| { | |
| "rules": [ | |
| { | |
| "id": "modal_transitive_axiom4", | |
| "domain": "modal_logic", | |
| "type": "theorem", | |
| "requires": ["frame:transitive"], | |
| "entails": ["□A->□□A", "□a->□□a", "公理4"], | |
| "evidence": 2 | |
| }, | |
| { | |
| "id": "modal_euclidean_axiom5", | |
| "domain": "modal_logic", | |
| "type": "theorem", | |
| "requires": ["frame:euclidean"], | |
| "entails": ["◊A->□◊A", "◊a->□◊a", "公理5"], | |
| "evidence": 2 | |
| }, | |
| { | |
| "id": "modal_reflexive_axiomT", | |
| "domain": "modal_logic", | |
| "type": "theorem", | |
| "requires": ["frame:reflexive"], | |
| "entails": ["□A->A", "□a->a", "公理T"], | |
| "evidence": 2 | |
| }, | |
| { | |
| "id": "topo_sphere_connectivity", | |
| "domain": "topology", | |
| "type": "theorem", | |
| "requires": ["condition:n<k"], | |
| "entails": ["π_n(S^k)=0", "pi_n(S^k)=0", "自明", "trivial"], | |
| "evidence": 3 | |
| }, | |
| { | |
| "id": "topo_refute_n_gt_k_always_finite", | |
| "domain": "topology", | |
| "type": "counter_example", | |
| "requires": ["condition:n>k"], | |
| "match_all": ["常に", "有限"], | |
| "note": "Counterexample: pi_3(S^2)=Z is not finite.", | |
| "evidence": -10 | |
| }, | |
| { | |
| "id": "topo_s1_higher_vanish", | |
| "domain": "topology", | |
| "type": "theorem", | |
| "requires": ["condition:n>1"], | |
| "entails": ["π_n(S^1)=0", "pi_n(S^1)=0", "trivial"], | |
| "evidence": 2 | |
| }, | |
| { | |
| "id": "topo_refute_s1_all_Z", | |
| "domain": "topology", | |
| "type": "counter_example", | |
| "requires": ["condition:all_n"], | |
| "match_all": ["すべて", "Z", "同型"], | |
| "note": "Counterexample: for n>1, pi_n(S^1)=0.", | |
| "evidence": -10 | |
| }, | |
| { | |
| "id": "hybrid_characterize_serial_trans_notrefl_noteucl", | |
| "domain": "modal_logic", | |
| "type": "theorem", | |
| "requires": ["logic:hybrid", "frame:serial", "frame:transitive", "frame:not_reflexive", "frame:not_euclidean"], | |
| "match_all": [ | |
| "@i", | |
| "◊", | |
| "□", | |
| "¬@i□i" | |
| ], | |
| "entails": ["@i□p->@i□□p"], | |
| "evidence": 5 | |
| }, | |
| { | |
| "id": "nec_serial_for_box_implies_diamond", | |
| "domain": "modal_logic", | |
| "type": "necessity_counterexample", | |
| "target_then": "□A->◊A", | |
| "missing_assumption": "frame:serial", | |
| "evidence": -10, | |
| "note": "Without seriality, □A can hold vacuously at a dead-end while ◊A fails." | |
| }, | |
| { | |
| "id": "modal_reflexive_axiomT", | |
| "domain": "modal_logic", | |
| "type": "theorem", | |
| "requires": ["frame:reflexive"], | |
| "entails": ["□A->A", "boxA->A"], | |
| "evidence": 2 | |
| }, | |
| { | |
| "id": "modal_transitive_axiom4", | |
| "domain": "modal_logic", | |
| "type": "theorem", | |
| "requires": ["frame:transitive"], | |
| "entails": ["□A->□□A", "boxA->boxboxA"], | |
| "evidence": 2 | |
| }, | |
| { | |
| "id": "modal_euclidean_axiom5", | |
| "domain": "modal_logic", | |
| "type": "theorem", | |
| "requires": ["frame:euclidean"], | |
| "entails": ["◊A->□◊A", "diamondA->boxdiamondA"], | |
| "evidence": 2 | |
| }, | |
| { | |
| "id": "modal_serial_axiomD", | |
| "domain": "modal_logic", | |
| "type": "theorem", | |
| "requires": ["frame:serial"], | |
| "entails": ["□A->◊A", "boxA->diamondA"], | |
| "evidence": 2 | |
| }, | |
| { | |
| "id": "refute_T_by_not_reflexive", | |
| "domain": "modal_logic", | |
| "type": "counter_example", | |
| "requires": ["frame:not_reflexive"], | |
| "match_all": ["□A->A"], | |
| "evidence": -10, | |
| "note": "T fails in non-reflexive frames." | |
| }, | |
| { | |
| "id": "refute_4_by_not_transitive", | |
| "domain": "modal_logic", | |
| "type": "counter_example", | |
| "requires": ["frame:not_transitive"], | |
| "match_all": ["□A->□□A"], | |
| "evidence": -10, | |
| "note": "4 fails in non-transitive frames." | |
| }, | |
| { | |
| "id": "refute_5_by_not_euclidean", | |
| "domain": "modal_logic", | |
| "type": "counter_example", | |
| "requires": ["frame:not_euclidean"], | |
| "match_all": ["◊A->□◊A"], | |
| "evidence": -10, | |
| "note": "5 fails in non-euclidean frames." | |
| }, | |
| { | |
| "id": "refute_D_by_not_serial", | |
| "domain": "modal_logic", | |
| "type": "counter_example", | |
| "requires": ["frame:not_serial"], | |
| "match_all": ["□A->◊A"], | |
| "evidence": -10, | |
| "note": "D fails in non-serial frames." | |
| }, | |
| { | |
| "id": "nec_reflexive_for_axiomT", | |
| "domain": "modal_logic", | |
| "type": "necessity_counterexample", | |
| "target_then": "□A->A", | |
| "missing_assumption": "frame:reflexive", | |
| "evidence": -10, | |
| "note": "Without reflexivity, axiom T fails." | |
| }, | |
| { | |
| "id": "nec_transitive_for_axiom4", | |
| "domain": "modal_logic", | |
| "type": "necessity_counterexample", | |
| "target_then": "□A->□□A", | |
| "missing_assumption": "frame:transitive", | |
| "evidence": -10, | |
| "note": "Without transitivity, axiom 4 fails." | |
| }, | |
| { | |
| "id": "nec_euclidean_for_axiom5", | |
| "domain": "modal_logic", | |
| "type": "necessity_counterexample", | |
| "target_then": "◊A->□◊A", | |
| "missing_assumption": "frame:euclidean", | |
| "evidence": -10, | |
| "note": "Without euclidean property, axiom 5 fails." | |
| }, | |
| { | |
| "id": "schema_describe_features_food", | |
| "domain": "schema", | |
| "type": "schema_theorem", | |
| "requires": ["intent:describe_features", "entity_type:food_item"], | |
| "output_schema": { | |
| "overview": "string", | |
| "category": "string", | |
| "taste": "string", | |
| "texture": "string", | |
| "nutrition": "string", | |
| "origin": "string", | |
| "uses": "string", | |
| "notes": "string" | |
| }, | |
| "evidence": 5 | |
| }, | |
| { | |
| "id": "schema_compare_general", | |
| "domain": "schema", | |
| "type": "schema_theorem", | |
| "requires": ["intent:compare", "comparison:true"], | |
| "output_schema": { | |
| "items_compared": "string", | |
| "key_differences": "string", | |
| "similarities": "string", | |
| "conclusion": "string" | |
| }, | |
| "evidence": 4 | |
| }, | |
| { | |
| "id": "schema_why_general", | |
| "domain": "schema", | |
| "type": "schema_theorem", | |
| "requires": ["intent:why"], | |
| "output_schema": { | |
| "core_reason": "string", | |
| "background_context": "string", | |
| "implications": "string" | |
| }, | |
| "evidence": 4 | |
| }, | |
| { | |
| "id": "schema_how_to_general", | |
| "domain": "schema", | |
| "type": "schema_theorem", | |
| "requires": ["intent:how_to"], | |
| "output_schema": { | |
| "goal": "string", | |
| "prerequisites": "string", | |
| "steps": "string", | |
| "tips": "string" | |
| }, | |
| "evidence": 4 | |
| }, | |
| { | |
| "id": "schema_explain_general", | |
| "domain": "schema", | |
| "type": "schema_theorem", | |
| "requires": ["intent:explain"], | |
| "output_schema": { | |
| "definition": "string", | |
| "details": "string", | |
| "key_points": "string" | |
| }, | |
| "evidence": 3 | |
| }, | |
| { | |
| "id": "schema_assumption_analysis_v1", | |
| "domain": "schema", | |
| "type": "schema_theorem", | |
| "requires": ["intent:assumption_analysis"], | |
| "output_schema": { | |
| "claim": "string", | |
| "assumptions": "string", | |
| "critical_assumptions": "string", | |
| "effect_if_dropped": "string", | |
| "counterexamples": "string", | |
| "final_verdict": "string" | |
| }, | |
| "evidence": 5 | |
| }, | |
| { | |
| "id": "schema_scope_and_limits_v1", | |
| "domain": "schema", | |
| "type": "schema_theorem", | |
| "requires": ["intent:scope_and_limits"], | |
| "output_schema": { | |
| "topic": "string", | |
| "scope": "string", | |
| "non_scope": "string", | |
| "assumptions": "string", | |
| "limits": "string", | |
| "unknowns": "string" | |
| }, | |
| "evidence": 5 | |
| }, | |
| { | |
| "id": "schema_failure_modes_v1", | |
| "domain": "schema", | |
| "type": "schema_theorem", | |
| "requires": ["intent:failure_modes"], | |
| "output_schema": { | |
| "system_or_topic": "string", | |
| "failure_modes": "string", | |
| "triggers": "string", | |
| "impact": "string", | |
| "mitigation": "string" | |
| }, | |
| "evidence": 5 | |
| }, | |
| { | |
| "id": "schema_why_not_v1", | |
| "domain": "schema", | |
| "type": "schema_theorem", | |
| "requires": ["intent:why_not"], | |
| "output_schema": { | |
| "question": "string", | |
| "reason": "string", | |
| "missing_conditions": "string", | |
| "conflicts": "string", | |
| "what_would_enable": "string", | |
| "summary": "string" | |
| }, | |
| "evidence": 5 | |
| } | |
| ] | |
| } |