{"record_id":"DCB-LIT-P1-001","paper_system_number":1,"task_type":"stability","problem_name":"Stability certification","tier":"literature","dimension":2,"time_domain":"continuous","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))","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2 \\end{pmatrix} = \\begin{pmatrix} -\\dfrac{1}{2}\\tan\\left(\\dfrac{\\pi x_1}{2} + \\dfrac{\\pi}{4}\\right) + x_2 + \\dfrac{1}{2}\\\\[1.5ex] x_1 - \\dfrac{1}{2}\\tan\\left(\\dfrac{\\pi x_2}{2} + \\dfrac{\\pi}{4}\\right) + \\dfrac{1}{2} \\end{pmatrix}. \\]","citations":["khalil2002nonlinear"],"problem_json":"{\"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\"]]},\"analysis_bounds\":[[\"-0.25\",\"0.25\"],[\"-0.25\",\"0.25\"]]}","executable_record_json":"{\"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\"]]},\"analysis_bounds\":[[\"-0.25\",\"0.25\"],[\"-0.25\",\"0.25\"]]},\"candidate\":null,\"paper_system_number\":1,\"paper_system_label\":\"System 1\",\"local_name\":\"S001__DCB-LIT-P1-001\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":1,\"problem_name\":\"Stability certification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0)$\",\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid0.01^2\\\\leq x_1^2+x_2^2\\\\leq0.25^2\\\\}$\",\"Original verified type: $\\\\epsilon$-asymptotic stability\",\"Verified method in the original paper: analytical Lyapunov analysis\",\"Our $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid x_1^2+x_2^2\\\\leq0.25^2\\\\}$\",\"Our verified type: local asymptotic stability\"],\"record_id\":\"DCB-LIT-P1-001\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=(0,0)$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid0.01^2\\\\leq x_1^2+x_2^2\\\\leq0.25^2\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"x^\\\\star=(0,0);\\\\quad \\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid0.01^2\\\\leq x_1^2+x_2^2\\\\leq0.25^2\\\\}\",\"note\":\"Audited catalogue wording; not silently treated as an Overleaf statement.\",\"source\":\"catalogue:OVL-EXT-001\",\"status\":\"documented\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":\"OVL-EXT-001\",\"citations\":[\"khalil2002nonlinear\"],\"label\":\"sys:khalil-4-30-shifted\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":285,\"start\":258},\"title_latex\":\"Exercise~4.30, \\\\cite{khalil2002nonlinear}\"},\"source_order\":1,\"system\":{\"dimension\":{\"source\":\"catalogue:OVL-EXT-001\",\"status\":\"documented\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} -\\\\dfrac{1}{2}\\\\tan\\\\left(\\\\dfrac{\\\\pi x_1}{2} + \\\\dfrac{\\\\pi}{4}\\\\right) + x_2 + \\\\dfrac{1}{2}\\\\\\\\[1.5ex] x_1 - \\\\dfrac{1}{2}\\\\tan\\\\left(\\\\dfrac{\\\\pi x_2}{2} + \\\\dfrac{\\\\pi}{4}\\\\right) + \\\\dfrac{1}{2} \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} -\\\\dfrac{1}{2}\\\\tan\\\\left(\\\\dfrac{\\\\pi x_1}{2} + \\\\dfrac{\\\\pi}{4}\\\\right) + x_2 + \\\\dfrac{1}{2}\\\\\\\\[1.5ex] x_1 - \\\\dfrac{1}{2}\\\\tan\\\\left(\\\\dfrac{\\\\pi x_2}{2} + \\\\dfrac{\\\\pi}{4}\\\\right) + \\\\dfrac{1}{2} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"Stability certification\"},\"reported_method_latex\":\"Verified method in the original paper: analytical Lyapunov analysis\",\"reported_type_latex\":\"Original verified type: $\\\\epsilon$-asymptotic stability\"}}"} {"record_id":"DCB-LIT-P1-002","paper_system_number":2,"task_type":"stability","problem_name":"Stability certification","tier":"literature","dimension":2,"time_domain":"continuous","dynamics":"dx1/dt = -x1+x1*x2, dx2/dt = -x2","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2 \\end{pmatrix} = \\begin{pmatrix} -x_1+x_1x_2\\\\ -x_2 \\end{pmatrix}. \\]","citations":["abate2021fossil"],"problem_json":"{\"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\"]]},\"analysis_bounds\":[[\"-1\",\"1\"],[\"-1\",\"1\"]]}","executable_record_json":"{\"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\"]]},\"analysis_bounds\":[[\"-1\",\"1\"],[\"-1\",\"1\"]]},\"candidate\":null,\"paper_system_number\":2,\"paper_system_label\":\"System 2\",\"local_name\":\"S002__DCB-LIT-P1-002\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":1,\"problem_name\":\"Stability certification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0)$\",\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid0.01^2\\\\leq x_1^2+x_2^2\\\\leq1^2\\\\}$\",\"Original verified type: local $\\\\epsilon$-asymptotic stability\",\"Verified method in the original paper: dReal\",\"Our $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid x_1^2+x_2^2\\\\leq1^2\\\\}$\",\"Our verified type: local asymptotic stability\"],\"record_id\":\"DCB-LIT-P1-002\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=(0,0)$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid0.01^2\\\\leq x_1^2+x_2^2\\\\leq1^2\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"X_D=\\\\{x:\\\\ 0<\\\\|x\\\\|_2\\\\le10\\\\}\\\\cap\\\\mathbb R_{\\\\ge0}^{2}\",\"note\":\"Audited catalogue wording; not silently treated as an Overleaf statement.\",\"source\":\"catalogue:SYS0016\",\"status\":\"documented\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":\"SYS0016\",\"citations\":[\"abate2021fossil\"],\"label\":\"sys:catalogue-sys0016\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":336,\"start\":310},\"title_latex\":\"Table~1 row~1 (NonPoly0), \\\\cite{abate2021fossil}\"},\"source_order\":2,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0016\",\"status\":\"documented\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} -x_1+x_1x_2\\\\\\\\ -x_2 \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} -x_1+x_1x_2\\\\\\\\ -x_2 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"Stability certification\"},\"reported_method_latex\":\"Verified method in the original paper: dReal\",\"reported_type_latex\":\"Original verified type: local $\\\\epsilon$-asymptotic stability\"}}"} {"record_id":"DCB-LIT-P1-003","paper_system_number":3,"task_type":"stability","problem_name":"Stability certification","tier":"literature","dimension":2,"time_domain":"continuous","dynamics":"dx1/dt = -x1+2*x1**2*x2, dx2/dt = -x2","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2 \\end{pmatrix} = \\begin{pmatrix} -x_1+2x_1^2x_2\\\\ -x_2 \\end{pmatrix}. \\]","citations":["abate2021fossil"],"problem_json":"{\"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\"},{\"lhs\":\"x1\",\"relation\":\"ge\",\"rhs\":\"0\"},{\"lhs\":\"x2\",\"relation\":\"ge\",\"rhs\":\"0\"}],\"bounds\":[[\"0\",\"10\"],[\"0\",\"10\"]]},\"analysis_bounds\":[[\"0\",\"10\"],[\"0\",\"10\"]]}","executable_record_json":"{\"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\"},{\"lhs\":\"x1\",\"relation\":\"ge\",\"rhs\":\"0\"},{\"lhs\":\"x2\",\"relation\":\"ge\",\"rhs\":\"0\"}],\"bounds\":[[\"0\",\"10\"],[\"0\",\"10\"]]},\"analysis_bounds\":[[\"0\",\"10\"],[\"0\",\"10\"]]},\"candidate\":null,\"paper_system_number\":3,\"paper_system_label\":\"System 3\",\"local_name\":\"S003__DCB-LIT-P1-003\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":1,\"problem_name\":\"Stability certification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0)$\",\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid x_1,x_2\\\\geq0,\\\\;0\",\"source_lines\":{\"end\":364,\"start\":338},\"title_latex\":\"Table~1 row~2 (NonPoly1), \\\\cite{abate2021fossil}\"},\"source_order\":3,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0017\",\"status\":\"documented\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} -x_1+2x_1^2x_2\\\\\\\\ -x_2 \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} -x_1+2x_1^2x_2\\\\\\\\ -x_2 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"Stability certification\"},\"reported_method_latex\":\"Verified method in the original paper: Z3\",\"reported_type_latex\":\"Original verified type: local asymptotic stability\"}}"} {"record_id":"DCB-LIT-P1-004","paper_system_number":4,"task_type":"stability","problem_name":"Stability certification","tier":"literature","dimension":3,"time_domain":"continuous","dynamics":"dx1/dt = -x1, dx2/dt = -2*x2+0.1*x1*x2**2+x3, dx3/dt = -x3-1.5*x2","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2\\\\ \\dot{x}_3 \\end{pmatrix} = \\begin{pmatrix} -x_1\\\\ -2x_2+0.1x_1x_2^2+x_3\\\\ -x_3-1.5x_2 \\end{pmatrix}. \\]","citations":["abate2021fossil"],"problem_json":"{\"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\":[[\"0\",\"10\"],[\"0\",\"10\"],[\"0\",\"10\"]]},\"analysis_bounds\":[[\"0\",\"10\"],[\"0\",\"10\"],[\"0\",\"10\"]]}","executable_record_json":"{\"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\":[[\"0\",\"10\"],[\"0\",\"10\"],[\"0\",\"10\"]]},\"analysis_bounds\":[[\"0\",\"10\"],[\"0\",\"10\"],[\"0\",\"10\"]]},\"candidate\":null,\"paper_system_number\":4,\"paper_system_label\":\"System 4\",\"local_name\":\"S004__DCB-LIT-P1-004\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":1,\"problem_name\":\"Stability certification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0,0)$\",\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{3}\\\\mid x_1,x_2,x_3\\\\geq0,\\\\;0\",\"source_lines\":{\"end\":394,\"start\":366},\"title_latex\":\"Table~1 row~3 (NonPoly2), \\\\cite{abate2021fossil}\"},\"source_order\":4,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0018\",\"status\":\"documented\",\"symbolic\":null,\"value\":3},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3 \\\\end{pmatrix} = \\\\begin{pmatrix} -x_1\\\\\\\\ -2x_2+0.1x_1x_2^2+x_3\\\\\\\\ -x_3-1.5x_2 \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3 \\\\end{pmatrix} = \\\\begin{pmatrix} -x_1\\\\\\\\ -2x_2+0.1x_1x_2^2+x_3\\\\\\\\ -x_3-1.5x_2 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"Stability certification\"},\"reported_method_latex\":\"Verified method in the original paper: Z3\",\"reported_type_latex\":\"Original verified type: local asymptotic stability\"}}"} {"record_id":"DCB-LIT-P1-005","paper_system_number":5,"task_type":"stability","problem_name":"Stability certification","tier":"literature","dimension":3,"time_domain":"continuous","dynamics":"dx1/dt = -3*x1-0.1*x1*x2**3, dx2/dt = -x2+x3, dx3/dt = -x3","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2\\\\ \\dot{x}_3 \\end{pmatrix} = \\begin{pmatrix} -3x_1-0.1x_1x_2^3\\\\ -x_2+x_3\\\\ -x_3 \\end{pmatrix}. \\]","citations":["abate2021fossil"],"problem_json":"{\"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\",\"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\":[[\"0\",\"10\"],[\"0\",\"10\"],[\"0\",\"10\"]]},\"analysis_bounds\":[[\"0\",\"10\"],[\"0\",\"10\"],[\"0\",\"10\"]]}","executable_record_json":"{\"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\",\"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\":[[\"0\",\"10\"],[\"0\",\"10\"],[\"0\",\"10\"]]},\"analysis_bounds\":[[\"0\",\"10\"],[\"0\",\"10\"],[\"0\",\"10\"]]},\"candidate\":null,\"paper_system_number\":5,\"paper_system_label\":\"System 5\",\"local_name\":\"S005__DCB-LIT-P1-005\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":1,\"problem_name\":\"Stability certification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0,0)$\",\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{3}\\\\mid x_1,x_2,x_3\\\\geq0,\\\\;0\",\"source_lines\":{\"end\":424,\"start\":396},\"title_latex\":\"Table~1 row~4 (NonPoly3), \\\\cite{abate2021fossil}\"},\"source_order\":5,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0019\",\"status\":\"documented\",\"symbolic\":null,\"value\":3},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3 \\\\end{pmatrix} = \\\\begin{pmatrix} -3x_1-0.1x_1x_2^3\\\\\\\\ -x_2+x_3\\\\\\\\ -x_3 \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3 \\\\end{pmatrix} = \\\\begin{pmatrix} -3x_1-0.1x_1x_2^3\\\\\\\\ -x_2+x_3\\\\\\\\ -x_3 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"Stability certification\"},\"reported_method_latex\":\"Verified method in the original paper: Z3\",\"reported_type_latex\":\"Original verified type: local asymptotic stability\"}}"} {"record_id":"DCB-LIT-P1-006","paper_system_number":6,"task_type":"stability","problem_name":"Stability certification","tier":"literature","dimension":3,"time_domain":"continuous","dynamics":"dx1/dt = -x1**3-x1*x3**2, dx2/dt = -x2-x1**2*x2, dx3/dt = -x3+3*x1**2*x3-3*x3","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2\\\\ \\dot{x}_3 \\end{pmatrix} = \\begin{pmatrix} -x_1^3-x_1x_3^2\\\\ -x_2-x_1^2x_2\\\\ -x_3+3x_1^2x_3-3x_3 \\end{pmatrix}. \\]","citations":["abate2021fossil"],"problem_json":"{\"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\"]]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":1,\"problem_name\":\"Stability certification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0,0)$\",\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{3}\\\\mid0.1^2\\\\leq x_1^2+x_2^2+x_3^2\\\\leq10^2\\\\}$\",\"Original verified type: local $\\\\epsilon$-asymptotic stability\",\"Verified method in the original paper: dReal\",\"Our $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{3}\\\\mid x_1^2+x_2^2+x_3^2\\\\leq10^2\\\\}$\",\"Our verified type: local asymptotic stability\"],\"record_id\":\"DCB-LIT-P1-006\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=(0,0,0)$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{3}\\\\mid0.1^2\\\\leq x_1^2+x_2^2+x_3^2\\\\leq10^2\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"X_D=\\\\{x:\\\\ 0<\\\\|x\\\\|_2\\\\le10\\\\}\",\"note\":\"Audited catalogue wording; not silently treated as an Overleaf statement.\",\"source\":\"catalogue:SYS0020\",\"status\":\"documented\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":\"SYS0020\",\"citations\":[\"abate2021fossil\"],\"label\":\"sys:catalogue-sys0020\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":454,\"start\":426},\"title_latex\":\"Table~1 row~5 (Poly1), \\\\cite{abate2021fossil}\"},\"source_order\":6,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0020\",\"status\":\"documented\",\"symbolic\":null,\"value\":3},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3 \\\\end{pmatrix} = \\\\begin{pmatrix} -x_1^3-x_1x_3^2\\\\\\\\ -x_2-x_1^2x_2\\\\\\\\ -x_3+3x_1^2x_3-3x_3 \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3 \\\\end{pmatrix} = \\\\begin{pmatrix} -x_1^3-x_1x_3^2\\\\\\\\ -x_2-x_1^2x_2\\\\\\\\ -x_3+3x_1^2x_3-3x_3 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"Stability certification\"},\"reported_method_latex\":\"Verified method in the original paper: dReal\",\"reported_type_latex\":\"Original verified type: local $\\\\epsilon$-asymptotic stability\"}}"} {"record_id":"DCB-LIT-P1-007","paper_system_number":7,"task_type":"stability","problem_name":"Stability certification","tier":"literature","dimension":2,"time_domain":"continuous","dynamics":"dx1/dt = -x1**3+x2, dx2/dt = -x1-x2","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2 \\end{pmatrix} = \\begin{pmatrix} -x_1^3+x_2\\\\ -x_1-x_2 \\end{pmatrix}. \\]","citations":["abate2021fossil"],"problem_json":"{\"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\"]]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":1,\"problem_name\":\"Stability certification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0)$\",\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid0.01^2\\\\leq x_1^2+x_2^2\\\\leq10^2\\\\}$\",\"Original verified type: local $\\\\epsilon$-asymptotic stability\",\"Verified method in the original paper: Z3\",\"Our $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid x_1^2+x_2^2\\\\leq10^2\\\\}$\",\"Our verified type: local asymptotic stability\"],\"record_id\":\"DCB-LIT-P1-007\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=(0,0)$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid0.01^2\\\\leq x_1^2+x_2^2\\\\leq10^2\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"X_D=\\\\{x:\\\\ 0<\\\\|x\\\\|_2\\\\le10\\\\}\",\"note\":\"Audited catalogue wording; not silently treated as an Overleaf statement.\",\"source\":\"catalogue:SYS0021\",\"status\":\"documented\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":\"SYS0021\",\"citations\":[\"abate2021fossil\"],\"label\":\"sys:catalogue-sys0021\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":482,\"start\":456},\"title_latex\":\"Table~1 row~6 (Poly2), \\\\cite{abate2021fossil}\"},\"source_order\":7,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0021\",\"status\":\"documented\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} -x_1^3+x_2\\\\\\\\ -x_1-x_2 \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} -x_1^3+x_2\\\\\\\\ -x_1-x_2 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"Stability certification\"},\"reported_method_latex\":\"Verified method in the original paper: Z3\",\"reported_type_latex\":\"Original verified type: local $\\\\epsilon$-asymptotic stability\"}}"} {"record_id":"DCB-LIT-P1-008","paper_system_number":8,"task_type":"stability","problem_name":"Stability certification","tier":"literature","dimension":2,"time_domain":"continuous","dynamics":"dx1/dt = -x1**3-x2**2, dx2/dt = x1*x2-x2**3","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2 \\end{pmatrix} = \\begin{pmatrix} -x_1^3-x_2^2\\\\ x_1x_2-x_2^3 \\end{pmatrix}. \\]","citations":["abate2021fossil"],"problem_json":"{\"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\"]]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":1,\"problem_name\":\"Stability certification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0)$\",\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid0.1^2\\\\leq x_1^2+x_2^2\\\\leq10^2\\\\}$\",\"Original verified type: local $\\\\epsilon$-asymptotic stability\",\"Verified method in the original paper: dReal\",\"Our $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid x_1^2+x_2^2\\\\leq10^2\\\\}$\",\"Our verified type: local asymptotic stability\"],\"record_id\":\"DCB-LIT-P1-008\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=(0,0)$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid0.1^2\\\\leq x_1^2+x_2^2\\\\leq10^2\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"X_D=\\\\{x:\\\\ 0<\\\\|x\\\\|_2\\\\le10\\\\}\",\"note\":\"Audited catalogue wording; not silently treated as an Overleaf statement.\",\"source\":\"catalogue:SYS0022\",\"status\":\"documented\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":\"SYS0022\",\"citations\":[\"abate2021fossil\"],\"label\":\"sys:catalogue-sys0022\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":510,\"start\":484},\"title_latex\":\"Table~1 row~7 (Poly3), \\\\cite{abate2021fossil}\"},\"source_order\":8,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0022\",\"status\":\"documented\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} -x_1^3-x_2^2\\\\\\\\ x_1x_2-x_2^3 \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} -x_1^3-x_2^2\\\\\\\\ x_1x_2-x_2^3 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"Stability certification\"},\"reported_method_latex\":\"Verified method in the original paper: dReal\",\"reported_type_latex\":\"Original verified type: local $\\\\epsilon$-asymptotic stability\"}}"} {"record_id":"DCB-LIT-P1-009","paper_system_number":9,"task_type":"stability","problem_name":"Stability certification","tier":"literature","dimension":2,"time_domain":"continuous","dynamics":"dx1/dt = -x1-1.5*x1**2*x2**3, dx2/dt = -x2**3+0.5*x1**3*x2**2","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2 \\end{pmatrix} = \\begin{pmatrix} -x_1-1.5x_1^2x_2^3\\\\ -x_2^3+0.5x_1^3x_2^2 \\end{pmatrix}. \\]","citations":["abate2021fossil"],"problem_json":"{\"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\"]]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":1,\"problem_name\":\"Stability certification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0)$\",\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid0.1^2\\\\leq x_1^2+x_2^2\\\\leq10^2\\\\}$\",\"Original verified type: local $\\\\epsilon$-asymptotic stability\",\"Verified method in the original paper: dReal\",\"Our $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid x_1^2+x_2^2\\\\leq10^2\\\\}$\",\"Our verified type: local asymptotic stability\"],\"record_id\":\"DCB-LIT-P1-009\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=(0,0)$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid0.1^2\\\\leq x_1^2+x_2^2\\\\leq10^2\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"X_D=\\\\{x:\\\\ 0<\\\\|x\\\\|_2\\\\le10\\\\}\",\"note\":\"Audited catalogue wording; not silently treated as an Overleaf statement.\",\"source\":\"catalogue:SYS0023\",\"status\":\"documented\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":\"SYS0023\",\"citations\":[\"abate2021fossil\"],\"label\":\"sys:catalogue-sys0023\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":538,\"start\":512},\"title_latex\":\"Table~1 row~8 (Poly4), \\\\cite{abate2021fossil}\"},\"source_order\":9,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0023\",\"status\":\"documented\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} -x_1-1.5x_1^2x_2^3\\\\\\\\ -x_2^3+0.5x_1^3x_2^2 \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} -x_1-1.5x_1^2x_2^3\\\\\\\\ -x_2^3+0.5x_1^3x_2^2 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"Stability certification\"},\"reported_method_latex\":\"Verified method in the original paper: dReal\",\"reported_type_latex\":\"Original verified type: local $\\\\epsilon$-asymptotic stability\"}}"} {"record_id":"DCB-LIT-P1-010","paper_system_number":10,"task_type":"stability","problem_name":"Stability certification","tier":"literature","dimension":2,"time_domain":"continuous","dynamics":"dx1/dt = x2+u1, dx2/dt = u2+((3/20)*(981/100)*(1/2)*sin(x1)-(1/10)*x2)/((3/20)*(1/2)**2)","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2 \\end{pmatrix} = \\begin{pmatrix} x_2+u_1\\\\ u_2+\\dfrac{mgL\\sin(x_1)-bx_2}{mL^2} \\end{pmatrix}, \\] where $m=0.15$, $g=9.81$, $L=0.5$, and $b=0.1$.","citations":["edwards2024fossil"],"problem_json":"{\"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\"]]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":1,\"problem_name\":\"Stability certification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0)$\",\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid0.1^2\\\\leq x_1^2+x_2^2\\\\leq1^2\\\\}$\",\"Original verified type: local $\\\\epsilon$-asymptotic stability\",\"Verified method in the original paper: dReal\",\"Our $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid x_1^2+x_2^2\\\\leq1^2\\\\}$\",\"Our verified type: local asymptotic stability\"],\"record_id\":\"DCB-LIT-P1-010\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=(0,0)$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid0.1^2\\\\leq x_1^2+x_2^2\\\\leq1^2\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"X_D=\\\\left\\\\{x\\\\mid 0.1\\\\le\\\\lVert x-\\\\left(0,0\\\\right)\\\\rVert_2\\\\le1\\\\right\\\\}\",\"note\":\"Audited catalogue wording; not silently treated as an Overleaf statement.\",\"source\":\"catalogue:SYS0081\",\"status\":\"documented\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":\"SYS0081\",\"citations\":[\"edwards2024fossil\"],\"label\":\"sys:fossil2-f2-02-sinemodel\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":567,\"start\":540},\"title_latex\":\"Table~4 \\\\#2 (SineModel), \\\\cite{edwards2024fossil}\"},\"source_order\":10,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0081\",\"status\":\"documented\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} x_2+u_1\\\\\\\\ u_2+\\\\dfrac{mgL\\\\sin(x_1)-bx_2}{mL^2} \\\\end{pmatrix},\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} x_2+u_1\\\\\\\\ u_2+\\\\dfrac{mgL\\\\sin(x_1)-bx_2}{mL^2} \\\\end{pmatrix}, \\\\] where $m=0.15$, $g=9.81$, $L=0.5$, and $b=0.1$.\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"Stability certification\"},\"reported_method_latex\":\"Verified method in the original paper: dReal\",\"reported_type_latex\":\"Original verified type: local $\\\\epsilon$-asymptotic stability\"}}"} {"record_id":"DCB-LIT-P1-011","paper_system_number":11,"task_type":"stability","problem_name":"Stability certification","tier":"literature","dimension":2,"time_domain":"discrete","dynamics":"x1[k+1] = 0.687*x1+0.558*x2-0.0001*x1*x2, x2[k+1] = -0.292*x1+0.773*x2","specification_latex":"The discrete-time transition map is: \\[ \\mathbf{x}^{+} = F(\\mathbf{x}) = \\begin{pmatrix} 0.687x_1+0.558x_2-0.0001x_1x_2\\\\ -0.292x_1+0.773x_2 \\end{pmatrix}. \\]","citations":["edwards2024fossil"],"problem_json":"{\"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\"]]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":1,\"problem_name\":\"Stability certification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0)$\",\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid0.1^2\\\\leq x_1^2+x_2^2\\\\leq1^2\\\\}$\",\"Original verified type: local $\\\\epsilon$-asymptotic stability\",\"Verified method in the original paper: dReal\",\"Our $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid x_1^2+x_2^2\\\\leq1^2\\\\}$\",\"Our verified type: local asymptotic stability\"],\"record_id\":\"DCB-LIT-P1-011\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=(0,0)$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid0.1^2\\\\leq x_1^2+x_2^2\\\\leq1^2\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"X_D=\\\\left\\\\{x\\\\mid 0.1\\\\le\\\\lVert x-\\\\left(0,0\\\\right)\\\\rVert_2\\\\le1\\\\right\\\\}\",\"note\":\"Audited catalogue wording; not silently treated as an Overleaf statement.\",\"source\":\"catalogue:SYS0093\",\"status\":\"documented\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":\"SYS0093\",\"citations\":[\"edwards2024fossil\"],\"label\":\"sys:fossil2-f2-14-dtahmadi2\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":592,\"start\":569},\"title_latex\":\"Table~4 \\\\#14 (DTAhmadi2), \\\\cite{edwards2024fossil}\"},\"source_order\":11,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0093\",\"status\":\"documented\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\mathbf{x}^{+} = F(\\\\mathbf{x}) = \\\\begin{pmatrix} 0.687x_1+0.558x_2-0.0001x_1x_2\\\\\\\\ -0.292x_1+0.773x_2 \\\\end{pmatrix}.\"],\"specification_latex\":\"The discrete-time transition map is: \\\\[ \\\\mathbf{x}^{+} = F(\\\\mathbf{x}) = \\\\begin{pmatrix} 0.687x_1+0.558x_2-0.0001x_1x_2\\\\\\\\ -0.292x_1+0.773x_2 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"discrete\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"Stability certification\"},\"reported_method_latex\":\"Verified method in the original paper: dReal\",\"reported_type_latex\":\"Original verified type: local $\\\\epsilon$-asymptotic stability\"}}"} {"record_id":"DCB-LIT-P1-012","paper_system_number":12,"task_type":"stability","problem_name":"Stability certification","tier":"literature","dimension":2,"time_domain":"discrete","dynamics":"x1[k+1] = 0.3*x1+u1, x2[k+1] = -x1+0.5*x2+((7)/(18))*x1**2+u2","specification_latex":"The discrete-time transition map is: \\[ \\mathbf{x}^{+} = F(\\mathbf{x},\\mathbf{u}) = \\begin{pmatrix} 0.3x_1+u_1\\\\ -x_1+0.5x_2+\\dfrac{7}{18}x_1^2+u_2 \\end{pmatrix}. \\]","citations":["edwards2024fossil"],"problem_json":"{\"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\"]]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":1,\"problem_name\":\"Stability certification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0)$\",\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid0.1^2\\\\leq x_1^2+x_2^2\\\\leq1^2\\\\}$\",\"Original verified type: local $\\\\epsilon$-asymptotic stability\",\"Verified method in the original paper: dReal\",\"Our $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid x_1^2+x_2^2\\\\leq1^2\\\\}$\",\"Our verified type: local asymptotic stability\"],\"record_id\":\"DCB-LIT-P1-012\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=(0,0)$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid0.1^2\\\\leq x_1^2+x_2^2\\\\leq1^2\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"X_D=\\\\left\\\\{x\\\\mid 0.1\\\\le\\\\lVert x-\\\\left(0,0\\\\right)\\\\rVert_2\\\\le1\\\\right\\\\}\",\"note\":\"Audited catalogue wording; not silently treated as an Overleaf statement.\",\"source\":\"catalogue:SYS0094\",\"status\":\"documented\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":\"SYS0094\",\"citations\":[\"edwards2024fossil\"],\"label\":\"sys:fossil2-f2-15-dtahmadi\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":617,\"start\":594},\"title_latex\":\"Table~4 \\\\#15 (DTAhmadi), \\\\cite{edwards2024fossil}\"},\"source_order\":12,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0094\",\"status\":\"documented\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\mathbf{x}^{+} = F(\\\\mathbf{x},\\\\mathbf{u}) = \\\\begin{pmatrix} 0.3x_1+u_1\\\\\\\\ -x_1+0.5x_2+\\\\dfrac{7}{18}x_1^2+u_2 \\\\end{pmatrix}.\"],\"specification_latex\":\"The discrete-time transition map is: \\\\[ \\\\mathbf{x}^{+} = F(\\\\mathbf{x},\\\\mathbf{u}) = \\\\begin{pmatrix} 0.3x_1+u_1\\\\\\\\ -x_1+0.5x_2+\\\\dfrac{7}{18}x_1^2+u_2 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"discrete\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"Stability certification\"},\"reported_method_latex\":\"Verified method in the original paper: dReal\",\"reported_type_latex\":\"Original verified type: local $\\\\epsilon$-asymptotic stability\"}}"} {"record_id":"DCB-LIT-P1-013","paper_system_number":13,"task_type":"stability","problem_name":"Stability certification","tier":"literature","dimension":3,"time_domain":"continuous","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","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2\\\\ \\dot{x}_3 \\end{pmatrix} = \\begin{pmatrix} -3x_1^3+3x_1x_3-9x_1\\\\ -x_1^3-5x_2+5x_3^3\\\\ -9x_3^3 \\end{pmatrix}. \\]","citations":["zou2025analytical"],"problem_json":"{\"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\"]]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":1,\"problem_name\":\"Stability certification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0,0)$\",\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{3}\\\\mid e^{-6}\\\\leq x_1^2+x_2^2+x_3^2,\\\\;|x_i|\\\\leq1,\\\\ i=1,2,3\\\\}$\",\"Original verified type: local $\\\\epsilon$-asymptotic stability\",\"Verified method in the original paper: dReal\",\"Our $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{3}\\\\mid |x_i|\\\\leq1,\\\\ i=1,2,3\\\\}$\",\"Our verified type: local asymptotic stability\"],\"record_id\":\"DCB-LIT-P1-013\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=(0,0,0)$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{3}\\\\mid e^{-6}\\\\leq x_1^2+x_2^2+x_3^2,\\\\;|x_i|\\\\leq1,\\\\ i=1,2,3\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"X=[-1,1]^3\",\"note\":\"Audited catalogue wording; not silently treated as an Overleaf statement.\",\"source\":\"catalogue:SYS0227\",\"status\":\"documented\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":\"SYS0227\",\"citations\":[\"zou2025analytical\"],\"label\":\"sys:catalogue-sys0227\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":647,\"start\":619},\"title_latex\":\"Table~1 row~4, \\\\cite{zou2025analytical}\"},\"source_order\":13,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0227\",\"status\":\"documented\",\"symbolic\":null,\"value\":3},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3 \\\\end{pmatrix} = \\\\begin{pmatrix} -3x_1^3+3x_1x_3-9x_1\\\\\\\\ -x_1^3-5x_2+5x_3^3\\\\\\\\ -9x_3^3 \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3 \\\\end{pmatrix} = \\\\begin{pmatrix} -3x_1^3+3x_1x_3-9x_1\\\\\\\\ -x_1^3-5x_2+5x_3^3\\\\\\\\ -9x_3^3 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"Stability certification\"},\"reported_method_latex\":\"Verified method in the original paper: dReal\",\"reported_type_latex\":\"Original verified type: local $\\\\epsilon$-asymptotic stability\"}}"} {"record_id":"DCB-LIT-P1-014","paper_system_number":14,"task_type":"stability","problem_name":"Stability certification","tier":"literature","dimension":3,"time_domain":"continuous","dynamics":"dx1/dt = x2, dx2/dt = -sin(x1)*cos(x1)-x2-sin(x3)*cos(x3), dx3/dt = x2-x3","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2\\\\ \\dot{x}_3 \\end{pmatrix} = \\begin{pmatrix} x_2\\\\ -\\sin(x_1)\\cos(x_1)-x_2-\\sin(x_3)\\cos(x_3)\\\\ x_2-x_3 \\end{pmatrix}. \\]","citations":["zou2025analytical"],"problem_json":"{\"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\"]]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":1,\"problem_name\":\"Stability certification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0,0)$\",\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{3}\\\\mid e^{-6}\\\\leq x_1^2+x_2^2+x_3^2,\\\\;|x_i|\\\\leq1.5,\\\\ i=1,2,3\\\\}$\",\"Original verified type: local $\\\\epsilon$-asymptotic stability\",\"Verified method in the original paper: dReal\",\"Our $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{3}\\\\mid |x_i|\\\\leq1.5,\\\\ i=1,2,3\\\\}$\",\"Our verified type: local asymptotic stability\"],\"record_id\":\"DCB-LIT-P1-014\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=(0,0,0)$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{3}\\\\mid e^{-6}\\\\leq x_1^2+x_2^2+x_3^2,\\\\;|x_i|\\\\leq1.5,\\\\ i=1,2,3\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"X=\\\\text{not declared in this configuration file}\",\"note\":\"Audited catalogue wording; not silently treated as an Overleaf statement.\",\"source\":\"catalogue:SYS0228\",\"status\":\"documented\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":\"SYS0228\",\"citations\":[\"zou2025analytical\"],\"label\":\"sys:catalogue-sys0228\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":678,\"start\":650},\"title_latex\":\"Table~1 row~5, \\\\cite{zou2025analytical}\"},\"source_order\":14,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0228\",\"status\":\"documented\",\"symbolic\":null,\"value\":3},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3 \\\\end{pmatrix} = \\\\begin{pmatrix} x_2\\\\\\\\ -\\\\sin(x_1)\\\\cos(x_1)-x_2-\\\\sin(x_3)\\\\cos(x_3)\\\\\\\\ x_2-x_3 \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3 \\\\end{pmatrix} = \\\\begin{pmatrix} x_2\\\\\\\\ -\\\\sin(x_1)\\\\cos(x_1)-x_2-\\\\sin(x_3)\\\\cos(x_3)\\\\\\\\ x_2-x_3 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"Stability certification\"},\"reported_method_latex\":\"Verified method in the original paper: dReal\",\"reported_type_latex\":\"Original verified type: local $\\\\epsilon$-asymptotic stability\"}}"} {"record_id":"DCB-LIT-P1-015","paper_system_number":15,"task_type":"stability","problem_name":"Stability certification","tier":"literature","dimension":4,"time_domain":"continuous","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))","specification_latex":"The source system is expressed in the center-of-inertia coordinates $(\\delta_1,\\delta_2,\\omega_1,\\omega_2)$, which satisfy $\\delta_1+\\delta_2=0$. We therefore remove the redundant common-angle coordinate and define $q:=\\delta_1-\\delta_2, \\delta_1=\\frac{q}{2}, \\delta_2=-\\frac{q}{2}$. Let $\\mathbf{x}=(q,\\omega_1,\\omega_2)\\in\\mathbb{R}^3$. The reduced vector field is \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot q\\\\ \\dot\\omega_1\\\\ \\dot\\omega_2 \\end{pmatrix} = \\begin{pmatrix} \\omega_1-\\omega_2\\\\[0.4ex] \\frac12\\bigl(-1-2\\omega_1-\\sin(q)+\\cos(q)\\bigr)\\\\[0.4ex] \\frac12\\bigl(-1-2\\omega_2+\\sin(q)+\\cos(q)\\bigr) \\end{pmatrix}. \\]","citations":["zou2025analytical"],"problem_json":"{\"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\"]]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":1,\"problem_name\":\"Stability certification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0,0)$\",\"Original $ \\\\mathcal{N} = \\\\left\\\\{ (q,\\\\omega_1,\\\\omega_2)\\\\in\\\\mathbb{R}^3 \\\\;\\\\middle|\\\\; e^{-6} \\\\le \\\\frac12 q^2+\\\\omega_1^2+\\\\omega_2^2,\\\\; |q|\\\\le1.5,\\\\; |\\\\omega_i|\\\\le2,\\\\ i=1,2 \\\\right\\\\}. $\",\"Original verified type: local $\\\\epsilon$-asymptotic stability\",\"Verified method in the original paper: dReal\",\"Our $ \\\\mathcal{N} = \\\\left\\\\{ (q,\\\\omega_1,\\\\omega_2)\\\\in\\\\mathbb{R}^3 \\\\;\\\\middle|\\\\; |q|\\\\le1.5,\\\\; |\\\\omega_i|\\\\le2,\\\\ i=1,2 \\\\right\\\\}. $\",\"Our target: local asymptotic stability of the origin\"],\"record_id\":\"DCB-LIT-P1-015\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=(0,0,0)$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":\"Original $ \\\\mathcal{N} = \\\\left\\\\{ (q,\\\\omega_1,\\\\omega_2)\\\\in\\\\mathbb{R}^3 \\\\;\\\\middle|\\\\; e^{-6} \\\\le \\\\frac12 q^2+\\\\omega_1^2+\\\\omega_2^2,\\\\; |q|\\\\le1.5,\\\\; |\\\\omega_i|\\\\le2,\\\\ i=1,2 \\\\right\\\\}. $\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"X=\\\\text{not declared in this configuration file}\",\"note\":\"Audited catalogue wording; not silently treated as an Overleaf statement.\",\"source\":\"catalogue:SYS0229\",\"status\":\"documented\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":\"SYS0229\",\"citations\":[\"zou2025analytical\"],\"label\":\"sys:catalogue-sys0229\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":742,\"start\":680},\"title_latex\":\"Table~1 row~6, \\\\cite{zou2025analytical}\"},\"source_order\":15,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0229\",\"status\":\"documented\",\"symbolic\":null,\"value\":4},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot q\\\\\\\\ \\\\dot\\\\omega_1\\\\\\\\ \\\\dot\\\\omega_2 \\\\end{pmatrix} = \\\\begin{pmatrix} \\\\omega_1-\\\\omega_2\\\\\\\\[0.4ex] \\\\frac12\\\\bigl(-1-2\\\\omega_1-\\\\sin(q)+\\\\cos(q)\\\\bigr)\\\\\\\\[0.4ex] \\\\frac12\\\\bigl(-1-2\\\\omega_2+\\\\sin(q)+\\\\cos(q)\\\\bigr) \\\\end{pmatrix}.\"],\"specification_latex\":\"The source system is expressed in the center-of-inertia coordinates $(\\\\delta_1,\\\\delta_2,\\\\omega_1,\\\\omega_2)$, which satisfy $\\\\delta_1+\\\\delta_2=0$. We therefore remove the redundant common-angle coordinate and define $q:=\\\\delta_1-\\\\delta_2, \\\\delta_1=\\\\frac{q}{2}, \\\\delta_2=-\\\\frac{q}{2}$. Let $\\\\mathbf{x}=(q,\\\\omega_1,\\\\omega_2)\\\\in\\\\mathbb{R}^3$. The reduced vector field is \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot q\\\\\\\\ \\\\dot\\\\omega_1\\\\\\\\ \\\\dot\\\\omega_2 \\\\end{pmatrix} = \\\\begin{pmatrix} \\\\omega_1-\\\\omega_2\\\\\\\\[0.4ex] \\\\frac12\\\\bigl(-1-2\\\\omega_1-\\\\sin(q)+\\\\cos(q)\\\\bigr)\\\\\\\\[0.4ex] \\\\frac12\\\\bigl(-1-2\\\\omega_2+\\\\sin(q)+\\\\cos(q)\\\\bigr) \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"Stability certification\"},\"reported_method_latex\":\"Verified method in the original paper: dReal\",\"reported_type_latex\":\"Original verified type: local $\\\\epsilon$-asymptotic stability\"}}"} {"record_id":"DCB-LIT-P1-016","paper_system_number":16,"task_type":"stability","problem_name":"Stability certification","tier":"literature","dimension":6,"time_domain":"continuous","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","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2\\\\ \\dot{x}_3\\\\ \\dot{x}_4\\\\ \\dot{x}_5\\\\ \\dot{x}_6 \\end{pmatrix} = \\begin{pmatrix} -x_1+0.5x_2-0.1x_5^2\\\\ -0.5x_1-x_2\\\\ -x_3+0.5x_4-0.1x_1^2\\\\ -0.5x_3-x_4\\\\ -x_5+0.5x_6\\\\ -0.5x_5-x_6+0.1x_2^2 \\end{pmatrix}. \\]","citations":["zou2025analytical"],"problem_json":"{\"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\"]]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":1,\"problem_name\":\"Stability certification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0,0,0,0,0)$\",\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{6}\\\\mid e^{-6}\\\\leq\\\\sum_{i=1}^{6}x_i^2,\\\\;|x_i|\\\\leq1,\\\\ i=1,\\\\ldots,6\\\\}$\",\"Original verified type: local $\\\\epsilon$-asymptotic stability\",\"Verified method in the original paper: dReal\",\"Our $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{6}\\\\mid |x_i|\\\\leq1,\\\\ i=1,\\\\ldots,6\\\\}$\",\"Our verified type: local asymptotic stability\"],\"record_id\":\"DCB-LIT-P1-016\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=(0,0,0,0,0,0)$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{6}\\\\mid e^{-6}\\\\leq\\\\sum_{i=1}^{6}x_i^2,\\\\;|x_i|\\\\leq1,\\\\ i=1,\\\\ldots,6\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"X=\\\\text{not declared in this configuration file}\",\"note\":\"Audited catalogue wording; not silently treated as an Overleaf statement.\",\"source\":\"catalogue:SYS0230\",\"status\":\"documented\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":\"SYS0230\",\"citations\":[\"zou2025analytical\"],\"label\":\"sys:catalogue-sys0230\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":808,\"start\":779},\"title_latex\":\"Table~1 row~7, \\\\cite{zou2025analytical}\"},\"source_order\":16,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0230\",\"status\":\"documented\",\"symbolic\":null,\"value\":6},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3\\\\\\\\ \\\\dot{x}_4\\\\\\\\ \\\\dot{x}_5\\\\\\\\ \\\\dot{x}_6 \\\\end{pmatrix} = \\\\begin{pmatrix} -x_1+0.5x_2-0.1x_5^2\\\\\\\\ -0.5x_1-x_2\\\\\\\\ -x_3+0.5x_4-0.1x_1^2\\\\\\\\ -0.5x_3-x_4\\\\\\\\ -x_5+0.5x_6\\\\\\\\ -0.5x_5-x_6+0.1x_2^2 \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3\\\\\\\\ \\\\dot{x}_4\\\\\\\\ \\\\dot{x}_5\\\\\\\\ \\\\dot{x}_6 \\\\end{pmatrix} = \\\\begin{pmatrix} -x_1+0.5x_2-0.1x_5^2\\\\\\\\ -0.5x_1-x_2\\\\\\\\ -x_3+0.5x_4-0.1x_1^2\\\\\\\\ -0.5x_3-x_4\\\\\\\\ -x_5+0.5x_6\\\\\\\\ -0.5x_5-x_6+0.1x_2^2 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"Stability certification\"},\"reported_method_latex\":\"Verified method in the original paper: dReal\",\"reported_type_latex\":\"Original verified type: local $\\\\epsilon$-asymptotic stability\"}}"} {"record_id":"DCB-LIT-P1-017","paper_system_number":17,"task_type":"stability","problem_name":"Stability certification","tier":"literature","dimension":6,"time_domain":"continuous","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","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2\\\\ \\dot{x}_3\\\\ \\dot{x}_4\\\\ \\dot{x}_5\\\\ \\dot{x}_6 \\end{pmatrix} = \\begin{pmatrix} x_2\\\\ -\\tfrac32x_6x_4-2x_1-\\tfrac52x_2-\\tfrac12x_4\\sin(x_2)\\cos(x_4)\\\\ x_4\\\\ \\tfrac32x_6x_2-2x_3-10x_4+\\tfrac12x_2\\sin(x_2)\\cos(x_4)\\\\ x_6\\\\ -5x_5-\\tfrac45x_6 \\end{pmatrix}. \\]","citations":["zou2025analytical"],"problem_json":"{\"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\"]]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":1,\"problem_name\":\"Stability certification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0,0,0,0,0)$\",\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{6}\\\\mid e^{-6}\\\\leq\\\\sum_{i=1}^{6}x_i^2,\\\\;|x_i|\\\\leq3,\\\\ i=1,\\\\ldots,6\\\\}$\",\"Original verified type: local $\\\\epsilon$-asymptotic stability\",\"Verified method in the original paper: dReal and LaSalle's invariance principle\",\"Our $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{6}\\\\mid |x_i|\\\\leq3,\\\\ i=1,\\\\ldots,6\\\\}$\",\"Our verified type: local asymptotic stability\"],\"record_id\":\"DCB-LIT-P1-017\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=(0,0,0,0,0,0)$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{6}\\\\mid e^{-6}\\\\leq\\\\sum_{i=1}^{6}x_i^2,\\\\;|x_i|\\\\leq3,\\\\ i=1,\\\\ldots,6\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"X=\\\\text{not declared in this configuration file}\",\"note\":\"Audited catalogue wording; not silently treated as an Overleaf statement.\",\"source\":\"catalogue:SYS0231\",\"status\":\"documented\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":\"SYS0231\",\"citations\":[\"zou2025analytical\"],\"label\":\"sys:catalogue-sys0231\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":839,\"start\":810},\"title_latex\":\"Table~1 row~8, \\\\cite{zou2025analytical}\"},\"source_order\":17,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0231\",\"status\":\"documented\",\"symbolic\":null,\"value\":6},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3\\\\\\\\ \\\\dot{x}_4\\\\\\\\ \\\\dot{x}_5\\\\\\\\ \\\\dot{x}_6 \\\\end{pmatrix} = \\\\begin{pmatrix} x_2\\\\\\\\ -\\\\tfrac32x_6x_4-2x_1-\\\\tfrac52x_2-\\\\tfrac12x_4\\\\sin(x_2)\\\\cos(x_4)\\\\\\\\ x_4\\\\\\\\ \\\\tfrac32x_6x_2-2x_3-10x_4+\\\\tfrac12x_2\\\\sin(x_2)\\\\cos(x_4)\\\\\\\\ x_6\\\\\\\\ -5x_5-\\\\tfrac45x_6 \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3\\\\\\\\ \\\\dot{x}_4\\\\\\\\ \\\\dot{x}_5\\\\\\\\ \\\\dot{x}_6 \\\\end{pmatrix} = \\\\begin{pmatrix} x_2\\\\\\\\ -\\\\tfrac32x_6x_4-2x_1-\\\\tfrac52x_2-\\\\tfrac12x_4\\\\sin(x_2)\\\\cos(x_4)\\\\\\\\ x_4\\\\\\\\ \\\\tfrac32x_6x_2-2x_3-10x_4+\\\\tfrac12x_2\\\\sin(x_2)\\\\cos(x_4)\\\\\\\\ x_6\\\\\\\\ -5x_5-\\\\tfrac45x_6 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"Stability certification\"},\"reported_method_latex\":\"Verified method in the original paper: dReal and LaSalle's invariance principle\",\"reported_type_latex\":\"Original verified type: local $\\\\epsilon$-asymptotic stability\"}}"} {"record_id":"DCB-LIT-P1-018","paper_system_number":18,"task_type":"stability","problem_name":"Stability certification","tier":"literature","dimension":6,"time_domain":"continuous","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))","specification_latex":"The source system is expressed in the center-of-inertia coordinates $(\\delta_1,\\delta_2,\\delta_3,\\omega_1,\\omega_2,\\omega_3)$, which satisfy $\\delta_1+\\delta_2+\\delta_3=0$. We therefore remove the redundant common-angle coordinate and define $\\delta_3:=-(\\delta_1+\\delta_2)$ and $\\bar{\\omega}:=\\frac13(\\omega_1+\\omega_2+\\omega_3)$. Let $\\mathbf{x}=(\\delta_1,\\delta_2,\\omega_1,\\omega_2,\\omega_3) \\in\\mathbb{R}^5$. The reduced vector field is \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{\\delta}_1\\\\ \\dot{\\delta}_2\\\\ \\dot{\\omega}_1\\\\ \\dot{\\omega}_2\\\\ \\dot{\\omega}_3 \\end{pmatrix} = \\begin{pmatrix} \\omega_1-\\bar{\\omega}\\\\[0.4ex] \\omega_2-\\bar{\\omega}\\\\[0.4ex] -\\omega_1-\\frac12\\sum_{j=1}^{3}\\sin(\\delta_1-\\delta_j)\\\\[0.4ex] -\\omega_2-\\frac12\\sum_{j=1}^{3}\\sin(\\delta_2-\\delta_j)\\\\[0.4ex] -\\omega_3-\\frac12\\sum_{j=1}^{3}\\sin(\\delta_3-\\delta_j) \\end{pmatrix}. \\]","citations":["zou2025analytical"],"problem_json":"{\"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\"]]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":1,\"problem_name\":\"Stability certification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0,0,0,0)$\",\"Original $\\\\mathcal{N} = \\\\{(\\\\delta_1,\\\\delta_2,\\\\omega_1,\\\\omega_2,\\\\omega_3)\\\\in\\\\mathbb{R}^5 \\\\mid e^{-6}\\\\le \\\\delta_1^2+\\\\delta_2^2+(\\\\delta_1+\\\\delta_2)^2 +\\\\sum_{i=1}^{3}\\\\omega_i^2, \\\\allowbreak\\\\; |\\\\delta_1|\\\\le0.75,\\\\; |\\\\delta_2|\\\\le0.75,\\\\; |\\\\delta_1+\\\\delta_2|\\\\le0.75, \\\\allowbreak\\\\; |\\\\omega_i|\\\\le1.2,\\\\ i=1,2,3\\\\}$\",\"Original verified type: local $\\\\epsilon$-asymptotic stability\",\"Verified method in the original paper: dReal and LaSalle's invariance principle\",\"Our $\\\\mathcal{N} = \\\\{(\\\\delta_1,\\\\delta_2,\\\\omega_1,\\\\omega_2,\\\\omega_3)\\\\in\\\\mathbb{R}^5 \\\\mid |\\\\delta_1|\\\\leq0.75,\\\\; \\\\allowbreak |\\\\delta_2|\\\\leq0.75,\\\\; \\\\allowbreak |\\\\delta_1+\\\\delta_2|\\\\leq0.75,\\\\; \\\\allowbreak |\\\\omega_i|\\\\leq1.2,\\\\ i=1,2,3 \\\\}$\",\"Our target: local asymptotic stability of the origin\"],\"record_id\":\"DCB-LIT-P1-018\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=(0,0,0,0,0)$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":\"Original $\\\\mathcal{N} = \\\\{(\\\\delta_1,\\\\delta_2,\\\\omega_1,\\\\omega_2,\\\\omega_3)\\\\in\\\\mathbb{R}^5 \\\\mid e^{-6}\\\\le \\\\delta_1^2+\\\\delta_2^2+(\\\\delta_1+\\\\delta_2)^2 +\\\\sum_{i=1}^{3}\\\\omega_i^2, \\\\allowbreak\\\\; |\\\\delta_1|\\\\le0.75,\\\\; |\\\\delta_2|\\\\le0.75,\\\\; |\\\\delta_1+\\\\delta_2|\\\\le0.75, \\\\allowbreak\\\\; |\\\\omega_i|\\\\le1.2,\\\\ i=1,2,3\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"|\\\\delta_i|\\\\le.75,\\\\ |\\\\omega_i|\\\\le1.2\",\"note\":\"Audited catalogue wording; not silently treated as an Overleaf statement.\",\"source\":\"catalogue:SYS0232\",\"status\":\"documented\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":\"SYS0232\",\"citations\":[\"zou2025analytical\"],\"label\":\"sys:catalogue-sys0232\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":953,\"start\":873},\"title_latex\":\"Table~1 row~9, \\\\cite{zou2025analytical}\"},\"source_order\":18,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0232\",\"status\":\"documented\",\"symbolic\":null,\"value\":6},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{\\\\delta}_1\\\\\\\\ \\\\dot{\\\\delta}_2\\\\\\\\ \\\\dot{\\\\omega}_1\\\\\\\\ \\\\dot{\\\\omega}_2\\\\\\\\ \\\\dot{\\\\omega}_3 \\\\end{pmatrix} = \\\\begin{pmatrix} \\\\omega_1-\\\\bar{\\\\omega}\\\\\\\\[0.4ex] \\\\omega_2-\\\\bar{\\\\omega}\\\\\\\\[0.4ex] -\\\\omega_1-\\\\frac12\\\\sum_{j=1}^{3}\\\\sin(\\\\delta_1-\\\\delta_j)\\\\\\\\[0.4ex] -\\\\omega_2-\\\\frac12\\\\sum_{j=1}^{3}\\\\sin(\\\\delta_2-\\\\delta_j)\\\\\\\\[0.4ex] -\\\\omega_3-\\\\frac12\\\\sum_{j=1}^{3}\\\\sin(\\\\delta_3-\\\\delta_j) \\\\end{pmatrix}.\"],\"specification_latex\":\"The source system is expressed in the center-of-inertia coordinates $(\\\\delta_1,\\\\delta_2,\\\\delta_3,\\\\omega_1,\\\\omega_2,\\\\omega_3)$, which satisfy $\\\\delta_1+\\\\delta_2+\\\\delta_3=0$. We therefore remove the redundant common-angle coordinate and define $\\\\delta_3:=-(\\\\delta_1+\\\\delta_2)$ and $\\\\bar{\\\\omega}:=\\\\frac13(\\\\omega_1+\\\\omega_2+\\\\omega_3)$. Let $\\\\mathbf{x}=(\\\\delta_1,\\\\delta_2,\\\\omega_1,\\\\omega_2,\\\\omega_3) \\\\in\\\\mathbb{R}^5$. The reduced vector field is \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{\\\\delta}_1\\\\\\\\ \\\\dot{\\\\delta}_2\\\\\\\\ \\\\dot{\\\\omega}_1\\\\\\\\ \\\\dot{\\\\omega}_2\\\\\\\\ \\\\dot{\\\\omega}_3 \\\\end{pmatrix} = \\\\begin{pmatrix} \\\\omega_1-\\\\bar{\\\\omega}\\\\\\\\[0.4ex] \\\\omega_2-\\\\bar{\\\\omega}\\\\\\\\[0.4ex] -\\\\omega_1-\\\\frac12\\\\sum_{j=1}^{3}\\\\sin(\\\\delta_1-\\\\delta_j)\\\\\\\\[0.4ex] -\\\\omega_2-\\\\frac12\\\\sum_{j=1}^{3}\\\\sin(\\\\delta_2-\\\\delta_j)\\\\\\\\[0.4ex] -\\\\omega_3-\\\\frac12\\\\sum_{j=1}^{3}\\\\sin(\\\\delta_3-\\\\delta_j) \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":\"\\\\begin{remark} The original system is stated in coordinates with $\\\\delta_1+\\\\delta_2+\\\\delta_3=0$. We substitute $\\\\delta_3 := -(\\\\delta_1+\\\\delta_2)$ and impose $|\\\\delta_1+\\\\delta_2|\\\\le0.75$ to eliminate the equilibrium line $\\\\{(c,c,c,0,0,0)\\\\}$ present in the six-dimensional formulation. This reduction makes the single-point stability problem well-defined. \\\\end{remark}\"},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"Stability certification\"},\"reported_method_latex\":\"Verified method in the original paper: dReal and LaSalle's invariance principle\",\"reported_type_latex\":\"Original verified type: local $\\\\epsilon$-asymptotic stability\"}}"} {"record_id":"DCB-LIT-P1-019","paper_system_number":19,"task_type":"stability","problem_name":"Stability certification","tier":"literature","dimension":9,"time_domain":"continuous","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","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2\\\\ \\dot{x}_3\\\\ \\dot{x}_4\\\\ \\dot{x}_5\\\\ \\dot{x}_6\\\\ \\dot{x}_7\\\\ \\dot{x}_8\\\\ \\dot{x}_9 \\end{pmatrix} = \\begin{pmatrix} -x_1+0.5x_2-0.1x_5^2\\\\ -0.5x_1-x_2+0.1x_8\\\\ -x_3+0.5x_4-0.1x_1^2\\\\ -0.5x_3-x_4\\\\ -x_5+0.5x_6\\\\ -0.5x_5-x_6+0.1x_2^2\\\\ x_8\\\\ -\\sin(x_7)\\cos(x_7)-x_8-\\sin(x_9)\\cos(x_9)-0.1x_2\\\\ x_8-x_9 \\end{pmatrix}. \\]","citations":["zou2025analytical"],"problem_json":"{\"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\"]]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":1,\"problem_name\":\"Stability certification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0,0,0,0,0,0,0,0)$\",\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{9}\\\\mid e^{-6}\\\\leq\\\\sum_{i=1}^{9}x_i^2,\\\\;|x_i|\\\\leq1.5,\\\\ i=1,\\\\ldots,9\\\\}$\",\"Original verified type: local $\\\\epsilon$-asymptotic stability\",\"Verified method in the original paper: dReal\",\"Our $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{9}\\\\mid |x_i|\\\\leq1.5,\\\\ i=1,\\\\ldots,9\\\\}$\",\"Our verified type: local asymptotic stability\"],\"record_id\":\"DCB-LIT-P1-019\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=(0,0,0,0,0,0,0,0,0)$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{9}\\\\mid e^{-6}\\\\leq\\\\sum_{i=1}^{9}x_i^2,\\\\;|x_i|\\\\leq1.5,\\\\ i=1,\\\\ldots,9\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"X\\\\text{ as defined in the source appendix}\",\"note\":\"Audited catalogue wording; not silently treated as an Overleaf statement.\",\"source\":\"catalogue:SYS0233\",\"status\":\"documented\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":\"SYS0233\",\"citations\":[\"zou2025analytical\"],\"label\":\"sys:catalogue-sys0233\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":994,\"start\":961},\"title_latex\":\"Table~1 row~10, \\\\cite{zou2025analytical}\"},\"source_order\":19,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0233\",\"status\":\"documented\",\"symbolic\":null,\"value\":9},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3\\\\\\\\ \\\\dot{x}_4\\\\\\\\ \\\\dot{x}_5\\\\\\\\ \\\\dot{x}_6\\\\\\\\ \\\\dot{x}_7\\\\\\\\ \\\\dot{x}_8\\\\\\\\ \\\\dot{x}_9 \\\\end{pmatrix} = \\\\begin{pmatrix} -x_1+0.5x_2-0.1x_5^2\\\\\\\\ -0.5x_1-x_2+0.1x_8\\\\\\\\ -x_3+0.5x_4-0.1x_1^2\\\\\\\\ -0.5x_3-x_4\\\\\\\\ -x_5+0.5x_6\\\\\\\\ -0.5x_5-x_6+0.1x_2^2\\\\\\\\ x_8\\\\\\\\ -\\\\sin(x_7)\\\\cos(x_7)-x_8-\\\\sin(x_9)\\\\cos(x_9)-0.1x_2\\\\\\\\ x_8-x_9 \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3\\\\\\\\ \\\\dot{x}_4\\\\\\\\ \\\\dot{x}_5\\\\\\\\ \\\\dot{x}_6\\\\\\\\ \\\\dot{x}_7\\\\\\\\ \\\\dot{x}_8\\\\\\\\ \\\\dot{x}_9 \\\\end{pmatrix} = \\\\begin{pmatrix} -x_1+0.5x_2-0.1x_5^2\\\\\\\\ -0.5x_1-x_2+0.1x_8\\\\\\\\ -x_3+0.5x_4-0.1x_1^2\\\\\\\\ -0.5x_3-x_4\\\\\\\\ -x_5+0.5x_6\\\\\\\\ -0.5x_5-x_6+0.1x_2^2\\\\\\\\ x_8\\\\\\\\ -\\\\sin(x_7)\\\\cos(x_7)-x_8-\\\\sin(x_9)\\\\cos(x_9)-0.1x_2\\\\\\\\ x_8-x_9 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"Stability certification\"},\"reported_method_latex\":\"Verified method in the original paper: dReal\",\"reported_type_latex\":\"Original verified type: local $\\\\epsilon$-asymptotic stability\"}}"} {"record_id":"DCB-LIT-P1-020","paper_system_number":20,"task_type":"stability","problem_name":"Stability certification","tier":"literature","dimension":3,"time_domain":"continuous","dynamics":"dx1/dt = -8*x1*x2**2-10*x2**4, dx2/dt = -8*x2**3+3*x2**2-8*x2, dx3/dt = -x3","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2\\\\ \\dot{x}_3 \\end{pmatrix} = \\begin{pmatrix} -8x_1x_2^2-10x_2^4\\\\ -8x_2^3+3x_2^2-8x_2\\\\ -x_3 \\end{pmatrix}. \\]","citations":["zou2025analytical"],"problem_json":"{\"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\"]]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":1,\"problem_name\":\"Stability certification\",\"raw_items_latex\":[\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{3}\\\\mid e^{-6}\\\\leq x_1^2+x_2^2+x_3^2,\\\\;|x_i|\\\\leq1,\\\\ i=1,2,3\\\\}$\",\"Original verified type: local $\\\\epsilon$-stability\",\"Verified method in the original paper: dReal\",\"Our $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{3}\\\\mid |x_i|\\\\leq1,\\\\ i=1,2,3\\\\}$\",\"Our verified type: local stability\"],\"record_id\":\"DCB-LIT-P1-020\",\"sets\":{\"equilibrium\":{\"latex\":\"x^\\\\star=(0,0,0)\",\"note\":\"Every displayed component of the vector field vanishes at the origin.\",\"source\":\"displayed_dynamics\",\"status\":\"inferred\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{3}\\\\mid e^{-6}\\\\leq x_1^2+x_2^2+x_3^2,\\\\;|x_i|\\\\leq1,\\\\ i=1,2,3\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"X=[-1,1]^3\",\"note\":\"Audited catalogue wording; not silently treated as an Overleaf statement.\",\"source\":\"catalogue:SYS0235\",\"status\":\"documented\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":\"SYS0235\",\"citations\":[\"zou2025analytical\"],\"label\":\"sys:catalogue-sys0235\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":1023,\"start\":996},\"title_latex\":\"Appendix Table~6, \\\\cite{zou2025analytical}\"},\"source_order\":20,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0235\",\"status\":\"documented\",\"symbolic\":null,\"value\":3},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3 \\\\end{pmatrix} = \\\\begin{pmatrix} -8x_1x_2^2-10x_2^4\\\\\\\\ -8x_2^3+3x_2^2-8x_2\\\\\\\\ -x_3 \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3 \\\\end{pmatrix} = \\\\begin{pmatrix} -8x_1x_2^2-10x_2^4\\\\\\\\ -8x_2^3+3x_2^2-8x_2\\\\\\\\ -x_3 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"Stability certification\"},\"reported_method_latex\":\"Verified method in the original paper: dReal\",\"reported_type_latex\":\"Original verified type: local $\\\\epsilon$-stability\"}}"} {"record_id":"DCB-LIT-P1-021","paper_system_number":21,"task_type":"stability","problem_name":"Stability certification","tier":"literature","dimension":2,"time_domain":"continuous","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","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}}= \\begin{pmatrix}\\dot{x}\\\\\\dot{y}\\end{pmatrix} = \\begin{pmatrix}4x^3-x^2y-6xy^2-y^3\\\\x^3+4x^2y+xy^2-6y^3\\end{pmatrix}. \\]","citations":["liu2026globally"],"problem_json":"{\"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\"}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":1,\"problem_name\":\"Stability certification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0)$\",\"Original $\\\\mathcal{N}=\\\\mathbb{R}^{2}$\",\"Original verified type: global asymptotic stability\",\"Verified method in the original paper: human verified by product\",\"Our $\\\\mathcal{N}=\\\\mathbb{R}^{2}$\",\"Our verified type: global asymptotic stability\"],\"record_id\":\"DCB-LIT-P1-021\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=(0,0)$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":\"Original $\\\\mathcal{N}=\\\\mathbb{R}^{2}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"x^\\\\star=(0,0);\\\\quad \\\\mathcal{N}=\\\\mathbb{R}^{2}\",\"note\":\"Audited catalogue wording; not silently treated as an Overleaf statement.\",\"source\":\"catalogue:OVL-EXT-003\",\"status\":\"documented\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":\"OVL-EXT-003\",\"citations\":[\"liu2026globally\"],\"label\":\"sys:liu2026-globally\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":1124,\"start\":1105},\"title_latex\":\"Main theorem, \\\\cite{liu2026globally}\"},\"source_order\":21,\"system\":{\"dimension\":{\"source\":\"catalogue:OVL-EXT-003\",\"status\":\"documented\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}}= \\\\begin{pmatrix}\\\\dot{x}\\\\\\\\\\\\dot{y}\\\\end{pmatrix} = \\\\begin{pmatrix}4x^3-x^2y-6xy^2-y^3\\\\\\\\x^3+4x^2y+xy^2-6y^3\\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}}= \\\\begin{pmatrix}\\\\dot{x}\\\\\\\\\\\\dot{y}\\\\end{pmatrix} = \\\\begin{pmatrix}4x^3-x^2y-6xy^2-y^3\\\\\\\\x^3+4x^2y+xy^2-6y^3\\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"Stability certification\"},\"reported_method_latex\":\"Verified method in the original paper: human verified by product\",\"reported_type_latex\":\"Original verified type: global asymptotic stability\"}}"} {"record_id":"DCB-LIT-P2-001","paper_system_number":22,"task_type":"roa","problem_name":"Region-of-attraction verification","tier":"literature","dimension":2,"time_domain":"continuous","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)","specification_latex":"The closed-loop vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{\\theta}\\\\ \\dot{\\omega} \\end{pmatrix} = \\begin{pmatrix} \\omega\\\\ \\dfrac{mg\\ell\\sin(\\theta)+u-0.1\\omega}{m\\ell^2} \\end{pmatrix}. \\] \\[ u=20\\tanh(-23.286329\\theta-5.27055\\omega), \\] where $m=0.15,\\ell=0.5$ and $g=9.81$.","citations":["zhou2022neural"],"problem_json":"{\"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\"]]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":2,\"problem_name\":\"Region-of-attraction verification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0)$\",\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid0.4^2\\\\leq x_1^2+x_2^2\\\\leq4^2\\\\}$\",\"Original verified type: $\\\\epsilon$-ROA\",\"Verified method in the original paper: dReal\",\"Our $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid x_1^2+x_2^2\\\\leq4^2\\\\}$\",\"Our verified type: ROA verification\"],\"record_id\":\"DCB-LIT-P2-001\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=(0,0)$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"No separate initial set is required to certify Omega_c itself.\",\"source\":\"exact_roa_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid0.4^2\\\\leq x_1^2+x_2^2\\\\leq4^2\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"X=\\\\{x:\\\\|x\\\\|_2\\\\le4\\\\}\",\"note\":\"Audited catalogue wording; not silently treated as an Overleaf statement.\",\"source\":\"catalogue:SYS0033\",\"status\":\"documented\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 2.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 2.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":\"SYS0033\",\"citations\":[\"zhou2022neural\"],\"label\":\"sys:catalogue-sys0033\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":1160,\"start\":1130},\"title_latex\":\"Section~5.3 and Appendix~C.3.3, \\\\cite{zhou2022neural}\"},\"source_order\":22,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0033\",\"status\":\"documented\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{\\\\theta}\\\\\\\\ \\\\dot{\\\\omega} \\\\end{pmatrix} = \\\\begin{pmatrix} \\\\omega\\\\\\\\ \\\\dfrac{mg\\\\ell\\\\sin(\\\\theta)+u-0.1\\\\omega}{m\\\\ell^2} \\\\end{pmatrix}.\",\"u=20\\\\tanh(-23.286329\\\\theta-5.27055\\\\omega),\"],\"specification_latex\":\"The closed-loop vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{\\\\theta}\\\\\\\\ \\\\dot{\\\\omega} \\\\end{pmatrix} = \\\\begin{pmatrix} \\\\omega\\\\\\\\ \\\\dfrac{mg\\\\ell\\\\sin(\\\\theta)+u-0.1\\\\omega}{m\\\\ell^2} \\\\end{pmatrix}. \\\\] \\\\[ u=20\\\\tanh(-23.286329\\\\theta-5.27055\\\\omega), \\\\] where $m=0.15,\\\\ell=0.5$ and $g=9.81$.\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"exact_roa\"},\"reported_method_latex\":\"Verified method in the original paper: dReal\",\"reported_type_latex\":\"Original verified type: $\\\\epsilon$-ROA\"}}"} {"record_id":"DCB-LIT-P2-002","paper_system_number":23,"task_type":"roa","problem_name":"Region-of-attraction verification","tier":"literature","dimension":2,"time_domain":"continuous","dynamics":"dx1/dt = -x2, dx2/dt = x1 + (x1**2 - 1)*x2","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2 \\end{pmatrix} = \\begin{pmatrix} -x_2\\\\ x_1+(x_1^2-1)x_2 \\end{pmatrix}. \\]","citations":["zhou2022neural"],"problem_json":"{\"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.\"]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":2,\"problem_name\":\"Region-of-attraction verification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0)$\",\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid0.2^2\\\\leq x_1^2+x_2^2\\\\leq1.2^2\\\\}$\",\"Original verified type: $\\\\epsilon$-ROA\",\"Verified method in the original paper: dReal\",\"Our $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid x_1^2+x_2^2\\\\leq1.2^2\\\\}$\",\"Our verified type: ROA verification\"],\"record_id\":\"DCB-LIT-P2-002\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=(0,0)$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"No separate initial set is required to certify Omega_c itself.\",\"source\":\"exact_roa_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid0.2^2\\\\leq x_1^2+x_2^2\\\\leq1.2^2\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"X=\\\\{x\\\\in\\\\mathbb R^2\\\\mid 0.2\\\\le\\\\lVert x\\\\rVert_2\\\\le1.2\\\\} [artifact] SMT verification annulus in the notebook\",\"note\":\"Audited catalogue wording; not silently treated as an Overleaf statement.\",\"source\":\"catalogue:SYS0034\",\"status\":\"documented\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 2.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 2.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":\"SYS0034\",\"citations\":[\"zhou2022neural\"],\"label\":\"sys:catalogue-sys0034\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":1188,\"start\":1162},\"title_latex\":\"Section~5.1 and Appendix~C.3.1, \\\\cite{zhou2022neural}\"},\"source_order\":23,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0034\",\"status\":\"documented\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} -x_2\\\\\\\\ x_1+(x_1^2-1)x_2 \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} -x_2\\\\\\\\ x_1+(x_1^2-1)x_2 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"exact_roa\"},\"reported_method_latex\":\"Verified method in the original paper: dReal\",\"reported_type_latex\":\"Original verified type: $\\\\epsilon$-ROA\"}}"} {"record_id":"DCB-LIT-P2-003","paper_system_number":24,"task_type":"roa","problem_name":"Region-of-attraction verification","tier":"literature","dimension":1,"time_domain":"continuous","dynamics":"dx/dt = -x + x**3","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x} \\end{pmatrix} = \\begin{pmatrix} -x+x^3 \\end{pmatrix}. \\]","citations":["liu2023towards"],"problem_json":"{\"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.\"]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":2,\"problem_name\":\"Region-of-attraction verification\",\"raw_items_latex\":[\"$x^\\\\star=0$\",\"Original $\\\\mathcal{N}=(-1,1)$\",\"Original verified type: analytical ROA construction\",\"Verified method in the original paper: analytical Zubov construction\",\"Our $\\\\mathcal{N}=(-1,1)$\",\"Our verified type: ROA verification\"],\"record_id\":\"DCB-LIT-P2-003\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=0$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"No separate initial set is required to certify Omega_c itself.\",\"source\":\"exact_roa_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":\"Original $\\\\mathcal{N}=(-1,1)$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"D(0)=(-1,1)\",\"note\":\"Audited catalogue wording; not silently treated as an Overleaf statement.\",\"source\":\"catalogue:SYS0036\",\"status\":\"documented\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 2.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 2.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":\"SYS0036\",\"citations\":[\"liu2023towards\"],\"label\":\"sys:catalogue-sys0036\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":1214,\"start\":1190},\"title_latex\":\"Example~3.1, \\\\cite{liu2023towards}\"},\"source_order\":24,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0036\",\"status\":\"documented\",\"symbolic\":null,\"value\":1},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x} \\\\end{pmatrix} = \\\\begin{pmatrix} -x+x^3 \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x} \\\\end{pmatrix} = \\\\begin{pmatrix} -x+x^3 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"exact_roa\"},\"reported_method_latex\":\"Verified method in the original paper: analytical Zubov construction\",\"reported_type_latex\":\"Original verified type: analytical ROA construction\"}}"} {"record_id":"DCB-LIT-P2-004","paper_system_number":25,"task_type":"roa","problem_name":"Region-of-attraction verification","tier":"literature","dimension":2,"time_domain":"continuous","dynamics":"dx1/dt = x2, dx2/dt = -2*x1 + 1/3*x1**3 - x2","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2 \\end{pmatrix} = \\begin{pmatrix} x_2\\\\ -2x_1+\\tfrac13x_1^3-x_2 \\end{pmatrix}. \\]","citations":["liu2023towards"],"problem_json":"{\"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.\"]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":2,\"problem_name\":\"Region-of-attraction verification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0)$\",\"Original $\\\\mathcal{N}=[-6,6]^2$\",\"Original verified type: ROA verification\",\"Verified method in the original paper: dReal with Jacobian-based local verification\",\"Our $\\\\mathcal{N}=[-6,6]^2$\",\"Our verified type: ROA verification\"],\"record_id\":\"DCB-LIT-P2-004\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=(0,0)$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"No separate initial set is required to certify Omega_c itself.\",\"source\":\"exact_roa_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":\"Original $\\\\mathcal{N}=[-6,6]^2$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"X=[-6,6]^2;\\\\ \\\\Omega_{1.05}=\\\\{x:x^TPx\\\\le1.05\\\\}\",\"note\":\"Audited catalogue wording; not silently treated as an Overleaf statement.\",\"source\":\"catalogue:SYS0047\",\"status\":\"documented\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 2.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 2.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":\"SYS0047\",\"citations\":[\"liu2023towards\"],\"label\":\"sys:catalogue-sys0047\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":1242,\"start\":1216},\"title_latex\":\"Example~6.2 and Table~II, \\\\cite{liu2023towards}\"},\"source_order\":25,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0047\",\"status\":\"documented\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} x_2\\\\\\\\ -2x_1+\\\\tfrac13x_1^3-x_2 \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} x_2\\\\\\\\ -2x_1+\\\\tfrac13x_1^3-x_2 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"exact_roa\"},\"reported_method_latex\":\"Verified method in the original paper: dReal with Jacobian-based local verification\",\"reported_type_latex\":\"Original verified type: ROA verification\"}}"} {"record_id":"DCB-LIT-P2-005","paper_system_number":26,"task_type":"roa","problem_name":"Region-of-attraction verification","tier":"literature","dimension":3,"time_domain":"continuous","dynamics":"dx1/dt = -10*(x1-x2) + u1, dx2/dt = 28*x1 - x2 - x1*x3 + u2, dx3/dt = x1*x2 - 8/3*x3 + u3","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2\\\\ \\dot{x}_3 \\end{pmatrix} = \\begin{pmatrix} -10(x_1-x_2)+u_1\\\\ 28x_1-x_2-x_1x_3+u_2\\\\ x_1x_2-\\dfrac83x_3+u_3 \\end{pmatrix}. \\]","citations":["edwards2024fossil"],"problem_json":"{\"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.\"]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"initial_set_X_I\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":2,\"problem_name\":\"Region-of-attraction verification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0,0)$\",\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{3}\\\\mid0.01^2\\\\leq x_1^2+x_2^2+x_3^2\\\\leq2^2\\\\}$\",\"$\\\\mathcal{X}_I=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{3}\\\\mid x_1^2+x_2^2+x_3^2\\\\leq0.3^2\\\\}$\",\"$u_1,u_2,u_3 \\\\in \\\\mathbb{R}$\\\\\",\"Original verified type: $\\\\epsilon$-ROA\",\"Verified method in the original paper: dReal\",\"Our $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{3}\\\\mid x_1^2+x_2^2+x_3^2\\\\leq2^2\\\\}$\",\"Our verified type: ROA verification\"],\"record_id\":\"DCB-LIT-P2-005\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=(0,0,0)$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_I=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{3}\\\\mid x_1^2+x_2^2+x_3^2\\\\leq0.3^2\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{3}\\\\mid0.01^2\\\\leq x_1^2+x_2^2+x_3^2\\\\leq2^2\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"X_D=\\\\mathbb{B}_{2}\\\\left(\\\\left(0,0,0\\\\right)\\\\right) X_I=\\\\mathbb{B}_{0.3}\\\\left(\\\\left(0,0,0\\\\right)\\\\right)\",\"note\":\"Audited catalogue wording; not silently treated as an Overleaf statement.\",\"source\":\"catalogue:SYS0082\",\"status\":\"documented\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 2.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 2.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":\"SYS0082\",\"citations\":[\"edwards2024fossil\"],\"label\":\"sys:fossil2-f2-03-lorenz-roa\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":1274,\"start\":1244},\"title_latex\":\"Table~4 \\\\#3 (LorenzSystem), \\\\cite{edwards2024fossil}\"},\"source_order\":26,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0082\",\"status\":\"documented\",\"symbolic\":null,\"value\":3},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3 \\\\end{pmatrix} = \\\\begin{pmatrix} -10(x_1-x_2)+u_1\\\\\\\\ 28x_1-x_2-x_1x_3+u_2\\\\\\\\ x_1x_2-\\\\dfrac83x_3+u_3 \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3 \\\\end{pmatrix} = \\\\begin{pmatrix} -10(x_1-x_2)+u_1\\\\\\\\ 28x_1-x_2-x_1x_3+u_2\\\\\\\\ x_1x_2-\\\\dfrac83x_3+u_3 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$u_1,u_2,u_3 \\\\in \\\\mathbb{R}$\\\\\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"initial_set_X_I\",\"neighborhood_N\"],\"requires_initial_set\":true,\"target\":\"exact_roa\"},\"reported_method_latex\":\"Verified method in the original paper: dReal\",\"reported_type_latex\":\"Original verified type: $\\\\epsilon$-ROA\"}}"} {"record_id":"DCB-LIT-P2-006","paper_system_number":27,"task_type":"roa","problem_name":"Region-of-attraction verification","tier":"literature","dimension":2,"time_domain":"continuous","dynamics":"dx1/dt = x2, dx2/dt = -1/2*x2-sin(x1+pi/3)+sin(pi/3)","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2 \\end{pmatrix} = \\begin{pmatrix} x_2\\\\ -0.5x_2-\\sin\\left(x_1+\\dfrac{\\pi}{3}\\right) +\\sin\\left(\\dfrac{\\pi}{3}\\right) \\end{pmatrix}. \\]","citations":["liu2024tool"],"problem_json":"{\"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\"]]}}","executable_record_json":"{\"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}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":2,\"problem_name\":\"Region-of-attraction verification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0)$\",\"Original $\\\\mathcal{N}=[-2,3]\\\\times[-3,1.5]$\",\"Original verified type: ROA verification\",\"Verified method in the original paper: dReal with Jacobian-based local verification\",\"Our $\\\\mathcal{N}=[-2,3]\\\\times[-3,1.5]$\",\"Our verified type: ROA verification\"],\"record_id\":\"DCB-LIT-P2-006\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=(0,0)$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"No separate initial set is required to certify Omega_c itself.\",\"source\":\"exact_roa_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":\"Original $\\\\mathcal{N}=[-2,3]\\\\times[-3,1.5]$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"X=[-2, 3]\\\\times [-3, 1.5]\",\"note\":\"Audited catalogue wording; not silently treated as an Overleaf statement.\",\"source\":\"catalogue:SYS0152\",\"status\":\"documented\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 2.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 2.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":\"SYS0152\",\"citations\":[\"liu2024tool\"],\"label\":\"sys:catalogue-sys0152\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":1303,\"start\":1276},\"title_latex\":\"Example~5.2 and Table~2, \\\\cite{liu2024tool}\"},\"source_order\":27,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0152\",\"status\":\"documented\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} x_2\\\\\\\\ -0.5x_2-\\\\sin\\\\left(x_1+\\\\dfrac{\\\\pi}{3}\\\\right) +\\\\sin\\\\left(\\\\dfrac{\\\\pi}{3}\\\\right) \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} x_2\\\\\\\\ -0.5x_2-\\\\sin\\\\left(x_1+\\\\dfrac{\\\\pi}{3}\\\\right) +\\\\sin\\\\left(\\\\dfrac{\\\\pi}{3}\\\\right) \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"exact_roa\"},\"reported_method_latex\":\"Verified method in the original paper: dReal with Jacobian-based local verification\",\"reported_type_latex\":\"Original verified type: ROA verification\"}}"} {"record_id":"DCB-LIT-P2-007","paper_system_number":28,"task_type":"roa","problem_name":"Region-of-attraction verification","tier":"literature","dimension":10,"time_domain":"continuous","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","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2\\\\ \\dot{x}_3\\\\ \\dot{x}_4\\\\ \\dot{x}_5\\\\ \\dot{x}_6\\\\ \\dot{x}_7\\\\ \\dot{x}_8\\\\ \\dot{x}_9\\\\ \\dot{x}_{10} \\end{pmatrix} = \\begin{pmatrix} -x_1+0.5x_2-0.1x_9^2\\\\ -0.5x_1-x_2\\\\ -x_3+0.5x_4-0.1x_1^2\\\\ -0.5x_3-x_4\\\\ -x_5+0.5x_6+0.1x_7^2\\\\ -0.5x_5-x_6\\\\ -x_7+0.5x_8\\\\ -0.5x_7-x_8\\\\ -x_9+0.5x_{10}\\\\ -0.5x_9-x_{10}+0.1x_2^2 \\end{pmatrix}. \\]","citations":["liu2024tool"],"problem_json":"{\"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.\"]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":2,\"problem_name\":\"Region-of-attraction verification\",\"raw_items_latex\":[\"$x^\\\\star=\\\\mathbf{0}$\",\"Original $\\\\mathcal{N}=[-1,1]^{10}$\",\"Original verified type: ROA verification\",\"Verified method in the original paper: dReal with Jacobian-based local verification\",\"Our $\\\\mathcal{N}=[-1,1]^{10}$\",\"Our verified type: ROA verification\"],\"record_id\":\"DCB-LIT-P2-007\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=\\\\mathbf{0}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"No separate initial set is required to certify Omega_c itself.\",\"source\":\"exact_roa_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":\"Original $\\\\mathcal{N}=[-1,1]^{10}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"X=[-1, 1]^{10}\",\"note\":\"Audited catalogue wording; not silently treated as an Overleaf statement.\",\"source\":\"catalogue:SYS0154\",\"status\":\"documented\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 2.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 2.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":\"SYS0154\",\"citations\":[\"liu2024tool\"],\"label\":\"sys:catalogue-sys0154\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":1339,\"start\":1305},\"title_latex\":\"Example~5.4 and Table~3, \\\\cite{liu2024tool}\"},\"source_order\":28,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0154\",\"status\":\"documented\",\"symbolic\":null,\"value\":10},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3\\\\\\\\ \\\\dot{x}_4\\\\\\\\ \\\\dot{x}_5\\\\\\\\ \\\\dot{x}_6\\\\\\\\ \\\\dot{x}_7\\\\\\\\ \\\\dot{x}_8\\\\\\\\ \\\\dot{x}_9\\\\\\\\ \\\\dot{x}_{10} \\\\end{pmatrix} = \\\\begin{pmatrix} -x_1+0.5x_2-0.1x_9^2\\\\\\\\ -0.5x_1-x_2\\\\\\\\ -x_3+0.5x_4-0.1x_1^2\\\\\\\\ -0.5x_3-x_4\\\\\\\\ -x_5+0.5x_6+0.1x_7^2\\\\\\\\ -0.5x_5-x_6\\\\\\\\ -x_7+0.5x_8\\\\\\\\ -0.5x_7-x_8\\\\\\\\ -x_9+0.5x_{10}\\\\\\\\ -0.5x_9-x_{10}+0.1x_2^2 \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3\\\\\\\\ \\\\dot{x}_4\\\\\\\\ \\\\dot{x}_5\\\\\\\\ \\\\dot{x}_6\\\\\\\\ \\\\dot{x}_7\\\\\\\\ \\\\dot{x}_8\\\\\\\\ \\\\dot{x}_9\\\\\\\\ \\\\dot{x}_{10} \\\\end{pmatrix} = \\\\begin{pmatrix} -x_1+0.5x_2-0.1x_9^2\\\\\\\\ -0.5x_1-x_2\\\\\\\\ -x_3+0.5x_4-0.1x_1^2\\\\\\\\ -0.5x_3-x_4\\\\\\\\ -x_5+0.5x_6+0.1x_7^2\\\\\\\\ -0.5x_5-x_6\\\\\\\\ -x_7+0.5x_8\\\\\\\\ -0.5x_7-x_8\\\\\\\\ -x_9+0.5x_{10}\\\\\\\\ -0.5x_9-x_{10}+0.1x_2^2 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"exact_roa\"},\"reported_method_latex\":\"Verified method in the original paper: dReal with Jacobian-based local verification\",\"reported_type_latex\":\"Original verified type: ROA verification\"}}"} {"record_id":"DCB-LIT-P2-008","paper_system_number":29,"task_type":"roa","problem_name":"Region-of-attraction verification","tier":"literature","dimension":1,"time_domain":"continuous","dynamics":"dx/dt = -x - x**3","specification_latex":"The closed-loop scalar dynamics are: \\[ \\dot{x}=-x-x^3, \\]","citations":["meng2024physics"],"problem_json":"{\"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.\"]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":2,\"problem_name\":\"Region-of-attraction verification\",\"raw_items_latex\":[\"Equilibrium: $x^\\\\star=0$\",\"Original $ \\\\mathcal{N} =\\\\{x\\\\in\\\\mathbb{R}\\\\mid 0.1\\\\le |x|\\\\le1\\\\}. $\",\"Original verified type: $\\\\epsilon$-ROA\",\"Verified method in the original paper: dReal\",\"Our $ \\\\mathcal{N} =\\\\{x\\\\in\\\\mathbb{R}\\\\mid |x|\\\\leq1\\\\}. $\",\"Our verified type: ROA verification\"],\"record_id\":\"DCB-LIT-P2-008\",\"sets\":{\"equilibrium\":{\"latex\":\"Equilibrium: $x^\\\\star=0$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"No separate initial set is required to certify Omega_c itself.\",\"source\":\"exact_roa_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":\"Original $ \\\\mathcal{N} =\\\\{x\\\\in\\\\mathbb{R}\\\\mid 0.1\\\\le |x|\\\\le1\\\\}. $\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"\\\\Omega=[-1,1];\\\\quad U_{\\\\varepsilon}=\\\\{x:\\\\lVert x\\\\rVert<0.1\\\\};\\\\quad c_1=0.01,\\\\ c_2=1.99,\\\\ \\\\mu=10^{-4}\",\"note\":\"Audited catalogue wording; not silently treated as an Overleaf statement.\",\"source\":\"catalogue:SYS0254\",\"status\":\"documented\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 2.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 2.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":\"SYS0254\",\"citations\":[\"meng2024physics\"],\"label\":\"sys:catalogue-sys0254\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":1366,\"start\":1341},\"title_latex\":\"Section~5.1, with verification details in Appendix~E.2 (\\\\(n=1\\\\)), \\\\cite{meng2024physics}\"},\"source_order\":29,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0254\",\"status\":\"documented\",\"symbolic\":null,\"value\":1},\"display_math_latex\":[\"\\\\dot{x}=-x-x^3,\"],\"specification_latex\":\"The closed-loop scalar dynamics are: \\\\[ \\\\dot{x}=-x-x^3, \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"exact_roa\"},\"reported_method_latex\":\"Verified method in the original paper: dReal\",\"reported_type_latex\":\"Original verified type: $\\\\epsilon$-ROA\"}}"} {"record_id":"DCB-LIT-P3-001","paper_system_number":30,"task_type":"invariant","problem_name":"Forward-invariant set synthesis","tier":"literature","dimension":6,"time_domain":"discrete","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))","specification_latex":"Let the state be ordered as \\[ \\mathbf{x} = (p_x,p_z,\\phi,v_x,v_z,\\omega)^\\top \\in \\mathbb{R}^{6}, \\] where $(p_x,p_z)$ are the planar position, $\\phi$ is the roll angle, $(v_x,v_z)$ are the body-frame velocities, and $\\omega$ is the angular rate. Using the sampling period $\\Delta t=0.05$, the discrete-time transition map is: \\[ \\mathbf{x}^{+} = F(\\mathbf{x},\\mathbf{u}) = \\begin{pmatrix} p_x + 0.05\\left(v_x\\cos\\phi-v_z\\sin\\phi\\right) \\\\[1ex] p_z + 0.05\\left(v_x\\sin\\phi+v_z\\cos\\phi\\right) \\\\[1ex] \\phi + 0.05\\,\\omega \\\\[1ex] v_x + 0.05\\left(v_z\\omega-9.8\\sin\\phi\\right) \\\\[1ex] v_z + 0.05\\left( -v_x\\omega-9.8\\cos\\phi+\\dfrac{u_1+u_2}{4} \\right) \\\\[1ex] \\omega + 0.05\\left( \\dfrac{0.25}{0.0475}(u_1-u_2) \\right) \\end{pmatrix}, \\] where $\\mathbf{u}=(u_1,u_2)^\\top\\in[0,39.2]^2.$","citations":["wu2023neural"],"problem_json":"{\"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\"}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":3,\"problem_name\":\"Forward-invariant set synthesis\",\"raw_items_latex\":[\"$ \\\\mathcal{X} = \\\\left\\\\{ \\\\mathbf{x}\\\\in\\\\mathbb{R}^{6} \\\\mid \\\\lVert\\\\mathbf{x}\\\\rVert_\\\\infty\\\\leq 1 \\\\right\\\\}. $\",\"Unsafe region: $ \\\\mathcal{X}_U = \\\\left\\\\{ \\\\mathbf{x}\\\\in\\\\mathbb{R}^{6} \\\\mid \\\\lVert\\\\mathbf{x}\\\\rVert_\\\\infty>1 \\\\right\\\\}. $\",\"Verification target: forward-invariance verification\"],\"record_id\":\"DCB-LIT-P3-001\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$ \\\\mathcal{X} = \\\\left\\\\{ \\\\mathbf{x}\\\\in\\\\mathbb{R}^{6} \\\\mid \\\\lVert\\\\mathbf{x}\\\\rVert_\\\\infty\\\\leq 1 \\\\right\\\\}. $\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"S is the requested nonempty forward-invariant output, not an input set.\",\"source\":\"problem_contract\",\"status\":\"output_to_synthesize\"},\"unsafe_set_X_U\":{\"latex\":\"Unsafe region: $ \\\\mathcal{X}_U = \\\\left\\\\{ \\\\mathbf{x}\\\\in\\\\mathbb{R}^{6} \\\\mid \\\\lVert\\\\mathbf{x}\\\\rVert_\\\\infty>1 \\\\right\\\\}. $\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":null,\"citations\":[\"wu2023neural\"],\"label\":\"sys:dt-pvtol6-invariance\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":1554,\"start\":1491},\"title_latex\":\"Appendix~A.1 (PVTOL), \\\\cite{wu2023neural}\"},\"source_order\":30,\"system\":{\"dimension\":{\"source\":\"overleaf_state_space_notation\",\"status\":\"inferred\",\"symbolic\":null,\"value\":6},\"display_math_latex\":[\"\\\\mathbf{x} = (p_x,p_z,\\\\phi,v_x,v_z,\\\\omega)^\\\\top \\\\in \\\\mathbb{R}^{6},\",\"\\\\mathbf{x}^{+} = F(\\\\mathbf{x},\\\\mathbf{u}) = \\\\begin{pmatrix} p_x + 0.05\\\\left(v_x\\\\cos\\\\phi-v_z\\\\sin\\\\phi\\\\right) \\\\\\\\[1ex] p_z + 0.05\\\\left(v_x\\\\sin\\\\phi+v_z\\\\cos\\\\phi\\\\right) \\\\\\\\[1ex] \\\\phi + 0.05\\\\,\\\\omega \\\\\\\\[1ex] v_x + 0.05\\\\left(v_z\\\\omega-9.8\\\\sin\\\\phi\\\\right) \\\\\\\\[1ex] v_z + 0.05\\\\left( -v_x\\\\omega-9.8\\\\cos\\\\phi+\\\\dfrac{u_1+u_2}{4} \\\\right) \\\\\\\\[1ex] \\\\omega + 0.05\\\\left( \\\\dfrac{0.25}{0.0475}(u_1-u_2) \\\\right) \\\\end{pmatrix},\"],\"specification_latex\":\"Let the state be ordered as \\\\[ \\\\mathbf{x} = (p_x,p_z,\\\\phi,v_x,v_z,\\\\omega)^\\\\top \\\\in \\\\mathbb{R}^{6}, \\\\] where $(p_x,p_z)$ are the planar position, $\\\\phi$ is the roll angle, $(v_x,v_z)$ are the body-frame velocities, and $\\\\omega$ is the angular rate. Using the sampling period $\\\\Delta t=0.05$, the discrete-time transition map is: \\\\[ \\\\mathbf{x}^{+} = F(\\\\mathbf{x},\\\\mathbf{u}) = \\\\begin{pmatrix} p_x + 0.05\\\\left(v_x\\\\cos\\\\phi-v_z\\\\sin\\\\phi\\\\right) \\\\\\\\[1ex] p_z + 0.05\\\\left(v_x\\\\sin\\\\phi+v_z\\\\cos\\\\phi\\\\right) \\\\\\\\[1ex] \\\\phi + 0.05\\\\,\\\\omega \\\\\\\\[1ex] v_x + 0.05\\\\left(v_z\\\\omega-9.8\\\\sin\\\\phi\\\\right) \\\\\\\\[1ex] v_z + 0.05\\\\left( -v_x\\\\omega-9.8\\\\cos\\\\phi+\\\\dfrac{u_1+u_2}{4} \\\\right) \\\\\\\\[1ex] \\\\omega + 0.05\\\\left( \\\\dfrac{0.25}{0.0475}(u_1-u_2) \\\\right) \\\\end{pmatrix}, \\\\] where $\\\\mathbf{u}=(u_1,u_2)^\\\\top\\\\in[0,39.2]^2.$\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"discrete\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"Unsafe region: $ \\\\mathcal{X}_U = \\\\left\\\\{ \\\\mathbf{x}\\\\in\\\\mathbb{R}^{6} \\\\mid \\\\lVert\\\\mathbf{x}\\\\rVert_\\\\infty>1 \\\\right\\\\}. $\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"unsafe_set_X_U\"],\"requires_initial_set\":false,\"target\":\"Forward-invariant set synthesis\"},\"reported_method_latex\":null,\"reported_type_latex\":null}}"} {"record_id":"DCB-LIT-P3-002","paper_system_number":31,"task_type":"invariant","problem_name":"Forward-invariant set synthesis","tier":"literature","dimension":2,"time_domain":"continuous","dynamics":"dx1/dt = x2, dx2/dt = sin(x1)-3*x1-3*x2","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2 \\end{pmatrix} = \\begin{pmatrix} x_2\\\\[1ex] \\sin(x_1) + u \\end{pmatrix}, \\] where $u = -3x_1 - 3x_2$.","citations":["samanipour2026relu"],"problem_json":"{\"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\"}","executable_record_json":"{\"record_id\":\"DCB-LIT-P3-002\",\"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,\"paper_system_number\":31,\"paper_system_label\":\"System 31\",\"local_name\":\"S031__DCB-LIT-P3-002\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":3,\"problem_name\":\"Forward-invariant set synthesis\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid \\\\lVert x\\\\rVert_\\\\infty \\\\le \\\\pi\\\\}$\",\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid \\\\lVert x\\\\rVert_\\\\infty > \\\\pi\\\\}$\",\"Verified type: forward-invariance verification\",\"Verified method in the original paper: linear programming and facet-wise verification\"],\"record_id\":\"DCB-LIT-P3-002\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid \\\\lVert x\\\\rVert_\\\\infty \\\\le \\\\pi\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"S is the requested nonempty forward-invariant output, not an input set.\",\"source\":\"problem_contract\",\"status\":\"output_to_synthesize\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid \\\\lVert x\\\\rVert_\\\\infty > \\\\pi\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"OVL-EXT-008\",\"citations\":[\"samanipour2026relu\"],\"label\":\"sys:inverted-pendulum-saturation\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":1582,\"start\":1556},\"title_latex\":\"Example~1, \\\\cite{samanipour2026relu}\"},\"source_order\":31,\"system\":{\"dimension\":{\"source\":\"catalogue:OVL-EXT-008\",\"status\":\"documented\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} x_2\\\\\\\\[1ex] \\\\sin(x_1) + u \\\\end{pmatrix},\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} x_2\\\\\\\\[1ex] \\\\sin(x_1) + u \\\\end{pmatrix}, \\\\] where $u = -3x_1 - 3x_2$.\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid \\\\lVert x\\\\rVert_\\\\infty > \\\\pi\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"unsafe_set_X_U\"],\"requires_initial_set\":false,\"target\":\"Forward-invariant set synthesis\"},\"reported_method_latex\":\"Verified method in the original paper: linear programming and facet-wise verification\",\"reported_type_latex\":\"Verified type: forward-invariance verification\"}}"} {"record_id":"DCB-LIT-P3-003","paper_system_number":32,"task_type":"invariant","problem_name":"Forward-invariant set synthesis","tier":"literature","dimension":2,"time_domain":"continuous","dynamics":"dx1/dt = x1**2+x1*x2+x1, dx2/dt = x1*x2+x2**2+x2","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_{1}\\\\ \\dot{x}_{2} \\end{pmatrix} = \\begin{pmatrix} x_{1}^2 + x_{1}x_{2} + x_{1}\\\\[1ex] x_{1}x_{2} + x_{2}^2 + x_{2} \\end{pmatrix}. \\]","citations":["samanipour2026relu"],"problem_json":"{\"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\"}","executable_record_json":"{\"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,\"paper_system_number\":32,\"paper_system_label\":\"System 32\",\"local_name\":\"S032__DCB-LIT-P3-003\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":3,\"problem_name\":\"Forward-invariant set synthesis\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid \\\\lVert x\\\\rVert_\\\\infty \\\\le 2\\\\}$\",\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid -1 \\\\le x_1, x_2 \\\\le 0\\\\}$\",\"Verified type: forward-invariance verification\",\"Verified method in the original paper: linear programming and facet-wise verification\"],\"record_id\":\"DCB-LIT-P3-003\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid \\\\lVert x\\\\rVert_\\\\infty \\\\le 2\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"S is the requested nonempty forward-invariant output, not an input set.\",\"source\":\"problem_contract\",\"status\":\"output_to_synthesize\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid -1 \\\\le x_1, x_2 \\\\le 0\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"OVL-EXT-014\",\"citations\":[\"samanipour2026relu\"],\"label\":\"sys:2d-quadratic-system\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":1610,\"start\":1584},\"title_latex\":\"Example~2, \\\\cite{samanipour2026relu}\"},\"source_order\":32,\"system\":{\"dimension\":{\"source\":\"catalogue:OVL-EXT-014\",\"status\":\"documented\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{1}^2 + x_{1}x_{2} + x_{1}\\\\\\\\[1ex] x_{1}x_{2} + x_{2}^2 + x_{2} \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{1}^2 + x_{1}x_{2} + x_{1}\\\\\\\\[1ex] x_{1}x_{2} + x_{2}^2 + x_{2} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid -1 \\\\le x_1, x_2 \\\\le 0\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"unsafe_set_X_U\"],\"requires_initial_set\":false,\"target\":\"Forward-invariant set synthesis\"},\"reported_method_latex\":\"Verified method in the original paper: linear programming and facet-wise verification\",\"reported_type_latex\":\"Verified type: forward-invariance verification\"}}"} {"record_id":"DCB-LIT-P3-004","paper_system_number":33,"task_type":"invariant","problem_name":"Forward-invariant set synthesis","tier":"literature","dimension":4,"time_domain":"continuous","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)","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2\\\\ \\dot{x}_3\\\\ \\dot{x}_4 \\end{pmatrix} = \\begin{pmatrix} x_2\\\\[1ex] \\dfrac{u + 0.1\\,x_4^2 \\sin x_3 - 0.981 \\sin x_3 \\cos x_3}{1 + 0.1\\sin^2 x_3}\\\\[2ex] x_4\\\\[1ex] \\dfrac{10.791 \\sin x_3 - u \\cos x_3 - 0.1\\,x_4^2 \\sin x_3 \\cos x_3}{1 + 0.1\\sin^2 x_3} \\end{pmatrix}, \\] where $u = x_1 + 2.4109\\,x_2 + 34.3620\\,x_3 + 10.7009\\,x_4$.","citations":["samanipour2026relu"],"problem_json":"{\"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},\"analysis_bounds\":[[\"-1\",\"1\"],[\"-1\",\"1\"],[\"-1\",\"1\"],[\"-1\",\"1\"]],\"safety_interpretation\":\"outside_X_is_unsafe\"}","executable_record_json":"{\"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},\"analysis_bounds\":[[\"-1\",\"1\"],[\"-1\",\"1\"],[\"-1\",\"1\"],[\"-1\",\"1\"]],\"safety_interpretation\":\"outside_X_is_unsafe\"},\"candidate\":null,\"paper_system_number\":33,\"paper_system_label\":\"System 33\",\"local_name\":\"S033__DCB-LIT-P3-004\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":3,\"problem_name\":\"Forward-invariant set synthesis\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_\\\\infty \\\\le 1\\\\}$\",\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_\\\\infty > 1\\\\}$\",\"Verified type: forward-invariance verification\",\"Verified method in the original paper: linear programming and facet-wise verification\"],\"record_id\":\"DCB-LIT-P3-004\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_\\\\infty \\\\le 1\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"S is the requested nonempty forward-invariant output, not an input set.\",\"source\":\"problem_contract\",\"status\":\"output_to_synthesize\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_\\\\infty > 1\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"OVL-EXT-009\",\"citations\":[\"samanipour2026relu\"],\"label\":\"sys:cartpole-saturation\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":1642,\"start\":1612},\"title_latex\":\"Example~3, \\\\cite{samanipour2026relu}\"},\"source_order\":33,\"system\":{\"dimension\":{\"source\":\"catalogue:OVL-EXT-009\",\"status\":\"documented\",\"symbolic\":null,\"value\":4},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3\\\\\\\\ \\\\dot{x}_4 \\\\end{pmatrix} = \\\\begin{pmatrix} x_2\\\\\\\\[1ex] \\\\dfrac{u + 0.1\\\\,x_4^2 \\\\sin x_3 - 0.981 \\\\sin x_3 \\\\cos x_3}{1 + 0.1\\\\sin^2 x_3}\\\\\\\\[2ex] x_4\\\\\\\\[1ex] \\\\dfrac{10.791 \\\\sin x_3 - u \\\\cos x_3 - 0.1\\\\,x_4^2 \\\\sin x_3 \\\\cos x_3}{1 + 0.1\\\\sin^2 x_3} \\\\end{pmatrix},\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3\\\\\\\\ \\\\dot{x}_4 \\\\end{pmatrix} = \\\\begin{pmatrix} x_2\\\\\\\\[1ex] \\\\dfrac{u + 0.1\\\\,x_4^2 \\\\sin x_3 - 0.981 \\\\sin x_3 \\\\cos x_3}{1 + 0.1\\\\sin^2 x_3}\\\\\\\\[2ex] x_4\\\\\\\\[1ex] \\\\dfrac{10.791 \\\\sin x_3 - u \\\\cos x_3 - 0.1\\\\,x_4^2 \\\\sin x_3 \\\\cos x_3}{1 + 0.1\\\\sin^2 x_3} \\\\end{pmatrix}, \\\\] where $u = x_1 + 2.4109\\\\,x_2 + 34.3620\\\\,x_3 + 10.7009\\\\,x_4$.\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_\\\\infty > 1\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"unsafe_set_X_U\"],\"requires_initial_set\":false,\"target\":\"Forward-invariant set synthesis\"},\"reported_method_latex\":\"Verified method in the original paper: linear programming and facet-wise verification\",\"reported_type_latex\":\"Verified type: forward-invariance verification\"}}"} {"record_id":"DCB-LIT-P3-005","paper_system_number":34,"task_type":"invariant","problem_name":"Forward-invariant set synthesis","tier":"literature","dimension":2,"time_domain":"discrete","dynamics":"x1[k+1] = x1+0.01*x2, x2[k+1] = x2+0.01*(9.8*sin(x1)+u)","specification_latex":"The discrete-time transition map is: \\[ \\mathbf{x}^{+} = F(\\mathbf{x}) = \\begin{pmatrix} x_1 + 0.01\\,x_2\\\\[1ex] x_2 + 0.01(9.8\\sin x_1 + u) \\end{pmatrix}, \\] where $u \\in [-10, 10]$.","citations":["vaisi2025extracting"],"problem_json":"{\"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]]}","executable_record_json":"{\"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,\"paper_system_number\":34,\"paper_system_label\":\"System 34\",\"local_name\":\"S034__DCB-LIT-P3-005\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":3,\"problem_name\":\"Forward-invariant set synthesis\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid \\\\lVert x\\\\rVert_\\\\infty \\\\le \\\\pi/4\\\\}$\",\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid \\\\lVert x\\\\rVert_\\\\infty > \\\\pi/4\\\\}$\",\"Verified type: forward-invariance verification\",\"Verified method in the original paper: neural-network reachability and hyperplane-region enumeration\"],\"record_id\":\"DCB-LIT-P3-005\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid \\\\lVert x\\\\rVert_\\\\infty \\\\le \\\\pi/4\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"S is the requested nonempty forward-invariant output, not an input set.\",\"source\":\"problem_contract\",\"status\":\"output_to_synthesize\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid \\\\lVert x\\\\rVert_\\\\infty > \\\\pi/4\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"OVL-EXT-011\",\"citations\":[\"vaisi2025extracting\"],\"label\":\"sys:dt-inverted-pendulum\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":1704,\"start\":1680},\"title_latex\":\"Section~6.1.1, \\\\cite{vaisi2025extracting}\"},\"source_order\":34,\"system\":{\"dimension\":{\"source\":\"catalogue:OVL-EXT-011\",\"status\":\"documented\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\mathbf{x}^{+} = F(\\\\mathbf{x}) = \\\\begin{pmatrix} x_1 + 0.01\\\\,x_2\\\\\\\\[1ex] x_2 + 0.01(9.8\\\\sin x_1 + u) \\\\end{pmatrix},\"],\"specification_latex\":\"The discrete-time transition map is: \\\\[ \\\\mathbf{x}^{+} = F(\\\\mathbf{x}) = \\\\begin{pmatrix} x_1 + 0.01\\\\,x_2\\\\\\\\[1ex] x_2 + 0.01(9.8\\\\sin x_1 + u) \\\\end{pmatrix}, \\\\] where $u \\\\in [-10, 10]$.\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"discrete\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid \\\\lVert x\\\\rVert_\\\\infty > \\\\pi/4\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"unsafe_set_X_U\"],\"requires_initial_set\":false,\"target\":\"Forward-invariant set synthesis\"},\"reported_method_latex\":\"Verified method in the original paper: neural-network reachability and hyperplane-region enumeration\",\"reported_type_latex\":\"Verified type: forward-invariance verification\"}}"} {"record_id":"DCB-LIT-P3-006","paper_system_number":35,"task_type":"invariant","problem_name":"Forward-invariant set synthesis","tier":"literature","dimension":3,"time_domain":"discrete","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","specification_latex":"The discrete-time transition map is: \\[ \\mathbf{x}^{+} = F(\\mathbf{x}) = \\begin{pmatrix} x_2\\\\[1ex] 29.4 \\sin x_1 + 300 \\cos x_1 \\tan x_3 + 15 u \\cos x_1 \\cos^2 x_3\\\\[1ex] u \\end{pmatrix}, \\] where $u \\in [-10, 10]$.","citations":["vaisi2025extracting"],"problem_json":"{\"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]]}","executable_record_json":"{\"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,\"paper_system_number\":35,\"paper_system_label\":\"System 35\",\"local_name\":\"S035__DCB-LIT-P3-006\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":3,\"problem_name\":\"Forward-invariant set synthesis\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_\\\\infty \\\\le 2.2\\\\}$\",\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_\\\\infty > 2.2\\\\}$\",\"Verified type: forward-invariance verification\",\"Verified method in the original paper: neural-network reachability and hyperplane-region enumeration\"],\"record_id\":\"DCB-LIT-P3-006\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_\\\\infty \\\\le 2.2\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"S is the requested nonempty forward-invariant output, not an input set.\",\"source\":\"problem_contract\",\"status\":\"output_to_synthesize\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_\\\\infty > 2.2\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"OVL-EXT-012\",\"citations\":[\"vaisi2025extracting\"],\"label\":\"sys:dt-steerable-bicycle\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":1731,\"start\":1706},\"title_latex\":\"Section~6.1.2, \\\\cite{vaisi2025extracting}\"},\"source_order\":35,\"system\":{\"dimension\":{\"source\":\"catalogue:OVL-EXT-012\",\"status\":\"documented\",\"symbolic\":null,\"value\":3},\"display_math_latex\":[\"\\\\mathbf{x}^{+} = F(\\\\mathbf{x}) = \\\\begin{pmatrix} x_2\\\\\\\\[1ex] 29.4 \\\\sin x_1 + 300 \\\\cos x_1 \\\\tan x_3 + 15 u \\\\cos x_1 \\\\cos^2 x_3\\\\\\\\[1ex] u \\\\end{pmatrix},\"],\"specification_latex\":\"The discrete-time transition map is: \\\\[ \\\\mathbf{x}^{+} = F(\\\\mathbf{x}) = \\\\begin{pmatrix} x_2\\\\\\\\[1ex] 29.4 \\\\sin x_1 + 300 \\\\cos x_1 \\\\tan x_3 + 15 u \\\\cos x_1 \\\\cos^2 x_3\\\\\\\\[1ex] u \\\\end{pmatrix}, \\\\] where $u \\\\in [-10, 10]$.\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"discrete\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_\\\\infty > 2.2\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"unsafe_set_X_U\"],\"requires_initial_set\":false,\"target\":\"Forward-invariant set synthesis\"},\"reported_method_latex\":\"Verified method in the original paper: neural-network reachability and hyperplane-region enumeration\",\"reported_type_latex\":\"Verified type: forward-invariance verification\"}}"} {"record_id":"DCB-LIT-P4-001","paper_system_number":36,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":2,"time_domain":"continuous","dynamics":"dx1/dt = x2, dx2/dt = x1","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2 \\end{pmatrix} = \\begin{pmatrix} x_2\\\\ x_1 \\end{pmatrix}. \\]","citations":["sogokon2018vector"],"problem_json":"{\"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\"}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X}=\\\\mathbb{R}^{2}$\",\"$\\\\mathcal{X}_I= \\\\{(x_1,x_2)\\\\mid x_1\\\\leq0\\\\wedge x_2\\\\leq0\\\\}$\",\"$\\\\mathcal{X}_U=\\\\{(x_1,x_2)\\\\mid x_1>0\\\\}$\",\"Verified type: barrier/safety verification\",\"Verified method in the original paper: vector barrier certificates\"],\"record_id\":\"DCB-LIT-P4-001\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_I= \\\\{(x_1,x_2)\\\\mid x_1\\\\leq0\\\\wedge x_2\\\\leq0\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X}=\\\\mathbb{R}^{2}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U=\\\\{(x_1,x_2)\\\\mid x_1>0\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"SYS0009\",\"citations\":[\"sogokon2018vector\"],\"label\":\"sys:catalogue-sys0009\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":1764,\"start\":1739},\"title_latex\":\"Theorem~4, \\\\cite{sogokon2018vector}\"},\"source_order\":36,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0009\",\"status\":\"documented\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} x_2\\\\\\\\ x_1 \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} x_2\\\\\\\\ x_1 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: vector barrier certificates\",\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-002","paper_system_number":37,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":3,"time_domain":"continuous","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","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2\\\\ \\dot{x}_3 \\end{pmatrix} = \\begin{pmatrix} 2x_1+x_2+2x_3-x_1^2+x_1x_3-x_3^2\\\\ -2+x_1-x_2-x_2^2\\\\ -2-x_2-x_3+x_1^2-x_1x_3 \\end{pmatrix}. \\]","citations":["sogokon2018vector"],"problem_json":"{\"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\"}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X}=\\\\mathbb{R}^{3}$\",\"$\\\\mathcal{X}_I= \\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{3}\\\\mid-x_i\\\\geq0,\\\\ i=1,2,3\\\\}$\",\"$\\\\mathcal{X}_U= \\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{3}\\\\mid x_1+x_2+x_3\\\\geq1\\\\}$\",\"Verified type: barrier/safety verification\",\"Verified method in the original paper: vector barrier certificates and SOS programming\"],\"record_id\":\"DCB-LIT-P4-002\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_I= \\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{3}\\\\mid-x_i\\\\geq0,\\\\ i=1,2,3\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X}=\\\\mathbb{R}^{3}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U= \\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{3}\\\\mid x_1+x_2+x_3\\\\geq1\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"SYS0010\",\"citations\":[\"sogokon2018vector\"],\"label\":\"sys:catalogue-sys0010\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":1793,\"start\":1766},\"title_latex\":\"Example~1, \\\\cite{sogokon2018vector}\"},\"source_order\":37,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0010\",\"status\":\"documented\",\"symbolic\":null,\"value\":3},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3 \\\\end{pmatrix} = \\\\begin{pmatrix} 2x_1+x_2+2x_3-x_1^2+x_1x_3-x_3^2\\\\\\\\ -2+x_1-x_2-x_2^2\\\\\\\\ -2-x_2-x_3+x_1^2-x_1x_3 \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3 \\\\end{pmatrix} = \\\\begin{pmatrix} 2x_1+x_2+2x_3-x_1^2+x_1x_3-x_3^2\\\\\\\\ -2+x_1-x_2-x_2^2\\\\\\\\ -2-x_2-x_3+x_1^2-x_1x_3 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: vector barrier certificates and SOS programming\",\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-003","paper_system_number":38,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":2,"time_domain":"continuous","dynamics":"dx1/dt = x2, dx2/dt = -x1+((1)/(3))*x1**3-x2","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2 \\end{pmatrix} = \\begin{pmatrix} x_2\\\\ -x_1+\\tfrac13x_1^3-x_2 \\end{pmatrix}. \\]","citations":["zhao2020synthesizing"],"problem_json":"{\"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\"]]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X}= \\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid-3.5\\\\leq x_1\\\\leq2,\\\\ -2\\\\leq x_2\\\\leq1\\\\}$\",\"$\\\\mathcal{X}_I= \\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid(x_1-1.5)^2+x_2^2\\\\leq0.25\\\\}$\",\"$\\\\mathcal{X}_U= \\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid(x_1+1)^2+(x_2+1)^2\\\\leq0.16\\\\}$\",\"Verified type: barrier/safety verification\",\"Verified method in the original paper: iSAT3\"],\"record_id\":\"DCB-LIT-P4-003\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_I= \\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid(x_1-1.5)^2+x_2^2\\\\leq0.25\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X}= \\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid-3.5\\\\leq x_1\\\\leq2,\\\\ -2\\\\leq x_2\\\\leq1\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U= \\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid(x_1+1)^2+(x_2+1)^2\\\\leq0.16\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"SYS0011\",\"citations\":[\"zhao2020synthesizing\"],\"label\":\"sys:catalogue-sys0011\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":1820,\"start\":1795},\"title_latex\":\"Example~3.1, \\\\cite{zhao2020synthesizing}\"},\"source_order\":38,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0011\",\"status\":\"documented\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} x_2\\\\\\\\ -x_1+\\\\tfrac13x_1^3-x_2 \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} x_2\\\\\\\\ -x_1+\\\\tfrac13x_1^3-x_2 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: iSAT3\",\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-004","paper_system_number":39,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":2,"time_domain":"continuous","dynamics":"dx1/dt = x2+2*x1*x2, dx2/dt = -x1-x2**2+2*x1**2","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2 \\end{pmatrix} = \\begin{pmatrix} x_2+2x_1x_2\\\\ -x_1-x_2^2+2x_1^2 \\end{pmatrix}. \\]","citations":["abate2021fossil"],"problem_json":"{\"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\"]]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X}=[-2,2]^2$\",\"$\\\\mathcal{X}_I=[0,1]\\\\times[1,2]$\",\"$\\\\mathcal{X}_U= \\\\{(x_1,x_2)\\\\in\\\\mathbb{R}^{2}\\\\mid x_1+x_2^2\\\\leq0\\\\}$\",\"Verified type: barrier/safety verification\",\"Verified method in the original paper: dReal\"],\"record_id\":\"DCB-LIT-P4-004\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_I=[0,1]\\\\times[1,2]$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X}=[-2,2]^2$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U= \\\\{(x_1,x_2)\\\\in\\\\mathbb{R}^{2}\\\\mid x_1+x_2^2\\\\leq0\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"SYS0024\",\"citations\":[\"abate2021fossil\"],\"label\":\"sys:catalogue-sys0024\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":1847,\"start\":1822},\"title_latex\":\"Table~1 row~9 (Barr1), \\\\cite{abate2021fossil}\"},\"source_order\":39,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0024\",\"status\":\"documented\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} x_2+2x_1x_2\\\\\\\\ -x_1-x_2^2+2x_1^2 \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} x_2+2x_1x_2\\\\\\\\ -x_1-x_2^2+2x_1^2 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: dReal\",\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-005","paper_system_number":40,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":3,"time_domain":"continuous","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))","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2\\\\ \\dot{\\phi} \\end{pmatrix} = \\begin{pmatrix} \\sin(\\phi)\\\\ \\cos(\\phi)\\\\ -\\sin(\\phi)+ \\dfrac{3(x_1\\sin(\\phi)+x_2\\cos(\\phi))} {0.5+x_1^2+x_2^2} \\end{pmatrix}. \\]","citations":["abate2021fossil"],"problem_json":"{\"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\"]]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X}=[-2,2]\\\\times[-2,2]\\\\times[-1.57,1.57]$\",\"$\\\\mathcal{X}_I=[-0.1,0.1]\\\\times[-2,-1.8]\\\\times[-0.52,0.52]$\",\"$\\\\mathcal{X}_U= \\\\{(x_1,x_2,\\\\phi)\\\\in\\\\mathbb{R}^{3}\\\\mid x_1^2+x_2^2\\\\leq0.04\\\\}$\",\"Verified type: barrier/safety verification\",\"Verified method in the original paper: dReal\"],\"record_id\":\"DCB-LIT-P4-005\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_I=[-0.1,0.1]\\\\times[-2,-1.8]\\\\times[-0.52,0.52]$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X}=[-2,2]\\\\times[-2,2]\\\\times[-1.57,1.57]$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U= \\\\{(x_1,x_2,\\\\phi)\\\\in\\\\mathbb{R}^{3}\\\\mid x_1^2+x_2^2\\\\leq0.04\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"SYS0027\",\"citations\":[\"abate2021fossil\"],\"label\":\"sys:catalogue-sys0027\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":1878,\"start\":1849},\"title_latex\":\"Table~1 row~12 (Barr4), \\\\cite{abate2021fossil}\"},\"source_order\":40,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0027\",\"status\":\"documented\",\"symbolic\":null,\"value\":3},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{\\\\phi} \\\\end{pmatrix} = \\\\begin{pmatrix} \\\\sin(\\\\phi)\\\\\\\\ \\\\cos(\\\\phi)\\\\\\\\ -\\\\sin(\\\\phi)+ \\\\dfrac{3(x_1\\\\sin(\\\\phi)+x_2\\\\cos(\\\\phi))} {0.5+x_1^2+x_2^2} \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{\\\\phi} \\\\end{pmatrix} = \\\\begin{pmatrix} \\\\sin(\\\\phi)\\\\\\\\ \\\\cos(\\\\phi)\\\\\\\\ -\\\\sin(\\\\phi)+ \\\\dfrac{3(x_1\\\\sin(\\\\phi)+x_2\\\\cos(\\\\phi))} {0.5+x_1^2+x_2^2} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: dReal\",\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-006","paper_system_number":41,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":8,"time_domain":"continuous","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)","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_0\\\\ \\dot{x}_1\\\\ \\dot{x}_2\\\\ \\dot{x}_3\\\\ \\dot{x}_4\\\\ \\dot{x}_5\\\\ \\dot{x}_6\\\\ \\dot{x}_7 \\end{pmatrix} = \\begin{pmatrix} x_1\\\\ x_2\\\\ x_3\\\\ x_4\\\\ x_5\\\\ x_6\\\\ x_7\\\\ -\\begin{aligned}[t] &20x_7+170x_6+800x_5+2273x_4\\\\[-0.15ex] &{}+3980x_3+4180x_2+2400x_1+576x_0 \\end{aligned} \\end{pmatrix}. \\]","citations":["abate2021fossil"],"problem_json":"{\"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.\"]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X}=[-2.2,2.2]^8$\",\"$\\\\mathcal{X}_I=[0.9,1.1]^8$\",\"$\\\\mathcal{X}_U=[-2.2,-1.8]^8$\",\"Verified type: barrier/safety verification\",\"Verified method in the original paper: dReal\"],\"record_id\":\"DCB-LIT-P4-006\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_I=[0.9,1.1]^8$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X}=[-2.2,2.2]^8$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U=[-2.2,-1.8]^8$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"SYS0032\",\"citations\":[\"abate2021fossil\"],\"label\":\"sys:catalogue-sys0032\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":1914,\"start\":1880},\"title_latex\":\"Table~1 row~17 (Hi-Ord8), \\\\cite{abate2021fossil}\"},\"source_order\":41,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0032\",\"status\":\"documented\",\"symbolic\":null,\"value\":8},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_0\\\\\\\\ \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3\\\\\\\\ \\\\dot{x}_4\\\\\\\\ \\\\dot{x}_5\\\\\\\\ \\\\dot{x}_6\\\\\\\\ \\\\dot{x}_7 \\\\end{pmatrix} = \\\\begin{pmatrix} x_1\\\\\\\\ x_2\\\\\\\\ x_3\\\\\\\\ x_4\\\\\\\\ x_5\\\\\\\\ x_6\\\\\\\\ x_7\\\\\\\\ -\\\\begin{aligned}[t] &20x_7+170x_6+800x_5+2273x_4\\\\\\\\[-0.15ex] &{}+3980x_3+4180x_2+2400x_1+576x_0 \\\\end{aligned} \\\\end{pmatrix}.\",\"[t] &20x_7+170x_6+800x_5+2273x_4\\\\\\\\[-0.15ex] &{}+3980x_3+4180x_2+2400x_1+576x_0\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_0\\\\\\\\ \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3\\\\\\\\ \\\\dot{x}_4\\\\\\\\ \\\\dot{x}_5\\\\\\\\ \\\\dot{x}_6\\\\\\\\ \\\\dot{x}_7 \\\\end{pmatrix} = \\\\begin{pmatrix} x_1\\\\\\\\ x_2\\\\\\\\ x_3\\\\\\\\ x_4\\\\\\\\ x_5\\\\\\\\ x_6\\\\\\\\ x_7\\\\\\\\ -\\\\begin{aligned}[t] &20x_7+170x_6+800x_5+2273x_4\\\\\\\\[-0.15ex] &{}+3980x_3+4180x_2+2400x_1+576x_0 \\\\end{aligned} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: dReal\",\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-007","paper_system_number":42,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":2,"time_domain":"continuous","dynamics":"dx1/dt = x1**2+x2**2-1, dx2/dt = 5*(x1*x2-1)","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2 \\end{pmatrix} = \\begin{pmatrix} x_1^2+x_2^2-1\\\\ 5(x_1x_2-1) \\end{pmatrix}. \\]","citations":["wu2024completeness"],"problem_json":"{\"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\"}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X}=\\\\mathbb{R}^{2}$\",\"$\\\\mathcal{X}_I= \\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid 2.25-x_1^2-(x_2-2)^2\\\\geq0\\\\}$\",\"$\\\\mathcal{X}_U= \\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid -1-x_1\\\\geq0\\\\wedge x_1+3\\\\geq0\\\\wedge -2-x_2\\\\geq0\\\\wedge x_2+3\\\\geq0\\\\}$\",\"Verified type: unbounded-domain barrier verification\",\"Verified method in the original paper: SDP-based barrier synthesis\"],\"record_id\":\"DCB-LIT-P4-007\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_I= \\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid 2.25-x_1^2-(x_2-2)^2\\\\geq0\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X}=\\\\mathbb{R}^{2}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U= \\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid -1-x_1\\\\geq0\\\\wedge x_1+3\\\\geq0\\\\wedge -2-x_2\\\\geq0\\\\wedge x_2+3\\\\geq0\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"SYS0060\",\"citations\":[\"wu2024completeness\"],\"label\":\"sys:catalogue-sys0060\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":1941,\"start\":1916},\"title_latex\":\"Table~1 (arch2[36], instance~1), \\\\cite{wu2024completeness}\"},\"source_order\":42,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0060\",\"status\":\"documented\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} x_1^2+x_2^2-1\\\\\\\\ 5(x_1x_2-1) \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} x_1^2+x_2^2-1\\\\\\\\ 5(x_1x_2-1) \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: SDP-based barrier synthesis\",\"reported_type_latex\":\"Verified type: unbounded-domain barrier verification\"}}"} {"record_id":"DCB-LIT-P4-008","paper_system_number":43,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":2,"time_domain":"continuous","dynamics":"dx1/dt = x1-x1**3+x2-x1*x2**2, dx2/dt = -x1+x2-x1**2*x2-x2**3","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2 \\end{pmatrix} = \\begin{pmatrix} x_1-x_1^3+x_2-x_1x_2^2\\\\ -x_1+x_2-x_1^2x_2-x_2^3 \\end{pmatrix}. \\]","citations":["wu2024completeness"],"problem_json":"{\"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\"}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X}=\\\\mathbb{R}^{2}$\",\"$\\\\mathcal{X}_I= \\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid0.04-x_1^2-x_2^2\\\\geq0\\\\}$\",\"$\\\\mathcal{X}_U= \\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid 0.25-(x_1-2.5)^2-(x_2-2.5)^2\\\\geq0\\\\}$\",\"Verified type: unbounded-domain barrier verification\",\"Verified method in the original paper: SDP-based barrier synthesis\"],\"record_id\":\"DCB-LIT-P4-008\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_I= \\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid0.04-x_1^2-x_2^2\\\\geq0\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X}=\\\\mathbb{R}^{2}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U= \\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid 0.25-(x_1-2.5)^2-(x_2-2.5)^2\\\\geq0\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"SYS0062\",\"citations\":[\"wu2024completeness\"],\"label\":\"sys:catalogue-sys0062\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":1968,\"start\":1943},\"title_latex\":\"Table~1 (arch3[36], instance~1), \\\\cite{wu2024completeness}\"},\"source_order\":43,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0062\",\"status\":\"documented\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} x_1-x_1^3+x_2-x_1x_2^2\\\\\\\\ -x_1+x_2-x_1^2x_2-x_2^3 \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} x_1-x_1^3+x_2-x_1x_2^2\\\\\\\\ -x_1+x_2-x_1^2x_2-x_2^3 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: SDP-based barrier synthesis\",\"reported_type_latex\":\"Verified type: unbounded-domain barrier verification\"}}"} {"record_id":"DCB-LIT-P4-009","paper_system_number":44,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":2,"time_domain":"continuous","dynamics":"dx1/dt = exp(-x1)+x2-1, dx2/dt = -sin(x1)**2","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2 \\end{pmatrix} = \\begin{pmatrix} e^{-x_1}+x_2-1\\\\ -\\sin^2(x_1) \\end{pmatrix}. \\]","citations":["liu2015abstraction"],"problem_json":"{\"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]]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X}=[-2,2]^2$\",\"$\\\\mathcal{X}_I= \\\\{(x_1,x_2)\\\\mid(x_1+0.5)^2+(x_2-0.5)^2\\\\leq0.16\\\\}$\",\"$\\\\mathcal{X}_U= \\\\{(x_1,x_2)\\\\mid(x_1-0.7)^2+(x_2+0.7)^2\\\\leq0.09\\\\}$\",\"Verified type: safety verification\",\"Verified method in the original paper: polynomial abstraction and SOS programming\"],\"record_id\":\"DCB-LIT-P4-009\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_I= \\\\{(x_1,x_2)\\\\mid(x_1+0.5)^2+(x_2-0.5)^2\\\\leq0.16\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X}=[-2,2]^2$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U= \\\\{(x_1,x_2)\\\\mid(x_1-0.7)^2+(x_2+0.7)^2\\\\leq0.09\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"SYS0005\",\"citations\":[\"liu2015abstraction\"],\"label\":\"sys:catalogue-sys0005\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":1995,\"start\":1970},\"title_latex\":\"Example~5 and Table~1, \\\\cite{liu2015abstraction}\"},\"source_order\":44,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0005\",\"status\":\"documented\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} e^{-x_1}+x_2-1\\\\\\\\ -\\\\sin^2(x_1) \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} e^{-x_1}+x_2-1\\\\\\\\ -\\\\sin^2(x_1) \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: polynomial abstraction and SOS programming\",\"reported_type_latex\":\"Verified type: safety verification\"}}"} {"record_id":"DCB-LIT-P4-010","paper_system_number":45,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":3,"time_domain":"continuous","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","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{u}} = \\begin{pmatrix} \\dot{u}_1\\\\ \\dot{u}_2\\\\ \\dot{u}_3 \\end{pmatrix} = \\begin{pmatrix} -\\dfrac{\\beta c u_1u_2}{u_1+u_2+u_3}-\\mu u_1\\\\ \\dfrac{\\beta c u_1u_2}{u_1+u_2+u_3}-(\\mu+\\nu)u_2\\\\ \\nu u_2-\\alpha u_3 \\end{pmatrix}, \\] where $\\beta=0.2$, $c=10$, $\\mu=0.008$, $\\alpha=0.95$, and $\\nu=0.1$.","citations":["liu2015abstraction"],"problem_json":"{\"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\"]]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X}=\\\\{\\\\mathbf{u}\\\\in\\\\mathbb{R}^{3}\\\\mid u_i\\\\geq0,\\\\ u_1+u_2+u_3\\\\leq10.013\\\\}$\",\"$\\\\mathcal{X}_I= [9.985,9.995]\\\\times[0.005,0.015]\\\\times[0,0.003]$\",\"$\\\\mathcal{X}_U=\\\\{\\\\mathbf{u}\\\\mid u_3>1\\\\}$\",\"Verified type: safety verification\",\"Verified method in the original paper: polynomial abstraction and SOS programming\"],\"record_id\":\"DCB-LIT-P4-010\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_I= [9.985,9.995]\\\\times[0.005,0.015]\\\\times[0,0.003]$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X}=\\\\{\\\\mathbf{u}\\\\in\\\\mathbb{R}^{3}\\\\mid u_i\\\\geq0,\\\\ u_1+u_2+u_3\\\\leq10.013\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U=\\\\{\\\\mathbf{u}\\\\mid u_3>1\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"SYS0006\",\"citations\":[\"liu2015abstraction\"],\"label\":\"sys:catalogue-sys0006\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":2025,\"start\":1997},\"title_latex\":\"Example~6 and Table~1, \\\\cite{liu2015abstraction}\"},\"source_order\":45,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0006\",\"status\":\"documented\",\"symbolic\":null,\"value\":3},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{u}} = \\\\begin{pmatrix} \\\\dot{u}_1\\\\\\\\ \\\\dot{u}_2\\\\\\\\ \\\\dot{u}_3 \\\\end{pmatrix} = \\\\begin{pmatrix} -\\\\dfrac{\\\\beta c u_1u_2}{u_1+u_2+u_3}-\\\\mu u_1\\\\\\\\ \\\\dfrac{\\\\beta c u_1u_2}{u_1+u_2+u_3}-(\\\\mu+\\\\nu)u_2\\\\\\\\ \\\\nu u_2-\\\\alpha u_3 \\\\end{pmatrix},\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{u}} = \\\\begin{pmatrix} \\\\dot{u}_1\\\\\\\\ \\\\dot{u}_2\\\\\\\\ \\\\dot{u}_3 \\\\end{pmatrix} = \\\\begin{pmatrix} -\\\\dfrac{\\\\beta c u_1u_2}{u_1+u_2+u_3}-\\\\mu u_1\\\\\\\\ \\\\dfrac{\\\\beta c u_1u_2}{u_1+u_2+u_3}-(\\\\mu+\\\\nu)u_2\\\\\\\\ \\\\nu u_2-\\\\alpha u_3 \\\\end{pmatrix}, \\\\] where $\\\\beta=0.2$, $c=10$, $\\\\mu=0.008$, $\\\\alpha=0.95$, and $\\\\nu=0.1$.\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}=\\\\{\\\\mathbf{u}\\\\in\\\\mathbb{R}^{3}\\\\mid u_i\\\\geq0,\\\\ u_1+u_2+u_3\\\\leq10.013\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: polynomial abstraction and SOS programming\",\"reported_type_latex\":\"Verified type: safety verification\"}}"} {"record_id":"DCB-LIT-P4-011","paper_system_number":46,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":3,"time_domain":"continuous","dynamics":"dx1/dt = x1*(1-x3), dx2/dt = x2*(1-2*x3), dx3/dt = x3*(x1+x2-1)","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2\\\\ \\dot{x}_3 \\end{pmatrix} = \\begin{pmatrix} x_1(1-x_3)\\\\ x_2(1-2x_3)\\\\ x_3(x_1+x_2-1) \\end{pmatrix}. \\]","citations":["wu2024completeness"],"problem_json":"{\"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\"},\"unsafe_set\":{\"type\":\"ball\",\"center\":[\"-1\",\"-1\",\"-3/2\"],\"radius\":\"3/10\"},\"analysis_bounds\":[[-3,3],[-3,3],[-3,3]],\"formal_domain\":\"unbounded\"}","executable_record_json":"{\"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\"},\"unsafe_set\":{\"type\":\"ball\",\"center\":[\"-1\",\"-1\",\"-3/2\"],\"radius\":\"3/10\"},\"analysis_bounds\":[[-3,3],[-3,3],[-3,3]],\"formal_domain\":\"unbounded\"},\"candidate\":null,\"paper_system_number\":46,\"paper_system_label\":\"System 46\",\"local_name\":\"S046__DCB-LIT-P4-011\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X}=\\\\mathbb{R}^{3}$\",\"$\\\\mathcal{X}_I= \\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{3}\\\\mid 0.09-(x_1-1)^2-(x_2-1)^2-(x_3-1.5)^2\\\\geq0\\\\}$\",\"$\\\\mathcal{X}_U= \\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{3}\\\\mid 0.09-(x_1+1)^2-(x_2+1)^2-(x_3+1.5)^2\\\\geq0\\\\}$\",\"Verified type: unbounded-domain barrier verification\",\"Verified method in the original paper: SDP-based barrier synthesis\"],\"record_id\":\"DCB-LIT-P4-011\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_I= \\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{3}\\\\mid 0.09-(x_1-1)^2-(x_2-1)^2-(x_3-1.5)^2\\\\geq0\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X}=\\\\mathbb{R}^{3}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U= \\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{3}\\\\mid 0.09-(x_1+1)^2-(x_2+1)^2-(x_3+1.5)^2\\\\geq0\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"SYS0070\",\"citations\":[\"wu2024completeness\"],\"label\":\"sys:catalogue-sys0070\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":2054,\"start\":2027},\"title_latex\":\"Table~1 (lotka[14], instance~1), \\\\cite{wu2024completeness}\"},\"source_order\":46,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0070\",\"status\":\"documented\",\"symbolic\":null,\"value\":3},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3 \\\\end{pmatrix} = \\\\begin{pmatrix} x_1(1-x_3)\\\\\\\\ x_2(1-2x_3)\\\\\\\\ x_3(x_1+x_2-1) \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3 \\\\end{pmatrix} = \\\\begin{pmatrix} x_1(1-x_3)\\\\\\\\ x_2(1-2x_3)\\\\\\\\ x_3(x_1+x_2-1) \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: SDP-based barrier synthesis\",\"reported_type_latex\":\"Verified type: unbounded-domain barrier verification\"}}"} {"record_id":"DCB-LIT-P4-012","paper_system_number":47,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":3,"time_domain":"continuous","dynamics":"dx1/dt = sin(x3), dx2/dt = cos(x3), dx3/dt = -sin(x3)+u1","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2\\\\ \\dot{\\phi} \\end{pmatrix} = \\begin{pmatrix} \\sin(\\phi)\\\\ \\cos(\\phi)\\\\ -\\sin(\\phi)+u_1 \\end{pmatrix}. \\]","citations":["edwards2024fossil"],"problem_json":"{\"problem_id\":\"dcb_lit_p4_012\",\"task_type\":\"safety\",\"name\":\"DynaCertBench System 047 · DCB-LIT-P4-012 · Table~4 \\\\#5 (ctrlbarr obstacle), \\\\cite{edwards2024fossil}\",\"dynamics\":\"dx1/dt = sin(x3), dx2/dt = cos(x3), dx3/dt = -sin(x3)+u1\",\"source\":\"paper System 47; canonical record DCB-LIT-P4-012\",\"assumptions\":[\"Deterministically normalized from canonical record DCB-LIT-P4-012.\",\"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.\"],\"controller_parameters\":\"u1\",\"initial_set\":{\"type\":\"box\",\"bounds\":[[\"4\",\"6\"],[\"4\",\"6\"],[\"-pi/2\",\"pi/2\"]]},\"unsafe_set\":{\"type\":\"box\",\"bounds\":[[\"-9\",\"-7\"],[\"-9\",\"-6\"],[\"-pi/2\",\"pi/2\"]]},\"analysis_bounds\":[[-10,10],[-10,10],[-4,4]]}","executable_record_json":"{\"record_id\":\"DCB-LIT-P4-012\",\"problem\":{\"problem_id\":\"dcb_lit_p4_012\",\"task_type\":\"safety\",\"name\":\"DynaCertBench System 047 · DCB-LIT-P4-012 · Table~4 \\\\#5 (ctrlbarr obstacle), \\\\cite{edwards2024fossil}\",\"dynamics\":\"dx1/dt = sin(x3), dx2/dt = cos(x3), dx3/dt = -sin(x3)+u1\",\"source\":\"paper System 47; canonical record DCB-LIT-P4-012\",\"assumptions\":[\"Deterministically normalized from canonical record DCB-LIT-P4-012.\",\"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.\"],\"controller_parameters\":\"u1\",\"initial_set\":{\"type\":\"box\",\"bounds\":[[\"4\",\"6\"],[\"4\",\"6\"],[\"-pi/2\",\"pi/2\"]]},\"unsafe_set\":{\"type\":\"box\",\"bounds\":[[\"-9\",\"-7\"],[\"-9\",\"-6\"],[\"-pi/2\",\"pi/2\"]]},\"analysis_bounds\":[[-10,10],[-10,10],[-4,4]]},\"candidate\":null,\"paper_system_number\":47,\"paper_system_label\":\"System 47\",\"local_name\":\"S047__DCB-LIT-P4-012\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X}=[-10,10]\\\\times[-10,10]\\\\times[-\\\\pi,\\\\pi]$\",\"$\\\\mathcal{X}_I=[4,6]\\\\times[4,6]\\\\times [-\\\\tfrac{\\\\pi}{2},\\\\tfrac{\\\\pi}{2}]$\",\"$\\\\mathcal{X}_U=[-9,-7]\\\\times[-9,-6]\\\\times [-\\\\tfrac{\\\\pi}{2},\\\\tfrac{\\\\pi}{2}]$\",\"Verified type: barrier/safety verification\",\"Verified method in the original paper: dReal\"],\"record_id\":\"DCB-LIT-P4-012\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_I=[4,6]\\\\times[4,6]\\\\times [-\\\\tfrac{\\\\pi}{2},\\\\tfrac{\\\\pi}{2}]$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X}=[-10,10]\\\\times[-10,10]\\\\times[-\\\\pi,\\\\pi]$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U=[-9,-7]\\\\times[-9,-6]\\\\times [-\\\\tfrac{\\\\pi}{2},\\\\tfrac{\\\\pi}{2}]$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"SYS0084\",\"citations\":[\"edwards2024fossil\"],\"label\":\"sys:catalogue-sys0084\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":2113,\"start\":2086},\"title_latex\":\"Table~4 \\\\#5 (ctrlbarr obstacle), \\\\cite{edwards2024fossil}\"},\"source_order\":47,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0084\",\"status\":\"documented\",\"symbolic\":null,\"value\":3},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{\\\\phi} \\\\end{pmatrix} = \\\\begin{pmatrix} \\\\sin(\\\\phi)\\\\\\\\ \\\\cos(\\\\phi)\\\\\\\\ -\\\\sin(\\\\phi)+u_1 \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{\\\\phi} \\\\end{pmatrix} = \\\\begin{pmatrix} \\\\sin(\\\\phi)\\\\\\\\ \\\\cos(\\\\phi)\\\\\\\\ -\\\\sin(\\\\phi)+u_1 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: dReal\",\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-013","paper_system_number":48,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":2,"time_domain":"discrete","dynamics":"x1[k+1] = 0.725*x1+0.25*x2+0.375+0.018*(55-x1)*u1, x2[k+1] = 0.25*x1+0.71*x2+0.6+0.018*(55-x2)*u2","specification_latex":"The discrete-time transition map is: \\[ \\mathbf{x}^{+} = F(\\mathbf{x},\\mathbf{u}) = \\begin{pmatrix} 0.725x_1+0.25x_2+0.375+0.018(55-x_1)u_1\\\\ 0.25x_1+0.71x_2+0.6+0.018(55-x_2)u_2 \\end{pmatrix}. \\]","citations":["edwards2024fossil"],"problem_json":"{\"problem_id\":\"dcb_lit_p4_013\",\"task_type\":\"safety\",\"name\":\"DynaCertBench System 048 · DCB-LIT-P4-013 · Table~4 \\\\#17 (ctrlbarr 2roomtemp DT), \\\\cite{edwards2024fossil}\",\"dynamics\":\"x1[k+1] = 0.725*x1+0.25*x2+0.375+0.018*(55-x1)*u1, x2[k+1] = 0.25*x1+0.71*x2+0.6+0.018*(55-x2)*u2\",\"source\":\"paper System 48; canonical record DCB-LIT-P4-013\",\"assumptions\":[\"Deterministically normalized from canonical record DCB-LIT-P4-013.\",\"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.\"],\"controller_parameters\":\"u1,u2\",\"initial_set\":{\"type\":\"box\",\"bounds\":[[\"17\",\"18\"],[\"17\",\"18\"]]},\"unsafe_set\":{\"type\":\"box\",\"bounds\":[[\"28\",\"30\"],[\"28\",\"30\"]]},\"analysis_bounds\":[[17,30],[17,30]]}","executable_record_json":"{\"record_id\":\"DCB-LIT-P4-013\",\"problem\":{\"problem_id\":\"dcb_lit_p4_013\",\"task_type\":\"safety\",\"name\":\"DynaCertBench System 048 · DCB-LIT-P4-013 · Table~4 \\\\#17 (ctrlbarr 2roomtemp DT), \\\\cite{edwards2024fossil}\",\"dynamics\":\"x1[k+1] = 0.725*x1+0.25*x2+0.375+0.018*(55-x1)*u1, x2[k+1] = 0.25*x1+0.71*x2+0.6+0.018*(55-x2)*u2\",\"source\":\"paper System 48; canonical record DCB-LIT-P4-013\",\"assumptions\":[\"Deterministically normalized from canonical record DCB-LIT-P4-013.\",\"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.\"],\"controller_parameters\":\"u1,u2\",\"initial_set\":{\"type\":\"box\",\"bounds\":[[\"17\",\"18\"],[\"17\",\"18\"]]},\"unsafe_set\":{\"type\":\"box\",\"bounds\":[[\"28\",\"30\"],[\"28\",\"30\"]]},\"analysis_bounds\":[[17,30],[17,30]]},\"candidate\":null,\"paper_system_number\":48,\"paper_system_label\":\"System 48\",\"local_name\":\"S048__DCB-LIT-P4-013\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X}=[17,30]^2$\",\"$\\\\mathcal{X}_I=[17,18]^2$\",\"$\\\\mathcal{X}_U=[28,30]^2$\",\"Verified type: barrier/safety verification\",\"Verified method in the original paper: dReal\"],\"record_id\":\"DCB-LIT-P4-013\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_I=[17,18]^2$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X}=[17,30]^2$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U=[28,30]^2$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"SYS0096\",\"citations\":[\"edwards2024fossil\"],\"label\":\"sys:catalogue-sys0096\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":2137,\"start\":2115},\"title_latex\":\"Table~4 \\\\#17 (ctrlbarr 2roomtemp DT), \\\\cite{edwards2024fossil}\"},\"source_order\":48,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0096\",\"status\":\"documented\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\mathbf{x}^{+} = F(\\\\mathbf{x},\\\\mathbf{u}) = \\\\begin{pmatrix} 0.725x_1+0.25x_2+0.375+0.018(55-x_1)u_1\\\\\\\\ 0.25x_1+0.71x_2+0.6+0.018(55-x_2)u_2 \\\\end{pmatrix}.\"],\"specification_latex\":\"The discrete-time transition map is: \\\\[ \\\\mathbf{x}^{+} = F(\\\\mathbf{x},\\\\mathbf{u}) = \\\\begin{pmatrix} 0.725x_1+0.25x_2+0.375+0.018(55-x_1)u_1\\\\\\\\ 0.25x_1+0.71x_2+0.6+0.018(55-x_2)u_2 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"discrete\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: dReal\",\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-014","paper_system_number":49,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":3,"time_domain":"continuous","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","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_{1}\\\\ \\dot{x}_{2}\\\\ \\dot{x}_{3} \\end{pmatrix} = \\begin{pmatrix} x_{2} + u_{0}\\\\ -0.8\\,x_{1} - 0.15\\,x_{2} + 0.2\\,x_{1}^{2} + u_{1}\\\\ -0.1\\,x_{3} + 0.1\\,x_{1} + u_{2} \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"problem_id\":\"dcb_lit_p4_014\",\"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\",\"source\":\"paper System 49; canonical record DCB-LIT-P4-014\",\"assumptions\":[\"Deterministically normalized from canonical record DCB-LIT-P4-014.\",\"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\":\"0.4\"},\"unsafe_set\":{\"type\":\"ball\",\"center\":[\"0\",\"0\",\"0\"],\"radius\":\"3\",\"complement\":true},\"analysis_bounds\":[[\"-4\",\"4\"],[\"-4\",\"4\"],[\"-4\",\"4\"]],\"formal_domain\":\"unbounded\"}","executable_record_json":"{\"record_id\":\"DCB-LIT-P4-014\",\"problem\":{\"problem_id\":\"dcb_lit_p4_014\",\"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\",\"source\":\"paper System 49; canonical record DCB-LIT-P4-014\",\"assumptions\":[\"Deterministically normalized from canonical record DCB-LIT-P4-014.\",\"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\":\"0.4\"},\"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\":49,\"paper_system_label\":\"System 49\",\"local_name\":\"S049__DCB-LIT-P4-014\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.4\\\\}$\",\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 3\\\\}$\",\"Verified type: barrier/safety verification\",\"Verified method in the original paper: Z3/Yices\"],\"record_id\":\"DCB-LIT-P4-014\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.4\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 3\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"BB-S001\",\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-s001\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":2167,\"start\":2139},\"title_latex\":\"~\\\\texttt{S001}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":49,\"system\":{\"dimension\":{\"source\":\"catalogue:BB-S001\",\"status\":\"documented\",\"symbolic\":null,\"value\":3},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{2} + u_{0}\\\\\\\\ -0.8\\\\,x_{1} - 0.15\\\\,x_{2} + 0.2\\\\,x_{1}^{2} + u_{1}\\\\\\\\ -0.1\\\\,x_{3} + 0.1\\\\,x_{1} + u_{2} \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{2} + u_{0}\\\\\\\\ -0.8\\\\,x_{1} - 0.15\\\\,x_{2} + 0.2\\\\,x_{1}^{2} + u_{1}\\\\\\\\ -0.1\\\\,x_{3} + 0.1\\\\,x_{1} + u_{2} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 3\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: Z3/Yices\",\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-015","paper_system_number":50,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":4,"time_domain":"continuous","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","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_{1}\\\\ \\dot{x}_{2}\\\\ \\dot{x}_{3}\\\\ \\dot{x}_{4} \\end{pmatrix} = \\begin{pmatrix} x_{2} + u_{0}\\\\ -0.95\\,\\sin(x_{1}) - 0.02\\,x_{2} + 0.01\\,x_{3} + u_{1}\\\\ -0.1\\,x_{3} + 0.02\\,x_{1} + u_{2}\\\\ -0.03\\,x_{4} + 0.01\\,x_{2} + u_{3} \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"problem_id\":\"dcb_lit_p4_015\",\"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.\"],\"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\"}","executable_record_json":"{\"record_id\":\"DCB-LIT-P4-015\",\"problem\":{\"problem_id\":\"dcb_lit_p4_015\",\"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.\"],\"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\"},\"candidate\":null,\"paper_system_number\":50,\"paper_system_label\":\"System 50\",\"local_name\":\"S050__DCB-LIT-P4-015\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{4}$\",\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.3\\\\}$\",\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{4}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 2.5\\\\}$\",\"Verified type: barrier/safety verification\",\"Verified method in the original paper: Z3/Yices with Taylor approximations\"],\"record_id\":\"DCB-LIT-P4-015\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.3\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{4}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{4}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 2.5\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"BB-S002\",\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-s002\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":2198,\"start\":2169},\"title_latex\":\"~\\\\texttt{S002}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":50,\"system\":{\"dimension\":{\"source\":\"catalogue:BB-S002\",\"status\":\"documented\",\"symbolic\":null,\"value\":4},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3}\\\\\\\\ \\\\dot{x}_{4} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{2} + u_{0}\\\\\\\\ -0.95\\\\,\\\\sin(x_{1}) - 0.02\\\\,x_{2} + 0.01\\\\,x_{3} + u_{1}\\\\\\\\ -0.1\\\\,x_{3} + 0.02\\\\,x_{1} + u_{2}\\\\\\\\ -0.03\\\\,x_{4} + 0.01\\\\,x_{2} + u_{3} \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3}\\\\\\\\ \\\\dot{x}_{4} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{2} + u_{0}\\\\\\\\ -0.95\\\\,\\\\sin(x_{1}) - 0.02\\\\,x_{2} + 0.01\\\\,x_{3} + u_{1}\\\\\\\\ -0.1\\\\,x_{3} + 0.02\\\\,x_{1} + u_{2}\\\\\\\\ -0.03\\\\,x_{4} + 0.01\\\\,x_{2} + u_{3} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{4}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 2.5\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: Z3/Yices with Taylor approximations\",\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-016","paper_system_number":51,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":4,"time_domain":"continuous","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","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_{1}\\\\ \\dot{x}_{2}\\\\ \\dot{x}_{3}\\\\ \\dot{x}_{4} \\end{pmatrix} = \\begin{pmatrix} 10\\,(x_{2} - x_{1}) + u_{0}\\\\ 28\\,x_{1} - x_{2} - x_{1}\\,x_{3} + u_{1}\\\\ x_{1}\\,x_{2} - \\frac{8}{3}\\,x_{3} + u_{2}\\\\ -x_{2}\\,x_{3} + 2.7\\,x_{4} + u_{3} \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"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\",\"source\":\"paper System 51; canonical record DCB-LIT-P4-016\",\"assumptions\":[\"Deterministically normalized from canonical record DCB-LIT-P4-016.\",\"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\":\"0.1\"},\"unsafe_set\":{\"type\":\"ball\",\"center\":[\"0\",\"0\",\"0\",\"0\"],\"radius\":\"25\",\"complement\":true},\"analysis_bounds\":[[\"-26\",\"26\"],[\"-26\",\"26\"],[\"-26\",\"26\"],[\"-26\",\"26\"]],\"formal_domain\":\"unbounded\"}","executable_record_json":"{\"record_id\":\"DCB-LIT-P4-016\",\"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\",\"source\":\"paper System 51; canonical record DCB-LIT-P4-016\",\"assumptions\":[\"Deterministically normalized from canonical record DCB-LIT-P4-016.\",\"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\":\"0.1\"},\"unsafe_set\":{\"type\":\"ball\",\"center\":[\"0\",\"0\",\"0\",\"0\"],\"radius\":\"25\",\"complement\":true},\"analysis_bounds\":[[\"-26\",\"26\"],[\"-26\",\"26\"],[\"-26\",\"26\"],[\"-26\",\"26\"]],\"formal_domain\":\"unbounded\"},\"candidate\":null,\"paper_system_number\":51,\"paper_system_label\":\"System 51\",\"local_name\":\"S051__DCB-LIT-P4-016\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{4}$\",\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.1\\\\}$\",\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{4}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 25\\\\}$\",\"Verified type: barrier/safety verification\",\"Verified method in the original paper: Z3/Yices\"],\"record_id\":\"DCB-LIT-P4-016\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.1\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{4}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{4}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 25\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"BB-S004\",\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-s004\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":2230,\"start\":2200},\"title_latex\":\"~\\\\texttt{S004}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":51,\"system\":{\"dimension\":{\"source\":\"catalogue:BB-S004\",\"status\":\"documented\",\"symbolic\":null,\"value\":4},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3}\\\\\\\\ \\\\dot{x}_{4} \\\\end{pmatrix} = \\\\begin{pmatrix} 10\\\\,(x_{2} - x_{1}) + u_{0}\\\\\\\\ 28\\\\,x_{1} - x_{2} - x_{1}\\\\,x_{3} + u_{1}\\\\\\\\ x_{1}\\\\,x_{2} - \\\\frac{8}{3}\\\\,x_{3} + u_{2}\\\\\\\\ -x_{2}\\\\,x_{3} + 2.7\\\\,x_{4} + u_{3} \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3}\\\\\\\\ \\\\dot{x}_{4} \\\\end{pmatrix} = \\\\begin{pmatrix} 10\\\\,(x_{2} - x_{1}) + u_{0}\\\\\\\\ 28\\\\,x_{1} - x_{2} - x_{1}\\\\,x_{3} + u_{1}\\\\\\\\ x_{1}\\\\,x_{2} - \\\\frac{8}{3}\\\\,x_{3} + u_{2}\\\\\\\\ -x_{2}\\\\,x_{3} + 2.7\\\\,x_{4} + u_{3} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{4}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 25\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: Z3/Yices\",\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-017","paper_system_number":52,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":3,"time_domain":"continuous","dynamics":"dx1/dt = -2*x1+6.7*x2-x2*x3+u0, dx2/dt = x1+u1, dx3/dt = -x3+x2**(2)+u2","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_{1}\\\\ \\dot{x}_{2}\\\\ \\dot{x}_{3} \\end{pmatrix} = \\begin{pmatrix} -2\\,x_{1} + 6.7\\,x_{2} - x_{2}\\,x_{3} + u_{0}\\\\ x_{1} + u_{1}\\\\ -x_{3} + x_{2}^{2} + u_{2} \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"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.\"],\"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\":\"5\",\"complement\":true},\"analysis_bounds\":[[\"-6\",\"6\"],[\"-6\",\"6\"],[\"-6\",\"6\"]],\"formal_domain\":\"unbounded\"}","executable_record_json":"{\"record_id\":\"DCB-LIT-P4-017\",\"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.\"],\"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\":\"5\",\"complement\":true},\"analysis_bounds\":[[\"-6\",\"6\"],[\"-6\",\"6\"],[\"-6\",\"6\"]],\"formal_domain\":\"unbounded\"},\"candidate\":null,\"paper_system_number\":52,\"paper_system_label\":\"System 52\",\"local_name\":\"S052__DCB-LIT-P4-017\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.2\\\\}$\",\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 5\\\\}$\",\"Verified type: barrier/safety verification\",\"Verified method in the original paper: Z3/Yices\"],\"record_id\":\"DCB-LIT-P4-017\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.2\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 5\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"BB-S008\",\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-s008\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":2260,\"start\":2232},\"title_latex\":\"~\\\\texttt{S008}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":52,\"system\":{\"dimension\":{\"source\":\"catalogue:BB-S008\",\"status\":\"documented\",\"symbolic\":null,\"value\":3},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3} \\\\end{pmatrix} = \\\\begin{pmatrix} -2\\\\,x_{1} + 6.7\\\\,x_{2} - x_{2}\\\\,x_{3} + u_{0}\\\\\\\\ x_{1} + u_{1}\\\\\\\\ -x_{3} + x_{2}^{2} + u_{2} \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3} \\\\end{pmatrix} = \\\\begin{pmatrix} -2\\\\,x_{1} + 6.7\\\\,x_{2} - x_{2}\\\\,x_{3} + u_{0}\\\\\\\\ x_{1} + u_{1}\\\\\\\\ -x_{3} + x_{2}^{2} + u_{2} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 5\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: Z3/Yices\",\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-018","paper_system_number":53,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":2,"time_domain":"continuous","dynamics":"dx1/dt = x2+u0, dx2/dt = -0.3*x1-0.2*(1+cos(x1))*x2-0.1*x1**(3)+u1","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_{1}\\\\ \\dot{x}_{2} \\end{pmatrix} = \\begin{pmatrix} x_{2} + u_{0}\\\\ -0.3\\,x_{1} - 0.2\\,(1 + \\cos(x_{1}))\\,x_{2} - 0.1\\,x_{1}^{3} + u_{1} \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"problem_id\":\"dcb_lit_p4_018\",\"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.\"],\"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\"}","executable_record_json":"{\"record_id\":\"DCB-LIT-P4-018\",\"problem\":{\"problem_id\":\"dcb_lit_p4_018\",\"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.\"],\"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\"},\"candidate\":null,\"paper_system_number\":53,\"paper_system_label\":\"System 53\",\"local_name\":\"S053__DCB-LIT-P4-018\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{2}$\",\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.6\\\\}$\",\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid \\\\lVert x-(2,2)^{\\\\mathsf T}\\\\rVert_2\\\\le 1\\\\}$\",\"Verified type: barrier/safety verification\",\"Verified method in the original paper: Z3/Yices with Taylor approximations\"],\"record_id\":\"DCB-LIT-P4-018\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.6\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{2}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid \\\\lVert x-(2,2)^{\\\\mathsf T}\\\\rVert_2\\\\le 1\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"BB-S011\",\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-s011\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":2287,\"start\":2262},\"title_latex\":\"~\\\\texttt{S011}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":53,\"system\":{\"dimension\":{\"source\":\"catalogue:BB-S011\",\"status\":\"documented\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{2} + u_{0}\\\\\\\\ -0.3\\\\,x_{1} - 0.2\\\\,(1 + \\\\cos(x_{1}))\\\\,x_{2} - 0.1\\\\,x_{1}^{3} + u_{1} \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{2} + u_{0}\\\\\\\\ -0.3\\\\,x_{1} - 0.2\\\\,(1 + \\\\cos(x_{1}))\\\\,x_{2} - 0.1\\\\,x_{1}^{3} + u_{1} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid \\\\lVert x-(2,2)^{\\\\mathsf T}\\\\rVert_2\\\\le 1\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: Z3/Yices with Taylor approximations\",\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-019","paper_system_number":54,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":3,"time_domain":"continuous","dynamics":"dx1/dt = -x1+x2*x3, dx2/dt = -x2+x1*x3, dx3/dt = -x3-x1*x2","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_{1}\\\\ \\dot{x}_{2}\\\\ \\dot{x}_{3} \\end{pmatrix} = \\begin{pmatrix} -x_{1} + x_{2}\\,x_{3}\\\\ -x_{2} + x_{1}\\,x_{3}\\\\ -x_{3} - x_{1}\\,x_{2} \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"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.\"],\"initial_set\":{\"type\":\"ball\",\"center\":[\"0\",\"0\",\"0\"],\"radius\":\"0.2\"},\"unsafe_set\":{\"type\":\"ball\",\"center\":[\"0\",\"0\",\"0\"],\"radius\":\"1\",\"complement\":true},\"analysis_bounds\":[[\"-2\",\"2\"],[\"-2\",\"2\"],[\"-2\",\"2\"]],\"formal_domain\":\"unbounded\"}","executable_record_json":"{\"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.\"],\"initial_set\":{\"type\":\"ball\",\"center\":[\"0\",\"0\",\"0\"],\"radius\":\"0.2\"},\"unsafe_set\":{\"type\":\"ball\",\"center\":[\"0\",\"0\",\"0\"],\"radius\":\"1\",\"complement\":true},\"analysis_bounds\":[[\"-2\",\"2\"],[\"-2\",\"2\"],[\"-2\",\"2\"]],\"formal_domain\":\"unbounded\"},\"candidate\":null,\"paper_system_number\":54,\"paper_system_label\":\"System 54\",\"local_name\":\"S054__DCB-LIT-P4-019\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.2\\\\}$\",\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 1\\\\}$\",\"Verified type: barrier/safety verification\",\"Verified method in the original paper: Z3/Yices\"],\"record_id\":\"DCB-LIT-P4-019\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.2\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 1\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"BB-S019\",\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-s019\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":2348,\"start\":2320},\"title_latex\":\"~\\\\texttt{S019}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":54,\"system\":{\"dimension\":{\"source\":\"catalogue:BB-S019\",\"status\":\"documented\",\"symbolic\":null,\"value\":3},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3} \\\\end{pmatrix} = \\\\begin{pmatrix} -x_{1} + x_{2}\\\\,x_{3}\\\\\\\\ -x_{2} + x_{1}\\\\,x_{3}\\\\\\\\ -x_{3} - x_{1}\\\\,x_{2} \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3} \\\\end{pmatrix} = \\\\begin{pmatrix} -x_{1} + x_{2}\\\\,x_{3}\\\\\\\\ -x_{2} + x_{1}\\\\,x_{3}\\\\\\\\ -x_{3} - x_{1}\\\\,x_{2} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 1\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: Z3/Yices\",\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-020","paper_system_number":55,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":4,"time_domain":"continuous","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","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_{1}\\\\ \\dot{x}_{2}\\\\ \\dot{x}_{3}\\\\ \\dot{x}_{4} \\end{pmatrix} = \\begin{pmatrix} x_{2} + u_{0}\\\\ -x_{1} - 0.4\\,x_{2} + 0.15\\,\\sin(x_{3}) + u_{1}\\\\ x_{4} + u_{2}\\\\ -0.6\\,x_{3} - 0.3\\,x_{4} + u_{3} \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"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\"]]},\"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\"}","executable_record_json":"{\"record_id\":\"DCB-LIT-P4-020\",\"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\"]]},\"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\"},\"candidate\":null,\"paper_system_number\":55,\"paper_system_label\":\"System 55\",\"local_name\":\"S055__DCB-LIT-P4-020\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{4}$\",\"$\\\\mathcal{X}_0 = [0.5,0.8]\\\\times [0.3,0.6]\\\\times [0.5,0.8]\\\\times [0.3,0.6]$\",\"$\\\\mathcal{X}_U = [2.5,3.5]\\\\times [2,3]\\\\times [2.5,3.5]\\\\times [2,3]\\\\cup [-3.5,-2.5]\\\\times [-3,-2]\\\\times [-3.5,-2.5]\\\\times [-3,-2]$\",\"Verified type: barrier/safety verification\",\"Verified method in the original paper: Z3/Yices with Taylor approximations\"],\"record_id\":\"DCB-LIT-P4-020\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = [0.5,0.8]\\\\times [0.3,0.6]\\\\times [0.5,0.8]\\\\times [0.3,0.6]$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{4}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = [2.5,3.5]\\\\times [2,3]\\\\times [2.5,3.5]\\\\times [2,3]\\\\cup [-3.5,-2.5]\\\\times [-3,-2]\\\\times [-3.5,-2.5]\\\\times [-3,-2]$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"BB-S024\",\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-s024\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":2411,\"start\":2382},\"title_latex\":\"~\\\\texttt{S024}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":55,\"system\":{\"dimension\":{\"source\":\"catalogue:BB-S024\",\"status\":\"documented\",\"symbolic\":null,\"value\":4},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3}\\\\\\\\ \\\\dot{x}_{4} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{2} + u_{0}\\\\\\\\ -x_{1} - 0.4\\\\,x_{2} + 0.15\\\\,\\\\sin(x_{3}) + u_{1}\\\\\\\\ x_{4} + u_{2}\\\\\\\\ -0.6\\\\,x_{3} - 0.3\\\\,x_{4} + u_{3} \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3}\\\\\\\\ \\\\dot{x}_{4} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{2} + u_{0}\\\\\\\\ -x_{1} - 0.4\\\\,x_{2} + 0.15\\\\,\\\\sin(x_{3}) + u_{1}\\\\\\\\ x_{4} + u_{2}\\\\\\\\ -0.6\\\\,x_{3} - 0.3\\\\,x_{4} + u_{3} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = [2.5,3.5]\\\\times [2,3]\\\\times [2.5,3.5]\\\\times [2,3]\\\\cup [-3.5,-2.5]\\\\times [-3,-2]\\\\times [-3.5,-2.5]\\\\times [-3,-2]$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: Z3/Yices with Taylor approximations\",\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-021","paper_system_number":56,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":3,"time_domain":"continuous","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","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_{1}\\\\ \\dot{x}_{2}\\\\ \\dot{x}_{3} \\end{pmatrix} = \\begin{pmatrix} -0.4\\,x_{1} + 0.25\\,x_{2}\\,x_{3} - 0.15\\,x_{1}^{3}\\\\ -0.5\\,x_{2} + 0.3\\,x_{1}\\,x_{3} - 0.2\\,x_{2}^{3}\\\\ -0.6\\,x_{3} + 0.2\\,x_{1}\\,x_{2} - 0.18\\,x_{1}^{2}\\,x_{2} \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"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\",\"source\":\"paper System 56; canonical record DCB-LIT-P4-021\",\"assumptions\":[\"Deterministically normalized from canonical record DCB-LIT-P4-021.\",\"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.\"],\"initial_set\":{\"type\":\"ball\",\"center\":[\"0\",\"0\",\"0\"],\"radius\":\"0.5\"},\"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\"}","executable_record_json":"{\"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\",\"source\":\"paper System 56; canonical record DCB-LIT-P4-021\",\"assumptions\":[\"Deterministically normalized from canonical record DCB-LIT-P4-021.\",\"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.\"],\"initial_set\":{\"type\":\"ball\",\"center\":[\"0\",\"0\",\"0\"],\"radius\":\"0.5\"},\"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\":56,\"paper_system_label\":\"System 56\",\"local_name\":\"S056__DCB-LIT-P4-021\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.5\\\\}$\",\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 3.5\\\\}$\",\"Verified type: barrier/safety verification\",\"Verified method in the original paper: Z3/Yices\"],\"record_id\":\"DCB-LIT-P4-021\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.5\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 3.5\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"BB-S028\",\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-s028\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":2471,\"start\":2443},\"title_latex\":\"~\\\\texttt{S028}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":56,\"system\":{\"dimension\":{\"source\":\"catalogue:BB-S028\",\"status\":\"documented\",\"symbolic\":null,\"value\":3},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3} \\\\end{pmatrix} = \\\\begin{pmatrix} -0.4\\\\,x_{1} + 0.25\\\\,x_{2}\\\\,x_{3} - 0.15\\\\,x_{1}^{3}\\\\\\\\ -0.5\\\\,x_{2} + 0.3\\\\,x_{1}\\\\,x_{3} - 0.2\\\\,x_{2}^{3}\\\\\\\\ -0.6\\\\,x_{3} + 0.2\\\\,x_{1}\\\\,x_{2} - 0.18\\\\,x_{1}^{2}\\\\,x_{2} \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3} \\\\end{pmatrix} = \\\\begin{pmatrix} -0.4\\\\,x_{1} + 0.25\\\\,x_{2}\\\\,x_{3} - 0.15\\\\,x_{1}^{3}\\\\\\\\ -0.5\\\\,x_{2} + 0.3\\\\,x_{1}\\\\,x_{3} - 0.2\\\\,x_{2}^{3}\\\\\\\\ -0.6\\\\,x_{3} + 0.2\\\\,x_{1}\\\\,x_{2} - 0.18\\\\,x_{1}^{2}\\\\,x_{2} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 3.5\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: Z3/Yices\",\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-022","paper_system_number":57,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":4,"time_domain":"continuous","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","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_{1}\\\\ \\dot{x}_{2}\\\\ \\dot{x}_{3}\\\\ \\dot{x}_{4} \\end{pmatrix} = \\begin{pmatrix} -0.3\\,x_{1} + 0.2\\,x_{2}\\,x_{4} - 0.15\\,x_{3}^{2}\\\\ -0.4\\,x_{2} + 0.25\\,x_{1}\\,x_{3} - 0.18\\,x_{2}\\,x_{4}\\\\ -0.5\\,x_{3} + 0.3\\,x_{1}\\,x_{4} + 0.12\\,x_{2}^{2}\\\\ -0.6\\,x_{4} + 0.2\\,x_{2}\\,x_{3} - 0.1\\,x_{1}^{2}\\,x_{4} \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"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.\"],\"initial_set\":{\"type\":\"ball\",\"center\":[\"0\",\"0\",\"0\",\"0\"],\"radius\":\"0.4\"},\"unsafe_set\":{\"type\":\"ball\",\"center\":[\"0\",\"0\",\"0\",\"0\"],\"radius\":\"3\",\"complement\":true},\"analysis_bounds\":[[\"-4\",\"4\"],[\"-4\",\"4\"],[\"-4\",\"4\"],[\"-4\",\"4\"]],\"formal_domain\":\"unbounded\"}","executable_record_json":"{\"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.\"],\"initial_set\":{\"type\":\"ball\",\"center\":[\"0\",\"0\",\"0\",\"0\"],\"radius\":\"0.4\"},\"unsafe_set\":{\"type\":\"ball\",\"center\":[\"0\",\"0\",\"0\",\"0\"],\"radius\":\"3\",\"complement\":true},\"analysis_bounds\":[[\"-4\",\"4\"],[\"-4\",\"4\"],[\"-4\",\"4\"],[\"-4\",\"4\"]],\"formal_domain\":\"unbounded\"},\"candidate\":null,\"paper_system_number\":57,\"paper_system_label\":\"System 57\",\"local_name\":\"S057__DCB-LIT-P4-022\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{4}$\",\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.4\\\\}$\",\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{4}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 3\\\\}$\",\"Verified type: barrier/safety verification\",\"Verified method in the original paper: Z3/Yices\"],\"record_id\":\"DCB-LIT-P4-022\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.4\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{4}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{4}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 3\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"BB-S031\",\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-s031\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":2503,\"start\":2473},\"title_latex\":\"~\\\\texttt{S031}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":57,\"system\":{\"dimension\":{\"source\":\"catalogue:BB-S031\",\"status\":\"documented\",\"symbolic\":null,\"value\":4},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3}\\\\\\\\ \\\\dot{x}_{4} \\\\end{pmatrix} = \\\\begin{pmatrix} -0.3\\\\,x_{1} + 0.2\\\\,x_{2}\\\\,x_{4} - 0.15\\\\,x_{3}^{2}\\\\\\\\ -0.4\\\\,x_{2} + 0.25\\\\,x_{1}\\\\,x_{3} - 0.18\\\\,x_{2}\\\\,x_{4}\\\\\\\\ -0.5\\\\,x_{3} + 0.3\\\\,x_{1}\\\\,x_{4} + 0.12\\\\,x_{2}^{2}\\\\\\\\ -0.6\\\\,x_{4} + 0.2\\\\,x_{2}\\\\,x_{3} - 0.1\\\\,x_{1}^{2}\\\\,x_{4} \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3}\\\\\\\\ \\\\dot{x}_{4} \\\\end{pmatrix} = \\\\begin{pmatrix} -0.3\\\\,x_{1} + 0.2\\\\,x_{2}\\\\,x_{4} - 0.15\\\\,x_{3}^{2}\\\\\\\\ -0.4\\\\,x_{2} + 0.25\\\\,x_{1}\\\\,x_{3} - 0.18\\\\,x_{2}\\\\,x_{4}\\\\\\\\ -0.5\\\\,x_{3} + 0.3\\\\,x_{1}\\\\,x_{4} + 0.12\\\\,x_{2}^{2}\\\\\\\\ -0.6\\\\,x_{4} + 0.2\\\\,x_{2}\\\\,x_{3} - 0.1\\\\,x_{1}^{2}\\\\,x_{4} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{4}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 3\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: Z3/Yices\",\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-023","paper_system_number":58,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":2,"time_domain":"discrete","dynamics":"x1[k+1] = 0.85*x1[k]+u0[k], x2[k+1] = 0.8*x2[k]+0.15*x1[k]+u1[k]","specification_latex":"The update map is: \\[ \\mathbf{x}_{k+1} = \\begin{pmatrix} x_{1,k+1}\\\\ x_{2,k+1} \\end{pmatrix} = \\begin{pmatrix} 0.85\\,x_{1,k} + u_{0,k}\\\\ 0.8\\,x_{2,k} + 0.15\\,x_{1,k} + u_{1,k} \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"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\",\"bounds\":[[\"-5.5\",\"-4.5\"],[\"-5\",\"-4\"]]}]},\"analysis_bounds\":[[\"1/2\",\"3\"],[\"1/5\",\"14/5\"]],\"formal_domain\":\"unbounded\"}","executable_record_json":"{\"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\",\"bounds\":[[\"-5.5\",\"-4.5\"],[\"-5\",\"-4\"]]}]},\"analysis_bounds\":[[\"1/2\",\"3\"],[\"1/5\",\"14/5\"]],\"formal_domain\":\"unbounded\"},\"candidate\":null,\"paper_system_number\":58,\"paper_system_label\":\"System 58\",\"local_name\":\"S058__DCB-LIT-P4-023\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{2}$\",\"$\\\\mathcal{X}_0 = [1.5,2]\\\\times [1.2,1.8]$\",\"$\\\\mathcal{X}_U = [4.5,5.5]\\\\times [4,5]\\\\cup [-5.5,-4.5]\\\\times [-5,-4]$\",\"Verified type: barrier/safety verification\"],\"record_id\":\"DCB-LIT-P4-023\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = [1.5,2]\\\\times [1.2,1.8]$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{2}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = [4.5,5.5]\\\\times [4,5]\\\\cup [-5.5,-4.5]\\\\times [-5,-4]$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"BB-S036\",\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-s036\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":2531,\"start\":2505},\"title_latex\":\"~\\\\texttt{S036}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":58,\"system\":{\"dimension\":{\"source\":\"catalogue:BB-S036\",\"status\":\"documented\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\mathbf{x}_{k+1} = \\\\begin{pmatrix} x_{1,k+1}\\\\\\\\ x_{2,k+1} \\\\end{pmatrix} = \\\\begin{pmatrix} 0.85\\\\,x_{1,k} + u_{0,k}\\\\\\\\ 0.8\\\\,x_{2,k} + 0.15\\\\,x_{1,k} + u_{1,k} \\\\end{pmatrix}.\"],\"specification_latex\":\"The update map is: \\\\[ \\\\mathbf{x}_{k+1} = \\\\begin{pmatrix} x_{1,k+1}\\\\\\\\ x_{2,k+1} \\\\end{pmatrix} = \\\\begin{pmatrix} 0.85\\\\,x_{1,k} + u_{0,k}\\\\\\\\ 0.8\\\\,x_{2,k} + 0.15\\\\,x_{1,k} + u_{1,k} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"discrete\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = [4.5,5.5]\\\\times [4,5]\\\\cup [-5.5,-4.5]\\\\times [-5,-4]$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":null,\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-024","paper_system_number":59,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":2,"time_domain":"continuous","dynamics":"dx1/dt = x2+u0, dx2/dt = -0.2*x1-0.1*(x1**(2)-1)*x2+u1","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_{1}\\\\ \\dot{x}_{2} \\end{pmatrix} = \\begin{pmatrix} x_{2} + u_{0}\\\\ -0.2\\,x_{1} - 0.1\\,(x_{1}^{2} - 1)\\,x_{2} + u_{1} \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"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\",\"complement\":true},\"analysis_bounds\":[[\"-19/5\",\"19/5\"],[\"-19/5\",\"19/5\"]],\"formal_domain\":\"unbounded\"}","executable_record_json":"{\"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\",\"complement\":true},\"analysis_bounds\":[[\"-19/5\",\"19/5\"],[\"-19/5\",\"19/5\"]],\"formal_domain\":\"unbounded\"},\"candidate\":null,\"paper_system_number\":59,\"paper_system_label\":\"System 59\",\"local_name\":\"S059__DCB-LIT-P4-024\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{2}$\",\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.5\\\\}$\",\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{2}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 2.8\\\\}$\",\"Verified type: barrier/safety verification\",\"Verified method in the original paper: Z3/Yices\"],\"record_id\":\"DCB-LIT-P4-024\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.5\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{2}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{2}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 2.8\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"BB-S037\",\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-s037\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":2559,\"start\":2533},\"title_latex\":\"~\\\\texttt{S037}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":59,\"system\":{\"dimension\":{\"source\":\"catalogue:BB-S037\",\"status\":\"documented\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{2} + u_{0}\\\\\\\\ -0.2\\\\,x_{1} - 0.1\\\\,(x_{1}^{2} - 1)\\\\,x_{2} + u_{1} \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{2} + u_{0}\\\\\\\\ -0.2\\\\,x_{1} - 0.1\\\\,(x_{1}^{2} - 1)\\\\,x_{2} + u_{1} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{2}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 2.8\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: Z3/Yices\",\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-025","paper_system_number":60,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":4,"time_domain":"continuous","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","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_{1}\\\\ \\dot{x}_{2}\\\\ \\dot{x}_{3}\\\\ \\dot{x}_{4} \\end{pmatrix} = \\begin{pmatrix} x_{2} + u_{0}\\\\ -0.6\\,x_{1} - 0.1\\,x_{2} + 0.05\\,x_{3} + u_{1}\\\\ x_{4} + u_{2}\\\\ -0.8\\,x_{3} - 0.15\\,x_{4} + 0.05\\,x_{1} + u_{3} \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"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},\"analysis_bounds\":[[\"-9/2\",\"9/2\"],[\"-9/2\",\"9/2\"],[\"-9/2\",\"9/2\"],[\"-9/2\",\"9/2\"]],\"formal_domain\":\"unbounded\"}","executable_record_json":"{\"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},\"analysis_bounds\":[[\"-9/2\",\"9/2\"],[\"-9/2\",\"9/2\"],[\"-9/2\",\"9/2\"],[\"-9/2\",\"9/2\"]],\"formal_domain\":\"unbounded\"},\"candidate\":null,\"paper_system_number\":60,\"paper_system_label\":\"System 60\",\"local_name\":\"S060__DCB-LIT-P4-025\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{4}$\",\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.6\\\\}$\",\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{4}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 3.5\\\\}$\",\"Verified type: barrier/safety verification\",\"Verified method in the original paper: Z3/Yices\"],\"record_id\":\"DCB-LIT-P4-025\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.6\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{4}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{4}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 3.5\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"BB-S042\",\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-s042\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":2622,\"start\":2592},\"title_latex\":\"~\\\\texttt{S042}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":60,\"system\":{\"dimension\":{\"source\":\"catalogue:BB-S042\",\"status\":\"documented\",\"symbolic\":null,\"value\":4},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3}\\\\\\\\ \\\\dot{x}_{4} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{2} + u_{0}\\\\\\\\ -0.6\\\\,x_{1} - 0.1\\\\,x_{2} + 0.05\\\\,x_{3} + u_{1}\\\\\\\\ x_{4} + u_{2}\\\\\\\\ -0.8\\\\,x_{3} - 0.15\\\\,x_{4} + 0.05\\\\,x_{1} + u_{3} \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3}\\\\\\\\ \\\\dot{x}_{4} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{2} + u_{0}\\\\\\\\ -0.6\\\\,x_{1} - 0.1\\\\,x_{2} + 0.05\\\\,x_{3} + u_{1}\\\\\\\\ x_{4} + u_{2}\\\\\\\\ -0.8\\\\,x_{3} - 0.15\\\\,x_{4} + 0.05\\\\,x_{1} + u_{3} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{4}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 3.5\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: Z3/Yices\",\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-026","paper_system_number":61,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":3,"time_domain":"continuous","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)","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_{1}\\\\ \\dot{x}_{2}\\\\ \\dot{x}_{3} \\end{pmatrix} = \\begin{pmatrix} -0.5\\,x_{1} + 0.3\\,x_{2}\\,\\sin(x_{3}) - 0.1\\,x_{1}^{2}\\,x_{2}\\\\ -0.6\\,x_{2} + 0.5\\,x_{1}\\,\\cos(x_{3}) + 0.15\\,x_{3}^{2}\\,x_{1}\\\\ -0.7\\,x_{3} - 0.1\\,x_{3}^{3} \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"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\",\"0\"],\"radius\":\"3.5\",\"complement\":true},\"analysis_bounds\":[[\"-9/2\",\"9/2\"],[\"-9/2\",\"9/2\"],[\"-9/2\",\"9/2\"]],\"formal_domain\":\"unbounded\"}","executable_record_json":"{\"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\",\"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\":61,\"paper_system_label\":\"System 61\",\"local_name\":\"S061__DCB-LIT-P4-026\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.4\\\\}$\",\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 3.5\\\\}$\",\"Verified type: barrier/safety verification\",\"Verified method in the original paper: Z3/Yices with Taylor approximations\"],\"record_id\":\"DCB-LIT-P4-026\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.4\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 3.5\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"BB-S048\",\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-s048\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":2682,\"start\":2655},\"title_latex\":\"~\\\\texttt{S048}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":61,\"system\":{\"dimension\":{\"source\":\"catalogue:BB-S048\",\"status\":\"documented\",\"symbolic\":null,\"value\":3},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3} \\\\end{pmatrix} = \\\\begin{pmatrix} -0.5\\\\,x_{1} + 0.3\\\\,x_{2}\\\\,\\\\sin(x_{3}) - 0.1\\\\,x_{1}^{2}\\\\,x_{2}\\\\\\\\ -0.6\\\\,x_{2} + 0.5\\\\,x_{1}\\\\,\\\\cos(x_{3}) + 0.15\\\\,x_{3}^{2}\\\\,x_{1}\\\\\\\\ -0.7\\\\,x_{3} - 0.1\\\\,x_{3}^{3} \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3} \\\\end{pmatrix} = \\\\begin{pmatrix} -0.5\\\\,x_{1} + 0.3\\\\,x_{2}\\\\,\\\\sin(x_{3}) - 0.1\\\\,x_{1}^{2}\\\\,x_{2}\\\\\\\\ -0.6\\\\,x_{2} + 0.5\\\\,x_{1}\\\\,\\\\cos(x_{3}) + 0.15\\\\,x_{3}^{2}\\\\,x_{1}\\\\\\\\ -0.7\\\\,x_{3} - 0.1\\\\,x_{3}^{3} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 3.5\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: Z3/Yices with Taylor approximations\",\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-027","paper_system_number":62,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":3,"time_domain":"continuous","dynamics":"dx1/dt = x2+u0, dx2/dt = x3+u1, dx3/dt = -6*x3-2.92*x2-1.2*x1+x1**(2)+u2","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_{1}\\\\ \\dot{x}_{2}\\\\ \\dot{x}_{3} \\end{pmatrix} = \\begin{pmatrix} x_{2} + u_{0}\\\\ x_{3} + u_{1}\\\\ -6\\,x_{3} - 2.92\\,x_{2} - 1.2\\,x_{1} + x_{1}^{2} + u_{2} \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"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\"}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.1\\\\}$\",\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 2\\\\}$\",\"Verified type: barrier/safety verification\",\"Verified method in the original paper: Z3/Yices\"],\"record_id\":\"DCB-LIT-P4-027\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.1\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 2\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"BB-S052\",\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-s052\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":2712,\"start\":2684},\"title_latex\":\"~\\\\texttt{S052}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":62,\"system\":{\"dimension\":{\"source\":\"catalogue:BB-S052\",\"status\":\"documented\",\"symbolic\":null,\"value\":3},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{2} + u_{0}\\\\\\\\ x_{3} + u_{1}\\\\\\\\ -6\\\\,x_{3} - 2.92\\\\,x_{2} - 1.2\\\\,x_{1} + x_{1}^{2} + u_{2} \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{2} + u_{0}\\\\\\\\ x_{3} + u_{1}\\\\\\\\ -6\\\\,x_{3} - 2.92\\\\,x_{2} - 1.2\\\\,x_{1} + x_{1}^{2} + u_{2} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 2\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: Z3/Yices\",\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-028","paper_system_number":63,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":2,"time_domain":"continuous","dynamics":"dx1/dt = -x1+0.08*x2+x1**(2)*x2+u0, dx2/dt = 0.6-0.08*x2-x1**(2)*x2+u1","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_{1}\\\\ \\dot{x}_{2} \\end{pmatrix} = \\begin{pmatrix} -x_{1} + 0.08\\,x_{2} + x_{1}^{2}\\,x_{2} + u_{0}\\\\ 0.6 - 0.08\\,x_{2} - x_{1}^{2}\\,x_{2} + u_{1} \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"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\"}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{2}$\",\"$\\\\mathcal{X}_0 = [0.1,0.5]\\\\times [1,2]$\",\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{2}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 5\\\\}$\",\"Verified type: barrier/safety verification\",\"Verified method in the original paper: Z3/Yices\"],\"record_id\":\"DCB-LIT-P4-028\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = [0.1,0.5]\\\\times [1,2]$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{2}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{2}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 5\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"BB-S057\",\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-s057\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":2773,\"start\":2747},\"title_latex\":\"~\\\\texttt{S057}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":63,\"system\":{\"dimension\":{\"source\":\"catalogue:BB-S057\",\"status\":\"documented\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2} \\\\end{pmatrix} = \\\\begin{pmatrix} -x_{1} + 0.08\\\\,x_{2} + x_{1}^{2}\\\\,x_{2} + u_{0}\\\\\\\\ 0.6 - 0.08\\\\,x_{2} - x_{1}^{2}\\\\,x_{2} + u_{1} \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2} \\\\end{pmatrix} = \\\\begin{pmatrix} -x_{1} + 0.08\\\\,x_{2} + x_{1}^{2}\\\\,x_{2} + u_{0}\\\\\\\\ 0.6 - 0.08\\\\,x_{2} - x_{1}^{2}\\\\,x_{2} + u_{1} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{2}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 5\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: Z3/Yices\",\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-029","paper_system_number":64,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":3,"time_domain":"continuous","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","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_{1}\\\\ \\dot{x}_{2}\\\\ \\dot{x}_{3} \\end{pmatrix} = \\begin{pmatrix} x_{2} + u_{0}\\\\ -0.1\\,x_{1} - 0.05\\,x_{2} + 0.02\\,x_{1}\\,x_{3} + u_{1}\\\\ -0.2\\,x_{3} + 0.1\\,x_{1} + u_{2} \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"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\"}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.5\\\\}$\",\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 3\\\\}$\",\"Verified type: barrier/safety verification\",\"Verified method in the original paper: Z3/Yices\"],\"record_id\":\"DCB-LIT-P4-029\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.5\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 3\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"BB-S058\",\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-s058\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":2803,\"start\":2775},\"title_latex\":\"~\\\\texttt{S058}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":64,\"system\":{\"dimension\":{\"source\":\"catalogue:BB-S058\",\"status\":\"documented\",\"symbolic\":null,\"value\":3},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{2} + u_{0}\\\\\\\\ -0.1\\\\,x_{1} - 0.05\\\\,x_{2} + 0.02\\\\,x_{1}\\\\,x_{3} + u_{1}\\\\\\\\ -0.2\\\\,x_{3} + 0.1\\\\,x_{1} + u_{2} \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{2} + u_{0}\\\\\\\\ -0.1\\\\,x_{1} - 0.05\\\\,x_{2} + 0.02\\\\,x_{1}\\\\,x_{3} + u_{1}\\\\\\\\ -0.2\\\\,x_{3} + 0.1\\\\,x_{1} + u_{2} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 3\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: Z3/Yices\",\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-030","paper_system_number":65,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":3,"time_domain":"continuous","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","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_{1}\\\\ \\dot{x}_{2}\\\\ \\dot{x}_{3} \\end{pmatrix} = \\begin{pmatrix} -0.25\\,x_{1} - x_{2}^{2} - x_{3}^{2} + 0.75 + u_{0}\\\\ -x_{2} + x_{1}\\,x_{3} + u_{1}\\\\ -x_{3} + x_{1}\\,x_{2} - 0.1\\,x_{1} + u_{2} \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"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\"}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.2\\\\}$\",\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 3\\\\}$\",\"Verified type: barrier/safety verification\",\"Verified method in the original paper: Z3/Yices\"],\"record_id\":\"DCB-LIT-P4-030\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.2\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 3\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"BB-S059\",\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-s059\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":2833,\"start\":2805},\"title_latex\":\"~\\\\texttt{S059}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":65,\"system\":{\"dimension\":{\"source\":\"catalogue:BB-S059\",\"status\":\"documented\",\"symbolic\":null,\"value\":3},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3} \\\\end{pmatrix} = \\\\begin{pmatrix} -0.25\\\\,x_{1} - x_{2}^{2} - x_{3}^{2} + 0.75 + u_{0}\\\\\\\\ -x_{2} + x_{1}\\\\,x_{3} + u_{1}\\\\\\\\ -x_{3} + x_{1}\\\\,x_{2} - 0.1\\\\,x_{1} + u_{2} \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3} \\\\end{pmatrix} = \\\\begin{pmatrix} -0.25\\\\,x_{1} - x_{2}^{2} - x_{3}^{2} + 0.75 + u_{0}\\\\\\\\ -x_{2} + x_{1}\\\\,x_{3} + u_{1}\\\\\\\\ -x_{3} + x_{1}\\\\,x_{2} - 0.1\\\\,x_{1} + u_{2} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 3\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: Z3/Yices\",\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-031","paper_system_number":66,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":2,"time_domain":"continuous","dynamics":"dx1/dt = x1*(x1-0.1)*(1-x1)-0.3*x1*x2+u0, dx2/dt = -0.2*x2+0.1*x1*x2+u1","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_{1}\\\\ \\dot{x}_{2} \\end{pmatrix} = \\begin{pmatrix} x_{1}\\,(x_{1} - 0.1)\\,(1 - x_{1}) - 0.3\\,x_{1}\\,x_{2} + u_{0}\\\\ -0.2\\,x_{2} + 0.1\\,x_{1}\\,x_{2} + u_{1} \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"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\"}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{2}$\",\"$\\\\mathcal{X}_0 = [0.2,0.8]\\\\times [0.1,0.6]$\",\"$\\\\mathcal{X}_U = [1.5,2]\\\\times [1,1.5]$\",\"Verified type: barrier/safety verification\",\"Verified method in the original paper: Z3/Yices\"],\"record_id\":\"DCB-LIT-P4-031\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = [0.2,0.8]\\\\times [0.1,0.6]$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{2}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = [1.5,2]\\\\times [1,1.5]$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"BB-S060\",\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-s060\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":2861,\"start\":2835},\"title_latex\":\"~\\\\texttt{S060}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":66,\"system\":{\"dimension\":{\"source\":\"catalogue:BB-S060\",\"status\":\"documented\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{1}\\\\,(x_{1} - 0.1)\\\\,(1 - x_{1}) - 0.3\\\\,x_{1}\\\\,x_{2} + u_{0}\\\\\\\\ -0.2\\\\,x_{2} + 0.1\\\\,x_{1}\\\\,x_{2} + u_{1} \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{1}\\\\,(x_{1} - 0.1)\\\\,(1 - x_{1}) - 0.3\\\\,x_{1}\\\\,x_{2} + u_{0}\\\\\\\\ -0.2\\\\,x_{2} + 0.1\\\\,x_{1}\\\\,x_{2} + u_{1} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = [1.5,2]\\\\times [1,1.5]$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: Z3/Yices\",\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-032","paper_system_number":67,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":3,"time_domain":"continuous","dynamics":"dx1/dt = x2+u0, dx2/dt = -x1+x2*x3+u1, dx3/dt = 1-x2**(2)+u2","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_{1}\\\\ \\dot{x}_{2}\\\\ \\dot{x}_{3} \\end{pmatrix} = \\begin{pmatrix} x_{2} + u_{0}\\\\ -x_{1} + x_{2}\\,x_{3} + u_{1}\\\\ 1 - x_{2}^{2} + u_{2} \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"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\"}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.2\\\\}$\",\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 2.5\\\\}$\",\"Verified type: barrier/safety verification\",\"Verified method in the original paper: Z3/Yices\"],\"record_id\":\"DCB-LIT-P4-032\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.2\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 2.5\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"BB-S062\",\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-s062\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":2891,\"start\":2863},\"title_latex\":\"~\\\\texttt{S062}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":67,\"system\":{\"dimension\":{\"source\":\"catalogue:BB-S062\",\"status\":\"documented\",\"symbolic\":null,\"value\":3},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{2} + u_{0}\\\\\\\\ -x_{1} + x_{2}\\\\,x_{3} + u_{1}\\\\\\\\ 1 - x_{2}^{2} + u_{2} \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{2} + u_{0}\\\\\\\\ -x_{1} + x_{2}\\\\,x_{3} + u_{1}\\\\\\\\ 1 - x_{2}^{2} + u_{2} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 2.5\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: Z3/Yices\",\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-033","paper_system_number":68,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":3,"time_domain":"continuous","dynamics":"dx1/dt = x3+x2*x1+u0, dx2/dt = -x2-x1**(2)+u1, dx3/dt = -x1-0.5*x3+u2","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_{1}\\\\ \\dot{x}_{2}\\\\ \\dot{x}_{3} \\end{pmatrix} = \\begin{pmatrix} x_{3} + x_{2}\\,x_{1} + u_{0}\\\\ -x_{2} - x_{1}^{2} + u_{1}\\\\ -x_{1} - 0.5\\,x_{3} + u_{2} \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"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\"}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.3\\\\}$\",\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 3.5\\\\}$\",\"Verified type: barrier/safety verification\",\"Verified method in the original paper: Z3/Yices\"],\"record_id\":\"DCB-LIT-P4-033\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.3\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 3.5\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"BB-S076\",\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-s076\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":2921,\"start\":2893},\"title_latex\":\"~\\\\texttt{S076}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":68,\"system\":{\"dimension\":{\"source\":\"catalogue:BB-S076\",\"status\":\"documented\",\"symbolic\":null,\"value\":3},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{3} + x_{2}\\\\,x_{1} + u_{0}\\\\\\\\ -x_{2} - x_{1}^{2} + u_{1}\\\\\\\\ -x_{1} - 0.5\\\\,x_{3} + u_{2} \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{3} + x_{2}\\\\,x_{1} + u_{0}\\\\\\\\ -x_{2} - x_{1}^{2} + u_{1}\\\\\\\\ -x_{1} - 0.5\\\\,x_{3} + u_{2} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 3.5\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: Z3/Yices\",\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-034","paper_system_number":69,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":4,"time_domain":"continuous","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","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_{1}\\\\ \\dot{x}_{2}\\\\ \\dot{x}_{3}\\\\ \\dot{x}_{4} \\end{pmatrix} = \\begin{pmatrix} x_{2} + u_{0}\\\\ -0.8\\,x_{1} - 0.3\\,x_{2}\\,(1 + x_{1}^{2}) + 0.1\\,x_{3} + u_{1}\\\\ x_{4} + u_{2}\\\\ -0.9\\,x_{3} - 0.25\\,x_{4}\\,(1 + x_{3}^{2}) + 0.1\\,x_{1} + u_{3} \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"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\"}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{4}$\",\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.35\\\\}$\",\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{4}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 2.8\\\\}$\",\"Verified type: barrier/safety verification\",\"Verified method in the original paper: Z3/Yices\"],\"record_id\":\"DCB-LIT-P4-034\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.35\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{4}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{4}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 2.8\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"BB-S077\",\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-s077\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":2953,\"start\":2923},\"title_latex\":\"~\\\\texttt{S077}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":69,\"system\":{\"dimension\":{\"source\":\"catalogue:BB-S077\",\"status\":\"documented\",\"symbolic\":null,\"value\":4},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3}\\\\\\\\ \\\\dot{x}_{4} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{2} + u_{0}\\\\\\\\ -0.8\\\\,x_{1} - 0.3\\\\,x_{2}\\\\,(1 + x_{1}^{2}) + 0.1\\\\,x_{3} + u_{1}\\\\\\\\ x_{4} + u_{2}\\\\\\\\ -0.9\\\\,x_{3} - 0.25\\\\,x_{4}\\\\,(1 + x_{3}^{2}) + 0.1\\\\,x_{1} + u_{3} \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3}\\\\\\\\ \\\\dot{x}_{4} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{2} + u_{0}\\\\\\\\ -0.8\\\\,x_{1} - 0.3\\\\,x_{2}\\\\,(1 + x_{1}^{2}) + 0.1\\\\,x_{3} + u_{1}\\\\\\\\ x_{4} + u_{2}\\\\\\\\ -0.9\\\\,x_{3} - 0.25\\\\,x_{4}\\\\,(1 + x_{3}^{2}) + 0.1\\\\,x_{1} + u_{3} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{4}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 2.8\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: Z3/Yices\",\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-035","paper_system_number":70,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":2,"time_domain":"discrete","dynamics":"x1[k+1] = -1/100*x2, x2[k+1] = 1/100*x1","specification_latex":"The update map is: \\[ \\mathbf{x}_{k+1} = \\begin{pmatrix} x_{1,k+1}\\\\ x_{2,k+1} \\end{pmatrix} = \\begin{pmatrix} x_{1,k} + 0.01\\,(-100\\,x_{1,k} - 1\\,x_{2,k})\\\\ x_{2,k} + 0.01\\,(1\\,x_{1,k} - 100\\,x_{2,k}) \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"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.\"]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{2}$\",\"$\\\\mathcal{X}_0 = [0.1,0.4]\\\\times [0.1,0.55]$\",\"$\\\\mathcal{X}_U = [0.45,0.5]\\\\times [0.6,1]$\",\"Verified type: barrier/safety verification\"],\"record_id\":\"DCB-LIT-P4-035\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = [0.1,0.4]\\\\times [0.1,0.55]$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{2}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = [0.45,0.5]\\\\times [0.6,1]$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"BB-S087\",\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-s087\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":2982,\"start\":2956},\"title_latex\":\"~\\\\texttt{S087}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":70,\"system\":{\"dimension\":{\"source\":\"catalogue:BB-S087\",\"status\":\"documented\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\mathbf{x}_{k+1} = \\\\begin{pmatrix} x_{1,k+1}\\\\\\\\ x_{2,k+1} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{1,k} + 0.01\\\\,(-100\\\\,x_{1,k} - 1\\\\,x_{2,k})\\\\\\\\ x_{2,k} + 0.01\\\\,(1\\\\,x_{1,k} - 100\\\\,x_{2,k}) \\\\end{pmatrix}.\"],\"specification_latex\":\"The update map is: \\\\[ \\\\mathbf{x}_{k+1} = \\\\begin{pmatrix} x_{1,k+1}\\\\\\\\ x_{2,k+1} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{1,k} + 0.01\\\\,(-100\\\\,x_{1,k} - 1\\\\,x_{2,k})\\\\\\\\ x_{2,k} + 0.01\\\\,(1\\\\,x_{1,k} - 100\\\\,x_{2,k}) \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"discrete\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = [0.45,0.5]\\\\times [0.6,1]$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":null,\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-036","paper_system_number":71,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":4,"time_domain":"continuous","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","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_{1}\\\\ \\dot{x}_{2}\\\\ \\dot{x}_{3}\\\\ \\dot{x}_{4} \\end{pmatrix} = \\begin{pmatrix} -0.5\\,x_{1} + 0.3\\,x_{2}\\\\ -0.4\\,x_{2} + 0.15\\,\\cos(x_{1})\\,\\sin(x_{3})\\\\ -0.6\\,x_{3} + 0.1\\,\\sin(x_{2})\\,\\cos(x_{4})\\\\ -0.5\\,x_{4} \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"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\"}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{4}$\",\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.3\\\\}$\",\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{4}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 2.5\\\\}$\",\"Verified type: barrier/safety verification\"],\"record_id\":\"DCB-LIT-P4-036\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.3\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{4}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{4}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 2.5\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"BB-U000\",\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-u000\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":3014,\"start\":2984},\"title_latex\":\"~\\\\texttt{U000}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":71,\"system\":{\"dimension\":{\"source\":\"catalogue:BB-U000\",\"status\":\"documented\",\"symbolic\":null,\"value\":4},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3}\\\\\\\\ \\\\dot{x}_{4} \\\\end{pmatrix} = \\\\begin{pmatrix} -0.5\\\\,x_{1} + 0.3\\\\,x_{2}\\\\\\\\ -0.4\\\\,x_{2} + 0.15\\\\,\\\\cos(x_{1})\\\\,\\\\sin(x_{3})\\\\\\\\ -0.6\\\\,x_{3} + 0.1\\\\,\\\\sin(x_{2})\\\\,\\\\cos(x_{4})\\\\\\\\ -0.5\\\\,x_{4} \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3}\\\\\\\\ \\\\dot{x}_{4} \\\\end{pmatrix} = \\\\begin{pmatrix} -0.5\\\\,x_{1} + 0.3\\\\,x_{2}\\\\\\\\ -0.4\\\\,x_{2} + 0.15\\\\,\\\\cos(x_{1})\\\\,\\\\sin(x_{3})\\\\\\\\ -0.6\\\\,x_{3} + 0.1\\\\,\\\\sin(x_{2})\\\\,\\\\cos(x_{4})\\\\\\\\ -0.5\\\\,x_{4} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{4}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 2.5\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":null,\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-037","paper_system_number":72,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":3,"time_domain":"continuous","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)","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_{1}\\\\ \\dot{x}_{2}\\\\ \\dot{x}_{3} \\end{pmatrix} = \\begin{pmatrix} -0.3\\,x_{1} + 0.5\\,\\sin(x_{2})\\\\ -0.4\\,x_{2} - 0.6\\,\\sin(x_{1}) + 0.1\\,\\sin(x_{3})\\\\ -0.5\\,x_{3} + 0.2\\,\\sin(x_{1})\\,\\sin(x_{2}) \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"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\"}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.4\\\\}$\",\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 3\\\\}$\",\"Verified type: barrier/safety verification\"],\"record_id\":\"DCB-LIT-P4-037\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.4\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 3\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"BB-U001\",\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-u001\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":3044,\"start\":3016},\"title_latex\":\"~\\\\texttt{U001}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":72,\"system\":{\"dimension\":{\"source\":\"catalogue:BB-U001\",\"status\":\"documented\",\"symbolic\":null,\"value\":3},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3} \\\\end{pmatrix} = \\\\begin{pmatrix} -0.3\\\\,x_{1} + 0.5\\\\,\\\\sin(x_{2})\\\\\\\\ -0.4\\\\,x_{2} - 0.6\\\\,\\\\sin(x_{1}) + 0.1\\\\,\\\\sin(x_{3})\\\\\\\\ -0.5\\\\,x_{3} + 0.2\\\\,\\\\sin(x_{1})\\\\,\\\\sin(x_{2}) \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3} \\\\end{pmatrix} = \\\\begin{pmatrix} -0.3\\\\,x_{1} + 0.5\\\\,\\\\sin(x_{2})\\\\\\\\ -0.4\\\\,x_{2} - 0.6\\\\,\\\\sin(x_{1}) + 0.1\\\\,\\\\sin(x_{3})\\\\\\\\ -0.5\\\\,x_{3} + 0.2\\\\,\\\\sin(x_{1})\\\\,\\\\sin(x_{2}) \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 3\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":null,\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-038","paper_system_number":73,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":3,"time_domain":"continuous","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)","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_{1}\\\\ \\dot{x}_{2}\\\\ \\dot{x}_{3} \\end{pmatrix} = \\begin{pmatrix} -0.3\\,x_{1} + 0.2\\,\\sin(x_{2})\\,\\exp(-x_{3}^{2})\\\\ -0.4\\,x_{2} + 0.15\\,\\cos(x_{1})\\,\\exp(-x_{1}^{2})\\\\ -0.5\\,x_{3} + 0.1\\,\\sin(x_{1} + x_{2}) \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"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\"}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.45\\\\}$\",\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 3.2\\\\}$\",\"Verified type: barrier/safety verification\"],\"record_id\":\"DCB-LIT-P4-038\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.45\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 3.2\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"BB-U002\",\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-u002\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":3074,\"start\":3046},\"title_latex\":\"~\\\\texttt{U002}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":73,\"system\":{\"dimension\":{\"source\":\"catalogue:BB-U002\",\"status\":\"documented\",\"symbolic\":null,\"value\":3},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3} \\\\end{pmatrix} = \\\\begin{pmatrix} -0.3\\\\,x_{1} + 0.2\\\\,\\\\sin(x_{2})\\\\,\\\\exp(-x_{3}^{2})\\\\\\\\ -0.4\\\\,x_{2} + 0.15\\\\,\\\\cos(x_{1})\\\\,\\\\exp(-x_{1}^{2})\\\\\\\\ -0.5\\\\,x_{3} + 0.1\\\\,\\\\sin(x_{1} + x_{2}) \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3} \\\\end{pmatrix} = \\\\begin{pmatrix} -0.3\\\\,x_{1} + 0.2\\\\,\\\\sin(x_{2})\\\\,\\\\exp(-x_{3}^{2})\\\\\\\\ -0.4\\\\,x_{2} + 0.15\\\\,\\\\cos(x_{1})\\\\,\\\\exp(-x_{1}^{2})\\\\\\\\ -0.5\\\\,x_{3} + 0.1\\\\,\\\\sin(x_{1} + x_{2}) \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 3.2\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":null,\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-039","paper_system_number":74,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":3,"time_domain":"discrete","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)","specification_latex":"The update map is: \\[ \\mathbf{x}_{k+1} = \\begin{pmatrix} x_{1,k+1}\\\\ x_{2,k+1}\\\\ x_{3,k+1} \\end{pmatrix} = \\begin{pmatrix} 0.8\\,x_{1,k} + 0.1\\,\\sin(x_{2,k})\\\\ 0.85\\,x_{2,k} + 0.08\\,x_{1,k}\\,x_{3,k}\\\\ 0.9\\,x_{3,k} + 0.08\\,\\cos(x_{1,k}\\,x_{2,k}) \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"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.\"]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"$\\\\mathcal{X}_0 = [0.1,0.4]\\\\times [0.1,0.3]\\\\times [0.1,0.4]$\",\"$\\\\mathcal{X}_U = [2,2.5]\\\\times [1.8,2.2]\\\\times [1.8,2.3]$\",\"Verified type: barrier/safety verification\"],\"record_id\":\"DCB-LIT-P4-039\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = [0.1,0.4]\\\\times [0.1,0.3]\\\\times [0.1,0.4]$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = [2,2.5]\\\\times [1.8,2.2]\\\\times [1.8,2.3]$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"BB-U003\",\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-u003\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":3105,\"start\":3077},\"title_latex\":\"~\\\\texttt{U003}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":74,\"system\":{\"dimension\":{\"source\":\"catalogue:BB-U003\",\"status\":\"documented\",\"symbolic\":null,\"value\":3},\"display_math_latex\":[\"\\\\mathbf{x}_{k+1} = \\\\begin{pmatrix} x_{1,k+1}\\\\\\\\ x_{2,k+1}\\\\\\\\ x_{3,k+1} \\\\end{pmatrix} = \\\\begin{pmatrix} 0.8\\\\,x_{1,k} + 0.1\\\\,\\\\sin(x_{2,k})\\\\\\\\ 0.85\\\\,x_{2,k} + 0.08\\\\,x_{1,k}\\\\,x_{3,k}\\\\\\\\ 0.9\\\\,x_{3,k} + 0.08\\\\,\\\\cos(x_{1,k}\\\\,x_{2,k}) \\\\end{pmatrix}.\"],\"specification_latex\":\"The update map is: \\\\[ \\\\mathbf{x}_{k+1} = \\\\begin{pmatrix} x_{1,k+1}\\\\\\\\ x_{2,k+1}\\\\\\\\ x_{3,k+1} \\\\end{pmatrix} = \\\\begin{pmatrix} 0.8\\\\,x_{1,k} + 0.1\\\\,\\\\sin(x_{2,k})\\\\\\\\ 0.85\\\\,x_{2,k} + 0.08\\\\,x_{1,k}\\\\,x_{3,k}\\\\\\\\ 0.9\\\\,x_{3,k} + 0.08\\\\,\\\\cos(x_{1,k}\\\\,x_{2,k}) \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"discrete\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = [2,2.5]\\\\times [1.8,2.2]\\\\times [1.8,2.3]$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":null,\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-040","paper_system_number":75,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":4,"time_domain":"continuous","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)","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_{1}\\\\ \\dot{x}_{2}\\\\ \\dot{x}_{3}\\\\ \\dot{x}_{4} \\end{pmatrix} = \\begin{pmatrix} -0.65\\,x_{1} + 0.1\\,x_{2}^{2}\\,\\cos(x_{3})\\\\ -0.7\\,x_{2} + 0.12\\,\\sin(x_{1})\\,\\tanh(x_{4})\\\\ -0.75\\,x_{3} + 0.08\\,x_{1}\\,x_{2}\\\\ -0.6\\,x_{4} + 0.06\\,(x_{1} + x_{2}) \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"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].\"]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{4}$\",\"$\\\\mathcal{X}_0 = [0.2,0.5]\\\\times [-0.3,0]\\\\times [0.1,0.4]\\\\times [-0.2,0.1]$\",\"$\\\\mathcal{X}_U = [-2.2,-1.7]\\\\times [1.6,2.1]\\\\times [-1.9,-1.4]\\\\times [1.3,1.8]$\",\"Verified type: barrier/safety verification\"],\"record_id\":\"DCB-LIT-P4-040\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = [0.2,0.5]\\\\times [-0.3,0]\\\\times [0.1,0.4]\\\\times [-0.2,0.1]$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{4}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = [-2.2,-1.7]\\\\times [1.6,2.1]\\\\times [-1.9,-1.4]\\\\times [1.3,1.8]$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"BB-U004\",\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-u004\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":3137,\"start\":3107},\"title_latex\":\"~\\\\texttt{U004}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":75,\"system\":{\"dimension\":{\"source\":\"catalogue:BB-U004\",\"status\":\"documented\",\"symbolic\":null,\"value\":4},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3}\\\\\\\\ \\\\dot{x}_{4} \\\\end{pmatrix} = \\\\begin{pmatrix} -0.65\\\\,x_{1} + 0.1\\\\,x_{2}^{2}\\\\,\\\\cos(x_{3})\\\\\\\\ -0.7\\\\,x_{2} + 0.12\\\\,\\\\sin(x_{1})\\\\,\\\\tanh(x_{4})\\\\\\\\ -0.75\\\\,x_{3} + 0.08\\\\,x_{1}\\\\,x_{2}\\\\\\\\ -0.6\\\\,x_{4} + 0.06\\\\,(x_{1} + x_{2}) \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3}\\\\\\\\ \\\\dot{x}_{4} \\\\end{pmatrix} = \\\\begin{pmatrix} -0.65\\\\,x_{1} + 0.1\\\\,x_{2}^{2}\\\\,\\\\cos(x_{3})\\\\\\\\ -0.7\\\\,x_{2} + 0.12\\\\,\\\\sin(x_{1})\\\\,\\\\tanh(x_{4})\\\\\\\\ -0.75\\\\,x_{3} + 0.08\\\\,x_{1}\\\\,x_{2}\\\\\\\\ -0.6\\\\,x_{4} + 0.06\\\\,(x_{1} + x_{2}) \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = [-2.2,-1.7]\\\\times [1.6,2.1]\\\\times [-1.9,-1.4]\\\\times [1.3,1.8]$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":null,\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-041","paper_system_number":76,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":3,"time_domain":"continuous","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))","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_{1}\\\\ \\dot{x}_{2}\\\\ \\dot{x}_{3} \\end{pmatrix} = \\begin{pmatrix} -0.58\\,x_{1} + 0.12\\,\\sin(x_{2})\\,\\cos(x_{3})\\\\ -0.62\\,x_{2} + 0.1\\,\\exp(-x_{1}^{2})\\,\\tanh(x_{3})\\\\ -0.66\\,x_{3} + 0.08\\,\\sin(x_{1}x_{2})\\,\\exp(-x_{3}^{2}) \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"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\"}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"$\\\\mathcal{X}_0 = [-0.3,0.2]\\\\times [0.1,0.4]\\\\times [-0.2,0.3]$\",\"$\\\\mathcal{X}_U = [-2.3,-1.8]\\\\times [1.7,2.2]\\\\times [-2,-1.5]$\",\"Verified type: barrier/safety verification\"],\"record_id\":\"DCB-LIT-P4-041\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = [-0.3,0.2]\\\\times [0.1,0.4]\\\\times [-0.2,0.3]$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = [-2.3,-1.8]\\\\times [1.7,2.2]\\\\times [-2,-1.5]$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"BB-U005\",\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-u005\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":3167,\"start\":3139},\"title_latex\":\"~\\\\texttt{U005}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":76,\"system\":{\"dimension\":{\"source\":\"catalogue:BB-U005\",\"status\":\"documented\",\"symbolic\":null,\"value\":3},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3} \\\\end{pmatrix} = \\\\begin{pmatrix} -0.58\\\\,x_{1} + 0.12\\\\,\\\\sin(x_{2})\\\\,\\\\cos(x_{3})\\\\\\\\ -0.62\\\\,x_{2} + 0.1\\\\,\\\\exp(-x_{1}^{2})\\\\,\\\\tanh(x_{3})\\\\\\\\ -0.66\\\\,x_{3} + 0.08\\\\,\\\\sin(x_{1}x_{2})\\\\,\\\\exp(-x_{3}^{2}) \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3} \\\\end{pmatrix} = \\\\begin{pmatrix} -0.58\\\\,x_{1} + 0.12\\\\,\\\\sin(x_{2})\\\\,\\\\cos(x_{3})\\\\\\\\ -0.62\\\\,x_{2} + 0.1\\\\,\\\\exp(-x_{1}^{2})\\\\,\\\\tanh(x_{3})\\\\\\\\ -0.66\\\\,x_{3} + 0.08\\\\,\\\\sin(x_{1}x_{2})\\\\,\\\\exp(-x_{3}^{2}) \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = [-2.3,-1.8]\\\\times [1.7,2.2]\\\\times [-2,-1.5]$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":null,\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-042","paper_system_number":77,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":3,"time_domain":"continuous","dynamics":"dx1/dt = u0*cos(x3), dx2/dt = u0*sin(x3), dx3/dt = u1","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_{1}\\\\ \\dot{x}_{2}\\\\ \\dot{x}_{3} \\end{pmatrix} = \\begin{pmatrix} u_{0}\\cos(x_{3})\\\\ u_{0}\\sin(x_{3})\\\\ u_{1} \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"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\"}","executable_record_json":"{\"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,\"paper_system_number\":77,\"paper_system_label\":\"System 77\",\"local_name\":\"S077__DCB-LIT-P4-042\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"$\\\\mathcal{X}_0 = [-1,1]\\\\times [-1,1]\\\\times [-0.5,0.5]$\",\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\bigl([-5,5]\\\\times[-5,5]\\\\times[-3.14,3.14]\\\\bigr)$\",\"Verified type: barrier/safety verification\"],\"record_id\":\"DCB-LIT-P4-042\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = [-1,1]\\\\times [-1,1]\\\\times [-0.5,0.5]$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\bigl([-5,5]\\\\times[-5,5]\\\\times[-3.14,3.14]\\\\bigr)$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"BB-U006\",\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-u006\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":3200,\"start\":3169},\"title_latex\":\"~\\\\texttt{U006}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":77,\"system\":{\"dimension\":{\"source\":\"catalogue:BB-U006\",\"status\":\"documented\",\"symbolic\":null,\"value\":3},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3} \\\\end{pmatrix} = \\\\begin{pmatrix} u_{0}\\\\cos(x_{3})\\\\\\\\ u_{0}\\\\sin(x_{3})\\\\\\\\ u_{1} \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3} \\\\end{pmatrix} = \\\\begin{pmatrix} u_{0}\\\\cos(x_{3})\\\\\\\\ u_{0}\\\\sin(x_{3})\\\\\\\\ u_{1} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{3}\\\\setminus \\\\bigl([-5,5]\\\\times[-5,5]\\\\times[-3.14,3.14]\\\\bigr)$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":null,\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-043","paper_system_number":78,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":3,"time_domain":"discrete","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])","specification_latex":"The update map is: \\[ \\mathbf{x}_{k+1} = \\begin{pmatrix} x_{1,k+1}\\\\ x_{2,k+1}\\\\ x_{3,k+1} \\end{pmatrix} = \\begin{pmatrix} 0.75\\,x_{1,k} + 0.1\\,\\tanh(x_{2,k})\\,\\tanh(x_{3,k})\\\\ 0.82\\,x_{2,k} + 0.09\\,\\sin(x_{1,k})\\,\\exp(-x_{3,k}^{2})\\\\ 0.79\\,x_{3,k} + 0.08\\,\\cos(x_{1,k})\\,\\cos(x_{2,k}) \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"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\"}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x-(1,-0.5,0.8)^{\\\\mathsf T}\\\\rVert_2\\\\le 0.5\\\\}$\",\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x-(-2.5,2,-2.2)^{\\\\mathsf T}\\\\rVert_2\\\\le 0.6\\\\}$\",\"Verified type: barrier/safety verification\"],\"record_id\":\"DCB-LIT-P4-043\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x-(1,-0.5,0.8)^{\\\\mathsf T}\\\\rVert_2\\\\le 0.5\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x-(-2.5,2,-2.2)^{\\\\mathsf T}\\\\rVert_2\\\\le 0.6\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"BB-U007\",\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-u007\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":3230,\"start\":3202},\"title_latex\":\"~\\\\texttt{U007}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":78,\"system\":{\"dimension\":{\"source\":\"catalogue:BB-U007\",\"status\":\"documented\",\"symbolic\":null,\"value\":3},\"display_math_latex\":[\"\\\\mathbf{x}_{k+1} = \\\\begin{pmatrix} x_{1,k+1}\\\\\\\\ x_{2,k+1}\\\\\\\\ x_{3,k+1} \\\\end{pmatrix} = \\\\begin{pmatrix} 0.75\\\\,x_{1,k} + 0.1\\\\,\\\\tanh(x_{2,k})\\\\,\\\\tanh(x_{3,k})\\\\\\\\ 0.82\\\\,x_{2,k} + 0.09\\\\,\\\\sin(x_{1,k})\\\\,\\\\exp(-x_{3,k}^{2})\\\\\\\\ 0.79\\\\,x_{3,k} + 0.08\\\\,\\\\cos(x_{1,k})\\\\,\\\\cos(x_{2,k}) \\\\end{pmatrix}.\"],\"specification_latex\":\"The update map is: \\\\[ \\\\mathbf{x}_{k+1} = \\\\begin{pmatrix} x_{1,k+1}\\\\\\\\ x_{2,k+1}\\\\\\\\ x_{3,k+1} \\\\end{pmatrix} = \\\\begin{pmatrix} 0.75\\\\,x_{1,k} + 0.1\\\\,\\\\tanh(x_{2,k})\\\\,\\\\tanh(x_{3,k})\\\\\\\\ 0.82\\\\,x_{2,k} + 0.09\\\\,\\\\sin(x_{1,k})\\\\,\\\\exp(-x_{3,k}^{2})\\\\\\\\ 0.79\\\\,x_{3,k} + 0.08\\\\,\\\\cos(x_{1,k})\\\\,\\\\cos(x_{2,k}) \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"discrete\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x-(-2.5,2,-2.2)^{\\\\mathsf T}\\\\rVert_2\\\\le 0.6\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":null,\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-044","paper_system_number":79,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":3,"time_domain":"discrete","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])","specification_latex":"The update map is: \\[ \\mathbf{x}_{k+1} = \\begin{pmatrix} x_{1,k+1}\\\\ x_{2,k+1}\\\\ x_{3,k+1} \\end{pmatrix} = \\begin{pmatrix} 0.73\\,x_{1,k} + 0.13\\,\\sin(x_{2,k})\\,\\tanh(x_{3,k})\\\\ 0.89\\,x_{2,k} + 0.17\\,\\cos(x_{1,k}^{2})\\,\\exp(-x_{3,k}^{2})\\\\ 0.64\\,x_{3,k} + 0.22\\,\\tanh(x_{1,k}\\,x_{2,k}) \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"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\"}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x-(0.9,-0.7,1.2)^{\\\\mathsf T}\\\\rVert_2\\\\le 0.55\\\\}$\",\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x-(-3.1,2.8,-2.6)^{\\\\mathsf T}\\\\rVert_2\\\\le 0.75\\\\}$\",\"Verified type: barrier/safety verification\"],\"record_id\":\"DCB-LIT-P4-044\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x-(0.9,-0.7,1.2)^{\\\\mathsf T}\\\\rVert_2\\\\le 0.55\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x-(-3.1,2.8,-2.6)^{\\\\mathsf T}\\\\rVert_2\\\\le 0.75\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"BB-U008\",\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-u008\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":3260,\"start\":3232},\"title_latex\":\"~\\\\texttt{U008}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":79,\"system\":{\"dimension\":{\"source\":\"catalogue:BB-U008\",\"status\":\"documented\",\"symbolic\":null,\"value\":3},\"display_math_latex\":[\"\\\\mathbf{x}_{k+1} = \\\\begin{pmatrix} x_{1,k+1}\\\\\\\\ x_{2,k+1}\\\\\\\\ x_{3,k+1} \\\\end{pmatrix} = \\\\begin{pmatrix} 0.73\\\\,x_{1,k} + 0.13\\\\,\\\\sin(x_{2,k})\\\\,\\\\tanh(x_{3,k})\\\\\\\\ 0.89\\\\,x_{2,k} + 0.17\\\\,\\\\cos(x_{1,k}^{2})\\\\,\\\\exp(-x_{3,k}^{2})\\\\\\\\ 0.64\\\\,x_{3,k} + 0.22\\\\,\\\\tanh(x_{1,k}\\\\,x_{2,k}) \\\\end{pmatrix}.\"],\"specification_latex\":\"The update map is: \\\\[ \\\\mathbf{x}_{k+1} = \\\\begin{pmatrix} x_{1,k+1}\\\\\\\\ x_{2,k+1}\\\\\\\\ x_{3,k+1} \\\\end{pmatrix} = \\\\begin{pmatrix} 0.73\\\\,x_{1,k} + 0.13\\\\,\\\\sin(x_{2,k})\\\\,\\\\tanh(x_{3,k})\\\\\\\\ 0.89\\\\,x_{2,k} + 0.17\\\\,\\\\cos(x_{1,k}^{2})\\\\,\\\\exp(-x_{3,k}^{2})\\\\\\\\ 0.64\\\\,x_{3,k} + 0.22\\\\,\\\\tanh(x_{1,k}\\\\,x_{2,k}) \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"discrete\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{3}\\\\mid \\\\lVert x-(-3.1,2.8,-2.6)^{\\\\mathsf T}\\\\rVert_2\\\\le 0.75\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":null,\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-LIT-P4-045","paper_system_number":80,"task_type":"safety","problem_name":"Safety verification","tier":"literature","dimension":3,"time_domain":"continuous","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)","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_{1}\\\\ \\dot{x}_{2}\\\\ \\dot{x}_{3} \\end{pmatrix} = \\begin{pmatrix} -0.62\\,x_{1} + 0.1\\,\\sin(x_{2} + x_{3})\\\\ -0.67\\,x_{2} + 0.08\\,\\cos(x_{1})\\,\\exp(-x_{3}^{2})\\\\ -0.64\\,x_{3} + 0.09\\,\\tanh(x_{1})\\,\\sin(x_{2}) \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"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\"}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"$\\\\mathcal{X}_0 = [-0.2,0.3]\\\\times [0.1,0.5]\\\\times [-0.3,0.2]$\",\"$\\\\mathcal{X}_U = [2,2.6]\\\\times [-2.4,-1.8]\\\\times [2.1,2.7]$\",\"Verified type: barrier/safety verification\"],\"record_id\":\"DCB-LIT-P4-045\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = [-0.2,0.3]\\\\times [0.1,0.5]\\\\times [-0.3,0.2]$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{3}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = [2,2.6]\\\\times [-2.4,-1.8]\\\\times [2.1,2.7]$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":\"BB-U009\",\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-u009\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":3290,\"start\":3262},\"title_latex\":\"~\\\\texttt{U009}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":80,\"system\":{\"dimension\":{\"source\":\"catalogue:BB-U009\",\"status\":\"documented\",\"symbolic\":null,\"value\":3},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3} \\\\end{pmatrix} = \\\\begin{pmatrix} -0.62\\\\,x_{1} + 0.1\\\\,\\\\sin(x_{2} + x_{3})\\\\\\\\ -0.67\\\\,x_{2} + 0.08\\\\,\\\\cos(x_{1})\\\\,\\\\exp(-x_{3}^{2})\\\\\\\\ -0.64\\\\,x_{3} + 0.09\\\\,\\\\tanh(x_{1})\\\\,\\\\sin(x_{2}) \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3} \\\\end{pmatrix} = \\\\begin{pmatrix} -0.62\\\\,x_{1} + 0.1\\\\,\\\\sin(x_{2} + x_{3})\\\\\\\\ -0.67\\\\,x_{2} + 0.08\\\\,\\\\cos(x_{1})\\\\,\\\\exp(-x_{3}^{2})\\\\\\\\ -0.64\\\\,x_{3} + 0.09\\\\,\\\\tanh(x_{1})\\\\,\\\\sin(x_{2}) \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"literature\",\"tier_name\":\"Literature-derived\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = [2,2.6]\\\\times [-2.4,-1.8]\\\\times [2.1,2.7]$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":null,\"reported_type_latex\":\"Verified type: barrier/safety verification\"}}"} {"record_id":"DCB-EXT-P1-001","paper_system_number":81,"task_type":"stability","problem_name":"Stability certification","tier":"extended","dimension":2,"time_domain":"continuous","dynamics":"dx1/dt = -x1+x1*x2, dx2/dt = -x2","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}\\\\ \\dot{y} \\end{pmatrix} = \\begin{pmatrix} -x+xy\\\\ -y \\end{pmatrix}. \\]","citations":["abate2021fossil"],"problem_json":"{\"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\"}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":1,\"problem_name\":\"Stability certification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0)$\",\"$\\\\mathcal{N}=\\\\mathbb{R}^{2}$\",\"Verified type: Global asymptotic stability\"],\"record_id\":\"DCB-EXT-P1-001\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=(0,0)$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":\"$\\\\mathcal{N}=\\\\mathbb{R}^{2}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"\\\\mathbb{R}^{2}\",\"note\":\"Ambient Euclidean state space inferred from the stated dimension; no finite working domain was added.\",\"source\":\"dimension_and_dynamics\",\"status\":\"inferred\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":null,\"citations\":[\"abate2021fossil\"],\"label\":\"sys:catalogue-sys0016-ext\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":3355,\"start\":3331},\"title_latex\":\"Extend-Table~1 row~1 (NonPoly0), \\\\cite{abate2021fossil}\"},\"source_order\":81,\"system\":{\"dimension\":{\"source\":\"overleaf_state_space_notation\",\"status\":\"inferred\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}\\\\\\\\ \\\\dot{y} \\\\end{pmatrix} = \\\\begin{pmatrix} -x+xy\\\\\\\\ -y \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}\\\\\\\\ \\\\dot{y} \\\\end{pmatrix} = \\\\begin{pmatrix} -x+xy\\\\\\\\ -y \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":\"We modify the local asymptotic stability to the global asymptotic stability.\"},\"tier\":\"extended\",\"tier_name\":\"Extended\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"Stability certification\"},\"reported_method_latex\":null,\"reported_type_latex\":\"Verified type: Global asymptotic stability\"}}"} {"record_id":"DCB-EXT-P1-002","paper_system_number":82,"task_type":"stability","problem_name":"Stability certification","tier":"extended","dimension":2,"time_domain":"continuous","dynamics":"dx1/dt = -x1**3+x2, dx2/dt = -x1-x2","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2 \\end{pmatrix} = \\begin{pmatrix} -x_1^3+x_2\\\\ -x_1-x_2 \\end{pmatrix}. \\]","citations":["abate2021fossil"],"problem_json":"{\"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\"}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":1,\"problem_name\":\"Stability certification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0)$\",\"$\\\\mathcal{N}=\\\\mathbb{R}^{2}$\",\"Verified type: Global exponential stability\"],\"record_id\":\"DCB-EXT-P1-002\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=(0,0)$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":\"$\\\\mathcal{N}=\\\\mathbb{R}^{2}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"\\\\mathbb{R}^{2}\",\"note\":\"Ambient Euclidean state space inferred from the stated dimension; no finite working domain was added.\",\"source\":\"dimension_and_dynamics\",\"status\":\"inferred\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":null,\"citations\":[\"abate2021fossil\"],\"label\":\"sys:catalogue-sys0021-ext\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":3411,\"start\":3387},\"title_latex\":\"Extend-Table~1 row~6 (Poly2), \\\\cite{abate2021fossil}\"},\"source_order\":82,\"system\":{\"dimension\":{\"source\":\"overleaf_state_space_notation\",\"status\":\"inferred\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} -x_1^3+x_2\\\\\\\\ -x_1-x_2 \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} -x_1^3+x_2\\\\\\\\ -x_1-x_2 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":\"We modify the local asymptotic stability to the global exponential stability.\"},\"tier\":\"extended\",\"tier_name\":\"Extended\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"Stability certification\"},\"reported_method_latex\":null,\"reported_type_latex\":\"Verified type: Global exponential stability\"}}"} {"record_id":"DCB-EXT-P1-003","paper_system_number":83,"task_type":"stability","problem_name":"Stability certification","tier":"extended","dimension":2,"time_domain":"continuous","dynamics":"dx1/dt = x2+u1, dx2/dt = u2+((3/20)*(981/100)*(1/2)*sin(x1)-(1/10)*x2)/((3/20)*(1/2)**2)","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2 \\end{pmatrix} = \\begin{pmatrix} x_2+u_1\\\\ u_2+\\dfrac{mgL\\sin(x_1)-bx_2}{mL^2} \\end{pmatrix}, \\] where $m=0.15$, $g=9.81$, $L=0.5$, and $b=0.1$.","citations":["edwards2024fossil"],"problem_json":"{\"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]]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":1,\"problem_name\":\"Stability certification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0)$\",\"$\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid x_1^2+x_2^2\\\\leq1^2\\\\}$\",\"$u_1,u_2 \\\\in [-5,5]$ when $\\\\mathbf{x}\\\\in \\\\mathcal{N}$ and $u_1(0)=u_2(0)=0$\",\"Verified type: Exponentially stable on $\\\\mathcal{N}$\",\"Verified method in the original paper: dReal\"],\"record_id\":\"DCB-EXT-P1-003\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=(0,0)$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":\"$\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid x_1^2+x_2^2\\\\leq1^2\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"\\\\mathbb{R}^{2}\",\"note\":\"Ambient Euclidean state space inferred from the stated dimension; no finite working domain was added.\",\"source\":\"dimension_and_dynamics\",\"status\":\"inferred\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":null,\"citations\":[\"edwards2024fossil\"],\"label\":\"sys:fossil2-f2-02-sinemodel-ext\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":3479,\"start\":3453},\"title_latex\":\"Extend-Table~4 \\\\#2 (SineModel), \\\\cite{edwards2024fossil}\"},\"source_order\":83,\"system\":{\"dimension\":{\"source\":\"overleaf_state_space_notation\",\"status\":\"inferred\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} x_2+u_1\\\\\\\\ u_2+\\\\dfrac{mgL\\\\sin(x_1)-bx_2}{mL^2} \\\\end{pmatrix},\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} x_2+u_1\\\\\\\\ u_2+\\\\dfrac{mgL\\\\sin(x_1)-bx_2}{mL^2} \\\\end{pmatrix}, \\\\] where $m=0.15$, $g=9.81$, $L=0.5$, and $b=0.1$.\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":\"We add control constrains $u_1,u_2 \\\\in [-5,5], u_1(0)=u_2(0)=0$ and change the local asymptotic stability to the local exponentially stability.\"},\"tier\":\"extended\",\"tier_name\":\"Extended\",\"verification\":{\"control_constraints_latex\":\"$u_1,u_2 \\\\in [-5,5]$ when $\\\\mathbf{x}\\\\in \\\\mathcal{N}$ and $u_1(0)=u_2(0)=0$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"Stability certification\"},\"reported_method_latex\":\"Verified method in the original paper: dReal\",\"reported_type_latex\":\"Verified type: Exponentially stable on $\\\\mathcal{N}$\"}}"} {"record_id":"DCB-EXT-P2-001","paper_system_number":84,"task_type":"roa","problem_name":"Region-of-attraction verification","tier":"extended","dimension":2,"time_domain":"continuous","dynamics":"dx1/dt = sin(x2), dx2/dt = u-cos(x2)/(1-x1)","specification_latex":"The closed-loop vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2 \\end{pmatrix} = \\begin{pmatrix} \\sin(x_2)\\\\ u-\\dfrac{\\cos(x_2)}{1-x_1} \\end{pmatrix} \\]","citations":["zhou2022neural"],"problem_json":"{\"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.\"]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":2,\"problem_name\":\"Region-of-attraction verification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0)$\",\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid0.1^2\\\\leq x_1^2+x_2^2\\\\leq0.8^2\\\\}$\",\"Original verified type: $\\\\epsilon$-ROA\",\"Verified method in the original paper: dReal\",\"Our $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid x_1^2+x_2^2\\\\leq0.8^2\\\\}$\",\"Our verified type: ROA verification\",\"$u \\\\in [-5,5]$ when $\\\\mathbf{x}\\\\in \\\\mathcal{N}$ and $u(0,0)=1$\"],\"record_id\":\"DCB-EXT-P2-001\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=(0,0)$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"No separate initial set is required to certify Omega_c itself.\",\"source\":\"exact_roa_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":\"Original $\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{2}\\\\mid0.1^2\\\\leq x_1^2+x_2^2\\\\leq0.8^2\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"X=\\\\{x\\\\in\\\\mathbb R^2\\\\mid 0.1\\\\le\\\\lVert x\\\\rVert_2\\\\le0.8\\\\} [artifact] SMT verification annulus in the notebook\",\"note\":\"Audited catalogue wording; not silently treated as an Overleaf statement.\",\"source\":\"catalogue:SYS0035\",\"status\":\"documented\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 2.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 2.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":\"SYS0035\",\"citations\":[\"zhou2022neural\"],\"label\":\"sys:catalogue-sys0035\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":3534,\"start\":3507},\"title_latex\":\"Extend-Section~5.2 and Appendix~C.3.2, \\\\cite{zhou2022neural}\"},\"source_order\":84,\"system\":{\"dimension\":{\"source\":\"catalogue:SYS0035\",\"status\":\"documented\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} \\\\sin(x_2)\\\\\\\\ u-\\\\dfrac{\\\\cos(x_2)}{1-x_1} \\\\end{pmatrix}\"],\"specification_latex\":\"The closed-loop vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2 \\\\end{pmatrix} = \\\\begin{pmatrix} \\\\sin(x_2)\\\\\\\\ u-\\\\dfrac{\\\\cos(x_2)}{1-x_1} \\\\end{pmatrix} \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":\"In the original paper, the control policy was $u=5\\\\tanh(q_1x_1+q_2x_2+b)$ which contributes to the $(0,0)^\\\\top$ is not the equilibrium point. So we just change the goal to find a policy $u\\\\in[-5,5]$ such that $u\\\\bigl((0,0)^\\\\top\\\\bigr)=1$.\"},\"tier\":\"extended\",\"tier_name\":\"Extended\",\"verification\":{\"control_constraints_latex\":\"$u \\\\in [-5,5]$ when $\\\\mathbf{x}\\\\in \\\\mathcal{N}$ and $u(0,0)=1$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"exact_roa\"},\"reported_method_latex\":\"Verified method in the original paper: dReal\",\"reported_type_latex\":\"Original verified type: $\\\\epsilon$-ROA\"}}"} {"record_id":"DCB-EXT-P3-001","paper_system_number":85,"task_type":"invariant","problem_name":"Forward-invariant set synthesis","tier":"extended","dimension":6,"time_domain":"continuous","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","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2\\\\ \\dot{x}_3\\\\ \\dot{x}_4\\\\ \\dot{x}_5\\\\ \\dot{x}_6 \\end{pmatrix} = \\begin{pmatrix} x_4\\\\[1ex] x_5\\\\[1ex] x_6\\\\[1ex] 9.81\\sin(x_3)+\\sin(x_3)u_1\\\\[1ex] -9.81+9.81\\cos(x_3)+\\cos(x_3)u_1\\\\[1ex] u_2 \\end{pmatrix}, \\] where $\\mathbf{u}=(u_1,u_2)^\\top$ is the control input to be synthesized.","citations":["meng2024physics"],"problem_json":"{\"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]]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":3,\"problem_name\":\"Forward-invariant set synthesis\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\{x\\\\in\\\\mathbb{R}^{6}\\\\mid \\\\lVert x\\\\rVert_\\\\infty \\\\le 0.2\\\\}$\",\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{6}\\\\mid \\\\lVert x\\\\rVert_\\\\infty > 0.2\\\\}$\",\"$u_1,u_2 \\\\in [-5,5]$ when $\\\\mathbf{x}\\\\in \\\\mathcal{X}$\",\"Verified type: controlled forward-invariant set synthesis\",\"Verified method in the original paper: no formal invariance certificate was reported for this instance\"],\"record_id\":\"DCB-EXT-P3-001\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\{x\\\\in\\\\mathbb{R}^{6}\\\\mid \\\\lVert x\\\\rVert_\\\\infty \\\\le 0.2\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"S is the requested nonempty forward-invariant output, not an input set.\",\"source\":\"problem_contract\",\"status\":\"output_to_synthesize\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{6}\\\\mid \\\\lVert x\\\\rVert_\\\\infty > 0.2\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":null,\"citations\":[\"meng2024physics\"],\"label\":\"sys:extend-meng-2d-quadrotor\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":3573,\"start\":3539},\"title_latex\":\"Extend--2D quadrotor, \\\\cite{meng2024physics}\"},\"source_order\":85,\"system\":{\"dimension\":{\"source\":\"overleaf_state_space_notation\",\"status\":\"inferred\",\"symbolic\":null,\"value\":6},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3\\\\\\\\ \\\\dot{x}_4\\\\\\\\ \\\\dot{x}_5\\\\\\\\ \\\\dot{x}_6 \\\\end{pmatrix} = \\\\begin{pmatrix} x_4\\\\\\\\[1ex] x_5\\\\\\\\[1ex] x_6\\\\\\\\[1ex] 9.81\\\\sin(x_3)+\\\\sin(x_3)u_1\\\\\\\\[1ex] -9.81+9.81\\\\cos(x_3)+\\\\cos(x_3)u_1\\\\\\\\[1ex] u_2 \\\\end{pmatrix},\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3\\\\\\\\ \\\\dot{x}_4\\\\\\\\ \\\\dot{x}_5\\\\\\\\ \\\\dot{x}_6 \\\\end{pmatrix} = \\\\begin{pmatrix} x_4\\\\\\\\[1ex] x_5\\\\\\\\[1ex] x_6\\\\\\\\[1ex] 9.81\\\\sin(x_3)+\\\\sin(x_3)u_1\\\\\\\\[1ex] -9.81+9.81\\\\cos(x_3)+\\\\cos(x_3)u_1\\\\\\\\[1ex] u_2 \\\\end{pmatrix}, \\\\] where $\\\\mathbf{u}=(u_1,u_2)^\\\\top$ is the control input to be synthesized.\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":\"We add control constrains $u_1,u_2 \\\\in [-5,5]$.\"},\"tier\":\"extended\",\"tier_name\":\"Extended\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{6}\\\\mid \\\\lVert x\\\\rVert_\\\\infty > 0.2\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"unsafe_set_X_U\"],\"requires_initial_set\":false,\"target\":\"Forward-invariant set synthesis\"},\"reported_method_latex\":\"Verified method in the original paper: no formal invariance certificate was reported for this instance\",\"reported_type_latex\":\"Verified type: controlled forward-invariant set synthesis\"}}"} {"record_id":"DCB-EXT-P3-002","paper_system_number":86,"task_type":"invariant","problem_name":"Forward-invariant set synthesis","tier":"extended","dimension":9,"time_domain":"continuous","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","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2\\\\ \\dot{x}_3\\\\ \\dot{x}_4\\\\ \\dot{x}_5\\\\ \\dot{x}_6\\\\ \\dot{x}_7\\\\ \\dot{x}_8\\\\ \\dot{x}_9 \\end{pmatrix} = \\begin{pmatrix} x_4\\\\[1ex] x_5\\\\[1ex] x_6\\\\[1ex] -9.81\\sin(x_8)-\\sin(x_8)u_1\\\\[1ex] 9.81\\cos(x_8)\\sin(x_7) +\\cos(x_8)\\sin(x_7)u_1\\\\[1ex] 9.81\\cos(x_8)\\cos(x_7)-9.81 +\\cos(x_8)\\cos(x_7)u_1\\\\[1ex] u_2\\\\[1ex] u_3\\\\[1ex] u_4 \\end{pmatrix}, \\] where $\\mathbf{u}=(u_1,u_2,u_3,u_4)^\\top$ is the control input to be synthesized.","citations":["meng2024physics"],"problem_json":"{\"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]]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":3,\"problem_name\":\"Forward-invariant set synthesis\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\{x\\\\in\\\\mathbb{R}^{9}\\\\mid \\\\lVert x\\\\rVert_\\\\infty \\\\le 0.2\\\\}$\",\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{9}\\\\mid \\\\lVert x\\\\rVert_\\\\infty > 0.2\\\\}$\",\"$u_i \\\\in [-5,5]$ when $\\\\mathbf{x}\\\\in \\\\mathcal{X}\\\\quad i=1,2,3,4$\",\"Verified type: controlled forward-invariant set synthesis\",\"Verified method in the original paper: no formal invariance certificate was reported for this instance\"],\"record_id\":\"DCB-EXT-P3-002\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\{x\\\\in\\\\mathbb{R}^{9}\\\\mid \\\\lVert x\\\\rVert_\\\\infty \\\\le 0.2\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"S is the requested nonempty forward-invariant output, not an input set.\",\"source\":\"problem_contract\",\"status\":\"output_to_synthesize\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{9}\\\\mid \\\\lVert x\\\\rVert_\\\\infty > 0.2\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":null,\"citations\":[\"meng2024physics\"],\"label\":\"sys:extend-meng-3d-quadrotor\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":3620,\"start\":3577},\"title_latex\":\"Extend--3D quadrotor, \\\\cite{meng2024physics}\"},\"source_order\":86,\"system\":{\"dimension\":{\"source\":\"overleaf_state_space_notation\",\"status\":\"inferred\",\"symbolic\":null,\"value\":9},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3\\\\\\\\ \\\\dot{x}_4\\\\\\\\ \\\\dot{x}_5\\\\\\\\ \\\\dot{x}_6\\\\\\\\ \\\\dot{x}_7\\\\\\\\ \\\\dot{x}_8\\\\\\\\ \\\\dot{x}_9 \\\\end{pmatrix} = \\\\begin{pmatrix} x_4\\\\\\\\[1ex] x_5\\\\\\\\[1ex] x_6\\\\\\\\[1ex] -9.81\\\\sin(x_8)-\\\\sin(x_8)u_1\\\\\\\\[1ex] 9.81\\\\cos(x_8)\\\\sin(x_7) +\\\\cos(x_8)\\\\sin(x_7)u_1\\\\\\\\[1ex] 9.81\\\\cos(x_8)\\\\cos(x_7)-9.81 +\\\\cos(x_8)\\\\cos(x_7)u_1\\\\\\\\[1ex] u_2\\\\\\\\[1ex] u_3\\\\\\\\[1ex] u_4 \\\\end{pmatrix},\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3\\\\\\\\ \\\\dot{x}_4\\\\\\\\ \\\\dot{x}_5\\\\\\\\ \\\\dot{x}_6\\\\\\\\ \\\\dot{x}_7\\\\\\\\ \\\\dot{x}_8\\\\\\\\ \\\\dot{x}_9 \\\\end{pmatrix} = \\\\begin{pmatrix} x_4\\\\\\\\[1ex] x_5\\\\\\\\[1ex] x_6\\\\\\\\[1ex] -9.81\\\\sin(x_8)-\\\\sin(x_8)u_1\\\\\\\\[1ex] 9.81\\\\cos(x_8)\\\\sin(x_7) +\\\\cos(x_8)\\\\sin(x_7)u_1\\\\\\\\[1ex] 9.81\\\\cos(x_8)\\\\cos(x_7)-9.81 +\\\\cos(x_8)\\\\cos(x_7)u_1\\\\\\\\[1ex] u_2\\\\\\\\[1ex] u_3\\\\\\\\[1ex] u_4 \\\\end{pmatrix}, \\\\] where $\\\\mathbf{u}=(u_1,u_2,u_3,u_4)^\\\\top$ is the control input to be synthesized.\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":\"We add the control constraints $u_i \\\\in [-5,5]$ when $\\\\mathbf{x}\\\\in \\\\mathcal{X}\\\\quad i=1,2,3,4$.\"},\"tier\":\"extended\",\"tier_name\":\"Extended\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{9}\\\\mid \\\\lVert x\\\\rVert_\\\\infty > 0.2\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"unsafe_set_X_U\"],\"requires_initial_set\":false,\"target\":\"Forward-invariant set synthesis\"},\"reported_method_latex\":\"Verified method in the original paper: no formal invariance certificate was reported for this instance\",\"reported_type_latex\":\"Verified type: controlled forward-invariant set synthesis\"}}"} {"record_id":"DCB-EXT-P4-001","paper_system_number":87,"task_type":"safety","problem_name":"Safety verification","tier":"extended","dimension":4,"time_domain":"continuous","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","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_{1}\\\\ \\dot{x}_{2}\\\\ \\dot{x}_{3}\\\\ \\dot{x}_{4} \\end{pmatrix} = \\begin{pmatrix} x_{2} + u_{0}\\\\ -0.95\\,\\sin(x_{1}) - 0.02\\,x_{2} + 0.01\\,x_{3} + u_{1}\\\\ -0.1\\,x_{3} + 0.02\\,x_{1} + u_{2}\\\\ -0.03\\,x_{4} + 0.01\\,x_{2} + u_{3} \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"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]]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{4}$\",\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.3\\\\}$\",\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{4}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 2.5\\\\}$\",\"$u_i\\\\in [-5,5](i=0,1,2,3)$ when $\\\\mathbf{x}\\\\in \\\\mathcal{X}$\",\"Verified type: safety verification\",\"Verified method in the original paper: Z3/Yices with Taylor approximations\"],\"record_id\":\"DCB-EXT-P4-001\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.3\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{4}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{4}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 2.5\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":null,\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-s002-ext\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":3762,\"start\":3732},\"title_latex\":\"Extend-\\\\texttt{S002}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":87,\"system\":{\"dimension\":{\"source\":\"overleaf_state_space_notation\",\"status\":\"inferred\",\"symbolic\":null,\"value\":4},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3}\\\\\\\\ \\\\dot{x}_{4} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{2} + u_{0}\\\\\\\\ -0.95\\\\,\\\\sin(x_{1}) - 0.02\\\\,x_{2} + 0.01\\\\,x_{3} + u_{1}\\\\\\\\ -0.1\\\\,x_{3} + 0.02\\\\,x_{1} + u_{2}\\\\\\\\ -0.03\\\\,x_{4} + 0.01\\\\,x_{2} + u_{3} \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3}\\\\\\\\ \\\\dot{x}_{4} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{2} + u_{0}\\\\\\\\ -0.95\\\\,\\\\sin(x_{1}) - 0.02\\\\,x_{2} + 0.01\\\\,x_{3} + u_{1}\\\\\\\\ -0.1\\\\,x_{3} + 0.02\\\\,x_{1} + u_{2}\\\\\\\\ -0.03\\\\,x_{4} + 0.01\\\\,x_{2} + u_{3} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":\"We add the control constrains $u_i\\\\in [-5,5]; i=0,1,2,3$.\"},\"tier\":\"extended\",\"tier_name\":\"Extended\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{4}\\\\setminus \\\\{x\\\\in\\\\mathbb{R}^{4}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 2.5\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: Z3/Yices with Taylor approximations\",\"reported_type_latex\":\"Verified type: safety verification\"}}"} {"record_id":"DCB-EXT-P4-002","paper_system_number":88,"task_type":"safety","problem_name":"Safety verification","tier":"extended","dimension":2,"time_domain":"continuous","dynamics":"dx1/dt = x2+u0, dx2/dt = -0.3*x1-0.2*(1+cos(x1))*x2-0.1*x1**(3)+u1","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_{1}\\\\ \\dot{x}_{2} \\end{pmatrix} = \\begin{pmatrix} x_{2} + u_{0}\\\\ -0.3\\,x_{1} - 0.2\\,(1 + \\cos(x_{1}))\\,x_{2} - 0.1\\,x_{1}^{3} + u_{1} \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"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]]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{2}$\",\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.6\\\\}$\",\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid \\\\lVert x-(2,2)^{\\\\mathsf T}\\\\rVert_2\\\\le 1\\\\}$\",\"$u_{0,1}\\\\in [-1,1]$ when $x \\\\in \\\\mathcal{X}$\",\"Verified type: safety verification\",\"Verified method in the original paper: Z3/Yices with Taylor approximations\"],\"record_id\":\"DCB-EXT-P4-002\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.6\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{2}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid \\\\lVert x-(2,2)^{\\\\mathsf T}\\\\rVert_2\\\\le 1\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":null,\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-s011-ext\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":3792,\"start\":3766},\"title_latex\":\"Extend-\\\\texttt{S011}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":88,\"system\":{\"dimension\":{\"source\":\"overleaf_state_space_notation\",\"status\":\"inferred\",\"symbolic\":null,\"value\":2},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{2} + u_{0}\\\\\\\\ -0.3\\\\,x_{1} - 0.2\\\\,(1 + \\\\cos(x_{1}))\\\\,x_{2} - 0.1\\\\,x_{1}^{3} + u_{1} \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{2} + u_{0}\\\\\\\\ -0.3\\\\,x_{1} - 0.2\\\\,(1 + \\\\cos(x_{1}))\\\\,x_{2} - 0.1\\\\,x_{1}^{3} + u_{1} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":\"We add the control constrains $u_{0,1}\\\\in [-1,1]^2$.\"},\"tier\":\"extended\",\"tier_name\":\"Extended\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\{x\\\\in\\\\mathbb{R}^{2}\\\\mid \\\\lVert x-(2,2)^{\\\\mathsf T}\\\\rVert_2\\\\le 1\\\\}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: Z3/Yices with Taylor approximations\",\"reported_type_latex\":\"Verified type: safety verification\"}}"} {"record_id":"DCB-EXT-P4-003","paper_system_number":89,"task_type":"safety","problem_name":"Safety verification","tier":"extended","dimension":4,"time_domain":"continuous","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","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_{1}\\\\ \\dot{x}_{2}\\\\ \\dot{x}_{3}\\\\ \\dot{x}_{4} \\end{pmatrix} = \\begin{pmatrix} x_{2} + u_{0}\\\\ -x_{1} - 0.4\\,x_{2} + 0.15\\,\\sin(x_{3}) + u_{1}\\\\ x_{4} + u_{2}\\\\ -0.6\\,x_{3} - 0.3\\,x_{4} + u_{3} \\end{pmatrix}. \\]","citations":["taheri2025barrierbench"],"problem_json":"{\"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]]}","executable_record_json":"{\"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\"}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X} = \\\\mathbb{R}^{4}$\",\"$\\\\mathcal{X}_0 = [0.5,0.8]\\\\times [0.3,0.6]\\\\times [0.5,0.8]\\\\times [0.3,0.6]$\",\"$\\\\mathcal{X}_U = [2.5,3.5]\\\\times [2,3]\\\\times [2.5,3.5]\\\\times [2,3]\\\\cup [-3.5,-2.5]\\\\times [-3,-2]\\\\times [-3.5,-2.5]\\\\times [-3,-2]$\",\"$u_{i}\\\\in [-1,1], i=0,1,2,3$ when $x \\\\in \\\\mathcal{X}$\",\"Verified type: safety verification\",\"Verified method in the original paper: Z3/Yices with Taylor approximations\"],\"record_id\":\"DCB-EXT-P4-003\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_0 = [0.5,0.8]\\\\times [0.3,0.6]\\\\times [0.5,0.8]\\\\times [0.3,0.6]$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X} = \\\\mathbb{R}^{4}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = [2.5,3.5]\\\\times [2,3]\\\\times [2.5,3.5]\\\\times [2,3]\\\\cup [-3.5,-2.5]\\\\times [-3,-2]\\\\times [-3.5,-2.5]\\\\times [-3,-2]$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":null,\"citations\":[\"taheri2025barrierbench\"],\"label\":\"sys:bb-s024-ext\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":3825,\"start\":3795},\"title_latex\":\"Extend-\\\\texttt{S024}, \\\\cite{taheri2025barrierbench}\"},\"source_order\":89,\"system\":{\"dimension\":{\"source\":\"overleaf_state_space_notation\",\"status\":\"inferred\",\"symbolic\":null,\"value\":4},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3}\\\\\\\\ \\\\dot{x}_{4} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{2} + u_{0}\\\\\\\\ -x_{1} - 0.4\\\\,x_{2} + 0.15\\\\,\\\\sin(x_{3}) + u_{1}\\\\\\\\ x_{4} + u_{2}\\\\\\\\ -0.6\\\\,x_{3} - 0.3\\\\,x_{4} + u_{3} \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_{1}\\\\\\\\ \\\\dot{x}_{2}\\\\\\\\ \\\\dot{x}_{3}\\\\\\\\ \\\\dot{x}_{4} \\\\end{pmatrix} = \\\\begin{pmatrix} x_{2} + u_{0}\\\\\\\\ -x_{1} - 0.4\\\\,x_{2} + 0.15\\\\,\\\\sin(x_{3}) + u_{1}\\\\\\\\ x_{4} + u_{2}\\\\\\\\ -0.6\\\\,x_{3} - 0.3\\\\,x_{4} + u_{3} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":\"We add the control constrains $u_{i}\\\\in [-1,1], i=0,1,2,3$.\"},\"tier\":\"extended\",\"tier_name\":\"Extended\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = [2.5,3.5]\\\\times [2,3]\\\\times [2.5,3.5]\\\\times [2,3]\\\\cup [-3.5,-2.5]\\\\times [-3,-2]\\\\times [-3.5,-2.5]\\\\times [-3,-2]$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":\"Verified method in the original paper: Z3/Yices with Taylor approximations\",\"reported_type_latex\":\"Verified type: safety verification\"}}"} {"record_id":"DCB-EXT-P4-004","paper_system_number":90,"task_type":"safety","problem_name":"Safety verification","tier":"extended","dimension":8,"time_domain":"continuous","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)","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_0\\\\ \\dot{x}_1\\\\ \\dot{x}_2\\\\ \\dot{x}_3\\\\ \\dot{x}_4\\\\ \\dot{x}_5\\\\ \\dot{x}_6\\\\ \\dot{x}_7 \\end{pmatrix} = \\begin{pmatrix} x_1\\\\ x_2\\\\ x_3\\\\ x_4\\\\ x_5\\\\ x_6\\\\ x_7\\\\ -\\begin{aligned}[t] &20x_7+170x_6+800x_5+2273x_4\\\\[-0.15ex] &{}+3980x_3+4180x_2+2400x_1+576x_0\\\\[-0.15ex] &{}+\\sin(\\omega x_0) \\end{aligned} \\end{pmatrix}. \\]","citations":["abate2021fossil"],"problem_json":"{\"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\",\"11/5\"]],\"fixed_parameters\":[{\"name\":\"omega\",\"bounds\":[\"1/2\",2]}],\"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\"]]}}","executable_record_json":"{\"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\",\"11/5\"]],\"fixed_parameters\":[{\"name\":\"omega\",\"bounds\":[\"1/2\",2]}],\"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\"]]}},\"candidate\":null}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X}=[-2.2,2.2]^8$\",\"$\\\\mathcal{X}_I=[0.9,1.1]^8$\",\"$\\\\mathcal{X}_U=[-2.2,-1.8]^8$\",\"$\\\\omega\\\\in[0.5,2]$\",\"Verified type: safety verification\"],\"record_id\":\"DCB-EXT-P4-004\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_I=[0.9,1.1]^8$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X}=[-2.2,2.2]^8$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U=[-2.2,-1.8]^8$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":null,\"citations\":[\"abate2021fossil\"],\"label\":\"sys:extend-hiord8-safety\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":3867,\"start\":3830},\"title_latex\":\"Extend--Hi-Ord8 with bounded sinusoidal disturbance, \\\\cite{abate2021fossil}\"},\"source_order\":90,\"system\":{\"dimension\":{\"source\":\"overleaf_state_space_notation\",\"status\":\"inferred\",\"symbolic\":null,\"value\":8},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_0\\\\\\\\ \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3\\\\\\\\ \\\\dot{x}_4\\\\\\\\ \\\\dot{x}_5\\\\\\\\ \\\\dot{x}_6\\\\\\\\ \\\\dot{x}_7 \\\\end{pmatrix} = \\\\begin{pmatrix} x_1\\\\\\\\ x_2\\\\\\\\ x_3\\\\\\\\ x_4\\\\\\\\ x_5\\\\\\\\ x_6\\\\\\\\ x_7\\\\\\\\ -\\\\begin{aligned}[t] &20x_7+170x_6+800x_5+2273x_4\\\\\\\\[-0.15ex] &{}+3980x_3+4180x_2+2400x_1+576x_0\\\\\\\\[-0.15ex] &{}+\\\\sin(\\\\omega x_0) \\\\end{aligned} \\\\end{pmatrix}.\",\"[t] &20x_7+170x_6+800x_5+2273x_4\\\\\\\\[-0.15ex] &{}+3980x_3+4180x_2+2400x_1+576x_0\\\\\\\\[-0.15ex] &{}+\\\\sin(\\\\omega x_0)\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_0\\\\\\\\ \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3\\\\\\\\ \\\\dot{x}_4\\\\\\\\ \\\\dot{x}_5\\\\\\\\ \\\\dot{x}_6\\\\\\\\ \\\\dot{x}_7 \\\\end{pmatrix} = \\\\begin{pmatrix} x_1\\\\\\\\ x_2\\\\\\\\ x_3\\\\\\\\ x_4\\\\\\\\ x_5\\\\\\\\ x_6\\\\\\\\ x_7\\\\\\\\ -\\\\begin{aligned}[t] &20x_7+170x_6+800x_5+2273x_4\\\\\\\\[-0.15ex] &{}+3980x_3+4180x_2+2400x_1+576x_0\\\\\\\\[-0.15ex] &{}+\\\\sin(\\\\omega x_0) \\\\end{aligned} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":\"The safety property is verified uniformly for all admissible values of $\\\\omega\\\\in[0.5,2]$ which is a time-invariant parameter. \\\\begin{remark} In this paper, we mainly consider the Barrier function is only related to the $x$, not with uncertain factor $w$, i.e. we require that, for any $w \\\\in [0.5,2]$, the barrier conditions hold. \\\\end{remark}\"},\"tier\":\"extended\",\"tier_name\":\"Extended\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":null,\"reported_type_latex\":\"Verified type: safety verification\"}}"} {"record_id":"DCB-EXT-P4-005","paper_system_number":91,"task_type":"safety","problem_name":"Safety verification","tier":"extended","dimension":10,"time_domain":"continuous","dynamics":"dx1/dt = -x1+1/2*x2-1/10*x9**2+sin(x2), dx2/dt = -1/2*x1-x2, dx3/dt = -x3+1/2*x4-1/10*x1**2+exp(-x3)-1, 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","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2\\\\ \\dot{x}_3\\\\ \\dot{x}_4\\\\ \\dot{x}_5\\\\ \\dot{x}_6\\\\ \\dot{x}_7\\\\ \\dot{x}_8\\\\ \\dot{x}_9\\\\ \\dot{x}_{10} \\end{pmatrix} = \\begin{pmatrix} -x_1+0.5x_2-0.1x_9^2+\\sin(x_2)\\\\ -0.5x_1-x_2\\\\ -x_3+0.5x_4-0.1x_1^2+e^{-x_3}-1\\\\ -0.5x_3-x_4\\\\ -x_5+0.5x_6+0.1x_7^2\\\\ -0.5x_5-x_6\\\\ -x_7+0.5x_8\\\\ -0.5x_7-x_8\\\\ -x_9+0.5x_{10}\\\\ -0.5x_9-x_{10}+0.1x_2^2 \\end{pmatrix}. \\]","citations":["liu2024tool"],"problem_json":"{\"problem_id\":\"dcb_ext_p4_005\",\"task_type\":\"safety\",\"name\":\"DynaCertBench System 091 · DCB-EXT-P4-005 · Safety extension of Example~5.4, \\\\cite{liu2024tool}\",\"dynamics\":\"dx1/dt = -x1+1/2*x2-1/10*x9**2+sin(x2), dx2/dt = -1/2*x1-x2, dx3/dt = -x3+1/2*x4-1/10*x1**2+exp(-x3)-1, 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\",\"source\":\"paper System 91; canonical record DCB-EXT-P4-005\",\"assumptions\":[\"Transcribed from paper-4-20-20260904 System 94.\",\"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.\",\"PDF 4-20 removes omega and d(t); both perturbations are deterministic state functions.\"],\"analysis_bounds\":[[-1,1],[-1,1],[-1,1],[-1,1],[-1,1],[-1,1],[-1,1],[-1,1],[-1,1],[-1,1]],\"initial_set\":{\"type\":\"box\",\"bounds\":[[\"-7/20\",\"7/20\"],[\"-7/20\",\"7/20\"],[\"-7/20\",\"7/20\"],[\"-7/20\",\"7/20\"],[\"-7/20\",\"7/20\"],[\"-7/20\",\"7/20\"],[\"-7/20\",\"7/20\"],[\"-7/20\",\"7/20\"],[\"-7/20\",\"7/20\"],[\"-7/20\",\"7/20\"]]},\"unsafe_set\":{\"type\":\"box\",\"bounds\":[[\"9/20\",\"4/5\"],[\"-1/5\",\"1/5\"],[\"-2/5\",\"2/5\"],[\"-2/5\",\"2/5\"],[\"-2/5\",\"2/5\"],[\"-2/5\",\"2/5\"],[\"-2/5\",\"2/5\"],[\"-2/5\",\"2/5\"],[\"-2/5\",\"2/5\"],[\"-2/5\",\"2/5\"]]}}","executable_record_json":"{\"record_id\":\"DCB-EXT-P4-005\",\"paper_system_number\":91,\"paper_system_label\":\"System 91\",\"local_name\":\"S091__DCB-EXT-P4-005\",\"problem\":{\"problem_id\":\"dcb_ext_p4_005\",\"task_type\":\"safety\",\"name\":\"DynaCertBench System 091 · DCB-EXT-P4-005 · Safety extension of Example~5.4, \\\\cite{liu2024tool}\",\"dynamics\":\"dx1/dt = -x1+1/2*x2-1/10*x9**2+sin(x2), dx2/dt = -1/2*x1-x2, dx3/dt = -x3+1/2*x4-1/10*x1**2+exp(-x3)-1, 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\",\"source\":\"paper System 91; canonical record DCB-EXT-P4-005\",\"assumptions\":[\"Transcribed from paper-4-20-20260904 System 94.\",\"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.\",\"PDF 4-20 removes omega and d(t); both perturbations are deterministic state functions.\"],\"analysis_bounds\":[[-1,1],[-1,1],[-1,1],[-1,1],[-1,1],[-1,1],[-1,1],[-1,1],[-1,1],[-1,1]],\"initial_set\":{\"type\":\"box\",\"bounds\":[[\"-7/20\",\"7/20\"],[\"-7/20\",\"7/20\"],[\"-7/20\",\"7/20\"],[\"-7/20\",\"7/20\"],[\"-7/20\",\"7/20\"],[\"-7/20\",\"7/20\"],[\"-7/20\",\"7/20\"],[\"-7/20\",\"7/20\"],[\"-7/20\",\"7/20\"],[\"-7/20\",\"7/20\"]]},\"unsafe_set\":{\"type\":\"box\",\"bounds\":[[\"9/20\",\"4/5\"],[\"-1/5\",\"1/5\"],[\"-2/5\",\"2/5\"],[\"-2/5\",\"2/5\"],[\"-2/5\",\"2/5\"],[\"-2/5\",\"2/5\"],[\"-2/5\",\"2/5\"],[\"-2/5\",\"2/5\"],[\"-2/5\",\"2/5\"],[\"-2/5\",\"2/5\"]]}},\"candidate\":null}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal{X}=[-1,1]^{10}$\",\"$\\\\mathcal{X}_I=[-0.35,0.35]^{10}$\",\"$\\\\mathcal{X}_U= [0.45,0.8]\\\\times[-0.2,0.2]\\\\times[-0.4,0.4]^8$\",\"Verified type: safety verification\"],\"record_id\":\"DCB-EXT-P4-005\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_I=[-0.35,0.35]^{10}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal{X}=[-1,1]^{10}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U= [0.45,0.8]\\\\times[-0.2,0.2]\\\\times[-0.4,0.4]^8$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":null,\"citations\":[\"liu2024tool\"],\"label\":\"sys:extend-liu2024-example54-safety\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":3919,\"start\":3875},\"title_latex\":\"Safety extension of Example~5.4, \\\\cite{liu2024tool}\"},\"source_order\":91,\"system\":{\"dimension\":{\"source\":\"overleaf_state_space_notation\",\"status\":\"inferred\",\"symbolic\":null,\"value\":10},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3\\\\\\\\ \\\\dot{x}_4\\\\\\\\ \\\\dot{x}_5\\\\\\\\ \\\\dot{x}_6\\\\\\\\ \\\\dot{x}_7\\\\\\\\ \\\\dot{x}_8\\\\\\\\ \\\\dot{x}_9\\\\\\\\ \\\\dot{x}_{10} \\\\end{pmatrix} = \\\\begin{pmatrix} -x_1+0.5x_2-0.1x_9^2+\\\\sin(x_2)\\\\\\\\ -0.5x_1-x_2\\\\\\\\ -x_3+0.5x_4-0.1x_1^2+e^{-x_3}-1\\\\\\\\ -0.5x_3-x_4\\\\\\\\ -x_5+0.5x_6+0.1x_7^2\\\\\\\\ -0.5x_5-x_6\\\\\\\\ -x_7+0.5x_8\\\\\\\\ -0.5x_7-x_8\\\\\\\\ -x_9+0.5x_{10}\\\\\\\\ -0.5x_9-x_{10}+0.1x_2^2 \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3\\\\\\\\ \\\\dot{x}_4\\\\\\\\ \\\\dot{x}_5\\\\\\\\ \\\\dot{x}_6\\\\\\\\ \\\\dot{x}_7\\\\\\\\ \\\\dot{x}_8\\\\\\\\ \\\\dot{x}_9\\\\\\\\ \\\\dot{x}_{10} \\\\end{pmatrix} = \\\\begin{pmatrix} -x_1+0.5x_2-0.1x_9^2+\\\\sin(x_2)\\\\\\\\ -0.5x_1-x_2\\\\\\\\ -x_3+0.5x_4-0.1x_1^2+e^{-x_3}-1\\\\\\\\ -0.5x_3-x_4\\\\\\\\ -x_5+0.5x_6+0.1x_7^2\\\\\\\\ -0.5x_5-x_6\\\\\\\\ -x_7+0.5x_8\\\\\\\\ -0.5x_7-x_8\\\\\\\\ -x_9+0.5x_{10}\\\\\\\\ -0.5x_9-x_{10}+0.1x_2^2 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":\"We augment the dynamics by introducing two perturbations, $\\\\sin(x_2)$ to the vector field of $x_1$ and $e^{-x_3}$ to the vector field of $x_3$.\"},\"tier\":\"extended\",\"tier_name\":\"Extended\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":null,\"reported_type_latex\":\"Verified type: safety verification\"}}"} {"record_id":"DCB-OURS-P1-001","paper_system_number":92,"task_type":"stability","problem_name":"Stability certification","tier":"ours","dimension":8,"time_domain":"continuous","dynamics":"dx1/dt = -3*x1+x2*cos(x1)/(1+x8**2), dx2/dt = -3*x2+x3*cos(x2)/(1+x1**2), dx3/dt = -3*x3+x4*cos(x3)/(1+x2**2), dx4/dt = -3*x4+x5*cos(x4)/(1+x3**2), dx5/dt = -3*x5+x6*cos(x1)/(1+x4**2), dx6/dt = -3*x6+x7*cos(x2)/(1+x5**2), dx7/dt = -3*x7+x8*cos(x3)/(1+x6**2), dx8/dt = -3*x8+x1*cos(x4)/(1+x7**2)","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2\\\\ \\dot{x}_3\\\\ \\dot{x}_4\\\\ \\dot{x}_5\\\\ \\dot{x}_6\\\\ \\dot{x}_7\\\\ \\dot{x}_8 \\end{pmatrix} = \\begin{pmatrix} -3 x_1 + x_2 \\frac{\\cos x_1}{1 + x_8^2}\\\\[6pt] -3 x_2 + x_3 \\frac{\\cos x_2}{1 + x_1^2}\\\\[6pt] -3 x_3 + x_4 \\frac{\\cos x_3}{1 + x_2^2}\\\\[6pt] -3 x_4 + x_5 \\frac{\\cos x_4}{1 + x_3^2}\\\\[6pt] -3 x_5 + x_6 \\frac{\\cos x_1}{1 + x_4^2}\\\\[6pt] -3 x_6 + x_7 \\frac{\\cos x_2}{1 + x_5^2}\\\\[6pt] -3 x_7 + x_8 \\frac{\\cos x_3}{1 + x_6^2}\\\\[6pt] -3 x_8 + x_1 \\frac{\\cos x_4}{1 + x_7^2} \\end{pmatrix}. \\]","citations":[],"problem_json":"{\"problem_id\":\"dcb_ours_p1_001\",\"task_type\":\"stability\",\"name\":\"DynaCertBench System 092 · DCB-OURS-P1-001 · 8D Rational-Cosine Ring\",\"dynamics\":\"dx1/dt = -3*x1+x2*cos(x1)/(1+x8**2), dx2/dt = -3*x2+x3*cos(x2)/(1+x1**2), dx3/dt = -3*x3+x4*cos(x3)/(1+x2**2), dx4/dt = -3*x4+x5*cos(x4)/(1+x3**2), dx5/dt = -3*x5+x6*cos(x1)/(1+x4**2), dx6/dt = -3*x6+x7*cos(x2)/(1+x5**2), dx7/dt = -3*x7+x8*cos(x3)/(1+x6**2), dx8/dt = -3*x8+x1*cos(x4)/(1+x7**2)\",\"source\":\"paper System 92; canonical record DCB-OURS-P1-001\",\"assumptions\":[\"Transcribed from active LaTeX System 92.\",\"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.\"],\"analysis_bounds\":[[-3,3],[-3,3],[-3,3],[-3,3],[-3,3],[-3,3],[-3,3],[-3,3]],\"equilibrium\":[0,0,0,0,0,0,0,0],\"stability_kind\":\"global_exponential\",\"neighborhood\":{\"type\":\"unbounded\"}}","executable_record_json":"{\"record_id\":\"DCB-OURS-P1-001\",\"paper_system_number\":92,\"paper_system_label\":\"System 92\",\"local_name\":\"S092__DCB-OURS-P1-001\",\"problem\":{\"problem_id\":\"dcb_ours_p1_001\",\"task_type\":\"stability\",\"name\":\"DynaCertBench System 092 · DCB-OURS-P1-001 · 8D Rational-Cosine Ring\",\"dynamics\":\"dx1/dt = -3*x1+x2*cos(x1)/(1+x8**2), dx2/dt = -3*x2+x3*cos(x2)/(1+x1**2), dx3/dt = -3*x3+x4*cos(x3)/(1+x2**2), dx4/dt = -3*x4+x5*cos(x4)/(1+x3**2), dx5/dt = -3*x5+x6*cos(x1)/(1+x4**2), dx6/dt = -3*x6+x7*cos(x2)/(1+x5**2), dx7/dt = -3*x7+x8*cos(x3)/(1+x6**2), dx8/dt = -3*x8+x1*cos(x4)/(1+x7**2)\",\"source\":\"paper System 92; canonical record DCB-OURS-P1-001\",\"assumptions\":[\"Transcribed from active LaTeX System 92.\",\"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.\"],\"analysis_bounds\":[[-3,3],[-3,3],[-3,3],[-3,3],[-3,3],[-3,3],[-3,3],[-3,3]],\"equilibrium\":[0,0,0,0,0,0,0,0],\"stability_kind\":\"global_exponential\",\"neighborhood\":{\"type\":\"unbounded\"}},\"candidate\":null}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[\"source_label_missing\"],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":1,\"problem_name\":\"Stability certification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0,0,0,0,0,0,0)$\",\"$\\\\mathcal{N}=\\\\mathbb{R}^{8}$\",\"Verified type: Global exponential stability\"],\"record_id\":\"DCB-OURS-P1-001\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=(0,0,0,0,0,0,0,0)$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":\"$\\\\mathcal{N}=\\\\mathbb{R}^{8}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"\\\\mathbb{R}^{8}\",\"note\":\"Ambient Euclidean state space inferred from the stated dimension; no finite working domain was added.\",\"source\":\"dimension_and_dynamics\",\"status\":\"inferred\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":null,\"citations\":[],\"label\":null,\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":3992,\"start\":3958},\"title_latex\":\"8D Rational--Cosine Ring\"},\"source_order\":92,\"system\":{\"dimension\":{\"source\":\"overleaf_state_space_notation\",\"status\":\"inferred\",\"symbolic\":null,\"value\":8},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3\\\\\\\\ \\\\dot{x}_4\\\\\\\\ \\\\dot{x}_5\\\\\\\\ \\\\dot{x}_6\\\\\\\\ \\\\dot{x}_7\\\\\\\\ \\\\dot{x}_8 \\\\end{pmatrix} = \\\\begin{pmatrix} -3 x_1 + x_2 \\\\frac{\\\\cos x_1}{1 + x_8^2}\\\\\\\\[6pt] -3 x_2 + x_3 \\\\frac{\\\\cos x_2}{1 + x_1^2}\\\\\\\\[6pt] -3 x_3 + x_4 \\\\frac{\\\\cos x_3}{1 + x_2^2}\\\\\\\\[6pt] -3 x_4 + x_5 \\\\frac{\\\\cos x_4}{1 + x_3^2}\\\\\\\\[6pt] -3 x_5 + x_6 \\\\frac{\\\\cos x_1}{1 + x_4^2}\\\\\\\\[6pt] -3 x_6 + x_7 \\\\frac{\\\\cos x_2}{1 + x_5^2}\\\\\\\\[6pt] -3 x_7 + x_8 \\\\frac{\\\\cos x_3}{1 + x_6^2}\\\\\\\\[6pt] -3 x_8 + x_1 \\\\frac{\\\\cos x_4}{1 + x_7^2} \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3\\\\\\\\ \\\\dot{x}_4\\\\\\\\ \\\\dot{x}_5\\\\\\\\ \\\\dot{x}_6\\\\\\\\ \\\\dot{x}_7\\\\\\\\ \\\\dot{x}_8 \\\\end{pmatrix} = \\\\begin{pmatrix} -3 x_1 + x_2 \\\\frac{\\\\cos x_1}{1 + x_8^2}\\\\\\\\[6pt] -3 x_2 + x_3 \\\\frac{\\\\cos x_2}{1 + x_1^2}\\\\\\\\[6pt] -3 x_3 + x_4 \\\\frac{\\\\cos x_3}{1 + x_2^2}\\\\\\\\[6pt] -3 x_4 + x_5 \\\\frac{\\\\cos x_4}{1 + x_3^2}\\\\\\\\[6pt] -3 x_5 + x_6 \\\\frac{\\\\cos x_1}{1 + x_4^2}\\\\\\\\[6pt] -3 x_6 + x_7 \\\\frac{\\\\cos x_2}{1 + x_5^2}\\\\\\\\[6pt] -3 x_7 + x_8 \\\\frac{\\\\cos x_3}{1 + x_6^2}\\\\\\\\[6pt] -3 x_8 + x_1 \\\\frac{\\\\cos x_4}{1 + x_7^2} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"ours\",\"tier_name\":\"Hard/Ours\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"Stability certification\"},\"reported_method_latex\":null,\"reported_type_latex\":\"Verified type: Global exponential stability\"}}"} {"record_id":"DCB-OURS-P1-002","paper_system_number":93,"task_type":"stability","problem_name":"Stability certification","tier":"ours","dimension":7,"time_domain":"continuous","dynamics":"dx1/dt = -5*x1+x2*cos(x1)/(1+x7**2), dx2/dt = -(9/2)*x2+(6/5)*x4*sin(2*x2)/(1+2*x1**2), dx3/dt = -6*x3+(4/5)*x6*cos(x3/2)/(1+(3/2)*x2**2), dx4/dt = -(11/2)*x4+(3/2)*x7*sin((3/2)*x4)/(1+3*x3**2), dx5/dt = -5*x5+(9/10)*x1*cos(2*x5)/(1+(4/3)*x4**2), dx6/dt = -(9/2)*x6+(7/5)*x5*sin(x6/3)/(1+(5/2)*x5**2), dx7/dt = -6*x7+(1/2)*x3*cos((5/2)*x7)/(1+(1/2)*x5**2)","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} -5x_1 + x_2\\dfrac{\\cos(x_1)}{1+x_7^2}\\\\[4pt] -\\dfrac{9}{2}x_2 +\\dfrac{6}{5}x_4\\dfrac{\\sin(2x_2)}{1+2x_1^2}\\\\[4pt] -6x_3 +\\dfrac{4}{5}x_6\\dfrac{\\cos(x_3/2)} {1+\\frac{3}{2}x_2^2}\\\\[4pt] -\\dfrac{11}{2}x_4 +\\dfrac{3}{2}x_7\\dfrac{\\sin(3x_4/2)} {1+3x_3^2}\\\\[4pt] -5x_5 +\\dfrac{9}{10}x_1\\dfrac{\\cos(2x_5)} {1+\\frac{4}{3}x_4^2}\\\\[4pt] -\\dfrac{9}{2}x_6 +\\dfrac{7}{5}x_5\\dfrac{\\sin(x_6/3)} {1+\\frac{5}{2}x_5^2}\\\\[4pt] -6x_7 +\\dfrac{1}{2}x_3\\dfrac{\\cos(5x_7/2)} {1+\\frac{1}{2}x_5^2} \\end{pmatrix}. \\]","citations":[],"problem_json":"{\"problem_id\":\"dcb_ours_p1_002\",\"task_type\":\"stability\",\"name\":\"DynaCertBench System 093 · DCB-OURS-P1-002 · 7D Heterogeneous Trigonometric-Rational Network\",\"dynamics\":\"dx1/dt = -5*x1+x2*cos(x1)/(1+x7**2), dx2/dt = -(9/2)*x2+(6/5)*x4*sin(2*x2)/(1+2*x1**2), dx3/dt = -6*x3+(4/5)*x6*cos(x3/2)/(1+(3/2)*x2**2), dx4/dt = -(11/2)*x4+(3/2)*x7*sin((3/2)*x4)/(1+3*x3**2), dx5/dt = -5*x5+(9/10)*x1*cos(2*x5)/(1+(4/3)*x4**2), dx6/dt = -(9/2)*x6+(7/5)*x5*sin(x6/3)/(1+(5/2)*x5**2), dx7/dt = -6*x7+(1/2)*x3*cos((5/2)*x7)/(1+(1/2)*x5**2)\",\"source\":\"paper System 93; canonical record DCB-OURS-P1-002\",\"assumptions\":[\"Transcribed from active LaTeX System 93.\",\"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.\"],\"analysis_bounds\":[[-3,3],[-3,3],[-3,3],[-3,3],[-3,3],[-3,3],[-3,3]],\"equilibrium\":[0,0,0,0,0,0,0],\"stability_kind\":\"global_exponential\",\"neighborhood\":{\"type\":\"unbounded\"}}","executable_record_json":"{\"record_id\":\"DCB-OURS-P1-002\",\"paper_system_number\":93,\"paper_system_label\":\"System 93\",\"local_name\":\"S093__DCB-OURS-P1-002\",\"problem\":{\"problem_id\":\"dcb_ours_p1_002\",\"task_type\":\"stability\",\"name\":\"DynaCertBench System 093 · DCB-OURS-P1-002 · 7D Heterogeneous Trigonometric-Rational Network\",\"dynamics\":\"dx1/dt = -5*x1+x2*cos(x1)/(1+x7**2), dx2/dt = -(9/2)*x2+(6/5)*x4*sin(2*x2)/(1+2*x1**2), dx3/dt = -6*x3+(4/5)*x6*cos(x3/2)/(1+(3/2)*x2**2), dx4/dt = -(11/2)*x4+(3/2)*x7*sin((3/2)*x4)/(1+3*x3**2), dx5/dt = -5*x5+(9/10)*x1*cos(2*x5)/(1+(4/3)*x4**2), dx6/dt = -(9/2)*x6+(7/5)*x5*sin(x6/3)/(1+(5/2)*x5**2), dx7/dt = -6*x7+(1/2)*x3*cos((5/2)*x7)/(1+(1/2)*x5**2)\",\"source\":\"paper System 93; canonical record DCB-OURS-P1-002\",\"assumptions\":[\"Transcribed from active LaTeX System 93.\",\"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.\"],\"analysis_bounds\":[[-3,3],[-3,3],[-3,3],[-3,3],[-3,3],[-3,3],[-3,3]],\"equilibrium\":[0,0,0,0,0,0,0],\"stability_kind\":\"global_exponential\",\"neighborhood\":{\"type\":\"unbounded\"}},\"candidate\":null}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[\"source_label_missing\"],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":1,\"problem_name\":\"Stability certification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0,0,0,0,0,0)$\",\"$\\\\mathcal{N}=\\\\mathbb{R}^{7}$\",\"Verified type: Global exponential stability\"],\"record_id\":\"DCB-OURS-P1-002\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=(0,0,0,0,0,0,0)$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":\"$\\\\mathcal{N}=\\\\mathbb{R}^{7}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"\\\\mathbb{R}^{7}\",\"note\":\"Ambient Euclidean state space inferred from the stated dimension; no finite working domain was added.\",\"source\":\"dimension_and_dynamics\",\"status\":\"inferred\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 1.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":null,\"citations\":[],\"label\":null,\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":4028,\"start\":3994},\"title_latex\":\"7D Heterogeneous Trigonometric--Rational Network\"},\"source_order\":93,\"system\":{\"dimension\":{\"source\":\"overleaf_state_space_notation\",\"status\":\"inferred\",\"symbolic\":null,\"value\":7},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} -5x_1 + x_2\\\\dfrac{\\\\cos(x_1)}{1+x_7^2}\\\\\\\\[4pt] -\\\\dfrac{9}{2}x_2 +\\\\dfrac{6}{5}x_4\\\\dfrac{\\\\sin(2x_2)}{1+2x_1^2}\\\\\\\\[4pt] -6x_3 +\\\\dfrac{4}{5}x_6\\\\dfrac{\\\\cos(x_3/2)} {1+\\\\frac{3}{2}x_2^2}\\\\\\\\[4pt] -\\\\dfrac{11}{2}x_4 +\\\\dfrac{3}{2}x_7\\\\dfrac{\\\\sin(3x_4/2)} {1+3x_3^2}\\\\\\\\[4pt] -5x_5 +\\\\dfrac{9}{10}x_1\\\\dfrac{\\\\cos(2x_5)} {1+\\\\frac{4}{3}x_4^2}\\\\\\\\[4pt] -\\\\dfrac{9}{2}x_6 +\\\\dfrac{7}{5}x_5\\\\dfrac{\\\\sin(x_6/3)} {1+\\\\frac{5}{2}x_5^2}\\\\\\\\[4pt] -6x_7 +\\\\dfrac{1}{2}x_3\\\\dfrac{\\\\cos(5x_7/2)} {1+\\\\frac{1}{2}x_5^2} \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} -5x_1 + x_2\\\\dfrac{\\\\cos(x_1)}{1+x_7^2}\\\\\\\\[4pt] -\\\\dfrac{9}{2}x_2 +\\\\dfrac{6}{5}x_4\\\\dfrac{\\\\sin(2x_2)}{1+2x_1^2}\\\\\\\\[4pt] -6x_3 +\\\\dfrac{4}{5}x_6\\\\dfrac{\\\\cos(x_3/2)} {1+\\\\frac{3}{2}x_2^2}\\\\\\\\[4pt] -\\\\dfrac{11}{2}x_4 +\\\\dfrac{3}{2}x_7\\\\dfrac{\\\\sin(3x_4/2)} {1+3x_3^2}\\\\\\\\[4pt] -5x_5 +\\\\dfrac{9}{10}x_1\\\\dfrac{\\\\cos(2x_5)} {1+\\\\frac{4}{3}x_4^2}\\\\\\\\[4pt] -\\\\dfrac{9}{2}x_6 +\\\\dfrac{7}{5}x_5\\\\dfrac{\\\\sin(x_6/3)} {1+\\\\frac{5}{2}x_5^2}\\\\\\\\[4pt] -6x_7 +\\\\dfrac{1}{2}x_3\\\\dfrac{\\\\cos(5x_7/2)} {1+\\\\frac{1}{2}x_5^2} \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"ours\",\"tier_name\":\"Hard/Ours\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"Stability certification\"},\"reported_method_latex\":null,\"reported_type_latex\":\"Verified type: Global exponential stability\"}}"} {"record_id":"DCB-OURS-P2-001","paper_system_number":94,"task_type":"roa","problem_name":"Region-of-attraction verification","tier":"ours","dimension":10,"time_domain":"continuous","dynamics":"dx1/dt = -x1+1/4*x2+1/20*(x3/(1+x3**2)), dx2/dt = -x2+1/4*x3+1/20*(x4/(1+x4**2)), dx3/dt = -x3+1/4*x4+1/20*(x5/(1+x5**2)), dx4/dt = -x4+1/4*x5+1/20*(x6/(1+x6**2)), dx5/dt = -x5+1/4*x6+1/20*(x7/(1+x7**2)), dx6/dt = -x6+1/4*x7+1/20*(x8/(1+x8**2)), dx7/dt = -x7+1/4*x8+1/20*(x9/(1+x9**2)), dx8/dt = -x8+1/4*x9+1/20*(x10/(1+x10**2)), dx9/dt = -x9+1/4*x10+1/20*(x1/(1+x1**2)), dx10/dt = -x10+1/4*x1+1/20*(x2/(1+x2**2))","specification_latex":"The vector flow field is \\[ \\dot{x}_i=-x_i+\\frac{1}{4}x_{i+1}+\\frac{1}{20}\\frac{x_{i+2}}{1+x_{i+2}^2},\\qquad i=1,\\ldots,10, \\] where the indices are interpreted cyclically, with $x_{11}=x_1$ and $x_{12}=x_2$.","citations":[],"problem_json":"{\"problem_id\":\"dcb_ours_p2_001\",\"task_type\":\"roa\",\"name\":\"DynaCertBench System 094 · DCB-OURS-P2-001 · Our cyclic ten-dimensional rational system\",\"dynamics\":\"dx1/dt = -x1+1/4*x2+1/20*(x3/(1+x3**2)), dx2/dt = -x2+1/4*x3+1/20*(x4/(1+x4**2)), dx3/dt = -x3+1/4*x4+1/20*(x5/(1+x5**2)), dx4/dt = -x4+1/4*x5+1/20*(x6/(1+x6**2)), dx5/dt = -x5+1/4*x6+1/20*(x7/(1+x7**2)), dx6/dt = -x6+1/4*x7+1/20*(x8/(1+x8**2)), dx7/dt = -x7+1/4*x8+1/20*(x9/(1+x9**2)), dx8/dt = -x8+1/4*x9+1/20*(x10/(1+x10**2)), dx9/dt = -x9+1/4*x10+1/20*(x1/(1+x1**2)), dx10/dt = -x10+1/4*x1+1/20*(x2/(1+x2**2))\",\"source\":\"paper System 94; canonical record DCB-OURS-P2-001\",\"assumptions\":[\"Transcribed from paper-4-20-20260904 System 96.\",\"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.\",\"The formal neighborhood is R^10; [-10,10]^10 is numerical screening only.\"],\"analysis_bounds\":[[-10,10],[-10,10],[-10,10],[-10,10],[-10,10],[-10,10],[-10,10],[-10,10],[-10,10],[-10,10]],\"equilibrium\":[0,0,0,0,0,0,0,0,0,0],\"stability_kind\":\"local_asymptotic\",\"neighborhood\":{\"type\":\"unbounded\"},\"formal_domain\":\"unbounded\"}","executable_record_json":"{\"record_id\":\"DCB-OURS-P2-001\",\"paper_system_number\":94,\"paper_system_label\":\"System 94\",\"local_name\":\"S094__DCB-OURS-P2-001\",\"problem\":{\"problem_id\":\"dcb_ours_p2_001\",\"task_type\":\"roa\",\"name\":\"DynaCertBench System 094 · DCB-OURS-P2-001 · Our cyclic ten-dimensional rational system\",\"dynamics\":\"dx1/dt = -x1+1/4*x2+1/20*(x3/(1+x3**2)), dx2/dt = -x2+1/4*x3+1/20*(x4/(1+x4**2)), dx3/dt = -x3+1/4*x4+1/20*(x5/(1+x5**2)), dx4/dt = -x4+1/4*x5+1/20*(x6/(1+x6**2)), dx5/dt = -x5+1/4*x6+1/20*(x7/(1+x7**2)), dx6/dt = -x6+1/4*x7+1/20*(x8/(1+x8**2)), dx7/dt = -x7+1/4*x8+1/20*(x9/(1+x9**2)), dx8/dt = -x8+1/4*x9+1/20*(x10/(1+x10**2)), dx9/dt = -x9+1/4*x10+1/20*(x1/(1+x1**2)), dx10/dt = -x10+1/4*x1+1/20*(x2/(1+x2**2))\",\"source\":\"paper System 94; canonical record DCB-OURS-P2-001\",\"assumptions\":[\"Transcribed from paper-4-20-20260904 System 96.\",\"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.\",\"The formal neighborhood is R^10; [-10,10]^10 is numerical screening only.\"],\"analysis_bounds\":[[-10,10],[-10,10],[-10,10],[-10,10],[-10,10],[-10,10],[-10,10],[-10,10],[-10,10],[-10,10]],\"equilibrium\":[0,0,0,0,0,0,0,0,0,0],\"stability_kind\":\"local_asymptotic\",\"neighborhood\":{\"type\":\"unbounded\"},\"formal_domain\":\"unbounded\"},\"candidate\":null}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[\"source_label_missing\"],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":2,\"problem_name\":\"Region-of-attraction verification\",\"raw_items_latex\":[\"$x^\\\\star=\\\\mathbf{0}\\\\in\\\\mathbb{R}^{10}$\",\"$\\\\mathcal{N}=\\\\mathbb{R}^{10}$\",\"Verified type: ROA\"],\"record_id\":\"DCB-OURS-P2-001\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=\\\\mathbf{0}\\\\in\\\\mathbb{R}^{10}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"No separate initial set is required to certify Omega_c itself.\",\"source\":\"exact_roa_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":\"$\\\\mathcal{N}=\\\\mathbb{R}^{10}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"\\\\mathbb{R}^{10}\",\"note\":\"Ambient Euclidean state space inferred from the stated dimension; no finite working domain was added.\",\"source\":\"dimension_and_dynamics\",\"status\":\"inferred\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 2.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 2.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":null,\"citations\":[],\"label\":null,\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":4047,\"start\":4033},\"title_latex\":\"Our cyclic ten-dimensional rational system\"},\"source_order\":94,\"system\":{\"dimension\":{\"source\":\"overleaf_state_space_notation\",\"status\":\"inferred\",\"symbolic\":null,\"value\":10},\"display_math_latex\":[\"\\\\dot{x}_i=-x_i+\\\\frac{1}{4}x_{i+1}+\\\\frac{1}{20}\\\\frac{x_{i+2}}{1+x_{i+2}^2},\\\\qquad i=1,\\\\ldots,10,\"],\"specification_latex\":\"The vector flow field is \\\\[ \\\\dot{x}_i=-x_i+\\\\frac{1}{4}x_{i+1}+\\\\frac{1}{20}\\\\frac{x_{i+2}}{1+x_{i+2}^2},\\\\qquad i=1,\\\\ldots,10, \\\\] where the indices are interpreted cyclically, with $x_{11}=x_1$ and $x_{12}=x_2$.\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"ours\",\"tier_name\":\"Hard/Ours\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"neighborhood_N\"],\"requires_initial_set\":false,\"target\":\"exact_roa\"},\"reported_method_latex\":null,\"reported_type_latex\":\"Verified type: ROA\"}}"} {"record_id":"DCB-OURS-P2-002","paper_system_number":95,"task_type":"roa","problem_name":"Region-of-attraction verification","tier":"ours","dimension":7,"time_domain":"continuous","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)","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2\\\\ \\dot{x}_3\\\\ \\dot{x}_4\\\\ \\dot{x}_5\\\\ \\dot{x}_6\\\\ \\dot{x}_7 \\end{pmatrix} = \\begin{pmatrix} x_2\\\\[1mm] -\\dfrac12x_2-\\dfrac32\\sin x_1-\\sin(x_1-x_3)-\\dfrac32\\sin(x_1-x_7)\\\\[2mm] x_4\\\\[1mm] -\\dfrac12x_4-\\dfrac12\\sin x_3-\\sin(x_3-x_1)-\\sin(x_3-x_5)\\\\[2mm] x_6\\\\[1mm] -\\dfrac12x_6-\\sin x_5-\\sin(x_5-x_3)-\\dfrac32\\sin(x_5-x_7)\\\\[2mm] -\\dfrac15\\sin x_7-\\dfrac32\\sin(x_7-x_1)-\\dfrac32\\sin(x_7-x_5) \\end{pmatrix}. \\]","citations":[],"problem_json":"{\"problem_id\":\"dcb_ours_p2_002\",\"task_type\":\"roa\",\"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)\",\"source\":\"paper System 95; canonical record DCB-OURS-P2-002\",\"assumptions\":[\"Transcribed from active LaTeX System 95.\",\"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.\"],\"analysis_bounds\":[[-3,3],[-3,3],[-3,3],[-3,3],[-3,3],[-3,3],[-3,3]],\"equilibrium\":[0,0,0,0,0,0,0],\"neighborhood\":{\"type\":\"box\",\"bounds\":[[\"-pi\",\"pi\"],[\"-pi\",\"pi\"],[\"-pi\",\"pi\"],[\"-pi\",\"pi\"],[\"-pi\",\"pi\"],[\"-pi\",\"pi\"],[\"-pi\",\"pi\"]]},\"initial_set\":{\"type\":\"box\",\"bounds\":[[\"-3/5\",\"3/5\"],[\"-3/5\",\"3/5\"],[\"-3/5\",\"3/5\"],[\"-3/5\",\"3/5\"],[\"-3/5\",\"3/5\"],[\"-3/5\",\"3/5\"],[\"-3/5\",\"3/5\"]]},\"stability_kind\":\"local_asymptotic\"}","executable_record_json":"{\"record_id\":\"DCB-OURS-P2-002\",\"paper_system_number\":95,\"paper_system_label\":\"System 95\",\"local_name\":\"S095__DCB-OURS-P2-002\",\"problem\":{\"problem_id\":\"dcb_ours_p2_002\",\"task_type\":\"roa\",\"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)\",\"source\":\"paper System 95; canonical record DCB-OURS-P2-002\",\"assumptions\":[\"Transcribed from active LaTeX System 95.\",\"Decimal source constants are interpreted as exact rationals; no floating approximation enters Lean.\"],\"analysis_bounds\":[[-3,3],[-3,3],[-3,3],[-3,3],[-3,3],[-3,3],[-3,3]],\"equilibrium\":[0,0,0,0,0,0,0],\"neighborhood\":{\"type\":\"box\",\"bounds\":[[\"-pi\",\"pi\"],[\"-pi\",\"pi\"],[\"-pi\",\"pi\"],[\"-pi\",\"pi\"],[\"-pi\",\"pi\"],[\"-pi\",\"pi\"],[\"-pi\",\"pi\"]]},\"initial_set\":{\"type\":\"box\",\"bounds\":[[\"-3/5\",\"3/5\"],[\"-3/5\",\"3/5\"],[\"-3/5\",\"3/5\"],[\"-3/5\",\"3/5\"],[\"-3/5\",\"3/5\"],[\"-3/5\",\"3/5\"],[\"-3/5\",\"3/5\"]]},\"stability_kind\":\"local_asymptotic\"},\"candidate\":null}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"initial_set_X_I\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":2,\"problem_name\":\"Region-of-attraction verification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0,0,0,0,0,0)$\",\"$\\\\mathcal{N}=[-\\\\pi,\\\\pi]^7$\",\"$\\\\mathcal{X}_I=[-0.6,0.6]^7$\",\"Our verified type: ROA verification\"],\"record_id\":\"DCB-OURS-P2-002\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=(0,0,0,0,0,0,0)$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_I=[-0.6,0.6]^7$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":\"$\\\\mathcal{N}=[-\\\\pi,\\\\pi]^7$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"\\\\mathbb{R}^{7}\",\"note\":\"Ambient Euclidean state space inferred from the stated dimension; no finite working domain was added.\",\"source\":\"dimension_and_dynamics\",\"status\":\"inferred\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 2.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 2.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":null,\"citations\":[],\"label\":\"sys:swing7-roa\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":4084,\"start\":4050},\"title_latex\":\"Seven-dimensional multi-machine swing system\"},\"source_order\":95,\"system\":{\"dimension\":{\"source\":\"overleaf_state_space_notation\",\"status\":\"inferred\",\"symbolic\":null,\"value\":7},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3\\\\\\\\ \\\\dot{x}_4\\\\\\\\ \\\\dot{x}_5\\\\\\\\ \\\\dot{x}_6\\\\\\\\ \\\\dot{x}_7 \\\\end{pmatrix} = \\\\begin{pmatrix} x_2\\\\\\\\[1mm] -\\\\dfrac12x_2-\\\\dfrac32\\\\sin x_1-\\\\sin(x_1-x_3)-\\\\dfrac32\\\\sin(x_1-x_7)\\\\\\\\[2mm] x_4\\\\\\\\[1mm] -\\\\dfrac12x_4-\\\\dfrac12\\\\sin x_3-\\\\sin(x_3-x_1)-\\\\sin(x_3-x_5)\\\\\\\\[2mm] x_6\\\\\\\\[1mm] -\\\\dfrac12x_6-\\\\sin x_5-\\\\sin(x_5-x_3)-\\\\dfrac32\\\\sin(x_5-x_7)\\\\\\\\[2mm] -\\\\dfrac15\\\\sin x_7-\\\\dfrac32\\\\sin(x_7-x_1)-\\\\dfrac32\\\\sin(x_7-x_5) \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3\\\\\\\\ \\\\dot{x}_4\\\\\\\\ \\\\dot{x}_5\\\\\\\\ \\\\dot{x}_6\\\\\\\\ \\\\dot{x}_7 \\\\end{pmatrix} = \\\\begin{pmatrix} x_2\\\\\\\\[1mm] -\\\\dfrac12x_2-\\\\dfrac32\\\\sin x_1-\\\\sin(x_1-x_3)-\\\\dfrac32\\\\sin(x_1-x_7)\\\\\\\\[2mm] x_4\\\\\\\\[1mm] -\\\\dfrac12x_4-\\\\dfrac12\\\\sin x_3-\\\\sin(x_3-x_1)-\\\\sin(x_3-x_5)\\\\\\\\[2mm] x_6\\\\\\\\[1mm] -\\\\dfrac12x_6-\\\\sin x_5-\\\\sin(x_5-x_3)-\\\\dfrac32\\\\sin(x_5-x_7)\\\\\\\\[2mm] -\\\\dfrac15\\\\sin x_7-\\\\dfrac32\\\\sin(x_7-x_1)-\\\\dfrac32\\\\sin(x_7-x_5) \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"ours\",\"tier_name\":\"Hard/Ours\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"initial_set_X_I\",\"neighborhood_N\"],\"requires_initial_set\":true,\"target\":\"exact_roa\"},\"reported_method_latex\":null,\"reported_type_latex\":\"Our verified type: ROA verification\"}}"} {"record_id":"DCB-OURS-P2-003","paper_system_number":96,"task_type":"roa","problem_name":"Region-of-attraction verification","tier":"ours","dimension":7,"time_domain":"continuous","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","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2\\\\ \\dot{x}_3\\\\ \\dot{x}_4\\\\ \\dot{x}_5\\\\ \\dot{x}_6\\\\ \\dot{x}_7 \\end{pmatrix} = \\begin{pmatrix} x_2\\\\[1mm] -\\dfrac{2x_1}{1+x_1^2}+\\dfrac12x_3-0.3\\,x_2+0.2\\,x_5\\\\[2mm] x_4\\\\[1mm] \\dfrac12x_1-\\dfrac32x_3-\\sin(x_3)-0.2\\,x_4+0.1\\,x_6\\\\[2mm] -0.5\\,x_5-0.2\\,x_2+0.1\\,x_6\\\\[1mm] -0.4\\,x_6-0.1\\,x_5-0.1\\,x_4+0.1\\,x_7\\\\[1mm] -0.3\\,x_7-0.1\\,x_6 \\end{pmatrix}. \\]","citations":[],"problem_json":"{\"problem_id\":\"dcb_ours_p2_003\",\"task_type\":\"roa\",\"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.\"],\"analysis_bounds\":[[-5,5],[-5,5],[-5,5],[-5,5],[-5,5],[-5,5],[-5,5]],\"equilibrium\":[0,0,0,0,0,0,0],\"neighborhood\":{\"type\":\"box\",\"bounds\":[[\"-5\",\"5\"],[\"-5\",\"5\"],[\"-5\",\"5\"],[\"-5\",\"5\"],[\"-5\",\"5\"],[\"-5\",\"5\"],[\"-5\",\"5\"]]},\"initial_set\":{\"type\":\"box\",\"bounds\":[[\"-3/5\",\"3/5\"],[\"-3/5\",\"3/5\"],[\"-3/5\",\"3/5\"],[\"-3/5\",\"3/5\"],[\"-3/5\",\"3/5\"],[\"-3/5\",\"3/5\"],[\"-3/5\",\"3/5\"]]},\"stability_kind\":\"local_asymptotic\"}","executable_record_json":"{\"record_id\":\"DCB-OURS-P2-003\",\"paper_system_number\":96,\"paper_system_label\":\"System 96\",\"local_name\":\"S096__DCB-OURS-P2-003\",\"problem\":{\"problem_id\":\"dcb_ours_p2_003\",\"task_type\":\"roa\",\"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.\"],\"analysis_bounds\":[[-5,5],[-5,5],[-5,5],[-5,5],[-5,5],[-5,5],[-5,5]],\"equilibrium\":[0,0,0,0,0,0,0],\"neighborhood\":{\"type\":\"box\",\"bounds\":[[\"-5\",\"5\"],[\"-5\",\"5\"],[\"-5\",\"5\"],[\"-5\",\"5\"],[\"-5\",\"5\"],[\"-5\",\"5\"],[\"-5\",\"5\"]]},\"initial_set\":{\"type\":\"box\",\"bounds\":[[\"-3/5\",\"3/5\"],[\"-3/5\",\"3/5\"],[\"-3/5\",\"3/5\"],[\"-3/5\",\"3/5\"],[\"-3/5\",\"3/5\"],[\"-3/5\",\"3/5\"],[\"-3/5\",\"3/5\"]]},\"stability_kind\":\"local_asymptotic\"},\"candidate\":null}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"equilibrium\",\"initial_set_X_I\",\"neighborhood_N\"],\"set_contract_complete\":true},\"problem\":2,\"problem_name\":\"Region-of-attraction verification\",\"raw_items_latex\":[\"$x^\\\\star=(0,0,0,0,0,0,0)$\",\"$\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{7}\\\\mid -5\\\\le x_i\\\\le 5,\\\\ i=1,\\\\ldots,7\\\\}$\",\"$\\\\mathcal{X}_I=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{7}\\\\mid -0.6\\\\le x_i\\\\le 0.6,\\\\ i=1,\\\\ldots,7\\\\}$\",\"Our verified type: ROA verification\"],\"record_id\":\"DCB-OURS-P2-003\",\"sets\":{\"equilibrium\":{\"latex\":\"$x^\\\\star=(0,0,0,0,0,0,0)$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal{X}_I=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{7}\\\\mid -0.6\\\\le x_i\\\\le 0.6,\\\\ i=1,\\\\ldots,7\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":\"$\\\\mathcal{N}=\\\\{\\\\mathbf{x}\\\\in\\\\mathbb{R}^{7}\\\\mid -5\\\\le x_i\\\\le 5,\\\\ i=1,\\\\ldots,7\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"state_space_X\":{\"latex\":\"\\\\mathbb{R}^{7}\",\"note\":\"Ambient Euclidean state space inferred from the stated dimension; no finite working domain was added.\",\"source\":\"dimension_and_dynamics\",\"status\":\"inferred\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 2.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":null,\"note\":\"Not an input set in Problem 2.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"}},\"source\":{\"catalogue_record_id\":null,\"citations\":[],\"label\":\"sys:ours-roa-sat-7d\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":4114,\"start\":4086},\"title_latex\":\"Seven-dimensional coupled saturating oscillator\"},\"source_order\":96,\"system\":{\"dimension\":{\"source\":\"overleaf_state_space_notation\",\"status\":\"inferred\",\"symbolic\":null,\"value\":7},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3\\\\\\\\ \\\\dot{x}_4\\\\\\\\ \\\\dot{x}_5\\\\\\\\ \\\\dot{x}_6\\\\\\\\ \\\\dot{x}_7 \\\\end{pmatrix} = \\\\begin{pmatrix} x_2\\\\\\\\[1mm] -\\\\dfrac{2x_1}{1+x_1^2}+\\\\dfrac12x_3-0.3\\\\,x_2+0.2\\\\,x_5\\\\\\\\[2mm] x_4\\\\\\\\[1mm] \\\\dfrac12x_1-\\\\dfrac32x_3-\\\\sin(x_3)-0.2\\\\,x_4+0.1\\\\,x_6\\\\\\\\[2mm] -0.5\\\\,x_5-0.2\\\\,x_2+0.1\\\\,x_6\\\\\\\\[1mm] -0.4\\\\,x_6-0.1\\\\,x_5-0.1\\\\,x_4+0.1\\\\,x_7\\\\\\\\[1mm] -0.3\\\\,x_7-0.1\\\\,x_6 \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3\\\\\\\\ \\\\dot{x}_4\\\\\\\\ \\\\dot{x}_5\\\\\\\\ \\\\dot{x}_6\\\\\\\\ \\\\dot{x}_7 \\\\end{pmatrix} = \\\\begin{pmatrix} x_2\\\\\\\\[1mm] -\\\\dfrac{2x_1}{1+x_1^2}+\\\\dfrac12x_3-0.3\\\\,x_2+0.2\\\\,x_5\\\\\\\\[2mm] x_4\\\\\\\\[1mm] \\\\dfrac12x_1-\\\\dfrac32x_3-\\\\sin(x_3)-0.2\\\\,x_4+0.1\\\\,x_6\\\\\\\\[2mm] -0.5\\\\,x_5-0.2\\\\,x_2+0.1\\\\,x_6\\\\\\\\[1mm] -0.4\\\\,x_6-0.1\\\\,x_5-0.1\\\\,x_4+0.1\\\\,x_7\\\\\\\\[1mm] -0.3\\\\,x_7-0.1\\\\,x_6 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"ours\",\"tier_name\":\"Hard/Ours\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"equilibrium\",\"initial_set_X_I\",\"neighborhood_N\"],\"requires_initial_set\":true,\"target\":\"exact_roa\"},\"reported_method_latex\":null,\"reported_type_latex\":\"Our verified type: ROA verification\"}}"} {"record_id":"DCB-OURS-P3-001","paper_system_number":97,"task_type":"invariant","problem_name":"Forward-invariant set synthesis","tier":"ours","dimension":6,"time_domain":"continuous","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","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2\\\\ \\dot{x}_3\\\\ \\dot{x}_4\\\\ \\dot{x}_5\\\\ \\dot{x}_6 \\end{pmatrix} = \\begin{pmatrix} x_2+u_0\\\\[1mm] -x_1-\\dfrac1{10}\\sin(x_1)-x_2+x_3\\\\[2mm] -2x_3+x_4\\\\[1mm] -\\dfrac32x_4+x_5\\\\[2mm] -2x_5+x_6\\\\[1mm] -\\dfrac32x_6+u_1 \\end{pmatrix}. \\]","citations":[],"problem_json":"{\"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.\"],\"analysis_bounds\":[[-1,1],[-1,1],[-1,1],[-1,1],[-1,1],[-1,1]],\"controller_parameters\":\"u0,u1\",\"control_bounds\":[[-1,1],[-1,1]],\"initial_set\":{\"type\":\"ball\",\"center\":[0,0,0,0,0,0],\"radius\":\"1/2\"},\"unsafe_set\":{\"type\":\"ball\",\"center\":[0,\"4/5\",0,0,0,0],\"radius\":\"1/10\"}}","executable_record_json":"{\"record_id\":\"DCB-OURS-P3-001\",\"paper_system_number\":97,\"paper_system_label\":\"System 97\",\"local_name\":\"S097__DCB-OURS-P3-001\",\"problem\":{\"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.\"],\"analysis_bounds\":[[-1,1],[-1,1],[-1,1],[-1,1],[-1,1],[-1,1]],\"controller_parameters\":\"u0,u1\",\"control_bounds\":[[-1,1],[-1,1]],\"initial_set\":{\"type\":\"ball\",\"center\":[0,0,0,0,0,0],\"radius\":\"1/2\"},\"unsafe_set\":{\"type\":\"ball\",\"center\":[0,\"4/5\",0,0,0,0],\"radius\":\"1/10\"}},\"candidate\":null}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":3,\"problem_name\":\"Forward-invariant set synthesis\",\"raw_items_latex\":[\"$\\\\mathcal X=[-1,1]^6$\",\"$\\\\mathcal X_{\\\\mathrm{seed}}=\\\\left\\\\{x\\\\in\\\\mathbb R^6\\\\mid \\\\lVert x\\\\rVert_2\\\\le\\\\frac12\\\\right\\\\}$\",\"$\\\\mathcal X_U=\\\\left\\\\{x\\\\in\\\\mathbb R^6\\\\mid \\\\left\\\\lVert x-\\\\left(0,\\\\frac45,0,0,0,0\\\\right)^{\\\\mathsf T}\\\\right\\\\rVert_2\\\\le\\\\frac1{10}\\\\right\\\\}$\",\"$u_0,u_1\\\\in[-1,1]$ for every $x\\\\in\\\\mathcal X$\",\"The requested output is a nonempty controlled forward-invariant set $\\\\mathcal S\\\\subseteq\\\\mathcal X\\\\setminus\\\\mathcal X_U$ containing $\\\\mathcal X_{\\\\mathrm{seed}}$\",\"Task type: controlled forward-invariant set synthesis\"],\"record_id\":\"DCB-OURS-P3-001\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal X=[-1,1]^6$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"S is the requested nonempty forward-invariant output, not an input set.\",\"source\":\"problem_contract\",\"status\":\"output_to_synthesize\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal X_U=\\\\left\\\\{x\\\\in\\\\mathbb R^6\\\\mid \\\\left\\\\lVert x-\\\\left(0,\\\\frac45,0,0,0,0\\\\right)^{\\\\mathsf T}\\\\right\\\\rVert_2\\\\le\\\\frac1{10}\\\\right\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":null,\"citations\":[],\"label\":\"sys:new-p3-cosine-cascade-bounded\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":4155,\"start\":4119},\"title_latex\":\"New-6D Sin Cascade Invariance\"},\"source_order\":97,\"system\":{\"dimension\":{\"source\":\"overleaf_state_space_notation\",\"status\":\"inferred\",\"symbolic\":null,\"value\":6},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3\\\\\\\\ \\\\dot{x}_4\\\\\\\\ \\\\dot{x}_5\\\\\\\\ \\\\dot{x}_6 \\\\end{pmatrix} = \\\\begin{pmatrix} x_2+u_0\\\\\\\\[1mm] -x_1-\\\\dfrac1{10}\\\\sin(x_1)-x_2+x_3\\\\\\\\[2mm] -2x_3+x_4\\\\\\\\[1mm] -\\\\dfrac32x_4+x_5\\\\\\\\[2mm] -2x_5+x_6\\\\\\\\[1mm] -\\\\dfrac32x_6+u_1 \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3\\\\\\\\ \\\\dot{x}_4\\\\\\\\ \\\\dot{x}_5\\\\\\\\ \\\\dot{x}_6 \\\\end{pmatrix} = \\\\begin{pmatrix} x_2+u_0\\\\\\\\[1mm] -x_1-\\\\dfrac1{10}\\\\sin(x_1)-x_2+x_3\\\\\\\\[2mm] -2x_3+x_4\\\\\\\\[1mm] -\\\\dfrac32x_4+x_5\\\\\\\\[2mm] -2x_5+x_6\\\\\\\\[1mm] -\\\\dfrac32x_6+u_1 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"ours\",\"tier_name\":\"Hard/Ours\",\"verification\":{\"control_constraints_latex\":\"$u_0,u_1\\\\in[-1,1]$ for every $x\\\\in\\\\mathcal X$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"unsafe_set_X_U\"],\"requires_initial_set\":false,\"target\":\"Forward-invariant set synthesis\"},\"reported_method_latex\":null,\"reported_type_latex\":null}}"} {"record_id":"DCB-OURS-P3-002","paper_system_number":98,"task_type":"invariant","problem_name":"Forward-invariant set synthesis","tier":"ours","dimension":6,"time_domain":"continuous","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","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}}= \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2\\\\ \\dot{x}_3\\\\ \\dot{x}_4\\\\ \\dot{x}_5\\\\ \\dot{x}_6 \\end{pmatrix} = \\begin{pmatrix} x_2+u\\\\[1mm] -x_1-\\dfrac1{20}\\left(\\exp(x_1)-\\exp(-x_1)\\right)-x_2+x_3\\\\[2mm] -2x_3+x_4\\\\ -\\dfrac32x_4+x_5\\\\ -2x_5+x_6\\\\ -\\dfrac32x_6 \\end{pmatrix}. \\]","citations":[],"problem_json":"{\"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.\"],\"analysis_bounds\":[[-1,1],[-1,1],[-1,1],[-1,1],[-1,1],[-1,1]],\"controller_parameters\":\"u\",\"control_bounds\":[[-1,1]],\"initial_set\":{\"type\":\"ball\",\"center\":[\"-1/4\",\"1/4\",0,0,0,0],\"radius\":\"1/8\"},\"unsafe_set\":{\"type\":\"ball\",\"center\":[\"7/10\",\"-7/10\",0,0,0,0],\"radius\":\"1/10\"}}","executable_record_json":"{\"record_id\":\"DCB-OURS-P3-002\",\"paper_system_number\":98,\"paper_system_label\":\"System 98\",\"local_name\":\"S098__DCB-OURS-P3-002\",\"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.\"],\"analysis_bounds\":[[-1,1],[-1,1],[-1,1],[-1,1],[-1,1],[-1,1]],\"controller_parameters\":\"u\",\"control_bounds\":[[-1,1]],\"initial_set\":{\"type\":\"ball\",\"center\":[\"-1/4\",\"1/4\",0,0,0,0],\"radius\":\"1/8\"},\"unsafe_set\":{\"type\":\"ball\",\"center\":[\"7/10\",\"-7/10\",0,0,0,0],\"radius\":\"1/10\"}},\"candidate\":null}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":3,\"problem_name\":\"Forward-invariant set synthesis\",\"raw_items_latex\":[\"$\\\\mathcal X=[-1,1]^6$\",\"$\\\\mathcal X_I=\\\\left\\\\{x\\\\in\\\\mathbb R^6\\\\mid \\\\left(x_1+\\\\frac14\\\\right)^2+\\\\left(x_2-\\\\frac14\\\\right)^2+x_3^2+x_4^2+x_5^2+x_6^2\\\\le\\\\frac1{64}\\\\right\\\\}$\",\"$\\\\mathcal X_U=\\\\left\\\\{x\\\\in\\\\mathbb R^6\\\\mid \\\\left(x_1-\\\\frac7{10}\\\\right)^2+\\\\left(x_2+\\\\frac7{10}\\\\right)^2+x_3^2+x_4^2+x_5^2+x_6^2\\\\le\\\\frac1{100}\\\\right\\\\}$\",\"$u\\\\in[-1,1]$ for every $\\\\mathbf{x}\\\\in\\\\mathcal X$\",\"The requested output is a controlled forward-invariant set $\\\\mathcal S\\\\subseteq\\\\mathcal X\\\\setminus\\\\mathcal X_U$ containing $\\\\mathcal X_I$\",\"Task type: controlled forward-invariant set synthesis\"],\"record_id\":\"DCB-OURS-P3-002\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal X_I=\\\\left\\\\{x\\\\in\\\\mathbb R^6\\\\mid \\\\left(x_1+\\\\frac14\\\\right)^2+\\\\left(x_2-\\\\frac14\\\\right)^2+x_3^2+x_4^2+x_5^2+x_6^2\\\\le\\\\frac1{64}\\\\right\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal X=[-1,1]^6$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"S is the requested nonempty forward-invariant output, not an input set.\",\"source\":\"problem_contract\",\"status\":\"output_to_synthesize\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal X_U=\\\\left\\\\{x\\\\in\\\\mathbb R^6\\\\mid \\\\left(x_1-\\\\frac7{10}\\\\right)^2+\\\\left(x_2+\\\\frac7{10}\\\\right)^2+x_3^2+x_4^2+x_5^2+x_6^2\\\\le\\\\frac1{100}\\\\right\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":null,\"citations\":[],\"label\":\"sys:new-p3-exponential-cascade-bounded\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":4190,\"start\":4157},\"title_latex\":\"New-6D Exponential Cascade Invariance\"},\"source_order\":98,\"system\":{\"dimension\":{\"source\":\"overleaf_state_space_notation\",\"status\":\"inferred\",\"symbolic\":null,\"value\":6},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}}= \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3\\\\\\\\ \\\\dot{x}_4\\\\\\\\ \\\\dot{x}_5\\\\\\\\ \\\\dot{x}_6 \\\\end{pmatrix} = \\\\begin{pmatrix} x_2+u\\\\\\\\[1mm] -x_1-\\\\dfrac1{20}\\\\left(\\\\exp(x_1)-\\\\exp(-x_1)\\\\right)-x_2+x_3\\\\\\\\[2mm] -2x_3+x_4\\\\\\\\ -\\\\dfrac32x_4+x_5\\\\\\\\ -2x_5+x_6\\\\\\\\ -\\\\dfrac32x_6 \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}}= \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3\\\\\\\\ \\\\dot{x}_4\\\\\\\\ \\\\dot{x}_5\\\\\\\\ \\\\dot{x}_6 \\\\end{pmatrix} = \\\\begin{pmatrix} x_2+u\\\\\\\\[1mm] -x_1-\\\\dfrac1{20}\\\\left(\\\\exp(x_1)-\\\\exp(-x_1)\\\\right)-x_2+x_3\\\\\\\\[2mm] -2x_3+x_4\\\\\\\\ -\\\\dfrac32x_4+x_5\\\\\\\\ -2x_5+x_6\\\\\\\\ -\\\\dfrac32x_6 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"ours\",\"tier_name\":\"Hard/Ours\",\"verification\":{\"control_constraints_latex\":\"$u\\\\in[-1,1]$ for every $\\\\mathbf{x}\\\\in\\\\mathcal X$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"unsafe_set_X_U\"],\"requires_initial_set\":false,\"target\":\"Forward-invariant set synthesis\"},\"reported_method_latex\":null,\"reported_type_latex\":null}}"} {"record_id":"DCB-OURS-P3-003","paper_system_number":99,"task_type":"invariant","problem_name":"Forward-invariant set synthesis","tier":"ours","dimension":5,"time_domain":"continuous","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","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}} = \\begin{pmatrix} \\dot{x}_1\\\\ \\dot{x}_2\\\\ \\dot{x}_3\\\\ \\dot{x}_4\\\\ \\dot{x}_5 \\end{pmatrix} = \\begin{pmatrix} x_2-\\dfrac12x_1\\\\[2mm] -x_1-\\dfrac{x_1}{5(1+x_1^2)}-2x_2+\\dfrac{x_3}{10(1+x_3^2)}\\\\[3mm] x_4-\\dfrac12x_3\\\\[2mm] -x_3-\\dfrac1{40}\\left(\\exp(x_3)-\\exp(-x_3)\\right)-2x_4+\\dfrac15x_5\\\\[2mm] -\\dfrac32x_5-\\dfrac1{10}x_5^3+u \\end{pmatrix}. \\]","citations":[],"problem_json":"{\"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.\"],\"analysis_bounds\":[[-1,1],[-1,1],[-1,1],[-1,1],[-1,1]],\"controller_parameters\":\"u\",\"control_bounds\":[[-1,1]],\"unsafe_set\":{\"type\":\"ball\",\"center\":[\"7/10\",\"-7/10\",0,0,0],\"radius\":\"1/10\"}}","executable_record_json":"{\"record_id\":\"DCB-OURS-P3-003\",\"paper_system_number\":99,\"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.\"],\"analysis_bounds\":[[-1,1],[-1,1],[-1,1],[-1,1],[-1,1]],\"controller_parameters\":\"u\",\"control_bounds\":[[-1,1]],\"unsafe_set\":{\"type\":\"ball\",\"center\":[\"7/10\",\"-7/10\",0,0,0],\"radius\":\"1/10\"}},\"candidate\":null}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":3,\"problem_name\":\"Forward-invariant set synthesis\",\"raw_items_latex\":[\"$\\\\mathcal X=[-1,1]^5$\",\"$\\\\mathcal X_U=\\\\left\\\\{x\\\\in\\\\mathbb R^5\\\\mid \\\\left(x_1-\\\\frac7{10}\\\\right)^2+\\\\left(x_2+\\\\frac7{10}\\\\right)^2+x_3^2+x_4^2+x_5^2\\\\le\\\\frac1{100}\\\\right\\\\}$\",\"$u\\\\in[-1,1]$ for every $x\\\\in\\\\mathcal X$\",\"Task type: controlled forward-invariant set synthesis\"],\"record_id\":\"DCB-OURS-P3-003\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal X=[-1,1]^5$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"S is the requested nonempty forward-invariant output, not an input set.\",\"source\":\"problem_contract\",\"status\":\"output_to_synthesize\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal X_U=\\\\left\\\\{x\\\\in\\\\mathbb R^5\\\\mid \\\\left(x_1-\\\\frac7{10}\\\\right)^2+\\\\left(x_2+\\\\frac7{10}\\\\right)^2+x_3^2+x_4^2+x_5^2\\\\le\\\\frac1{100}\\\\right\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":null,\"citations\":[],\"label\":\"sys:new-p3-mixed-nonlinear-cascade-5d\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":4226,\"start\":4192},\"title_latex\":\"New-5D Mixed Nonlinear Cascade Invariance\"},\"source_order\":99,\"system\":{\"dimension\":{\"source\":\"overleaf_state_space_notation\",\"status\":\"inferred\",\"symbolic\":null,\"value\":5},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3\\\\\\\\ \\\\dot{x}_4\\\\\\\\ \\\\dot{x}_5 \\\\end{pmatrix} = \\\\begin{pmatrix} x_2-\\\\dfrac12x_1\\\\\\\\[2mm] -x_1-\\\\dfrac{x_1}{5(1+x_1^2)}-2x_2+\\\\dfrac{x_3}{10(1+x_3^2)}\\\\\\\\[3mm] x_4-\\\\dfrac12x_3\\\\\\\\[2mm] -x_3-\\\\dfrac1{40}\\\\left(\\\\exp(x_3)-\\\\exp(-x_3)\\\\right)-2x_4+\\\\dfrac15x_5\\\\\\\\[2mm] -\\\\dfrac32x_5-\\\\dfrac1{10}x_5^3+u \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}} = \\\\begin{pmatrix} \\\\dot{x}_1\\\\\\\\ \\\\dot{x}_2\\\\\\\\ \\\\dot{x}_3\\\\\\\\ \\\\dot{x}_4\\\\\\\\ \\\\dot{x}_5 \\\\end{pmatrix} = \\\\begin{pmatrix} x_2-\\\\dfrac12x_1\\\\\\\\[2mm] -x_1-\\\\dfrac{x_1}{5(1+x_1^2)}-2x_2+\\\\dfrac{x_3}{10(1+x_3^2)}\\\\\\\\[3mm] x_4-\\\\dfrac12x_3\\\\\\\\[2mm] -x_3-\\\\dfrac1{40}\\\\left(\\\\exp(x_3)-\\\\exp(-x_3)\\\\right)-2x_4+\\\\dfrac15x_5\\\\\\\\[2mm] -\\\\dfrac32x_5-\\\\dfrac1{10}x_5^3+u \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"ours\",\"tier_name\":\"Hard/Ours\",\"verification\":{\"control_constraints_latex\":\"$u\\\\in[-1,1]$ for every $x\\\\in\\\\mathcal X$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"unsafe_set_X_U\"],\"requires_initial_set\":false,\"target\":\"Forward-invariant set synthesis\"},\"reported_method_latex\":null,\"reported_type_latex\":null}}"} {"record_id":"DCB-OURS-P3-004","paper_system_number":100,"task_type":"invariant","problem_name":"Forward-invariant set synthesis","tier":"ours","dimension":6,"time_domain":"discrete","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))","specification_latex":"Consider an undirected network $\\mathcal G=(\\mathcal V,\\mathcal E)$ with $N=6$ nodes and adjacency matrix \\[ A= \\begin{pmatrix} 0&1&1&0&0&0\\\\ 1&0&1&1&0&0\\\\ 1&1&0&0&1&0\\\\ 0&1&0&0&1&0\\\\ 0&0&1&1&0&1\\\\ 0&0&0&0&1&0 \\end{pmatrix}, \\] where $A_{ij}=1$ means that nodes $i$ and $j$ are connected. The discrete-time dynamics are \\[ x_i(k+1)=x_i(k)+\\frac{1}{4}\\sum_{j=1}^{6}A_{ij}\\tanh\\bigl(x_j(k)-x_i(k)\\bigr), \\qquad i=1,\\ldots,6. \\]","citations":[],"problem_json":"{\"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.\"],\"analysis_bounds\":[[-1,1],[-1,1],[-1,1],[-1,1],[-1,1],[-1,1]],\"unsafe_set\":{\"type\":\"union\",\"sets\":[{\"type\":\"constraint\",\"lhs\":\"x1\",\"relation\":\"ge\",\"rhs\":\"4/5\"},{\"type\":\"constraint\",\"lhs\":\"x1\",\"relation\":\"le\",\"rhs\":\"-4/5\"},{\"type\":\"constraint\",\"lhs\":\"x2\",\"relation\":\"ge\",\"rhs\":\"4/5\"},{\"type\":\"constraint\",\"lhs\":\"x2\",\"relation\":\"le\",\"rhs\":\"-4/5\"},{\"type\":\"constraint\",\"lhs\":\"x3\",\"relation\":\"ge\",\"rhs\":\"4/5\"},{\"type\":\"constraint\",\"lhs\":\"x3\",\"relation\":\"le\",\"rhs\":\"-4/5\"},{\"type\":\"constraint\",\"lhs\":\"x4\",\"relation\":\"ge\",\"rhs\":\"4/5\"},{\"type\":\"constraint\",\"lhs\":\"x4\",\"relation\":\"le\",\"rhs\":\"-4/5\"},{\"type\":\"constraint\",\"lhs\":\"x5\",\"relation\":\"ge\",\"rhs\":\"4/5\"},{\"type\":\"constraint\",\"lhs\":\"x5\",\"relation\":\"le\",\"rhs\":\"-4/5\"},{\"type\":\"constraint\",\"lhs\":\"x6\",\"relation\":\"ge\",\"rhs\":\"4/5\"},{\"type\":\"constraint\",\"lhs\":\"x6\",\"relation\":\"le\",\"rhs\":\"-4/5\"}]}}","executable_record_json":"{\"record_id\":\"DCB-OURS-P3-004\",\"paper_system_number\":100,\"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.\"],\"analysis_bounds\":[[-1,1],[-1,1],[-1,1],[-1,1],[-1,1],[-1,1]],\"unsafe_set\":{\"type\":\"union\",\"sets\":[{\"type\":\"constraint\",\"lhs\":\"x1\",\"relation\":\"ge\",\"rhs\":\"4/5\"},{\"type\":\"constraint\",\"lhs\":\"x1\",\"relation\":\"le\",\"rhs\":\"-4/5\"},{\"type\":\"constraint\",\"lhs\":\"x2\",\"relation\":\"ge\",\"rhs\":\"4/5\"},{\"type\":\"constraint\",\"lhs\":\"x2\",\"relation\":\"le\",\"rhs\":\"-4/5\"},{\"type\":\"constraint\",\"lhs\":\"x3\",\"relation\":\"ge\",\"rhs\":\"4/5\"},{\"type\":\"constraint\",\"lhs\":\"x3\",\"relation\":\"le\",\"rhs\":\"-4/5\"},{\"type\":\"constraint\",\"lhs\":\"x4\",\"relation\":\"ge\",\"rhs\":\"4/5\"},{\"type\":\"constraint\",\"lhs\":\"x4\",\"relation\":\"le\",\"rhs\":\"-4/5\"},{\"type\":\"constraint\",\"lhs\":\"x5\",\"relation\":\"ge\",\"rhs\":\"4/5\"},{\"type\":\"constraint\",\"lhs\":\"x5\",\"relation\":\"le\",\"rhs\":\"-4/5\"},{\"type\":\"constraint\",\"lhs\":\"x6\",\"relation\":\"ge\",\"rhs\":\"4/5\"},{\"type\":\"constraint\",\"lhs\":\"x6\",\"relation\":\"le\",\"rhs\":\"-4/5\"}]}},\"candidate\":null}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":3,\"problem_name\":\"Forward-invariant set synthesis\",\"raw_items_latex\":[\"$\\\\mathcal X=[-1,1]^6$\",\"$\\\\mathcal X_U=\\\\{x\\\\in\\\\mathcal X\\\\mid \\\\exists i\\\\in\\\\{1,\\\\ldots,6\\\\},\\\\ |x_i|\\\\ge\\\\frac45\\\\}$\",\"Verified type: forward invariance verification\"],\"record_id\":\"DCB-OURS-P3-004\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal X=[-1,1]^6$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"S is the requested nonempty forward-invariant output, not an input set.\",\"source\":\"problem_contract\",\"status\":\"output_to_synthesize\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal X_U=\\\\{x\\\\in\\\\mathcal X\\\\mid \\\\exists i\\\\in\\\\{1,\\\\ldots,6\\\\},\\\\ |x_i|\\\\ge\\\\frac45\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":null,\"citations\":[],\"label\":\"sys:new-6d-opinion-dynamics\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":4277,\"start\":4247},\"title_latex\":\"New-6D Opinion Dynamics\"},\"source_order\":100,\"system\":{\"dimension\":{\"source\":\"overleaf_state_space_notation\",\"status\":\"inferred\",\"symbolic\":null,\"value\":6},\"display_math_latex\":[\"A= \\\\begin{pmatrix} 0&1&1&0&0&0\\\\\\\\ 1&0&1&1&0&0\\\\\\\\ 1&1&0&0&1&0\\\\\\\\ 0&1&0&0&1&0\\\\\\\\ 0&0&1&1&0&1\\\\\\\\ 0&0&0&0&1&0 \\\\end{pmatrix},\",\"x_i(k+1)=x_i(k)+\\\\frac{1}{4}\\\\sum_{j=1}^{6}A_{ij}\\\\tanh\\\\bigl(x_j(k)-x_i(k)\\\\bigr), \\\\qquad i=1,\\\\ldots,6.\"],\"specification_latex\":\"Consider an undirected network $\\\\mathcal G=(\\\\mathcal V,\\\\mathcal E)$ with $N=6$ nodes and adjacency matrix \\\\[ A= \\\\begin{pmatrix} 0&1&1&0&0&0\\\\\\\\ 1&0&1&1&0&0\\\\\\\\ 1&1&0&0&1&0\\\\\\\\ 0&1&0&0&1&0\\\\\\\\ 0&0&1&1&0&1\\\\\\\\ 0&0&0&0&1&0 \\\\end{pmatrix}, \\\\] where $A_{ij}=1$ means that nodes $i$ and $j$ are connected. The discrete-time dynamics are \\\\[ x_i(k+1)=x_i(k)+\\\\frac{1}{4}\\\\sum_{j=1}^{6}A_{ij}\\\\tanh\\\\bigl(x_j(k)-x_i(k)\\\\bigr), \\\\qquad i=1,\\\\ldots,6. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"discrete\"},\"variant_postamble_latex\":null},\"tier\":\"ours\",\"tier_name\":\"Hard/Ours\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"unsafe_set_X_U\"],\"requires_initial_set\":false,\"target\":\"Forward-invariant set synthesis\"},\"reported_method_latex\":null,\"reported_type_latex\":\"Verified type: forward invariance verification\"}}"} {"record_id":"DCB-OURS-P3-005","paper_system_number":101,"task_type":"invariant","problem_name":"Forward-invariant set synthesis","tier":"ours","dimension":10,"time_domain":"discrete","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)","specification_latex":"Consider the closed-loop $10$-dimensional Hopfield network $u(k)=\\pi(x(k))$. For every $i\\in\\{1,\\dots,10\\}$, \\[ x_i(k+1)=x_i(k) +0.05\\bigl(1-x_i(k)^2\\bigr) \\left[ -\\operatorname{arctanh}\\bigl(x_i(k)\\bigr) +\\sum_{j=1}^{10} W_{ij}x_j(k) +\\pi_{i}(x(k)) \\right]. \\] where the symmetric weight matrix is \\[ W= \\begin{pmatrix} -\\frac32 & \\frac14 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\ \\frac14 & -\\frac32 & \\frac14 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\ 0 & \\frac14 & -\\frac32 & \\frac14 & 0 & 0 & 0 & 0 & 0 & 0\\\\ 0 & 0 & \\frac14 & -\\frac32 & \\frac14 & 0 & 0 & 0 & 0 & 0\\\\ 0 & 0 & 0 & \\frac14 & -\\frac32 & \\frac14 & 0 & 0 & 0 & 0\\\\ 0 & 0 & 0 & 0 & \\frac14 & -\\frac32 & \\frac14 & 0 & 0 & 0\\\\ 0 & 0 & 0 & 0 & 0 & \\frac14 & -\\frac32 & \\frac14 & 0 & 0\\\\ 0 & 0 & 0 & 0 & 0 & 0 & \\frac14 & -\\frac32 & \\frac14 & 0\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac14 & -\\frac32 & \\frac14\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac14 & -\\frac32 \\end{pmatrix}. \\]","citations":[],"problem_json":"{\"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.\"],\"analysis_bounds\":[[\"-9/10\",\"9/10\"],[\"-9/10\",\"9/10\"],[\"-9/10\",\"9/10\"],[\"-9/10\",\"9/10\"],[\"-9/10\",\"9/10\"],[\"-9/10\",\"9/10\"],[\"-9/10\",\"9/10\"],[\"-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],[-2,2],[-2,2],[-2,2],[-2,2],[-2,2],[-2,2],[-2,2],[-2,2],[-2,2]],\"safety_interpretation\":\"outside_X_is_unsafe\"}","executable_record_json":"{\"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.\"],\"analysis_bounds\":[[\"-9/10\",\"9/10\"],[\"-9/10\",\"9/10\"],[\"-9/10\",\"9/10\"],[\"-9/10\",\"9/10\"],[\"-9/10\",\"9/10\"],[\"-9/10\",\"9/10\"],[\"-9/10\",\"9/10\"],[\"-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],[-2,2],[-2,2],[-2,2],[-2,2],[-2,2],[-2,2],[-2,2],[-2,2],[-2,2]],\"safety_interpretation\":\"outside_X_is_unsafe\"},\"candidate\":null}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":3,\"problem_name\":\"Forward-invariant set synthesis\",\"raw_items_latex\":[\"Domain set: $ \\\\mathcal{X}=[-0.9,0.9]^{10}. $\",\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{10}\\\\setminus \\\\mathcal{X}$\",\"Control constraint: $ \\\\pi(x)\\\\in[-2,2]. $\",\"Verified type: forward invariance verification\"],\"record_id\":\"DCB-OURS-P3-005\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 3.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"Domain set: $ \\\\mathcal{X}=[-0.9,0.9]^{10}. $\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"S is the requested nonempty forward-invariant output, not an input set.\",\"source\":\"problem_contract\",\"status\":\"output_to_synthesize\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{10}\\\\setminus \\\\mathcal{X}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":null,\"citations\":[],\"label\":\"sys:catalogue-hopfield-invariance\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":4323,\"start\":4279},\"title_latex\":\"Invariance Verification of a Discrete-Time Controlled Hopfield Network\"},\"source_order\":101,\"system\":{\"dimension\":{\"source\":\"overleaf_state_space_notation\",\"status\":\"inferred\",\"symbolic\":null,\"value\":10},\"display_math_latex\":[\"x_i(k+1)=x_i(k) +0.05\\\\bigl(1-x_i(k)^2\\\\bigr) \\\\left[ -\\\\operatorname{arctanh}\\\\bigl(x_i(k)\\\\bigr) +\\\\sum_{j=1}^{10} W_{ij}x_j(k) +\\\\pi_{i}(x(k)) \\\\right].\",\"W= \\\\begin{pmatrix} -\\\\frac32 & \\\\frac14 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\\\\ \\\\frac14 & -\\\\frac32 & \\\\frac14 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\\\\ 0 & \\\\frac14 & -\\\\frac32 & \\\\frac14 & 0 & 0 & 0 & 0 & 0 & 0\\\\\\\\ 0 & 0 & \\\\frac14 & -\\\\frac32 & \\\\frac14 & 0 & 0 & 0 & 0 & 0\\\\\\\\ 0 & 0 & 0 & \\\\frac14 & -\\\\frac32 & \\\\frac14 & 0 & 0 & 0 & 0\\\\\\\\ 0 & 0 & 0 & 0 & \\\\frac14 & -\\\\frac32 & \\\\frac14 & 0 & 0 & 0\\\\\\\\ 0 & 0 & 0 & 0 & 0 & \\\\frac14 & -\\\\frac32 & \\\\frac14 & 0 & 0\\\\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & \\\\frac14 & -\\\\frac32 & \\\\frac14 & 0\\\\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\\\frac14 & -\\\\frac32 & \\\\frac14\\\\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\\\frac14 & -\\\\frac32 \\\\end{pmatrix}.\"],\"specification_latex\":\"Consider the closed-loop $10$-dimensional Hopfield network $u(k)=\\\\pi(x(k))$. For every $i\\\\in\\\\{1,\\\\dots,10\\\\}$, \\\\[ x_i(k+1)=x_i(k) +0.05\\\\bigl(1-x_i(k)^2\\\\bigr) \\\\left[ -\\\\operatorname{arctanh}\\\\bigl(x_i(k)\\\\bigr) +\\\\sum_{j=1}^{10} W_{ij}x_j(k) +\\\\pi_{i}(x(k)) \\\\right]. \\\\] where the symmetric weight matrix is \\\\[ W= \\\\begin{pmatrix} -\\\\frac32 & \\\\frac14 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\\\\ \\\\frac14 & -\\\\frac32 & \\\\frac14 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\\\\ 0 & \\\\frac14 & -\\\\frac32 & \\\\frac14 & 0 & 0 & 0 & 0 & 0 & 0\\\\\\\\ 0 & 0 & \\\\frac14 & -\\\\frac32 & \\\\frac14 & 0 & 0 & 0 & 0 & 0\\\\\\\\ 0 & 0 & 0 & \\\\frac14 & -\\\\frac32 & \\\\frac14 & 0 & 0 & 0 & 0\\\\\\\\ 0 & 0 & 0 & 0 & \\\\frac14 & -\\\\frac32 & \\\\frac14 & 0 & 0 & 0\\\\\\\\ 0 & 0 & 0 & 0 & 0 & \\\\frac14 & -\\\\frac32 & \\\\frac14 & 0 & 0\\\\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & \\\\frac14 & -\\\\frac32 & \\\\frac14 & 0\\\\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\\\frac14 & -\\\\frac32 & \\\\frac14\\\\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\\\frac14 & -\\\\frac32 \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"discrete\"},\"variant_postamble_latex\":null},\"tier\":\"ours\",\"tier_name\":\"Hard/Ours\",\"verification\":{\"control_constraints_latex\":\"$\\\\mathcal{X}_U = \\\\mathbb{R}^{10}\\\\setminus \\\\mathcal{X}$\",\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"unsafe_set_X_U\"],\"requires_initial_set\":false,\"target\":\"Forward-invariant set synthesis\"},\"reported_method_latex\":null,\"reported_type_latex\":\"Verified type: forward invariance verification\"}}"} {"record_id":"DCB-OURS-P4-001","paper_system_number":102,"task_type":"safety","problem_name":"Safety verification","tier":"ours","dimension":8,"time_domain":"discrete","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)","specification_latex":"The discrete-time state update is: \\[ \\mathbf{x}(k+1)= \\begin{pmatrix} x_1(k)+\\dfrac1{200}\\left( -\\dfrac{x_1(k)}{1+x_1(k)^2}-4x_2(k) \\right) \\\\[2mm] x_2(k)+\\dfrac1{200}\\left( \\dfrac{4x_1(k)}{1+x_1(k)^2} -x_2(k)-\\dfrac{3x_3(k)}{1+x_3(k)^2} \\right) \\\\[2mm] x_3(k)+\\dfrac1{200}\\left( 3x_2(k)-\\dfrac{x_3(k)}{1+x_3(k)^2}-5x_4(k) \\right) \\\\[2mm] x_4(k)+\\dfrac1{200}\\left( \\dfrac{5x_3(k)}{1+x_3(k)^2} -x_4(k)-\\dfrac{2x_5(k)}{1+x_5(k)^2} \\right) \\\\[2mm] x_5(k)+\\dfrac1{200}\\left( 2x_4(k)-\\dfrac{x_5(k)}{1+x_5(k)^2}-4x_6(k) \\right) \\\\[2mm] x_6(k)+\\dfrac1{200}\\left( \\dfrac{4x_5(k)}{1+x_5(k)^2} -x_6(k)-\\dfrac{3x_7(k)}{1+x_7(k)^2} \\right) \\\\[2mm] x_7(k)+\\dfrac1{200}\\left( 3x_6(k)-\\dfrac{x_7(k)}{1+x_7(k)^2}-6x_8(k) \\right) \\\\[2mm] x_8(k)+\\dfrac1{200}\\left( \\dfrac{6x_7(k)}{1+x_7(k)^2}-x_8(k) \\right) \\end{pmatrix}. \\]","citations":[],"problem_json":"{\"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],[-1,1],[-1,1],[-1,1],[-1,1],[-1,1],[-1,1],[-1,1]],\"initial_set\":{\"type\":\"box\",\"bounds\":[[\"-3/20\",\"3/20\"],[\"-3/20\",\"3/20\"],[\"-3/20\",\"3/20\"],[\"-3/20\",\"3/20\"],[\"-3/20\",\"3/20\"],[\"-3/20\",\"3/20\"],[\"-3/20\",\"3/20\"],[\"-3/20\",\"3/20\"]]},\"unsafe_set\":{\"type\":\"box\",\"bounds\":[[\"3/4\",\"9/10\"],[\"-1/10\",\"1/10\"],[\"-1/10\",\"1/10\"],[\"-1/10\",\"1/10\"],[\"-1/10\",\"1/10\"],[\"-1/10\",\"1/10\"],[\"-1/10\",\"1/10\"],[\"-1/10\",\"1/10\"]]}}","executable_record_json":"{\"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],[-1,1],[-1,1],[-1,1],[-1,1],[-1,1],[-1,1],[-1,1]],\"initial_set\":{\"type\":\"box\",\"bounds\":[[\"-3/20\",\"3/20\"],[\"-3/20\",\"3/20\"],[\"-3/20\",\"3/20\"],[\"-3/20\",\"3/20\"],[\"-3/20\",\"3/20\"],[\"-3/20\",\"3/20\"],[\"-3/20\",\"3/20\"],[\"-3/20\",\"3/20\"]]},\"unsafe_set\":{\"type\":\"box\",\"bounds\":[[\"3/4\",\"9/10\"],[\"-1/10\",\"1/10\"],[\"-1/10\",\"1/10\"],[\"-1/10\",\"1/10\"],[\"-1/10\",\"1/10\"],[\"-1/10\",\"1/10\"],[\"-1/10\",\"1/10\"],[\"-1/10\",\"1/10\"]]}},\"candidate\":null}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal X=[-1,1]^8$\",\"$\\\\mathcal X_I=[-\\\\frac3{20},\\\\frac3{20}]^8$\",\"$\\\\mathcal X_U=[\\\\frac34,\\\\frac9{10}] \\\\times[-\\\\frac1{10},\\\\frac1{10}]^7$\",\"Task type: safety verification\"],\"record_id\":\"DCB-OURS-P4-001\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal X_I=[-\\\\frac3{20},\\\\frac3{20}]^8$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal X=[-1,1]^8$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal X_U=[\\\\frac34,\\\\frac9{10}] \\\\times[-\\\\frac1{10},\\\\frac1{10}]^7$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":null,\"citations\":[],\"label\":\"sys:new-8d-logarithmic-chain-safety\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":4391,\"start\":4328},\"title_latex\":\"New-8D Chain Safety\"},\"source_order\":102,\"system\":{\"dimension\":{\"source\":\"overleaf_state_space_notation\",\"status\":\"inferred\",\"symbolic\":null,\"value\":8},\"display_math_latex\":[\"\\\\mathbf{x}(k+1)= \\\\begin{pmatrix} x_1(k)+\\\\dfrac1{200}\\\\left( -\\\\dfrac{x_1(k)}{1+x_1(k)^2}-4x_2(k) \\\\right) \\\\\\\\[2mm] x_2(k)+\\\\dfrac1{200}\\\\left( \\\\dfrac{4x_1(k)}{1+x_1(k)^2} -x_2(k)-\\\\dfrac{3x_3(k)}{1+x_3(k)^2} \\\\right) \\\\\\\\[2mm] x_3(k)+\\\\dfrac1{200}\\\\left( 3x_2(k)-\\\\dfrac{x_3(k)}{1+x_3(k)^2}-5x_4(k) \\\\right) \\\\\\\\[2mm] x_4(k)+\\\\dfrac1{200}\\\\left( \\\\dfrac{5x_3(k)}{1+x_3(k)^2} -x_4(k)-\\\\dfrac{2x_5(k)}{1+x_5(k)^2} \\\\right) \\\\\\\\[2mm] x_5(k)+\\\\dfrac1{200}\\\\left( 2x_4(k)-\\\\dfrac{x_5(k)}{1+x_5(k)^2}-4x_6(k) \\\\right) \\\\\\\\[2mm] x_6(k)+\\\\dfrac1{200}\\\\left( \\\\dfrac{4x_5(k)}{1+x_5(k)^2} -x_6(k)-\\\\dfrac{3x_7(k)}{1+x_7(k)^2} \\\\right) \\\\\\\\[2mm] x_7(k)+\\\\dfrac1{200}\\\\left( 3x_6(k)-\\\\dfrac{x_7(k)}{1+x_7(k)^2}-6x_8(k) \\\\right) \\\\\\\\[2mm] x_8(k)+\\\\dfrac1{200}\\\\left( \\\\dfrac{6x_7(k)}{1+x_7(k)^2}-x_8(k) \\\\right) \\\\end{pmatrix}.\"],\"specification_latex\":\"The discrete-time state update is: \\\\[ \\\\mathbf{x}(k+1)= \\\\begin{pmatrix} x_1(k)+\\\\dfrac1{200}\\\\left( -\\\\dfrac{x_1(k)}{1+x_1(k)^2}-4x_2(k) \\\\right) \\\\\\\\[2mm] x_2(k)+\\\\dfrac1{200}\\\\left( \\\\dfrac{4x_1(k)}{1+x_1(k)^2} -x_2(k)-\\\\dfrac{3x_3(k)}{1+x_3(k)^2} \\\\right) \\\\\\\\[2mm] x_3(k)+\\\\dfrac1{200}\\\\left( 3x_2(k)-\\\\dfrac{x_3(k)}{1+x_3(k)^2}-5x_4(k) \\\\right) \\\\\\\\[2mm] x_4(k)+\\\\dfrac1{200}\\\\left( \\\\dfrac{5x_3(k)}{1+x_3(k)^2} -x_4(k)-\\\\dfrac{2x_5(k)}{1+x_5(k)^2} \\\\right) \\\\\\\\[2mm] x_5(k)+\\\\dfrac1{200}\\\\left( 2x_4(k)-\\\\dfrac{x_5(k)}{1+x_5(k)^2}-4x_6(k) \\\\right) \\\\\\\\[2mm] x_6(k)+\\\\dfrac1{200}\\\\left( \\\\dfrac{4x_5(k)}{1+x_5(k)^2} -x_6(k)-\\\\dfrac{3x_7(k)}{1+x_7(k)^2} \\\\right) \\\\\\\\[2mm] x_7(k)+\\\\dfrac1{200}\\\\left( 3x_6(k)-\\\\dfrac{x_7(k)}{1+x_7(k)^2}-6x_8(k) \\\\right) \\\\\\\\[2mm] x_8(k)+\\\\dfrac1{200}\\\\left( \\\\dfrac{6x_7(k)}{1+x_7(k)^2}-x_8(k) \\\\right) \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"discrete\"},\"variant_postamble_latex\":null},\"tier\":\"ours\",\"tier_name\":\"Hard/Ours\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":null,\"reported_type_latex\":null}}"} {"record_id":"DCB-OURS-P4-002","paper_system_number":103,"task_type":"safety","problem_name":"Safety verification","tier":"ours","dimension":6,"time_domain":"discrete","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)","specification_latex":"The discrete-time state update is: \\[ \\mathbf{x}(k+1)= \\begin{pmatrix} x_1(k)+\\dfrac1{200}\\left(-\\arctan(x_1(k))-8x_2(k)\\right)\\\\[2mm] x_2(k)+\\dfrac1{200}\\left(8\\arctan(x_1(k))-x_2(k)-6\\arctan(x_3(k))\\right)\\\\[2mm] x_3(k)+\\dfrac1{200}\\left(6x_2(k)-\\arctan(x_3(k))-10x_4(k)\\right)\\\\[2mm] x_4(k)+\\dfrac1{200}\\left(10\\arctan(x_3(k))-x_4(k)-4\\arctan(x_5(k))\\right)\\\\[2mm] x_5(k)+\\dfrac1{200}\\left(4x_4(k)-\\arctan(x_5(k))-8x_6(k)\\right)\\\\[2mm] x_6(k)+\\dfrac1{200}\\left(8\\arctan(x_5(k))-x_6(k)\\right) \\end{pmatrix}. \\]","citations":[],"problem_json":"{\"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],[-1,1],[-1,1],[-1,1],[-1,1],[-1,1]],\"initial_set\":{\"type\":\"box\",\"bounds\":[[\"-3/20\",\"3/20\"],[\"-3/20\",\"3/20\"],[\"-3/20\",\"3/20\"],[\"-3/20\",\"3/20\"],[\"-3/20\",\"3/20\"],[\"-3/20\",\"3/20\"]]},\"unsafe_set\":{\"type\":\"box\",\"bounds\":[[\"3/4\",\"9/10\"],[\"-1/10\",\"1/10\"],[\"-1/10\",\"1/10\"],[\"-1/10\",\"1/10\"],[\"-1/10\",\"1/10\"],[\"-1/10\",\"1/10\"]]}}","executable_record_json":"{\"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],[-1,1],[-1,1],[-1,1],[-1,1],[-1,1]],\"initial_set\":{\"type\":\"box\",\"bounds\":[[\"-3/20\",\"3/20\"],[\"-3/20\",\"3/20\"],[\"-3/20\",\"3/20\"],[\"-3/20\",\"3/20\"],[\"-3/20\",\"3/20\"],[\"-3/20\",\"3/20\"]]},\"unsafe_set\":{\"type\":\"box\",\"bounds\":[[\"3/4\",\"9/10\"],[\"-1/10\",\"1/10\"],[\"-1/10\",\"1/10\"],[\"-1/10\",\"1/10\"],[\"-1/10\",\"1/10\"],[\"-1/10\",\"1/10\"]]}},\"candidate\":null}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal X=[-1,1]^6$\",\"$\\\\mathcal X_I=[-\\\\frac3{20},\\\\frac3{20}]^6$\",\"$\\\\mathcal X_U=[\\\\frac34,\\\\frac9{10}]\\\\times[-\\\\frac1{10},\\\\frac1{10}]^5$\",\"Task type: safety verification\"],\"record_id\":\"DCB-OURS-P4-002\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal X_I=[-\\\\frac3{20},\\\\frac3{20}]^6$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal X=[-1,1]^6$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal X_U=[\\\\frac34,\\\\frac9{10}]\\\\times[-\\\\frac1{10},\\\\frac1{10}]^5$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":null,\"citations\":[],\"label\":\"sys:new-6d-arctangent-chain-safety\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":4417,\"start\":4393},\"title_latex\":\"New-6D Arctan Chain Safety\"},\"source_order\":103,\"system\":{\"dimension\":{\"source\":\"overleaf_state_space_notation\",\"status\":\"inferred\",\"symbolic\":null,\"value\":6},\"display_math_latex\":[\"\\\\mathbf{x}(k+1)= \\\\begin{pmatrix} x_1(k)+\\\\dfrac1{200}\\\\left(-\\\\arctan(x_1(k))-8x_2(k)\\\\right)\\\\\\\\[2mm] x_2(k)+\\\\dfrac1{200}\\\\left(8\\\\arctan(x_1(k))-x_2(k)-6\\\\arctan(x_3(k))\\\\right)\\\\\\\\[2mm] x_3(k)+\\\\dfrac1{200}\\\\left(6x_2(k)-\\\\arctan(x_3(k))-10x_4(k)\\\\right)\\\\\\\\[2mm] x_4(k)+\\\\dfrac1{200}\\\\left(10\\\\arctan(x_3(k))-x_4(k)-4\\\\arctan(x_5(k))\\\\right)\\\\\\\\[2mm] x_5(k)+\\\\dfrac1{200}\\\\left(4x_4(k)-\\\\arctan(x_5(k))-8x_6(k)\\\\right)\\\\\\\\[2mm] x_6(k)+\\\\dfrac1{200}\\\\left(8\\\\arctan(x_5(k))-x_6(k)\\\\right) \\\\end{pmatrix}.\"],\"specification_latex\":\"The discrete-time state update is: \\\\[ \\\\mathbf{x}(k+1)= \\\\begin{pmatrix} x_1(k)+\\\\dfrac1{200}\\\\left(-\\\\arctan(x_1(k))-8x_2(k)\\\\right)\\\\\\\\[2mm] x_2(k)+\\\\dfrac1{200}\\\\left(8\\\\arctan(x_1(k))-x_2(k)-6\\\\arctan(x_3(k))\\\\right)\\\\\\\\[2mm] x_3(k)+\\\\dfrac1{200}\\\\left(6x_2(k)-\\\\arctan(x_3(k))-10x_4(k)\\\\right)\\\\\\\\[2mm] x_4(k)+\\\\dfrac1{200}\\\\left(10\\\\arctan(x_3(k))-x_4(k)-4\\\\arctan(x_5(k))\\\\right)\\\\\\\\[2mm] x_5(k)+\\\\dfrac1{200}\\\\left(4x_4(k)-\\\\arctan(x_5(k))-8x_6(k)\\\\right)\\\\\\\\[2mm] x_6(k)+\\\\dfrac1{200}\\\\left(8\\\\arctan(x_5(k))-x_6(k)\\\\right) \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"discrete\"},\"variant_postamble_latex\":null},\"tier\":\"ours\",\"tier_name\":\"Hard/Ours\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":null,\"reported_type_latex\":null}}"} {"record_id":"DCB-OURS-P4-003","paper_system_number":104,"task_type":"safety","problem_name":"Safety verification","tier":"ours","dimension":8,"time_domain":"continuous","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)","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}}= \\begin{pmatrix} x_2 \\\\[1mm] -x_1-0.1\\sin(\\omega x_1)-x_2+x_3 \\\\[2mm] -2x_3+x_4 \\\\[1mm] -1.5x_4+x_5 \\\\[2mm] -2x_5+x_6 \\\\[1mm] -1.5x_6+0.2x_7 \\\\[2mm] -2x_7+0.2x_6+0.2x_8 -0.1x_7\\sin(x_8) \\\\[1mm] -1.5x_8+0.2x_7 -0.1x_8\\cos(x_7) \\end{pmatrix}. \\]","citations":[],"problem_json":"{\"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],[-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\"}}","executable_record_json":"{\"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],[-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}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal X=[-1,1]^8$\",\"$\\\\mathcal X_I= \\\\left\\\\{x\\\\in\\\\mathbb R^8\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.5\\\\right\\\\}$\",\"$\\\\mathcal X_U= \\\\left\\\\{x\\\\in\\\\mathbb R^8\\\\mid \\\\left\\\\|x-(0,0.8,0,0,0,0,0,0)^{\\\\mathsf T}\\\\right\\\\|_2\\\\le 0.1 \\\\right\\\\}$\",\"$\\\\omega\\\\in[0.5,2]$ is a fixed uncertain parameter\",\"The barrier certificate is state-only and uniform for every fixed $\\\\omega\\\\in[0.5,2]$\",\"Task type: robust safety verification\"],\"record_id\":\"DCB-OURS-P4-003\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal X_I= \\\\left\\\\{x\\\\in\\\\mathbb R^8\\\\mid \\\\lVert x\\\\rVert_2\\\\le 0.5\\\\right\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal X=[-1,1]^8$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal X_U= \\\\left\\\\{x\\\\in\\\\mathbb R^8\\\\mid \\\\left\\\\|x-(0,0.8,0,0,0,0,0,0)^{\\\\mathsf T}\\\\right\\\\|_2\\\\le 0.1 \\\\right\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":null,\"citations\":[],\"label\":\"sys:new-8d-parametric-nonlinear-safety\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":4455,\"start\":4420},\"title_latex\":\"New-8D Parametric Nonlinear Safety\"},\"source_order\":104,\"system\":{\"dimension\":{\"source\":\"overleaf_state_space_notation\",\"status\":\"inferred\",\"symbolic\":null,\"value\":8},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}}= \\\\begin{pmatrix} x_2 \\\\\\\\[1mm] -x_1-0.1\\\\sin(\\\\omega x_1)-x_2+x_3 \\\\\\\\[2mm] -2x_3+x_4 \\\\\\\\[1mm] -1.5x_4+x_5 \\\\\\\\[2mm] -2x_5+x_6 \\\\\\\\[1mm] -1.5x_6+0.2x_7 \\\\\\\\[2mm] -2x_7+0.2x_6+0.2x_8 -0.1x_7\\\\sin(x_8) \\\\\\\\[1mm] -1.5x_8+0.2x_7 -0.1x_8\\\\cos(x_7) \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}}= \\\\begin{pmatrix} x_2 \\\\\\\\[1mm] -x_1-0.1\\\\sin(\\\\omega x_1)-x_2+x_3 \\\\\\\\[2mm] -2x_3+x_4 \\\\\\\\[1mm] -1.5x_4+x_5 \\\\\\\\[2mm] -2x_5+x_6 \\\\\\\\[1mm] -1.5x_6+0.2x_7 \\\\\\\\[2mm] -2x_7+0.2x_6+0.2x_8 -0.1x_7\\\\sin(x_8) \\\\\\\\[1mm] -1.5x_8+0.2x_7 -0.1x_8\\\\cos(x_7) \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"ours\",\"tier_name\":\"Hard/Ours\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":null,\"reported_type_latex\":null}}"} {"record_id":"DCB-OURS-P4-004","paper_system_number":105,"task_type":"safety","problem_name":"Safety verification","tier":"ours","dimension":5,"time_domain":"continuous","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)","specification_latex":"The vector flow field is: \\[ \\dot{\\mathbf{x}}= \\begin{pmatrix} x_2+\\dfrac12x_5\\\\[2mm] -x_1-\\omega x_2+2x_4\\\\[1mm] x_4\\\\[1mm] -2x_3-\\dfrac18\\arctan(x_3)-\\dfrac32x_4-2x_2+x_5\\\\[2mm] -\\dfrac12x_1-x_4-x_5-\\dfrac18\\arctan(x_5) \\end{pmatrix}. \\]","citations":[],"problem_json":"{\"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],[-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\"}}","executable_record_json":"{\"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],[-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}","descriptive_record_json":"{\"candidate\":null,\"completeness\":{\"flags\":[],\"llm_input_ready\":true,\"missing_required_set_roles\":[],\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"set_contract_complete\":true},\"problem\":4,\"problem_name\":\"Safety verification\",\"raw_items_latex\":[\"$\\\\mathcal X=[-1,1]^5$\",\"$\\\\mathcal X_I=\\\\left\\\\{x\\\\in\\\\mathbb R^5\\\\mid\\\\lVert x\\\\rVert_2\\\\le\\\\frac12\\\\right\\\\}$\",\"$\\\\mathcal X_U=\\\\left\\\\{x\\\\in\\\\mathbb R^5\\\\mid\\\\left\\\\lVert x-\\\\left(0,0,\\\\frac45,0,0\\\\right)^{\\\\mathsf T}\\\\right\\\\rVert_2\\\\le\\\\frac1{10}\\\\right\\\\}$\",\"$\\\\omega\\\\in[1,2]$\",\"Task type: robust safety verification\"],\"record_id\":\"DCB-OURS-P4-004\",\"sets\":{\"equilibrium\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"initial_set_X_I\":{\"latex\":\"$\\\\mathcal X_I=\\\\left\\\\{x\\\\in\\\\mathbb R^5\\\\mid\\\\lVert x\\\\rVert_2\\\\le\\\\frac12\\\\right\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"neighborhood_N\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"state_space_X\":{\"latex\":\"$\\\\mathcal X=[-1,1]^5$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"},\"target_invariant_set_S\":{\"latex\":null,\"note\":\"Not an input set in Problem 4.\",\"source\":\"problem_contract\",\"status\":\"not_applicable\"},\"unsafe_set_X_U\":{\"latex\":\"$\\\\mathcal X_U=\\\\left\\\\{x\\\\in\\\\mathbb R^5\\\\mid\\\\left\\\\lVert x-\\\\left(0,0,\\\\frac45,0,0\\\\right)^{\\\\mathsf T}\\\\right\\\\rVert_2\\\\le\\\\frac1{10}\\\\right\\\\}$\",\"note\":\"\",\"source\":\"overleaf_item\",\"status\":\"specified\"}},\"source\":{\"catalogue_record_id\":null,\"citations\":[],\"label\":\"sys:new-5d-arctangent-perturbation-safety\",\"local_pdf\":null,\"overleaf_tex\":\"\",\"source_lines\":{\"end\":4480,\"start\":4457},\"title_latex\":\"New-5D Arctangent Perturbation Safety\"},\"source_order\":105,\"system\":{\"dimension\":{\"source\":\"overleaf_state_space_notation\",\"status\":\"inferred\",\"symbolic\":null,\"value\":5},\"display_math_latex\":[\"\\\\dot{\\\\mathbf{x}}= \\\\begin{pmatrix} x_2+\\\\dfrac12x_5\\\\\\\\[2mm] -x_1-\\\\omega x_2+2x_4\\\\\\\\[1mm] x_4\\\\\\\\[1mm] -2x_3-\\\\dfrac18\\\\arctan(x_3)-\\\\dfrac32x_4-2x_2+x_5\\\\\\\\[2mm] -\\\\dfrac12x_1-x_4-x_5-\\\\dfrac18\\\\arctan(x_5) \\\\end{pmatrix}.\"],\"specification_latex\":\"The vector flow field is: \\\\[ \\\\dot{\\\\mathbf{x}}= \\\\begin{pmatrix} x_2+\\\\dfrac12x_5\\\\\\\\[2mm] -x_1-\\\\omega x_2+2x_4\\\\\\\\[1mm] x_4\\\\\\\\[1mm] -2x_3-\\\\dfrac18\\\\arctan(x_3)-\\\\dfrac32x_4-2x_2+x_5\\\\\\\\[2mm] -\\\\dfrac12x_1-x_4-x_5-\\\\dfrac18\\\\arctan(x_5) \\\\end{pmatrix}. \\\\]\",\"time_domain\":{\"source\":\"dynamics_notation\",\"status\":\"inferred\",\"value\":\"continuous\"},\"variant_postamble_latex\":null},\"tier\":\"ours\",\"tier_name\":\"Hard/Ours\",\"verification\":{\"control_constraints_latex\":null,\"normalized_contract\":{\"current_framework_profile\":\"supported\",\"required_set_roles\":[\"state_space_X\",\"initial_set_X_I\",\"unsafe_set_X_U\"],\"requires_initial_set\":true,\"target\":\"Safety verification\"},\"reported_method_latex\":null,\"reported_type_latex\":null}}"}