",
"name": "DynaCertBench System 072 · DCB-LIT-P4-037 · ~\\texttt{U001}, \\cite{taheri2025barrierbench}",
"dynamics": "dx1/dt = -0.3*x1+0.5*sin(x2), dx2/dt = -0.4*x2-0.6*sin(x1)+0.1*sin(x3), dx3/dt = -0.5*x3+0.2*sin(x1)*sin(x2)",
"source": "paper System 72; canonical record DCB-LIT-P4-037",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-037.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"initial_set": {
"type": "ball",
"center": [
"0",
"0",
"0"
],
"radius": "0.4"
},
"unsafe_set": {
"type": "ball",
"center": [
"0",
"0",
"0"
],
"radius": "3",
"complement": true
},
"analysis_bounds": [
[
"-4",
"4"
],
[
"-4",
"4"
],
[
"-4",
"4"
]
],
"formal_domain": "unbounded"
},
"candidate": null,
"paper_system_number": 72,
"paper_system_label": "System 72",
"local_name": "S072__DCB-LIT-P4-037"
},
{
"record_id": "DCB-LIT-P4-038",
"problem": {
"problem_id": "dcb_lit_p4_038",
"task_type": "safety",
"name": "DynaCertBench System 073 · DCB-LIT-P4-038 · ~\\texttt{U002}, \\cite{taheri2025barrierbench}",
"dynamics": "dx1/dt = -0.3*x1+0.2*sin(x2)*exp(-x3**(2)), dx2/dt = -0.4*x2+0.15*cos(x1)*exp(-x1**(2)), dx3/dt = -0.5*x3+0.1*sin(x1+x2)",
"source": "paper System 73; canonical record DCB-LIT-P4-038",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-038.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"initial_set": {
"type": "ball",
"center": [
"0",
"0",
"0"
],
"radius": "0.45"
},
"unsafe_set": {
"type": "ball",
"center": [
"0",
"0",
"0"
],
"radius": "3.2",
"complement": true
},
"analysis_bounds": [
[
"-21/5",
"21/5"
],
[
"-21/5",
"21/5"
],
[
"-21/5",
"21/5"
]
],
"formal_domain": "unbounded"
},
"candidate": null,
"paper_system_number": 73,
"paper_system_label": "System 73",
"local_name": "S073__DCB-LIT-P4-038"
},
{
"record_id": "DCB-LIT-P4-039",
"problem": {
"problem_id": "dcb_lit_p4_039",
"task_type": "safety",
"name": "DynaCertBench System 074 · DCB-LIT-P4-039 · ~\\texttt{U003}, \\cite{taheri2025barrierbench}",
"dynamics": "x1[k+1] = 4/5*x1 + 1/10*sin(x2), x2[k+1] = 17/20*x2 + 2/25*x1*x3, x3[k+1] = 9/10*x3 + 2/25*cos(x1*x2)",
"formal_domain": "unbounded",
"initial_set": {
"type": "box",
"bounds": [
[
"1/10",
"2/5"
],
[
"1/10",
"3/10"
],
[
"1/10",
"2/5"
]
]
},
"unsafe_set": {
"type": "box",
"bounds": [
[
2,
"5/2"
],
[
"9/5",
"11/5"
],
[
"9/5",
"23/10"
]
]
},
"analysis_bounds": [
[
-3,
3
],
[
-3,
3
],
[
-3,
3
]
],
"source": "paper System 74; canonical record DCB-LIT-P4-039",
"assumptions": [
"The source declares X = R^3, so all formal conditions are global.",
"The finite analysis box is only a FastCheck screening box.",
"BarrierBench reports no successful source certificate for U003."
]
},
"candidate": null,
"paper_system_number": 74,
"paper_system_label": "System 74",
"local_name": "S074__DCB-LIT-P4-039"
},
{
"record_id": "DCB-LIT-P4-040",
"problem": {
"problem_id": "dcb_lit_p4_040",
"task_type": "safety",
"name": "DynaCertBench System 075 · DCB-LIT-P4-040 · ~\\texttt{U004}, \\cite{taheri2025barrierbench}",
"dynamics": "dx1/dt = -0.65*x1 + 0.1*x2**2*cos(x3), dx2/dt = -0.7*x2 + 0.12*sin(x1)*tanh(x4), dx3/dt = -0.75*x3 + 0.08*x1*x2, dx4/dt = -0.6*x4 + 0.06*(x1 + x2)",
"initial_set": {
"type": "box",
"bounds": [
[
"1/5",
"1/2"
],
[
"-3/10",
"0"
],
[
"1/10",
"2/5"
],
[
"-1/5",
"1/10"
]
]
},
"unsafe_set": {
"type": "box",
"bounds": [
[
"-11/5",
"-17/10"
],
[
"8/5",
"21/10"
],
[
"-19/10",
"-7/5"
],
[
"13/10",
"9/5"
]
]
},
"analysis_bounds": [
[
-3,
3
],
[
-3,
3
],
[
-3,
3
],
[
-3,
3
]
],
"formal_domain": "unbounded",
"source": "paper System 75; canonical record DCB-LIT-P4-040",
"assumptions": [
"The catalogue states X = R^4; formal verification is global. The box [-3,3]^4 is used only for FastCheck screening.",
"The initial set is [1/5,1/2] x [-3/10,0] x [1/10,2/5] x [-1/5,1/10].",
"The unsafe set is [-11/5,-17/10] x [8/5,21/10] x [-19/10,-7/5] x [13/10,9/5]."
]
},
"candidate": null,
"paper_system_number": 75,
"paper_system_label": "System 75",
"local_name": "S075__DCB-LIT-P4-040"
},
{
"record_id": "DCB-LIT-P4-041",
"problem": {
"problem_id": "dcb_lit_p4_041",
"task_type": "safety",
"name": "DynaCertBench System 076 · DCB-LIT-P4-041 · ~\\texttt{U005}, \\cite{taheri2025barrierbench}",
"dynamics": "dx1/dt = -0.58*x1+0.12*sin(x2)*cos(x3), dx2/dt = -0.62*x2+0.1*exp(-x1**(2))*tanh(x3), dx3/dt = -0.66*x3+0.08*sin(x1*x2)*exp(-x3**(2))",
"source": "paper System 76; canonical record DCB-LIT-P4-041",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-041.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"initial_set": {
"type": "box",
"bounds": [
[
"-0.3",
"0.2"
],
[
"0.1",
"0.4"
],
[
"-0.2",
"0.3"
]
]
},
"unsafe_set": {
"type": "box",
"bounds": [
[
"-2.3",
"-1.8"
],
[
"1.7",
"2.2"
],
[
"-2",
"-1.5"
]
]
},
"analysis_bounds": [
[
"-33/10",
"6/5"
],
[
"-9/10",
"16/5"
],
[
"-3",
"13/10"
]
],
"formal_domain": "unbounded"
},
"candidate": null,
"paper_system_number": 76,
"paper_system_label": "System 76",
"local_name": "S076__DCB-LIT-P4-041"
},
{
"record_id": "DCB-LIT-P4-042",
"problem": {
"problem_id": "dcb_lit_p4_042",
"task_type": "safety",
"name": "DynaCertBench System 077 · DCB-LIT-P4-042 · ~\\texttt{U006}, \\cite{taheri2025barrierbench}",
"dynamics": "dx1/dt = u0*cos(x3), dx2/dt = u0*sin(x3), dx3/dt = u1",
"source": "paper System 77; canonical record DCB-LIT-P4-042",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-042.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.",
"The unsafe set is the complement of the outer box, as specified by complement=true in the source JSON."
],
"controller_parameters": "u0,u1",
"initial_set": {
"type": "box",
"bounds": [
[
"-1",
"1"
],
[
"-1",
"1"
],
[
"-0.5",
"0.5"
]
]
},
"unsafe_set": {
"type": "box",
"bounds": [
[
"-5",
"5"
],
[
"-5",
"5"
],
[
"-3.14",
"3.14"
]
],
"complement": true
},
"analysis_bounds": [
[
"-6",
"6"
],
[
"-6",
"6"
],
[
"-207/50",
"207/50"
]
],
"formal_domain": "unbounded"
},
"candidate": null,
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"paper_system_label": "System 77",
"local_name": "S077__DCB-LIT-P4-042"
},
{
"record_id": "DCB-LIT-P4-043",
"problem": {
"problem_id": "dcb_lit_p4_043",
"task_type": "safety",
"name": "DynaCertBench System 078 · DCB-LIT-P4-043 · ~\\texttt{U007}, \\cite{taheri2025barrierbench}",
"dynamics": "x1[k+1] = 0.75*x1[k]+0.1*tanh(x2[k])*tanh(x3[k]), x2[k+1] = 0.82*x2[k]+0.09*sin(x1[k])*exp(-x3[k]**(2)), x3[k+1] = 0.79*x3[k]+0.08*cos(x1[k])*cos(x2[k])",
"source": "paper System 78; canonical record DCB-LIT-P4-043",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-043.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"initial_set": {
"type": "ball",
"center": [
"1",
"-0.5",
"0.8"
],
"radius": "0.5"
},
"unsafe_set": {
"type": "ball",
"center": [
"-2.5",
"2",
"-2.2"
],
"radius": "0.6"
},
"analysis_bounds": [
[
"-41/10",
"5/2"
],
[
"-2",
"18/5"
],
[
"-19/5",
"23/10"
]
],
"formal_domain": "unbounded"
},
"candidate": null,
"paper_system_number": 78,
"paper_system_label": "System 78",
"local_name": "S078__DCB-LIT-P4-043"
},
{
"record_id": "DCB-LIT-P4-044",
"problem": {
"problem_id": "dcb_lit_p4_044",
"task_type": "safety",
"name": "DynaCertBench System 079 · DCB-LIT-P4-044 · ~\\texttt{U008}, \\cite{taheri2025barrierbench}",
"dynamics": "x1[k+1] = 0.73*x1[k]+0.13*sin(x2[k])*tanh(x3[k]), x2[k+1] = 0.89*x2[k]+0.17*cos(x1[k]**(2))*exp(-x3[k]**(2)), x3[k+1] = 0.64*x3[k]+0.22*tanh(x1[k]*x2[k])",
"source": "paper System 79; canonical record DCB-LIT-P4-044",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-044.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"initial_set": {
"type": "ball",
"center": [
"0.9",
"-0.7",
"1.2"
],
"radius": "0.55"
},
"unsafe_set": {
"type": "ball",
"center": [
"-3.1",
"2.8",
"-2.6"
],
"radius": "0.75"
},
"analysis_bounds": [
[
"-97/20",
"49/20"
],
[
"-9/4",
"91/20"
],
[
"-87/20",
"11/4"
]
],
"formal_domain": "unbounded"
},
"candidate": null,
"paper_system_number": 79,
"paper_system_label": "System 79",
"local_name": "S079__DCB-LIT-P4-044"
},
{
"record_id": "DCB-LIT-P4-045",
"problem": {
"problem_id": "dcb_lit_p4_045",
"task_type": "safety",
"name": "DynaCertBench System 080 · DCB-LIT-P4-045 · ~\\texttt{U009}, \\cite{taheri2025barrierbench}",
"dynamics": "dx1/dt = -0.62*x1+0.1*sin(x2+x3), dx2/dt = -0.67*x2+0.08*cos(x1)*exp(-x3**(2)), dx3/dt = -0.64*x3+0.09*tanh(x1)*sin(x2)",
"source": "paper System 80; canonical record DCB-LIT-P4-045",
"assumptions": [
"Deterministically normalized from canonical record DCB-LIT-P4-045.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"initial_set": {
"type": "box",
"bounds": [
[
"-0.2",
"0.3"
],
[
"0.1",
"0.5"
],
[
"-0.3",
"0.2"
]
]
},
"unsafe_set": {
"type": "box",
"bounds": [
[
"2",
"2.6"
],
[
"-2.4",
"-1.8"
],
[
"2.1",
"2.7"
]
]
},
"analysis_bounds": [
[
"-6/5",
"18/5"
],
[
"-17/5",
"3/2"
],
[
"-13/10",
"37/10"
]
],
"formal_domain": "unbounded"
},
"candidate": null,
"paper_system_number": 80,
"paper_system_label": "System 80",
"local_name": "S080__DCB-LIT-P4-045"
},
{
"record_id": "DCB-EXT-P1-001",
"problem": {
"problem_id": "dcb_ext_p1_001",
"task_type": "stability",
"name": "DynaCertBench System 081 · DCB-EXT-P1-001 · Extend-Table~1 row~1 (NonPoly0), \\cite{abate2021fossil}",
"dynamics": "dx1/dt = -x1+x1*x2, dx2/dt = -x2",
"source": "paper System 81; canonical record DCB-EXT-P1-001",
"assumptions": [
"Deterministically normalized from canonical record DCB-EXT-P1-001.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.",
"The target is global asymptotic stability; uniform Euclidean global exponential stability is not claimed."
],
"equilibrium": [
0,
0
],
"stability_kind": "global_asymptotic",
"neighborhood": {
"type": "unbounded"
},
"analysis_bounds": [
[
-10,
10
],
[
-10,
10
]
],
"formal_domain": "unbounded"
},
"candidate": null,
"paper_system_number": 81,
"paper_system_label": "System 81",
"local_name": "S081__DCB-EXT-P1-001"
},
{
"record_id": "DCB-EXT-P1-002",
"problem": {
"problem_id": "dcb_ext_p1_002",
"task_type": "stability",
"name": "DynaCertBench System 082 · DCB-EXT-P1-002 · Extend-Table~1 row~6 (Poly2), \\cite{abate2021fossil}",
"dynamics": "dx1/dt = -x1**3+x2, dx2/dt = -x1-x2",
"source": "paper System 82; canonical record DCB-EXT-P1-002",
"assumptions": [
"Deterministically normalized from canonical record DCB-EXT-P1-002.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"equilibrium": [
0,
0
],
"stability_kind": "global_exponential",
"neighborhood": {
"type": "unbounded"
},
"analysis_bounds": [
[
-10,
10
],
[
-10,
10
]
],
"formal_domain": "unbounded"
},
"candidate": null,
"paper_system_number": 82,
"paper_system_label": "System 82",
"local_name": "S082__DCB-EXT-P1-002"
},
{
"record_id": "DCB-EXT-P1-003",
"problem": {
"problem_id": "dcb_ext_p1_003",
"task_type": "stability",
"name": "DynaCertBench System 083 · DCB-EXT-P1-003 · Extend-Table~4 \\#2 (SineModel), \\cite{edwards2024fossil}",
"dynamics": "dx1/dt = x2+u1, dx2/dt = u2+((3/20)*(981/100)*(1/2)*sin(x1)-(1/10)*x2)/((3/20)*(1/2)**2)",
"source": "paper System 83; canonical record DCB-EXT-P1-003",
"assumptions": [
"Deterministically normalized from canonical record DCB-EXT-P1-005.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"controller_parameters": "u1,u2",
"equilibrium": [
0,
0
],
"stability_kind": "local_exponential",
"neighborhood": {
"type": "constraints",
"constraints": [
{
"lhs": "x1**2 + x2**2",
"relation": "le",
"rhs": "(1)**2"
}
],
"bounds": [
[
"-1",
"1"
],
[
"-1",
"1"
]
]
},
"analysis_bounds": [
[
"-1",
"1"
],
[
"-1",
"1"
]
],
"control_bounds": [
[
-5,
5
],
[
-5,
5
]
]
},
"candidate": null,
"paper_system_number": 83,
"paper_system_label": "System 83",
"local_name": "S083__DCB-EXT-P1-003"
},
{
"record_id": "DCB-EXT-P2-001",
"problem": {
"problem_id": "dcb_ext_p2_001",
"task_type": "roa",
"name": "DynaCertBench System 084 · DCB-EXT-P2-001 · Extend-Section~5.2 and Appendix~C.3.2, \\cite{zhou2022neural}",
"dynamics": "dx1/dt = sin(x2), dx2/dt = u-cos(x2)/(1-x1)",
"controller_parameters": "u",
"controller_observations": {
"u": [
"x1",
"x2"
]
},
"control_bounds": [
[
-5,
5
]
],
"equilibrium": [
0,
0
],
"stability_kind": "local_asymptotic",
"neighborhood": {
"type": "ball",
"center": [
0,
0
],
"radius": "4/5"
},
"analysis_bounds": [
[
"-4/5",
"4/5"
],
[
"-4/5",
"4/5"
]
],
"source": "paper System 84; canonical record DCB-EXT-P2-001",
"assumptions": [
"The source controller and annular epsilon-ROA result are provenance only; this extension jointly synthesizes a new full-state feedback controller, Lyapunov function, and exact level c.",
"The controller must remain in [-5,5] on the certified sublevel set Omega_c.",
"The closed-loop equilibrium obligation is checked independently; for these dynamics it forces u(0,0)=1, not 0.",
"No separate initial set is supplied; the target is to certify Omega_c itself inside x1^2+x2^2 <= (4/5)^2."
]
},
"candidate": null,
"paper_system_number": 84,
"paper_system_label": "System 84",
"local_name": "S084__DCB-EXT-P2-001"
},
{
"record_id": "DCB-EXT-P3-001",
"problem": {
"problem_id": "dcb_ext_p3_001",
"task_type": "invariant",
"name": "DynaCertBench System 085 · DCB-EXT-P3-001 · Extend--2D quadrotor, \\cite{meng2024physics}",
"dynamics": "dx1/dt = x4, dx2/dt = x5, dx3/dt = x6, dx4/dt = 9.81*sin(x3)+sin(x3)*u1, dx5/dt = -9.81+9.81*cos(x3)+cos(x3)*u1, dx6/dt = u2",
"source": "paper System 85; canonical record DCB-EXT-P3-001",
"assumptions": [
"Deterministically normalized from canonical record DCB-EXT-P3-001.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"controller_parameters": "u1,u2",
"unsafe_set": {
"type": "box",
"bounds": [
[
"-(1/5)",
"1/5"
],
[
"-(1/5)",
"1/5"
],
[
"-(1/5)",
"1/5"
],
[
"-(1/5)",
"1/5"
],
[
"-(1/5)",
"1/5"
],
[
"-(1/5)",
"1/5"
]
],
"complement": true
},
"analysis_bounds": [
[
"-1/5",
"1/5"
],
[
"-1/5",
"1/5"
],
[
"-1/5",
"1/5"
],
[
"-1/5",
"1/5"
],
[
"-1/5",
"1/5"
],
[
"-1/5",
"1/5"
]
],
"safety_interpretation": "outside_X_is_unsafe",
"control_bounds": [
[
-5,
5
],
[
-5,
5
]
]
},
"candidate": null,
"paper_system_number": 85,
"paper_system_label": "System 85",
"local_name": "S085__DCB-EXT-P3-001"
},
{
"record_id": "DCB-EXT-P3-002",
"problem": {
"problem_id": "dcb_ext_p3_002",
"task_type": "invariant",
"name": "DynaCertBench System 086 · DCB-EXT-P3-002 · Extend--3D quadrotor, \\cite{meng2024physics}",
"dynamics": "dx1/dt = x4, dx2/dt = x5, dx3/dt = x6, dx4/dt = -9.81*sin(x8)-sin(x8)*u1, dx5/dt = 9.81*cos(x8)*sin(x7)+cos(x8)*sin(x7)*u1, dx6/dt = 9.81*cos(x8)*cos(x7)-9.81+cos(x8)*cos(x7)*u1, dx7/dt = u2, dx8/dt = u3, dx9/dt = u4",
"source": "paper System 86; canonical record DCB-EXT-P3-002",
"assumptions": [
"Deterministically normalized from canonical record DCB-EXT-P3-002.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"controller_parameters": "u1,u2,u3,u4",
"unsafe_set": {
"type": "box",
"bounds": [
[
"-(1/5)",
"1/5"
],
[
"-(1/5)",
"1/5"
],
[
"-(1/5)",
"1/5"
],
[
"-(1/5)",
"1/5"
],
[
"-(1/5)",
"1/5"
],
[
"-(1/5)",
"1/5"
],
[
"-(1/5)",
"1/5"
],
[
"-(1/5)",
"1/5"
],
[
"-(1/5)",
"1/5"
]
],
"complement": true
},
"analysis_bounds": [
[
"-1/5",
"1/5"
],
[
"-1/5",
"1/5"
],
[
"-1/5",
"1/5"
],
[
"-1/5",
"1/5"
],
[
"-1/5",
"1/5"
],
[
"-1/5",
"1/5"
],
[
"-1/5",
"1/5"
],
[
"-1/5",
"1/5"
],
[
"-1/5",
"1/5"
]
],
"safety_interpretation": "outside_X_is_unsafe",
"control_bounds": [
[
-5,
5
],
[
-5,
5
],
[
-5,
5
],
[
-5,
5
]
]
},
"candidate": null,
"paper_system_number": 86,
"paper_system_label": "System 86",
"local_name": "S086__DCB-EXT-P3-002"
},
{
"record_id": "DCB-EXT-P4-001",
"problem": {
"problem_id": "dcb_ext_p4_001",
"task_type": "safety",
"name": "DynaCertBench System 087 · DCB-EXT-P4-001 · Extend-\\texttt{S002}, \\cite{taheri2025barrierbench}",
"dynamics": "dx1/dt = x2+u0, dx2/dt = -0.95*sin(x1)-0.02*x2+0.01*x3+u1, dx3/dt = -0.1*x3+0.02*x1+u2, dx4/dt = -0.03*x4+0.01*x2+u3",
"source": "paper System 87; canonical record DCB-EXT-P4-001",
"assumptions": [
"Deterministically normalized from canonical record DCB-EXT-P4-001.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"controller_parameters": "u0,u1,u2,u3",
"initial_set": {
"type": "ball",
"center": [
"0",
"0",
"0",
"0"
],
"radius": "0.3"
},
"unsafe_set": {
"type": "ball",
"center": [
"0",
"0",
"0",
"0"
],
"radius": "2.5",
"complement": true
},
"analysis_bounds": [
[
"-7/2",
"7/2"
],
[
"-7/2",
"7/2"
],
[
"-7/2",
"7/2"
],
[
"-7/2",
"7/2"
]
],
"formal_domain": "unbounded",
"control_bounds": [
[
-5,
5
],
[
-5,
5
],
[
-5,
5
],
[
-5,
5
]
]
},
"candidate": null,
"paper_system_number": 87,
"paper_system_label": "System 87",
"local_name": "S087__DCB-EXT-P4-001"
},
{
"record_id": "DCB-EXT-P4-002",
"problem": {
"problem_id": "dcb_ext_p4_002",
"task_type": "safety",
"name": "DynaCertBench System 088 · DCB-EXT-P4-002 · Extend-\\texttt{S011}, \\cite{taheri2025barrierbench}",
"dynamics": "dx1/dt = x2+u0, dx2/dt = -0.3*x1-0.2*(1+cos(x1))*x2-0.1*x1**(3)+u1",
"source": "paper System 88; canonical record DCB-EXT-P4-002",
"assumptions": [
"Deterministically normalized from canonical record DCB-EXT-P4-002.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"controller_parameters": "u0,u1",
"initial_set": {
"type": "ball",
"center": [
"0",
"0"
],
"radius": "0.6"
},
"unsafe_set": {
"type": "ball",
"center": [
"2",
"2"
],
"radius": "1"
},
"analysis_bounds": [
[
"-8/5",
"4"
],
[
"-8/5",
"4"
]
],
"formal_domain": "unbounded",
"control_bounds": [
[
-1,
1
],
[
-1,
1
]
]
},
"candidate": null,
"paper_system_number": 88,
"paper_system_label": "System 88",
"local_name": "S088__DCB-EXT-P4-002"
},
{
"record_id": "DCB-EXT-P4-003",
"problem": {
"problem_id": "dcb_ext_p4_003",
"task_type": "safety",
"name": "DynaCertBench System 089 · DCB-EXT-P4-003 · Extend-\\texttt{S024}, \\cite{taheri2025barrierbench}",
"dynamics": "dx1/dt = x2+u0, dx2/dt = -x1-0.4*x2+0.15*sin(x3)+u1, dx3/dt = x4+u2, dx4/dt = -0.6*x3-0.3*x4+u3",
"source": "paper System 89; canonical record DCB-EXT-P4-003",
"assumptions": [
"Deterministically normalized from canonical record DCB-EXT-P4-003.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"controller_parameters": "u0,u1,u2,u3",
"initial_set": {
"type": "box",
"bounds": [
[
"0.5",
"0.8"
],
[
"0.3",
"0.6"
],
[
"0.5",
"0.8"
],
[
"0.3",
"0.6"
]
]
},
"unsafe_set": {
"type": "union",
"sets": [
{
"type": "box",
"bounds": [
[
"2.5",
"3.5"
],
[
"2",
"3"
],
[
"2.5",
"3.5"
],
[
"2",
"3"
]
]
},
{
"type": "box",
"bounds": [
[
"-3.5",
"-2.5"
],
[
"-3",
"-2"
],
[
"-3.5",
"-2.5"
],
[
"-3",
"-2"
]
]
}
]
},
"analysis_bounds": [
[
"-1/2",
"9/5"
],
[
"-7/10",
"8/5"
],
[
"-1/2",
"9/5"
],
[
"-7/10",
"8/5"
]
],
"formal_domain": "unbounded",
"control_bounds": [
[
-1,
1
],
[
-1,
1
],
[
-1,
1
],
[
-1,
1
]
]
},
"candidate": null,
"paper_system_number": 89,
"paper_system_label": "System 89",
"local_name": "S089__DCB-EXT-P4-003"
},
{
"record_id": "DCB-EXT-P4-004",
"paper_system_number": 90,
"paper_system_label": "System 90",
"local_name": "S090__DCB-EXT-P4-004",
"problem": {
"problem_id": "dcb_ext_p4_004",
"task_type": "safety",
"name": "DynaCertBench System 090 · DCB-EXT-P4-004 · Extend--Hi-Ord8 with bounded sinusoidal disturbance, \\cite{abate2021fossil}",
"dynamics": "dx1/dt = x2, dx2/dt = x3, dx3/dt = x4, dx4/dt = x5, dx5/dt = x6, dx6/dt = x7, dx7/dt = x8, dx8/dt = -20*x8+170*x7+800*x6+2273*x5+3980*x4+4180*x3+2400*x2+576*x1+sin(omega*x1)",
"source": "paper System 90; canonical record DCB-EXT-P4-004",
"assumptions": [
"Transcribed from paper-4-20-20260904 System 93.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.",
"Safety is required for every fixed omega in [1/2,2].",
"The synthesized barrier is one common state-only function B(x), independent of omega.",
"PDF 4-20 removes d(t) and corrects the signs in the last flow component."
],
"analysis_bounds": [
[
"-11/5",
"11/5"
],
[
"-11/5",
"11/5"
],
[
"-11/5",
"11/5"
],
[
"-11/5",
"11/5"
],
[
"-11/5",
"11/5"
],
[
"-11/5",
"11/5"
],
[
"-11/5",
"11/5"
],
[
"-11/5",
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"name": "DynaCertBench System 092 · DCB-OURS-P1-001 · 8D Rational-Cosine Ring",
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"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
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"name": "DynaCertBench System 094 · DCB-OURS-P2-001 · Our cyclic ten-dimensional rational system",
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"Transcribed from paper-4-20-20260904 System 96.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.",
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"name": "DynaCertBench System 095 · DCB-OURS-P2-002 · Seven-dimensional multi-machine swing system",
"dynamics": "dx1/dt = x2, dx2/dt = -(1/2)*x2-(3/2)*sin(x1)-sin(x1-x3)-(3/2)*sin(x1-x7), dx3/dt = x4, dx4/dt = -(1/2)*x4-(1/2)*sin(x3)-sin(x3-x1)-sin(x3-x5), dx5/dt = x6, dx6/dt = -(1/2)*x6-sin(x5)-sin(x5-x3)-(3/2)*sin(x5-x7), dx7/dt = -(1/5)*sin(x7)-(3/2)*sin(x7-x1)-(3/2)*sin(x7-x5)",
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"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
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"name": "DynaCertBench System 096 · DCB-OURS-P2-003 · Seven-dimensional coupled saturating oscillator",
"dynamics": "dx1/dt = x2, dx2/dt = -2*x1/(1+x1**2)+(1/2)*x3-(3/10)*x2+(1/5)*x5, dx3/dt = x4, dx4/dt = (1/2)*x1-(3/2)*x3-sin(x3)-(1/5)*x4+(1/10)*x6, dx5/dt = -(1/2)*x5-(1/5)*x2+(1/10)*x6, dx6/dt = -(2/5)*x6-(1/10)*x5-(1/10)*x4+(1/10)*x7, dx7/dt = -(3/10)*x7-(1/10)*x6",
"source": "paper System 96; canonical record DCB-OURS-P2-003",
"assumptions": [
"Transcribed from active LaTeX System 96.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
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"problem_id": "dcb_ours_p3_001",
"task_type": "invariant",
"name": "DynaCertBench System 097 · DCB-OURS-P3-001 · New-6D Sin Cascade Invariance",
"dynamics": "dx1/dt = x2+u0, dx2/dt = -x1-1/10*sin(x1)-x2+x3, dx3/dt = -2*x3+x4, dx4/dt = -3/2*x4+x5, dx5/dt = -2*x5+x6, dx6/dt = -3/2*x6+u1",
"source": "paper System 97; canonical record DCB-OURS-P3-001",
"assumptions": [
"Transcribed from paper-4-20-20260904 System 97.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.",
"X_seed is the paper's required nontrivial seed set."
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"problem": {
"problem_id": "dcb_ours_p3_002",
"task_type": "invariant",
"name": "DynaCertBench System 098 · DCB-OURS-P3-002 · New-6D Exponential Cascade Invariance",
"dynamics": "dx1/dt = x2+u, dx2/dt = -x1-1/20*(exp(x1)-exp(-x1))-x2+x3, dx3/dt = -2*x3+x4, dx4/dt = -3/2*x4+x5, dx5/dt = -2*x5+x6, dx6/dt = -3/2*x6",
"source": "paper System 98; canonical record DCB-OURS-P3-002",
"assumptions": [
"Transcribed from paper-4-20-20260904 System 98.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.",
"The source X_I ball has radius 1/8."
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"paper_system_label": "System 99",
"local_name": "S099__DCB-OURS-P3-003",
"problem": {
"problem_id": "dcb_ours_p3_003",
"task_type": "invariant",
"name": "DynaCertBench System 099 · DCB-OURS-P3-003 · New-5D Mixed Nonlinear Cascade Invariance",
"dynamics": "dx1/dt = x2-1/2*x1, dx2/dt = -x1-x1/(5*(1+x1**2))-2*x2+x3/(10*(1+x3**2)), dx3/dt = x4-1/2*x3, dx4/dt = -x3-1/40*(exp(x3)-exp(-x3))-2*x4+1/5*x5, dx5/dt = -3/2*x5-1/10*x5**3+u",
"source": "paper System 99; canonical record DCB-OURS-P3-003",
"assumptions": [
"Transcribed from paper-4-20-20260904 System 99.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.",
"The paper supplies no X_I/X_seed; no seed set is invented locally."
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"paper_system_label": "System 100",
"local_name": "S100__DCB-OURS-P3-004",
"problem": {
"problem_id": "dcb_ours_p3_004",
"task_type": "invariant",
"name": "DynaCertBench System 100 · DCB-OURS-P3-004 · New-6D Opinion Dynamics",
"dynamics": "x1[k+1] = x1+1/4*(tanh(x2-x1) + tanh(x3-x1)), x2[k+1] = x2+1/4*(tanh(x1-x2) + tanh(x3-x2) + tanh(x4-x2)), x3[k+1] = x3+1/4*(tanh(x1-x3) + tanh(x2-x3) + tanh(x5-x3)), x4[k+1] = x4+1/4*(tanh(x2-x4) + tanh(x5-x4)), x5[k+1] = x5+1/4*(tanh(x3-x5) + tanh(x4-x5) + tanh(x6-x5)), x6[k+1] = x6+1/4*(tanh(x5-x6))",
"source": "paper System 100; canonical record DCB-OURS-P3-004",
"assumptions": [
"Transcribed from paper-4-20-20260904 System 100.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.",
"PDF 4-20 fixes N=6, alpha=1/4, and the displayed adjacency matrix.",
"The paper supplies no X_I/X_seed; no seed set is invented locally."
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[
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{
"type": "constraint",
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{
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"type": "constraint",
"lhs": "x6",
"relation": "le",
"rhs": "-4/5"
}
]
}
},
"candidate": null
},
{
"record_id": "DCB-OURS-P3-005",
"paper_system_number": 101,
"paper_system_label": "System 101",
"local_name": "S101__DCB-OURS-P3-005",
"problem": {
"problem_id": "dcb_ours_p3_005",
"task_type": "invariant",
"name": "DynaCertBench System 101 · DCB-OURS-P3-005 · Invariance Verification of a Discrete-Time Controlled Hopfield Network",
"dynamics": "x1[k+1] = x1+1/20*(1-x1**2)*(-1/2*log(1+x1)+1/2*log(1-x1)+-3/2*x1 + 1/4*x2+u1), x2[k+1] = x2+1/20*(1-x2**2)*(-1/2*log(1+x2)+1/2*log(1-x2)+-3/2*x2 + 1/4*x1 + 1/4*x3+u2), x3[k+1] = x3+1/20*(1-x3**2)*(-1/2*log(1+x3)+1/2*log(1-x3)+-3/2*x3 + 1/4*x2 + 1/4*x4+u3), x4[k+1] = x4+1/20*(1-x4**2)*(-1/2*log(1+x4)+1/2*log(1-x4)+-3/2*x4 + 1/4*x3 + 1/4*x5+u4), x5[k+1] = x5+1/20*(1-x5**2)*(-1/2*log(1+x5)+1/2*log(1-x5)+-3/2*x5 + 1/4*x4 + 1/4*x6+u5), x6[k+1] = x6+1/20*(1-x6**2)*(-1/2*log(1+x6)+1/2*log(1-x6)+-3/2*x6 + 1/4*x5 + 1/4*x7+u6), x7[k+1] = x7+1/20*(1-x7**2)*(-1/2*log(1+x7)+1/2*log(1-x7)+-3/2*x7 + 1/4*x6 + 1/4*x8+u7), x8[k+1] = x8+1/20*(1-x8**2)*(-1/2*log(1+x8)+1/2*log(1-x8)+-3/2*x8 + 1/4*x7 + 1/4*x9+u8), x9[k+1] = x9+1/20*(1-x9**2)*(-1/2*log(1+x9)+1/2*log(1-x9)+-3/2*x9 + 1/4*x8 + 1/4*x10+u9), x10[k+1] = x10+1/20*(1-x10**2)*(-1/2*log(1+x10)+1/2*log(1-x10)+-3/2*x10 + 1/4*x9+u10)",
"source": "paper System 101; canonical record DCB-OURS-P3-005",
"assumptions": [
"Transcribed from paper-4-20-20260904 System 101.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.",
"atanh(x) is represented exactly by (log(1+x)-log(1-x))/2 on X.",
"W is the tridiagonal matrix displayed in PDF 4-20: diagonal -3/2 and adjacent entries 1/4.",
"The paper supplies no X_I/X_seed; no seed set is invented locally."
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"analysis_bounds": [
[
"-9/10",
"9/10"
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[
"-9/10",
"9/10"
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[
"-9/10",
"9/10"
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[
"-9/10",
"9/10"
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[
"-9/10",
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[
"-9/10",
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[
"-9/10",
"9/10"
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[
"-9/10",
"9/10"
],
[
"-9/10",
"9/10"
],
[
"-9/10",
"9/10"
]
],
"controller_parameters": "u1,u2,u3,u4,u5,u6,u7,u8,u9,u10",
"control_bounds": [
[
-2,
2
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[
-2,
2
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[
-2,
2
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[
-2,
2
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[
-2,
2
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[
-2,
2
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[
-2,
2
],
[
-2,
2
],
[
-2,
2
],
[
-2,
2
]
],
"safety_interpretation": "outside_X_is_unsafe"
},
"candidate": null
},
{
"record_id": "DCB-OURS-P4-001",
"paper_system_number": 102,
"paper_system_label": "System 102",
"local_name": "S102__DCB-OURS-P4-001",
"problem": {
"problem_id": "dcb_ours_p4_001",
"task_type": "safety",
"name": "DynaCertBench System 102 · DCB-OURS-P4-001 · New-8D Chain Safety",
"dynamics": "x1[k+1] = x1+1/200*(-x1/(1+x1**2)-4*x2), x2[k+1] = x2+1/200*(4*x1/(1+x1**2)-x2-3*x3/(1+x3**2)), x3[k+1] = x3+1/200*(3*x2-x3/(1+x3**2)-5*x4), x4[k+1] = x4+1/200*(5*x3/(1+x3**2)-x4-2*x5/(1+x5**2)), x5[k+1] = x5+1/200*(2*x4-x5/(1+x5**2)-4*x6), x6[k+1] = x6+1/200*(4*x5/(1+x5**2)-x6-3*x7/(1+x7**2)), x7[k+1] = x7+1/200*(3*x6-x7/(1+x7**2)-6*x8), x8[k+1] = x8+1/200*(6*x7/(1+x7**2)-x8)",
"source": "paper System 102; canonical record DCB-OURS-P4-001",
"assumptions": [
"Transcribed from active LaTeX System 102.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"analysis_bounds": [
[
-1,
1
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[
-1,
1
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[
-1,
1
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[
-1,
1
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[
-1,
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[
-1,
1
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[
-1,
1
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[
-1,
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"initial_set": {
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"bounds": [
[
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[
"-3/20",
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[
"-3/20",
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[
"-3/20",
"3/20"
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[
"-3/20",
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[
"-3/20",
"3/20"
],
[
"-3/20",
"3/20"
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[
"-3/20",
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},
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"bounds": [
[
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[
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[
"-1/10",
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[
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[
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[
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[
"-1/10",
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[
"-1/10",
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]
}
},
"candidate": null
},
{
"record_id": "DCB-OURS-P4-002",
"paper_system_number": 103,
"paper_system_label": "System 103",
"local_name": "S103__DCB-OURS-P4-002",
"problem": {
"problem_id": "dcb_ours_p4_002",
"task_type": "safety",
"name": "DynaCertBench System 103 · DCB-OURS-P4-002 · New-6D Arctan Chain Safety",
"dynamics": "x1[k+1] = x1+1/200*(-atan(x1)-8*x2), x2[k+1] = x2+1/200*(8*atan(x1)-x2-6*atan(x3)), x3[k+1] = x3+1/200*(6*x2-atan(x3)-10*x4), x4[k+1] = x4+1/200*(10*atan(x3)-x4-4*atan(x5)), x5[k+1] = x5+1/200*(4*x4-atan(x5)-8*x6), x6[k+1] = x6+1/200*(8*atan(x5)-x6)",
"source": "paper System 103; canonical record DCB-OURS-P4-002",
"assumptions": [
"Transcribed from paper-4-20-20260904 System 103.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"analysis_bounds": [
[
-1,
1
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[
-1,
1
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[
-1,
1
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[
-1,
1
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[
-1,
1
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[
-1,
1
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],
"initial_set": {
"type": "box",
"bounds": [
[
"-3/20",
"3/20"
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[
"-3/20",
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[
"-3/20",
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[
"-3/20",
"3/20"
],
[
"-3/20",
"3/20"
],
[
"-3/20",
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},
"unsafe_set": {
"type": "box",
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[
"-1/10",
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[
"-1/10",
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[
"-1/10",
"1/10"
],
[
"-1/10",
"1/10"
],
[
"-1/10",
"1/10"
]
]
}
},
"candidate": null
},
{
"record_id": "DCB-OURS-P4-003",
"paper_system_number": 104,
"paper_system_label": "System 104",
"local_name": "S104__DCB-OURS-P4-003",
"problem": {
"problem_id": "dcb_ours_p4_003",
"task_type": "safety",
"name": "DynaCertBench System 104 · DCB-OURS-P4-003 · New-8D Parametric Nonlinear Safety",
"dynamics": "dx1/dt = x2, dx2/dt = -x1-(1/10)*sin(omega*x1)-x2+x3, dx3/dt = -2*x3+x4, dx4/dt = -(3/2)*x4+x5, dx5/dt = -2*x5+x6, dx6/dt = -(3/2)*x6+(1/5)*x7, dx7/dt = -2*x7+(1/5)*x6+(1/5)*x8-(1/10)*x7*sin(x8), dx8/dt = -(3/2)*x8+(1/5)*x7-(1/10)*x8*cos(x7)",
"source": "paper System 104; canonical record DCB-OURS-P4-003",
"assumptions": [
"Transcribed from active LaTeX System 104.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"analysis_bounds": [
[
-1,
1
],
[
-1,
1
],
[
-1,
1
],
[
-1,
1
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[
-1,
1
],
[
-1,
1
],
[
-1,
1
],
[
-1,
1
]
],
"fixed_parameters": [
{
"name": "omega",
"bounds": [
"1/2",
2
]
}
],
"initial_set": {
"type": "ball",
"center": [
0,
0,
0,
0,
0,
0,
0,
0
],
"radius": "1/2"
},
"unsafe_set": {
"type": "ball",
"center": [
0,
"4/5",
0,
0,
0,
0,
0,
0
],
"radius": "1/10"
}
},
"candidate": null
},
{
"record_id": "DCB-OURS-P4-004",
"paper_system_number": 105,
"paper_system_label": "System 105",
"local_name": "S105__DCB-OURS-P4-004",
"problem": {
"problem_id": "dcb_ours_p4_004",
"task_type": "safety",
"name": "DynaCertBench System 105 · DCB-OURS-P4-004 · New-5D Arctangent Perturbation Safety",
"dynamics": "dx1/dt = x2+(1/2)*x5, dx2/dt = -x1-omega*x2+2*x4, dx3/dt = x4, dx4/dt = -2*x3-(1/8)*atan(x3)-(3/2)*x4-2*x2+x5, dx5/dt = -(1/2)*x1-x4-x5-(1/8)*atan(x5)",
"source": "paper System 105; canonical record DCB-OURS-P4-004",
"assumptions": [
"Transcribed from active LaTeX System 105.",
"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean."
],
"analysis_bounds": [
[
-1,
1
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[
-1,
1
],
[
-1,
1
],
[
-1,
1
],
[
-1,
1
]
],
"fixed_parameters": [
{
"name": "omega",
"bounds": [
1,
2
]
}
],
"initial_set": {
"type": "ball",
"center": [
0,
0,
0,
0,
0
],
"radius": "1/2"
},
"unsafe_set": {
"type": "ball",
"center": [
0,
0,
"4/5",
0,
0
],
"radius": "1/10"
}
},
"candidate": null
}
],
"revision_notes": [
"Generated from active LaTeX systeminstance order.",
"All candidates are null; proof answers are not part of the dataset."
]
}