{"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 4): 2 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 4)\n (=> (= split_index 0) (and (>= 2 3) true))\n (=> (= split_index 1) (and (>= 2 3) (< 2 3)))\n (=> (= split_index 2) (and (>= 3 3) (< 2 3) (< 2 3)))\n (=> (= split_index 3) (and (>= 3 3) (< 2 3) (< 2 3) (< 3 3))))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 4): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "arrays", "id": "arrays-monotonic_split-f01898bbe39e7059", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 2, "type": "integer"}, "maxItems": 4, "minItems": 4, "type": "array"}, "threshold": {"maximum": 9, "minimum": 2, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 2, "array": [2, 2, 3, 3], "difficulty": "easy", "generator": "array_invariants", "length": 4, "max_value": 9, "min_value": 2, "seed": 20260824, "threshold": 3, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 3, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 4 with\n2 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=4, T=3)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-f01898bbe39e7059-t000", "input": {"array": [2, 2, 3, 3], "threshold": 3}, "is_public": true, "problem_id": "arrays-monotonic_split-f01898bbe39e7059"}], "verification": {"duration_ms": 1.3677930110134184, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 4): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 4)\n (=> (= equilibrium_index 0) (= 0 11))\n (=> (= equilibrium_index 1) (= 6 6))\n (=> (= equilibrium_index 2) (= 11 2))\n (=> (= equilibrium_index 3) (= 15 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 4): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "arrays", "id": "arrays-equilibrium_index-fcaf408ef5765438", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 4, "minItems": 4, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 1, "array": [6, 5, 4, 2], "difficulty": "easy", "generator": "array_invariants", "length": 4, "max_value": 9, "min_value": 1, "pivot": 1, "seed": 20260825, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 3, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 4. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=4)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-fcaf408ef5765438-t000", "input": {"array": [6, 5, 4, 2]}, "is_public": true, "problem_id": "arrays-equilibrium_index-fcaf408ef5765438"}], "verification": {"duration_ms": 0.9125840151682496, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 5): 2 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 5)\n (=> (= split_index 0) (and (>= 2 2) true))\n (=> (= split_index 1) (and (>= 2 2) (< 2 2)))\n (=> (= split_index 2) (and (>= 2 2) (< 2 2) (< 2 2)))\n (=> (= split_index 3) (and (>= 4 2) (< 2 2) (< 2 2) (< 2 2)))\n (=> (= split_index 4) (and (>= 5 2) (< 2 2) (< 2 2) (< 2 2) (< 4 2))))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 5): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "arrays", "id": "arrays-monotonic_split-350c2705f9c69f79", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 2, "type": "integer"}, "maxItems": 5, "minItems": 5, "type": "array"}, "threshold": {"maximum": 9, "minimum": 2, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 0, "array": [2, 2, 2, 4, 5], "difficulty": "easy", "generator": "array_invariants", "length": 5, "max_value": 9, "min_value": 2, "seed": 20260826, "threshold": 2, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 4, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 5 with\n2 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=5, T=2)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 0, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-350c2705f9c69f79-t000", "input": {"array": [2, 2, 2, 4, 5], "threshold": 2}, "is_public": true, "problem_id": "arrays-monotonic_split-350c2705f9c69f79"}], "verification": {"duration_ms": 0.9551250259391963, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 3): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 3)\n (=> (= equilibrium_index 0) (= 0 8))\n (=> (= equilibrium_index 1) (= 4 4))\n (=> (= equilibrium_index 2) (= 8 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 3): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "arrays", "id": "arrays-equilibrium_index-3d7cbc7fe35c7c48", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 3, "minItems": 3, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 1, "array": [4, 4, 4], "difficulty": "easy", "generator": "array_invariants", "length": 3, "max_value": 9, "min_value": 1, "pivot": 1, "seed": 20260827, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 2, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 3. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=3)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-3d7cbc7fe35c7c48-t000", "input": {"array": [4, 4, 4]}, "is_public": true, "problem_id": "arrays-equilibrium_index-3d7cbc7fe35c7c48"}], "verification": {"duration_ms": 0.8890419849194586, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 4): 0 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 4)\n (=> (= split_index 0) (and (>= 0 1) true))\n (=> (= split_index 1) (and (>= 1 1) (< 0 1)))\n (=> (= split_index 2) (and (>= 2 1) (< 0 1) (< 1 1)))\n (=> (= split_index 3) (and (>= 4 1) (< 0 1) (< 1 1) (< 2 1))))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 4): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "arrays", "id": "arrays-monotonic_split-23989c1c06cd8564", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 0, "type": "integer"}, "maxItems": 4, "minItems": 4, "type": "array"}, "threshold": {"maximum": 9, "minimum": 0, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 1, "array": [0, 1, 2, 4], "difficulty": "easy", "generator": "array_invariants", "length": 4, "max_value": 9, "min_value": 0, "seed": 20260828, "threshold": 1, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 3, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 4 with\n0 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=4, T=1)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-23989c1c06cd8564-t000", "input": {"array": [0, 1, 2, 4], "threshold": 1}, "is_public": true, "problem_id": "arrays-monotonic_split-23989c1c06cd8564"}], "verification": {"duration_ms": 0.889291986823082, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 4): 2 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 4)\n (=> (= split_index 0) (and (>= 2 2) true))\n (=> (= split_index 1) (and (>= 4 2) (< 2 2)))\n (=> (= split_index 2) (and (>= 4 2) (< 2 2) (< 4 2)))\n (=> (= split_index 3) (and (>= 4 2) (< 2 2) (< 4 2) (< 4 2))))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 4): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "arrays", "id": "arrays-monotonic_split-f83ad4c3b3a4945b", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 2, "type": "integer"}, "maxItems": 4, "minItems": 4, "type": "array"}, "threshold": {"maximum": 9, "minimum": 2, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 0, "array": [2, 4, 4, 4], "difficulty": "easy", "generator": "array_invariants", "length": 4, "max_value": 9, "min_value": 2, "seed": 20260829, "threshold": 2, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 3, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 4 with\n2 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=4, T=2)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 0, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-f83ad4c3b3a4945b-t000", "input": {"array": [2, 4, 4, 4], "threshold": 2}, "is_public": true, "problem_id": "arrays-monotonic_split-f83ad4c3b3a4945b"}], "verification": {"duration_ms": 0.9175420273095369, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 3): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 3)\n (=> (= equilibrium_index 0) (= 0 11))\n (=> (= equilibrium_index 1) (= 2 2))\n (=> (= equilibrium_index 2) (= 11 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 3): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "arrays", "id": "arrays-equilibrium_index-5c25118b81b05c85", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 3, "minItems": 3, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 1, "array": [2, 9, 2], "difficulty": "easy", "generator": "array_invariants", "length": 3, "max_value": 9, "min_value": 1, "pivot": 1, "seed": 20260830, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 2, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 3. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=3)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-5c25118b81b05c85-t000", "input": {"array": [2, 9, 2]}, "is_public": true, "problem_id": "arrays-equilibrium_index-5c25118b81b05c85"}], "verification": {"duration_ms": 0.8659999584779143, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 3): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 3)\n (=> (= equilibrium_index 0) (= 0 15))\n (=> (= equilibrium_index 1) (= 8 8))\n (=> (= equilibrium_index 2) (= 15 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 3): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "arrays", "id": "arrays-equilibrium_index-effa7d045cc99980", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 3, "minItems": 3, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 1, "array": [8, 7, 8], "difficulty": "easy", "generator": "array_invariants", "length": 3, "max_value": 9, "min_value": 1, "pivot": 1, "seed": 20260831, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 2, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 3. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=3)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-effa7d045cc99980-t000", "input": {"array": [8, 7, 8]}, "is_public": true, "problem_id": "arrays-equilibrium_index-effa7d045cc99980"}], "verification": {"duration_ms": 0.8572919759899378, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 3): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 3)\n (=> (= equilibrium_index 0) (= 0 12))\n (=> (= equilibrium_index 1) (= 8 8))\n (=> (= equilibrium_index 2) (= 12 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 3): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "arrays", "id": "arrays-equilibrium_index-88beb1835466c78f", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 3, "minItems": 3, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 1, "array": [8, 4, 8], "difficulty": "easy", "generator": "array_invariants", "length": 3, "max_value": 9, "min_value": 1, "pivot": 1, "seed": 20260832, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 2, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 3. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=3)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-88beb1835466c78f-t000", "input": {"array": [8, 4, 8]}, "is_public": true, "problem_id": "arrays-equilibrium_index-88beb1835466c78f"}], "verification": {"duration_ms": 0.8485830039717257, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 5): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 5)\n (=> (= equilibrium_index 0) (= 0 32))\n (=> (= equilibrium_index 1) (= 9 25))\n (=> (= equilibrium_index 2) (= 16 16))\n (=> (= equilibrium_index 3) (= 25 9))\n (=> (= equilibrium_index 4) (= 32 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 5): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "arrays", "id": "arrays-equilibrium_index-16232be3b50ce037", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 5, "minItems": 5, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 2, "array": [9, 7, 9, 7, 9], "difficulty": "easy", "generator": "array_invariants", "length": 5, "max_value": 9, "min_value": 1, "pivot": 2, "seed": 20260833, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 4, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 5. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=5)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-16232be3b50ce037-t000", "input": {"array": [9, 7, 9, 7, 9]}, "is_public": true, "problem_id": "arrays-equilibrium_index-16232be3b50ce037"}], "verification": {"duration_ms": 0.8824160322546959, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 5): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 5)\n (=> (= equilibrium_index 0) (= 0 11))\n (=> (= equilibrium_index 1) (= 7 7))\n (=> (= equilibrium_index 2) (= 11 6))\n (=> (= equilibrium_index 3) (= 12 3))\n (=> (= equilibrium_index 4) (= 15 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 5): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "arrays", "id": "arrays-equilibrium_index-2f71bdcf418d921f", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 5, "minItems": 5, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 1, "array": [7, 4, 1, 3, 3], "difficulty": "easy", "generator": "array_invariants", "length": 5, "max_value": 9, "min_value": 1, "pivot": 1, "seed": 20260834, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 4, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 5. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=5)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-2f71bdcf418d921f-t000", "input": {"array": [7, 4, 1, 3, 3]}, "is_public": true, "problem_id": "arrays-equilibrium_index-2f71bdcf418d921f"}], "verification": {"duration_ms": 0.9207919938489795, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 3): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 3)\n (=> (= split_index 0) (and (>= 1 5) true))\n (=> (= split_index 1) (and (>= 3 5) (< 1 5)))\n (=> (= split_index 2) (and (>= 5 5) (< 1 5) (< 3 5))))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 3): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "arrays", "id": "arrays-monotonic_split-ec4a226bd5d49775", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 3, "minItems": 3, "type": "array"}, "threshold": {"maximum": 9, "minimum": 1, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 2, "array": [1, 3, 5], "difficulty": "easy", "generator": "array_invariants", "length": 3, "max_value": 9, "min_value": 1, "seed": 20260835, "threshold": 5, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 2, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 3 with\n1 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=3, T=5)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-ec4a226bd5d49775-t000", "input": {"array": [1, 3, 5], "threshold": 5}, "is_public": true, "problem_id": "arrays-monotonic_split-ec4a226bd5d49775"}], "verification": {"duration_ms": 0.9512079996056855, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 4): 2 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 4)\n (=> (= split_index 0) (and (>= 2 6) true))\n (=> (= split_index 1) (and (>= 3 6) (< 2 6)))\n (=> (= split_index 2) (and (>= 4 6) (< 2 6) (< 3 6)))\n (=> (= split_index 3) (and (>= 6 6) (< 2 6) (< 3 6) (< 4 6))))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 4): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "arrays", "id": "arrays-monotonic_split-530fb97887b10d6a", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 2, "type": "integer"}, "maxItems": 4, "minItems": 4, "type": "array"}, "threshold": {"maximum": 9, "minimum": 2, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 3, "array": [2, 3, 4, 6], "difficulty": "easy", "generator": "array_invariants", "length": 4, "max_value": 9, "min_value": 2, "seed": 20260836, "threshold": 6, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 3, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 4 with\n2 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=4, T=6)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 3, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-530fb97887b10d6a-t000", "input": {"array": [2, 3, 4, 6], "threshold": 6}, "is_public": true, "problem_id": "arrays-monotonic_split-530fb97887b10d6a"}], "verification": {"duration_ms": 0.9209589916281402, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 3): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 3)\n (=> (= equilibrium_index 0) (= 0 16))\n (=> (= equilibrium_index 1) (= 8 8))\n (=> (= equilibrium_index 2) (= 16 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 3): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "arrays", "id": "arrays-equilibrium_index-5f2b8929f7be3577", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 3, "minItems": 3, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 1, "array": [8, 8, 8], "difficulty": "easy", "generator": "array_invariants", "length": 3, "max_value": 9, "min_value": 1, "pivot": 1, "seed": 20260837, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 2, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 3. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=3)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-5f2b8929f7be3577-t000", "input": {"array": [8, 8, 8]}, "is_public": true, "problem_id": "arrays-equilibrium_index-5f2b8929f7be3577"}], "verification": {"duration_ms": 0.8921670378185809, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 5): 2 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 5)\n (=> (= split_index 0) (and (>= 2 3) true))\n (=> (= split_index 1) (and (>= 2 3) (< 2 3)))\n (=> (= split_index 2) (and (>= 3 3) (< 2 3) (< 2 3)))\n (=> (= split_index 3) (and (>= 4 3) (< 2 3) (< 2 3) (< 3 3)))\n (=> (= split_index 4) (and (>= 6 3) (< 2 3) (< 2 3) (< 3 3) (< 4 3))))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 5): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "arrays", "id": "arrays-monotonic_split-8883bccc01f9282a", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 2, "type": "integer"}, "maxItems": 5, "minItems": 5, "type": "array"}, "threshold": {"maximum": 9, "minimum": 2, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 2, "array": [2, 2, 3, 4, 6], "difficulty": "easy", "generator": "array_invariants", "length": 5, "max_value": 9, "min_value": 2, "seed": 20260838, "threshold": 3, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 4, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 5 with\n2 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=5, T=3)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-8883bccc01f9282a-t000", "input": {"array": [2, 2, 3, 4, 6], "threshold": 3}, "is_public": true, "problem_id": "arrays-monotonic_split-8883bccc01f9282a"}], "verification": {"duration_ms": 0.9295420022681355, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 5): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 5)\n (=> (= equilibrium_index 0) (= 0 21))\n (=> (= equilibrium_index 1) (= 5 14))\n (=> (= equilibrium_index 2) (= 12 12))\n (=> (= equilibrium_index 3) (= 14 5))\n (=> (= equilibrium_index 4) (= 21 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 5): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "arrays", "id": "arrays-equilibrium_index-13520bb36d44b68e", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 5, "minItems": 5, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 2, "array": [5, 7, 2, 7, 5], "difficulty": "easy", "generator": "array_invariants", "length": 5, "max_value": 9, "min_value": 1, "pivot": 2, "seed": 20260839, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 4, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 5. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=5)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-13520bb36d44b68e-t000", "input": {"array": [5, 7, 2, 7, 5]}, "is_public": true, "problem_id": "arrays-equilibrium_index-13520bb36d44b68e"}], "verification": {"duration_ms": 0.898124068044126, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 3): 0 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 3)\n (=> (= split_index 0) (and (>= 0 2) true))\n (=> (= split_index 1) (and (>= 2 2) (< 0 2)))\n (=> (= split_index 2) (and (>= 2 2) (< 0 2) (< 2 2))))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 3): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "arrays", "id": "arrays-monotonic_split-c6974a828fa7a876", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 0, "type": "integer"}, "maxItems": 3, "minItems": 3, "type": "array"}, "threshold": {"maximum": 9, "minimum": 0, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 1, "array": [0, 2, 2], "difficulty": "easy", "generator": "array_invariants", "length": 3, "max_value": 9, "min_value": 0, "seed": 20260840, "threshold": 2, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 2, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 3 with\n0 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=3, T=2)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-c6974a828fa7a876-t000", "input": {"array": [0, 2, 2], "threshold": 2}, "is_public": true, "problem_id": "arrays-monotonic_split-c6974a828fa7a876"}], "verification": {"duration_ms": 1.265708007849753, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 3): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 3)\n (=> (= equilibrium_index 0) (= 0 10))\n (=> (= equilibrium_index 1) (= 8 8))\n (=> (= equilibrium_index 2) (= 10 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 3): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "arrays", "id": "arrays-equilibrium_index-8f1eb9e46bd017b5", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 3, "minItems": 3, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 1, "array": [8, 2, 8], "difficulty": "easy", "generator": "array_invariants", "length": 3, "max_value": 9, "min_value": 1, "pivot": 1, "seed": 20260841, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 2, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 3. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=3)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-8f1eb9e46bd017b5-t000", "input": {"array": [8, 2, 8]}, "is_public": true, "problem_id": "arrays-equilibrium_index-8f1eb9e46bd017b5"}], "verification": {"duration_ms": 0.9983740164898336, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 5): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 5)\n (=> (= equilibrium_index 0) (= 0 20))\n (=> (= equilibrium_index 1) (= 6 17))\n (=> (= equilibrium_index 2) (= 9 9))\n (=> (= equilibrium_index 3) (= 17 3))\n (=> (= equilibrium_index 4) (= 23 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 5): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "arrays", "id": "arrays-equilibrium_index-f6855a420b49fdbf", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 5, "minItems": 5, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 2, "array": [6, 3, 8, 6, 3], "difficulty": "easy", "generator": "array_invariants", "length": 5, "max_value": 9, "min_value": 1, "pivot": 2, "seed": 20260842, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 4, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 5. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=5)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-f6855a420b49fdbf-t000", "input": {"array": [6, 3, 8, 6, 3]}, "is_public": true, "problem_id": "arrays-equilibrium_index-f6855a420b49fdbf"}], "verification": {"duration_ms": 0.9305819403380156, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 4): 2 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 4)\n (=> (= split_index 0) (and (>= 2 2) true))\n (=> (= split_index 1) (and (>= 2 2) (< 2 2)))\n (=> (= split_index 2) (and (>= 2 2) (< 2 2) (< 2 2)))\n (=> (= split_index 3) (and (>= 2 2) (< 2 2) (< 2 2) (< 2 2))))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 4): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "arrays", "id": "arrays-monotonic_split-2abbcd0d577e18ff", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 2, "type": "integer"}, "maxItems": 4, "minItems": 4, "type": "array"}, "threshold": {"maximum": 9, "minimum": 2, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 0, "array": [2, 2, 2, 2], "difficulty": "easy", "generator": "array_invariants", "length": 4, "max_value": 9, "min_value": 2, "seed": 20260843, "threshold": 2, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 3, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 4 with\n2 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=4, T=2)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 0, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-2abbcd0d577e18ff-t000", "input": {"array": [2, 2, 2, 2], "threshold": 2}, "is_public": true, "problem_id": "arrays-monotonic_split-2abbcd0d577e18ff"}], "verification": {"duration_ms": 0.9706250275485218, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 3): 0 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 3)\n (=> (= split_index 0) (and (>= 0 3) true))\n (=> (= split_index 1) (and (>= 2 3) (< 0 3)))\n (=> (= split_index 2) (and (>= 3 3) (< 0 3) (< 2 3))))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 3): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "arrays", "id": "arrays-monotonic_split-db26a75d6e8d799e", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 0, "type": "integer"}, "maxItems": 3, "minItems": 3, "type": "array"}, "threshold": {"maximum": 9, "minimum": 0, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 2, "array": [0, 2, 3], "difficulty": "easy", "generator": "array_invariants", "length": 3, "max_value": 9, "min_value": 0, "seed": 20260844, "threshold": 3, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 2, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 3 with\n0 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=3, T=3)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-db26a75d6e8d799e-t000", "input": {"array": [0, 2, 3], "threshold": 3}, "is_public": true, "problem_id": "arrays-monotonic_split-db26a75d6e8d799e"}], "verification": {"duration_ms": 0.8764160447753966, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 3): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 3)\n (=> (= equilibrium_index 0) (= 0 13))\n (=> (= equilibrium_index 1) (= 5 5))\n (=> (= equilibrium_index 2) (= 13 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 3): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "arrays", "id": "arrays-equilibrium_index-1a8307b31d48fd6e", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 3, "minItems": 3, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 1, "array": [5, 8, 5], "difficulty": "easy", "generator": "array_invariants", "length": 3, "max_value": 9, "min_value": 1, "pivot": 1, "seed": 20260845, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 2, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 3. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=3)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-1a8307b31d48fd6e-t000", "input": {"array": [5, 8, 5]}, "is_public": true, "problem_id": "arrays-equilibrium_index-1a8307b31d48fd6e"}], "verification": {"duration_ms": 0.8657909347675741, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 3): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 3)\n (=> (= equilibrium_index 0) (= 0 4))\n (=> (= equilibrium_index 1) (= 2 2))\n (=> (= equilibrium_index 2) (= 4 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 3): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "arrays", "id": "arrays-equilibrium_index-12c829027b103838", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 3, "minItems": 3, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 1, "array": [2, 2, 2], "difficulty": "easy", "generator": "array_invariants", "length": 3, "max_value": 9, "min_value": 1, "pivot": 1, "seed": 20260846, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 2, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 3. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=3)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-12c829027b103838-t000", "input": {"array": [2, 2, 2]}, "is_public": true, "problem_id": "arrays-equilibrium_index-12c829027b103838"}], "verification": {"duration_ms": 0.8664590422995389, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 5): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 5)\n (=> (= equilibrium_index 0) (= 0 9))\n (=> (= equilibrium_index 1) (= 1 7))\n (=> (= equilibrium_index 2) (= 3 3))\n (=> (= equilibrium_index 3) (= 7 1))\n (=> (= equilibrium_index 4) (= 9 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 5): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "arrays", "id": "arrays-equilibrium_index-0a076e6cac693065", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 5, "minItems": 5, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 2, "array": [1, 2, 4, 2, 1], "difficulty": "easy", "generator": "array_invariants", "length": 5, "max_value": 9, "min_value": 1, "pivot": 2, "seed": 20260847, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 4, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 5. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=5)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-0a076e6cac693065-t000", "input": {"array": [1, 2, 4, 2, 1]}, "is_public": true, "problem_id": "arrays-equilibrium_index-0a076e6cac693065"}], "verification": {"duration_ms": 0.8913740166462958, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 5): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 5)\n (=> (= equilibrium_index 0) (= 0 19))\n (=> (= equilibrium_index 1) (= 5 13))\n (=> (= equilibrium_index 2) (= 11 11))\n (=> (= equilibrium_index 3) (= 13 5))\n (=> (= equilibrium_index 4) (= 19 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 5): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "arrays", "id": "arrays-equilibrium_index-a4a86fc34e0f4e60", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 5, "minItems": 5, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 2, "array": [5, 6, 2, 6, 5], "difficulty": "easy", "generator": "array_invariants", "length": 5, "max_value": 9, "min_value": 1, "pivot": 2, "seed": 20260848, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 4, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 5. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=5)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-a4a86fc34e0f4e60-t000", "input": {"array": [5, 6, 2, 6, 5]}, "is_public": true, "problem_id": "arrays-equilibrium_index-a4a86fc34e0f4e60"}], "verification": {"duration_ms": 0.8927499875426292, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 11): 2 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 11)\n (=> (= split_index 0) (and (>= 2 4) true))\n (=> (= split_index 1) (and (>= 4 4) (< 2 4)))\n (=> (= split_index 2) (and (>= 6 4) (< 2 4) (< 4 4)))\n (=> (= split_index 3) (and (>= 8 4) (< 2 4) (< 4 4) (< 6 4)))\n (=> (= split_index 4) (and (>= 9 4) (< 2 4) (< 4 4) (< 6 4) (< 8 4)))\n (=> (= split_index 5)\n (and (>= 9 4) (< 2 4) (< 4 4) (< 6 4) (< 8 4) (< 9 4)))\n (=> (= split_index 6)\n (and (>= 9 4) (< 2 4) (< 4 4) (< 6 4) (< 8 4) (< 9 4) (< 9 4)))\n (=> (= split_index 7)\n (and (>= 9 4) (< 2 4)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 11): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "arrays", "id": "arrays-monotonic_split-e54d5a6c1d256858", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 2, "type": "integer"}, "maxItems": 11, "minItems": 11, "type": "array"}, "threshold": {"maximum": 9, "minimum": 2, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 1, "array": [2, 4, 6, 8, 9, 9, 9, 9, 9, 9, 9], "difficulty": "medium", "generator": "array_invariants", "length": 11, "max_value": 9, "min_value": 2, "seed": 20260849, "threshold": 4, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 10, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 11 with\n2 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=11, T=4)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-e54d5a6c1d256858-t000", "input": {"array": [2, 4, 6, 8, 9, 9, 9, 9, 9, 9, 9], "threshold": 4}, "is_public": true, "problem_id": "arrays-monotonic_split-e54d5a6c1d256858"}], "verification": {"duration_ms": 2.6683739852160215, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 10): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 10)\n (=> (= equilibrium_index 0) (= 0 54))\n (=> (= equilibrium_index 1) (= 2 53))\n (=> (= equilibrium_index 2) (= 3 50))\n (=> (= equilibrium_index 3) (= 6 42))\n (=> (= equilibrium_index 4) (= 14 37))\n (=> (= equilibrium_index 5) (= 19 32))\n (=> (= equilibrium_index 6) (= 24 24))\n (=> (= equilibrium_index 7) (= 32 16))\n (=> (= equilibrium_index 8) (= 40 7))\n (=> (= equilibrium_index 9) (= 49 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 10): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "arrays", "id": "arrays-equilibrium_index-54bbd88f3bc19d84", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 10, "minItems": 10, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 6, "array": [2, 1, 3, 8, 5, 5, 8, 8, 9, 7], "difficulty": "medium", "generator": "array_invariants", "length": 10, "max_value": 9, "min_value": 1, "pivot": 6, "seed": 20260850, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 9, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 10. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=10)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 6, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-54bbd88f3bc19d84-t000", "input": {"array": [2, 1, 3, 8, 5, 5, 8, 8, 9, 7]}, "is_public": true, "problem_id": "arrays-equilibrium_index-54bbd88f3bc19d84"}], "verification": {"duration_ms": 1.1579590500332415, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 7): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 7)\n (=> (= equilibrium_index 0) (= 0 11))\n (=> (= equilibrium_index 1) (= 9 9))\n (=> (= equilibrium_index 2) (= 11 5))\n (=> (= equilibrium_index 3) (= 15 3))\n (=> (= equilibrium_index 4) (= 17 2))\n (=> (= equilibrium_index 5) (= 18 1))\n (=> (= equilibrium_index 6) (= 19 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 7): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "arrays", "id": "arrays-equilibrium_index-04dccbf2b76a15b1", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 7, "minItems": 7, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 1, "array": [9, 2, 4, 2, 1, 1, 1], "difficulty": "medium", "generator": "array_invariants", "length": 7, "max_value": 9, "min_value": 1, "pivot": 1, "seed": 20260851, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 6, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 7. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=7)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-04dccbf2b76a15b1-t000", "input": {"array": [9, 2, 4, 2, 1, 1, 1]}, "is_public": true, "problem_id": "arrays-equilibrium_index-04dccbf2b76a15b1"}], "verification": {"duration_ms": 1.7022090032696724, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 7): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 7)\n (=> (= equilibrium_index 0) (= 0 9))\n (=> (= equilibrium_index 1) (= 8 8))\n (=> (= equilibrium_index 2) (= 9 5))\n (=> (= equilibrium_index 3) (= 12 3))\n (=> (= equilibrium_index 4) (= 14 2))\n (=> (= equilibrium_index 5) (= 15 1))\n (=> (= equilibrium_index 6) (= 16 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 7): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "arrays", "id": "arrays-equilibrium_index-a78343f951c33c77", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 7, "minItems": 7, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 1, "array": [8, 1, 3, 2, 1, 1, 1], "difficulty": "medium", "generator": "array_invariants", "length": 7, "max_value": 9, "min_value": 1, "pivot": 1, "seed": 20260852, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 6, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 7. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=7)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-a78343f951c33c77-t000", "input": {"array": [8, 1, 3, 2, 1, 1, 1]}, "is_public": true, "problem_id": "arrays-equilibrium_index-a78343f951c33c77"}], "verification": {"duration_ms": 1.2402490247040987, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 7): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 7)\n (=> (= equilibrium_index 0) (= 0 26))\n (=> (= equilibrium_index 1) (= 1 17))\n (=> (= equilibrium_index 2) (= 10 14))\n (=> (= equilibrium_index 3) (= 13 13))\n (=> (= equilibrium_index 4) (= 14 8))\n (=> (= equilibrium_index 5) (= 19 4))\n (=> (= equilibrium_index 6) (= 23 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 7): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "arrays", "id": "arrays-equilibrium_index-34b12989982d3303", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 7, "minItems": 7, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 3, "array": [1, 9, 3, 1, 5, 4, 4], "difficulty": "medium", "generator": "array_invariants", "length": 7, "max_value": 9, "min_value": 1, "pivot": 3, "seed": 20260853, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 6, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 7. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=7)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 3, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-34b12989982d3303-t000", "input": {"array": [1, 9, 3, 1, 5, 4, 4]}, "is_public": true, "problem_id": "arrays-equilibrium_index-34b12989982d3303"}], "verification": {"duration_ms": 1.0570409940555692, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 9): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 9)\n (=> (= split_index 0) (and (>= 1 5) true))\n (=> (= split_index 1) (and (>= 1 5) (< 1 5)))\n (=> (= split_index 2) (and (>= 1 5) (< 1 5) (< 1 5)))\n (=> (= split_index 3) (and (>= 2 5) (< 1 5) (< 1 5) (< 1 5)))\n (=> (= split_index 4) (and (>= 4 5) (< 1 5) (< 1 5) (< 1 5) (< 2 5)))\n (=> (= split_index 5)\n (and (>= 5 5) (< 1 5) (< 1 5) (< 1 5) (< 2 5) (< 4 5)))\n (=> (= split_index 6)\n (and (>= 5 5) (< 1 5) (< 1 5) (< 1 5) (< 2 5) (< 4 5) (< 5 5)))\n (=> (= split_index 7)\n (and (>= 6 5) (< 1 5) ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 9): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "arrays", "id": "arrays-monotonic_split-198839695f931f0b", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 9, "minItems": 9, "type": "array"}, "threshold": {"maximum": 9, "minimum": 1, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 5, "array": [1, 1, 1, 2, 4, 5, 5, 6, 8], "difficulty": "medium", "generator": "array_invariants", "length": 9, "max_value": 9, "min_value": 1, "seed": 20260854, "threshold": 5, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 8, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 9 with\n1 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=9, T=5)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-198839695f931f0b-t000", "input": {"array": [1, 1, 1, 2, 4, 5, 5, 6, 8], "threshold": 5}, "is_public": true, "problem_id": "arrays-monotonic_split-198839695f931f0b"}], "verification": {"duration_ms": 1.124250004068017, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 9): 0 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 9)\n (=> (= split_index 0) (and (>= 0 0) true))\n (=> (= split_index 1) (and (>= 0 0) (< 0 0)))\n (=> (= split_index 2) (and (>= 1 0) (< 0 0) (< 0 0)))\n (=> (= split_index 3) (and (>= 1 0) (< 0 0) (< 0 0) (< 1 0)))\n (=> (= split_index 4) (and (>= 1 0) (< 0 0) (< 0 0) (< 1 0) (< 1 0)))\n (=> (= split_index 5)\n (and (>= 3 0) (< 0 0) (< 0 0) (< 1 0) (< 1 0) (< 1 0)))\n (=> (= split_index 6)\n (and (>= 5 0) (< 0 0) (< 0 0) (< 1 0) (< 1 0) (< 1 0) (< 3 0)))\n (=> (= split_index 7)\n (and (>= 6 0) (< 0 0) ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 9): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "arrays", "id": "arrays-monotonic_split-13397d7253845128", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 0, "type": "integer"}, "maxItems": 9, "minItems": 9, "type": "array"}, "threshold": {"maximum": 9, "minimum": 0, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 0, "array": [0, 0, 1, 1, 1, 3, 5, 6, 7], "difficulty": "medium", "generator": "array_invariants", "length": 9, "max_value": 9, "min_value": 0, "seed": 20260855, "threshold": 0, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 8, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 9 with\n0 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=9, T=0)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 0, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-13397d7253845128-t000", "input": {"array": [0, 0, 1, 1, 1, 3, 5, 6, 7], "threshold": 0}, "is_public": true, "problem_id": "arrays-monotonic_split-13397d7253845128"}], "verification": {"duration_ms": 1.7339159967377782, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 12): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 12)\n (=> (= split_index 0) (and (>= 1 8) true))\n (=> (= split_index 1) (and (>= 3 8) (< 1 8)))\n (=> (= split_index 2) (and (>= 5 8) (< 1 8) (< 3 8)))\n (=> (= split_index 3) (and (>= 5 8) (< 1 8) (< 3 8) (< 5 8)))\n (=> (= split_index 4) (and (>= 6 8) (< 1 8) (< 3 8) (< 5 8) (< 5 8)))\n (=> (= split_index 5)\n (and (>= 8 8) (< 1 8) (< 3 8) (< 5 8) (< 5 8) (< 6 8)))\n (=> (= split_index 6)\n (and (>= 8 8) (< 1 8) (< 3 8) (< 5 8) (< 5 8) (< 6 8) (< 8 8)))\n (=> (= split_index 7)\n (and (>= 9 8) (< 1 8)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 12): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "arrays", "id": "arrays-monotonic_split-4079e4e1ea3805b3", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 12, "minItems": 12, "type": "array"}, "threshold": {"maximum": 9, "minimum": 1, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 5, "array": [1, 3, 5, 5, 6, 8, 8, 9, 9, 9, 9, 9], "difficulty": "medium", "generator": "array_invariants", "length": 12, "max_value": 9, "min_value": 1, "seed": 20260856, "threshold": 8, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 11, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 12 with\n1 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=12, T=8)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-4079e4e1ea3805b3-t000", "input": {"array": [1, 3, 5, 5, 6, 8, 8, 9, 9, 9, 9, 9], "threshold": 8}, "is_public": true, "problem_id": "arrays-monotonic_split-4079e4e1ea3805b3"}], "verification": {"duration_ms": 1.6992500168271363, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 6): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 6)\n (=> (= equilibrium_index 0) (= 0 13))\n (=> (= equilibrium_index 1) (= 4 4))\n (=> (= equilibrium_index 2) (= 13 3))\n (=> (= equilibrium_index 3) (= 14 2))\n (=> (= equilibrium_index 4) (= 15 1))\n (=> (= equilibrium_index 5) (= 16 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 6): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "arrays", "id": "arrays-equilibrium_index-caae4b1548b4d261", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 6, "minItems": 6, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 1, "array": [4, 9, 1, 1, 1, 1], "difficulty": "medium", "generator": "array_invariants", "length": 6, "max_value": 9, "min_value": 1, "pivot": 1, "seed": 20260857, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 5, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 6. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=6)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-caae4b1548b4d261-t000", "input": {"array": [4, 9, 1, 1, 1, 1]}, "is_public": true, "problem_id": "arrays-equilibrium_index-caae4b1548b4d261"}], "verification": {"duration_ms": 1.0063749505206943, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 11): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 11)\n (=> (= equilibrium_index 0) (= 0 36))\n (=> (= equilibrium_index 1) (= 2 33))\n (=> (= equilibrium_index 2) (= 5 29))\n (=> (= equilibrium_index 3) (= 9 23))\n (=> (= equilibrium_index 4) (= 15 20))\n (=> (= equilibrium_index 5) (= 18 18))\n (=> (= equilibrium_index 6) (= 20 14))\n (=> (= equilibrium_index 7) (= 24 12))\n (=> (= equilibrium_index 8) (= 26 11))\n (=> (= equilibrium_index 9) (= 27 5))\n (=> (= equilibrium_index 10) (= 33 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 11): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "arrays", "id": "arrays-equilibrium_index-99751fe0100f3d4a", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 11, "minItems": 11, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 5, "array": [2, 3, 4, 6, 3, 2, 4, 2, 1, 6, 5], "difficulty": "medium", "generator": "array_invariants", "length": 11, "max_value": 9, "min_value": 1, "pivot": 5, "seed": 20260858, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 10, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 11. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=11)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-99751fe0100f3d4a-t000", "input": {"array": [2, 3, 4, 6, 3, 2, 4, 2, 1, 6, 5]}, "is_public": true, "problem_id": "arrays-equilibrium_index-99751fe0100f3d4a"}], "verification": {"duration_ms": 1.225499960128218, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 12): 2 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 12)\n (=> (= split_index 0) (and (>= 2 9) true))\n (=> (= split_index 1) (and (>= 4 9) (< 2 9)))\n (=> (= split_index 2) (and (>= 4 9) (< 2 9) (< 4 9)))\n (=> (= split_index 3) (and (>= 5 9) (< 2 9) (< 4 9) (< 4 9)))\n (=> (= split_index 4) (and (>= 6 9) (< 2 9) (< 4 9) (< 4 9) (< 5 9)))\n (=> (= split_index 5)\n (and (>= 6 9) (< 2 9) (< 4 9) (< 4 9) (< 5 9) (< 6 9)))\n (=> (= split_index 6)\n (and (>= 6 9) (< 2 9) (< 4 9) (< 4 9) (< 5 9) (< 6 9) (< 6 9)))\n (=> (= split_index 7)\n (and (>= 8 9) (< 2 9)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 12): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "arrays", "id": "arrays-monotonic_split-f2f7f7e9a99bb4a7", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 2, "type": "integer"}, "maxItems": 12, "minItems": 12, "type": "array"}, "threshold": {"maximum": 9, "minimum": 2, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 8, "array": [2, 4, 4, 5, 6, 6, 6, 8, 9, 9, 9, 9], "difficulty": "medium", "generator": "array_invariants", "length": 12, "max_value": 9, "min_value": 2, "seed": 20260859, "threshold": 9, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 11, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 12 with\n2 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=12, T=9)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 8, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-f2f7f7e9a99bb4a7-t000", "input": {"array": [2, 4, 4, 5, 6, 6, 6, 8, 9, 9, 9, 9], "threshold": 9}, "is_public": true, "problem_id": "arrays-monotonic_split-f2f7f7e9a99bb4a7"}], "verification": {"duration_ms": 1.0650839540176094, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 8): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 8)\n (=> (= equilibrium_index 0) (= 0 19))\n (=> (= equilibrium_index 1) (= 8 17))\n (=> (= equilibrium_index 2) (= 10 10))\n (=> (= equilibrium_index 3) (= 17 8))\n (=> (= equilibrium_index 4) (= 19 6))\n (=> (= equilibrium_index 5) (= 21 4))\n (=> (= equilibrium_index 6) (= 23 1))\n (=> (= equilibrium_index 7) (= 26 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 8): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "arrays", "id": "arrays-equilibrium_index-c6edcf2f27612cfa", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 8, "minItems": 8, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 2, "array": [8, 2, 7, 2, 2, 2, 3, 1], "difficulty": "medium", "generator": "array_invariants", "length": 8, "max_value": 9, "min_value": 1, "pivot": 2, "seed": 20260860, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 7, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 8. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=8)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-c6edcf2f27612cfa-t000", "input": {"array": [8, 2, 7, 2, 2, 2, 3, 1]}, "is_public": true, "problem_id": "arrays-equilibrium_index-c6edcf2f27612cfa"}], "verification": {"duration_ms": 1.6420410247519612, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 6): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 6)\n (=> (= equilibrium_index 0) (= 0 8))\n (=> (= equilibrium_index 1) (= 5 5))\n (=> (= equilibrium_index 2) (= 8 3))\n (=> (= equilibrium_index 3) (= 10 2))\n (=> (= equilibrium_index 4) (= 11 1))\n (=> (= equilibrium_index 5) (= 12 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 6): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "arrays", "id": "arrays-equilibrium_index-0da61b37b643ef9f", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 6, "minItems": 6, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 1, "array": [5, 3, 2, 1, 1, 1], "difficulty": "medium", "generator": "array_invariants", "length": 6, "max_value": 9, "min_value": 1, "pivot": 1, "seed": 20260861, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 5, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 6. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=6)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-0da61b37b643ef9f-t000", "input": {"array": [5, 3, 2, 1, 1, 1]}, "is_public": true, "problem_id": "arrays-equilibrium_index-0da61b37b643ef9f"}], "verification": {"duration_ms": 0.970416993368417, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 10): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 10)\n (=> (= equilibrium_index 0) (= 0 54))\n (=> (= equilibrium_index 1) (= 7 46))\n (=> (= equilibrium_index 2) (= 15 40))\n (=> (= equilibrium_index 3) (= 21 37))\n (=> (= equilibrium_index 4) (= 24 35))\n (=> (= equilibrium_index 5) (= 26 26))\n (=> (= equilibrium_index 6) (= 35 24))\n (=> (= equilibrium_index 7) (= 37 15))\n (=> (= equilibrium_index 8) (= 46 8))\n (=> (= equilibrium_index 9) (= 53 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 10): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "arrays", "id": "arrays-equilibrium_index-6763f4daf0f079d8", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 10, "minItems": 10, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 5, "array": [7, 8, 6, 3, 2, 9, 2, 9, 7, 8], "difficulty": "medium", "generator": "array_invariants", "length": 10, "max_value": 9, "min_value": 1, "pivot": 5, "seed": 20260862, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 9, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 10. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=10)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-6763f4daf0f079d8-t000", "input": {"array": [7, 8, 6, 3, 2, 9, 2, 9, 7, 8]}, "is_public": true, "problem_id": "arrays-equilibrium_index-6763f4daf0f079d8"}], "verification": {"duration_ms": 0.9812499629333615, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 6): 0 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 6)\n (=> (= split_index 0) (and (>= 0 4) true))\n (=> (= split_index 1) (and (>= 1 4) (< 0 4)))\n (=> (= split_index 2) (and (>= 3 4) (< 0 4) (< 1 4)))\n (=> (= split_index 3) (and (>= 4 4) (< 0 4) (< 1 4) (< 3 4)))\n (=> (= split_index 4) (and (>= 5 4) (< 0 4) (< 1 4) (< 3 4) (< 4 4)))\n (=> (= split_index 5)\n (and (>= 6 4) (< 0 4) (< 1 4) (< 3 4) (< 4 4) (< 5 4))))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 6): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "arrays", "id": "arrays-monotonic_split-3cefc25db0cf238a", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 0, "type": "integer"}, "maxItems": 6, "minItems": 6, "type": "array"}, "threshold": {"maximum": 9, "minimum": 0, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 3, "array": [0, 1, 3, 4, 5, 6], "difficulty": "medium", "generator": "array_invariants", "length": 6, "max_value": 9, "min_value": 0, "seed": 20260863, "threshold": 4, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 5, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 6 with\n0 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=6, T=4)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 3, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-3cefc25db0cf238a-t000", "input": {"array": [0, 1, 3, 4, 5, 6], "threshold": 4}, "is_public": true, "problem_id": "arrays-monotonic_split-3cefc25db0cf238a"}], "verification": {"duration_ms": 0.9107499499805272, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 5): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 5)\n (=> (= split_index 0) (and (>= 1 4) true))\n (=> (= split_index 1) (and (>= 1 4) (< 1 4)))\n (=> (= split_index 2) (and (>= 1 4) (< 1 4) (< 1 4)))\n (=> (= split_index 3) (and (>= 3 4) (< 1 4) (< 1 4) (< 1 4)))\n (=> (= split_index 4) (and (>= 4 4) (< 1 4) (< 1 4) (< 1 4) (< 3 4))))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 5): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "arrays", "id": "arrays-monotonic_split-a2ff9b933d8ec567", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 5, "minItems": 5, "type": "array"}, "threshold": {"maximum": 9, "minimum": 1, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 4, "array": [1, 1, 1, 3, 4], "difficulty": "medium", "generator": "array_invariants", "length": 5, "max_value": 9, "min_value": 1, "seed": 20260864, "threshold": 4, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 4, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 5 with\n1 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=5, T=4)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-a2ff9b933d8ec567-t000", "input": {"array": [1, 1, 1, 3, 4], "threshold": 4}, "is_public": true, "problem_id": "arrays-monotonic_split-a2ff9b933d8ec567"}], "verification": {"duration_ms": 0.8928339811973274, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 11): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 11)\n (=> (= equilibrium_index 0) (= 0 12))\n (=> (= equilibrium_index 1) (= 9 11))\n (=> (= equilibrium_index 2) (= 10 10))\n (=> (= equilibrium_index 3) (= 11 9))\n (=> (= equilibrium_index 4) (= 12 7))\n (=> (= equilibrium_index 5) (= 14 5))\n (=> (= equilibrium_index 6) (= 16 4))\n (=> (= equilibrium_index 7) (= 17 3))\n (=> (= equilibrium_index 8) (= 18 2))\n (=> (= equilibrium_index 9) (= 19 1))\n (=> (= equilibrium_index 10) (= 20 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 11): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "arrays", "id": "arrays-equilibrium_index-d6e82277e737de1c", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 11, "minItems": 11, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 2, "array": [9, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1], "difficulty": "medium", "generator": "array_invariants", "length": 11, "max_value": 9, "min_value": 1, "pivot": 2, "seed": 20260865, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 10, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 11. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=11)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-d6e82277e737de1c-t000", "input": {"array": [9, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1]}, "is_public": true, "problem_id": "arrays-equilibrium_index-d6e82277e737de1c"}], "verification": {"duration_ms": 1.657750050071627, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 5): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 5)\n (=> (= equilibrium_index 0) (= 0 15))\n (=> (= equilibrium_index 1) (= 3 10))\n (=> (= equilibrium_index 2) (= 8 8))\n (=> (= equilibrium_index 3) (= 10 4))\n (=> (= equilibrium_index 4) (= 14 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 5): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "arrays", "id": "arrays-equilibrium_index-1020b6d4abf74e86", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 5, "minItems": 5, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 2, "array": [3, 5, 2, 4, 4], "difficulty": "medium", "generator": "array_invariants", "length": 5, "max_value": 9, "min_value": 1, "pivot": 2, "seed": 20260866, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 4, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 5. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=5)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-1020b6d4abf74e86-t000", "input": {"array": [3, 5, 2, 4, 4]}, "is_public": true, "problem_id": "arrays-equilibrium_index-1020b6d4abf74e86"}], "verification": {"duration_ms": 0.9346660226583481, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 7): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 7)\n (=> (= equilibrium_index 0) (= 0 37))\n (=> (= equilibrium_index 1) (= 9 31))\n (=> (= equilibrium_index 2) (= 15 24))\n (=> (= equilibrium_index 3) (= 22 22))\n (=> (= equilibrium_index 4) (= 24 14))\n (=> (= equilibrium_index 5) (= 32 8))\n (=> (= equilibrium_index 6) (= 38 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 7): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "arrays", "id": "arrays-equilibrium_index-2c49907e4f5193e0", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 7, "minItems": 7, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 3, "array": [9, 6, 7, 2, 8, 6, 8], "difficulty": "medium", "generator": "array_invariants", "length": 7, "max_value": 9, "min_value": 1, "pivot": 3, "seed": 20260867, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 6, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 7. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=7)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 3, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-2c49907e4f5193e0-t000", "input": {"array": [9, 6, 7, 2, 8, 6, 8]}, "is_public": true, "problem_id": "arrays-equilibrium_index-2c49907e4f5193e0"}], "verification": {"duration_ms": 0.932416005525738, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 11): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 11)\n (=> (= equilibrium_index 0) (= 0 26))\n (=> (= equilibrium_index 1) (= 5 21))\n (=> (= equilibrium_index 2) (= 10 18))\n (=> (= equilibrium_index 3) (= 13 13))\n (=> (= equilibrium_index 4) (= 18 12))\n (=> (= equilibrium_index 5) (= 19 9))\n (=> (= equilibrium_index 6) (= 22 8))\n (=> (= equilibrium_index 7) (= 23 6))\n (=> (= equilibrium_index 8) (= 25 3))\n (=> (= equilibrium_index 9) (= 28 1))\n (=> (= equilibrium_index 10) (= 30 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 11): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "arrays", "id": "arrays-equilibrium_index-e12fef724e3dc50b", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 11, "minItems": 11, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 3, "array": [5, 5, 3, 5, 1, 3, 1, 2, 3, 2, 1], "difficulty": "medium", "generator": "array_invariants", "length": 11, "max_value": 9, "min_value": 1, "pivot": 3, "seed": 20260868, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 10, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 11. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=11)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 3, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-e12fef724e3dc50b-t000", "input": {"array": [5, 5, 3, 5, 1, 3, 1, 2, 3, 2, 1]}, "is_public": true, "problem_id": "arrays-equilibrium_index-e12fef724e3dc50b"}], "verification": {"duration_ms": 1.6188339795917273, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 6): 0 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 6)\n (=> (= split_index 0) (and (>= 0 0) true))\n (=> (= split_index 1) (and (>= 1 0) (< 0 0)))\n (=> (= split_index 2) (and (>= 1 0) (< 0 0) (< 1 0)))\n (=> (= split_index 3) (and (>= 2 0) (< 0 0) (< 1 0) (< 1 0)))\n (=> (= split_index 4) (and (>= 3 0) (< 0 0) (< 1 0) (< 1 0) (< 2 0)))\n (=> (= split_index 5)\n (and (>= 5 0) (< 0 0) (< 1 0) (< 1 0) (< 2 0) (< 3 0))))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 6): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "arrays", "id": "arrays-monotonic_split-a3ec287d97f08132", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 0, "type": "integer"}, "maxItems": 6, "minItems": 6, "type": "array"}, "threshold": {"maximum": 9, "minimum": 0, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 0, "array": [0, 1, 1, 2, 3, 5], "difficulty": "medium", "generator": "array_invariants", "length": 6, "max_value": 9, "min_value": 0, "seed": 20260869, "threshold": 0, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 5, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 6 with\n0 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=6, T=0)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 0, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-a3ec287d97f08132-t000", "input": {"array": [0, 1, 1, 2, 3, 5], "threshold": 0}, "is_public": true, "problem_id": "arrays-monotonic_split-a3ec287d97f08132"}], "verification": {"duration_ms": 1.5304579865187407, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 11): 0 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 11)\n (=> (= split_index 0) (and (>= 0 1) true))\n (=> (= split_index 1) (and (>= 1 1) (< 0 1)))\n (=> (= split_index 2) (and (>= 2 1) (< 0 1) (< 1 1)))\n (=> (= split_index 3) (and (>= 4 1) (< 0 1) (< 1 1) (< 2 1)))\n (=> (= split_index 4) (and (>= 6 1) (< 0 1) (< 1 1) (< 2 1) (< 4 1)))\n (=> (= split_index 5)\n (and (>= 6 1) (< 0 1) (< 1 1) (< 2 1) (< 4 1) (< 6 1)))\n (=> (= split_index 6)\n (and (>= 6 1) (< 0 1) (< 1 1) (< 2 1) (< 4 1) (< 6 1) (< 6 1)))\n (=> (= split_index 7)\n (and (>= 6 1) (< 0 1)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 11): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "arrays", "id": "arrays-monotonic_split-cdee758b5d47088f", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 0, "type": "integer"}, "maxItems": 11, "minItems": 11, "type": "array"}, "threshold": {"maximum": 9, "minimum": 0, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 1, "array": [0, 1, 2, 4, 6, 6, 6, 6, 8, 9, 9], "difficulty": "medium", "generator": "array_invariants", "length": 11, "max_value": 9, "min_value": 0, "seed": 20260870, "threshold": 1, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 10, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 11 with\n0 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=11, T=1)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-cdee758b5d47088f-t000", "input": {"array": [0, 1, 2, 4, 6, 6, 6, 6, 8, 9, 9], "threshold": 1}, "is_public": true, "problem_id": "arrays-monotonic_split-cdee758b5d47088f"}], "verification": {"duration_ms": 1.6620010137557983, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 11): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 11)\n (=> (= equilibrium_index 0) (= 0 42))\n (=> (= equilibrium_index 1) (= 1 34))\n (=> (= equilibrium_index 2) (= 9 25))\n (=> (= equilibrium_index 3) (= 18 18))\n (=> (= equilibrium_index 4) (= 25 17))\n (=> (= equilibrium_index 5) (= 26 16))\n (=> (= equilibrium_index 6) (= 27 13))\n (=> (= equilibrium_index 7) (= 30 11))\n (=> (= equilibrium_index 8) (= 32 6))\n (=> (= equilibrium_index 9) (= 37 3))\n (=> (= equilibrium_index 10) (= 40 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 11): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "arrays", "id": "arrays-equilibrium_index-008890f9528f9ce1", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 11, "minItems": 11, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 3, "array": [1, 8, 9, 7, 1, 1, 3, 2, 5, 3, 3], "difficulty": "medium", "generator": "array_invariants", "length": 11, "max_value": 9, "min_value": 1, "pivot": 3, "seed": 20260871, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 10, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 11. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=11)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 3, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-008890f9528f9ce1-t000", "input": {"array": [1, 8, 9, 7, 1, 1, 3, 2, 5, 3, 3]}, "is_public": true, "problem_id": "arrays-equilibrium_index-008890f9528f9ce1"}], "verification": {"duration_ms": 1.6426669899374247, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 10): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 10)\n (=> (= equilibrium_index 0) (= 0 52))\n (=> (= equilibrium_index 1) (= 9 48))\n (=> (= equilibrium_index 2) (= 13 45))\n (=> (= equilibrium_index 3) (= 16 39))\n (=> (= equilibrium_index 4) (= 22 33))\n (=> (= equilibrium_index 5) (= 28 28))\n (=> (= equilibrium_index 6) (= 33 23))\n (=> (= equilibrium_index 7) (= 38 17))\n (=> (= equilibrium_index 8) (= 44 8))\n (=> (= equilibrium_index 9) (= 53 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 10): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "arrays", "id": "arrays-equilibrium_index-125cd5bfa7e38eb9", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 10, "minItems": 10, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 5, "array": [9, 4, 3, 6, 6, 5, 5, 6, 9, 8], "difficulty": "medium", "generator": "array_invariants", "length": 10, "max_value": 9, "min_value": 1, "pivot": 5, "seed": 20260872, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 9, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 10. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=10)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-125cd5bfa7e38eb9-t000", "input": {"array": [9, 4, 3, 6, 6, 5, 5, 6, 9, 8]}, "is_public": true, "problem_id": "arrays-equilibrium_index-125cd5bfa7e38eb9"}], "verification": {"duration_ms": 1.585374993737787, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 12): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 12)\n (=> (= equilibrium_index 0) (= 0 45))\n (=> (= equilibrium_index 1) (= 8 39))\n (=> (= equilibrium_index 2) (= 14 32))\n (=> (= equilibrium_index 3) (= 21 29))\n (=> (= equilibrium_index 4) (= 24 24))\n (=> (= equilibrium_index 5) (= 29 19))\n (=> (= equilibrium_index 6) (= 34 15))\n (=> (= equilibrium_index 7) (= 38 11))\n (=> (= equilibrium_index 8) (= 42 7))\n (=> (= equilibrium_index 9) (= 46 2))\n (=> (= equilibrium_index 10) (= 51 1))\n (=> (= equilibrium_index 11) (= 52 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 12): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "arrays", "id": "arrays-equilibrium_index-426b24ab0339af38", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 12, "minItems": 12, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 4, "array": [8, 6, 7, 3, 5, 5, 4, 4, 4, 5, 1, 1], "difficulty": "medium", "generator": "array_invariants", "length": 12, "max_value": 9, "min_value": 1, "pivot": 4, "seed": 20260873, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 11, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 12. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=12)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-426b24ab0339af38-t000", "input": {"array": [8, 6, 7, 3, 5, 5, 4, 4, 4, 5, 1, 1]}, "is_public": true, "problem_id": "arrays-equilibrium_index-426b24ab0339af38"}], "verification": {"duration_ms": 1.62808399181813, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 14): 2 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 14)\n (=> (= split_index 0) (and (>= 2 5) true))\n (=> (= split_index 1) (and (>= 2 5) (< 2 5)))\n (=> (= split_index 2) (and (>= 4 5) (< 2 5) (< 2 5)))\n (=> (= split_index 3) (and (>= 4 5) (< 2 5) (< 2 5) (< 4 5)))\n (=> (= split_index 4) (and (>= 5 5) (< 2 5) (< 2 5) (< 4 5) (< 4 5)))\n (=> (= split_index 5)\n (and (>= 5 5) (< 2 5) (< 2 5) (< 4 5) (< 4 5) (< 5 5)))\n (=> (= split_index 6)\n (and (>= 5 5) (< 2 5) (< 2 5) (< 4 5) (< 4 5) (< 5 5) (< 5 5)))\n (=> (= split_index 7)\n (and (>= 7 5) (< 2 5)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 14): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "arrays", "id": "arrays-monotonic_split-11e3ee5c5783aee6", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 2, "type": "integer"}, "maxItems": 14, "minItems": 14, "type": "array"}, "threshold": {"maximum": 9, "minimum": 2, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 4, "array": [2, 2, 4, 4, 5, 5, 5, 7, 9, 9, 9, 9, 9, 9], "difficulty": "hard", "generator": "array_invariants", "length": 14, "max_value": 9, "min_value": 2, "seed": 20260874, "threshold": 5, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 13, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 14 with\n2 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=14, T=5)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-11e3ee5c5783aee6-t000", "input": {"array": [2, 2, 4, 4, 5, 5, 5, 7, 9, 9, 9, 9, 9, 9], "threshold": 5}, "is_public": true, "problem_id": "arrays-monotonic_split-11e3ee5c5783aee6"}], "verification": {"duration_ms": 1.8037089612334967, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 13): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 13)\n (=> (= split_index 0) (and (>= 1 9) true))\n (=> (= split_index 1) (and (>= 3 9) (< 1 9)))\n (=> (= split_index 2) (and (>= 5 9) (< 1 9) (< 3 9)))\n (=> (= split_index 3) (and (>= 6 9) (< 1 9) (< 3 9) (< 5 9)))\n (=> (= split_index 4) (and (>= 8 9) (< 1 9) (< 3 9) (< 5 9) (< 6 9)))\n (=> (= split_index 5)\n (and (>= 9 9) (< 1 9) (< 3 9) (< 5 9) (< 6 9) (< 8 9)))\n (=> (= split_index 6)\n (and (>= 9 9) (< 1 9) (< 3 9) (< 5 9) (< 6 9) (< 8 9) (< 9 9)))\n (=> (= split_index 7)\n (and (>= 9 9) (< 1 9)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 13): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "arrays", "id": "arrays-monotonic_split-70400772a5fbe0d8", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 13, "minItems": 13, "type": "array"}, "threshold": {"maximum": 9, "minimum": 1, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 5, "array": [1, 3, 5, 6, 8, 9, 9, 9, 9, 9, 9, 9, 9], "difficulty": "hard", "generator": "array_invariants", "length": 13, "max_value": 9, "min_value": 1, "seed": 20260875, "threshold": 9, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 12, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 13 with\n1 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=13, T=9)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-70400772a5fbe0d8-t000", "input": {"array": [1, 3, 5, 6, 8, 9, 9, 9, 9, 9, 9, 9, 9], "threshold": 9}, "is_public": true, "problem_id": "arrays-monotonic_split-70400772a5fbe0d8"}], "verification": {"duration_ms": 1.7070010071620345, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 13): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 13)\n (=> (= equilibrium_index 0) (= 0 67))\n (=> (= equilibrium_index 1) (= 8 63))\n (=> (= equilibrium_index 2) (= 12 58))\n (=> (= equilibrium_index 3) (= 17 49))\n (=> (= equilibrium_index 4) (= 26 42))\n (=> (= equilibrium_index 5) (= 33 33))\n (=> (= equilibrium_index 6) (= 42 29))\n (=> (= equilibrium_index 7) (= 46 25))\n (=> (= equilibrium_index 8) (= 50 18))\n (=> (= equilibrium_index 9) (= 57 11))\n (=> (= equilibrium_index 10) (= 64 6))\n (=> (= equilibrium_index 11) (= 69 3))\n (=> (=...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 13): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "arrays", "id": "arrays-equilibrium_index-37816d47496e83ff", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 13, "minItems": 13, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 5, "array": [8, 4, 5, 9, 7, 9, 4, 4, 7, 7, 5, 3, 3], "difficulty": "hard", "generator": "array_invariants", "length": 13, "max_value": 9, "min_value": 1, "pivot": 5, "seed": 20260876, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 12, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 13. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=13)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-37816d47496e83ff-t000", "input": {"array": [8, 4, 5, 9, 7, 9, 4, 4, 7, 7, 5, 3, 3]}, "is_public": true, "problem_id": "arrays-equilibrium_index-37816d47496e83ff"}], "verification": {"duration_ms": 1.7005830304697156, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 13): 0 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 13)\n (=> (= split_index 0) (and (>= 0 9) true))\n (=> (= split_index 1) (and (>= 2 9) (< 0 9)))\n (=> (= split_index 2) (and (>= 4 9) (< 0 9) (< 2 9)))\n (=> (= split_index 3) (and (>= 6 9) (< 0 9) (< 2 9) (< 4 9)))\n (=> (= split_index 4) (and (>= 7 9) (< 0 9) (< 2 9) (< 4 9) (< 6 9)))\n (=> (= split_index 5)\n (and (>= 7 9) (< 0 9) (< 2 9) (< 4 9) (< 6 9) (< 7 9)))\n (=> (= split_index 6)\n (and (>= 9 9) (< 0 9) (< 2 9) (< 4 9) (< 6 9) (< 7 9) (< 7 9)))\n (=> (= split_index 7)\n (and (>= 9 9) (< 0 9)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 13): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "arrays", "id": "arrays-monotonic_split-cf6fa647d393b7f4", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 0, "type": "integer"}, "maxItems": 13, "minItems": 13, "type": "array"}, "threshold": {"maximum": 9, "minimum": 0, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 6, "array": [0, 2, 4, 6, 7, 7, 9, 9, 9, 9, 9, 9, 9], "difficulty": "hard", "generator": "array_invariants", "length": 13, "max_value": 9, "min_value": 0, "seed": 20260877, "threshold": 9, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 12, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 13 with\n0 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=13, T=9)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 6, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-cf6fa647d393b7f4-t000", "input": {"array": [0, 2, 4, 6, 7, 7, 9, 9, 9, 9, 9, 9, 9], "threshold": 9}, "is_public": true, "problem_id": "arrays-monotonic_split-cf6fa647d393b7f4"}], "verification": {"duration_ms": 1.780043006874621, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 12): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 12)\n (=> (= split_index 0) (and (>= 1 3) true))\n (=> (= split_index 1) (and (>= 3 3) (< 1 3)))\n (=> (= split_index 2) (and (>= 3 3) (< 1 3) (< 3 3)))\n (=> (= split_index 3) (and (>= 3 3) (< 1 3) (< 3 3) (< 3 3)))\n (=> (= split_index 4) (and (>= 5 3) (< 1 3) (< 3 3) (< 3 3) (< 3 3)))\n (=> (= split_index 5)\n (and (>= 5 3) (< 1 3) (< 3 3) (< 3 3) (< 3 3) (< 5 3)))\n (=> (= split_index 6)\n (and (>= 6 3) (< 1 3) (< 3 3) (< 3 3) (< 3 3) (< 5 3) (< 5 3)))\n (=> (= split_index 7)\n (and (>= 7 3) (< 1 3)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 12): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "arrays", "id": "arrays-monotonic_split-c1460316651a1049", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 12, "minItems": 12, "type": "array"}, "threshold": {"maximum": 9, "minimum": 1, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 1, "array": [1, 3, 3, 3, 5, 5, 6, 7, 7, 9, 9, 9], "difficulty": "hard", "generator": "array_invariants", "length": 12, "max_value": 9, "min_value": 1, "seed": 20260878, "threshold": 3, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 11, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 12 with\n1 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=12, T=3)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-c1460316651a1049-t000", "input": {"array": [1, 3, 3, 3, 5, 5, 6, 7, 7, 9, 9, 9], "threshold": 3}, "is_public": true, "problem_id": "arrays-monotonic_split-c1460316651a1049"}], "verification": {"duration_ms": 1.7153340158984065, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 11): 2 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 11)\n (=> (= split_index 0) (and (>= 2 8) true))\n (=> (= split_index 1) (and (>= 4 8) (< 2 8)))\n (=> (= split_index 2) (and (>= 6 8) (< 2 8) (< 4 8)))\n (=> (= split_index 3) (and (>= 6 8) (< 2 8) (< 4 8) (< 6 8)))\n (=> (= split_index 4) (and (>= 8 8) (< 2 8) (< 4 8) (< 6 8) (< 6 8)))\n (=> (= split_index 5)\n (and (>= 8 8) (< 2 8) (< 4 8) (< 6 8) (< 6 8) (< 8 8)))\n (=> (= split_index 6)\n (and (>= 8 8) (< 2 8) (< 4 8) (< 6 8) (< 6 8) (< 8 8) (< 8 8)))\n (=> (= split_index 7)\n (and (>= 9 8) (< 2 8)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 11): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "arrays", "id": "arrays-monotonic_split-dacaa16bd44b6e3f", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 2, "type": "integer"}, "maxItems": 11, "minItems": 11, "type": "array"}, "threshold": {"maximum": 9, "minimum": 2, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 4, "array": [2, 4, 6, 6, 8, 8, 8, 9, 9, 9, 9], "difficulty": "hard", "generator": "array_invariants", "length": 11, "max_value": 9, "min_value": 2, "seed": 20260879, "threshold": 8, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 10, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 11 with\n2 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=11, T=8)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-dacaa16bd44b6e3f-t000", "input": {"array": [2, 4, 6, 6, 8, 8, 8, 9, 9, 9, 9], "threshold": 8}, "is_public": true, "problem_id": "arrays-monotonic_split-dacaa16bd44b6e3f"}], "verification": {"duration_ms": 1.6657080268487334, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 10): 0 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 10)\n (=> (= split_index 0) (and (>= 0 6) true))\n (=> (= split_index 1) (and (>= 1 6) (< 0 6)))\n (=> (= split_index 2) (and (>= 3 6) (< 0 6) (< 1 6)))\n (=> (= split_index 3) (and (>= 4 6) (< 0 6) (< 1 6) (< 3 6)))\n (=> (= split_index 4) (and (>= 6 6) (< 0 6) (< 1 6) (< 3 6) (< 4 6)))\n (=> (= split_index 5)\n (and (>= 7 6) (< 0 6) (< 1 6) (< 3 6) (< 4 6) (< 6 6)))\n (=> (= split_index 6)\n (and (>= 7 6) (< 0 6) (< 1 6) (< 3 6) (< 4 6) (< 6 6) (< 7 6)))\n (=> (= split_index 7)\n (and (>= 8 6) (< 0 6)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 10): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "arrays", "id": "arrays-monotonic_split-bcf34a422244a5ee", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 0, "type": "integer"}, "maxItems": 10, "minItems": 10, "type": "array"}, "threshold": {"maximum": 9, "minimum": 0, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 4, "array": [0, 1, 3, 4, 6, 7, 7, 8, 9, 9], "difficulty": "hard", "generator": "array_invariants", "length": 10, "max_value": 9, "min_value": 0, "seed": 20260880, "threshold": 6, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 9, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 10 with\n0 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=10, T=6)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-bcf34a422244a5ee-t000", "input": {"array": [0, 1, 3, 4, 6, 7, 7, 8, 9, 9], "threshold": 6}, "is_public": true, "problem_id": "arrays-monotonic_split-bcf34a422244a5ee"}], "verification": {"duration_ms": 1.629832957405597, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 11): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 11)\n (=> (= split_index 0) (and (>= 1 1) true))\n (=> (= split_index 1) (and (>= 1 1) (< 1 1)))\n (=> (= split_index 2) (and (>= 1 1) (< 1 1) (< 1 1)))\n (=> (= split_index 3) (and (>= 3 1) (< 1 1) (< 1 1) (< 1 1)))\n (=> (= split_index 4) (and (>= 3 1) (< 1 1) (< 1 1) (< 1 1) (< 3 1)))\n (=> (= split_index 5)\n (and (>= 3 1) (< 1 1) (< 1 1) (< 1 1) (< 3 1) (< 3 1)))\n (=> (= split_index 6)\n (and (>= 4 1) (< 1 1) (< 1 1) (< 1 1) (< 3 1) (< 3 1) (< 3 1)))\n (=> (= split_index 7)\n (and (>= 5 1) (< 1 1)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 11): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "arrays", "id": "arrays-monotonic_split-034077ba86402903", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 11, "minItems": 11, "type": "array"}, "threshold": {"maximum": 9, "minimum": 1, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 0, "array": [1, 1, 1, 3, 3, 3, 4, 5, 5, 5, 7], "difficulty": "hard", "generator": "array_invariants", "length": 11, "max_value": 9, "min_value": 1, "seed": 20260881, "threshold": 1, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 10, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 11 with\n1 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=11, T=1)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 0, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-034077ba86402903-t000", "input": {"array": [1, 1, 1, 3, 3, 3, 4, 5, 5, 5, 7], "threshold": 1}, "is_public": true, "problem_id": "arrays-monotonic_split-034077ba86402903"}], "verification": {"duration_ms": 1.7428749706596136, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 10): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 10)\n (=> (= split_index 0) (and (>= 1 1) true))\n (=> (= split_index 1) (and (>= 1 1) (< 1 1)))\n (=> (= split_index 2) (and (>= 1 1) (< 1 1) (< 1 1)))\n (=> (= split_index 3) (and (>= 3 1) (< 1 1) (< 1 1) (< 1 1)))\n (=> (= split_index 4) (and (>= 4 1) (< 1 1) (< 1 1) (< 1 1) (< 3 1)))\n (=> (= split_index 5)\n (and (>= 5 1) (< 1 1) (< 1 1) (< 1 1) (< 3 1) (< 4 1)))\n (=> (= split_index 6)\n (and (>= 5 1) (< 1 1) (< 1 1) (< 1 1) (< 3 1) (< 4 1) (< 5 1)))\n (=> (= split_index 7)\n (and (>= 6 1) (< 1 1)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 10): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "arrays", "id": "arrays-monotonic_split-7d278960360d9948", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 10, "minItems": 10, "type": "array"}, "threshold": {"maximum": 9, "minimum": 1, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 0, "array": [1, 1, 1, 3, 4, 5, 5, 6, 6, 7], "difficulty": "hard", "generator": "array_invariants", "length": 10, "max_value": 9, "min_value": 1, "seed": 20260882, "threshold": 1, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 9, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 10 with\n1 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=10, T=1)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 0, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-7d278960360d9948-t000", "input": {"array": [1, 1, 1, 3, 4, 5, 5, 6, 6, 7], "threshold": 1}, "is_public": true, "problem_id": "arrays-monotonic_split-7d278960360d9948"}], "verification": {"duration_ms": 1.693999976851046, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 10): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 10)\n (=> (= equilibrium_index 0) (= 0 46))\n (=> (= equilibrium_index 1) (= 9 43))\n (=> (= equilibrium_index 2) (= 12 38))\n (=> (= equilibrium_index 3) (= 17 37))\n (=> (= equilibrium_index 4) (= 18 33))\n (=> (= equilibrium_index 5) (= 22 28))\n (=> (= equilibrium_index 6) (= 27 27))\n (=> (= equilibrium_index 7) (= 28 18))\n (=> (= equilibrium_index 8) (= 37 9))\n (=> (= equilibrium_index 9) (= 46 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 10): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "arrays", "id": "arrays-equilibrium_index-b355d994056c98ae", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 10, "minItems": 10, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 6, "array": [9, 3, 5, 1, 4, 5, 1, 9, 9, 9], "difficulty": "hard", "generator": "array_invariants", "length": 10, "max_value": 9, "min_value": 1, "pivot": 6, "seed": 20260883, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 9, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 10. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=10)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 6, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-b355d994056c98ae-t000", "input": {"array": [9, 3, 5, 1, 4, 5, 1, 9, 9, 9]}, "is_public": true, "problem_id": "arrays-equilibrium_index-b355d994056c98ae"}], "verification": {"duration_ms": 1.0459160548634827, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 10): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 10)\n (=> (= equilibrium_index 0) (= 0 32))\n (=> (= equilibrium_index 1) (= 5 30))\n (=> (= equilibrium_index 2) (= 7 22))\n (=> (= equilibrium_index 3) (= 15 19))\n (=> (= equilibrium_index 4) (= 18 18))\n (=> (= equilibrium_index 5) (= 19 16))\n (=> (= equilibrium_index 6) (= 21 13))\n (=> (= equilibrium_index 7) (= 24 8))\n (=> (= equilibrium_index 8) (= 29 5))\n (=> (= equilibrium_index 9) (= 32 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 10): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "arrays", "id": "arrays-equilibrium_index-44c6b4057d850891", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 10, "minItems": 10, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 4, "array": [5, 2, 8, 3, 1, 2, 3, 5, 3, 5], "difficulty": "hard", "generator": "array_invariants", "length": 10, "max_value": 9, "min_value": 1, "pivot": 4, "seed": 20260884, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 9, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 10. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=10)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-44c6b4057d850891-t000", "input": {"array": [5, 2, 8, 3, 1, 2, 3, 5, 3, 5]}, "is_public": true, "problem_id": "arrays-equilibrium_index-44c6b4057d850891"}], "verification": {"duration_ms": 1.0256239911541343, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 10): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 10)\n (=> (= equilibrium_index 0) (= 0 26))\n (=> (= equilibrium_index 1) (= 3 18))\n (=> (= equilibrium_index 2) (= 11 11))\n (=> (= equilibrium_index 3) (= 18 10))\n (=> (= equilibrium_index 4) (= 19 9))\n (=> (= equilibrium_index 5) (= 20 7))\n (=> (= equilibrium_index 6) (= 22 5))\n (=> (= equilibrium_index 7) (= 24 4))\n (=> (= equilibrium_index 8) (= 25 3))\n (=> (= equilibrium_index 9) (= 26 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 10): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "arrays", "id": "arrays-equilibrium_index-e7d9275950b5cbe0", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 10, "minItems": 10, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 2, "array": [3, 8, 7, 1, 1, 2, 2, 1, 1, 3], "difficulty": "hard", "generator": "array_invariants", "length": 10, "max_value": 9, "min_value": 1, "pivot": 2, "seed": 20260885, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 9, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 10. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=10)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-e7d9275950b5cbe0-t000", "input": {"array": [3, 8, 7, 1, 1, 2, 2, 1, 1, 3]}, "is_public": true, "problem_id": "arrays-equilibrium_index-e7d9275950b5cbe0"}], "verification": {"duration_ms": 1.576333015691489, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 14): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 14)\n (=> (= equilibrium_index 0) (= 0 72))\n (=> (= equilibrium_index 1) (= 3 67))\n (=> (= equilibrium_index 2) (= 8 63))\n (=> (= equilibrium_index 3) (= 12 54))\n (=> (= equilibrium_index 4) (= 21 48))\n (=> (= equilibrium_index 5) (= 27 41))\n (=> (= equilibrium_index 6) (= 34 34))\n (=> (= equilibrium_index 7) (= 41 28))\n (=> (= equilibrium_index 8) (= 47 25))\n (=> (= equilibrium_index 9) (= 50 21))\n (=> (= equilibrium_index 10) (= 54 13))\n (=> (= equilibrium_index 11) (= 62 11))\n (=> (...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 14): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "arrays", "id": "arrays-equilibrium_index-ff283d7f2eb793eb", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 14, "minItems": 14, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 6, "array": [3, 5, 4, 9, 6, 7, 7, 6, 3, 4, 8, 2, 5, 6], "difficulty": "hard", "generator": "array_invariants", "length": 14, "max_value": 9, "min_value": 1, "pivot": 6, "seed": 20260886, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 13, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 14. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=14)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 6, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-ff283d7f2eb793eb-t000", "input": {"array": [3, 5, 4, 9, 6, 7, 7, 6, 3, 4, 8, 2, 5, 6]}, "is_public": true, "problem_id": "arrays-equilibrium_index-ff283d7f2eb793eb"}], "verification": {"duration_ms": 1.6519579803571105, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 12): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 12)\n (=> (= split_index 0) (and (>= 1 9) true))\n (=> (= split_index 1) (and (>= 1 9) (< 1 9)))\n (=> (= split_index 2) (and (>= 1 9) (< 1 9) (< 1 9)))\n (=> (= split_index 3) (and (>= 3 9) (< 1 9) (< 1 9) (< 1 9)))\n (=> (= split_index 4) (and (>= 3 9) (< 1 9) (< 1 9) (< 1 9) (< 3 9)))\n (=> (= split_index 5)\n (and (>= 3 9) (< 1 9) (< 1 9) (< 1 9) (< 3 9) (< 3 9)))\n (=> (= split_index 6)\n (and (>= 4 9) (< 1 9) (< 1 9) (< 1 9) (< 3 9) (< 3 9) (< 3 9)))\n (=> (= split_index 7)\n (and (>= 5 9) (< 1 9)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 12): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "arrays", "id": "arrays-monotonic_split-3cc2ea460d4bbc17", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 12, "minItems": 12, "type": "array"}, "threshold": {"maximum": 9, "minimum": 1, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 11, "array": [1, 1, 1, 3, 3, 3, 4, 5, 5, 6, 8, 9], "difficulty": "hard", "generator": "array_invariants", "length": 12, "max_value": 9, "min_value": 1, "seed": 20260887, "threshold": 9, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 11, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 12 with\n1 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=12, T=9)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 11, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-3cc2ea460d4bbc17-t000", "input": {"array": [1, 1, 1, 3, 3, 3, 4, 5, 5, 6, 8, 9], "threshold": 9}, "is_public": true, "problem_id": "arrays-monotonic_split-3cc2ea460d4bbc17"}], "verification": {"duration_ms": 1.0161670506931841, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 11): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 11)\n (=> (= equilibrium_index 0) (= 0 68))\n (=> (= equilibrium_index 1) (= 7 61))\n (=> (= equilibrium_index 2) (= 14 59))\n (=> (= equilibrium_index 3) (= 16 51))\n (=> (= equilibrium_index 4) (= 24 42))\n (=> (= equilibrium_index 5) (= 33 33))\n (=> (= equilibrium_index 6) (= 42 28))\n (=> (= equilibrium_index 7) (= 47 22))\n (=> (= equilibrium_index 8) (= 53 13))\n (=> (= equilibrium_index 9) (= 62 9))\n (=> (= equilibrium_index 10) (= 66 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 11): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "arrays", "id": "arrays-equilibrium_index-d45f0617f126d82e", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 11, "minItems": 11, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 5, "array": [7, 7, 2, 8, 9, 9, 5, 6, 9, 4, 9], "difficulty": "hard", "generator": "array_invariants", "length": 11, "max_value": 9, "min_value": 1, "pivot": 5, "seed": 20260888, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 10, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 11. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=11)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-d45f0617f126d82e-t000", "input": {"array": [7, 7, 2, 8, 9, 9, 5, 6, 9, 4, 9]}, "is_public": true, "problem_id": "arrays-equilibrium_index-d45f0617f126d82e"}], "verification": {"duration_ms": 1.6026669763959944, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 12): 0 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 12)\n (=> (= split_index 0) (and (>= 0 9) true))\n (=> (= split_index 1) (and (>= 0 9) (< 0 9)))\n (=> (= split_index 2) (and (>= 1 9) (< 0 9) (< 0 9)))\n (=> (= split_index 3) (and (>= 3 9) (< 0 9) (< 0 9) (< 1 9)))\n (=> (= split_index 4) (and (>= 5 9) (< 0 9) (< 0 9) (< 1 9) (< 3 9)))\n (=> (= split_index 5)\n (and (>= 5 9) (< 0 9) (< 0 9) (< 1 9) (< 3 9) (< 5 9)))\n (=> (= split_index 6)\n (and (>= 7 9) (< 0 9) (< 0 9) (< 1 9) (< 3 9) (< 5 9) (< 5 9)))\n (=> (= split_index 7)\n (and (>= 9 9) (< 0 9)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 12): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "arrays", "id": "arrays-monotonic_split-99744826f756ba4f", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 0, "type": "integer"}, "maxItems": 12, "minItems": 12, "type": "array"}, "threshold": {"maximum": 9, "minimum": 0, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 7, "array": [0, 0, 1, 3, 5, 5, 7, 9, 9, 9, 9, 9], "difficulty": "hard", "generator": "array_invariants", "length": 12, "max_value": 9, "min_value": 0, "seed": 20260889, "threshold": 9, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 11, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 12 with\n0 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=12, T=9)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 7, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-99744826f756ba4f-t000", "input": {"array": [0, 0, 1, 3, 5, 5, 7, 9, 9, 9, 9, 9], "threshold": 9}, "is_public": true, "problem_id": "arrays-monotonic_split-99744826f756ba4f"}], "verification": {"duration_ms": 1.0819999733939767, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 11): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 11)\n (=> (= split_index 0) (and (>= 1 4) true))\n (=> (= split_index 1) (and (>= 2 4) (< 1 4)))\n (=> (= split_index 2) (and (>= 3 4) (< 1 4) (< 2 4)))\n (=> (= split_index 3) (and (>= 3 4) (< 1 4) (< 2 4) (< 3 4)))\n (=> (= split_index 4) (and (>= 3 4) (< 1 4) (< 2 4) (< 3 4) (< 3 4)))\n (=> (= split_index 5)\n (and (>= 4 4) (< 1 4) (< 2 4) (< 3 4) (< 3 4) (< 3 4)))\n (=> (= split_index 6)\n (and (>= 4 4) (< 1 4) (< 2 4) (< 3 4) (< 3 4) (< 3 4) (< 4 4)))\n (=> (= split_index 7)\n (and (>= 5 4) (< 1 4)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 11): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "arrays", "id": "arrays-monotonic_split-2544d0466732da89", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 11, "minItems": 11, "type": "array"}, "threshold": {"maximum": 9, "minimum": 1, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 5, "array": [1, 2, 3, 3, 3, 4, 4, 5, 6, 7, 9], "difficulty": "hard", "generator": "array_invariants", "length": 11, "max_value": 9, "min_value": 1, "seed": 20260890, "threshold": 4, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 10, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 11 with\n1 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=11, T=4)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-2544d0466732da89-t000", "input": {"array": [1, 2, 3, 3, 3, 4, 4, 5, 6, 7, 9], "threshold": 4}, "is_public": true, "problem_id": "arrays-monotonic_split-2544d0466732da89"}], "verification": {"duration_ms": 1.694332982879132, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 12): 2 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 12)\n (=> (= split_index 0) (and (>= 2 7) true))\n (=> (= split_index 1) (and (>= 3 7) (< 2 7)))\n (=> (= split_index 2) (and (>= 3 7) (< 2 7) (< 3 7)))\n (=> (= split_index 3) (and (>= 4 7) (< 2 7) (< 3 7) (< 3 7)))\n (=> (= split_index 4) (and (>= 6 7) (< 2 7) (< 3 7) (< 3 7) (< 4 7)))\n (=> (= split_index 5)\n (and (>= 7 7) (< 2 7) (< 3 7) (< 3 7) (< 4 7) (< 6 7)))\n (=> (= split_index 6)\n (and (>= 7 7) (< 2 7) (< 3 7) (< 3 7) (< 4 7) (< 6 7) (< 7 7)))\n (=> (= split_index 7)\n (and (>= 7 7) (< 2 7)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 12): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "arrays", "id": "arrays-monotonic_split-2280e407d4942a84", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 2, "type": "integer"}, "maxItems": 12, "minItems": 12, "type": "array"}, "threshold": {"maximum": 9, "minimum": 2, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 5, "array": [2, 3, 3, 4, 6, 7, 7, 7, 9, 9, 9, 9], "difficulty": "hard", "generator": "array_invariants", "length": 12, "max_value": 9, "min_value": 2, "seed": 20260891, "threshold": 7, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 11, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 12 with\n2 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=12, T=7)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-2280e407d4942a84-t000", "input": {"array": [2, 3, 3, 4, 6, 7, 7, 7, 9, 9, 9, 9], "threshold": 7}, "is_public": true, "problem_id": "arrays-monotonic_split-2280e407d4942a84"}], "verification": {"duration_ms": 1.8071660306304693, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 12): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 12)\n (=> (= equilibrium_index 0) (= 0 56))\n (=> (= equilibrium_index 1) (= 5 52))\n (=> (= equilibrium_index 2) (= 9 46))\n (=> (= equilibrium_index 3) (= 15 39))\n (=> (= equilibrium_index 4) (= 22 37))\n (=> (= equilibrium_index 5) (= 24 35))\n (=> (= equilibrium_index 6) (= 26 34))\n (=> (= equilibrium_index 7) (= 27 27))\n (=> (= equilibrium_index 8) (= 34 20))\n (=> (= equilibrium_index 9) (= 41 11))\n (=> (= equilibrium_index 10) (= 50 6))\n (=> (= equilibrium_index 11) (= 55 0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 12): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "arrays", "id": "arrays-equilibrium_index-c92873ee1519d1ba", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 12, "minItems": 12, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 7, "array": [5, 4, 6, 7, 2, 2, 1, 7, 7, 9, 5, 6], "difficulty": "hard", "generator": "array_invariants", "length": 12, "max_value": 9, "min_value": 1, "pivot": 7, "seed": 20260892, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 11, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 12. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=12)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 7, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-c92873ee1519d1ba-t000", "input": {"array": [5, 4, 6, 7, 2, 2, 1, 7, 7, 9, 5, 6]}, "is_public": true, "problem_id": "arrays-equilibrium_index-c92873ee1519d1ba"}], "verification": {"duration_ms": 1.6680839471518993, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 13): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 13)\n (=> (= equilibrium_index 0) (= 0 45))\n (=> (= equilibrium_index 1) (= 6 43))\n (=> (= equilibrium_index 2) (= 8 37))\n (=> (= equilibrium_index 3) (= 14 32))\n (=> (= equilibrium_index 4) (= 19 29))\n (=> (= equilibrium_index 5) (= 22 22))\n (=> (= equilibrium_index 6) (= 29 21))\n (=> (= equilibrium_index 7) (= 30 16))\n (=> (= equilibrium_index 8) (= 35 12))\n (=> (= equilibrium_index 9) (= 39 8))\n (=> (= equilibrium_index 10) (= 43 5))\n (=> (= equilibrium_index 11) (= 46 3))\n (=> (= e...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 13): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "arrays", "id": "arrays-equilibrium_index-e0a70216cc771b4d", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 13, "minItems": 13, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 5, "array": [6, 2, 6, 5, 3, 7, 1, 5, 4, 4, 3, 2, 3], "difficulty": "hard", "generator": "array_invariants", "length": 13, "max_value": 9, "min_value": 1, "pivot": 5, "seed": 20260893, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 12, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 13. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=13)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-e0a70216cc771b4d-t000", "input": {"array": [6, 2, 6, 5, 3, 7, 1, 5, 4, 4, 3, 2, 3]}, "is_public": true, "problem_id": "arrays-equilibrium_index-e0a70216cc771b4d"}], "verification": {"duration_ms": 1.656751031987369, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 10): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 10)\n (=> (= split_index 0) (and (>= 1 4) true))\n (=> (= split_index 1) (and (>= 3 4) (< 1 4)))\n (=> (= split_index 2) (and (>= 4 4) (< 1 4) (< 3 4)))\n (=> (= split_index 3) (and (>= 4 4) (< 1 4) (< 3 4) (< 4 4)))\n (=> (= split_index 4) (and (>= 6 4) (< 1 4) (< 3 4) (< 4 4) (< 4 4)))\n (=> (= split_index 5)\n (and (>= 8 4) (< 1 4) (< 3 4) (< 4 4) (< 4 4) (< 6 4)))\n (=> (= split_index 6)\n (and (>= 9 4) (< 1 4) (< 3 4) (< 4 4) (< 4 4) (< 6 4) (< 8 4)))\n (=> (= split_index 7)\n (and (>= 9 4) (< 1 4)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 10): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "arrays", "id": "arrays-monotonic_split-f4872ecd0c88919e", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 10, "minItems": 10, "type": "array"}, "threshold": {"maximum": 9, "minimum": 1, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 2, "array": [1, 3, 4, 4, 6, 8, 9, 9, 9, 9], "difficulty": "hard", "generator": "array_invariants", "length": 10, "max_value": 9, "min_value": 1, "seed": 20260894, "threshold": 4, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 9, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 10 with\n1 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=10, T=4)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-f4872ecd0c88919e-t000", "input": {"array": [1, 3, 4, 4, 6, 8, 9, 9, 9, 9], "threshold": 4}, "is_public": true, "problem_id": "arrays-monotonic_split-f4872ecd0c88919e"}], "verification": {"duration_ms": 1.6099170316010714, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 14): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 14)\n (=> (= split_index 0) (and (>= 1 3) true))\n (=> (= split_index 1) (and (>= 3 3) (< 1 3)))\n (=> (= split_index 2) (and (>= 3 3) (< 1 3) (< 3 3)))\n (=> (= split_index 3) (and (>= 4 3) (< 1 3) (< 3 3) (< 3 3)))\n (=> (= split_index 4) (and (>= 5 3) (< 1 3) (< 3 3) (< 3 3) (< 4 3)))\n (=> (= split_index 5)\n (and (>= 7 3) (< 1 3) (< 3 3) (< 3 3) (< 4 3) (< 5 3)))\n (=> (= split_index 6)\n (and (>= 9 3) (< 1 3) (< 3 3) (< 3 3) (< 4 3) (< 5 3) (< 7 3)))\n (=> (= split_index 7)\n (and (>= 9 3) (< 1 3)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 14): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "arrays", "id": "arrays-monotonic_split-5969fed8db60de4c", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 14, "minItems": 14, "type": "array"}, "threshold": {"maximum": 9, "minimum": 1, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 1, "array": [1, 3, 3, 4, 5, 7, 9, 9, 9, 9, 9, 9, 9, 9], "difficulty": "hard", "generator": "array_invariants", "length": 14, "max_value": 9, "min_value": 1, "seed": 20260895, "threshold": 3, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 13, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 14 with\n1 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=14, T=3)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-5969fed8db60de4c-t000", "input": {"array": [1, 3, 3, 4, 5, 7, 9, 9, 9, 9, 9, 9, 9, 9], "threshold": 3}, "is_public": true, "problem_id": "arrays-monotonic_split-5969fed8db60de4c"}], "verification": {"duration_ms": 1.774375035893172, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 10): 0 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 10)\n (=> (= split_index 0) (and (>= 0 9) true))\n (=> (= split_index 1) (and (>= 1 9) (< 0 9)))\n (=> (= split_index 2) (and (>= 3 9) (< 0 9) (< 1 9)))\n (=> (= split_index 3) (and (>= 4 9) (< 0 9) (< 1 9) (< 3 9)))\n (=> (= split_index 4) (and (>= 6 9) (< 0 9) (< 1 9) (< 3 9) (< 4 9)))\n (=> (= split_index 5)\n (and (>= 7 9) (< 0 9) (< 1 9) (< 3 9) (< 4 9) (< 6 9)))\n (=> (= split_index 6)\n (and (>= 9 9) (< 0 9) (< 1 9) (< 3 9) (< 4 9) (< 6 9) (< 7 9)))\n (=> (= split_index 7)\n (and (>= 9 9) (< 0 9)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 10): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "arrays", "id": "arrays-monotonic_split-f1fb23a19f251a75", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 0, "type": "integer"}, "maxItems": 10, "minItems": 10, "type": "array"}, "threshold": {"maximum": 9, "minimum": 0, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 6, "array": [0, 1, 3, 4, 6, 7, 9, 9, 9, 9], "difficulty": "hard", "generator": "array_invariants", "length": 10, "max_value": 9, "min_value": 0, "seed": 20260896, "threshold": 9, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 9, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 10 with\n0 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=10, T=9)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 6, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-f1fb23a19f251a75-t000", "input": {"array": [0, 1, 3, 4, 6, 7, 9, 9, 9, 9], "threshold": 9}, "is_public": true, "problem_id": "arrays-monotonic_split-f1fb23a19f251a75"}], "verification": {"duration_ms": 1.1619159486144781, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 13): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 13)\n (=> (= equilibrium_index 0) (= 0 83))\n (=> (= equilibrium_index 1) (= 8 75))\n (=> (= equilibrium_index 2) (= 16 69))\n (=> (= equilibrium_index 3) (= 22 67))\n (=> (= equilibrium_index 4) (= 24 63))\n (=> (= equilibrium_index 5) (= 28 54))\n (=> (= equilibrium_index 6) (= 37 47))\n (=> (= equilibrium_index 7) (= 44 44))\n (=> (= equilibrium_index 8) (= 47 36))\n (=> (= equilibrium_index 9) (= 55 27))\n (=> (= equilibrium_index 10) (= 64 18))\n (=> (= equilibrium_index 11) (= 73 9))\n (=> (...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 13): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "arrays", "id": "arrays-equilibrium_index-6ca4e7375128bdec", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 13, "minItems": 13, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 7, "array": [8, 8, 6, 2, 4, 9, 7, 3, 8, 9, 9, 9, 9], "difficulty": "hard", "generator": "array_invariants", "length": 13, "max_value": 9, "min_value": 1, "pivot": 7, "seed": 20260897, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 12, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 13. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=13)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 7, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-6ca4e7375128bdec-t000", "input": {"array": [8, 8, 6, 2, 4, 9, 7, 3, 8, 9, 9, 9, 9]}, "is_public": true, "problem_id": "arrays-equilibrium_index-6ca4e7375128bdec"}], "verification": {"duration_ms": 1.8766660359688103, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 14): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 14)\n (=> (= split_index 0) (and (>= 1 9) true))\n (=> (= split_index 1) (and (>= 3 9) (< 1 9)))\n (=> (= split_index 2) (and (>= 5 9) (< 1 9) (< 3 9)))\n (=> (= split_index 3) (and (>= 6 9) (< 1 9) (< 3 9) (< 5 9)))\n (=> (= split_index 4) (and (>= 8 9) (< 1 9) (< 3 9) (< 5 9) (< 6 9)))\n (=> (= split_index 5)\n (and (>= 8 9) (< 1 9) (< 3 9) (< 5 9) (< 6 9) (< 8 9)))\n (=> (= split_index 6)\n (and (>= 8 9) (< 1 9) (< 3 9) (< 5 9) (< 6 9) (< 8 9) (< 8 9)))\n (=> (= split_index 7)\n (and (>= 8 9) (< 1 9)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 14): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "arrays", "id": "arrays-monotonic_split-d3a76130217daee5", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 14, "minItems": 14, "type": "array"}, "threshold": {"maximum": 9, "minimum": 1, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 8, "array": [1, 3, 5, 6, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9], "difficulty": "hard", "generator": "array_invariants", "length": 14, "max_value": 9, "min_value": 1, "seed": 20260898, "threshold": 9, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 13, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 14 with\n1 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=14, T=9)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 8, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-d3a76130217daee5-t000", "input": {"array": [1, 3, 5, 6, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9], "threshold": 9}, "is_public": true, "problem_id": "arrays-monotonic_split-d3a76130217daee5"}], "verification": {"duration_ms": 1.2931250385008752, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 15): 2 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 15)\n (=> (= split_index 0) (and (>= 2 9) true))\n (=> (= split_index 1) (and (>= 4 9) (< 2 9)))\n (=> (= split_index 2) (and (>= 6 9) (< 2 9) (< 4 9)))\n (=> (= split_index 3) (and (>= 6 9) (< 2 9) (< 4 9) (< 6 9)))\n (=> (= split_index 4) (and (>= 7 9) (< 2 9) (< 4 9) (< 6 9) (< 6 9)))\n (=> (= split_index 5)\n (and (>= 7 9) (< 2 9) (< 4 9) (< 6 9) (< 6 9) (< 7 9)))\n (=> (= split_index 6)\n (and (>= 9 9) (< 2 9) (< 4 9) (< 6 9) (< 6 9) (< 7 9) (< 7 9)))\n (=> (= split_index 7)\n (and (>= 9 9) (< 2 9)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 15): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "arrays", "id": "arrays-monotonic_split-c777d4554acd8392", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 2, "type": "integer"}, "maxItems": 15, "minItems": 15, "type": "array"}, "threshold": {"maximum": 9, "minimum": 2, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 6, "array": [2, 4, 6, 6, 7, 7, 9, 9, 9, 9, 9, 9, 9, 9, 9], "difficulty": "extreme", "generator": "array_invariants", "length": 15, "max_value": 9, "min_value": 2, "seed": 20260899, "threshold": 9, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 14, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 15 with\n2 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=15, T=9)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 6, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-c777d4554acd8392-t000", "input": {"array": [2, 4, 6, 6, 7, 7, 9, 9, 9, 9, 9, 9, 9, 9, 9], "threshold": 9}, "is_public": true, "problem_id": "arrays-monotonic_split-c777d4554acd8392"}], "verification": {"duration_ms": 2.041166997514665, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 21): 0 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 21)\n (=> (= split_index 0) (and (>= 0 6) true))\n (=> (= split_index 1) (and (>= 2 6) (< 0 6)))\n (=> (= split_index 2) (and (>= 3 6) (< 0 6) (< 2 6)))\n (=> (= split_index 3) (and (>= 3 6) (< 0 6) (< 2 6) (< 3 6)))\n (=> (= split_index 4) (and (>= 5 6) (< 0 6) (< 2 6) (< 3 6) (< 3 6)))\n (=> (= split_index 5)\n (and (>= 6 6) (< 0 6) (< 2 6) (< 3 6) (< 3 6) (< 5 6)))\n (=> (= split_index 6)\n (and (>= 8 6) (< 0 6) (< 2 6) (< 3 6) (< 3 6) (< 5 6) (< 6 6)))\n (=> (= split_index 7)\n (and (>= 9 6) (< 0 6)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 21): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "arrays", "id": "arrays-monotonic_split-df90c32312844e96", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 0, "type": "integer"}, "maxItems": 21, "minItems": 21, "type": "array"}, "threshold": {"maximum": 9, "minimum": 0, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 5, "array": [0, 2, 3, 3, 5, 6, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9], "difficulty": "extreme", "generator": "array_invariants", "length": 21, "max_value": 9, "min_value": 0, "seed": 20260900, "threshold": 6, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 20, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 21 with\n0 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=21, T=6)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-df90c32312844e96-t000", "input": {"array": [0, 2, 3, 3, 5, 6, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9], "threshold": 6}, "is_public": true, "problem_id": "arrays-monotonic_split-df90c32312844e96"}], "verification": {"duration_ms": 2.040668041445315, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 15): 2 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 15)\n (=> (= split_index 0) (and (>= 2 9) true))\n (=> (= split_index 1) (and (>= 4 9) (< 2 9)))\n (=> (= split_index 2) (and (>= 6 9) (< 2 9) (< 4 9)))\n (=> (= split_index 3) (and (>= 8 9) (< 2 9) (< 4 9) (< 6 9)))\n (=> (= split_index 4) (and (>= 9 9) (< 2 9) (< 4 9) (< 6 9) (< 8 9)))\n (=> (= split_index 5)\n (and (>= 9 9) (< 2 9) (< 4 9) (< 6 9) (< 8 9) (< 9 9)))\n (=> (= split_index 6)\n (and (>= 9 9) (< 2 9) (< 4 9) (< 6 9) (< 8 9) (< 9 9) (< 9 9)))\n (=> (= split_index 7)\n (and (>= 9 9) (< 2 9)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 15): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "arrays", "id": "arrays-monotonic_split-9fb7019c937b1ab0", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 2, "type": "integer"}, "maxItems": 15, "minItems": 15, "type": "array"}, "threshold": {"maximum": 9, "minimum": 2, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 4, "array": [2, 4, 6, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9], "difficulty": "extreme", "generator": "array_invariants", "length": 15, "max_value": 9, "min_value": 2, "seed": 20260901, "threshold": 9, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 14, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 15 with\n2 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=15, T=9)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-9fb7019c937b1ab0-t000", "input": {"array": [2, 4, 6, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9], "threshold": 9}, "is_public": true, "problem_id": "arrays-monotonic_split-9fb7019c937b1ab0"}], "verification": {"duration_ms": 2.004874055273831, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 24): 2 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 24)\n (=> (= split_index 0) (and (>= 2 9) true))\n (=> (= split_index 1) (and (>= 3 9) (< 2 9)))\n (=> (= split_index 2) (and (>= 3 9) (< 2 9) (< 3 9)))\n (=> (= split_index 3) (and (>= 5 9) (< 2 9) (< 3 9) (< 3 9)))\n (=> (= split_index 4) (and (>= 5 9) (< 2 9) (< 3 9) (< 3 9) (< 5 9)))\n (=> (= split_index 5)\n (and (>= 7 9) (< 2 9) (< 3 9) (< 3 9) (< 5 9) (< 5 9)))\n (=> (= split_index 6)\n (and (>= 8 9) (< 2 9) (< 3 9) (< 3 9) (< 5 9) (< 5 9) (< 7 9)))\n (=> (= split_index 7)\n (and (>= 8 9) (< 2 9)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 24): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "arrays", "id": "arrays-monotonic_split-3eb9c250bd36cebc", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 2, "type": "integer"}, "maxItems": 24, "minItems": 24, "type": "array"}, "threshold": {"maximum": 9, "minimum": 2, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 8, "array": [2, 3, 3, 5, 5, 7, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9], "difficulty": "extreme", "generator": "array_invariants", "length": 24, "max_value": 9, "min_value": 2, "seed": 20260902, "threshold": 9, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 23, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 24 with\n2 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=24, T=9)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 8, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-3eb9c250bd36cebc-t000", "input": {"array": [2, 3, 3, 5, 5, 7, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9], "threshold": 9}, "is_public": true, "problem_id": "arrays-monotonic_split-3eb9c250bd36cebc"}], "verification": {"duration_ms": 2.0839159842580557, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 19): 0 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 19)\n (=> (= split_index 0) (and (>= 0 3) true))\n (=> (= split_index 1) (and (>= 2 3) (< 0 3)))\n (=> (= split_index 2) (and (>= 2 3) (< 0 3) (< 2 3)))\n (=> (= split_index 3) (and (>= 3 3) (< 0 3) (< 2 3) (< 2 3)))\n (=> (= split_index 4) (and (>= 4 3) (< 0 3) (< 2 3) (< 2 3) (< 3 3)))\n (=> (= split_index 5)\n (and (>= 4 3) (< 0 3) (< 2 3) (< 2 3) (< 3 3) (< 4 3)))\n (=> (= split_index 6)\n (and (>= 5 3) (< 0 3) (< 2 3) (< 2 3) (< 3 3) (< 4 3) (< 4 3)))\n (=> (= split_index 7)\n (and (>= 6 3) (< 0 3)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 19): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "arrays", "id": "arrays-monotonic_split-06fe2a5441d16699", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 0, "type": "integer"}, "maxItems": 19, "minItems": 19, "type": "array"}, "threshold": {"maximum": 9, "minimum": 0, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 3, "array": [0, 2, 2, 3, 4, 4, 5, 6, 6, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9], "difficulty": "extreme", "generator": "array_invariants", "length": 19, "max_value": 9, "min_value": 0, "seed": 20260903, "threshold": 3, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 18, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 19 with\n0 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=19, T=3)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 3, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-06fe2a5441d16699-t000", "input": {"array": [0, 2, 2, 3, 4, 4, 5, 6, 6, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9], "threshold": 3}, "is_public": true, "problem_id": "arrays-monotonic_split-06fe2a5441d16699"}], "verification": {"duration_ms": 1.879167975857854, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 23): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 23)\n (=> (= equilibrium_index 0) (= 0 82))\n (=> (= equilibrium_index 1) (= 1 79))\n (=> (= equilibrium_index 2) (= 4 77))\n (=> (= equilibrium_index 3) (= 6 71))\n (=> (= equilibrium_index 4) (= 12 66))\n (=> (= equilibrium_index 5) (= 17 58))\n (=> (= equilibrium_index 6) (= 25 57))\n (=> (= equilibrium_index 7) (= 26 53))\n (=> (= equilibrium_index 8) (= 30 46))\n (=> (= equilibrium_index 9) (= 37 37))\n (=> (= equilibrium_index 10) (= 46 33))\n (=> (= equilibrium_index 11) (= 50 30))\n (=> (=...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 23): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "arrays", "id": "arrays-equilibrium_index-1113d3dc978d43bc", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 23, "minItems": 23, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 9, "array": [1, 3, 2, 6, 5, 8, 1, 4, 7, 9, 4, 3, 2, 1, 4, 2, 3, 3, 4, 3, 3, 1, 4], "difficulty": "extreme", "generator": "array_invariants", "length": 23, "max_value": 9, "min_value": 1, "pivot": 9, "seed": 20260904, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 22, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 23. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=23)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 9, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-1113d3dc978d43bc-t000", "input": {"array": [1, 3, 2, 6, 5, 8, 1, 4, 7, 9, 4, 3, 2, 1, 4, 2, 3, 3, 4, 3, 3, 1, 4]}, "is_public": true, "problem_id": "arrays-equilibrium_index-1113d3dc978d43bc"}], "verification": {"duration_ms": 1.8898749840445817, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 16): 2 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 16)\n (=> (= split_index 0) (and (>= 2 8) true))\n (=> (= split_index 1) (and (>= 4 8) (< 2 8)))\n (=> (= split_index 2) (and (>= 6 8) (< 2 8) (< 4 8)))\n (=> (= split_index 3) (and (>= 7 8) (< 2 8) (< 4 8) (< 6 8)))\n (=> (= split_index 4) (and (>= 8 8) (< 2 8) (< 4 8) (< 6 8) (< 7 8)))\n (=> (= split_index 5)\n (and (>= 8 8) (< 2 8) (< 4 8) (< 6 8) (< 7 8) (< 8 8)))\n (=> (= split_index 6)\n (and (>= 8 8) (< 2 8) (< 4 8) (< 6 8) (< 7 8) (< 8 8) (< 8 8)))\n (=> (= split_index 7)\n (and (>= 8 8) (< 2 8)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 16): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "arrays", "id": "arrays-monotonic_split-6bccdc5744cf17c3", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 2, "type": "integer"}, "maxItems": 16, "minItems": 16, "type": "array"}, "threshold": {"maximum": 9, "minimum": 2, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 4, "array": [2, 4, 6, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9], "difficulty": "extreme", "generator": "array_invariants", "length": 16, "max_value": 9, "min_value": 2, "seed": 20260905, "threshold": 8, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 15, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 16 with\n2 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=16, T=8)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-6bccdc5744cf17c3-t000", "input": {"array": [2, 4, 6, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9], "threshold": 8}, "is_public": true, "problem_id": "arrays-monotonic_split-6bccdc5744cf17c3"}], "verification": {"duration_ms": 1.7582079162821174, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 19): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 19)\n (=> (= split_index 0) (and (>= 1 9) true))\n (=> (= split_index 1) (and (>= 3 9) (< 1 9)))\n (=> (= split_index 2) (and (>= 3 9) (< 1 9) (< 3 9)))\n (=> (= split_index 3) (and (>= 3 9) (< 1 9) (< 3 9) (< 3 9)))\n (=> (= split_index 4) (and (>= 5 9) (< 1 9) (< 3 9) (< 3 9) (< 3 9)))\n (=> (= split_index 5)\n (and (>= 7 9) (< 1 9) (< 3 9) (< 3 9) (< 3 9) (< 5 9)))\n (=> (= split_index 6)\n (and (>= 8 9) (< 1 9) (< 3 9) (< 3 9) (< 3 9) (< 5 9) (< 7 9)))\n (=> (= split_index 7)\n (and (>= 8 9) (< 1 9)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 19): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "arrays", "id": "arrays-monotonic_split-3b05210bd69bcaa6", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 19, "minItems": 19, "type": "array"}, "threshold": {"maximum": 9, "minimum": 1, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 10, "array": [1, 3, 3, 3, 5, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9], "difficulty": "extreme", "generator": "array_invariants", "length": 19, "max_value": 9, "min_value": 1, "seed": 20260906, "threshold": 9, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 18, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 19 with\n1 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=19, T=9)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 10, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-3b05210bd69bcaa6-t000", "input": {"array": [1, 3, 3, 3, 5, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9], "threshold": 9}, "is_public": true, "problem_id": "arrays-monotonic_split-3b05210bd69bcaa6"}], "verification": {"duration_ms": 1.7122499994002283, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 16): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 16)\n (=> (= equilibrium_index 0) (= 0 35))\n (=> (= equilibrium_index 1) (= 6 32))\n (=> (= equilibrium_index 2) (= 9 28))\n (=> (= equilibrium_index 3) (= 13 22))\n (=> (= equilibrium_index 4) (= 19 19))\n (=> (= equilibrium_index 5) (= 22 17))\n (=> (= equilibrium_index 6) (= 24 15))\n (=> (= equilibrium_index 7) (= 26 14))\n (=> (= equilibrium_index 8) (= 27 13))\n (=> (= equilibrium_index 9) (= 28 10))\n (=> (= equilibrium_index 10) (= 31 9))\n (=> (= equilibrium_index 11) (= 32 7))\n (=> (= ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 16): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "arrays", "id": "arrays-equilibrium_index-6eefb7375c4e8266", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 16, "minItems": 16, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 4, "array": [6, 3, 4, 6, 3, 2, 2, 1, 1, 3, 1, 2, 3, 2, 1, 1], "difficulty": "extreme", "generator": "array_invariants", "length": 16, "max_value": 9, "min_value": 1, "pivot": 4, "seed": 20260907, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 15, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 16. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=16)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-6eefb7375c4e8266-t000", "input": {"array": [6, 3, 4, 6, 3, 2, 2, 1, 1, 3, 1, 2, 3, 2, 1, 1]}, "is_public": true, "problem_id": "arrays-equilibrium_index-6eefb7375c4e8266"}], "verification": {"duration_ms": 1.8387489835731685, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 21): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 21)\n (=> (= equilibrium_index 0) (= 0 68))\n (=> (= equilibrium_index 1) (= 4 61))\n (=> (= equilibrium_index 2) (= 11 54))\n (=> (= equilibrium_index 3) (= 18 49))\n (=> (= equilibrium_index 4) (= 23 42))\n (=> (= equilibrium_index 5) (= 30 40))\n (=> (= equilibrium_index 6) (= 32 39))\n (=> (= equilibrium_index 7) (= 33 33))\n (=> (= equilibrium_index 8) (= 39 31))\n (=> (= equilibrium_index 9) (= 41 26))\n (=> (= equilibrium_index 10) (= 46 24))\n (=> (= equilibrium_index 11) (= 48 21))\n (=> ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 21): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "arrays", "id": "arrays-equilibrium_index-8f0d6f1f8b1ff691", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 21, "minItems": 21, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 7, "array": [4, 7, 7, 5, 7, 2, 1, 6, 2, 5, 2, 3, 4, 3, 2, 2, 2, 1, 1, 3, 3], "difficulty": "extreme", "generator": "array_invariants", "length": 21, "max_value": 9, "min_value": 1, "pivot": 7, "seed": 20260908, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 20, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 21. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=21)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 7, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-8f0d6f1f8b1ff691-t000", "input": {"array": [4, 7, 7, 5, 7, 2, 1, 6, 2, 5, 2, 3, 4, 3, 2, 2, 2, 1, 1, 3, 3]}, "is_public": true, "problem_id": "arrays-equilibrium_index-8f0d6f1f8b1ff691"}], "verification": {"duration_ms": 1.8545000348240137, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 16): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 16)\n (=> (= equilibrium_index 0) (= 0 37))\n (=> (= equilibrium_index 1) (= 9 35))\n (=> (= equilibrium_index 2) (= 11 27))\n (=> (= equilibrium_index 3) (= 19 19))\n (=> (= equilibrium_index 4) (= 27 16))\n (=> (= equilibrium_index 5) (= 30 15))\n (=> (= equilibrium_index 6) (= 31 14))\n (=> (= equilibrium_index 7) (= 32 12))\n (=> (= equilibrium_index 8) (= 34 10))\n (=> (= equilibrium_index 9) (= 36 9))\n (=> (= equilibrium_index 10) (= 37 8))\n (=> (= equilibrium_index 11) (= 38 7))\n (=> (= ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 16): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "arrays", "id": "arrays-equilibrium_index-f4847eabe416e5db", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 16, "minItems": 16, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 3, "array": [9, 2, 8, 8, 3, 1, 1, 2, 2, 1, 1, 1, 1, 1, 3, 2], "difficulty": "extreme", "generator": "array_invariants", "length": 16, "max_value": 9, "min_value": 1, "pivot": 3, "seed": 20260909, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 15, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 16. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=16)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 3, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-f4847eabe416e5db-t000", "input": {"array": [9, 2, 8, 8, 3, 1, 1, 2, 2, 1, 1, 1, 1, 1, 3, 2]}, "is_public": true, "problem_id": "arrays-equilibrium_index-f4847eabe416e5db"}], "verification": {"duration_ms": 1.7913750489242375, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 19): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 19)\n (=> (= equilibrium_index 0) (= 0 32))\n (=> (= equilibrium_index 1) (= 3 26))\n (=> (= equilibrium_index 2) (= 9 20))\n (=> (= equilibrium_index 3) (= 15 15))\n (=> (= equilibrium_index 4) (= 20 14))\n (=> (= equilibrium_index 5) (= 21 13))\n (=> (= equilibrium_index 6) (= 22 12))\n (=> (= equilibrium_index 7) (= 23 11))\n (=> (= equilibrium_index 8) (= 24 10))\n (=> (= equilibrium_index 9) (= 25 9))\n (=> (= equilibrium_index 10) (= 26 8))\n (=> (= equilibrium_index 11) (= 27 7))\n (=> (= e...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 19): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "arrays", "id": "arrays-equilibrium_index-baaf5d4886802bcf", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 19, "minItems": 19, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 3, "array": [3, 6, 6, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], "difficulty": "extreme", "generator": "array_invariants", "length": 19, "max_value": 9, "min_value": 1, "pivot": 3, "seed": 20260910, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 18, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 19. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=19)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 3, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-baaf5d4886802bcf-t000", "input": {"array": [3, 6, 6, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]}, "is_public": true, "problem_id": "arrays-equilibrium_index-baaf5d4886802bcf"}], "verification": {"duration_ms": 2.003417001105845, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 15): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 15)\n (=> (= split_index 0) (and (>= 1 9) true))\n (=> (= split_index 1) (and (>= 1 9) (< 1 9)))\n (=> (= split_index 2) (and (>= 3 9) (< 1 9) (< 1 9)))\n (=> (= split_index 3) (and (>= 3 9) (< 1 9) (< 1 9) (< 3 9)))\n (=> (= split_index 4) (and (>= 5 9) (< 1 9) (< 1 9) (< 3 9) (< 3 9)))\n (=> (= split_index 5)\n (and (>= 6 9) (< 1 9) (< 1 9) (< 3 9) (< 3 9) (< 5 9)))\n (=> (= split_index 6)\n (and (>= 7 9) (< 1 9) (< 1 9) (< 3 9) (< 3 9) (< 5 9) (< 6 9)))\n (=> (= split_index 7)\n (and (>= 8 9) (< 1 9)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 15): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "arrays", "id": "arrays-monotonic_split-6d182a6e1e8bb38a", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 15, "minItems": 15, "type": "array"}, "threshold": {"maximum": 9, "minimum": 1, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 8, "array": [1, 1, 3, 3, 5, 6, 7, 8, 9, 9, 9, 9, 9, 9, 9], "difficulty": "extreme", "generator": "array_invariants", "length": 15, "max_value": 9, "min_value": 1, "seed": 20260911, "threshold": 9, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 14, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 15 with\n1 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=15, T=9)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 8, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-6d182a6e1e8bb38a-t000", "input": {"array": [1, 1, 3, 3, 5, 6, 7, 8, 9, 9, 9, 9, 9, 9, 9], "threshold": 9}, "is_public": true, "problem_id": "arrays-monotonic_split-6d182a6e1e8bb38a"}], "verification": {"duration_ms": 1.7403329838998616, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 21): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 21)\n (=> (= equilibrium_index 0) (= 0 53))\n (=> (= equilibrium_index 1) (= 2 52))\n (=> (= equilibrium_index 2) (= 3 44))\n (=> (= equilibrium_index 3) (= 11 42))\n (=> (= equilibrium_index 4) (= 13 38))\n (=> (= equilibrium_index 5) (= 17 36))\n (=> (= equilibrium_index 6) (= 19 28))\n (=> (= equilibrium_index 7) (= 27 27))\n (=> (= equilibrium_index 8) (= 28 25))\n (=> (= equilibrium_index 9) (= 30 23))\n (=> (= equilibrium_index 10) (= 32 22))\n (=> (= equilibrium_index 11) (= 33 21))\n (=> (...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 21): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "arrays", "id": "arrays-equilibrium_index-f49f4a16245ca727", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 21, "minItems": 21, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 7, "array": [2, 1, 8, 2, 4, 2, 8, 1, 2, 2, 1, 1, 1, 2, 1, 3, 3, 3, 2, 3, 3], "difficulty": "extreme", "generator": "array_invariants", "length": 21, "max_value": 9, "min_value": 1, "pivot": 7, "seed": 20260912, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 20, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 21. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=21)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 7, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-f49f4a16245ca727-t000", "input": {"array": [2, 1, 8, 2, 4, 2, 8, 1, 2, 2, 1, 1, 1, 2, 1, 3, 3, 3, 2, 3, 3]}, "is_public": true, "problem_id": "arrays-equilibrium_index-f49f4a16245ca727"}], "verification": {"duration_ms": 1.8438330153003335, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 24): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 24)\n (=> (= equilibrium_index 0) (= 0 76))\n (=> (= equilibrium_index 1) (= 4 70))\n (=> (= equilibrium_index 2) (= 10 63))\n (=> (= equilibrium_index 3) (= 17 56))\n (=> (= equilibrium_index 4) (= 24 50))\n (=> (= equilibrium_index 5) (= 30 49))\n (=> (= equilibrium_index 6) (= 31 43))\n (=> (= equilibrium_index 7) (= 37 37))\n (=> (= equilibrium_index 8) (= 43 36))\n (=> (= equilibrium_index 9) (= 44 33))\n (=> (= equilibrium_index 10) (= 47 31))\n (=> (= equilibrium_index 11) (= 49 29))\n (=> ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 24): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "arrays", "id": "arrays-equilibrium_index-52e1c7927db90b2c", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 24, "minItems": 24, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 7, "array": [4, 6, 7, 7, 6, 1, 6, 6, 1, 3, 2, 2, 3, 2, 4, 1, 2, 3, 1, 2, 3, 2, 2, 4], "difficulty": "extreme", "generator": "array_invariants", "length": 24, "max_value": 9, "min_value": 1, "pivot": 7, "seed": 20260913, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 23, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 24. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=24)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 7, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-52e1c7927db90b2c-t000", "input": {"array": [4, 6, 7, 7, 6, 1, 6, 6, 1, 3, 2, 2, 3, 2, 4, 1, 2, 3, 1, 2, 3, 2, 2, 4]}, "is_public": true, "problem_id": "arrays-equilibrium_index-52e1c7927db90b2c"}], "verification": {"duration_ms": 2.198083035182208, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 24): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 24)\n (=> (= split_index 0) (and (>= 1 9) true))\n (=> (= split_index 1) (and (>= 3 9) (< 1 9)))\n (=> (= split_index 2) (and (>= 4 9) (< 1 9) (< 3 9)))\n (=> (= split_index 3) (and (>= 6 9) (< 1 9) (< 3 9) (< 4 9)))\n (=> (= split_index 4) (and (>= 8 9) (< 1 9) (< 3 9) (< 4 9) (< 6 9)))\n (=> (= split_index 5)\n (and (>= 8 9) (< 1 9) (< 3 9) (< 4 9) (< 6 9) (< 8 9)))\n (=> (= split_index 6)\n (and (>= 9 9) (< 1 9) (< 3 9) (< 4 9) (< 6 9) (< 8 9) (< 8 9)))\n (=> (= split_index 7)\n (and (>= 9 9) (< 1 9)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 24): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "arrays", "id": "arrays-monotonic_split-12d78280b716c643", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 24, "minItems": 24, "type": "array"}, "threshold": {"maximum": 9, "minimum": 1, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 6, "array": [1, 3, 4, 6, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9], "difficulty": "extreme", "generator": "array_invariants", "length": 24, "max_value": 9, "min_value": 1, "seed": 20260914, "threshold": 9, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 23, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 24 with\n1 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=24, T=9)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 6, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-12d78280b716c643-t000", "input": {"array": [1, 3, 4, 6, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9], "threshold": 9}, "is_public": true, "problem_id": "arrays-monotonic_split-12d78280b716c643"}], "verification": {"duration_ms": 2.2758329869247973, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 18): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 18)\n (=> (= equilibrium_index 0) (= 0 33))\n (=> (= equilibrium_index 1) (= 6 32))\n (=> (= equilibrium_index 2) (= 7 25))\n (=> (= equilibrium_index 3) (= 14 20))\n (=> (= equilibrium_index 4) (= 19 19))\n (=> (= equilibrium_index 5) (= 20 17))\n (=> (= equilibrium_index 6) (= 22 16))\n (=> (= equilibrium_index 7) (= 23 13))\n (=> (= equilibrium_index 8) (= 26 12))\n (=> (= equilibrium_index 9) (= 27 11))\n (=> (= equilibrium_index 10) (= 28 10))\n (=> (= equilibrium_index 11) (= 29 9))\n (=> (=...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 18): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "arrays", "id": "arrays-equilibrium_index-858735848919498b", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 18, "minItems": 18, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 4, "array": [6, 1, 7, 5, 1, 2, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1], "difficulty": "extreme", "generator": "array_invariants", "length": 18, "max_value": 9, "min_value": 1, "pivot": 4, "seed": 20260915, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 17, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 18. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=18)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-858735848919498b-t000", "input": {"array": [6, 1, 7, 5, 1, 2, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1]}, "is_public": true, "problem_id": "arrays-equilibrium_index-858735848919498b"}], "verification": {"duration_ms": 1.936415967065841, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 21): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 21)\n (=> (= equilibrium_index 0) (= 0 77))\n (=> (= equilibrium_index 1) (= 2 71))\n (=> (= equilibrium_index 2) (= 8 62))\n (=> (= equilibrium_index 3) (= 17 58))\n (=> (= equilibrium_index 4) (= 21 50))\n (=> (= equilibrium_index 5) (= 29 41))\n (=> (= equilibrium_index 6) (= 38 38))\n (=> (= equilibrium_index 7) (= 41 36))\n (=> (= equilibrium_index 8) (= 43 35))\n (=> (= equilibrium_index 9) (= 44 32))\n (=> (= equilibrium_index 10) (= 47 29))\n (=> (= equilibrium_index 11) (= 50 27))\n (=> (...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 21): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "arrays", "id": "arrays-equilibrium_index-a2fe3e3b0a2a0c97", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 21, "minItems": 21, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 6, "array": [2, 6, 9, 4, 8, 9, 3, 2, 1, 3, 3, 2, 1, 5, 3, 5, 2, 3, 3, 3, 2], "difficulty": "extreme", "generator": "array_invariants", "length": 21, "max_value": 9, "min_value": 1, "pivot": 6, "seed": 20260916, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 20, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 21. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=21)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 6, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-a2fe3e3b0a2a0c97-t000", "input": {"array": [2, 6, 9, 4, 8, 9, 3, 2, 1, 3, 3, 2, 1, 5, 3, 5, 2, 3, 3, 3, 2]}, "is_public": true, "problem_id": "arrays-equilibrium_index-a2fe3e3b0a2a0c97"}], "verification": {"duration_ms": 2.082916966173798, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 19): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 19)\n (=> (= equilibrium_index 0) (= 0 108))\n (=> (= equilibrium_index 1) (= 9 104))\n (=> (= equilibrium_index 2) (= 13 103))\n (=> (= equilibrium_index 3) (= 14 100))\n (=> (= equilibrium_index 4) (= 17 94))\n (=> (= equilibrium_index 5) (= 23 85))\n (=> (= equilibrium_index 6) (= 32 79))\n (=> (= equilibrium_index 7) (= 38 70))\n (=> (= equilibrium_index 8) (= 47 61))\n (=> (= equilibrium_index 9) (= 56 56))\n (=> (= equilibrium_index 10) (= 61 47))\n (=> (= equilibrium_index 11) (= 70 42))\n ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 19): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "arrays", "id": "arrays-equilibrium_index-d86846a4332ee097", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 19, "minItems": 19, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 9, "array": [9, 4, 1, 3, 6, 9, 6, 9, 9, 5, 9, 5, 4, 9, 7, 5, 4, 7, 6], "difficulty": "extreme", "generator": "array_invariants", "length": 19, "max_value": 9, "min_value": 1, "pivot": 9, "seed": 20260917, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 18, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 19. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=19)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 9, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-d86846a4332ee097-t000", "input": {"array": [9, 4, 1, 3, 6, 9, 6, 9, 9, 5, 9, 5, 4, 9, 7, 5, 4, 7, 6]}, "is_public": true, "problem_id": "arrays-equilibrium_index-d86846a4332ee097"}], "verification": {"duration_ms": 1.8433330114930868, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 17): 2 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 17)\n (=> (= split_index 0) (and (>= 2 8) true))\n (=> (= split_index 1) (and (>= 2 8) (< 2 8)))\n (=> (= split_index 2) (and (>= 4 8) (< 2 8) (< 2 8)))\n (=> (= split_index 3) (and (>= 5 8) (< 2 8) (< 2 8) (< 4 8)))\n (=> (= split_index 4) (and (>= 6 8) (< 2 8) (< 2 8) (< 4 8) (< 5 8)))\n (=> (= split_index 5)\n (and (>= 8 8) (< 2 8) (< 2 8) (< 4 8) (< 5 8) (< 6 8)))\n (=> (= split_index 6)\n (and (>= 9 8) (< 2 8) (< 2 8) (< 4 8) (< 5 8) (< 6 8) (< 8 8)))\n (=> (= split_index 7)\n (and (>= 9 8) (< 2 8)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 17): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "arrays", "id": "arrays-monotonic_split-4bcce6f584f0cc4f", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 2, "type": "integer"}, "maxItems": 17, "minItems": 17, "type": "array"}, "threshold": {"maximum": 9, "minimum": 2, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 5, "array": [2, 2, 4, 5, 6, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9], "difficulty": "extreme", "generator": "array_invariants", "length": 17, "max_value": 9, "min_value": 2, "seed": 20260918, "threshold": 8, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 16, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 17 with\n2 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=17, T=8)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-4bcce6f584f0cc4f-t000", "input": {"array": [2, 2, 4, 5, 6, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9], "threshold": 8}, "is_public": true, "problem_id": "arrays-monotonic_split-4bcce6f584f0cc4f"}], "verification": {"duration_ms": 1.8770419992506504, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 20): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 20)\n (=> (= equilibrium_index 0) (= 0 58))\n (=> (= equilibrium_index 1) (= 6 51))\n (=> (= equilibrium_index 2) (= 13 42))\n (=> (= equilibrium_index 3) (= 22 37))\n (=> (= equilibrium_index 4) (= 27 36))\n (=> (= equilibrium_index 5) (= 28 28))\n (=> (= equilibrium_index 6) (= 36 23))\n (=> (= equilibrium_index 7) (= 41 20))\n (=> (= equilibrium_index 8) (= 44 19))\n (=> (= equilibrium_index 9) (= 45 18))\n (=> (= equilibrium_index 10) (= 46 17))\n (=> (= equilibrium_index 11) (= 47 15))\n (=> ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 20): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "arrays", "id": "arrays-equilibrium_index-d1cdbb0150804b26", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 20, "minItems": 20, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 5, "array": [6, 7, 9, 5, 1, 8, 5, 3, 1, 1, 1, 2, 3, 1, 2, 2, 4, 1, 1, 1], "difficulty": "extreme", "generator": "array_invariants", "length": 20, "max_value": 9, "min_value": 1, "pivot": 5, "seed": 20260919, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 19, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 20. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=20)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-d1cdbb0150804b26-t000", "input": {"array": [6, 7, 9, 5, 1, 8, 5, 3, 1, 1, 1, 2, 3, 1, 2, 2, 4, 1, 1, 1]}, "is_public": true, "problem_id": "arrays-equilibrium_index-d1cdbb0150804b26"}], "verification": {"duration_ms": 1.9080000347457826, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 17): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 17)\n (=> (= split_index 0) (and (>= 1 5) true))\n (=> (= split_index 1) (and (>= 2 5) (< 1 5)))\n (=> (= split_index 2) (and (>= 3 5) (< 1 5) (< 2 5)))\n (=> (= split_index 3) (and (>= 3 5) (< 1 5) (< 2 5) (< 3 5)))\n (=> (= split_index 4) (and (>= 4 5) (< 1 5) (< 2 5) (< 3 5) (< 3 5)))\n (=> (= split_index 5)\n (and (>= 5 5) (< 1 5) (< 2 5) (< 3 5) (< 3 5) (< 4 5)))\n (=> (= split_index 6)\n (and (>= 5 5) (< 1 5) (< 2 5) (< 3 5) (< 3 5) (< 4 5) (< 5 5)))\n (=> (= split_index 7)\n (and (>= 5 5) (< 1 5)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 17): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "arrays", "id": "arrays-monotonic_split-f12c85dd99d5f318", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 17, "minItems": 17, "type": "array"}, "threshold": {"maximum": 9, "minimum": 1, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 5, "array": [1, 2, 3, 3, 4, 5, 5, 5, 7, 8, 9, 9, 9, 9, 9, 9, 9], "difficulty": "extreme", "generator": "array_invariants", "length": 17, "max_value": 9, "min_value": 1, "seed": 20260920, "threshold": 5, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 16, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 17 with\n1 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=17, T=5)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-f12c85dd99d5f318-t000", "input": {"array": [1, 2, 3, 3, 4, 5, 5, 5, 7, 8, 9, 9, 9, 9, 9, 9, 9], "threshold": 5}, "is_public": true, "problem_id": "arrays-monotonic_split-f12c85dd99d5f318"}], "verification": {"duration_ms": 1.9567079725675285, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 15): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= equilibrium_index 0)\n (< equilibrium_index 15)\n (=> (= equilibrium_index 0) (= 0 30))\n (=> (= equilibrium_index 1) (= 2 29))\n (=> (= equilibrium_index 2) (= 3 26))\n (=> (= equilibrium_index 3) (= 6 19))\n (=> (= equilibrium_index 4) (= 13 13))\n (=> (= equilibrium_index 5) (= 19 12))\n (=> (= equilibrium_index 6) (= 20 11))\n (=> (= equilibrium_index 7) (= 21 10))\n (=> (= equilibrium_index 8) (= 22 8))\n (=> (= equilibrium_index 9) (= 24 7))\n (=> (= equilibrium_index 10) (= 25 4))\n (=> (= equilibrium_index 11) (= 28 3))\n (=> (= equ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 15): equilibrium(i) & equilibrium(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "arrays", "id": "arrays-equilibrium_index-79342dd7ca953ba8", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 15, "minItems": 15, "type": "array"}}, "required": ["array"], "type": "object"}, "metadata": {"answer": 4, "array": [2, 1, 3, 7, 6, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1], "difficulty": "extreme", "generator": "array_invariants", "length": 15, "max_value": 9, "min_value": 1, "pivot": 4, "seed": 20260921, "variant": "equilibrium_index"}, "output_schema": {"description": "the unique equilibrium index", "maximum": 14, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\ntotal = sum(array)\nprefix = 0\nanswer = -1\nfor index, value in enumerate(array):\n suffix = total - prefix - value\n if prefix == suffix:\n answer = index\n break\n prefix += value\nprint(answer)\n", "statement": "You are given a 0-indexed integer array A of length 15. Every element is a\nstrictly positive integer with 1 <= A[i] <= 9.\n\nAn index i is an equilibrium index when the elements strictly before i sum to\nthe same value as the elements strictly after i:\n\n A[0] + ... + A[i-1] == A[i+1] + ... + A[n-1]\n\nThe provided array contains exactly one equilibrium index; existence and\nuniqueness are formally proven for this instance with the Z3 SMT solver.\nPrint that index.\n", "title": "Unique Equilibrium Index (n=15)", "variant": "equilibrium_index"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-equilibrium_index-79342dd7ca953ba8-t000", "input": {"array": [2, 1, 3, 7, 6, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1]}, "is_public": true, "problem_id": "arrays-equilibrium_index-79342dd7ca953ba8"}], "verification": {"duration_ms": 2.0549160544760525, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 16): 2 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 16)\n (=> (= split_index 0) (and (>= 2 8) true))\n (=> (= split_index 1) (and (>= 2 8) (< 2 8)))\n (=> (= split_index 2) (and (>= 4 8) (< 2 8) (< 2 8)))\n (=> (= split_index 3) (and (>= 4 8) (< 2 8) (< 2 8) (< 4 8)))\n (=> (= split_index 4) (and (>= 6 8) (< 2 8) (< 2 8) (< 4 8) (< 4 8)))\n (=> (= split_index 5)\n (and (>= 8 8) (< 2 8) (< 2 8) (< 4 8) (< 4 8) (< 6 8)))\n (=> (= split_index 6)\n (and (>= 8 8) (< 2 8) (< 2 8) (< 4 8) (< 4 8) (< 6 8) (< 8 8)))\n (=> (= split_index 7)\n (and (>= 8 8) (< 2 8)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 16): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "arrays", "id": "arrays-monotonic_split-a3471f77dd6a4794", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 2, "type": "integer"}, "maxItems": 16, "minItems": 16, "type": "array"}, "threshold": {"maximum": 9, "minimum": 2, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 5, "array": [2, 2, 4, 4, 6, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9], "difficulty": "extreme", "generator": "array_invariants", "length": 16, "max_value": 9, "min_value": 2, "seed": 20260922, "threshold": 8, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 15, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 16 with\n2 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=16, T=8)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-a3471f77dd6a4794-t000", "input": {"array": [2, 2, 4, 4, 6, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9], "threshold": 8}, "is_public": true, "problem_id": "arrays-monotonic_split-a3471f77dd6a4794"}], "verification": {"duration_ms": 1.8199170008301735, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Every array element lies inside the declared inclusive range.", "expression": "forall i in [0, 18): 1 <= A[i] <= 9", "name": "value_bounds"}, {"description": "The declarative encoding of the task over the concrete array; Z3 reports this system satisfiable, so an answer exists.", "expression": "(and (>= split_index 0)\n (< split_index 18)\n (=> (= split_index 0) (and (>= 1 9) true))\n (=> (= split_index 1) (and (>= 2 9) (< 1 9)))\n (=> (= split_index 2) (and (>= 2 9) (< 1 9) (< 2 9)))\n (=> (= split_index 3) (and (>= 3 9) (< 1 9) (< 2 9) (< 2 9)))\n (=> (= split_index 4) (and (>= 4 9) (< 1 9) (< 2 9) (< 2 9) (< 3 9)))\n (=> (= split_index 5)\n (and (>= 4 9) (< 1 9) (< 2 9) (< 2 9) (< 3 9) (< 4 9)))\n (=> (= split_index 6)\n (and (>= 4 9) (< 1 9) (< 2 9) (< 2 9) (< 3 9) (< 4 9) (< 4 9)))\n (=> (= split_index 7)\n (and (>= 6 9) (< 1 9)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "forall i, j in [0, 18): first_ge(i) & first_ge(j) -> i == j", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "arrays", "id": "arrays-monotonic_split-df9dba154b9caec2", "input_schema": {"additionalProperties": false, "properties": {"array": {"items": {"maximum": 9, "minimum": 1, "type": "integer"}, "maxItems": 18, "minItems": 18, "type": "array"}, "threshold": {"maximum": 9, "minimum": 1, "type": "integer"}}, "required": ["array", "threshold"], "type": "object"}, "metadata": {"answer": 12, "array": [1, 2, 2, 3, 4, 4, 4, 6, 6, 8, 8, 8, 9, 9, 9, 9, 9, 9], "difficulty": "extreme", "generator": "array_invariants", "length": 18, "max_value": 9, "min_value": 1, "seed": 20260923, "threshold": 9, "variant": "monotonic_split"}, "output_schema": {"description": "the smallest index whose value reaches the threshold", "maximum": 17, "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\narray = data[\"array\"]\nthreshold = data[\"threshold\"]\nanswer = -1\nfor index, value in enumerate(array):\n if value >= threshold:\n answer = index\n break\nprint(answer)\n", "statement": "You are given a 0-indexed non-decreasing integer array A of length 18 with\n1 <= A[i] <= 9, together with an integer threshold T drawn from the\nelements of A itself.\n\nPrint the smallest index k such that A[k] >= T. Because the predicate is\ndefined by minimality there is at most one such index, and because T occurs\nin A at least one exists; both facts are formally proven for this instance\nwith the Z3 SMT solver.\n", "title": "Bounded Monotonic Split (n=18, T=9)", "variant": "monotonic_split"}, "test_cases": [{"expected_output": 12, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "arrays-monotonic_split-df9dba154b9caec2-t000", "input": {"array": [1, 2, 2, 3, 4, 4, 4, 6, 6, 8, 8, 8, 9, 9, 9, 9, 9, 9], "threshold": 9}, "is_public": true, "problem_id": "arrays-monotonic_split-df9dba154b9caec2"}], "verification": {"duration_ms": 1.1767919640988111, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_1 member_2)\n (=> member_1 (or member_0))\n (=> member_2 (or member_1))\n (= reachable_count\n (+ (ite member_0 1 0) (ite member_1 1 0) (ite member_2 1 0))))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "graphs", "id": "graphs-dag_reachability-41f06cdaba76efc3", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1], [2], []], "answer": 3, "difficulty": "easy", "generator": "graph_invariants", "num_vertices": 3, "seed": 20260924, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 3 vertices labelled 0 to 2,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=3)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 3, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-41f06cdaba76efc3-t000", "input": {"adjacency": [[1], [2], []], "num_vertices": 3, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-41f06cdaba76efc3"}], "verification": {"duration_ms": 0.9221669752150774, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and (= dist_1 (+ dist_0 2)) (<= dist_1 (+ 2 dist_0)))))\n (a!2 (or (and (= dist_2 (+ dist_0 4)) (<= dist_2 (+ 4 dist_0))))))\n (and (= dist_0 0) a!1 a!2))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "graphs", "id": "graphs-dag_shortest_path-cd442f68ac61be8c", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 4, "difficulty": "easy", "edges": [[0, 1], [0, 2]], "generator": "graph_invariants", "num_vertices": 3, "seed": 20260925, "source": 0, "target": 2, "variant": "dag_shortest_path", "weighted_edges": [[0, 1, 2], [0, 2, 4]]}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\nsource = data[\"source\"]\ntarget = data[\"target\"]\n\ninfinity = float(\"inf\")\ndist = [infinity] * n\ndist[source] = 0\nfor u, v, w in edges:\n if dist[u] != infinity and dist[u] + w < dist[v]:\n dist[v] = dist[u] + w\nprint(dist[target])\n", "statement": "You are given a directed acyclic graph with 3 vertices labelled 0 to\n2. Every edge [u, v, w] goes from a lower to a higher label and has\ninteger weight 1 <= w <= 9; edges are listed in ascending (u, v) order.\n\nPrint the minimum possible total weight of a path from source s = 0 to\ntarget t = 2. The target is guaranteed reachable, the optimum is\nunique, and both facts have been formally proven for this instance with the\nZ3 SMT solver.\n", "title": "DAG Shortest Path (n=3, weight 4)", "variant": "dag_shortest_path"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_shortest_path-cd442f68ac61be8c-t000", "input": {"edges": [[0, 1, 2], [0, 2, 4]], "num_vertices": 3, "source": 0, "target": 2}, "is_public": true, "problem_id": "graphs-dag_shortest_path-cd442f68ac61be8c"}], "verification": {"duration_ms": 0.7897920440882444, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_0 member_2)\n (=> member_1 (or member_0))\n (=> member_2 (or member_0))\n (= reachable_count\n (+ (ite member_0 1 0) (ite member_1 1 0) (ite member_2 1 0))))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "graphs", "id": "graphs-dag_reachability-ec15e8cffe9d9fb5", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1, 2], [], []], "answer": 3, "difficulty": "easy", "generator": "graph_invariants", "num_vertices": 3, "seed": 20260926, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 3 vertices labelled 0 to 2,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=3)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 3, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-ec15e8cffe9d9fb5-t000", "input": {"adjacency": [[1, 2], [], []], "num_vertices": 3, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-ec15e8cffe9d9fb5"}], "verification": {"duration_ms": 0.7410839898511767, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_0 member_2)\n (=> member_1 member_3)\n (=> member_1 (or member_0))\n (=> member_2 (or member_0))\n (=> member_3 (or member_1))\n (= reachable_count\n (+ (ite member_0 1 0)\n (ite member_1 1 0)\n (ite member_2 1 0)\n (ite member_3 1 0))))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "graphs", "id": "graphs-dag_reachability-9b2017b5aaebf0eb", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1, 2], [3], [], []], "answer": 4, "difficulty": "easy", "generator": "graph_invariants", "num_vertices": 4, "seed": 20260927, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 4 vertices labelled 0 to 3,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=4)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-9b2017b5aaebf0eb-t000", "input": {"adjacency": [[1, 2], [3], [], []], "num_vertices": 4, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-9b2017b5aaebf0eb"}], "verification": {"duration_ms": 0.7517929770983756, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (=> (and (> pos_1 0) (and (< pos_0 0))) (< ord_0 1)))\n (a!2 (=> (and (> pos_2 0) (and (< pos_0 0))) (< ord_0 2)))\n (a!3 (=> (and (> pos_3 0) (and (< pos_0 0))) (< ord_0 3)))\n (a!4 (=> (and (> pos_1 1) (and (< pos_0 1))) (< ord_1 1)))\n (a!5 (=> (and (> pos_2 1) (and (< pos_0 1))) (< ord_1 2)))\n (a!6 (=> (and (> pos_3 1) (and (< pos_0 1))) (< ord_1 3)))\n (a!7 (=> (and (> pos_1 2) (and (< pos_0 2))) (< ord_2 1)))\n (a!8 (=> (and (> pos_2 2) (and (< pos_0 2))) (< ord_2 2)))\n (a!9 (=> (and (> pos_3 2) (and (< pos_0 2))) (< ord_2 3)))\n (a!10 ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "graphs", "id": "graphs-dag_topological_position-ddfe1598cd12be23", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 1, "difficulty": "easy", "edges": [[0, 1], [0, 2], [0, 3]], "generator": "graph_invariants", "num_vertices": 4, "seed": 20260928, "target_vertex": 1, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 4 vertices labelled 0 to\n3 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 1 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=4, vertex 1)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-ddfe1598cd12be23-t000", "input": {"edges": [[0, 1], [0, 2], [0, 3]], "num_vertices": 4, "target_vertex": 1}, "is_public": true, "problem_id": "graphs-dag_topological_position-ddfe1598cd12be23"}], "verification": {"duration_ms": 2.7644580113701522, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and (= dist_1 (+ dist_0 8)) (<= dist_1 (+ 8 dist_0)))))\n (a!2 (or (and (= dist_2 (+ dist_0 4)) (<= dist_2 (+ 4 dist_0)))))\n (a!3 (or (and (= dist_3 (+ dist_1 1)) (<= dist_3 (+ 1 dist_1))))))\n (and (= dist_0 0) a!1 a!2 a!3))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "graphs", "id": "graphs-dag_shortest_path-0648f726af1e0ed4", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 9, "difficulty": "easy", "edges": [[0, 1], [0, 2], [1, 3]], "generator": "graph_invariants", "num_vertices": 4, "seed": 20260929, "source": 0, "target": 3, "variant": "dag_shortest_path", "weighted_edges": [[0, 1, 8], [0, 2, 4], [1, 3, 1]]}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\nsource = data[\"source\"]\ntarget = data[\"target\"]\n\ninfinity = float(\"inf\")\ndist = [infinity] * n\ndist[source] = 0\nfor u, v, w in edges:\n if dist[u] != infinity and dist[u] + w < dist[v]:\n dist[v] = dist[u] + w\nprint(dist[target])\n", "statement": "You are given a directed acyclic graph with 4 vertices labelled 0 to\n3. Every edge [u, v, w] goes from a lower to a higher label and has\ninteger weight 1 <= w <= 9; edges are listed in ascending (u, v) order.\n\nPrint the minimum possible total weight of a path from source s = 0 to\ntarget t = 3. The target is guaranteed reachable, the optimum is\nunique, and both facts have been formally proven for this instance with the\nZ3 SMT solver.\n", "title": "DAG Shortest Path (n=4, weight 9)", "variant": "dag_shortest_path"}, "test_cases": [{"expected_output": 9, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_shortest_path-0648f726af1e0ed4-t000", "input": {"edges": [[0, 1, 8], [0, 2, 4], [1, 3, 1]], "num_vertices": 4, "source": 0, "target": 3}, "is_public": true, "problem_id": "graphs-dag_shortest_path-0648f726af1e0ed4"}], "verification": {"duration_ms": 0.8192509994842112, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and (= dist_1 (+ dist_0 1)) (<= dist_1 (+ 1 dist_0)))))\n (a!2 (or (and (= dist_2 (+ dist_1 8)) (<= dist_2 (+ 8 dist_1)))))\n (a!3 (or (and (= dist_3 (+ dist_2 7)) (<= dist_3 (+ 7 dist_2))))))\n (and (= dist_0 0) a!1 a!2 a!3))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "graphs", "id": "graphs-dag_shortest_path-27da35b6bebc0f87", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 16, "difficulty": "easy", "edges": [[0, 1], [1, 2], [2, 3]], "generator": "graph_invariants", "num_vertices": 4, "seed": 20260930, "source": 0, "target": 3, "variant": "dag_shortest_path", "weighted_edges": [[0, 1, 1], [1, 2, 8], [2, 3, 7]]}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\nsource = data[\"source\"]\ntarget = data[\"target\"]\n\ninfinity = float(\"inf\")\ndist = [infinity] * n\ndist[source] = 0\nfor u, v, w in edges:\n if dist[u] != infinity and dist[u] + w < dist[v]:\n dist[v] = dist[u] + w\nprint(dist[target])\n", "statement": "You are given a directed acyclic graph with 4 vertices labelled 0 to\n3. Every edge [u, v, w] goes from a lower to a higher label and has\ninteger weight 1 <= w <= 9; edges are listed in ascending (u, v) order.\n\nPrint the minimum possible total weight of a path from source s = 0 to\ntarget t = 3. The target is guaranteed reachable, the optimum is\nunique, and both facts have been formally proven for this instance with the\nZ3 SMT solver.\n", "title": "DAG Shortest Path (n=4, weight 16)", "variant": "dag_shortest_path"}, "test_cases": [{"expected_output": 16, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_shortest_path-27da35b6bebc0f87-t000", "input": {"edges": [[0, 1, 1], [1, 2, 8], [2, 3, 7]], "num_vertices": 4, "source": 0, "target": 3}, "is_public": true, "problem_id": "graphs-dag_shortest_path-27da35b6bebc0f87"}], "verification": {"duration_ms": 0.748125952668488, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_1 member_2)\n (=> member_1 (or member_0))\n (=> member_2 (or member_1))\n (= reachable_count\n (+ (ite member_0 1 0) (ite member_1 1 0) (ite member_2 1 0))))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "graphs", "id": "graphs-dag_reachability-ca6aeeaff16bbd97", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1], [2], []], "answer": 3, "difficulty": "easy", "generator": "graph_invariants", "num_vertices": 3, "seed": 20260931, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 3 vertices labelled 0 to 2,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=3)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 3, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-ca6aeeaff16bbd97-t000", "input": {"adjacency": [[1], [2], []], "num_vertices": 3, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-ca6aeeaff16bbd97"}], "verification": {"duration_ms": 0.7346670608967543, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and (= dist_1 (+ dist_0 7)) (<= dist_1 (+ 7 dist_0)))))\n (a!2 (or (and (= dist_2 (+ dist_1 2)) (<= dist_2 (+ 2 dist_1)))))\n (a!3 (or (and (= dist_3 (+ dist_2 1)) (<= dist_3 (+ 1 dist_2))))))\n (and (= dist_0 0) a!1 a!2 a!3))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "graphs", "id": "graphs-dag_shortest_path-ae666465328ad6b5", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 10, "difficulty": "easy", "edges": [[0, 1], [1, 2], [2, 3]], "generator": "graph_invariants", "num_vertices": 4, "seed": 20260932, "source": 0, "target": 3, "variant": "dag_shortest_path", "weighted_edges": [[0, 1, 7], [1, 2, 2], [2, 3, 1]]}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\nsource = data[\"source\"]\ntarget = data[\"target\"]\n\ninfinity = float(\"inf\")\ndist = [infinity] * n\ndist[source] = 0\nfor u, v, w in edges:\n if dist[u] != infinity and dist[u] + w < dist[v]:\n dist[v] = dist[u] + w\nprint(dist[target])\n", "statement": "You are given a directed acyclic graph with 4 vertices labelled 0 to\n3. Every edge [u, v, w] goes from a lower to a higher label and has\ninteger weight 1 <= w <= 9; edges are listed in ascending (u, v) order.\n\nPrint the minimum possible total weight of a path from source s = 0 to\ntarget t = 3. The target is guaranteed reachable, the optimum is\nunique, and both facts have been formally proven for this instance with the\nZ3 SMT solver.\n", "title": "DAG Shortest Path (n=4, weight 10)", "variant": "dag_shortest_path"}, "test_cases": [{"expected_output": 10, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_shortest_path-ae666465328ad6b5-t000", "input": {"edges": [[0, 1, 7], [1, 2, 2], [2, 3, 1]], "num_vertices": 4, "source": 0, "target": 3}, "is_public": true, "problem_id": "graphs-dag_shortest_path-ae666465328ad6b5"}], "verification": {"duration_ms": 0.8384570246562362, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_1 member_2)\n (=> member_1 member_3)\n (=> member_1 (or member_0))\n (=> member_2 (or member_1))\n (=> member_3 (or member_1))\n (= reachable_count\n (+ (ite member_0 1 0)\n (ite member_1 1 0)\n (ite member_2 1 0)\n (ite member_3 1 0))))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "graphs", "id": "graphs-dag_reachability-94a39ca9bcf5f7d0", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1], [2, 3], [], []], "answer": 4, "difficulty": "easy", "generator": "graph_invariants", "num_vertices": 4, "seed": 20260933, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 4 vertices labelled 0 to 3,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=4)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-94a39ca9bcf5f7d0-t000", "input": {"adjacency": [[1], [2, 3], [], []], "num_vertices": 4, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-94a39ca9bcf5f7d0"}], "verification": {"duration_ms": 0.7401250186376274, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_0 member_2)\n (=> member_1 (or member_0))\n (=> member_2 (or member_0))\n (= reachable_count\n (+ (ite member_0 1 0) (ite member_1 1 0) (ite member_2 1 0))))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "graphs", "id": "graphs-dag_reachability-54f09b6250c8117b", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1, 2], [], []], "answer": 3, "difficulty": "easy", "generator": "graph_invariants", "num_vertices": 3, "seed": 20260934, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 3 vertices labelled 0 to 2,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=3)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 3, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-54f09b6250c8117b-t000", "input": {"adjacency": [[1, 2], [], []], "num_vertices": 3, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-54f09b6250c8117b"}], "verification": {"duration_ms": 0.7277089753188193, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and (= dist_1 (+ dist_0 5)) (<= dist_1 (+ 5 dist_0)))))\n (a!2 (or (and (= dist_2 (+ dist_1 6)) (<= dist_2 (+ 6 dist_1))))))\n (and (= dist_0 0) a!1 a!2))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "graphs", "id": "graphs-dag_shortest_path-f5c7fba38957f658", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 11, "difficulty": "easy", "edges": [[0, 1], [1, 2]], "generator": "graph_invariants", "num_vertices": 3, "seed": 20260935, "source": 0, "target": 2, "variant": "dag_shortest_path", "weighted_edges": [[0, 1, 5], [1, 2, 6]]}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\nsource = data[\"source\"]\ntarget = data[\"target\"]\n\ninfinity = float(\"inf\")\ndist = [infinity] * n\ndist[source] = 0\nfor u, v, w in edges:\n if dist[u] != infinity and dist[u] + w < dist[v]:\n dist[v] = dist[u] + w\nprint(dist[target])\n", "statement": "You are given a directed acyclic graph with 3 vertices labelled 0 to\n2. Every edge [u, v, w] goes from a lower to a higher label and has\ninteger weight 1 <= w <= 9; edges are listed in ascending (u, v) order.\n\nPrint the minimum possible total weight of a path from source s = 0 to\ntarget t = 2. The target is guaranteed reachable, the optimum is\nunique, and both facts have been formally proven for this instance with the\nZ3 SMT solver.\n", "title": "DAG Shortest Path (n=3, weight 11)", "variant": "dag_shortest_path"}, "test_cases": [{"expected_output": 11, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_shortest_path-f5c7fba38957f658-t000", "input": {"edges": [[0, 1, 5], [1, 2, 6]], "num_vertices": 3, "source": 0, "target": 2}, "is_public": true, "problem_id": "graphs-dag_shortest_path-f5c7fba38957f658"}], "verification": {"duration_ms": 0.7289160275831819, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (=> (and (> pos_1 0) (and (< pos_0 0))) (< ord_0 1)))\n (a!2 (=> (and (> pos_2 0) (and (< pos_0 0))) (< ord_0 2)))\n (a!3 (=> (and (> pos_1 1) (and (< pos_0 1))) (< ord_1 1)))\n (a!4 (=> (and (> pos_2 1) (and (< pos_0 1))) (< ord_1 2)))\n (a!5 (=> (and (> pos_1 2) (and (< pos_0 2))) (< ord_2 1)))\n (a!6 (=> (and (> pos_2 2) (and (< pos_0 2))) (< ord_2 2))))\n (and (distinct ord_0 ord_1 ord_2)\n (>= ord_0 0)\n (< ord_0 3)\n (>= pos_0 0)\n (< pos_0 3)\n (>= ord_1 0)\n (< ord_1 3)\n (>= pos_1 0)\n (< pos_1 3)\n (>= ord...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "graphs", "id": "graphs-dag_topological_position-0f34dcb977d61896", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 2, "difficulty": "easy", "edges": [[0, 1], [0, 2]], "generator": "graph_invariants", "num_vertices": 3, "seed": 20260936, "target_vertex": 2, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 3 vertices labelled 0 to\n2 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 2 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=3, vertex 2)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-0f34dcb977d61896-t000", "input": {"edges": [[0, 1], [0, 2]], "num_vertices": 3, "target_vertex": 2}, "is_public": true, "problem_id": "graphs-dag_topological_position-0f34dcb977d61896"}], "verification": {"duration_ms": 1.9817919819615781, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (=> (and (> pos_1 0) (and (< pos_0 0))) (< ord_0 1)))\n (a!2 (=> (and (> pos_3 0) (and (< pos_0 0))) (< ord_0 3)))\n (a!3 (=> (and (> pos_1 1) (and (< pos_0 1))) (< ord_1 1)))\n (a!4 (=> (and (> pos_3 1) (and (< pos_0 1))) (< ord_1 3)))\n (a!5 (=> (and (> pos_1 2) (and (< pos_0 2))) (< ord_2 1)))\n (a!6 (=> (and (> pos_3 2) (and (< pos_0 2))) (< ord_2 3)))\n (a!7 (=> (and (> pos_1 3) (and (< pos_0 3))) (< ord_3 1)))\n (a!8 (=> (and (> pos_3 3) (and (< pos_0 3))) (< ord_3 3))))\n (and (distinct ord_0 ord_1 ord_2 ord_3)\n (>= ord_0 0)\n (< ord_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "graphs", "id": "graphs-dag_topological_position-f5df026b6031eb5b", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 1, "difficulty": "easy", "edges": [[0, 1], [0, 3], [1, 2]], "generator": "graph_invariants", "num_vertices": 4, "seed": 20260937, "target_vertex": 1, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 4 vertices labelled 0 to\n3 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 1 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=4, vertex 1)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-f5df026b6031eb5b-t000", "input": {"edges": [[0, 1], [0, 3], [1, 2]], "num_vertices": 4, "target_vertex": 1}, "is_public": true, "problem_id": "graphs-dag_topological_position-f5df026b6031eb5b"}], "verification": {"duration_ms": 2.4316670023836195, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (=> (and (> pos_1 0) (and (< pos_0 0))) (< ord_0 1)))\n (a!2 (=> (and (> pos_2 0) (and (< pos_0 0))) (< ord_0 2)))\n (a!3 (=> (and (> pos_1 1) (and (< pos_0 1))) (< ord_1 1)))\n (a!4 (=> (and (> pos_2 1) (and (< pos_0 1))) (< ord_1 2)))\n (a!5 (=> (and (> pos_1 2) (and (< pos_0 2))) (< ord_2 1)))\n (a!6 (=> (and (> pos_2 2) (and (< pos_0 2))) (< ord_2 2)))\n (a!7 (=> (and (> pos_1 3) (and (< pos_0 3))) (< ord_3 1)))\n (a!8 (=> (and (> pos_2 3) (and (< pos_0 3))) (< ord_3 2))))\n (and (distinct ord_0 ord_1 ord_2 ord_3)\n (>= ord_0 0)\n (< ord_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "graphs", "id": "graphs-dag_topological_position-8d03d3cf41b021f7", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 2, "difficulty": "easy", "edges": [[0, 1], [0, 2], [2, 3]], "generator": "graph_invariants", "num_vertices": 4, "seed": 20260938, "target_vertex": 2, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 4 vertices labelled 0 to\n3 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 2 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=4, vertex 2)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-8d03d3cf41b021f7-t000", "input": {"edges": [[0, 1], [0, 2], [2, 3]], "num_vertices": 4, "target_vertex": 2}, "is_public": true, "problem_id": "graphs-dag_topological_position-8d03d3cf41b021f7"}], "verification": {"duration_ms": 2.6662919553928077, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (=> (and (> pos_1 0) (and (< pos_0 0))) (< ord_0 1)))\n (a!2 (=> (and (> pos_2 0) (and (< pos_0 0))) (< ord_0 2)))\n (a!3 (=> (and (> pos_1 1) (and (< pos_0 1))) (< ord_1 1)))\n (a!4 (=> (and (> pos_2 1) (and (< pos_0 1))) (< ord_1 2)))\n (a!5 (=> (and (> pos_1 2) (and (< pos_0 2))) (< ord_2 1)))\n (a!6 (=> (and (> pos_2 2) (and (< pos_0 2))) (< ord_2 2)))\n (a!7 (=> (and (> pos_1 3) (and (< pos_0 3))) (< ord_3 1)))\n (a!8 (=> (and (> pos_2 3) (and (< pos_0 3))) (< ord_3 2))))\n (and (distinct ord_0 ord_1 ord_2 ord_3)\n (>= ord_0 0)\n (< ord_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "graphs", "id": "graphs-dag_topological_position-ed4529d3bef5463b", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 1, "difficulty": "easy", "edges": [[0, 1], [0, 2], [2, 3]], "generator": "graph_invariants", "num_vertices": 4, "seed": 20260939, "target_vertex": 1, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 4 vertices labelled 0 to\n3 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 1 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=4, vertex 1)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-ed4529d3bef5463b-t000", "input": {"edges": [[0, 1], [0, 2], [2, 3]], "num_vertices": 4, "target_vertex": 1}, "is_public": true, "problem_id": "graphs-dag_topological_position-ed4529d3bef5463b"}], "verification": {"duration_ms": 2.387000946328044, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (=> (and (> pos_1 0) (and (< pos_0 0))) (< ord_0 1)))\n (a!2 (=> (and (> pos_2 0) (and (< pos_0 0))) (< ord_0 2)))\n (a!3 (=> (and (> pos_3 0) (and (< pos_0 0))) (< ord_0 3)))\n (a!4 (=> (and (> pos_1 1) (and (< pos_0 1))) (< ord_1 1)))\n (a!5 (=> (and (> pos_2 1) (and (< pos_0 1))) (< ord_1 2)))\n (a!6 (=> (and (> pos_3 1) (and (< pos_0 1))) (< ord_1 3)))\n (a!7 (=> (and (> pos_1 2) (and (< pos_0 2))) (< ord_2 1)))\n (a!8 (=> (and (> pos_2 2) (and (< pos_0 2))) (< ord_2 2)))\n (a!9 (=> (and (> pos_3 2) (and (< pos_0 2))) (< ord_2 3)))\n (a!10 ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "graphs", "id": "graphs-dag_topological_position-90960c0d51fb67e2", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 3, "difficulty": "easy", "edges": [[0, 1], [0, 2], [0, 3]], "generator": "graph_invariants", "num_vertices": 4, "seed": 20260940, "target_vertex": 3, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 4 vertices labelled 0 to\n3 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 3 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=4, vertex 3)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 3, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-90960c0d51fb67e2-t000", "input": {"edges": [[0, 1], [0, 2], [0, 3]], "num_vertices": 4, "target_vertex": 3}, "is_public": true, "problem_id": "graphs-dag_topological_position-90960c0d51fb67e2"}], "verification": {"duration_ms": 2.3322919732891023, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (=> (and (> pos_1 0) (and (< pos_0 0))) (< ord_0 1)))\n (a!2 (=> (and (> pos_2 0) (and (< pos_0 0))) (< ord_0 2)))\n (a!3 (=> (and (> pos_1 1) (and (< pos_0 1))) (< ord_1 1)))\n (a!4 (=> (and (> pos_2 1) (and (< pos_0 1))) (< ord_1 2)))\n (a!5 (=> (and (> pos_1 2) (and (< pos_0 2))) (< ord_2 1)))\n (a!6 (=> (and (> pos_2 2) (and (< pos_0 2))) (< ord_2 2)))\n (a!7 (=> (and (> pos_1 3) (and (< pos_0 3))) (< ord_3 1)))\n (a!8 (=> (and (> pos_2 3) (and (< pos_0 3))) (< ord_3 2))))\n (and (distinct ord_0 ord_1 ord_2 ord_3)\n (>= ord_0 0)\n (< ord_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "graphs", "id": "graphs-dag_topological_position-82ca4564ff87ab4c", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 2, "difficulty": "easy", "edges": [[0, 1], [0, 2], [1, 3]], "generator": "graph_invariants", "num_vertices": 4, "seed": 20260941, "target_vertex": 2, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 4 vertices labelled 0 to\n3 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 2 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=4, vertex 2)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-82ca4564ff87ab4c-t000", "input": {"edges": [[0, 1], [0, 2], [1, 3]], "num_vertices": 4, "target_vertex": 2}, "is_public": true, "problem_id": "graphs-dag_topological_position-82ca4564ff87ab4c"}], "verification": {"duration_ms": 2.3829169804230332, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (=> (and (> pos_1 0) (and (< pos_0 0))) (< ord_0 1)))\n (a!2 (=> (and (> pos_2 0) (and (< pos_0 0))) (< ord_0 2)))\n (a!3 (=> (and (> pos_1 1) (and (< pos_0 1))) (< ord_1 1)))\n (a!4 (=> (and (> pos_2 1) (and (< pos_0 1))) (< ord_1 2)))\n (a!5 (=> (and (> pos_1 2) (and (< pos_0 2))) (< ord_2 1)))\n (a!6 (=> (and (> pos_2 2) (and (< pos_0 2))) (< ord_2 2))))\n (and (distinct ord_0 ord_1 ord_2)\n (>= ord_0 0)\n (< ord_0 3)\n (>= pos_0 0)\n (< pos_0 3)\n (>= ord_1 0)\n (< ord_1 3)\n (>= pos_1 0)\n (< pos_1 3)\n (>= ord...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "graphs", "id": "graphs-dag_topological_position-de39aa8866092fc6", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 2, "difficulty": "easy", "edges": [[0, 1], [0, 2]], "generator": "graph_invariants", "num_vertices": 3, "seed": 20260942, "target_vertex": 2, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 3 vertices labelled 0 to\n2 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 2 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=3, vertex 2)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-de39aa8866092fc6-t000", "input": {"edges": [[0, 1], [0, 2]], "num_vertices": 3, "target_vertex": 2}, "is_public": true, "problem_id": "graphs-dag_topological_position-de39aa8866092fc6"}], "verification": {"duration_ms": 1.853875000961125, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and (= dist_1 (+ dist_0 6)) (<= dist_1 (+ 6 dist_0)))))\n (a!2 (and (<= dist_2 (+ 4 dist_0)) (<= dist_2 (+ 2 dist_1)))))\n(let ((a!3 (or (and (= dist_2 (+ dist_0 4)) a!2)\n (and (= dist_2 (+ dist_1 2)) a!2))))\n (and (= dist_0 0) a!1 a!3)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "graphs", "id": "graphs-dag_shortest_path-fa1ca7718a1abd90", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 4, "difficulty": "easy", "edges": [[0, 1], [0, 2], [1, 2]], "generator": "graph_invariants", "num_vertices": 3, "seed": 20260943, "source": 0, "target": 2, "variant": "dag_shortest_path", "weighted_edges": [[0, 1, 6], [0, 2, 4], [1, 2, 2]]}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\nsource = data[\"source\"]\ntarget = data[\"target\"]\n\ninfinity = float(\"inf\")\ndist = [infinity] * n\ndist[source] = 0\nfor u, v, w in edges:\n if dist[u] != infinity and dist[u] + w < dist[v]:\n dist[v] = dist[u] + w\nprint(dist[target])\n", "statement": "You are given a directed acyclic graph with 3 vertices labelled 0 to\n2. Every edge [u, v, w] goes from a lower to a higher label and has\ninteger weight 1 <= w <= 9; edges are listed in ascending (u, v) order.\n\nPrint the minimum possible total weight of a path from source s = 0 to\ntarget t = 2. The target is guaranteed reachable, the optimum is\nunique, and both facts have been formally proven for this instance with the\nZ3 SMT solver.\n", "title": "DAG Shortest Path (n=3, weight 4)", "variant": "dag_shortest_path"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_shortest_path-fa1ca7718a1abd90-t000", "input": {"edges": [[0, 1, 6], [0, 2, 4], [1, 2, 2]], "num_vertices": 3, "source": 0, "target": 2}, "is_public": true, "problem_id": "graphs-dag_shortest_path-fa1ca7718a1abd90"}], "verification": {"duration_ms": 0.8376669720746577, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (=> (and (> pos_1 0) (and (< pos_0 0))) (< ord_0 1)))\n (a!2 (=> (and (> pos_2 0) (and (< pos_0 0))) (< ord_0 2)))\n (a!3 (=> (and (> pos_1 1) (and (< pos_0 1))) (< ord_1 1)))\n (a!4 (=> (and (> pos_2 1) (and (< pos_0 1))) (< ord_1 2)))\n (a!5 (=> (and (> pos_1 2) (and (< pos_0 2))) (< ord_2 1)))\n (a!6 (=> (and (> pos_2 2) (and (< pos_0 2))) (< ord_2 2))))\n (and (distinct ord_0 ord_1 ord_2)\n (>= ord_0 0)\n (< ord_0 3)\n (>= pos_0 0)\n (< pos_0 3)\n (>= ord_1 0)\n (< ord_1 3)\n (>= pos_1 0)\n (< pos_1 3)\n (>= ord...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "graphs", "id": "graphs-dag_topological_position-da530810614b8875", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 2, "difficulty": "easy", "edges": [[0, 1], [0, 2]], "generator": "graph_invariants", "num_vertices": 3, "seed": 20260944, "target_vertex": 2, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 3 vertices labelled 0 to\n2 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 2 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=3, vertex 2)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-da530810614b8875-t000", "input": {"edges": [[0, 1], [0, 2]], "num_vertices": 3, "target_vertex": 2}, "is_public": true, "problem_id": "graphs-dag_topological_position-da530810614b8875"}], "verification": {"duration_ms": 1.8484169850125909, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and (= dist_1 (+ dist_0 2)) (<= dist_1 (+ 2 dist_0)))))\n (a!2 (or (and (= dist_2 (+ dist_0 2)) (<= dist_2 (+ 2 dist_0))))))\n (and (= dist_0 0) a!1 a!2))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "graphs", "id": "graphs-dag_shortest_path-f46ec5157c327e1a", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 2, "difficulty": "easy", "edges": [[0, 1], [0, 2]], "generator": "graph_invariants", "num_vertices": 3, "seed": 20260945, "source": 0, "target": 2, "variant": "dag_shortest_path", "weighted_edges": [[0, 1, 2], [0, 2, 2]]}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\nsource = data[\"source\"]\ntarget = data[\"target\"]\n\ninfinity = float(\"inf\")\ndist = [infinity] * n\ndist[source] = 0\nfor u, v, w in edges:\n if dist[u] != infinity and dist[u] + w < dist[v]:\n dist[v] = dist[u] + w\nprint(dist[target])\n", "statement": "You are given a directed acyclic graph with 3 vertices labelled 0 to\n2. Every edge [u, v, w] goes from a lower to a higher label and has\ninteger weight 1 <= w <= 9; edges are listed in ascending (u, v) order.\n\nPrint the minimum possible total weight of a path from source s = 0 to\ntarget t = 2. The target is guaranteed reachable, the optimum is\nunique, and both facts have been formally proven for this instance with the\nZ3 SMT solver.\n", "title": "DAG Shortest Path (n=3, weight 2)", "variant": "dag_shortest_path"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_shortest_path-f46ec5157c327e1a-t000", "input": {"edges": [[0, 1, 2], [0, 2, 2]], "num_vertices": 3, "source": 0, "target": 2}, "is_public": true, "problem_id": "graphs-dag_shortest_path-f46ec5157c327e1a"}], "verification": {"duration_ms": 0.7724579772911966, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (=> (and (> pos_1 0) (and (< pos_0 0))) (< ord_0 1)))\n (a!2 (=> (and (> pos_2 0) (and (< pos_0 0))) (< ord_0 2)))\n (a!3 (=> (and (> pos_1 1) (and (< pos_0 1))) (< ord_1 1)))\n (a!4 (=> (and (> pos_2 1) (and (< pos_0 1))) (< ord_1 2)))\n (a!5 (=> (and (> pos_1 2) (and (< pos_0 2))) (< ord_2 1)))\n (a!6 (=> (and (> pos_2 2) (and (< pos_0 2))) (< ord_2 2)))\n (a!7 (=> (and (> pos_1 3) (and (< pos_0 3))) (< ord_3 1)))\n (a!8 (=> (and (> pos_2 3) (and (< pos_0 3))) (< ord_3 2))))\n (and (distinct ord_0 ord_1 ord_2 ord_3)\n (>= ord_0 0)\n (< ord_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "graphs", "id": "graphs-dag_topological_position-9bf21bf793e59e6a", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 1, "difficulty": "easy", "edges": [[0, 1], [0, 2], [2, 3]], "generator": "graph_invariants", "num_vertices": 4, "seed": 20260946, "target_vertex": 1, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 4 vertices labelled 0 to\n3 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 1 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=4, vertex 1)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-9bf21bf793e59e6a-t000", "input": {"edges": [[0, 1], [0, 2], [2, 3]], "num_vertices": 4, "target_vertex": 1}, "is_public": true, "problem_id": "graphs-dag_topological_position-9bf21bf793e59e6a"}], "verification": {"duration_ms": 2.311041986104101, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_0 member_3)\n (=> member_1 member_2)\n (=> member_1 (or member_0))\n (=> member_2 (or member_1))\n (=> member_3 (or member_0))\n (= reachable_count\n (+ (ite member_0 1 0)\n (ite member_1 1 0)\n (ite member_2 1 0)\n (ite member_3 1 0))))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "graphs", "id": "graphs-dag_reachability-6ab48e0cdbd94720", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1, 3], [2], [], []], "answer": 4, "difficulty": "easy", "generator": "graph_invariants", "num_vertices": 4, "seed": 20260947, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 4 vertices labelled 0 to 3,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=4)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-6ab48e0cdbd94720-t000", "input": {"adjacency": [[1, 3], [2], [], []], "num_vertices": 4, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-6ab48e0cdbd94720"}], "verification": {"duration_ms": 0.7482919609174132, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_0 member_2)\n (=> member_2 member_3)\n (=> member_1 (or member_0))\n (=> member_2 (or member_0))\n (=> member_3 (or member_2))\n (= reachable_count\n (+ (ite member_0 1 0)\n (ite member_1 1 0)\n (ite member_2 1 0)\n (ite member_3 1 0))))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "graphs", "id": "graphs-dag_reachability-8fc07e2283439593", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1, 2], [], [3], []], "answer": 4, "difficulty": "easy", "generator": "graph_invariants", "num_vertices": 4, "seed": 20260948, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 4 vertices labelled 0 to 3,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=4)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-8fc07e2283439593-t000", "input": {"adjacency": [[1, 2], [], [3], []], "num_vertices": 4, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-8fc07e2283439593"}], "verification": {"duration_ms": 0.734834058675915, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and (= dist_1 (+ dist_0 9)) (<= dist_1 (+ 9 dist_0)))))\n (a!2 (or (and (= dist_2 (+ dist_0 8)) (<= dist_2 (+ 8 dist_0)))))\n (a!3 (or (and (= dist_3 (+ dist_0 3)) (<= dist_3 (+ 3 dist_0)))))\n (a!4 (or (and (= dist_4 (+ dist_0 3)) (<= dist_4 (+ 3 dist_0)))))\n (a!5 (and (<= dist_5 (+ 2 dist_0))\n (<= dist_5 (+ 1 dist_1))\n (<= dist_5 (+ 1 dist_3))\n (<= dist_5 (+ 8 dist_4)))))\n(let ((a!6 (or (and (= dist_5 (+ dist_0 2)) a!5)\n (and (= dist_5 (+ dist_1 1)) a!5)\n (and (= dist_5 (+ dist_3 1)) ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "graphs", "id": "graphs-dag_shortest_path-df21eb3f76ef573d", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 2, "difficulty": "medium", "edges": [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 5], [3, 5], [4, 5]], "generator": "graph_invariants", "num_vertices": 6, "seed": 20260949, "source": 0, "target": 5, "variant": "dag_shortest_path", "weighted_edges": [[0, 1, 9], [0, 2, 8], [0, 3, 3], [0, 4, 3], [0, 5, 2], [1, 5, 1], [3, 5, 1], [4, 5, 8]]}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\nsource = data[\"source\"]\ntarget = data[\"target\"]\n\ninfinity = float(\"inf\")\ndist = [infinity] * n\ndist[source] = 0\nfor u, v, w in edges:\n if dist[u] != infinity and dist[u] + w < dist[v]:\n dist[v] = dist[u] + w\nprint(dist[target])\n", "statement": "You are given a directed acyclic graph with 6 vertices labelled 0 to\n5. Every edge [u, v, w] goes from a lower to a higher label and has\ninteger weight 1 <= w <= 9; edges are listed in ascending (u, v) order.\n\nPrint the minimum possible total weight of a path from source s = 0 to\ntarget t = 5. The target is guaranteed reachable, the optimum is\nunique, and both facts have been formally proven for this instance with the\nZ3 SMT solver.\n", "title": "DAG Shortest Path (n=6, weight 2)", "variant": "dag_shortest_path"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_shortest_path-df21eb3f76ef573d-t000", "input": {"edges": [[0, 1, 9], [0, 2, 8], [0, 3, 3], [0, 4, 3], [0, 5, 2], [1, 5, 1], [3, 5, 1], [4, 5, 8]], "num_vertices": 6, "source": 0, "target": 5}, "is_public": true, "problem_id": "graphs-dag_shortest_path-df21eb3f76ef573d"}], "verification": {"duration_ms": 0.981583958491683, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and (= dist_1 (+ dist_0 3)) (<= dist_1 (+ 3 dist_0)))))\n (a!2 (or (and (= dist_2 (+ dist_0 9)) (<= dist_2 (+ 9 dist_0)))))\n (a!3 (or (and (= dist_3 (+ dist_1 4)) (<= dist_3 (+ 4 dist_1))))))\n (and (= dist_0 0) a!1 a!2 a!3))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "graphs", "id": "graphs-dag_shortest_path-e9d67cdd0b3b0b97", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 7, "difficulty": "medium", "edges": [[0, 1], [0, 2], [1, 3]], "generator": "graph_invariants", "num_vertices": 4, "seed": 20260950, "source": 0, "target": 3, "variant": "dag_shortest_path", "weighted_edges": [[0, 1, 3], [0, 2, 9], [1, 3, 4]]}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\nsource = data[\"source\"]\ntarget = data[\"target\"]\n\ninfinity = float(\"inf\")\ndist = [infinity] * n\ndist[source] = 0\nfor u, v, w in edges:\n if dist[u] != infinity and dist[u] + w < dist[v]:\n dist[v] = dist[u] + w\nprint(dist[target])\n", "statement": "You are given a directed acyclic graph with 4 vertices labelled 0 to\n3. Every edge [u, v, w] goes from a lower to a higher label and has\ninteger weight 1 <= w <= 9; edges are listed in ascending (u, v) order.\n\nPrint the minimum possible total weight of a path from source s = 0 to\ntarget t = 3. The target is guaranteed reachable, the optimum is\nunique, and both facts have been formally proven for this instance with the\nZ3 SMT solver.\n", "title": "DAG Shortest Path (n=4, weight 7)", "variant": "dag_shortest_path"}, "test_cases": [{"expected_output": 7, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_shortest_path-e9d67cdd0b3b0b97-t000", "input": {"edges": [[0, 1, 3], [0, 2, 9], [1, 3, 4]], "num_vertices": 4, "source": 0, "target": 3}, "is_public": true, "problem_id": "graphs-dag_shortest_path-e9d67cdd0b3b0b97"}], "verification": {"duration_ms": 0.8272500126622617, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (=> (and (> pos_3 0) (and (< pos_2 0))) (< ord_0 3)))\n (a!2 (=> (and (> pos_4 0) (and (< pos_2 0))) (< ord_0 4)))\n (a!3 (=> (and (> pos_3 1) (and (< pos_2 1))) (< ord_1 3)))\n (a!4 (=> (and (> pos_4 1) (and (< pos_2 1))) (< ord_1 4)))\n (a!5 (=> (and (> pos_3 2) (and (< pos_2 2))) (< ord_2 3)))\n (a!6 (=> (and (> pos_4 2) (and (< pos_2 2))) (< ord_2 4)))\n (a!7 (=> (and (> pos_3 3) (and (< pos_2 3))) (< ord_3 3)))\n (a!8 (=> (and (> pos_4 3) (and (< pos_2 3))) (< ord_3 4)))\n (a!9 (=> (and (> pos_3 4) (and (< pos_2 4))) (< ord_4 3)))\n (a!10 ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "graphs", "id": "graphs-dag_topological_position-683d94700f64e76d", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 2, "difficulty": "medium", "edges": [[0, 1], [1, 2], [2, 3], [2, 4]], "generator": "graph_invariants", "num_vertices": 5, "seed": 20260951, "target_vertex": 2, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 5 vertices labelled 0 to\n4 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 2 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=5, vertex 2)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-683d94700f64e76d-t000", "input": {"edges": [[0, 1], [1, 2], [2, 3], [2, 4]], "num_vertices": 5, "target_vertex": 2}, "is_public": true, "problem_id": "graphs-dag_topological_position-683d94700f64e76d"}], "verification": {"duration_ms": 3.2669989741407335, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_0 member_3)\n (=> member_0 member_4)\n (=> member_1 member_2)\n (=> member_2 member_5)\n (=> member_4 member_5)\n (=> member_1 (or member_0))\n (=> member_2 (or member_1))\n (=> member_3 (or member_0))\n (=> member_4 (or member_0))\n (=> member_5 (or member_2 member_4))\n (= reachable_count\n (+ (ite member_0 1 0)\n (ite member_1 1 0)\n (ite member_2 1 0)\n (ite member_3 1 0)\n (ite member_4 1 0)\n (ite member_5 1 0))))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "graphs", "id": "graphs-dag_reachability-d65340acf38f499e", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1, 3, 4], [2], [5], [], [5], []], "answer": 6, "difficulty": "medium", "generator": "graph_invariants", "num_vertices": 6, "seed": 20260952, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 6 vertices labelled 0 to 5,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=6)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 6, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-d65340acf38f499e-t000", "input": {"adjacency": [[1, 3, 4], [2], [5], [], [5], []], "num_vertices": 6, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-d65340acf38f499e"}], "verification": {"duration_ms": 0.9685829863883555, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and (= dist_1 (+ dist_0 6)) (<= dist_1 (+ 6 dist_0)))))\n (a!2 (or (and (= dist_2 (+ dist_1 1)) (<= dist_2 (+ 1 dist_1)))))\n (a!3 (or (and (= dist_3 (+ dist_0 4)) (<= dist_3 (+ 4 dist_0)))))\n (a!4 (and (<= dist_4 (+ 2 dist_2)) (<= dist_4 (+ 6 dist_3))))\n (a!6 (or (and (= dist_5 (+ dist_4 6)) (<= dist_5 (+ 6 dist_4)))))\n (a!7 (or (and (= dist_6 (+ dist_2 8)) (<= dist_6 (+ 8 dist_2))))))\n(let ((a!5 (or (and (= dist_4 (+ dist_2 2)) a!4)\n (and (= dist_4 (+ dist_3 6)) a!4))))\n (and (= dist_0 0) a!1 a!2 a!3 a!5 a!6 a!7)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "graphs", "id": "graphs-dag_shortest_path-f3c3e58a9006661f", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 15, "difficulty": "medium", "edges": [[0, 1], [0, 3], [1, 2], [2, 4], [2, 6], [3, 4], [4, 5]], "generator": "graph_invariants", "num_vertices": 7, "seed": 20260953, "source": 0, "target": 6, "variant": "dag_shortest_path", "weighted_edges": [[0, 1, 6], [0, 3, 4], [1, 2, 1], [2, 4, 2], [2, 6, 8], [3, 4, 6], [4, 5, 6]]}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\nsource = data[\"source\"]\ntarget = data[\"target\"]\n\ninfinity = float(\"inf\")\ndist = [infinity] * n\ndist[source] = 0\nfor u, v, w in edges:\n if dist[u] != infinity and dist[u] + w < dist[v]:\n dist[v] = dist[u] + w\nprint(dist[target])\n", "statement": "You are given a directed acyclic graph with 7 vertices labelled 0 to\n6. Every edge [u, v, w] goes from a lower to a higher label and has\ninteger weight 1 <= w <= 9; edges are listed in ascending (u, v) order.\n\nPrint the minimum possible total weight of a path from source s = 0 to\ntarget t = 6. The target is guaranteed reachable, the optimum is\nunique, and both facts have been formally proven for this instance with the\nZ3 SMT solver.\n", "title": "DAG Shortest Path (n=7, weight 15)", "variant": "dag_shortest_path"}, "test_cases": [{"expected_output": 15, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_shortest_path-f3c3e58a9006661f-t000", "input": {"edges": [[0, 1, 6], [0, 3, 4], [1, 2, 1], [2, 4, 2], [2, 6, 8], [3, 4, 6], [4, 5, 6]], "num_vertices": 7, "source": 0, "target": 6}, "is_public": true, "problem_id": "graphs-dag_shortest_path-f3c3e58a9006661f"}], "verification": {"duration_ms": 1.0185829596593976, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and (distinct ord_0 ord_1 ord_2 ord_3)\n (>= ord_0 0)\n (< ord_0 4)\n (>= pos_0 0)\n (< pos_0 4)\n (>= ord_1 0)\n (< ord_1 4)\n (>= pos_1 0)\n (< pos_1 4)\n (>= ord_2 0)\n (< ord_2 4)\n (>= pos_2 0)\n (< pos_2 4)\n (>= ord_3 0)\n (< ord_3 4)\n (>= pos_3 0)\n (< pos_3 4)\n (=> (= ord_0 0) (= pos_0 0))\n (=> (= ord_0 1) (= pos_1 0))\n (=> (= ord_0 2) (= pos_2 0))\n (=> (= ord_0 3) (= pos_3 0))\n (=> (= ord_1 0) (= pos_0 1))\n (=> (= ord_1 1) (= pos_1 1))\n (=> (= ord_1 2) (= pos_2 1))\n (=> (= ord_1 3) (= pos_3 1))\n ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "graphs", "id": "graphs-dag_topological_position-0f51db1d89f5638b", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 1, "difficulty": "medium", "edges": [[0, 1], [1, 2], [2, 3]], "generator": "graph_invariants", "num_vertices": 4, "seed": 20260954, "target_vertex": 1, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 4 vertices labelled 0 to\n3 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 1 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=4, vertex 1)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-0f51db1d89f5638b-t000", "input": {"edges": [[0, 1], [1, 2], [2, 3]], "num_vertices": 4, "target_vertex": 1}, "is_public": true, "problem_id": "graphs-dag_topological_position-0f51db1d89f5638b"}], "verification": {"duration_ms": 2.504708943888545, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_0 member_2)\n (=> member_0 member_4)\n (=> member_2 member_3)\n (=> member_2 member_5)\n (=> member_3 member_5)\n (=> member_4 member_5)\n (=> member_1 (or member_0))\n (=> member_2 (or member_0))\n (=> member_3 (or member_2))\n (=> member_4 (or member_0))\n (=> member_5 (or member_2 member_3 member_4))\n (= reachable_count\n (+ (ite member_0 1 0)\n (ite member_1 1 0)\n (ite member_2 1 0)\n (ite member_3 1 0)\n (ite member_4 1 0)\n (ite member_5 1 0))))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "graphs", "id": "graphs-dag_reachability-085cb6a9532a6b05", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1, 2, 4], [], [3, 5], [5], [5], []], "answer": 6, "difficulty": "medium", "generator": "graph_invariants", "num_vertices": 6, "seed": 20260955, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 6 vertices labelled 0 to 5,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=6)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 6, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-085cb6a9532a6b05-t000", "input": {"adjacency": [[1, 2, 4], [], [3, 5], [5], [5], []], "num_vertices": 6, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-085cb6a9532a6b05"}], "verification": {"duration_ms": 0.7810419192537665, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (=> (and (> pos_3 0) (and (< pos_2 0))) (< ord_0 3)))\n (a!2 (=> (and (> pos_4 0) (and (< pos_2 0))) (< ord_0 4)))\n (a!3 (=> (and (> pos_3 1) (and (< pos_2 1))) (< ord_1 3)))\n (a!4 (=> (and (> pos_4 1) (and (< pos_2 1))) (< ord_1 4)))\n (a!5 (=> (and (> pos_3 2) (and (< pos_2 2))) (< ord_2 3)))\n (a!6 (=> (and (> pos_4 2) (and (< pos_2 2))) (< ord_2 4)))\n (a!7 (=> (and (> pos_3 3) (and (< pos_2 3))) (< ord_3 3)))\n (a!8 (=> (and (> pos_4 3) (and (< pos_2 3))) (< ord_3 4)))\n (a!9 (=> (and (> pos_3 4) (and (< pos_2 4))) (< ord_4 3)))\n (a!10 ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "graphs", "id": "graphs-dag_topological_position-46eaf7726585b7a5", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 1, "difficulty": "medium", "edges": [[0, 1], [0, 5], [0, 6], [1, 2], [1, 6], [2, 3], [2, 4], [2, 5], [4, 5], [5, 6]], "generator": "graph_invariants", "num_vertices": 7, "seed": 20260956, "target_vertex": 1, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 7 vertices labelled 0 to\n6 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 1 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=7, vertex 1)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-46eaf7726585b7a5-t000", "input": {"edges": [[0, 1], [0, 5], [0, 6], [1, 2], [1, 6], [2, 3], [2, 4], [2, 5], [4, 5], [5, 6]], "num_vertices": 7, "target_vertex": 1}, "is_public": true, "problem_id": "graphs-dag_topological_position-46eaf7726585b7a5"}], "verification": {"duration_ms": 4.857665975578129, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_1 member_2)\n (=> member_1 member_4)\n (=> member_2 member_3)\n (=> member_2 member_4)\n (=> member_3 member_4)\n (=> member_1 (or member_0))\n (=> member_2 (or member_1))\n (=> member_3 (or member_2))\n (=> member_4 (or member_1 member_2 member_3))\n (= reachable_count\n (+ (ite member_0 1 0)\n (ite member_1 1 0)\n (ite member_2 1 0)\n (ite member_3 1 0)\n (ite member_4 1 0))))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "graphs", "id": "graphs-dag_reachability-273066c87685b035", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1], [2, 4], [3, 4], [4], []], "answer": 5, "difficulty": "medium", "generator": "graph_invariants", "num_vertices": 5, "seed": 20260957, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 5 vertices labelled 0 to 4,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=5)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-273066c87685b035-t000", "input": {"adjacency": [[1], [2, 4], [3, 4], [4], []], "num_vertices": 5, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-273066c87685b035"}], "verification": {"duration_ms": 0.789084006100893, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_0 member_2)\n (=> member_1 member_2)\n (=> member_2 member_3)\n (=> member_1 (or member_0))\n (=> member_2 (or member_0 member_1))\n (=> member_3 (or member_2))\n (= reachable_count\n (+ (ite member_0 1 0)\n (ite member_1 1 0)\n (ite member_2 1 0)\n (ite member_3 1 0))))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "graphs", "id": "graphs-dag_reachability-77ae9376b4760fbc", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1, 2], [2], [3], []], "answer": 4, "difficulty": "medium", "generator": "graph_invariants", "num_vertices": 4, "seed": 20260958, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 4 vertices labelled 0 to 3,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=4)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-77ae9376b4760fbc-t000", "input": {"adjacency": [[1, 2], [2], [3], []], "num_vertices": 4, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-77ae9376b4760fbc"}], "verification": {"duration_ms": 0.7702500442974269, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (=> (and (> pos_1 0) (and (< pos_0 0))) (< ord_0 1)))\n (a!2 (=> (and (> pos_4 0) (and (< pos_0 0))) (< ord_0 4)))\n (a!3 (=> (and (> pos_1 1) (and (< pos_0 1))) (< ord_1 1)))\n (a!4 (=> (and (> pos_4 1) (and (< pos_0 1))) (< ord_1 4)))\n (a!5 (=> (and (> pos_1 2) (and (< pos_0 2))) (< ord_2 1)))\n (a!6 (=> (and (> pos_4 2) (and (< pos_0 2))) (< ord_2 4)))\n (a!7 (=> (and (> pos_1 3) (and (< pos_0 3))) (< ord_3 1)))\n (a!8 (=> (and (> pos_4 3) (and (< pos_0 3))) (< ord_3 4)))\n (a!9 (=> (and (> pos_1 4) (and (< pos_0 4))) (< ord_4 1)))\n (a!10 ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "graphs", "id": "graphs-dag_topological_position-cc78e12e03cb620d", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 5, "difficulty": "medium", "edges": [[0, 1], [0, 3], [0, 4], [0, 6], [1, 2], [1, 3], [1, 5], [3, 5], [3, 6], [4, 5]], "generator": "graph_invariants", "num_vertices": 7, "seed": 20260959, "target_vertex": 5, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 7 vertices labelled 0 to\n6 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 5 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=7, vertex 5)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-cc78e12e03cb620d-t000", "input": {"edges": [[0, 1], [0, 3], [0, 4], [0, 6], [1, 2], [1, 3], [1, 5], [3, 5], [3, 6], [4, 5]], "num_vertices": 7, "target_vertex": 5}, "is_public": true, "problem_id": "graphs-dag_topological_position-cc78e12e03cb620d"}], "verification": {"duration_ms": 5.308041989337653, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (=> (and (> pos_1 0) (and (< pos_0 0))) (< ord_0 1)))\n (a!2 (=> (and (> pos_2 0) (and (< pos_0 0))) (< ord_0 2)))\n (a!3 (=> (and (> pos_1 1) (and (< pos_0 1))) (< ord_1 1)))\n (a!4 (=> (and (> pos_2 1) (and (< pos_0 1))) (< ord_1 2)))\n (a!5 (=> (and (> pos_1 2) (and (< pos_0 2))) (< ord_2 1)))\n (a!6 (=> (and (> pos_2 2) (and (< pos_0 2))) (< ord_2 2)))\n (a!7 (=> (and (> pos_1 3) (and (< pos_0 3))) (< ord_3 1)))\n (a!8 (=> (and (> pos_2 3) (and (< pos_0 3))) (< ord_3 2)))\n (a!9 (=> (and (> pos_1 4) (and (< pos_0 4))) (< ord_4 1)))\n (a!10 ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "graphs", "id": "graphs-dag_topological_position-dd6e74e598916d54", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 2, "difficulty": "medium", "edges": [[0, 1], [0, 2], [0, 3], [2, 3], [2, 5], [3, 4], [3, 5]], "generator": "graph_invariants", "num_vertices": 6, "seed": 20260960, "target_vertex": 2, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 6 vertices labelled 0 to\n5 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 2 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=6, vertex 2)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-dd6e74e598916d54-t000", "input": {"edges": [[0, 1], [0, 2], [0, 3], [2, 3], [2, 5], [3, 4], [3, 5]], "num_vertices": 6, "target_vertex": 2}, "is_public": true, "problem_id": "graphs-dag_topological_position-dd6e74e598916d54"}], "verification": {"duration_ms": 4.318249993957579, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_0 member_2)\n (=> member_0 member_4)\n (=> member_1 member_4)\n (=> member_2 member_3)\n (=> member_3 member_4)\n (=> member_1 (or member_0))\n (=> member_2 (or member_0))\n (=> member_3 (or member_2))\n (=> member_4 (or member_0 member_1 member_3))\n (= reachable_count\n (+ (ite member_0 1 0)\n (ite member_1 1 0)\n (ite member_2 1 0)\n (ite member_3 1 0)\n (ite member_4 1 0))))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "graphs", "id": "graphs-dag_reachability-b0a44def826c1fa7", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1, 2, 4], [4], [3], [4], []], "answer": 5, "difficulty": "medium", "generator": "graph_invariants", "num_vertices": 5, "seed": 20260961, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 5 vertices labelled 0 to 4,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=5)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-b0a44def826c1fa7-t000", "input": {"adjacency": [[1, 2, 4], [4], [3], [4], []], "num_vertices": 5, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-b0a44def826c1fa7"}], "verification": {"duration_ms": 0.7967489655129611, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_0 member_2)\n (=> member_0 member_3)\n (=> member_1 member_3)\n (=> member_2 member_3)\n (=> member_3 member_4)\n (=> member_4 member_5)\n (=> member_1 (or member_0))\n (=> member_2 (or member_0))\n (=> member_3 (or member_0 member_1 member_2))\n (=> member_4 (or member_3))\n (=> member_5 (or member_4))\n (= reachable_count\n (+ (ite member_0 1 0)\n (ite member_1 1 0)\n (ite member_2 1 0)\n (ite member_3 1 0)\n (ite member_4 1 0)\n (ite member_5 1 0))))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "graphs", "id": "graphs-dag_reachability-aceebb54b66b53fd", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1, 2, 3], [3], [3], [4], [5], []], "answer": 6, "difficulty": "medium", "generator": "graph_invariants", "num_vertices": 6, "seed": 20260962, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 6 vertices labelled 0 to 5,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=6)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 6, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-aceebb54b66b53fd-t000", "input": {"adjacency": [[1, 2, 3], [3], [3], [4], [5], []], "num_vertices": 6, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-aceebb54b66b53fd"}], "verification": {"duration_ms": 0.7702909642830491, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_0 member_2)\n (=> member_1 member_2)\n (=> member_1 member_3)\n (=> member_1 (or member_0))\n (=> member_2 (or member_0 member_1))\n (=> member_3 (or member_1))\n (= reachable_count\n (+ (ite member_0 1 0)\n (ite member_1 1 0)\n (ite member_2 1 0)\n (ite member_3 1 0))))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "graphs", "id": "graphs-dag_reachability-e0723a563db1157c", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1, 2], [2, 3], [], []], "answer": 4, "difficulty": "medium", "generator": "graph_invariants", "num_vertices": 4, "seed": 20260963, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 4 vertices labelled 0 to 3,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=4)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-e0723a563db1157c-t000", "input": {"adjacency": [[1, 2], [2, 3], [], []], "num_vertices": 4, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-e0723a563db1157c"}], "verification": {"duration_ms": 0.7530429866164923, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (=> (and (> pos_1 0) (and (< pos_0 0))) (< ord_0 1)))\n (a!2 (=> (and (> pos_2 0) (and (< pos_0 0))) (< ord_0 2)))\n (a!3 (=> (and (> pos_1 1) (and (< pos_0 1))) (< ord_1 1)))\n (a!4 (=> (and (> pos_2 1) (and (< pos_0 1))) (< ord_1 2)))\n (a!5 (=> (and (> pos_1 2) (and (< pos_0 2))) (< ord_2 1)))\n (a!6 (=> (and (> pos_2 2) (and (< pos_0 2))) (< ord_2 2)))\n (a!7 (=> (and (> pos_1 3) (and (< pos_0 3))) (< ord_3 1)))\n (a!8 (=> (and (> pos_2 3) (and (< pos_0 3))) (< ord_3 2)))\n (a!9 (=> (and (> pos_1 4) (and (< pos_0 4))) (< ord_4 1)))\n (a!10 ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "graphs", "id": "graphs-dag_topological_position-9a184be87bf2f1b9", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 1, "difficulty": "medium", "edges": [[0, 1], [0, 2], [0, 4], [1, 3], [2, 4], [3, 4]], "generator": "graph_invariants", "num_vertices": 5, "seed": 20260964, "target_vertex": 1, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 5 vertices labelled 0 to\n4 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 1 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=5, vertex 1)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-9a184be87bf2f1b9-t000", "input": {"edges": [[0, 1], [0, 2], [0, 4], [1, 3], [2, 4], [3, 4]], "num_vertices": 5, "target_vertex": 1}, "is_public": true, "problem_id": "graphs-dag_topological_position-9a184be87bf2f1b9"}], "verification": {"duration_ms": 3.170749987475574, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and (distinct ord_0 ord_1 ord_2 ord_3 ord_4 ord_5)\n (>= ord_0 0)\n (< ord_0 6)\n (>= pos_0 0)\n (< pos_0 6)\n (>= ord_1 0)\n (< ord_1 6)\n (>= pos_1 0)\n (< pos_1 6)\n (>= ord_2 0)\n (< ord_2 6)\n (>= pos_2 0)\n (< pos_2 6)\n (>= ord_3 0)\n (< ord_3 6)\n (>= pos_3 0)\n (< pos_3 6)\n (>= ord_4 0)\n (< ord_4 6)\n (>= pos_4 0)\n (< pos_4 6)\n (>= ord_5 0)\n (< ord_5 6)\n (>= pos_5 0)\n (< pos_5 6)\n (=> (= ord_0 0) (= pos_0 0))\n (=> (= ord_0 1) (= pos_1 0))\n (=> (= ord_0 2) (= pos_2 0))\n (=> (= ord_0 3) (=...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "graphs", "id": "graphs-dag_topological_position-a971678ae0acfd93", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 1, "difficulty": "medium", "edges": [[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 4], [2, 3], [2, 4], [2, 5], [3, 5]], "generator": "graph_invariants", "num_vertices": 6, "seed": 20260965, "target_vertex": 1, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 6 vertices labelled 0 to\n5 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 1 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=6, vertex 1)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-a971678ae0acfd93-t000", "input": {"edges": [[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 4], [2, 3], [2, 4], [2, 5], [3, 5]], "num_vertices": 6, "target_vertex": 1}, "is_public": true, "problem_id": "graphs-dag_topological_position-a971678ae0acfd93"}], "verification": {"duration_ms": 3.9716659812256694, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_0 member_4)\n (=> member_1 member_2)\n (=> member_1 member_3)\n (=> member_2 member_4)\n (=> member_1 (or member_0))\n (=> member_2 (or member_1))\n (=> member_3 (or member_1))\n (=> member_4 (or member_0 member_2))\n (= reachable_count\n (+ (ite member_0 1 0)\n (ite member_1 1 0)\n (ite member_2 1 0)\n (ite member_3 1 0)\n (ite member_4 1 0))))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "graphs", "id": "graphs-dag_reachability-2a84202bfc6724e9", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1, 4], [2, 3], [4], [], []], "answer": 5, "difficulty": "medium", "generator": "graph_invariants", "num_vertices": 5, "seed": 20260966, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 5 vertices labelled 0 to 4,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=5)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-2a84202bfc6724e9-t000", "input": {"adjacency": [[1, 4], [2, 3], [4], [], []], "num_vertices": 5, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-2a84202bfc6724e9"}], "verification": {"duration_ms": 0.7649160106666386, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and (= dist_1 (+ dist_0 4)) (<= dist_1 (+ 4 dist_0)))))\n (a!2 (or (and (= dist_2 (+ dist_1 7)) (<= dist_2 (+ 7 dist_1)))))\n (a!3 (or (and (= dist_3 (+ dist_2 4)) (<= dist_3 (+ 4 dist_2))))))\n (and (= dist_0 0) a!1 a!2 a!3))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "graphs", "id": "graphs-dag_shortest_path-c4a9cc94fb937d97", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 15, "difficulty": "medium", "edges": [[0, 1], [1, 2], [2, 3]], "generator": "graph_invariants", "num_vertices": 4, "seed": 20260967, "source": 0, "target": 3, "variant": "dag_shortest_path", "weighted_edges": [[0, 1, 4], [1, 2, 7], [2, 3, 4]]}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\nsource = data[\"source\"]\ntarget = data[\"target\"]\n\ninfinity = float(\"inf\")\ndist = [infinity] * n\ndist[source] = 0\nfor u, v, w in edges:\n if dist[u] != infinity and dist[u] + w < dist[v]:\n dist[v] = dist[u] + w\nprint(dist[target])\n", "statement": "You are given a directed acyclic graph with 4 vertices labelled 0 to\n3. Every edge [u, v, w] goes from a lower to a higher label and has\ninteger weight 1 <= w <= 9; edges are listed in ascending (u, v) order.\n\nPrint the minimum possible total weight of a path from source s = 0 to\ntarget t = 3. The target is guaranteed reachable, the optimum is\nunique, and both facts have been formally proven for this instance with the\nZ3 SMT solver.\n", "title": "DAG Shortest Path (n=4, weight 15)", "variant": "dag_shortest_path"}, "test_cases": [{"expected_output": 15, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_shortest_path-c4a9cc94fb937d97-t000", "input": {"edges": [[0, 1, 4], [1, 2, 7], [2, 3, 4]], "num_vertices": 4, "source": 0, "target": 3}, "is_public": true, "problem_id": "graphs-dag_shortest_path-c4a9cc94fb937d97"}], "verification": {"duration_ms": 0.8454159833490849, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (=> (and (> pos_1 0) (and (< pos_0 0))) (< ord_0 1)))\n (a!2 (=> (and (> pos_2 0) (and (< pos_0 0))) (< ord_0 2)))\n (a!3 (=> (and (> pos_3 0) (and (< pos_0 0))) (< ord_0 3)))\n (a!4 (=> (and (> pos_4 0) (and (< pos_0 0))) (< ord_0 4)))\n (a!5 (=> (and (> pos_6 0) (and (< pos_0 0))) (< ord_0 6)))\n (a!6 (=> (and (> pos_1 1) (and (< pos_0 1))) (< ord_1 1)))\n (a!7 (=> (and (> pos_2 1) (and (< pos_0 1))) (< ord_1 2)))\n (a!8 (=> (and (> pos_3 1) (and (< pos_0 1))) (< ord_1 3)))\n (a!9 (=> (and (> pos_4 1) (and (< pos_0 1))) (< ord_1 4)))\n (a!10 ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "graphs", "id": "graphs-dag_topological_position-1f632943269ad322", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 1, "difficulty": "medium", "edges": [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [0, 6], [3, 5], [4, 5]], "generator": "graph_invariants", "num_vertices": 7, "seed": 20260968, "target_vertex": 1, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 7 vertices labelled 0 to\n6 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 1 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=7, vertex 1)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-1f632943269ad322-t000", "input": {"edges": [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [0, 6], [3, 5], [4, 5]], "num_vertices": 7, "target_vertex": 1}, "is_public": true, "problem_id": "graphs-dag_topological_position-1f632943269ad322"}], "verification": {"duration_ms": 4.848707991186529, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and (= dist_1 (+ dist_0 7)) (<= dist_1 (+ 7 dist_0)))))\n (a!2 (or (and (= dist_2 (+ dist_1 3)) (<= dist_2 (+ 3 dist_1)))))\n (a!3 (or (and (= dist_3 (+ dist_2 8)) (<= dist_3 (+ 8 dist_2)))))\n (a!4 (or (and (= dist_4 (+ dist_1 9)) (<= dist_4 (+ 9 dist_1))))))\n (and (= dist_0 0) a!1 a!2 a!3 a!4))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "graphs", "id": "graphs-dag_shortest_path-eea3c4de67d77e08", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 16, "difficulty": "medium", "edges": [[0, 1], [1, 2], [1, 4], [2, 3]], "generator": "graph_invariants", "num_vertices": 5, "seed": 20260969, "source": 0, "target": 4, "variant": "dag_shortest_path", "weighted_edges": [[0, 1, 7], [1, 2, 3], [1, 4, 9], [2, 3, 8]]}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\nsource = data[\"source\"]\ntarget = data[\"target\"]\n\ninfinity = float(\"inf\")\ndist = [infinity] * n\ndist[source] = 0\nfor u, v, w in edges:\n if dist[u] != infinity and dist[u] + w < dist[v]:\n dist[v] = dist[u] + w\nprint(dist[target])\n", "statement": "You are given a directed acyclic graph with 5 vertices labelled 0 to\n4. Every edge [u, v, w] goes from a lower to a higher label and has\ninteger weight 1 <= w <= 9; edges are listed in ascending (u, v) order.\n\nPrint the minimum possible total weight of a path from source s = 0 to\ntarget t = 4. The target is guaranteed reachable, the optimum is\nunique, and both facts have been formally proven for this instance with the\nZ3 SMT solver.\n", "title": "DAG Shortest Path (n=5, weight 16)", "variant": "dag_shortest_path"}, "test_cases": [{"expected_output": 16, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_shortest_path-eea3c4de67d77e08-t000", "input": {"edges": [[0, 1, 7], [1, 2, 3], [1, 4, 9], [2, 3, 8]], "num_vertices": 5, "source": 0, "target": 4}, "is_public": true, "problem_id": "graphs-dag_shortest_path-eea3c4de67d77e08"}], "verification": {"duration_ms": 0.8392490562982857, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (=> (and (> pos_2 0) (and (< pos_1 0))) (< ord_0 2)))\n (a!2 (=> (and (> pos_3 0) (and (< pos_1 0))) (< ord_0 3)))\n (a!3 (=> (and (> pos_2 1) (and (< pos_1 1))) (< ord_1 2)))\n (a!4 (=> (and (> pos_3 1) (and (< pos_1 1))) (< ord_1 3)))\n (a!5 (=> (and (> pos_2 2) (and (< pos_1 2))) (< ord_2 2)))\n (a!6 (=> (and (> pos_3 2) (and (< pos_1 2))) (< ord_2 3)))\n (a!7 (=> (and (> pos_2 3) (and (< pos_1 3))) (< ord_3 2)))\n (a!8 (=> (and (> pos_3 3) (and (< pos_1 3))) (< ord_3 3))))\n (and (distinct ord_0 ord_1 ord_2 ord_3)\n (>= ord_0 0)\n (< ord_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "graphs", "id": "graphs-dag_topological_position-bae2f8bd88b95130", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 1, "difficulty": "medium", "edges": [[0, 1], [1, 2], [1, 3]], "generator": "graph_invariants", "num_vertices": 4, "seed": 20260970, "target_vertex": 1, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 4 vertices labelled 0 to\n3 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 1 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=4, vertex 1)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-bae2f8bd88b95130-t000", "input": {"edges": [[0, 1], [1, 2], [1, 3]], "num_vertices": 4, "target_vertex": 1}, "is_public": true, "problem_id": "graphs-dag_topological_position-bae2f8bd88b95130"}], "verification": {"duration_ms": 2.4494589306414127, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and (distinct ord_0 ord_1 ord_2 ord_3)\n (>= ord_0 0)\n (< ord_0 4)\n (>= pos_0 0)\n (< pos_0 4)\n (>= ord_1 0)\n (< ord_1 4)\n (>= pos_1 0)\n (< pos_1 4)\n (>= ord_2 0)\n (< ord_2 4)\n (>= pos_2 0)\n (< pos_2 4)\n (>= ord_3 0)\n (< ord_3 4)\n (>= pos_3 0)\n (< pos_3 4)\n (=> (= ord_0 0) (= pos_0 0))\n (=> (= ord_0 1) (= pos_1 0))\n (=> (= ord_0 2) (= pos_2 0))\n (=> (= ord_0 3) (= pos_3 0))\n (=> (= ord_1 0) (= pos_0 1))\n (=> (= ord_1 1) (= pos_1 1))\n (=> (= ord_1 2) (= pos_2 1))\n (=> (= ord_1 3) (= pos_3 1))\n ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "graphs", "id": "graphs-dag_topological_position-5021f666f2c40a46", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 1, "difficulty": "medium", "edges": [[0, 1], [0, 3], [1, 2], [2, 3]], "generator": "graph_invariants", "num_vertices": 4, "seed": 20260971, "target_vertex": 1, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 4 vertices labelled 0 to\n3 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 1 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=4, vertex 1)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-5021f666f2c40a46-t000", "input": {"edges": [[0, 1], [0, 3], [1, 2], [2, 3]], "num_vertices": 4, "target_vertex": 1}, "is_public": true, "problem_id": "graphs-dag_topological_position-5021f666f2c40a46"}], "verification": {"duration_ms": 2.3841249640099704, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_0 member_3)\n (=> member_1 member_2)\n (=> member_1 member_3)\n (=> member_3 member_4)\n (=> member_1 (or member_0))\n (=> member_2 (or member_1))\n (=> member_3 (or member_0 member_1))\n (=> member_4 (or member_3))\n (= reachable_count\n (+ (ite member_0 1 0)\n (ite member_1 1 0)\n (ite member_2 1 0)\n (ite member_3 1 0)\n (ite member_4 1 0))))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "graphs", "id": "graphs-dag_reachability-3c3fbbfdea18dd91", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1, 3], [2, 3], [], [4], []], "answer": 5, "difficulty": "medium", "generator": "graph_invariants", "num_vertices": 5, "seed": 20260972, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 5 vertices labelled 0 to 4,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=5)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-3c3fbbfdea18dd91-t000", "input": {"adjacency": [[1, 3], [2, 3], [], [4], []], "num_vertices": 5, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-3c3fbbfdea18dd91"}], "verification": {"duration_ms": 0.9006240288726985, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (=> (and (> pos_1 0) (and (< pos_0 0))) (< ord_0 1)))\n (a!2 (=> (and (> pos_2 0) (and (< pos_1 0))) (< ord_0 2)))\n (a!3 (=> (and (> pos_3 0) (and (< pos_0 0))) (< ord_0 3)))\n (a!4 (=> (and (> pos_6 0) (and (< pos_1 0))) (< ord_0 6)))\n (a!5 (=> (and (> pos_1 1) (and (< pos_0 1))) (< ord_1 1)))\n (a!6 (=> (and (> pos_2 1) (and (< pos_1 1))) (< ord_1 2)))\n (a!7 (=> (and (> pos_3 1) (and (< pos_0 1))) (< ord_1 3)))\n (a!8 (=> (and (> pos_6 1) (and (< pos_1 1))) (< ord_1 6)))\n (a!9 (=> (and (> pos_1 2) (and (< pos_0 2))) (< ord_2 1)))\n (a!10 ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "graphs", "id": "graphs-dag_topological_position-1c224d81305b2e15", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 3, "difficulty": "medium", "edges": [[0, 1], [0, 3], [0, 4], [1, 2], [1, 5], [1, 6], [2, 4], [3, 4], [4, 5]], "generator": "graph_invariants", "num_vertices": 7, "seed": 20260973, "target_vertex": 3, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 7 vertices labelled 0 to\n6 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 3 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=7, vertex 3)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 3, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-1c224d81305b2e15-t000", "input": {"edges": [[0, 1], [0, 3], [0, 4], [1, 2], [1, 5], [1, 6], [2, 4], [3, 4], [4, 5]], "num_vertices": 7, "target_vertex": 3}, "is_public": true, "problem_id": "graphs-dag_topological_position-1c224d81305b2e15"}], "verification": {"duration_ms": 5.165416980162263, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_0 member_2)\n (=> member_0 member_3)\n (=> member_0 member_5)\n (=> member_1 member_3)\n (=> member_1 member_5)\n (=> member_2 member_4)\n (=> member_2 member_5)\n (=> member_2 member_6)\n (=> member_4 member_5)\n (=> member_5 member_6)\n (=> member_1 (or member_0))\n (=> member_2 (or member_0))\n (=> member_3 (or member_0 member_1))\n (=> member_4 (or member_2))\n (=> member_5 (or member_0 member_1 member_2 member_4))\n (=> member_6 (or member_2 member_5))\n (= reachable_count\n ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "graphs", "id": "graphs-dag_reachability-ef1199086f83bf90", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1, 2, 3, 5], [3, 5], [4, 5, 6], [], [5], [6], []], "answer": 7, "difficulty": "hard", "generator": "graph_invariants", "num_vertices": 7, "seed": 20260974, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 7 vertices labelled 0 to 6,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=7)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 7, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-ef1199086f83bf90-t000", "input": {"adjacency": [[1, 2, 3, 5], [3, 5], [4, 5, 6], [], [5], [6], []], "num_vertices": 7, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-ef1199086f83bf90"}], "verification": {"duration_ms": 0.8338330080732703, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and (= dist_1 (+ dist_0 2)) (<= dist_1 (+ 2 dist_0)))))\n (a!2 (or (and (= dist_2 (+ dist_1 3)) (<= dist_2 (+ 3 dist_1)))))\n (a!3 (or (and (= dist_3 (+ dist_0 2)) (<= dist_3 (+ 2 dist_0)))))\n (a!4 (and (<= dist_4 (+ 5 dist_0)) (<= dist_4 (+ 3 dist_1))))\n (a!6 (and (<= dist_5 (+ 6 dist_0)) (<= dist_5 (+ 8 dist_1))))\n (a!8 (and (<= dist_6 (+ 8 dist_2)) (<= dist_6 (+ 8 dist_4)))))\n(let ((a!5 (or (and (= dist_4 (+ dist_0 5)) a!4)\n (and (= dist_4 (+ dist_1 3)) a!4)))\n (a!7 (or (and (= dist_5 (+ dist_0 6)) a!6)\n (and (= dist...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "graphs", "id": "graphs-dag_shortest_path-6296e2cfb2e809e1", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 13, "difficulty": "hard", "edges": [[0, 1], [0, 3], [0, 4], [0, 5], [1, 2], [1, 4], [1, 5], [2, 6], [4, 6]], "generator": "graph_invariants", "num_vertices": 7, "seed": 20260975, "source": 0, "target": 6, "variant": "dag_shortest_path", "weighted_edges": [[0, 1, 2], [0, 3, 2], [0, 4, 5], [0, 5, 6], [1, 2, 3], [1, 4, 3], [1, 5, 8], [2, 6, 8], [4, 6, 8]]}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\nsource = data[\"source\"]\ntarget = data[\"target\"]\n\ninfinity = float(\"inf\")\ndist = [infinity] * n\ndist[source] = 0\nfor u, v, w in edges:\n if dist[u] != infinity and dist[u] + w < dist[v]:\n dist[v] = dist[u] + w\nprint(dist[target])\n", "statement": "You are given a directed acyclic graph with 7 vertices labelled 0 to\n6. Every edge [u, v, w] goes from a lower to a higher label and has\ninteger weight 1 <= w <= 9; edges are listed in ascending (u, v) order.\n\nPrint the minimum possible total weight of a path from source s = 0 to\ntarget t = 6. The target is guaranteed reachable, the optimum is\nunique, and both facts have been formally proven for this instance with the\nZ3 SMT solver.\n", "title": "DAG Shortest Path (n=7, weight 13)", "variant": "dag_shortest_path"}, "test_cases": [{"expected_output": 13, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_shortest_path-6296e2cfb2e809e1-t000", "input": {"edges": [[0, 1, 2], [0, 3, 2], [0, 4, 5], [0, 5, 6], [1, 2, 3], [1, 4, 3], [1, 5, 8], [2, 6, 8], [4, 6, 8]], "num_vertices": 7, "source": 0, "target": 6}, "is_public": true, "problem_id": "graphs-dag_shortest_path-6296e2cfb2e809e1"}], "verification": {"duration_ms": 1.073917024768889, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and (= dist_1 (+ dist_0 5)) (<= dist_1 (+ 5 dist_0)))))\n (a!2 (or (and (= dist_2 (+ dist_1 2)) (<= dist_2 (+ 2 dist_1)))))\n (a!3 (or (and (= dist_3 (+ dist_2 1)) (<= dist_3 (+ 1 dist_2)))))\n (a!4 (or (and (= dist_4 (+ dist_2 5)) (<= dist_4 (+ 5 dist_2)))))\n (a!5 (or (and (= dist_5 (+ dist_2 9)) (<= dist_5 (+ 9 dist_2))))))\n (and (= dist_0 0) a!1 a!2 a!3 a!4 a!5))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "graphs", "id": "graphs-dag_shortest_path-eccdefbb8fc79459", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 16, "difficulty": "hard", "edges": [[0, 1], [1, 2], [2, 3], [2, 4], [2, 5]], "generator": "graph_invariants", "num_vertices": 6, "seed": 20260976, "source": 0, "target": 5, "variant": "dag_shortest_path", "weighted_edges": [[0, 1, 5], [1, 2, 2], [2, 3, 1], [2, 4, 5], [2, 5, 9]]}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\nsource = data[\"source\"]\ntarget = data[\"target\"]\n\ninfinity = float(\"inf\")\ndist = [infinity] * n\ndist[source] = 0\nfor u, v, w in edges:\n if dist[u] != infinity and dist[u] + w < dist[v]:\n dist[v] = dist[u] + w\nprint(dist[target])\n", "statement": "You are given a directed acyclic graph with 6 vertices labelled 0 to\n5. Every edge [u, v, w] goes from a lower to a higher label and has\ninteger weight 1 <= w <= 9; edges are listed in ascending (u, v) order.\n\nPrint the minimum possible total weight of a path from source s = 0 to\ntarget t = 5. The target is guaranteed reachable, the optimum is\nunique, and both facts have been formally proven for this instance with the\nZ3 SMT solver.\n", "title": "DAG Shortest Path (n=6, weight 16)", "variant": "dag_shortest_path"}, "test_cases": [{"expected_output": 16, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_shortest_path-eccdefbb8fc79459-t000", "input": {"edges": [[0, 1, 5], [1, 2, 2], [2, 3, 1], [2, 4, 5], [2, 5, 9]], "num_vertices": 6, "source": 0, "target": 5}, "is_public": true, "problem_id": "graphs-dag_shortest_path-eccdefbb8fc79459"}], "verification": {"duration_ms": 0.8948330068960786, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_0 member_4)\n (=> member_0 member_5)\n (=> member_0 member_6)\n (=> member_1 member_2)\n (=> member_1 member_3)\n (=> member_1 member_5)\n (=> member_1 member_6)\n (=> member_2 member_7)\n (=> member_3 member_4)\n (=> member_3 member_6)\n (=> member_3 member_7)\n (=> member_4 member_5)\n (=> member_4 member_7)\n (=> member_1 (or member_0))\n (=> member_2 (or member_1))\n (=> member_3 (or member_1))\n (=> member_4 (or member_0 member_3))\n (=> member_5 (or member_0 member_1 member_4))...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "graphs", "id": "graphs-dag_reachability-0616c3bfff8e2140", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1, 4, 5, 6], [2, 3, 5, 6], [7], [4, 6, 7], [5, 7], [], [], []], "answer": 8, "difficulty": "hard", "generator": "graph_invariants", "num_vertices": 8, "seed": 20260977, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 8 vertices labelled 0 to 7,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=8)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 8, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-0616c3bfff8e2140-t000", "input": {"adjacency": [[1, 4, 5, 6], [2, 3, 5, 6], [7], [4, 6, 7], [5, 7], [], [], []], "num_vertices": 8, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-0616c3bfff8e2140"}], "verification": {"duration_ms": 0.7831249386072159, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_0 member_2)\n (=> member_0 member_4)\n (=> member_0 member_5)\n (=> member_1 member_3)\n (=> member_2 member_3)\n (=> member_2 member_4)\n (=> member_2 member_5)\n (=> member_4 member_5)\n (=> member_1 (or member_0))\n (=> member_2 (or member_0))\n (=> member_3 (or member_1 member_2))\n (=> member_4 (or member_0 member_2))\n (=> member_5 (or member_0 member_2 member_4))\n (= reachable_count\n (+ (ite member_0 1 0)\n (ite member_1 1 0)\n (ite member_2 1 0)\n (ite ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "graphs", "id": "graphs-dag_reachability-a8b12b497f49dca2", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1, 2, 4, 5], [3], [3, 4, 5], [], [5], []], "answer": 6, "difficulty": "hard", "generator": "graph_invariants", "num_vertices": 6, "seed": 20260978, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 6 vertices labelled 0 to 5,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=6)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 6, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-a8b12b497f49dca2-t000", "input": {"adjacency": [[1, 2, 4, 5], [3], [3, 4, 5], [], [5], []], "num_vertices": 6, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-a8b12b497f49dca2"}], "verification": {"duration_ms": 0.7599999662488699, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (=> (and (> pos_1 0) (and (< pos_0 0))) (< ord_0 1)))\n (a!2 (=> (and (> pos_2 0) (and (< pos_0 0))) (< ord_0 2)))\n (a!3 (=> (and (> pos_3 0) (and (< pos_0 0))) (< ord_0 3)))\n (a!4 (=> (and (> pos_1 1) (and (< pos_0 1))) (< ord_1 1)))\n (a!5 (=> (and (> pos_2 1) (and (< pos_0 1))) (< ord_1 2)))\n (a!6 (=> (and (> pos_3 1) (and (< pos_0 1))) (< ord_1 3)))\n (a!7 (=> (and (> pos_1 2) (and (< pos_0 2))) (< ord_2 1)))\n (a!8 (=> (and (> pos_2 2) (and (< pos_0 2))) (< ord_2 2)))\n (a!9 (=> (and (> pos_3 2) (and (< pos_0 2))) (< ord_2 3)))\n (a!10 ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "graphs", "id": "graphs-dag_topological_position-9ecc10f02569fa56", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 3, "difficulty": "hard", "edges": [[0, 1], [0, 2], [0, 3], [1, 4], [1, 5], [2, 5], [4, 5]], "generator": "graph_invariants", "num_vertices": 6, "seed": 20260979, "target_vertex": 3, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 6 vertices labelled 0 to\n5 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 3 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=6, vertex 3)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 3, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-9ecc10f02569fa56-t000", "input": {"edges": [[0, 1], [0, 2], [0, 3], [1, 4], [1, 5], [2, 5], [4, 5]], "num_vertices": 6, "target_vertex": 3}, "is_public": true, "problem_id": "graphs-dag_topological_position-9ecc10f02569fa56"}], "verification": {"duration_ms": 4.238541005179286, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_0 member_3)\n (=> member_0 member_5)\n (=> member_0 member_7)\n (=> member_1 member_2)\n (=> member_1 member_4)\n (=> member_2 member_3)\n (=> member_2 member_5)\n (=> member_2 member_6)\n (=> member_3 member_5)\n (=> member_3 member_7)\n (=> member_5 member_7)\n (=> member_6 member_7)\n (=> member_1 (or member_0))\n (=> member_2 (or member_1))\n (=> member_3 (or member_0 member_2))\n (=> member_4 (or member_1))\n (=> member_5 (or member_0 member_2 member_3))\n (=> member_6 (or membe...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "graphs", "id": "graphs-dag_reachability-7f046f0478b1c2e4", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1, 3, 5, 7], [2, 4], [3, 5, 6], [5, 7], [], [7], [7], []], "answer": 8, "difficulty": "hard", "generator": "graph_invariants", "num_vertices": 8, "seed": 20260980, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 8 vertices labelled 0 to 7,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=8)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 8, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-7f046f0478b1c2e4-t000", "input": {"adjacency": [[1, 3, 5, 7], [2, 4], [3, 5, 6], [5, 7], [], [7], [7], []], "num_vertices": 8, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-7f046f0478b1c2e4"}], "verification": {"duration_ms": 0.8393329917453229, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (=> (and (> pos_2 0) (and (< pos_0 0) (< pos_1 0))) (< ord_0 2)))\n (a!2 (=> (and (> pos_4 0) (and (< pos_0 0) (< pos_1 0))) (< ord_0 4)))\n (a!3 (=> (and (> pos_2 1) (and (< pos_0 1) (< pos_1 1))) (< ord_1 2)))\n (a!4 (=> (and (> pos_4 1) (and (< pos_0 1) (< pos_1 1))) (< ord_1 4)))\n (a!5 (=> (and (> pos_2 2) (and (< pos_0 2) (< pos_1 2))) (< ord_2 2)))\n (a!6 (=> (and (> pos_4 2) (and (< pos_0 2) (< pos_1 2))) (< ord_2 4)))\n (a!7 (=> (and (> pos_2 3) (and (< pos_0 3) (< pos_1 3))) (< ord_3 2)))\n (a!8 (=> (and (> pos_4 3) (and (< pos_0 3) (< pos_1 ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "graphs", "id": "graphs-dag_topological_position-13f5e6ca099ec6b8", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 4, "difficulty": "hard", "edges": [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [0, 6], [1, 2], [1, 3], [1, 4], [1, 5], [1, 6], [2, 3], [2, 5], [3, 5], [3, 6], [4, 5]], "generator": "graph_invariants", "num_vertices": 7, "seed": 20260981, "target_vertex": 4, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 7 vertices labelled 0 to\n6 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 4 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=7, vertex 4)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-13f5e6ca099ec6b8-t000", "input": {"edges": [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [0, 6], [1, 2], [1, 3], [1, 4], [1, 5], [1, 6], [2, 3], [2, 5], [3, 5], [3, 6], [4, 5]], "num_vertices": 7, "target_vertex": 4}, "is_public": true, "problem_id": "graphs-dag_topological_position-13f5e6ca099ec6b8"}], "verification": {"duration_ms": 5.482332955580205, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and (= dist_1 (+ dist_0 7)) (<= dist_1 (+ 7 dist_0)))))\n (a!2 (or (and (= dist_2 (+ dist_1 6)) (<= dist_2 (+ 6 dist_1)))))\n (a!3 (and (<= dist_3 (+ 2 dist_0)) (<= dist_3 (+ 9 dist_2))))\n (a!5 (and (<= dist_4 (+ 5 dist_1)) (<= dist_4 (+ 4 dist_3))))\n (a!7 (and (<= dist_5 (+ 4 dist_2)) (<= dist_5 (+ 3 dist_4))))\n (a!9 (and (<= dist_6 (+ 8 dist_0))\n (<= dist_6 (+ 5 dist_2))\n (<= dist_6 (+ 8 dist_3))))\n (a!11 (and (<= dist_7 (+ 3 dist_1))\n (<= dist_7 (+ 6 dist_3))\n (<= dist_7 (+ 1 dist_6)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "graphs", "id": "graphs-dag_shortest_path-bbd4839f43efb68e", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 8, "difficulty": "hard", "edges": [[0, 1], [0, 3], [0, 6], [1, 2], [1, 4], [1, 7], [2, 3], [2, 5], [2, 6], [3, 4], [3, 6], [3, 7], [4, 5], [6, 7]], "generator": "graph_invariants", "num_vertices": 8, "seed": 20260982, "source": 0, "target": 7, "variant": "dag_shortest_path", "weighted_edges": [[0, 1, 7], [0, 3, 2], [0, 6, 8], [1, 2, 6], [1, 4, 5], [1, 7, 3], [2, 3, 9], [2, 5, 4], [2, 6, 5], [3, 4, 4], [3, 6, 8], [3, 7, 6], [4, 5, 3], [6, 7, 1]]}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\nsource = data[\"source\"]\ntarget = data[\"target\"]\n\ninfinity = float(\"inf\")\ndist = [infinity] * n\ndist[source] = 0\nfor u, v, w in edges:\n if dist[u] != infinity and dist[u] + w < dist[v]:\n dist[v] = dist[u] + w\nprint(dist[target])\n", "statement": "You are given a directed acyclic graph with 8 vertices labelled 0 to\n7. Every edge [u, v, w] goes from a lower to a higher label and has\ninteger weight 1 <= w <= 9; edges are listed in ascending (u, v) order.\n\nPrint the minimum possible total weight of a path from source s = 0 to\ntarget t = 7. The target is guaranteed reachable, the optimum is\nunique, and both facts have been formally proven for this instance with the\nZ3 SMT solver.\n", "title": "DAG Shortest Path (n=8, weight 8)", "variant": "dag_shortest_path"}, "test_cases": [{"expected_output": 8, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_shortest_path-bbd4839f43efb68e-t000", "input": {"edges": [[0, 1, 7], [0, 3, 2], [0, 6, 8], [1, 2, 6], [1, 4, 5], [1, 7, 3], [2, 3, 9], [2, 5, 4], [2, 6, 5], [3, 4, 4], [3, 6, 8], [3, 7, 6], [4, 5, 3], [6, 7, 1]], "num_vertices": 8, "source": 0, "target": 7}, "is_public": true, "problem_id": "graphs-dag_shortest_path-bbd4839f43efb68e"}], "verification": {"duration_ms": 1.390875957440585, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and (= dist_1 (+ dist_0 5)) (<= dist_1 (+ 5 dist_0)))))\n (a!2 (or (and (= dist_2 (+ dist_1 1)) (<= dist_2 (+ 1 dist_1)))))\n (a!3 (or (and (= dist_3 (+ dist_0 5)) (<= dist_3 (+ 5 dist_0)))))\n (a!4 (or (and (= dist_4 (+ dist_1 3)) (<= dist_4 (+ 3 dist_1)))))\n (a!5 (or (and (= dist_5 (+ dist_0 4)) (<= dist_5 (+ 4 dist_0)))))\n (a!6 (and (<= dist_6 (+ 3 dist_3))\n (<= dist_6 (+ 6 dist_4))\n (<= dist_6 (+ 7 dist_5))))\n (a!8 (and (<= dist_7 (+ 6 dist_0))\n (<= dist_7 (+ 4 dist_1))\n (<= dist_7 (+ ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "graphs", "id": "graphs-dag_shortest_path-22a9a740d1868381", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 6, "difficulty": "hard", "edges": [[0, 1], [0, 3], [0, 5], [0, 7], [1, 2], [1, 4], [1, 7], [3, 6], [3, 7], [4, 6], [5, 6], [5, 7]], "generator": "graph_invariants", "num_vertices": 8, "seed": 20260983, "source": 0, "target": 7, "variant": "dag_shortest_path", "weighted_edges": [[0, 1, 5], [0, 3, 5], [0, 5, 4], [0, 7, 6], [1, 2, 1], [1, 4, 3], [1, 7, 4], [3, 6, 3], [3, 7, 8], [4, 6, 6], [5, 6, 7], [5, 7, 9]]}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\nsource = data[\"source\"]\ntarget = data[\"target\"]\n\ninfinity = float(\"inf\")\ndist = [infinity] * n\ndist[source] = 0\nfor u, v, w in edges:\n if dist[u] != infinity and dist[u] + w < dist[v]:\n dist[v] = dist[u] + w\nprint(dist[target])\n", "statement": "You are given a directed acyclic graph with 8 vertices labelled 0 to\n7. Every edge [u, v, w] goes from a lower to a higher label and has\ninteger weight 1 <= w <= 9; edges are listed in ascending (u, v) order.\n\nPrint the minimum possible total weight of a path from source s = 0 to\ntarget t = 7. The target is guaranteed reachable, the optimum is\nunique, and both facts have been formally proven for this instance with the\nZ3 SMT solver.\n", "title": "DAG Shortest Path (n=8, weight 6)", "variant": "dag_shortest_path"}, "test_cases": [{"expected_output": 6, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_shortest_path-22a9a740d1868381-t000", "input": {"edges": [[0, 1, 5], [0, 3, 5], [0, 5, 4], [0, 7, 6], [1, 2, 1], [1, 4, 3], [1, 7, 4], [3, 6, 3], [3, 7, 8], [4, 6, 6], [5, 6, 7], [5, 7, 9]], "num_vertices": 8, "source": 0, "target": 7}, "is_public": true, "problem_id": "graphs-dag_shortest_path-22a9a740d1868381"}], "verification": {"duration_ms": 1.0414999560452998, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (=> (and (> pos_2 0) (and (< pos_1 0))) (< ord_0 2)))\n (a!2 (=> (and (> pos_3 0) (and (< pos_1 0))) (< ord_0 3)))\n (a!3 (=> (and (> pos_2 1) (and (< pos_1 1))) (< ord_1 2)))\n (a!4 (=> (and (> pos_3 1) (and (< pos_1 1))) (< ord_1 3)))\n (a!5 (=> (and (> pos_2 2) (and (< pos_1 2))) (< ord_2 2)))\n (a!6 (=> (and (> pos_3 2) (and (< pos_1 2))) (< ord_2 3)))\n (a!7 (=> (and (> pos_2 3) (and (< pos_1 3))) (< ord_3 2)))\n (a!8 (=> (and (> pos_3 3) (and (< pos_1 3))) (< ord_3 3)))\n (a!9 (=> (and (> pos_2 4) (and (< pos_1 4))) (< ord_4 2)))\n (a!10 ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "graphs", "id": "graphs-dag_topological_position-c95783b4e7c6e06a", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 1, "difficulty": "hard", "edges": [[0, 1], [0, 5], [1, 2], [1, 3], [1, 4], [2, 4], [2, 5], [3, 5], [4, 5]], "generator": "graph_invariants", "num_vertices": 6, "seed": 20260984, "target_vertex": 1, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 6 vertices labelled 0 to\n5 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 1 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=6, vertex 1)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-c95783b4e7c6e06a-t000", "input": {"edges": [[0, 1], [0, 5], [1, 2], [1, 3], [1, 4], [2, 4], [2, 5], [3, 5], [4, 5]], "num_vertices": 6, "target_vertex": 1}, "is_public": true, "problem_id": "graphs-dag_topological_position-c95783b4e7c6e06a"}], "verification": {"duration_ms": 4.01091598905623, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (=> (and (> pos_1 0) (and (< pos_0 0))) (< ord_0 1)))\n (a!2 (=> (and (> pos_2 0) (and (< pos_0 0))) (< ord_0 2)))\n (a!3 (=> (and (> pos_3 0) (and (< pos_0 0) (< pos_2 0))) (< ord_0 3)))\n (a!4 (=> (and (> pos_4 0) (and (< pos_0 0) (< pos_2 0))) (< ord_0 4)))\n (a!5 (=> (and (> pos_1 1) (and (< pos_0 1))) (< ord_1 1)))\n (a!6 (=> (and (> pos_2 1) (and (< pos_0 1))) (< ord_1 2)))\n (a!7 (=> (and (> pos_3 1) (and (< pos_0 1) (< pos_2 1))) (< ord_1 3)))\n (a!8 (=> (and (> pos_4 1) (and (< pos_0 1) (< pos_2 1))) (< ord_1 4)))\n (a!9 (=> (and (> pos_1 ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "graphs", "id": "graphs-dag_topological_position-092fd207b04a36ee", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 2, "difficulty": "hard", "edges": [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [0, 7], [2, 3], [2, 4], [4, 5], [4, 6], [6, 7]], "generator": "graph_invariants", "num_vertices": 8, "seed": 20260985, "target_vertex": 2, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 8 vertices labelled 0 to\n7 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 2 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=8, vertex 2)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-092fd207b04a36ee-t000", "input": {"edges": [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [0, 7], [2, 3], [2, 4], [4, 5], [4, 6], [6, 7]], "num_vertices": 8, "target_vertex": 2}, "is_public": true, "problem_id": "graphs-dag_topological_position-092fd207b04a36ee"}], "verification": {"duration_ms": 6.119665922597051, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (=> (and (> pos_2 0) (and (< pos_0 0) (< pos_1 0))) (< ord_0 2)))\n (a!2 (=> (and (> pos_3 0) (and (< pos_0 0) (< pos_1 0))) (< ord_0 3)))\n (a!3 (=> (and (> pos_2 1) (and (< pos_0 1) (< pos_1 1))) (< ord_1 2)))\n (a!4 (=> (and (> pos_3 1) (and (< pos_0 1) (< pos_1 1))) (< ord_1 3)))\n (a!5 (=> (and (> pos_2 2) (and (< pos_0 2) (< pos_1 2))) (< ord_2 2)))\n (a!6 (=> (and (> pos_3 2) (and (< pos_0 2) (< pos_1 2))) (< ord_2 3)))\n (a!7 (=> (and (> pos_2 3) (and (< pos_0 3) (< pos_1 3))) (< ord_3 2)))\n (a!8 (=> (and (> pos_3 3) (and (< pos_0 3) (< pos_1 ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "graphs", "id": "graphs-dag_topological_position-8a8a744173953a2d", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 4, "difficulty": "hard", "edges": [[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [1, 5], [1, 6], [2, 4], [2, 5], [4, 5], [4, 6]], "generator": "graph_invariants", "num_vertices": 7, "seed": 20260986, "target_vertex": 4, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 7 vertices labelled 0 to\n6 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 4 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=7, vertex 4)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-8a8a744173953a2d-t000", "input": {"edges": [[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [1, 5], [1, 6], [2, 4], [2, 5], [4, 5], [4, 6]], "num_vertices": 7, "target_vertex": 4}, "is_public": true, "problem_id": "graphs-dag_topological_position-8a8a744173953a2d"}], "verification": {"duration_ms": 5.379624955821782, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and (= dist_1 (+ dist_0 4)) (<= dist_1 (+ 4 dist_0)))))\n (a!2 (or (and (= dist_2 (+ dist_1 8)) (<= dist_2 (+ 8 dist_1)))))\n (a!3 (or (and (= dist_3 (+ dist_1 7)) (<= dist_3 (+ 7 dist_1)))))\n (a!4 (and (<= dist_4 (+ 5 dist_0))\n (<= dist_4 (+ 2 dist_1))\n (<= dist_4 (+ 1 dist_3))))\n (a!6 (and (<= dist_5 (+ 5 dist_1)) (<= dist_5 (+ 2 dist_3))))\n (a!8 (or (and (= dist_6 (+ dist_4 5)) (<= dist_6 (+ 5 dist_4))))))\n(let ((a!5 (or (and (= dist_4 (+ dist_0 5)) a!4)\n (and (= dist_4 (+ dist_1 2)) a!4)\n (a...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "graphs", "id": "graphs-dag_shortest_path-ad097c5e0342e460", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 10, "difficulty": "hard", "edges": [[0, 1], [0, 4], [1, 2], [1, 3], [1, 4], [1, 5], [3, 4], [3, 5], [4, 6]], "generator": "graph_invariants", "num_vertices": 7, "seed": 20260987, "source": 0, "target": 6, "variant": "dag_shortest_path", "weighted_edges": [[0, 1, 4], [0, 4, 5], [1, 2, 8], [1, 3, 7], [1, 4, 2], [1, 5, 5], [3, 4, 1], [3, 5, 2], [4, 6, 5]]}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\nsource = data[\"source\"]\ntarget = data[\"target\"]\n\ninfinity = float(\"inf\")\ndist = [infinity] * n\ndist[source] = 0\nfor u, v, w in edges:\n if dist[u] != infinity and dist[u] + w < dist[v]:\n dist[v] = dist[u] + w\nprint(dist[target])\n", "statement": "You are given a directed acyclic graph with 7 vertices labelled 0 to\n6. Every edge [u, v, w] goes from a lower to a higher label and has\ninteger weight 1 <= w <= 9; edges are listed in ascending (u, v) order.\n\nPrint the minimum possible total weight of a path from source s = 0 to\ntarget t = 6. The target is guaranteed reachable, the optimum is\nunique, and both facts have been formally proven for this instance with the\nZ3 SMT solver.\n", "title": "DAG Shortest Path (n=7, weight 10)", "variant": "dag_shortest_path"}, "test_cases": [{"expected_output": 10, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_shortest_path-ad097c5e0342e460-t000", "input": {"edges": [[0, 1, 4], [0, 4, 5], [1, 2, 8], [1, 3, 7], [1, 4, 2], [1, 5, 5], [3, 4, 1], [3, 5, 2], [4, 6, 5]], "num_vertices": 7, "source": 0, "target": 6}, "is_public": true, "problem_id": "graphs-dag_shortest_path-ad097c5e0342e460"}], "verification": {"duration_ms": 1.1688339291140437, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and (= dist_1 (+ dist_0 8)) (<= dist_1 (+ 8 dist_0)))))\n (a!2 (or (and (= dist_2 (+ dist_0 8)) (<= dist_2 (+ 8 dist_0)))))\n (a!3 (and (<= dist_3 (+ 7 dist_0))\n (<= dist_3 (+ 6 dist_1))\n (<= dist_3 (+ 6 dist_2))))\n (a!5 (and (<= dist_4 (+ 5 dist_2)) (<= dist_4 (+ 9 dist_3))))\n (a!7 (and (<= dist_5 (+ 2 dist_0))\n (<= dist_5 (+ 5 dist_1))\n (<= dist_5 (+ 4 dist_3)))))\n(let ((a!4 (or (and (= dist_3 (+ dist_0 7)) a!3)\n (and (= dist_3 (+ dist_1 6)) a!3)\n (and (= dist_3 (+ dis...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "graphs", "id": "graphs-dag_shortest_path-adb258dded730d16", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 2, "difficulty": "hard", "edges": [[0, 1], [0, 2], [0, 3], [0, 5], [1, 3], [1, 5], [2, 3], [2, 4], [3, 4], [3, 5]], "generator": "graph_invariants", "num_vertices": 6, "seed": 20260988, "source": 0, "target": 5, "variant": "dag_shortest_path", "weighted_edges": [[0, 1, 8], [0, 2, 8], [0, 3, 7], [0, 5, 2], [1, 3, 6], [1, 5, 5], [2, 3, 6], [2, 4, 5], [3, 4, 9], [3, 5, 4]]}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\nsource = data[\"source\"]\ntarget = data[\"target\"]\n\ninfinity = float(\"inf\")\ndist = [infinity] * n\ndist[source] = 0\nfor u, v, w in edges:\n if dist[u] != infinity and dist[u] + w < dist[v]:\n dist[v] = dist[u] + w\nprint(dist[target])\n", "statement": "You are given a directed acyclic graph with 6 vertices labelled 0 to\n5. Every edge [u, v, w] goes from a lower to a higher label and has\ninteger weight 1 <= w <= 9; edges are listed in ascending (u, v) order.\n\nPrint the minimum possible total weight of a path from source s = 0 to\ntarget t = 5. The target is guaranteed reachable, the optimum is\nunique, and both facts have been formally proven for this instance with the\nZ3 SMT solver.\n", "title": "DAG Shortest Path (n=6, weight 2)", "variant": "dag_shortest_path"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_shortest_path-adb258dded730d16-t000", "input": {"edges": [[0, 1, 8], [0, 2, 8], [0, 3, 7], [0, 5, 2], [1, 3, 6], [1, 5, 5], [2, 3, 6], [2, 4, 5], [3, 4, 9], [3, 5, 4]], "num_vertices": 6, "source": 0, "target": 5}, "is_public": true, "problem_id": "graphs-dag_shortest_path-adb258dded730d16"}], "verification": {"duration_ms": 1.0732499649748206, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and (= dist_1 (+ dist_0 6)) (<= dist_1 (+ 6 dist_0)))))\n (a!2 (and (<= dist_2 (+ 1 dist_0)) (<= dist_2 (+ 9 dist_1))))\n (a!4 (or (and (= dist_3 (+ dist_0 3)) (<= dist_3 (+ 3 dist_0)))))\n (a!5 (and (<= dist_4 (+ 2 dist_1)) (<= dist_4 (+ 2 dist_2))))\n (a!7 (and (<= dist_5 (+ 6 dist_0))\n (<= dist_5 (+ 2 dist_1))\n (<= dist_5 (+ 2 dist_4))))\n (a!9 (and (<= dist_6 (+ 4 dist_1))\n (<= dist_6 (+ 5 dist_2))\n (<= dist_6 (+ 5 dist_3))\n (<= dist_6 (+ 3 dist_5)))))\n(let ((a!3 (or (and (= di...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "graphs", "id": "graphs-dag_shortest_path-be5dd6b2fdf44d93", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 6, "difficulty": "hard", "edges": [[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 4], [1, 5], [1, 6], [2, 4], [2, 6], [3, 6], [4, 5], [5, 6]], "generator": "graph_invariants", "num_vertices": 7, "seed": 20260989, "source": 0, "target": 6, "variant": "dag_shortest_path", "weighted_edges": [[0, 1, 6], [0, 2, 1], [0, 3, 3], [0, 5, 6], [1, 2, 9], [1, 4, 2], [1, 5, 2], [1, 6, 4], [2, 4, 2], [2, 6, 5], [3, 6, 5], [4, 5, 2], [5, 6, 3]]}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\nsource = data[\"source\"]\ntarget = data[\"target\"]\n\ninfinity = float(\"inf\")\ndist = [infinity] * n\ndist[source] = 0\nfor u, v, w in edges:\n if dist[u] != infinity and dist[u] + w < dist[v]:\n dist[v] = dist[u] + w\nprint(dist[target])\n", "statement": "You are given a directed acyclic graph with 7 vertices labelled 0 to\n6. Every edge [u, v, w] goes from a lower to a higher label and has\ninteger weight 1 <= w <= 9; edges are listed in ascending (u, v) order.\n\nPrint the minimum possible total weight of a path from source s = 0 to\ntarget t = 6. The target is guaranteed reachable, the optimum is\nunique, and both facts have been formally proven for this instance with the\nZ3 SMT solver.\n", "title": "DAG Shortest Path (n=7, weight 6)", "variant": "dag_shortest_path"}, "test_cases": [{"expected_output": 6, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_shortest_path-be5dd6b2fdf44d93-t000", "input": {"edges": [[0, 1, 6], [0, 2, 1], [0, 3, 3], [0, 5, 6], [1, 2, 9], [1, 4, 2], [1, 5, 2], [1, 6, 4], [2, 4, 2], [2, 6, 5], [3, 6, 5], [4, 5, 2], [5, 6, 3]], "num_vertices": 7, "source": 0, "target": 6}, "is_public": true, "problem_id": "graphs-dag_shortest_path-be5dd6b2fdf44d93"}], "verification": {"duration_ms": 1.3150830054655671, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and (distinct ord_0 ord_1 ord_2 ord_3 ord_4 ord_5 ord_6)\n (>= ord_0 0)\n (< ord_0 7)\n (>= pos_0 0)\n (< pos_0 7)\n (>= ord_1 0)\n (< ord_1 7)\n (>= pos_1 0)\n (< pos_1 7)\n (>= ord_2 0)\n (< ord_2 7)\n (>= pos_2 0)\n (< pos_2 7)\n (>= ord_3 0)\n (< ord_3 7)\n (>= pos_3 0)\n (< pos_3 7)\n (>= ord_4 0)\n (< ord_4 7)\n (>= pos_4 0)\n (< pos_4 7)\n (>= ord_5 0)\n (< ord_5 7)\n (>= pos_5 0)\n (< pos_5 7)\n (>= ord_6 0)\n (< ord_6 7)\n (>= pos_6 0)\n (< pos_6 7)\n (=> (= ord_0 0) (= pos_0 0))\n (=> (= ord...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "graphs", "id": "graphs-dag_topological_position-518e8fbcbdcce5dd", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 5, "difficulty": "hard", "edges": [[0, 1], [0, 2], [0, 3], [0, 4], [0, 6], [1, 2], [1, 3], [1, 5], [2, 3], [2, 5], [2, 6], [3, 4], [3, 5], [3, 6], [4, 6]], "generator": "graph_invariants", "num_vertices": 7, "seed": 20260990, "target_vertex": 5, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 7 vertices labelled 0 to\n6 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 5 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=7, vertex 5)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-518e8fbcbdcce5dd-t000", "input": {"edges": [[0, 1], [0, 2], [0, 3], [0, 4], [0, 6], [1, 2], [1, 3], [1, 5], [2, 3], [2, 5], [2, 6], [3, 4], [3, 5], [3, 6], [4, 6]], "num_vertices": 7, "target_vertex": 5}, "is_public": true, "problem_id": "graphs-dag_topological_position-518e8fbcbdcce5dd"}], "verification": {"duration_ms": 5.239749967586249, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (=> (and (> pos_4 0) (and (< pos_2 0))) (< ord_0 4)))\n (a!2 (=> (and (> pos_5 0) (and (< pos_2 0))) (< ord_0 5)))\n (a!3 (=> (and (> pos_4 1) (and (< pos_2 1))) (< ord_1 4)))\n (a!4 (=> (and (> pos_5 1) (and (< pos_2 1))) (< ord_1 5)))\n (a!5 (=> (and (> pos_4 2) (and (< pos_2 2))) (< ord_2 4)))\n (a!6 (=> (and (> pos_5 2) (and (< pos_2 2))) (< ord_2 5)))\n (a!7 (=> (and (> pos_4 3) (and (< pos_2 3))) (< ord_3 4)))\n (a!8 (=> (and (> pos_5 3) (and (< pos_2 3))) (< ord_3 5)))\n (a!9 (=> (and (> pos_4 4) (and (< pos_2 4))) (< ord_4 4)))\n (a!10 ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "graphs", "id": "graphs-dag_topological_position-9815de3f5b22dffa", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 4, "difficulty": "hard", "edges": [[0, 1], [0, 7], [1, 2], [1, 3], [1, 7], [2, 3], [2, 4], [2, 5], [3, 6], [3, 7], [6, 7]], "generator": "graph_invariants", "num_vertices": 8, "seed": 20260991, "target_vertex": 4, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 8 vertices labelled 0 to\n7 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 4 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=8, vertex 4)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-9815de3f5b22dffa-t000", "input": {"edges": [[0, 1], [0, 7], [1, 2], [1, 3], [1, 7], [2, 3], [2, 4], [2, 5], [3, 6], [3, 7], [6, 7]], "num_vertices": 8, "target_vertex": 4}, "is_public": true, "problem_id": "graphs-dag_topological_position-9815de3f5b22dffa"}], "verification": {"duration_ms": 6.463708006776869, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_0 member_2)\n (=> member_0 member_3)\n (=> member_1 member_3)\n (=> member_2 member_3)\n (=> member_2 member_4)\n (=> member_3 member_5)\n (=> member_3 member_6)\n (=> member_4 member_6)\n (=> member_1 (or member_0))\n (=> member_2 (or member_0))\n (=> member_3 (or member_0 member_1 member_2))\n (=> member_4 (or member_2))\n (=> member_5 (or member_3))\n (=> member_6 (or member_3 member_4))\n (= reachable_count\n (+ (ite member_0 1 0)\n (ite member_1 1 0)\n (ite member...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "graphs", "id": "graphs-dag_reachability-a25f66b2e18ad66b", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1, 2, 3], [3], [3, 4], [5, 6], [6], [], []], "answer": 7, "difficulty": "hard", "generator": "graph_invariants", "num_vertices": 7, "seed": 20260992, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 7 vertices labelled 0 to 6,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=7)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 7, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-a25f66b2e18ad66b-t000", "input": {"adjacency": [[1, 2, 3], [3], [3, 4], [5, 6], [6], [], []], "num_vertices": 7, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-a25f66b2e18ad66b"}], "verification": {"duration_ms": 0.805708987172693, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and (= dist_1 (+ dist_0 5)) (<= dist_1 (+ 5 dist_0)))))\n (a!2 (or (and (= dist_2 (+ dist_0 7)) (<= dist_2 (+ 7 dist_0)))))\n (a!3 (and (<= dist_3 (+ 3 dist_0)) (<= dist_3 (+ 6 dist_2))))\n (a!5 (or (and (= dist_4 (+ dist_0 2)) (<= dist_4 (+ 2 dist_0)))))\n (a!6 (and (<= dist_5 (+ 1 dist_1))\n (<= dist_5 (+ 7 dist_2))\n (<= dist_5 (+ 4 dist_3))\n (<= dist_5 (+ 3 dist_4))))\n (a!8 (and (<= dist_6 (+ 7 dist_0))\n (<= dist_6 (+ 5 dist_1))\n (<= dist_6 (+ 8 dist_3))\n (<= dist...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "graphs", "id": "graphs-dag_shortest_path-e1fc02cdb0de773f", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 4, "difficulty": "hard", "edges": [[0, 1], [0, 2], [0, 3], [0, 4], [0, 6], [1, 5], [1, 6], [2, 3], [2, 5], [3, 5], [3, 6], [4, 5], [4, 6]], "generator": "graph_invariants", "num_vertices": 7, "seed": 20260993, "source": 0, "target": 6, "variant": "dag_shortest_path", "weighted_edges": [[0, 1, 5], [0, 2, 7], [0, 3, 3], [0, 4, 2], [0, 6, 7], [1, 5, 1], [1, 6, 5], [2, 3, 6], [2, 5, 7], [3, 5, 4], [3, 6, 8], [4, 5, 3], [4, 6, 2]]}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\nsource = data[\"source\"]\ntarget = data[\"target\"]\n\ninfinity = float(\"inf\")\ndist = [infinity] * n\ndist[source] = 0\nfor u, v, w in edges:\n if dist[u] != infinity and dist[u] + w < dist[v]:\n dist[v] = dist[u] + w\nprint(dist[target])\n", "statement": "You are given a directed acyclic graph with 7 vertices labelled 0 to\n6. Every edge [u, v, w] goes from a lower to a higher label and has\ninteger weight 1 <= w <= 9; edges are listed in ascending (u, v) order.\n\nPrint the minimum possible total weight of a path from source s = 0 to\ntarget t = 6. The target is guaranteed reachable, the optimum is\nunique, and both facts have been formally proven for this instance with the\nZ3 SMT solver.\n", "title": "DAG Shortest Path (n=7, weight 4)", "variant": "dag_shortest_path"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_shortest_path-e1fc02cdb0de773f-t000", "input": {"edges": [[0, 1, 5], [0, 2, 7], [0, 3, 3], [0, 4, 2], [0, 6, 7], [1, 5, 1], [1, 6, 5], [2, 3, 6], [2, 5, 7], [3, 5, 4], [3, 6, 8], [4, 5, 3], [4, 6, 2]], "num_vertices": 7, "source": 0, "target": 6}, "is_public": true, "problem_id": "graphs-dag_shortest_path-e1fc02cdb0de773f"}], "verification": {"duration_ms": 1.2109580566175282, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_0 member_4)\n (=> member_1 member_2)\n (=> member_1 member_4)\n (=> member_1 member_5)\n (=> member_2 member_3)\n (=> member_2 member_4)\n (=> member_3 member_4)\n (=> member_1 (or member_0))\n (=> member_2 (or member_1))\n (=> member_3 (or member_2))\n (=> member_4 (or member_0 member_1 member_2 member_3))\n (=> member_5 (or member_1))\n (= reachable_count\n (+ (ite member_0 1 0)\n (ite member_1 1 0)\n (ite member_2 1 0)\n (ite member_3 1 0)\n (ite member_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "graphs", "id": "graphs-dag_reachability-45fd4753df54b8da", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1, 4], [2, 4, 5], [3, 4], [4], [], []], "answer": 6, "difficulty": "hard", "generator": "graph_invariants", "num_vertices": 6, "seed": 20260994, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 6 vertices labelled 0 to 5,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=6)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 6, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-45fd4753df54b8da-t000", "input": {"adjacency": [[1, 4], [2, 4, 5], [3, 4], [4], [], []], "num_vertices": 6, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-45fd4753df54b8da"}], "verification": {"duration_ms": 0.8286670199595392, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_0 member_2)\n (=> member_0 member_3)\n (=> member_1 member_3)\n (=> member_2 member_4)\n (=> member_2 member_5)\n (=> member_2 member_6)\n (=> member_3 member_4)\n (=> member_4 member_6)\n (=> member_1 (or member_0))\n (=> member_2 (or member_0))\n (=> member_3 (or member_0 member_1))\n (=> member_4 (or member_2 member_3))\n (=> member_5 (or member_2))\n (=> member_6 (or member_2 member_4))\n (= reachable_count\n (+ (ite member_0 1 0)\n (ite member_1 1 0)\n (ite member...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "graphs", "id": "graphs-dag_reachability-c5ee475c2f2b7b97", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1, 2, 3], [3], [4, 5, 6], [4], [6], [], []], "answer": 7, "difficulty": "hard", "generator": "graph_invariants", "num_vertices": 7, "seed": 20260995, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 7 vertices labelled 0 to 6,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=7)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 7, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-c5ee475c2f2b7b97-t000", "input": {"adjacency": [[1, 2, 3], [3], [4, 5, 6], [4], [6], [], []], "num_vertices": 7, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-c5ee475c2f2b7b97"}], "verification": {"duration_ms": 0.7743739988654852, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and (= dist_1 (+ dist_0 6)) (<= dist_1 (+ 6 dist_0)))))\n (a!2 (or (and (= dist_2 (+ dist_1 2)) (<= dist_2 (+ 2 dist_1)))))\n (a!3 (and (<= dist_3 (+ 7 dist_0)) (<= dist_3 (+ 2 dist_2))))\n (a!5 (or (and (= dist_4 (+ dist_1 1)) (<= dist_4 (+ 1 dist_1)))))\n (a!6 (and (<= dist_5 (+ 4 dist_1)) (<= dist_5 (+ 2 dist_3)))))\n(let ((a!4 (or (and (= dist_3 (+ dist_0 7)) a!3)\n (and (= dist_3 (+ dist_2 2)) a!3)))\n (a!7 (or (and (= dist_5 (+ dist_1 4)) a!6)\n (and (= dist_5 (+ dist_3 2)) a!6))))\n (and (= dist_0 0) a!1 a!2 a!4 a!5 a!7)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "graphs", "id": "graphs-dag_shortest_path-df5a9068e3c8e025", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 9, "difficulty": "hard", "edges": [[0, 1], [0, 3], [1, 2], [1, 4], [1, 5], [2, 3], [3, 5]], "generator": "graph_invariants", "num_vertices": 6, "seed": 20260996, "source": 0, "target": 5, "variant": "dag_shortest_path", "weighted_edges": [[0, 1, 6], [0, 3, 7], [1, 2, 2], [1, 4, 1], [1, 5, 4], [2, 3, 2], [3, 5, 2]]}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\nsource = data[\"source\"]\ntarget = data[\"target\"]\n\ninfinity = float(\"inf\")\ndist = [infinity] * n\ndist[source] = 0\nfor u, v, w in edges:\n if dist[u] != infinity and dist[u] + w < dist[v]:\n dist[v] = dist[u] + w\nprint(dist[target])\n", "statement": "You are given a directed acyclic graph with 6 vertices labelled 0 to\n5. Every edge [u, v, w] goes from a lower to a higher label and has\ninteger weight 1 <= w <= 9; edges are listed in ascending (u, v) order.\n\nPrint the minimum possible total weight of a path from source s = 0 to\ntarget t = 5. The target is guaranteed reachable, the optimum is\nunique, and both facts have been formally proven for this instance with the\nZ3 SMT solver.\n", "title": "DAG Shortest Path (n=6, weight 9)", "variant": "dag_shortest_path"}, "test_cases": [{"expected_output": 9, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_shortest_path-df5a9068e3c8e025-t000", "input": {"edges": [[0, 1, 6], [0, 3, 7], [1, 2, 2], [1, 4, 1], [1, 5, 4], [2, 3, 2], [3, 5, 2]], "num_vertices": 6, "source": 0, "target": 5}, "is_public": true, "problem_id": "graphs-dag_shortest_path-df5a9068e3c8e025"}], "verification": {"duration_ms": 0.9982089977711439, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_0 member_3)\n (=> member_0 member_5)\n (=> member_1 member_2)\n (=> member_1 member_4)\n (=> member_1 member_6)\n (=> member_2 member_5)\n (=> member_2 member_6)\n (=> member_3 member_4)\n (=> member_5 member_6)\n (=> member_1 (or member_0))\n (=> member_2 (or member_1))\n (=> member_3 (or member_0))\n (=> member_4 (or member_1 member_3))\n (=> member_5 (or member_0 member_2))\n (=> member_6 (or member_1 member_2 member_5))\n (= reachable_count\n (+ (ite member_0 1 0)\n (ite...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "graphs", "id": "graphs-dag_reachability-32949ad2b80e50b9", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1, 3, 5], [2, 4, 6], [5, 6], [4], [], [6], []], "answer": 7, "difficulty": "hard", "generator": "graph_invariants", "num_vertices": 7, "seed": 20260997, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 7 vertices labelled 0 to 6,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=7)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 7, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-32949ad2b80e50b9-t000", "input": {"adjacency": [[1, 3, 5], [2, 4, 6], [5, 6], [4], [], [6], []], "num_vertices": 7, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-32949ad2b80e50b9"}], "verification": {"duration_ms": 0.7670839549973607, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (=> (and (> pos_1 0) (and (< pos_0 0))) (< ord_0 1)))\n (a!2 (=> (and (> pos_2 0) (and (< pos_0 0))) (< ord_0 2)))\n (a!3 (=> (and (> pos_1 1) (and (< pos_0 1))) (< ord_1 1)))\n (a!4 (=> (and (> pos_2 1) (and (< pos_0 1))) (< ord_1 2)))\n (a!5 (=> (and (> pos_1 2) (and (< pos_0 2))) (< ord_2 1)))\n (a!6 (=> (and (> pos_2 2) (and (< pos_0 2))) (< ord_2 2)))\n (a!7 (=> (and (> pos_1 3) (and (< pos_0 3))) (< ord_3 1)))\n (a!8 (=> (and (> pos_2 3) (and (< pos_0 3))) (< ord_3 2)))\n (a!9 (=> (and (> pos_1 4) (and (< pos_0 4))) (< ord_4 1)))\n (a!10 ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "graphs", "id": "graphs-dag_topological_position-4521f2132eab9f11", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 3, "difficulty": "hard", "edges": [[0, 1], [0, 2], [0, 3], [0, 4], [1, 3], [1, 5], [3, 4], [4, 5]], "generator": "graph_invariants", "num_vertices": 6, "seed": 20260998, "target_vertex": 3, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 6 vertices labelled 0 to\n5 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 3 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=6, vertex 3)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 3, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-4521f2132eab9f11-t000", "input": {"edges": [[0, 1], [0, 2], [0, 3], [0, 4], [1, 3], [1, 5], [3, 4], [4, 5]], "num_vertices": 6, "target_vertex": 3}, "is_public": true, "problem_id": "graphs-dag_topological_position-4521f2132eab9f11"}], "verification": {"duration_ms": 4.068000009283423, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_0 member_4)\n (=> member_0 member_5)\n (=> member_0 member_6)\n (=> member_1 member_2)\n (=> member_1 member_3)\n (=> member_1 member_4)\n (=> member_2 member_4)\n (=> member_2 member_5)\n (=> member_3 member_4)\n (=> member_3 member_5)\n (=> member_3 member_6)\n (=> member_4 member_5)\n (=> member_4 member_6)\n (=> member_1 (or member_0))\n (=> member_2 (or member_1))\n (=> member_3 (or member_1))\n (=> member_4 (or member_0 member_1 member_2 member_3))\n (=> member_5 (or member_0 m...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "graphs", "id": "graphs-dag_reachability-395b23a9ddda7737", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1, 4, 5, 6], [2, 3, 4], [4, 5], [4, 5, 6], [5, 6], [], []], "answer": 7, "difficulty": "extreme", "generator": "graph_invariants", "num_vertices": 7, "seed": 20260999, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 7 vertices labelled 0 to 6,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=7)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 7, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-395b23a9ddda7737-t000", "input": {"adjacency": [[1, 4, 5, 6], [2, 3, 4], [4, 5], [4, 5, 6], [5, 6], [], []], "num_vertices": 7, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-395b23a9ddda7737"}], "verification": {"duration_ms": 0.8229590021073818, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_0 member_2)\n (=> member_0 member_3)\n (=> member_0 member_4)\n (=> member_0 member_6)\n (=> member_1 member_2)\n (=> member_1 member_3)\n (=> member_1 member_4)\n (=> member_1 member_6)\n (=> member_2 member_3)\n (=> member_2 member_4)\n (=> member_2 member_5)\n (=> member_2 member_6)\n (=> member_3 member_4)\n (=> member_3 member_5)\n (=> member_4 member_5)\n (=> member_4 member_6)\n (=> member_5 member_6)\n (=> member_1 (or member_0))\n (=> member_2 (or member_0 member_1))\n ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "graphs", "id": "graphs-dag_reachability-4fe8d92c0e7e75a7", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1, 2, 3, 4, 6], [2, 3, 4, 6], [3, 4, 5, 6], [4, 5], [5, 6], [6], []], "answer": 7, "difficulty": "extreme", "generator": "graph_invariants", "num_vertices": 7, "seed": 20261000, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 7 vertices labelled 0 to 6,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=7)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 7, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-4fe8d92c0e7e75a7-t000", "input": {"adjacency": [[1, 2, 3, 4, 6], [2, 3, 4, 6], [3, 4, 5, 6], [4, 5], [5, 6], [6], []], "num_vertices": 7, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-4fe8d92c0e7e75a7"}], "verification": {"duration_ms": 0.8074590587057173, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_0 member_2)\n (=> member_0 member_7)\n (=> member_1 member_3)\n (=> member_1 member_4)\n (=> member_1 member_5)\n (=> member_1 member_6)\n (=> member_2 member_4)\n (=> member_2 member_6)\n (=> member_2 member_7)\n (=> member_3 member_4)\n (=> member_3 member_7)\n (=> member_4 member_5)\n (=> member_4 member_6)\n (=> member_4 member_7)\n (=> member_5 member_7)\n (=> member_6 member_7)\n (=> member_1 (or member_0))\n (=> member_2 (or member_0))\n (=> member_3 (or member_1))\n (=>...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "graphs", "id": "graphs-dag_reachability-f1bc714e52b49a3e", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1, 2, 7], [3, 4, 5, 6], [4, 6, 7], [4, 7], [5, 6, 7], [7], [7], []], "answer": 8, "difficulty": "extreme", "generator": "graph_invariants", "num_vertices": 8, "seed": 20261001, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 8 vertices labelled 0 to 7,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=8)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 8, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-f1bc714e52b49a3e-t000", "input": {"adjacency": [[1, 2, 7], [3, 4, 5, 6], [4, 6, 7], [4, 7], [5, 6, 7], [7], [7], []], "num_vertices": 8, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-f1bc714e52b49a3e"}], "verification": {"duration_ms": 0.784706964623183, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_0 member_2)\n (=> member_0 member_6)\n (=> member_1 member_2)\n (=> member_1 member_3)\n (=> member_1 member_5)\n (=> member_1 member_6)\n (=> member_2 member_3)\n (=> member_3 member_4)\n (=> member_3 member_6)\n (=> member_4 member_5)\n (=> member_4 member_6)\n (=> member_5 member_6)\n (=> member_1 (or member_0))\n (=> member_2 (or member_0 member_1))\n (=> member_3 (or member_1 member_2))\n (=> member_4 (or member_3))\n (=> member_5 (or member_1 member_4))\n (=> member_6 (or membe...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "graphs", "id": "graphs-dag_reachability-0a03dfeffed20c3d", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1, 2, 6], [2, 3, 5, 6], [3], [4, 6], [5, 6], [6], []], "answer": 7, "difficulty": "extreme", "generator": "graph_invariants", "num_vertices": 7, "seed": 20261002, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 7 vertices labelled 0 to 6,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=7)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 7, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-0a03dfeffed20c3d-t000", "input": {"adjacency": [[1, 2, 6], [2, 3, 5, 6], [3], [4, 6], [5, 6], [6], []], "num_vertices": 7, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-0a03dfeffed20c3d"}], "verification": {"duration_ms": 0.772416009567678, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_0 member_3)\n (=> member_0 member_5)\n (=> member_0 member_6)\n (=> member_1 member_2)\n (=> member_1 member_6)\n (=> member_2 member_4)\n (=> member_2 member_5)\n (=> member_2 member_6)\n (=> member_3 member_4)\n (=> member_3 member_5)\n (=> member_3 member_6)\n (=> member_4 member_5)\n (=> member_1 (or member_0))\n (=> member_2 (or member_1))\n (=> member_3 (or member_0))\n (=> member_4 (or member_2 member_3))\n (=> member_5 (or member_0 member_2 member_3 member_4))\n (=> member_6 ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "graphs", "id": "graphs-dag_reachability-5a3211768a5ccf71", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1, 3, 5, 6], [2, 6], [4, 5, 6], [4, 5, 6], [5], [], []], "answer": 7, "difficulty": "extreme", "generator": "graph_invariants", "num_vertices": 7, "seed": 20261003, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 7 vertices labelled 0 to 6,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=7)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 7, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-5a3211768a5ccf71-t000", "input": {"adjacency": [[1, 3, 5, 6], [2, 6], [4, 5, 6], [4, 5, 6], [5], [], []], "num_vertices": 7, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-5a3211768a5ccf71"}], "verification": {"duration_ms": 0.7897500181570649, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_0 member_2)\n (=> member_0 member_3)\n (=> member_0 member_4)\n (=> member_0 member_6)\n (=> member_1 member_2)\n (=> member_1 member_3)\n (=> member_1 member_4)\n (=> member_1 member_6)\n (=> member_2 member_3)\n (=> member_2 member_4)\n (=> member_2 member_6)\n (=> member_3 member_6)\n (=> member_4 member_5)\n (=> member_1 (or member_0))\n (=> member_2 (or member_0 member_1))\n (=> member_3 (or member_0 member_1 member_2))\n (=> member_4 (or member_0 member_1 member_2))\n (=> membe...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "graphs", "id": "graphs-dag_reachability-9320aa3d7b728ffd", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1, 2, 3, 4, 6], [2, 3, 4, 6], [3, 4, 6], [6], [5], [], []], "answer": 7, "difficulty": "extreme", "generator": "graph_invariants", "num_vertices": 7, "seed": 20261004, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 7 vertices labelled 0 to 6,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=7)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 7, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-9320aa3d7b728ffd-t000", "input": {"adjacency": [[1, 2, 3, 4, 6], [2, 3, 4, 6], [3, 4, 6], [6], [5], [], []], "num_vertices": 7, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-9320aa3d7b728ffd"}], "verification": {"duration_ms": 0.7953750318847597, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and (= dist_1 (+ dist_0 2)) (<= dist_1 (+ 2 dist_0)))))\n (a!2 (or (and (= dist_2 (+ dist_1 2)) (<= dist_2 (+ 2 dist_1)))))\n (a!3 (and (<= dist_3 (+ 2 dist_0)) (<= dist_3 (+ 1 dist_1))))\n (a!5 (or (and (= dist_4 (+ dist_3 1)) (<= dist_4 (+ 1 dist_3)))))\n (a!6 (and (<= dist_5 (+ 8 dist_0))\n (<= dist_5 (+ 6 dist_1))\n (<= dist_5 (+ 5 dist_2))\n (<= dist_5 (+ 6 dist_4))))\n (a!8 (and (<= dist_6 (+ 5 dist_1))\n (<= dist_6 (+ 9 dist_2))\n (<= dist_6 (+ 8 dist_3))\n (<= dist...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "graphs", "id": "graphs-dag_shortest_path-63e96aed9d6beba7", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 3, "difficulty": "extreme", "edges": [[0, 1], [0, 3], [0, 5], [1, 2], [1, 3], [1, 5], [1, 6], [1, 7], [2, 5], [2, 6], [3, 4], [3, 6], [4, 5], [4, 6], [4, 7], [5, 7]], "generator": "graph_invariants", "num_vertices": 8, "seed": 20261005, "source": 0, "target": 7, "variant": "dag_shortest_path", "weighted_edges": [[0, 1, 2], [0, 3, 2], [0, 5, 8], [1, 2, 2], [1, 3, 1], [1, 5, 6], [1, 6, 5], [1, 7, 1], [2, 5, 5], [2, 6, 9], [3, 4, 1], [3, 6, 8], [4, 5, 6], [4, 6, 5], [4, 7, 1], [5, 7, 1]]}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\nsource = data[\"source\"]\ntarget = data[\"target\"]\n\ninfinity = float(\"inf\")\ndist = [infinity] * n\ndist[source] = 0\nfor u, v, w in edges:\n if dist[u] != infinity and dist[u] + w < dist[v]:\n dist[v] = dist[u] + w\nprint(dist[target])\n", "statement": "You are given a directed acyclic graph with 8 vertices labelled 0 to\n7. Every edge [u, v, w] goes from a lower to a higher label and has\ninteger weight 1 <= w <= 9; edges are listed in ascending (u, v) order.\n\nPrint the minimum possible total weight of a path from source s = 0 to\ntarget t = 7. The target is guaranteed reachable, the optimum is\nunique, and both facts have been formally proven for this instance with the\nZ3 SMT solver.\n", "title": "DAG Shortest Path (n=8, weight 3)", "variant": "dag_shortest_path"}, "test_cases": [{"expected_output": 3, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_shortest_path-63e96aed9d6beba7-t000", "input": {"edges": [[0, 1, 2], [0, 3, 2], [0, 5, 8], [1, 2, 2], [1, 3, 1], [1, 5, 6], [1, 6, 5], [1, 7, 1], [2, 5, 5], [2, 6, 9], [3, 4, 1], [3, 6, 8], [4, 5, 6], [4, 6, 5], [4, 7, 1], [5, 7, 1]], "num_vertices": 8, "source": 0, "target": 7}, "is_public": true, "problem_id": "graphs-dag_shortest_path-63e96aed9d6beba7"}], "verification": {"duration_ms": 1.448251015972346, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and (= dist_1 (+ dist_0 4)) (<= dist_1 (+ 4 dist_0)))))\n (a!2 (or (and (= dist_2 (+ dist_0 2)) (<= dist_2 (+ 2 dist_0)))))\n (a!3 (and (<= dist_3 (+ 9 dist_0))\n (<= dist_3 (+ 4 dist_1))\n (<= dist_3 (+ 3 dist_2))))\n (a!5 (and (<= dist_4 (+ 9 dist_0))\n (<= dist_4 (+ 5 dist_1))\n (<= dist_4 (+ 4 dist_2))))\n (a!7 (and (<= dist_5 (+ 7 dist_1))\n (<= dist_5 (+ 8 dist_2))\n (<= dist_5 (+ 5 dist_3))\n (<= dist_5 (+ 6 dist_4))))\n (a!9 (and (<= dist_6 (+ 4 dist_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "graphs", "id": "graphs-dag_shortest_path-4806c9274713e91a", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 4, "difficulty": "extreme", "edges": [[0, 1], [0, 2], [0, 3], [0, 4], [1, 3], [1, 4], [1, 5], [1, 6], [2, 3], [2, 4], [2, 5], [2, 6], [3, 5], [3, 6], [4, 5], [4, 6]], "generator": "graph_invariants", "num_vertices": 7, "seed": 20261006, "source": 0, "target": 6, "variant": "dag_shortest_path", "weighted_edges": [[0, 1, 4], [0, 2, 2], [0, 3, 9], [0, 4, 9], [1, 3, 4], [1, 4, 5], [1, 5, 7], [1, 6, 4], [2, 3, 3], [2, 4, 4], [2, 5, 8], [2, 6, 2], [3, 5, 5], [3, 6, 8], [4, 5, 6], [4, 6, 9]]}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\nsource = data[\"source\"]\ntarget = data[\"target\"]\n\ninfinity = float(\"inf\")\ndist = [infinity] * n\ndist[source] = 0\nfor u, v, w in edges:\n if dist[u] != infinity and dist[u] + w < dist[v]:\n dist[v] = dist[u] + w\nprint(dist[target])\n", "statement": "You are given a directed acyclic graph with 7 vertices labelled 0 to\n6. Every edge [u, v, w] goes from a lower to a higher label and has\ninteger weight 1 <= w <= 9; edges are listed in ascending (u, v) order.\n\nPrint the minimum possible total weight of a path from source s = 0 to\ntarget t = 6. The target is guaranteed reachable, the optimum is\nunique, and both facts have been formally proven for this instance with the\nZ3 SMT solver.\n", "title": "DAG Shortest Path (n=7, weight 4)", "variant": "dag_shortest_path"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_shortest_path-4806c9274713e91a-t000", "input": {"edges": [[0, 1, 4], [0, 2, 2], [0, 3, 9], [0, 4, 9], [1, 3, 4], [1, 4, 5], [1, 5, 7], [1, 6, 4], [2, 3, 3], [2, 4, 4], [2, 5, 8], [2, 6, 2], [3, 5, 5], [3, 6, 8], [4, 5, 6], [4, 6, 9]], "num_vertices": 7, "source": 0, "target": 6}, "is_public": true, "problem_id": "graphs-dag_shortest_path-4806c9274713e91a"}], "verification": {"duration_ms": 1.2433340307325125, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_0 member_2)\n (=> member_0 member_3)\n (=> member_0 member_5)\n (=> member_1 member_2)\n (=> member_1 member_3)\n (=> member_1 member_4)\n (=> member_2 member_3)\n (=> member_2 member_4)\n (=> member_2 member_6)\n (=> member_2 member_7)\n (=> member_3 member_4)\n (=> member_3 member_5)\n (=> member_3 member_6)\n (=> member_3 member_7)\n (=> member_4 member_6)\n (=> member_4 member_7)\n (=> member_5 member_6)\n (=> member_5 member_7)\n (=> member_1 (or member_0))\n (=> member_2 ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "graphs", "id": "graphs-dag_reachability-80329e1bba012462", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1, 2, 3, 5], [2, 3, 4], [3, 4, 6, 7], [4, 5, 6, 7], [6, 7], [6, 7], [], []], "answer": 8, "difficulty": "extreme", "generator": "graph_invariants", "num_vertices": 8, "seed": 20261007, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 8 vertices labelled 0 to 7,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=8)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 8, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-80329e1bba012462-t000", "input": {"adjacency": [[1, 2, 3, 5], [2, 3, 4], [3, 4, 6, 7], [4, 5, 6, 7], [6, 7], [6, 7], [], []], "num_vertices": 8, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-80329e1bba012462"}], "verification": {"duration_ms": 0.7929999264888465, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and (= dist_1 (+ dist_0 3)) (<= dist_1 (+ 3 dist_0)))))\n (a!2 (and (<= dist_2 (+ 2 dist_0)) (<= dist_2 (+ 5 dist_1))))\n (a!4 (or (and (= dist_3 (+ dist_2 5)) (<= dist_3 (+ 5 dist_2)))))\n (a!5 (and (<= dist_4 (+ 4 dist_0))\n (<= dist_4 (+ 2 dist_1))\n (<= dist_4 (+ 8 dist_2))\n (<= dist_4 (+ 8 dist_3))))\n (a!7 (and (<= dist_5 (+ 8 dist_0))\n (<= dist_5 (+ 9 dist_3))\n (<= dist_5 (+ 6 dist_4))))\n (a!9 (and (<= dist_6 (+ 6 dist_0))\n (<= dist_6 (+ 6 dist_2))\n ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "graphs", "id": "graphs-dag_shortest_path-52bb6ed9b1869df1", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 6, "difficulty": "extreme", "edges": [[0, 1], [0, 2], [0, 4], [0, 5], [0, 6], [1, 2], [1, 4], [2, 3], [2, 4], [2, 6], [3, 4], [3, 5], [4, 5], [5, 6]], "generator": "graph_invariants", "num_vertices": 7, "seed": 20261008, "source": 0, "target": 6, "variant": "dag_shortest_path", "weighted_edges": [[0, 1, 3], [0, 2, 2], [0, 4, 4], [0, 5, 8], [0, 6, 6], [1, 2, 5], [1, 4, 2], [2, 3, 5], [2, 4, 8], [2, 6, 6], [3, 4, 8], [3, 5, 9], [4, 5, 6], [5, 6, 2]]}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\nsource = data[\"source\"]\ntarget = data[\"target\"]\n\ninfinity = float(\"inf\")\ndist = [infinity] * n\ndist[source] = 0\nfor u, v, w in edges:\n if dist[u] != infinity and dist[u] + w < dist[v]:\n dist[v] = dist[u] + w\nprint(dist[target])\n", "statement": "You are given a directed acyclic graph with 7 vertices labelled 0 to\n6. Every edge [u, v, w] goes from a lower to a higher label and has\ninteger weight 1 <= w <= 9; edges are listed in ascending (u, v) order.\n\nPrint the minimum possible total weight of a path from source s = 0 to\ntarget t = 6. The target is guaranteed reachable, the optimum is\nunique, and both facts have been formally proven for this instance with the\nZ3 SMT solver.\n", "title": "DAG Shortest Path (n=7, weight 6)", "variant": "dag_shortest_path"}, "test_cases": [{"expected_output": 6, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_shortest_path-52bb6ed9b1869df1-t000", "input": {"edges": [[0, 1, 3], [0, 2, 2], [0, 4, 4], [0, 5, 8], [0, 6, 6], [1, 2, 5], [1, 4, 2], [2, 3, 5], [2, 4, 8], [2, 6, 6], [3, 4, 8], [3, 5, 9], [4, 5, 6], [5, 6, 2]], "num_vertices": 7, "source": 0, "target": 6}, "is_public": true, "problem_id": "graphs-dag_shortest_path-52bb6ed9b1869df1"}], "verification": {"duration_ms": 1.290624961256981, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (=> (and (> pos_1 0) (and (< pos_0 0))) (< ord_0 1)))\n (a!2 (=> (and (> pos_2 0) (and (< pos_0 0))) (< ord_0 2)))\n (a!3 (=> (and (> pos_1 1) (and (< pos_0 1))) (< ord_1 1)))\n (a!4 (=> (and (> pos_2 1) (and (< pos_0 1))) (< ord_1 2)))\n (a!5 (=> (and (> pos_1 2) (and (< pos_0 2))) (< ord_2 1)))\n (a!6 (=> (and (> pos_2 2) (and (< pos_0 2))) (< ord_2 2)))\n (a!7 (=> (and (> pos_1 3) (and (< pos_0 3))) (< ord_3 1)))\n (a!8 (=> (and (> pos_2 3) (and (< pos_0 3))) (< ord_3 2)))\n (a!9 (=> (and (> pos_1 4) (and (< pos_0 4))) (< ord_4 1)))\n (a!10 ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "graphs", "id": "graphs-dag_topological_position-6d1b2c8689f2e9b4", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 5, "difficulty": "extreme", "edges": [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [0, 6], [1, 3], [1, 4], [1, 6], [2, 3], [3, 6], [4, 5]], "generator": "graph_invariants", "num_vertices": 7, "seed": 20261009, "target_vertex": 5, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 7 vertices labelled 0 to\n6 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 5 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=7, vertex 5)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-6d1b2c8689f2e9b4-t000", "input": {"edges": [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [0, 6], [1, 3], [1, 4], [1, 6], [2, 3], [3, 6], [4, 5]], "num_vertices": 7, "target_vertex": 5}, "is_public": true, "problem_id": "graphs-dag_topological_position-6d1b2c8689f2e9b4"}], "verification": {"duration_ms": 5.101333954371512, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and (= dist_1 (+ dist_0 2)) (<= dist_1 (+ 2 dist_0)))))\n (a!2 (and (<= dist_2 (+ 5 dist_0)) (<= dist_2 (+ 8 dist_1))))\n (a!4 (or (and (= dist_3 (+ dist_0 6)) (<= dist_3 (+ 6 dist_0)))))\n (a!5 (and (<= dist_4 (+ 3 dist_0))\n (<= dist_4 (+ 6 dist_1))\n (<= dist_4 (+ 6 dist_2))\n (<= dist_4 (+ 1 dist_3))))\n (a!7 (and (<= dist_5 (+ 8 dist_0))\n (<= dist_5 (+ 2 dist_1))\n (<= dist_5 (+ 3 dist_3))\n (<= dist_5 (+ 9 dist_4))))\n (a!9 (and (<= dist_6 (+ 9 dist_0)) (<= dist_6 (...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "graphs", "id": "graphs-dag_shortest_path-38bfaff94879df79", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 7, "difficulty": "extreme", "edges": [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [0, 6], [1, 2], [1, 4], [1, 5], [2, 4], [3, 4], [3, 5], [4, 5], [5, 6]], "generator": "graph_invariants", "num_vertices": 7, "seed": 20261010, "source": 0, "target": 6, "variant": "dag_shortest_path", "weighted_edges": [[0, 1, 2], [0, 2, 5], [0, 3, 6], [0, 4, 3], [0, 5, 8], [0, 6, 9], [1, 2, 8], [1, 4, 6], [1, 5, 2], [2, 4, 6], [3, 4, 1], [3, 5, 3], [4, 5, 9], [5, 6, 3]]}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\nsource = data[\"source\"]\ntarget = data[\"target\"]\n\ninfinity = float(\"inf\")\ndist = [infinity] * n\ndist[source] = 0\nfor u, v, w in edges:\n if dist[u] != infinity and dist[u] + w < dist[v]:\n dist[v] = dist[u] + w\nprint(dist[target])\n", "statement": "You are given a directed acyclic graph with 7 vertices labelled 0 to\n6. Every edge [u, v, w] goes from a lower to a higher label and has\ninteger weight 1 <= w <= 9; edges are listed in ascending (u, v) order.\n\nPrint the minimum possible total weight of a path from source s = 0 to\ntarget t = 6. The target is guaranteed reachable, the optimum is\nunique, and both facts have been formally proven for this instance with the\nZ3 SMT solver.\n", "title": "DAG Shortest Path (n=7, weight 7)", "variant": "dag_shortest_path"}, "test_cases": [{"expected_output": 7, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_shortest_path-38bfaff94879df79-t000", "input": {"edges": [[0, 1, 2], [0, 2, 5], [0, 3, 6], [0, 4, 3], [0, 5, 8], [0, 6, 9], [1, 2, 8], [1, 4, 6], [1, 5, 2], [2, 4, 6], [3, 4, 1], [3, 5, 3], [4, 5, 9], [5, 6, 3]], "num_vertices": 7, "source": 0, "target": 6}, "is_public": true, "problem_id": "graphs-dag_shortest_path-38bfaff94879df79"}], "verification": {"duration_ms": 1.249832974281162, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and (= dist_1 (+ dist_0 5)) (<= dist_1 (+ 5 dist_0)))))\n (a!2 (and (<= dist_2 (+ 6 dist_0)) (<= dist_2 (+ 1 dist_1))))\n (a!4 (and (<= dist_3 (+ 1 dist_0))\n (<= dist_3 (+ 4 dist_1))\n (<= dist_3 (+ 5 dist_2))))\n (a!6 (or (and (= dist_4 (+ dist_1 6)) (<= dist_4 (+ 6 dist_1)))))\n (a!7 (and (<= dist_5 (+ 3 dist_3)) (<= dist_5 (+ 6 dist_4))))\n (a!9 (and (<= dist_6 (+ 9 dist_0)) (<= dist_6 (+ 8 dist_4))))\n (a!11 (and (<= dist_7 (+ 4 dist_2))\n (<= dist_7 (+ 8 dist_3))\n (<= dist_7 (+ 2 dist_5)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "graphs", "id": "graphs-dag_shortest_path-c5cfe220fff633ad", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 6, "difficulty": "extreme", "edges": [[0, 1], [0, 2], [0, 3], [0, 6], [1, 2], [1, 3], [1, 4], [2, 3], [2, 7], [3, 5], [3, 7], [4, 5], [4, 6], [5, 7], [6, 7]], "generator": "graph_invariants", "num_vertices": 8, "seed": 20261011, "source": 0, "target": 7, "variant": "dag_shortest_path", "weighted_edges": [[0, 1, 5], [0, 2, 6], [0, 3, 1], [0, 6, 9], [1, 2, 1], [1, 3, 4], [1, 4, 6], [2, 3, 5], [2, 7, 4], [3, 5, 3], [3, 7, 8], [4, 5, 6], [4, 6, 8], [5, 7, 2], [6, 7, 1]]}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\nsource = data[\"source\"]\ntarget = data[\"target\"]\n\ninfinity = float(\"inf\")\ndist = [infinity] * n\ndist[source] = 0\nfor u, v, w in edges:\n if dist[u] != infinity and dist[u] + w < dist[v]:\n dist[v] = dist[u] + w\nprint(dist[target])\n", "statement": "You are given a directed acyclic graph with 8 vertices labelled 0 to\n7. Every edge [u, v, w] goes from a lower to a higher label and has\ninteger weight 1 <= w <= 9; edges are listed in ascending (u, v) order.\n\nPrint the minimum possible total weight of a path from source s = 0 to\ntarget t = 7. The target is guaranteed reachable, the optimum is\nunique, and both facts have been formally proven for this instance with the\nZ3 SMT solver.\n", "title": "DAG Shortest Path (n=8, weight 6)", "variant": "dag_shortest_path"}, "test_cases": [{"expected_output": 6, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_shortest_path-c5cfe220fff633ad-t000", "input": {"edges": [[0, 1, 5], [0, 2, 6], [0, 3, 1], [0, 6, 9], [1, 2, 1], [1, 3, 4], [1, 4, 6], [2, 3, 5], [2, 7, 4], [3, 5, 3], [3, 7, 8], [4, 5, 6], [4, 6, 8], [5, 7, 2], [6, 7, 1]], "num_vertices": 8, "source": 0, "target": 7}, "is_public": true, "problem_id": "graphs-dag_shortest_path-c5cfe220fff633ad"}], "verification": {"duration_ms": 1.3522080262191594, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and (= dist_1 (+ dist_0 5)) (<= dist_1 (+ 5 dist_0)))))\n (a!2 (and (<= dist_2 (+ 1 dist_0)) (<= dist_2 (+ 3 dist_1))))\n (a!4 (or (and (= dist_3 (+ dist_1 9)) (<= dist_3 (+ 9 dist_1)))))\n (a!5 (and (<= dist_4 (+ 9 dist_1)) (<= dist_4 (+ 6 dist_2))))\n (a!7 (and (<= dist_5 (+ 2 dist_3)) (<= dist_5 (+ 8 dist_4))))\n (a!9 (and (<= dist_6 (+ 2 dist_1))\n (<= dist_6 (+ 2 dist_3))\n (<= dist_6 (+ 6 dist_4))\n (<= dist_6 (+ 1 dist_5)))))\n(let ((a!3 (or (and (= dist_2 (+ dist_0 1)) a!2)\n (and (= dist_2 (+ ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "graphs", "id": "graphs-dag_shortest_path-78541386265a5fb8", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 7, "difficulty": "extreme", "edges": [[0, 1], [0, 2], [1, 2], [1, 3], [1, 4], [1, 6], [2, 4], [3, 5], [3, 6], [4, 5], [4, 6], [5, 6]], "generator": "graph_invariants", "num_vertices": 7, "seed": 20261012, "source": 0, "target": 6, "variant": "dag_shortest_path", "weighted_edges": [[0, 1, 5], [0, 2, 1], [1, 2, 3], [1, 3, 9], [1, 4, 9], [1, 6, 2], [2, 4, 6], [3, 5, 2], [3, 6, 2], [4, 5, 8], [4, 6, 6], [5, 6, 1]]}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\nsource = data[\"source\"]\ntarget = data[\"target\"]\n\ninfinity = float(\"inf\")\ndist = [infinity] * n\ndist[source] = 0\nfor u, v, w in edges:\n if dist[u] != infinity and dist[u] + w < dist[v]:\n dist[v] = dist[u] + w\nprint(dist[target])\n", "statement": "You are given a directed acyclic graph with 7 vertices labelled 0 to\n6. Every edge [u, v, w] goes from a lower to a higher label and has\ninteger weight 1 <= w <= 9; edges are listed in ascending (u, v) order.\n\nPrint the minimum possible total weight of a path from source s = 0 to\ntarget t = 6. The target is guaranteed reachable, the optimum is\nunique, and both facts have been formally proven for this instance with the\nZ3 SMT solver.\n", "title": "DAG Shortest Path (n=7, weight 7)", "variant": "dag_shortest_path"}, "test_cases": [{"expected_output": 7, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_shortest_path-78541386265a5fb8-t000", "input": {"edges": [[0, 1, 5], [0, 2, 1], [1, 2, 3], [1, 3, 9], [1, 4, 9], [1, 6, 2], [2, 4, 6], [3, 5, 2], [3, 6, 2], [4, 5, 8], [4, 6, 6], [5, 6, 1]], "num_vertices": 7, "source": 0, "target": 6}, "is_public": true, "problem_id": "graphs-dag_shortest_path-78541386265a5fb8"}], "verification": {"duration_ms": 1.1744580697268248, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (=> (and (> pos_1 0) (and (< pos_0 0))) (< ord_0 1)))\n (a!2 (=> (and (> pos_3 0) (and (< pos_0 0))) (< ord_0 3)))\n (a!3 (=> (and (> pos_1 1) (and (< pos_0 1))) (< ord_1 1)))\n (a!4 (=> (and (> pos_3 1) (and (< pos_0 1))) (< ord_1 3)))\n (a!5 (=> (and (> pos_1 2) (and (< pos_0 2))) (< ord_2 1)))\n (a!6 (=> (and (> pos_3 2) (and (< pos_0 2))) (< ord_2 3)))\n (a!7 (=> (and (> pos_1 3) (and (< pos_0 3))) (< ord_3 1)))\n (a!8 (=> (and (> pos_3 3) (and (< pos_0 3))) (< ord_3 3)))\n (a!9 (=> (and (> pos_1 4) (and (< pos_0 4))) (< ord_4 1)))\n (a!10 ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "graphs", "id": "graphs-dag_topological_position-d89a19f3b12f8500", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 4, "difficulty": "extreme", "edges": [[0, 1], [0, 3], [0, 7], [1, 2], [1, 7], [2, 4], [2, 5], [2, 6], [3, 4], [3, 5], [3, 6], [3, 7], [4, 5], [5, 6], [6, 7]], "generator": "graph_invariants", "num_vertices": 8, "seed": 20261013, "target_vertex": 4, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 8 vertices labelled 0 to\n7 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 4 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=8, vertex 4)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-d89a19f3b12f8500-t000", "input": {"edges": [[0, 1], [0, 3], [0, 7], [1, 2], [1, 7], [2, 4], [2, 5], [2, 6], [3, 4], [3, 5], [3, 6], [3, 7], [4, 5], [5, 6], [6, 7]], "num_vertices": 8, "target_vertex": 4}, "is_public": true, "problem_id": "graphs-dag_topological_position-d89a19f3b12f8500"}], "verification": {"duration_ms": 6.50029192911461, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (=> (and (> pos_1 0) (and (< pos_0 0))) (< ord_0 1)))\n (a!2 (=> (and (> pos_2 0) (and (< pos_0 0))) (< ord_0 2)))\n (a!3 (=> (and (> pos_4 0) (and (< pos_0 0))) (< ord_0 4)))\n (a!4 (=> (and (> pos_1 1) (and (< pos_0 1))) (< ord_1 1)))\n (a!5 (=> (and (> pos_2 1) (and (< pos_0 1))) (< ord_1 2)))\n (a!6 (=> (and (> pos_4 1) (and (< pos_0 1))) (< ord_1 4)))\n (a!7 (=> (and (> pos_1 2) (and (< pos_0 2))) (< ord_2 1)))\n (a!8 (=> (and (> pos_2 2) (and (< pos_0 2))) (< ord_2 2)))\n (a!9 (=> (and (> pos_4 2) (and (< pos_0 2))) (< ord_2 4)))\n (a!10 ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "graphs", "id": "graphs-dag_topological_position-ea6f8949aae3b276", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 4, "difficulty": "extreme", "edges": [[0, 1], [0, 2], [0, 4], [0, 5], [1, 3], [1, 5], [1, 6], [2, 5], [2, 6], [4, 6], [5, 6]], "generator": "graph_invariants", "num_vertices": 7, "seed": 20261014, "target_vertex": 4, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 7 vertices labelled 0 to\n6 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 4 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=7, vertex 4)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-ea6f8949aae3b276-t000", "input": {"edges": [[0, 1], [0, 2], [0, 4], [0, 5], [1, 3], [1, 5], [1, 6], [2, 5], [2, 6], [4, 6], [5, 6]], "num_vertices": 7, "target_vertex": 4}, "is_public": true, "problem_id": "graphs-dag_topological_position-ea6f8949aae3b276"}], "verification": {"duration_ms": 5.553875002078712, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_0 member_2)\n (=> member_0 member_3)\n (=> member_0 member_4)\n (=> member_1 member_6)\n (=> member_2 member_4)\n (=> member_3 member_4)\n (=> member_3 member_5)\n (=> member_3 member_6)\n (=> member_4 member_5)\n (=> member_4 member_6)\n (=> member_5 member_6)\n (=> member_1 (or member_0))\n (=> member_2 (or member_0))\n (=> member_3 (or member_0))\n (=> member_4 (or member_0 member_2 member_3))\n (=> member_5 (or member_3 member_4))\n (=> member_6 (or member_1 member_3 member_4 member...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "graphs", "id": "graphs-dag_reachability-d6b907fc40f58dc5", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1, 2, 3, 4], [6], [4], [4, 5, 6], [5, 6], [6], []], "answer": 7, "difficulty": "extreme", "generator": "graph_invariants", "num_vertices": 7, "seed": 20261015, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 7 vertices labelled 0 to 6,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=7)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 7, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-d6b907fc40f58dc5-t000", "input": {"adjacency": [[1, 2, 3, 4], [6], [4], [4, 5, 6], [5, 6], [6], []], "num_vertices": 7, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-d6b907fc40f58dc5"}], "verification": {"duration_ms": 0.8388329879380763, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_0 member_2)\n (=> member_0 member_3)\n (=> member_0 member_4)\n (=> member_1 member_2)\n (=> member_1 member_5)\n (=> member_1 member_6)\n (=> member_2 member_3)\n (=> member_2 member_6)\n (=> member_3 member_4)\n (=> member_3 member_6)\n (=> member_3 member_7)\n (=> member_4 member_7)\n (=> member_5 member_6)\n (=> member_6 member_7)\n (=> member_1 (or member_0))\n (=> member_2 (or member_0 member_1))\n (=> member_3 (or member_0 member_2))\n (=> member_4 (or member_0 member_3))\n ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "graphs", "id": "graphs-dag_reachability-885146443bb68393", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1, 2, 3, 4], [2, 5, 6], [3, 6], [4, 6, 7], [7], [6], [7], []], "answer": 8, "difficulty": "extreme", "generator": "graph_invariants", "num_vertices": 8, "seed": 20261016, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 8 vertices labelled 0 to 7,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=8)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 8, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-885146443bb68393-t000", "input": {"adjacency": [[1, 2, 3, 4], [2, 5, 6], [3, 6], [4, 6, 7], [7], [6], [7], []], "num_vertices": 8, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-885146443bb68393"}], "verification": {"duration_ms": 0.9674999746493995, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and (= dist_1 (+ dist_0 2)) (<= dist_1 (+ 2 dist_0)))))\n (a!2 (or (and (= dist_2 (+ dist_0 9)) (<= dist_2 (+ 9 dist_0)))))\n (a!3 (or (and (= dist_3 (+ dist_1 9)) (<= dist_3 (+ 9 dist_1)))))\n (a!4 (and (<= dist_4 (+ 9 dist_0)) (<= dist_4 (+ 2 dist_1))))\n (a!6 (and (<= dist_5 (+ 3 dist_1))\n (<= dist_5 (+ 1 dist_2))\n (<= dist_5 (+ 9 dist_4))))\n (a!8 (and (<= dist_6 (+ 6 dist_1))\n (<= dist_6 (+ 3 dist_2))\n (<= dist_6 (+ 7 dist_4)))))\n(let ((a!5 (or (and (= dist_4 (+ dist_0 9)) a!4)\n ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "graphs", "id": "graphs-dag_shortest_path-f4ffa58a0e19c59b", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 8, "difficulty": "extreme", "edges": [[0, 1], [0, 2], [0, 4], [1, 3], [1, 4], [1, 5], [1, 6], [2, 5], [2, 6], [4, 5], [4, 6]], "generator": "graph_invariants", "num_vertices": 7, "seed": 20261017, "source": 0, "target": 6, "variant": "dag_shortest_path", "weighted_edges": [[0, 1, 2], [0, 2, 9], [0, 4, 9], [1, 3, 9], [1, 4, 2], [1, 5, 3], [1, 6, 6], [2, 5, 1], [2, 6, 3], [4, 5, 9], [4, 6, 7]]}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\nsource = data[\"source\"]\ntarget = data[\"target\"]\n\ninfinity = float(\"inf\")\ndist = [infinity] * n\ndist[source] = 0\nfor u, v, w in edges:\n if dist[u] != infinity and dist[u] + w < dist[v]:\n dist[v] = dist[u] + w\nprint(dist[target])\n", "statement": "You are given a directed acyclic graph with 7 vertices labelled 0 to\n6. Every edge [u, v, w] goes from a lower to a higher label and has\ninteger weight 1 <= w <= 9; edges are listed in ascending (u, v) order.\n\nPrint the minimum possible total weight of a path from source s = 0 to\ntarget t = 6. The target is guaranteed reachable, the optimum is\nunique, and both facts have been formally proven for this instance with the\nZ3 SMT solver.\n", "title": "DAG Shortest Path (n=7, weight 8)", "variant": "dag_shortest_path"}, "test_cases": [{"expected_output": 8, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_shortest_path-f4ffa58a0e19c59b-t000", "input": {"edges": [[0, 1, 2], [0, 2, 9], [0, 4, 9], [1, 3, 9], [1, 4, 2], [1, 5, 3], [1, 6, 6], [2, 5, 1], [2, 6, 3], [4, 5, 9], [4, 6, 7]], "num_vertices": 7, "source": 0, "target": 6}, "is_public": true, "problem_id": "graphs-dag_shortest_path-f4ffa58a0e19c59b"}], "verification": {"duration_ms": 1.1147079640068114, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and member_0\n (=> member_0 member_1)\n (=> member_0 member_3)\n (=> member_1 member_2)\n (=> member_1 member_4)\n (=> member_1 member_5)\n (=> member_1 member_7)\n (=> member_2 member_6)\n (=> member_3 member_5)\n (=> member_5 member_7)\n (=> member_1 (or member_0))\n (=> member_2 (or member_1))\n (=> member_3 (or member_0))\n (=> member_4 (or member_1))\n (=> member_5 (or member_1 member_3))\n (=> member_6 (or member_2))\n (=> member_7 (or member_1 member_5))\n (= reachable_count\n (+ (ite member_0 1 0)\n (ite member_1 1 0...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "graphs", "id": "graphs-dag_reachability-2494de2e72ca2c8d", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"adjacency": [[1, 3], [2, 4, 5, 7], [6], [5], [], [7], [], []], "answer": 8, "difficulty": "extreme", "generator": "graph_invariants", "num_vertices": 8, "seed": 20261018, "source": 0, "variant": "dag_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nadjacency = data[\"adjacency\"]\nsource = data[\"source\"]\n\nseen = [0] * n\nseen[source] = 1\nstack = [source]\nwhile stack:\n vertex = stack.pop()\n for nxt in adjacency[vertex]:\n if not seen[nxt]:\n seen[nxt] = 1\n stack.append(nxt)\ntotal = 0\nfor flag in seen:\n if flag:\n total += 1\nprint(total)\n", "statement": "You are given a directed acyclic graph with 8 vertices labelled 0 to 7,\ndescribed by adjacency lists. Starting from the source vertex s = 0, a\nvertex is reachable when some directed path leads to it; the source itself is\nalways reachable.\n\nPrint how many vertices are reachable from s. The exact count is unique and\nits uniqueness has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DAG Reachability Count (n=8)", "variant": "dag_reachability"}, "test_cases": [{"expected_output": 8, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_reachability-2494de2e72ca2c8d-t000", "input": {"adjacency": [[1, 3], [2, 4, 5, 7], [6], [5], [], [7], [], []], "num_vertices": 8, "source": 0}, "is_public": true, "problem_id": "graphs-dag_reachability-2494de2e72ca2c8d"}], "verification": {"duration_ms": 0.8149999775923789, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(and (distinct ord_0 ord_1 ord_2 ord_3 ord_4 ord_5 ord_6 ord_7)\n (>= ord_0 0)\n (< ord_0 8)\n (>= pos_0 0)\n (< pos_0 8)\n (>= ord_1 0)\n (< ord_1 8)\n (>= pos_1 0)\n (< pos_1 8)\n (>= ord_2 0)\n (< ord_2 8)\n (>= pos_2 0)\n (< pos_2 8)\n (>= ord_3 0)\n (< ord_3 8)\n (>= pos_3 0)\n (< pos_3 8)\n (>= ord_4 0)\n (< ord_4 8)\n (>= pos_4 0)\n (< pos_4 8)\n (>= ord_5 0)\n (< ord_5 8)\n (>= pos_5 0)\n (< pos_5 8)\n (>= ord_6 0)\n (< ord_6 8)\n (>= pos_6 0)\n (< pos_6 8)\n (>= ord_7 0)\n (< ord_7 8)\n (>=...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "graphs", "id": "graphs-dag_topological_position-8a5519b52c75b1db", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 5, "difficulty": "extreme", "edges": [[0, 1], [0, 3], [0, 4], [0, 5], [0, 7], [1, 2], [1, 3], [1, 5], [1, 6], [1, 7], [2, 5], [3, 4], [3, 6], [4, 5], [4, 6], [4, 7], [5, 7], [6, 7]], "generator": "graph_invariants", "num_vertices": 8, "seed": 20261019, "target_vertex": 5, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 8 vertices labelled 0 to\n7 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 5 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=8, vertex 5)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-8a5519b52c75b1db-t000", "input": {"edges": [[0, 1], [0, 3], [0, 4], [0, 5], [0, 7], [1, 2], [1, 3], [1, 5], [1, 6], [1, 7], [2, 5], [3, 4], [3, 6], [4, 5], [4, 6], [4, 7], [5, 7], [6, 7]], "num_vertices": 8, "target_vertex": 5}, "is_public": true, "problem_id": "graphs-dag_topological_position-8a5519b52c75b1db"}], "verification": {"duration_ms": 6.8752919905819, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and (= dist_1 (+ dist_0 1)) (<= dist_1 (+ 1 dist_0)))))\n (a!2 (or (and (= dist_2 (+ dist_0 7)) (<= dist_2 (+ 7 dist_0)))))\n (a!3 (or (and (= dist_3 (+ dist_1 3)) (<= dist_3 (+ 3 dist_1)))))\n (a!4 (and (<= dist_4 (+ 1 dist_0)) (<= dist_4 (+ 3 dist_1))))\n (a!6 (and (<= dist_5 (+ 4 dist_0))\n (<= dist_5 (+ 8 dist_2))\n (<= dist_5 (+ 4 dist_4))))\n (a!8 (and (<= dist_6 (+ 7 dist_0))\n (<= dist_6 (+ 5 dist_1))\n (<= dist_6 (+ 3 dist_5))))\n (a!10 (and (<= dist_7 (+ 4 dist_0))\n (<= ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "graphs", "id": "graphs-dag_shortest_path-bdc358d9d7c1b7fb", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 4, "difficulty": "extreme", "edges": [[0, 1], [0, 2], [0, 4], [0, 5], [0, 6], [0, 7], [1, 3], [1, 4], [1, 6], [1, 7], [2, 5], [3, 7], [4, 5], [4, 7], [5, 6], [5, 7], [6, 7]], "generator": "graph_invariants", "num_vertices": 8, "seed": 20261020, "source": 0, "target": 7, "variant": "dag_shortest_path", "weighted_edges": [[0, 1, 1], [0, 2, 7], [0, 4, 1], [0, 5, 4], [0, 6, 7], [0, 7, 4], [1, 3, 3], [1, 4, 3], [1, 6, 5], [1, 7, 5], [2, 5, 8], [3, 7, 1], [4, 5, 4], [4, 7, 6], [5, 6, 3], [5, 7, 1], [6, 7, 4]]}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\nsource = data[\"source\"]\ntarget = data[\"target\"]\n\ninfinity = float(\"inf\")\ndist = [infinity] * n\ndist[source] = 0\nfor u, v, w in edges:\n if dist[u] != infinity and dist[u] + w < dist[v]:\n dist[v] = dist[u] + w\nprint(dist[target])\n", "statement": "You are given a directed acyclic graph with 8 vertices labelled 0 to\n7. Every edge [u, v, w] goes from a lower to a higher label and has\ninteger weight 1 <= w <= 9; edges are listed in ascending (u, v) order.\n\nPrint the minimum possible total weight of a path from source s = 0 to\ntarget t = 7. The target is guaranteed reachable, the optimum is\nunique, and both facts have been formally proven for this instance with the\nZ3 SMT solver.\n", "title": "DAG Shortest Path (n=8, weight 4)", "variant": "dag_shortest_path"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_shortest_path-bdc358d9d7c1b7fb-t000", "input": {"edges": [[0, 1, 1], [0, 2, 7], [0, 4, 1], [0, 5, 4], [0, 6, 7], [0, 7, 4], [1, 3, 3], [1, 4, 3], [1, 6, 5], [1, 7, 5], [2, 5, 8], [3, 7, 1], [4, 5, 4], [4, 7, 6], [5, 6, 3], [5, 7, 1], [6, 7, 4]], "num_vertices": 8, "source": 0, "target": 7}, "is_public": true, "problem_id": "graphs-dag_shortest_path-bdc358d9d7c1b7fb"}], "verification": {"duration_ms": 1.4647089992649853, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (=> (and (> pos_1 0) (and (< pos_0 0))) (< ord_0 1)))\n (a!2 (=> (and (> pos_2 0) (and (< pos_1 0))) (< ord_0 2)))\n (a!3 (=> (and (> pos_3 0) (and (< pos_1 0))) (< ord_0 3)))\n (a!4 (=> (and (> pos_4 0) (and (< pos_0 0))) (< ord_0 4)))\n (a!5 (=> (and (> pos_1 1) (and (< pos_0 1))) (< ord_1 1)))\n (a!6 (=> (and (> pos_2 1) (and (< pos_1 1))) (< ord_1 2)))\n (a!7 (=> (and (> pos_3 1) (and (< pos_1 1))) (< ord_1 3)))\n (a!8 (=> (and (> pos_4 1) (and (< pos_0 1))) (< ord_1 4)))\n (a!9 (=> (and (> pos_1 2) (and (< pos_0 2))) (< ord_2 1)))\n (a!10 ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "graphs", "id": "graphs-dag_topological_position-71824f774ca68152", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 7, "difficulty": "extreme", "edges": [[0, 1], [0, 4], [0, 5], [0, 6], [0, 7], [1, 2], [1, 3], [1, 7], [2, 5], [2, 7], [3, 5], [3, 6], [3, 7], [4, 5], [4, 6]], "generator": "graph_invariants", "num_vertices": 8, "seed": 20261021, "target_vertex": 7, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 8 vertices labelled 0 to\n7 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 7 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=8, vertex 7)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 7, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-71824f774ca68152-t000", "input": {"edges": [[0, 1], [0, 4], [0, 5], [0, 6], [0, 7], [1, 2], [1, 3], [1, 7], [2, 5], [2, 7], [3, 5], [3, 6], [3, 7], [4, 5], [4, 6]], "num_vertices": 8, "target_vertex": 7}, "is_public": true, "problem_id": "graphs-dag_topological_position-71824f774ca68152"}], "verification": {"duration_ms": 6.275374966207892, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (=> (and (> pos_1 0) (and (< pos_0 0))) (< ord_0 1)))\n (a!2 (=> (and (> pos_2 0) (and (< pos_0 0))) (< ord_0 2)))\n (a!3 (=> (and (> pos_3 0) (and (< pos_0 0))) (< ord_0 3)))\n (a!4 (=> (and (> pos_1 1) (and (< pos_0 1))) (< ord_1 1)))\n (a!5 (=> (and (> pos_2 1) (and (< pos_0 1))) (< ord_1 2)))\n (a!6 (=> (and (> pos_3 1) (and (< pos_0 1))) (< ord_1 3)))\n (a!7 (=> (and (> pos_1 2) (and (< pos_0 2))) (< ord_2 1)))\n (a!8 (=> (and (> pos_2 2) (and (< pos_0 2))) (< ord_2 2)))\n (a!9 (=> (and (> pos_3 2) (and (< pos_0 2))) (< ord_2 3)))\n (a!10 ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "graphs", "id": "graphs-dag_topological_position-291e9c02e8621441", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 1, "difficulty": "extreme", "edges": [[0, 1], [0, 2], [0, 3], [0, 4], [0, 6], [0, 7], [1, 7], [2, 4], [2, 5], [2, 6], [3, 4], [3, 5], [3, 7], [4, 5], [4, 6], [5, 6], [5, 7]], "generator": "graph_invariants", "num_vertices": 8, "seed": 20261022, "target_vertex": 1, "variant": "dag_topological_position"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\ntarget_vertex = data[\"target_vertex\"]\n\nsuccessors = [[] for _ in range(n)]\npredecessors = [[] for _ in range(n)]\nfor u, v in edges:\n successors[u].append(v)\n predecessors[v].append(u)\n\nindegree = [len(preds) for preds in predecessors]\norder = []\nready = [vertex for vertex in range(n) if indegree[vertex] < 1]\nwhile ready:\n ready.sort()\n current = ready.pop(0)\n order.append(current)\n if current == target_vertex:\n break\n for nxt in successors[current]:\n indegree[nxt] -= 1\n if indegree[nxt] < 1:\n ready.append(nxt)\n\nprint(order.index(target_vertex))\n", "statement": "You are given a directed acyclic graph with 8 vertices labelled 0 to\n7 and edge list [u, v].\n\nRun Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the\nsmallest-labelled vertex whose remaining in-degree is zero, appending it to\nthe removal order (rounds are numbered from 0). Print the round at which the\ndesignated vertex 1 is removed. The resulting removal order is unique;\nexistence and uniqueness of the answer have been formally proven with the\nZ3 SMT solver.\n", "title": "DAG Topological Position (n=8, vertex 1)", "variant": "dag_topological_position"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_topological_position-291e9c02e8621441-t000", "input": {"edges": [[0, 1], [0, 2], [0, 3], [0, 4], [0, 6], [0, 7], [1, 7], [2, 4], [2, 5], [2, 6], [3, 4], [3, 5], [3, 7], [4, 5], [4, 6], [5, 6], [5, 7]], "num_vertices": 8, "target_vertex": 1}, "is_public": true, "problem_id": "graphs-dag_topological_position-291e9c02e8621441"}], "verification": {"duration_ms": 6.2713329680264, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Label-increasing edges guarantee acyclicity by construction.", "expression": "the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]", "name": "graph_shape"}, {"description": "The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and (= dist_1 (+ dist_0 6)) (<= dist_1 (+ 6 dist_0)))))\n (a!2 (or (and (= dist_2 (+ dist_0 4)) (<= dist_2 (+ 4 dist_0)))))\n (a!3 (and (<= dist_3 (+ 6 dist_1)) (<= dist_3 (+ 5 dist_2))))\n (a!5 (and (<= dist_4 (+ 2 dist_0)) (<= dist_4 (+ 7 dist_1))))\n (a!7 (and (<= dist_5 (+ 8 dist_1))\n (<= dist_5 (+ 4 dist_2))\n (<= dist_5 (+ 1 dist_3))\n (<= dist_5 (+ 3 dist_4))))\n (a!9 (or (and (= dist_6 (+ dist_4 5)) (<= dist_6 (+ 5 dist_4)))))\n (a!10 (and (<= dist_7 (+ 5 dist_0)) (<= dist_7 (+ 6 dist_2)))))\n(let ((a!...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "graphs", "id": "graphs-dag_shortest_path-22646f17b2870668", "input_schema": {"additionalProperties": true, "properties": {"num_vertices": {"maximum": 8, "minimum": 3, "type": "integer"}}, "required": ["num_vertices"], "type": "object"}, "metadata": {"answer": 5, "difficulty": "extreme", "edges": [[0, 1], [0, 2], [0, 4], [0, 7], [1, 3], [1, 4], [1, 5], [2, 3], [2, 5], [2, 7], [3, 5], [4, 5], [4, 6]], "generator": "graph_invariants", "num_vertices": 8, "seed": 20261023, "source": 0, "target": 7, "variant": "dag_shortest_path", "weighted_edges": [[0, 1, 6], [0, 2, 4], [0, 4, 2], [0, 7, 5], [1, 3, 6], [1, 4, 7], [1, 5, 8], [2, 3, 5], [2, 5, 4], [2, 7, 6], [3, 5, 1], [4, 5, 3], [4, 6, 5]]}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nn = data[\"num_vertices\"]\nedges = data[\"edges\"]\nsource = data[\"source\"]\ntarget = data[\"target\"]\n\ninfinity = float(\"inf\")\ndist = [infinity] * n\ndist[source] = 0\nfor u, v, w in edges:\n if dist[u] != infinity and dist[u] + w < dist[v]:\n dist[v] = dist[u] + w\nprint(dist[target])\n", "statement": "You are given a directed acyclic graph with 8 vertices labelled 0 to\n7. Every edge [u, v, w] goes from a lower to a higher label and has\ninteger weight 1 <= w <= 9; edges are listed in ascending (u, v) order.\n\nPrint the minimum possible total weight of a path from source s = 0 to\ntarget t = 7. The target is guaranteed reachable, the optimum is\nunique, and both facts have been formally proven for this instance with the\nZ3 SMT solver.\n", "title": "DAG Shortest Path (n=8, weight 5)", "variant": "dag_shortest_path"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "graphs-dag_shortest_path-22646f17b2870668-t000", "input": {"edges": [[0, 1, 6], [0, 2, 4], [0, 4, 2], [0, 7, 5], [1, 3, 6], [1, 4, 7], [1, 5, 8], [2, 3, 5], [2, 5, 4], [2, 7, 6], [3, 5, 1], [4, 5, 3], [4, 6, 5]], "num_vertices": 8, "source": 0, "target": 7}, "is_public": true, "problem_id": "graphs-dag_shortest_path-22646f17b2870668"}], "verification": {"duration_ms": 1.1616249685175717, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 1)\n (= count_0_1 0)\n (= count_1_0 (+ count_0_0 count_0_1 count_0_1))\n (= count_1_1 (+ count_0_0))\n (= count_2_0 (+ count_1_0 count_1_1 count_1_1))\n (= count_2_1 (+ count_1_0))\n (= accepted_count (+ count_2_0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "automata", "id": "automata-dfa_string_acceptance-703fcf53a3e50734", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, false], "answer": 3, "difficulty": "easy", "generator": "state_machine", "length": 2, "num_states": 2, "seed": 20261024, "start": 0, "transitions": [[1, 0], [0, 0]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 2 states numbered 0 to 1. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 0.\n\nPrint how many distinct binary strings of length exactly 2 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=2, L=2)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 3, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-703fcf53a3e50734-t000", "input": {"accepting": [true, false], "length": 2, "start": 0, "transitions": [[1, 0], [0, 0]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-703fcf53a3e50734"}], "verification": {"duration_ms": 0.7821240578778088, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_0_2 (= depth_0_0 (+ depth_0_2 1)))))\n (a!2 (or (and pair_0_0 (= depth_0_2 (+ depth_0_0 1)))\n (and pair_0_0 (= depth_0_2 (+ depth_0_0 1)))\n (and pair_0_2 (= depth_0_2 (+ depth_0_2 1)))))\n (a!3 (or (and pair_2_2 (= depth_1_0 (+ depth_2_2 1)))))\n (a!4 (or (and pair_2_1 (= depth_1_1 (+ depth_2_1 1)))))\n (a!5 (or (and pair_2_0 (= depth_1_2 (+ depth_2_0 1)))))\n (a!6 (or (and pair_1_2 (= depth_2_0 (+ depth_1_2 1)))))\n (a!7 (or (and pair_1_1 (= depth_2_1 (+ depth_1_1 1)))\n (and pair_1_1 (= depth_2_1 (+ ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "automata", "id": "automata-dfa_product_reachability-0497002392814fca", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 1, "difficulty": "easy", "equivalent_by_construction": true, "first": {"accepting": [false, false, false], "start": 0, "transitions": [[0, 0], [2, 2], [2, 1]]}, "generator": "state_machine", "num_states": 3, "second": {"accepting": [false, false, false], "start": 1, "transitions": [[2, 2], [1, 1], [2, 0]]}, "seed": 20261025, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(0, 1) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (3x3 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-0497002392814fca-t000", "input": {"first": {"accepting": [false, false, false], "start": 0, "transitions": [[0, 0], [2, 2], [2, 1]]}, "second": {"accepting": [false, false, false], "start": 1, "transitions": [[2, 2], [1, 1], [2, 0]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-0497002392814fca"}], "verification": {"duration_ms": 1.9280000124126673, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 1)\n (= count_0_1 0)\n (= count_1_0 (+ count_0_0 count_0_1))\n (= count_1_1 (+ count_0_0 count_0_1))\n (= count_2_0 (+ count_1_0 count_1_1))\n (= count_2_1 (+ count_1_0 count_1_1))\n (= accepted_count (+ count_2_0 count_2_1)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "automata", "id": "automata-dfa_string_acceptance-60a578d5377bff46", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, true], "answer": 4, "difficulty": "easy", "generator": "state_machine", "length": 2, "num_states": 2, "seed": 20261026, "start": 0, "transitions": [[0, 1], [1, 0]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 2 states numbered 0 to 1. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 0.\n\nPrint how many distinct binary strings of length exactly 2 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=2, L=2)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-60a578d5377bff46-t000", "input": {"accepting": [true, true], "length": 2, "start": 0, "transitions": [[0, 1], [1, 0]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-60a578d5377bff46"}], "verification": {"duration_ms": 0.9057500283233821, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 1)\n (= count_0_1 0)\n (= count_1_0 (+ count_0_0 count_0_0 count_0_1))\n (= count_1_1 (+ count_0_1))\n (= count_2_0 (+ count_1_0 count_1_0 count_1_1))\n (= count_2_1 (+ count_1_1))\n (= accepted_count (+ count_2_0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "automata", "id": "automata-dfa_string_acceptance-001fc9addf6cf3d9", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, false], "answer": 4, "difficulty": "easy", "generator": "state_machine", "length": 2, "num_states": 2, "seed": 20261027, "start": 0, "transitions": [[0, 0], [0, 1]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 2 states numbered 0 to 1. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 0.\n\nPrint how many distinct binary strings of length exactly 2 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=2, L=2)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-001fc9addf6cf3d9-t000", "input": {"accepting": [true, false], "length": 2, "start": 0, "transitions": [[0, 0], [0, 1]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-001fc9addf6cf3d9"}], "verification": {"duration_ms": 0.7828749367035925, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 1)\n (= count_0_1 0)\n (= count_1_0 (+ count_0_0 count_0_1))\n (= count_1_1 (+ count_0_0 count_0_1))\n (= count_2_0 (+ count_1_0 count_1_1))\n (= count_2_1 (+ count_1_0 count_1_1))\n (= accepted_count (+ count_2_0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "automata", "id": "automata-dfa_string_acceptance-dc013fccba032373", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, false], "answer": 2, "difficulty": "easy", "generator": "state_machine", "length": 2, "num_states": 2, "seed": 20261028, "start": 0, "transitions": [[0, 1], [0, 1]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 2 states numbered 0 to 1. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 0.\n\nPrint how many distinct binary strings of length exactly 2 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=2, L=2)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-dc013fccba032373-t000", "input": {"accepting": [true, false], "length": 2, "start": 0, "transitions": [[0, 1], [0, 1]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-dc013fccba032373"}], "verification": {"duration_ms": 0.7735840044915676, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_1_1 (= depth_0_0 (+ depth_1_1 1)))))\n (a!2 (or (and pair_1_0 (= depth_0_1 (+ depth_1_0 1)))))\n (a!3 (or (and pair_1_2 (= depth_0_2 (+ depth_1_2 1)))))\n (a!4 (or (and pair_0_0 (= depth_1_1 (+ depth_0_0 1)))))\n (a!5 (or (and pair_0_2 (= depth_1_2 (+ depth_0_2 1)))\n (and pair_0_2 (= depth_1_2 (+ depth_0_2 1)))\n (and pair_1_2 (= depth_1_2 (+ depth_1_2 1))))))\n (and pair_1_0\n (= depth_1_0 0)\n (=> pair_0_0 pair_1_0)\n (=> pair_0_0 pair_1_1)\n (=> pair_0_1 pair_1_0)\n (=> pair_0_1 pair_1_0)\n ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "automata", "id": "automata-dfa_product_reachability-6ed4fa9bb8294aff", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 2, "difficulty": "easy", "equivalent_by_construction": true, "first": {"accepting": [false, false], "start": 1, "transitions": [[1, 1], [1, 0]]}, "generator": "state_machine", "num_states": 2, "second": {"accepting": [false, false, false], "start": 0, "transitions": [[0, 1], [0, 0], [2, 2]]}, "seed": 20261029, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(1, 0) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (2x3 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-6ed4fa9bb8294aff-t000", "input": {"first": {"accepting": [false, false], "start": 1, "transitions": [[1, 1], [1, 0]]}, "second": {"accepting": [false, false, false], "start": 0, "transitions": [[0, 1], [0, 0], [2, 2]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-6ed4fa9bb8294aff"}], "verification": {"duration_ms": 1.4983339933678508, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 0)\n (= count_0_2 1)\n (= count_1_0 (+ count_0_0))\n (= count_1_1 (+ count_0_0 count_0_2))\n (= count_1_2 (+ count_0_1 count_0_1 count_0_2))\n (= count_2_0 (+ count_1_0))\n (= count_2_1 (+ count_1_0 count_1_2))\n (= count_2_2 (+ count_1_1 count_1_1 count_1_2))\n (= accepted_count (+ count_2_0 count_2_1 count_2_2)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "automata", "id": "automata-dfa_string_acceptance-0535d6384d471ce7", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, true, true], "answer": 4, "difficulty": "easy", "generator": "state_machine", "length": 2, "num_states": 3, "seed": 20261030, "start": 2, "transitions": [[1, 0], [2, 2], [2, 1]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 3 states numbered 0 to 2. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 2.\n\nPrint how many distinct binary strings of length exactly 2 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=3, L=2)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-0535d6384d471ce7-t000", "input": {"accepting": [true, true, true], "length": 2, "start": 2, "transitions": [[1, 0], [2, 2], [2, 1]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-0535d6384d471ce7"}], "verification": {"duration_ms": 0.7948330021463335, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_1_1 (= depth_0_0 (+ depth_1_1 1)))))\n (a!2 (or (and pair_0_2 (= depth_0_2 (+ depth_0_2 1)))\n (and pair_1_2 (= depth_0_2 (+ depth_1_2 1)))\n (and pair_1_2 (= depth_0_2 (+ depth_1_2 1)))))\n (a!3 (or (and pair_0_3 (= depth_0_3 (+ depth_0_3 1)))\n (and pair_1_3 (= depth_0_3 (+ depth_1_3 1)))\n (and pair_1_3 (= depth_0_3 (+ depth_1_3 1)))))\n (a!4 (or (and pair_0_1 (= depth_1_0 (+ depth_0_1 1)))))\n (a!5 (or (and pair_0_0 (= depth_1_1 (+ depth_0_0 1)))))\n (a!6 (or (and pair_0_2 (= depth_1_2 (+ de...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "automata", "id": "automata-dfa_product_reachability-7b5bfaf3dbd387f8", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 2, "difficulty": "easy", "equivalent_by_construction": true, "first": {"accepting": [false, false], "start": 0, "transitions": [[1, 0], [0, 0]]}, "generator": "state_machine", "num_states": 2, "second": {"accepting": [false, false, false, false], "start": 1, "transitions": [[1, 1], [0, 1], [2, 2], [3, 3]]}, "seed": 20261031, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(0, 1) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (2x4 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-7b5bfaf3dbd387f8-t000", "input": {"first": {"accepting": [false, false], "start": 0, "transitions": [[1, 0], [0, 0]]}, "second": {"accepting": [false, false, false, false], "start": 1, "transitions": [[1, 1], [0, 1], [2, 2], [3, 3]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-7b5bfaf3dbd387f8"}], "verification": {"duration_ms": 1.8069999641738832, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 1)\n (= count_1_0 (+ count_0_0 count_0_0 count_0_1))\n (= count_1_1 (+ count_0_1))\n (= count_2_0 (+ count_1_0 count_1_0 count_1_1))\n (= count_2_1 (+ count_1_1))\n (= accepted_count (+ count_2_0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "automata", "id": "automata-dfa_string_acceptance-1ff789d8e3655b15", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, false], "answer": 3, "difficulty": "easy", "generator": "state_machine", "length": 2, "num_states": 2, "seed": 20261032, "start": 1, "transitions": [[0, 0], [1, 0]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 2 states numbered 0 to 1. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 1.\n\nPrint how many distinct binary strings of length exactly 2 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=2, L=2)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 3, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-1ff789d8e3655b15-t000", "input": {"accepting": [true, false], "length": 2, "start": 1, "transitions": [[0, 0], [1, 0]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-1ff789d8e3655b15"}], "verification": {"duration_ms": 0.7885000086389482, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_0_0 (= depth_0_1 (+ depth_0_0 1)))\n (and pair_1_0 (= depth_0_1 (+ depth_1_0 1)))))\n (a!2 (or (and pair_0_1 (= depth_0_2 (+ depth_0_1 1)))\n (and pair_0_2 (= depth_0_2 (+ depth_0_2 1)))\n (and pair_1_1 (= depth_0_2 (+ depth_1_1 1)))\n (and pair_1_2 (= depth_0_2 (+ depth_1_2 1)))))\n (a!3 (or (and pair_0_3 (= depth_0_3 (+ depth_0_3 1)))\n (and pair_1_3 (= depth_0_3 (+ depth_1_3 1)))))\n (a!4 (or (and pair_0_4 (= depth_0_4 (+ depth_0_4 1)))\n (and pair_1_4 (= depth_0_4 (+ depth_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "automata", "id": "automata-dfa_product_reachability-8ddb1440912213eb", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 3, "difficulty": "easy", "equivalent_by_construction": true, "first": {"accepting": [false, false, false], "start": 2, "transitions": [[0, 1], [0, 2], [1, 1]]}, "generator": "state_machine", "num_states": 3, "second": {"accepting": [false, false, false, false, false], "start": 0, "transitions": [[1, 1], [2, 0], [2, 1], [3, 3], [4, 4]]}, "seed": 20261033, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(2, 0) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (3x5 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 3, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-8ddb1440912213eb-t000", "input": {"first": {"accepting": [false, false, false], "start": 2, "transitions": [[0, 1], [0, 2], [1, 1]]}, "second": {"accepting": [false, false, false, false, false], "start": 0, "transitions": [[1, 1], [2, 0], [2, 1], [3, 3], [4, 4]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-8ddb1440912213eb"}], "verification": {"duration_ms": 2.7754170005209744, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 0)\n (= count_0_2 1)\n (= count_1_0 (+ count_0_1 count_0_1 count_0_2))\n (= count_1_1 (+ count_0_0 count_0_2))\n (= count_1_2 (+ count_0_0))\n (= count_2_0 (+ count_1_1 count_1_1 count_1_2))\n (= count_2_1 (+ count_1_0 count_1_2))\n (= count_2_2 (+ count_1_0))\n (= accepted_count (+ count_2_0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "automata", "id": "automata-dfa_string_acceptance-d772bd3f31f369de", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, false, false], "answer": 2, "difficulty": "easy", "generator": "state_machine", "length": 2, "num_states": 3, "seed": 20261034, "start": 2, "transitions": [[2, 1], [0, 0], [0, 1]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 3 states numbered 0 to 2. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 2.\n\nPrint how many distinct binary strings of length exactly 2 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=3, L=2)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-d772bd3f31f369de-t000", "input": {"accepting": [true, false, false], "length": 2, "start": 2, "transitions": [[2, 1], [0, 0], [0, 1]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-d772bd3f31f369de"}], "verification": {"duration_ms": 0.8191660745069385, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_1_1 (= depth_0_0 (+ depth_1_1 1)))))\n (a!2 (or (and pair_0_2 (= depth_0_2 (+ depth_0_2 1)))\n (and pair_1_2 (= depth_0_2 (+ depth_1_2 1)))\n (and pair_1_2 (= depth_0_2 (+ depth_1_2 1)))))\n (a!3 (or (and pair_0_1 (= depth_1_0 (+ depth_0_1 1)))))\n (a!4 (or (and pair_0_0 (= depth_1_1 (+ depth_0_0 1)))))\n (a!5 (or (and pair_0_2 (= depth_1_2 (+ depth_0_2 1))))))\n (and pair_0_1\n (= depth_0_1 0)\n (=> pair_0_0 pair_0_1)\n (=> pair_0_0 pair_1_1)\n (=> pair_0_1 pair_0_1)\n (=> pair_0_1 pair_1_0)\n ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "automata", "id": "automata-dfa_product_reachability-6fae8278592ad277", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 2, "difficulty": "easy", "equivalent_by_construction": true, "first": {"accepting": [false, true], "start": 0, "transitions": [[0, 1], [0, 0]]}, "generator": "state_machine", "num_states": 2, "second": {"accepting": [true, false, false], "start": 1, "transitions": [[1, 1], [1, 0], [2, 2]]}, "seed": 20261035, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(0, 1) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (2x3 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-6fae8278592ad277-t000", "input": {"first": {"accepting": [false, true], "start": 0, "transitions": [[0, 1], [0, 0]]}, "second": {"accepting": [true, false, false], "start": 1, "transitions": [[1, 1], [1, 0], [2, 2]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-6fae8278592ad277"}], "verification": {"duration_ms": 1.756250043399632, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 0)\n (= count_0_2 1)\n (= count_1_0 (+ count_0_0 count_0_1 count_0_2))\n (= count_1_1 (+ count_0_1))\n (= count_1_2 (+ count_0_0 count_0_2))\n (= count_2_0 (+ count_1_0 count_1_1 count_1_2))\n (= count_2_1 (+ count_1_1))\n (= count_2_2 (+ count_1_0 count_1_2))\n (= accepted_count (+ count_2_0 count_2_2)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "automata", "id": "automata-dfa_string_acceptance-02d5f441cdd9aabc", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, false, true], "answer": 4, "difficulty": "easy", "generator": "state_machine", "length": 2, "num_states": 3, "seed": 20261036, "start": 2, "transitions": [[2, 0], [0, 1], [0, 2]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 3 states numbered 0 to 2. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 2.\n\nPrint how many distinct binary strings of length exactly 2 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=3, L=2)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-02d5f441cdd9aabc-t000", "input": {"accepting": [true, false, true], "length": 2, "start": 2, "transitions": [[2, 0], [0, 1], [0, 2]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-02d5f441cdd9aabc"}], "verification": {"duration_ms": 0.954125018324703, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_0_0 (= depth_0_1 (+ depth_0_0 1)))\n (and pair_0_1 (= depth_0_1 (+ depth_0_1 1)))\n (and pair_1_0 (= depth_0_1 (+ depth_1_0 1)))\n (and pair_1_1 (= depth_0_1 (+ depth_1_1 1)))))\n (a!2 (or (and pair_0_2 (= depth_0_2 (+ depth_0_2 1)))\n (and pair_1_2 (= depth_0_2 (+ depth_1_2 1)))))\n (a!3 (or (and pair_0_2 (= depth_1_2 (+ depth_0_2 1)))\n (and pair_1_2 (= depth_1_2 (+ depth_1_2 1))))))\n (and pair_1_0\n (= depth_1_0 0)\n (=> pair_0_0 pair_1_0)\n (=> pair_0_0 pair_0_1)\n (=> ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "automata", "id": "automata-dfa_product_reachability-27ee6dd07101a8ed", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 2, "difficulty": "easy", "equivalent_by_construction": true, "first": {"accepting": [true, true], "start": 1, "transitions": [[1, 0], [1, 0]]}, "generator": "state_machine", "num_states": 2, "second": {"accepting": [true, true, false], "start": 0, "transitions": [[0, 1], [0, 1], [2, 2]]}, "seed": 20261037, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(1, 0) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (2x3 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-27ee6dd07101a8ed-t000", "input": {"first": {"accepting": [true, true], "start": 1, "transitions": [[1, 0], [1, 0]]}, "second": {"accepting": [true, true, false], "start": 0, "transitions": [[0, 1], [0, 1], [2, 2]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-27ee6dd07101a8ed"}], "verification": {"duration_ms": 1.3368329964578152, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_0_0 (= depth_1_0 (+ depth_0_0 1)))\n (and pair_0_1 (= depth_1_0 (+ depth_0_1 1)))\n (and pair_1_0 (= depth_1_0 (+ depth_1_0 1)))\n (and pair_1_1 (= depth_1_0 (+ depth_1_1 1))))))\n (and pair_0_1\n (= depth_0_1 0)\n (=> pair_0_0 pair_0_1)\n (=> pair_0_0 pair_1_0)\n (=> pair_0_1 pair_0_1)\n (=> pair_0_1 pair_1_0)\n (=> pair_1_0 pair_0_1)\n (=> pair_1_0 pair_1_0)\n (=> pair_1_1 pair_0_1)\n (=> pair_1_1 pair_1_0)\n (=> pair_0_0 false)\n (=> pair_1_0 (and (>= depth_1_0 1) a!1))\n...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "automata", "id": "automata-dfa_product_reachability-c9ab7a89653f1bb2", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 2, "difficulty": "easy", "equivalent_by_construction": true, "first": {"accepting": [false, true], "start": 0, "transitions": [[0, 1], [0, 1]]}, "generator": "state_machine", "num_states": 2, "second": {"accepting": [true, false], "start": 1, "transitions": [[1, 0], [1, 0]]}, "seed": 20261038, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(0, 1) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (2x2 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-c9ab7a89653f1bb2-t000", "input": {"first": {"accepting": [false, true], "start": 0, "transitions": [[0, 1], [0, 1]]}, "second": {"accepting": [true, false], "start": 1, "transitions": [[1, 0], [1, 0]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-c9ab7a89653f1bb2"}], "verification": {"duration_ms": 0.8452910115011036, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 1)\n (= count_0_1 0)\n (= count_0_2 0)\n (= count_1_0 (+ count_0_1 count_0_2))\n (= count_1_1 (+ count_0_0 count_0_1))\n (= count_1_2 (+ count_0_0 count_0_2))\n (= count_2_0 (+ count_1_1 count_1_2))\n (= count_2_1 (+ count_1_0 count_1_1))\n (= count_2_2 (+ count_1_0 count_1_2))\n (= accepted_count (+ count_2_0 count_2_2)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "automata", "id": "automata-dfa_string_acceptance-46fc5d8d3d880304", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, false, true], "answer": 3, "difficulty": "easy", "generator": "state_machine", "length": 2, "num_states": 3, "seed": 20261039, "start": 0, "transitions": [[2, 1], [0, 1], [0, 2]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 3 states numbered 0 to 2. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 0.\n\nPrint how many distinct binary strings of length exactly 2 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=3, L=2)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 3, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-46fc5d8d3d880304-t000", "input": {"accepting": [true, false, true], "length": 2, "start": 0, "transitions": [[2, 1], [0, 1], [0, 2]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-46fc5d8d3d880304"}], "verification": {"duration_ms": 0.9134580031968653, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 1)\n (= count_0_2 0)\n (= count_1_0 (+ count_0_1))\n (= count_1_1 (+ count_0_0 count_0_2))\n (= count_1_2 (+ count_0_0 count_0_1 count_0_2))\n (= count_2_0 (+ count_1_1))\n (= count_2_1 (+ count_1_0 count_1_2))\n (= count_2_2 (+ count_1_0 count_1_1 count_1_2))\n (= accepted_count (+ count_2_0 count_2_1 count_2_2)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "automata", "id": "automata-dfa_string_acceptance-0eadd350bca6b2a0", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, true, true], "answer": 4, "difficulty": "easy", "generator": "state_machine", "length": 2, "num_states": 3, "seed": 20261040, "start": 1, "transitions": [[2, 1], [2, 0], [1, 2]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 3 states numbered 0 to 2. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 1.\n\nPrint how many distinct binary strings of length exactly 2 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=3, L=2)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-0eadd350bca6b2a0-t000", "input": {"accepting": [true, true, true], "length": 2, "start": 1, "transitions": [[2, 1], [2, 0], [1, 2]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-0eadd350bca6b2a0"}], "verification": {"duration_ms": 0.8749159751459956, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 0)\n (= count_0_2 1)\n (= count_1_0 (+ count_0_0))\n (= count_1_1 (+ count_0_0 count_0_1))\n (= count_1_2 (+ count_0_1 count_0_2 count_0_2))\n (= count_2_0 (+ count_1_0))\n (= count_2_1 (+ count_1_0 count_1_1))\n (= count_2_2 (+ count_1_1 count_1_2 count_1_2))\n (= accepted_count (+ count_2_0 count_2_2)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "automata", "id": "automata-dfa_string_acceptance-48f3cdfe0bffc64b", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, false, true], "answer": 4, "difficulty": "easy", "generator": "state_machine", "length": 2, "num_states": 3, "seed": 20261041, "start": 2, "transitions": [[0, 1], [1, 2], [2, 2]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 3 states numbered 0 to 2. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 2.\n\nPrint how many distinct binary strings of length exactly 2 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=3, L=2)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-48f3cdfe0bffc64b-t000", "input": {"accepting": [true, false, true], "length": 2, "start": 2, "transitions": [[0, 1], [1, 2], [2, 2]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-48f3cdfe0bffc64b"}], "verification": {"duration_ms": 0.8604579488746822, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 1)\n (= count_0_1 0)\n (= count_0_2 0)\n (= count_1_0 (+ count_0_2))\n (= count_1_1 (+ count_0_0 count_0_1 count_0_1 count_0_2))\n (= count_1_2 (+ count_0_0))\n (= count_2_0 (+ count_1_2))\n (= count_2_1 (+ count_1_0 count_1_1 count_1_1 count_1_2))\n (= count_2_2 (+ count_1_0))\n (= accepted_count (+ count_2_0 count_2_1)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "automata", "id": "automata-dfa_string_acceptance-db3126aafd5f09a6", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, true, false], "answer": 4, "difficulty": "easy", "generator": "state_machine", "length": 2, "num_states": 3, "seed": 20261042, "start": 0, "transitions": [[1, 2], [1, 1], [1, 0]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 3 states numbered 0 to 2. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 0.\n\nPrint how many distinct binary strings of length exactly 2 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=3, L=2)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-db3126aafd5f09a6-t000", "input": {"accepting": [true, true, false], "length": 2, "start": 0, "transitions": [[1, 2], [1, 1], [1, 0]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-db3126aafd5f09a6"}], "verification": {"duration_ms": 0.7694169180467725, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_0_2 (= depth_0_2 (+ depth_0_2 1)))\n (and pair_1_2 (= depth_0_2 (+ depth_1_2 1)))))\n (a!2 (or (and pair_0_0 (= depth_1_0 (+ depth_0_0 1)))\n (and pair_0_1 (= depth_1_0 (+ depth_0_1 1)))\n (and pair_1_0 (= depth_1_0 (+ depth_1_0 1)))\n (and pair_1_1 (= depth_1_0 (+ depth_1_1 1)))))\n (a!3 (or (and pair_0_2 (= depth_1_2 (+ depth_0_2 1)))\n (and pair_1_2 (= depth_1_2 (+ depth_1_2 1))))))\n (and pair_0_1\n (= depth_0_1 0)\n (=> pair_0_0 pair_1_0)\n (=> pair_0_0 pair_0_1)\n (=> ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "automata", "id": "automata-dfa_product_reachability-8284bdfebb8bf8a8", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 2, "difficulty": "easy", "equivalent_by_construction": true, "first": {"accepting": [false, true], "start": 0, "transitions": [[1, 0], [1, 0]]}, "generator": "state_machine", "num_states": 2, "second": {"accepting": [true, false, false], "start": 1, "transitions": [[0, 1], [0, 1], [2, 2]]}, "seed": 20261043, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(0, 1) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (2x3 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-8284bdfebb8bf8a8-t000", "input": {"first": {"accepting": [false, true], "start": 0, "transitions": [[1, 0], [1, 0]]}, "second": {"accepting": [true, false, false], "start": 1, "transitions": [[0, 1], [0, 1], [2, 2]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-8284bdfebb8bf8a8"}], "verification": {"duration_ms": 1.3122070231474936, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 1)\n (= count_1_0 (+ count_0_0 count_0_1 count_0_1))\n (= count_1_1 (+ count_0_0))\n (= count_2_0 (+ count_1_0 count_1_1 count_1_1))\n (= count_2_1 (+ count_1_0))\n (= accepted_count (+ count_2_0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "automata", "id": "automata-dfa_string_acceptance-68a0e87116af921a", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, false], "answer": 2, "difficulty": "easy", "generator": "state_machine", "length": 2, "num_states": 2, "seed": 20261044, "start": 1, "transitions": [[1, 0], [0, 0]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 2 states numbered 0 to 1. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 1.\n\nPrint how many distinct binary strings of length exactly 2 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=2, L=2)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-68a0e87116af921a-t000", "input": {"accepting": [true, false], "length": 2, "start": 1, "transitions": [[1, 0], [0, 0]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-68a0e87116af921a"}], "verification": {"duration_ms": 0.775249907746911, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_0_0 (= depth_0_0 (+ depth_0_0 1)))\n (and pair_0_1 (= depth_0_0 (+ depth_0_1 1)))\n (and pair_1_0 (= depth_0_0 (+ depth_1_0 1)))\n (and pair_1_1 (= depth_0_0 (+ depth_1_1 1)))\n (and pair_1_1 (= depth_0_0 (+ depth_1_1 1)))))\n (a!2 (or (and pair_1_0 (= depth_0_1 (+ depth_1_0 1)))))\n (a!3 (or (and pair_0_2 (= depth_0_2 (+ depth_0_2 1)))\n (and pair_1_2 (= depth_0_2 (+ depth_1_2 1)))\n (and pair_1_2 (= depth_0_2 (+ depth_1_2 1)))))\n (a!4 (or (and pair_0_1 (= depth_1_0 (+ depth_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "automata", "id": "automata-dfa_product_reachability-55ecd696f397025b", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 2, "difficulty": "easy", "equivalent_by_construction": true, "first": {"accepting": [false, true], "start": 1, "transitions": [[1, 0], [0, 0]]}, "generator": "state_machine", "num_states": 2, "second": {"accepting": [false, true, false], "start": 1, "transitions": [[1, 0], [0, 0], [2, 2]]}, "seed": 20261045, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(1, 1) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (2x3 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-55ecd696f397025b-t000", "input": {"first": {"accepting": [false, true], "start": 1, "transitions": [[1, 0], [0, 0]]}, "second": {"accepting": [false, true, false], "start": 1, "transitions": [[1, 0], [0, 0], [2, 2]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-55ecd696f397025b"}], "verification": {"duration_ms": 1.6687509487383068, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 1)\n (= count_1_0 (+ count_0_0 count_0_1))\n (= count_1_1 (+ count_0_0 count_0_1))\n (= count_2_0 (+ count_1_0 count_1_1))\n (= count_2_1 (+ count_1_0 count_1_1))\n (= accepted_count (+ count_2_0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "automata", "id": "automata-dfa_string_acceptance-f9261e02347d55bb", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, false], "answer": 2, "difficulty": "easy", "generator": "state_machine", "length": 2, "num_states": 2, "seed": 20261046, "start": 1, "transitions": [[1, 0], [0, 1]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 2 states numbered 0 to 1. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 1.\n\nPrint how many distinct binary strings of length exactly 2 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=2, L=2)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-f9261e02347d55bb-t000", "input": {"accepting": [true, false], "length": 2, "start": 1, "transitions": [[1, 0], [0, 1]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-f9261e02347d55bb"}], "verification": {"duration_ms": 0.8533329819329083, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 0)\n (= count_0_2 1)\n (= count_1_0 (+ count_0_0 count_0_2))\n (= count_1_1 (+ count_0_1 count_0_1 count_0_2))\n (= count_1_2 (+ count_0_0))\n (= count_2_0 (+ count_1_0 count_1_2))\n (= count_2_1 (+ count_1_1 count_1_1 count_1_2))\n (= count_2_2 (+ count_1_0))\n (= accepted_count (+ count_2_1 count_2_2)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "automata", "id": "automata-dfa_string_acceptance-b95d9d5f89176874", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [false, true, true], "answer": 3, "difficulty": "easy", "generator": "state_machine", "length": 2, "num_states": 3, "seed": 20261047, "start": 2, "transitions": [[2, 0], [1, 1], [0, 1]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 3 states numbered 0 to 2. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 2.\n\nPrint how many distinct binary strings of length exactly 2 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=3, L=2)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 3, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-b95d9d5f89176874-t000", "input": {"accepting": [false, true, true], "length": 2, "start": 2, "transitions": [[2, 0], [1, 1], [0, 1]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-b95d9d5f89176874"}], "verification": {"duration_ms": 0.7757919956929982, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 0)\n (= count_0_2 1)\n (= count_1_0 (+ count_0_0 count_0_1))\n (= count_1_1 (+ count_0_2))\n (= count_1_2 (+ count_0_0 count_0_1 count_0_2))\n (= count_2_0 (+ count_1_0 count_1_1))\n (= count_2_1 (+ count_1_2))\n (= count_2_2 (+ count_1_0 count_1_1 count_1_2))\n (= accepted_count (+ count_2_2)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "easy", "domain": "automata", "id": "automata-dfa_string_acceptance-4ab62e5c5fbfbe74", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [false, false, true], "answer": 2, "difficulty": "easy", "generator": "state_machine", "length": 2, "num_states": 3, "seed": 20261048, "start": 2, "transitions": [[0, 2], [0, 2], [1, 2]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 3 states numbered 0 to 2. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 2.\n\nPrint how many distinct binary strings of length exactly 2 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=3, L=2)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-4ab62e5c5fbfbe74-t000", "input": {"accepting": [false, false, true], "length": 2, "start": 2, "transitions": [[0, 2], [0, 2], [1, 2]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-4ab62e5c5fbfbe74"}], "verification": {"duration_ms": 0.8547499892301857, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 0)\n (= count_0_2 1)\n (= count_0_3 0)\n (= count_1_0 (+ count_0_0 count_0_1 count_0_3))\n (= count_1_1 (+ count_0_1))\n (= count_1_2 (+ count_0_0 count_0_2 count_0_3))\n (= count_1_3 (+ count_0_2))\n (= count_2_0 (+ count_1_0 count_1_1 count_1_3))\n (= count_2_1 (+ count_1_1))\n (= count_2_2 (+ count_1_0 count_1_2 count_1_3))\n (= count_2_3 (+ count_1_2))\n (= count_3_0 (+ count_2_0 count_2_1 count_2_3))\n (= count_3_1 (+ count_2_1))\n (= count_3_2 (+ count_2_0 count_2_2 count_2_3))\n (= count_3_3 (+ count_2_2...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "automata", "id": "automata-dfa_string_acceptance-b944edf7cf29c4c6", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [false, false, true, false], "answer": 8, "difficulty": "medium", "generator": "state_machine", "length": 4, "num_states": 4, "seed": 20261049, "start": 2, "transitions": [[2, 0], [0, 1], [3, 2], [2, 0]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 4 states numbered 0 to 3. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 2.\n\nPrint how many distinct binary strings of length exactly 4 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=4, L=4)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 8, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-b944edf7cf29c4c6-t000", "input": {"accepting": [false, false, true, false], "length": 4, "start": 2, "transitions": [[2, 0], [0, 1], [3, 2], [2, 0]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-b944edf7cf29c4c6"}], "verification": {"duration_ms": 0.8263750351034105, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 0)\n (= count_0_2 1)\n (= count_1_0 (+ count_0_0 count_0_0 count_0_1 count_0_2 count_0_2))\n (= count_1_1 (+ count_0_1))\n (= count_1_2 0)\n (= count_2_0 (+ count_1_0 count_1_0 count_1_1 count_1_2 count_1_2))\n (= count_2_1 (+ count_1_1))\n (= count_2_2 0)\n (= count_3_0 (+ count_2_0 count_2_0 count_2_1 count_2_2 count_2_2))\n (= count_3_1 (+ count_2_1))\n (= count_3_2 0)\n (= count_4_0 (+ count_3_0 count_3_0 count_3_1 count_3_2 count_3_2))\n (= count_4_1 (+ count_3_1))\n (= count_4_2 0)\n (= accepted_count (+ ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "automata", "id": "automata-dfa_string_acceptance-388a127a9ee9dd6f", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, false, false], "answer": 16, "difficulty": "medium", "generator": "state_machine", "length": 4, "num_states": 3, "seed": 20261050, "start": 2, "transitions": [[0, 0], [1, 0], [0, 0]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 3 states numbered 0 to 2. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 2.\n\nPrint how many distinct binary strings of length exactly 4 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=3, L=4)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 16, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-388a127a9ee9dd6f-t000", "input": {"accepting": [true, false, false], "length": 4, "start": 2, "transitions": [[0, 0], [1, 0], [0, 0]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-388a127a9ee9dd6f"}], "verification": {"duration_ms": 0.8031260222196579, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_0_0 (= depth_0_0 (+ depth_0_0 1)))\n (and pair_1_0 (= depth_0_0 (+ depth_1_0 1)))\n (and pair_2_0 (= depth_0_0 (+ depth_2_0 1)))))\n (a!2 (or (and pair_0_0 (= depth_0_1 (+ depth_0_0 1)))\n (and pair_1_0 (= depth_0_1 (+ depth_1_0 1)))\n (and pair_2_0 (= depth_0_1 (+ depth_2_0 1)))))\n (a!3 (or (and pair_0_1 (= depth_0_3 (+ depth_0_1 1)))\n (and pair_0_1 (= depth_0_3 (+ depth_0_1 1)))\n (and pair_0_2 (= depth_0_3 (+ depth_0_2 1)))\n (and pair_0_2 (= depth_0_3 (+ depth_0_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "automata", "id": "automata-dfa_product_reachability-0e982924d9111c54", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 3, "difficulty": "medium", "equivalent_by_construction": true, "first": {"accepting": [false, true, false, true], "start": 3, "transitions": [[0, 0], [0, 0], [0, 0], [3, 1]]}, "generator": "state_machine", "num_states": 4, "second": {"accepting": [true, true, false, false, false], "start": 0, "transitions": [[0, 1], [3, 3], [3, 3], [3, 3], [4, 4]]}, "seed": 20261051, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(3, 0) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (4x5 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 3, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-0e982924d9111c54-t000", "input": {"first": {"accepting": [false, true, false, true], "start": 3, "transitions": [[0, 0], [0, 0], [0, 0], [3, 1]]}, "second": {"accepting": [true, true, false, false, false], "start": 0, "transitions": [[0, 1], [3, 3], [3, 3], [3, 3], [4, 4]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-0e982924d9111c54"}], "verification": {"duration_ms": 1.288167026359588, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_0_1 (= depth_0_1 (+ depth_0_1 1)))\n (and pair_0_2 (= depth_0_1 (+ depth_0_2 1)))\n (and pair_1_1 (= depth_0_1 (+ depth_1_1 1)))\n (and pair_1_2 (= depth_0_1 (+ depth_1_2 1)))))\n (a!2 (or (and pair_0_0 (= depth_0_2 (+ depth_0_0 1)))\n (and pair_1_0 (= depth_0_2 (+ depth_1_0 1)))))\n (a!3 (or (and pair_0_3 (= depth_0_3 (+ depth_0_3 1)))\n (and pair_1_3 (= depth_0_3 (+ depth_1_3 1)))))\n (a!4 (or (and pair_2_1 (= depth_1_0 (+ depth_2_1 1)))\n (and pair_2_2 (= depth_1_0 (+ depth_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "automata", "id": "automata-dfa_product_reachability-5c5a38e2c6545a96", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 3, "difficulty": "medium", "equivalent_by_construction": true, "first": {"accepting": [true, true, true], "start": 1, "transitions": [[0, 2], [0, 2], [1, 1]]}, "generator": "state_machine", "num_states": 3, "second": {"accepting": [true, true, true, false], "start": 2, "transitions": [[2, 2], [1, 0], [1, 0], [3, 3]]}, "seed": 20261052, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(1, 2) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (3x4 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 3, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-5c5a38e2c6545a96-t000", "input": {"first": {"accepting": [true, true, true], "start": 1, "transitions": [[0, 2], [0, 2], [1, 1]]}, "second": {"accepting": [true, true, true, false], "start": 2, "transitions": [[2, 2], [1, 0], [1, 0], [3, 3]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-5c5a38e2c6545a96"}], "verification": {"duration_ms": 2.164791978430003, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_0_0 (= depth_0_0 (+ depth_0_0 1)))\n (and pair_0_2 (= depth_0_0 (+ depth_0_2 1)))\n (and pair_3_0 (= depth_0_0 (+ depth_3_0 1)))\n (and pair_3_2 (= depth_0_0 (+ depth_3_2 1)))))\n (a!2 (or (and pair_0_1 (= depth_0_2 (+ depth_0_1 1)))\n (and pair_3_1 (= depth_0_2 (+ depth_3_1 1)))))\n (a!3 (or (and pair_0_3 (= depth_0_3 (+ depth_0_3 1)))\n (and pair_3_3 (= depth_0_3 (+ depth_3_3 1)))))\n (a!4 (or (and pair_2_2 (= depth_1_1 (+ depth_2_2 1)))\n (and pair_2_3 (= depth_1_1 (+ depth_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "automata", "id": "automata-dfa_product_reachability-2651710e79ea21bf", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 4, "difficulty": "medium", "equivalent_by_construction": true, "first": {"accepting": [true, false, false, false], "start": 2, "transitions": [[0, 2], [3, 2], [2, 1], [0, 1]]}, "generator": "state_machine", "num_states": 4, "second": {"accepting": [true, false, false, false], "start": 3, "transitions": [[0, 3], [2, 3], [0, 1], [3, 1]]}, "seed": 20261053, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(2, 3) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (4x4 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-2651710e79ea21bf-t000", "input": {"first": {"accepting": [true, false, false, false], "start": 2, "transitions": [[0, 2], [3, 2], [2, 1], [0, 1]]}, "second": {"accepting": [true, false, false, false], "start": 3, "transitions": [[0, 3], [2, 3], [0, 1], [3, 1]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-2651710e79ea21bf"}], "verification": {"duration_ms": 2.3892089957371354, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_0_1 (= depth_0_0 (+ depth_0_1 1)))))\n (a!2 (or (and pair_0_0 (= depth_0_1 (+ depth_0_0 1)))))\n (a!3 (or (and pair_0_4 (= depth_0_3 (+ depth_0_4 1)))))\n (a!4 (or (and pair_0_3 (= depth_0_4 (+ depth_0_3 1)))))\n (a!5 (or (and pair_0_5 (= depth_0_5 (+ depth_0_5 1)))))\n (a!6 (or (and pair_0_6 (= depth_0_6 (+ depth_0_6 1)))))\n (a!7 (or (and pair_2_1 (= depth_1_0 (+ depth_2_1 1)))))\n (a!8 (or (and pair_2_0 (= depth_1_1 (+ depth_2_0 1)))))\n (a!9 (or (and pair_2_2 (= depth_1_2 (+ depth_2_2 1)))))\n (a!10 (or (and pair_0_0 (= depth_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "automata", "id": "automata-dfa_product_reachability-67eb7ef532d95234", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 3, "difficulty": "medium", "equivalent_by_construction": true, "first": {"accepting": [true, true, false, true, false], "start": 0, "transitions": [[1, 0], [1, 2], [2, 1], [1, 4], [1, 3]]}, "generator": "state_machine", "num_states": 5, "second": {"accepting": [true, false, true, true, false, false, false], "start": 2, "transitions": [[3, 1], [3, 0], [3, 2], [3, 4], [4, 3], [5, 5], [6, 6]]}, "seed": 20261054, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(0, 2) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (5x7 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 3, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-67eb7ef532d95234-t000", "input": {"first": {"accepting": [true, true, false, true, false], "start": 0, "transitions": [[1, 0], [1, 2], [2, 1], [1, 4], [1, 3]]}, "second": {"accepting": [true, false, true, true, false, false, false], "start": 2, "transitions": [[3, 1], [3, 0], [3, 2], [3, 4], [4, 3], [5, 5], [6, 6]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-67eb7ef532d95234"}], "verification": {"duration_ms": 4.866959003265947, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_1_0 (= depth_0_0 (+ depth_1_0 1)))\n (and pair_1_2 (= depth_0_0 (+ depth_1_2 1)))\n (and pair_3_0 (= depth_0_0 (+ depth_3_0 1)))\n (and pair_3_2 (= depth_0_0 (+ depth_3_2 1)))))\n (a!2 (or (and pair_1_1 (= depth_0_2 (+ depth_1_1 1)))\n (and pair_1_3 (= depth_0_2 (+ depth_1_3 1)))\n (and pair_3_1 (= depth_0_2 (+ depth_3_1 1)))\n (and pair_3_3 (= depth_0_2 (+ depth_3_3 1)))))\n (a!3 (or (and pair_1_4 (= depth_0_4 (+ depth_1_4 1)))\n (and pair_3_4 (= depth_0_4 (+ depth_3_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "automata", "id": "automata-dfa_product_reachability-bf2267537cad3c64", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 1, "difficulty": "medium", "equivalent_by_construction": true, "first": {"accepting": [false, true, false, true], "start": 2, "transitions": [[2, 1], [0, 3], [2, 2], [0, 3]]}, "generator": "state_machine", "num_states": 4, "second": {"accepting": [false, true, false, true, false], "start": 0, "transitions": [[0, 0], [2, 1], [0, 3], [2, 1], [4, 4]]}, "seed": 20261055, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(2, 0) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (4x5 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-bf2267537cad3c64-t000", "input": {"first": {"accepting": [false, true, false, true], "start": 2, "transitions": [[2, 1], [0, 3], [2, 2], [0, 3]]}, "second": {"accepting": [false, true, false, true, false], "start": 0, "transitions": [[0, 0], [2, 1], [0, 3], [2, 1], [4, 4]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-bf2267537cad3c64"}], "verification": {"duration_ms": 2.7128750225529075, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_2_1 (= depth_0_0 (+ depth_2_1 1)))))\n (a!2 (or (and pair_1_0 (= depth_0_1 (+ depth_1_0 1)))))\n (a!3 (or (and pair_1_2 (= depth_0_3 (+ depth_1_2 1)))\n (and pair_1_3 (= depth_0_3 (+ depth_1_3 1)))\n (and pair_2_0 (= depth_0_3 (+ depth_2_0 1)))\n (and pair_2_2 (= depth_0_3 (+ depth_2_2 1)))))\n (a!4 (or (and pair_3_0 (= depth_1_1 (+ depth_3_0 1)))))\n (a!5 (or (and pair_3_1 (= depth_1_2 (+ depth_3_1 1)))))\n (a!6 (or (and pair_3_2 (= depth_1_3 (+ depth_3_2 1)))\n (and pair_3_3 (= depth_1_3 (+ de...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "automata", "id": "automata-dfa_product_reachability-3dae2c76f0a7d8ec", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 2, "difficulty": "medium", "equivalent_by_construction": true, "first": {"accepting": [false, false, false, false], "start": 0, "transitions": [[2, 2], [3, 0], [0, 2], [2, 1]]}, "generator": "state_machine", "num_states": 4, "second": {"accepting": [false, false, false, false], "start": 2, "transitions": [[3, 1], [0, 2], [3, 3], [2, 3]]}, "seed": 20261056, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(0, 2) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (4x4 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-3dae2c76f0a7d8ec-t000", "input": {"first": {"accepting": [false, false, false, false], "start": 0, "transitions": [[2, 2], [3, 0], [0, 2], [2, 1]]}, "second": {"accepting": [false, false, false, false], "start": 2, "transitions": [[3, 1], [0, 2], [3, 3], [2, 3]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-3dae2c76f0a7d8ec"}], "verification": {"duration_ms": 2.4898750707507133, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_2_2 (= depth_0_0 (+ depth_2_2 1)))))\n (a!2 (or (and pair_2_0 (= depth_0_1 (+ depth_2_0 1)))))\n (a!3 (or (and pair_2_4 (= depth_0_2 (+ depth_2_4 1)))))\n (a!4 (or (and pair_2_1 (= depth_0_4 (+ depth_2_1 1)))\n (and pair_2_3 (= depth_0_4 (+ depth_2_3 1)))))\n (a!5 (or (and pair_2_5 (= depth_0_5 (+ depth_2_5 1)))))\n (a!6 (or (and pair_2_6 (= depth_0_6 (+ depth_2_6 1)))))\n (a!7 (or (and pair_0_2 (= depth_1_0 (+ depth_0_2 1)))))\n (a!8 (or (and pair_0_0 (= depth_1_1 (+ depth_0_0 1)))))\n (a!9 (or (and pair_0_4 (= depth_1_2...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "automata", "id": "automata-dfa_product_reachability-3801262df274e835", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 5, "difficulty": "medium", "equivalent_by_construction": true, "first": {"accepting": [false, true, true, true, true], "start": 2, "transitions": [[3, 1], [3, 4], [3, 0], [4, 4], [3, 2]]}, "generator": "state_machine", "num_states": 5, "second": {"accepting": [false, true, true, true, true, false, false], "start": 2, "transitions": [[3, 1], [3, 4], [3, 0], [4, 4], [3, 2], [5, 5], [6, 6]]}, "seed": 20261057, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(2, 2) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (5x7 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-3801262df274e835-t000", "input": {"first": {"accepting": [false, true, true, true, true], "start": 2, "transitions": [[3, 1], [3, 4], [3, 0], [4, 4], [3, 2]]}, "second": {"accepting": [false, true, true, true, true, false, false], "start": 2, "transitions": [[3, 1], [3, 4], [3, 0], [4, 4], [3, 2], [5, 5], [6, 6]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-3801262df274e835"}], "verification": {"duration_ms": 5.0377920269966125, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 0)\n (= count_0_2 1)\n (= count_0_3 0)\n (= count_0_4 0)\n (= count_1_0 (+ count_0_4))\n (= count_1_1 (+ count_0_2))\n (= count_1_2 (+ count_0_0))\n (= count_1_3 (+ count_0_0 count_0_1 count_0_2 count_0_3 count_0_4))\n (= count_1_4 (+ count_0_1 count_0_3))\n (= count_2_0 (+ count_1_4))\n (= count_2_1 (+ count_1_2))\n (= count_2_2 (+ count_1_0))\n (= count_2_3 (+ count_1_0 count_1_1 count_1_2 count_1_3 count_1_4))\n (= count_2_4 (+ count_1_1 count_1_3))\n (= count_3_0 (+ count_2_4))\n (= count_3_1 (+ count_2_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "automata", "id": "automata-dfa_string_acceptance-58512f3a75132bf8", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [false, false, false, true, false], "answer": 4, "difficulty": "medium", "generator": "state_machine", "length": 3, "num_states": 5, "seed": 20261058, "start": 2, "transitions": [[2, 3], [3, 4], [1, 3], [3, 4], [0, 3]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 5 states numbered 0 to 4. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 2.\n\nPrint how many distinct binary strings of length exactly 3 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=5, L=3)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-58512f3a75132bf8-t000", "input": {"accepting": [false, false, false, true, false], "length": 3, "start": 2, "transitions": [[2, 3], [3, 4], [1, 3], [3, 4], [0, 3]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-58512f3a75132bf8"}], "verification": {"duration_ms": 0.9794989600777626, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_0_2 (= depth_1_0 (+ depth_0_2 1)))\n (and pair_2_3 (= depth_1_0 (+ depth_2_3 1)))))\n (a!2 (or (and pair_0_0 (= depth_1_1 (+ depth_0_0 1)))\n (and pair_0_3 (= depth_1_1 (+ depth_0_3 1)))\n (and pair_2_0 (= depth_1_1 (+ depth_2_0 1)))\n (and pair_2_1 (= depth_1_1 (+ depth_2_1 1)))\n (and pair_2_4 (= depth_1_1 (+ depth_2_4 1)))))\n (a!3 (or (and pair_0_4 (= depth_1_3 (+ depth_0_4 1)))))\n (a!4 (or (and pair_0_1 (= depth_1_4 (+ depth_0_1 1)))\n (and pair_2_2 (= depth_1_4 (+ depth_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "automata", "id": "automata-dfa_product_reachability-480552be20777ff5", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 4, "difficulty": "medium", "equivalent_by_construction": true, "first": {"accepting": [false, true, true, true, false], "start": 4, "transitions": [[1, 3], [4, 4], [4, 1], [2, 4], [3, 4]]}, "generator": "state_machine", "num_states": 5, "second": {"accepting": [true, false, false, true, true, false, false], "start": 1, "transitions": [[1, 1], [4, 1], [0, 4], [1, 0], [3, 1], [5, 5], [6, 6]]}, "seed": 20261059, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(4, 1) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (5x7 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-480552be20777ff5-t000", "input": {"first": {"accepting": [false, true, true, true, false], "start": 4, "transitions": [[1, 3], [4, 4], [4, 1], [2, 4], [3, 4]]}, "second": {"accepting": [true, false, false, true, true, false, false], "start": 1, "transitions": [[1, 1], [4, 1], [0, 4], [1, 0], [3, 1], [5, 5], [6, 6]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-480552be20777ff5"}], "verification": {"duration_ms": 3.9736259495839477, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_0_0 (= depth_0_0 (+ depth_0_0 1)))))\n (a!2 (or (and pair_0_1 (= depth_0_1 (+ depth_0_1 1)))\n (and pair_0_2 (= depth_0_1 (+ depth_0_2 1)))))\n (a!3 (or (and pair_0_3 (= depth_0_3 (+ depth_0_3 1)))))\n (a!4 (or (and pair_0_4 (= depth_0_4 (+ depth_0_4 1)))))\n (a!5 (or (and pair_0_1 (= depth_1_1 (+ depth_0_1 1)))\n (and pair_0_2 (= depth_1_1 (+ depth_0_2 1)))))\n (a!6 (or (and pair_0_3 (= depth_1_3 (+ depth_0_3 1)))))\n (a!7 (or (and pair_0_4 (= depth_1_4 (+ depth_0_4 1)))))\n (a!8 (or (and pair_1_0 (= depth_2_0 (...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "automata", "id": "automata-dfa_product_reachability-c41c0e9f97ba4604", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 2, "difficulty": "medium", "equivalent_by_construction": true, "first": {"accepting": [true, true, false], "start": 1, "transitions": [[0, 1], [2, 2], [2, 2]]}, "generator": "state_machine", "num_states": 3, "second": {"accepting": [true, false, true, false, false], "start": 2, "transitions": [[0, 2], [1, 1], [1, 1], [3, 3], [4, 4]]}, "seed": 20261060, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(1, 2) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (3x5 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-c41c0e9f97ba4604-t000", "input": {"first": {"accepting": [true, true, false], "start": 1, "transitions": [[0, 1], [2, 2], [2, 2]]}, "second": {"accepting": [true, false, true, false, false], "start": 2, "transitions": [[0, 2], [1, 1], [1, 1], [3, 3], [4, 4]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-c41c0e9f97ba4604"}], "verification": {"duration_ms": 1.151791017036885, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 0)\n (= count_0_2 0)\n (= count_0_3 0)\n (= count_0_4 1)\n (= count_1_0 (+ count_0_1 count_0_3))\n (= count_1_1 (+ count_0_2 count_0_4))\n (= count_1_2 (+ count_0_0 count_0_0 count_0_1 count_0_4))\n (= count_1_3 (+ count_0_2))\n (= count_1_4 (+ count_0_3))\n (= count_2_0 (+ count_1_1 count_1_3))\n (= count_2_1 (+ count_1_2 count_1_4))\n (= count_2_2 (+ count_1_0 count_1_0 count_1_1 count_1_4))\n (= count_2_3 (+ count_1_2))\n (= count_2_4 (+ count_1_3))\n (= count_3_0 (+ count_2_1 count_2_3))\n (= count_3_1 (...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "automata", "id": "automata-dfa_string_acceptance-6f1a339481679ba9", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [false, false, true, false, false], "answer": 3, "difficulty": "medium", "generator": "state_machine", "length": 3, "num_states": 5, "seed": 20261061, "start": 4, "transitions": [[2, 2], [0, 2], [3, 1], [0, 4], [1, 2]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 5 states numbered 0 to 4. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 4.\n\nPrint how many distinct binary strings of length exactly 3 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=5, L=3)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 3, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-6f1a339481679ba9-t000", "input": {"accepting": [false, false, true, false, false], "length": 3, "start": 4, "transitions": [[2, 2], [0, 2], [3, 1], [0, 4], [1, 2]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-6f1a339481679ba9"}], "verification": {"duration_ms": 0.8652490214444697, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_0_1 (= depth_0_0 (+ depth_0_1 1)))\n (and pair_2_1 (= depth_0_0 (+ depth_2_1 1)))))\n (a!2 (or (and pair_0_2 (= depth_0_2 (+ depth_0_2 1)))\n (and pair_0_3 (= depth_0_2 (+ depth_0_3 1)))\n (and pair_2_2 (= depth_0_2 (+ depth_2_2 1)))\n (and pair_2_3 (= depth_0_2 (+ depth_2_3 1)))))\n (a!3 (or (and pair_0_0 (= depth_0_4 (+ depth_0_0 1)))\n (and pair_0_4 (= depth_0_4 (+ depth_0_4 1)))\n (and pair_2_0 (= depth_0_4 (+ depth_2_0 1)))\n (and pair_2_4 (= depth_0_4 (+ depth_2_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "automata", "id": "automata-dfa_product_reachability-740387bf288e00d5", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 4, "difficulty": "medium", "equivalent_by_construction": true, "first": {"accepting": [false, true, false, true, false], "start": 1, "transitions": [[0, 3], [3, 2], [0, 3], [3, 3], [2, 2]]}, "generator": "state_machine", "num_states": 5, "second": {"accepting": [false, false, true, true, false, false], "start": 3, "transitions": [[4, 2], [0, 0], [2, 2], [2, 0], [4, 2], [5, 5]]}, "seed": 20261062, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(1, 3) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (5x6 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-740387bf288e00d5-t000", "input": {"first": {"accepting": [false, true, false, true, false], "start": 1, "transitions": [[0, 3], [3, 2], [0, 3], [3, 3], [2, 2]]}, "second": {"accepting": [false, false, true, true, false, false], "start": 3, "transitions": [[4, 2], [0, 0], [2, 2], [2, 0], [4, 2], [5, 5]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-740387bf288e00d5"}], "verification": {"duration_ms": 2.5344580644741654, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_1_3 (= depth_0_0 (+ depth_1_3 1)))\n (and pair_2_3 (= depth_0_0 (+ depth_2_3 1)))\n (and pair_3_3 (= depth_0_0 (+ depth_3_3 1)))))\n (a!2 (or (and pair_1_0 (= depth_0_1 (+ depth_1_0 1)))\n (and pair_1_2 (= depth_0_1 (+ depth_1_2 1)))\n (and pair_1_4 (= depth_0_1 (+ depth_1_4 1)))\n (and pair_2_0 (= depth_0_1 (+ depth_2_0 1)))\n (and pair_2_2 (= depth_0_1 (+ depth_2_2 1)))\n (and pair_2_4 (= depth_0_1 (+ depth_2_4 1)))\n (and pair_3_0 (= depth_0_1 (+ depth_3_0 ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "automata", "id": "automata-dfa_product_reachability-21f9b05e95071d02", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 4, "difficulty": "medium", "equivalent_by_construction": true, "first": {"accepting": [false, true, true, true, false], "start": 3, "transitions": [[3, 3], [1, 0], [4, 0], [4, 0], [3, 1]]}, "generator": "state_machine", "num_states": 5, "second": {"accepting": [true, false, true, false, true, false], "start": 4, "transitions": [[0, 1], [4, 4], [3, 1], [4, 0], [3, 1], [5, 5]]}, "seed": 20261063, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(3, 4) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (5x6 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-21f9b05e95071d02-t000", "input": {"first": {"accepting": [false, true, true, true, false], "start": 3, "transitions": [[3, 3], [1, 0], [4, 0], [4, 0], [3, 1]]}, "second": {"accepting": [true, false, true, false, true, false], "start": 4, "transitions": [[0, 1], [4, 4], [3, 1], [4, 0], [3, 1], [5, 5]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-21f9b05e95071d02"}], "verification": {"duration_ms": 4.181457974482328, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 1)\n (= count_0_1 0)\n (= count_0_2 0)\n (= count_0_3 0)\n (= count_1_0\n (+ count_0_0 count_0_0 count_0_1 count_0_1 count_0_2 count_0_3))\n (= count_1_1 0)\n (= count_1_2 (+ count_0_3))\n (= count_1_3 (+ count_0_2))\n (= count_2_0\n (+ count_1_0 count_1_0 count_1_1 count_1_1 count_1_2 count_1_3))\n (= count_2_1 0)\n (= count_2_2 (+ count_1_3))\n (= count_2_3 (+ count_1_2))\n (= count_3_0\n (+ count_2_0 count_2_0 count_2_1 count_2_1 count_2_2 count_2_3))\n (= count_3_1 0)\n (= count_3_2 (+ count_2_3))\n (= coun...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "automata", "id": "automata-dfa_string_acceptance-7bce65ce17b3b2e3", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, true, true, true], "answer": 8, "difficulty": "medium", "generator": "state_machine", "length": 3, "num_states": 4, "seed": 20261064, "start": 0, "transitions": [[0, 0], [0, 0], [0, 3], [2, 0]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 4 states numbered 0 to 3. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 0.\n\nPrint how many distinct binary strings of length exactly 3 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=4, L=3)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 8, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-7bce65ce17b3b2e3-t000", "input": {"accepting": [true, true, true, true], "length": 3, "start": 0, "transitions": [[0, 0], [0, 0], [0, 3], [2, 0]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-7bce65ce17b3b2e3"}], "verification": {"duration_ms": 0.8638330036774278, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_1_3 (= depth_0_0 (+ depth_1_3 1)))\n (and pair_3_2 (= depth_0_0 (+ depth_3_2 1)))))\n (a!2 (or (and pair_1_2 (= depth_0_1 (+ depth_1_2 1)))))\n (a!3 (or (and pair_3_1 (= depth_0_2 (+ depth_3_1 1)))\n (and pair_3_4 (= depth_0_2 (+ depth_3_4 1)))))\n (a!4 (or (and pair_1_0 (= depth_0_3 (+ depth_1_0 1)))\n (and pair_1_1 (= depth_0_3 (+ depth_1_1 1)))\n (and pair_3_3 (= depth_0_3 (+ depth_3_3 1)))))\n (a!5 (or (and pair_1_4 (= depth_0_4 (+ depth_1_4 1)))\n (and pair_3_0 (= depth_0_4 (+ dept...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "automata", "id": "automata-dfa_product_reachability-30c4b9f3dd3dc4a2", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 5, "difficulty": "medium", "equivalent_by_construction": true, "first": {"accepting": [false, false, true, false, false], "start": 3, "transitions": [[4, 1], [1, 0], [3, 1], [0, 2], [3, 4]]}, "generator": "state_machine", "num_states": 5, "second": {"accepting": [false, true, false, false, false, false], "start": 2, "transitions": [[4, 3], [2, 3], [0, 1], [3, 0], [2, 4], [5, 5]]}, "seed": 20261065, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(3, 2) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (5x6 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-30c4b9f3dd3dc4a2-t000", "input": {"first": {"accepting": [false, false, true, false, false], "start": 3, "transitions": [[4, 1], [1, 0], [3, 1], [0, 2], [3, 4]]}, "second": {"accepting": [false, true, false, false, false, false], "start": 2, "transitions": [[4, 3], [2, 3], [0, 1], [3, 0], [2, 4], [5, 5]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-30c4b9f3dd3dc4a2"}], "verification": {"duration_ms": 8.513665932696313, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 1)\n (= count_0_2 0)\n (= count_0_3 0)\n (= count_1_0 (+ count_0_0))\n (= count_1_1 (+ count_0_1 count_0_2))\n (= count_1_2 (+ count_0_0 count_0_1))\n (= count_1_3 (+ count_0_2 count_0_3 count_0_3))\n (= count_2_0 (+ count_1_0))\n (= count_2_1 (+ count_1_1 count_1_2))\n (= count_2_2 (+ count_1_0 count_1_1))\n (= count_2_3 (+ count_1_2 count_1_3 count_1_3))\n (= accepted_count (+ count_2_1)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "automata", "id": "automata-dfa_string_acceptance-f81742ba54f3ae61", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [false, true, false, false], "answer": 2, "difficulty": "medium", "generator": "state_machine", "length": 2, "num_states": 4, "seed": 20261066, "start": 1, "transitions": [[0, 2], [1, 2], [1, 3], [3, 3]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 4 states numbered 0 to 3. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 1.\n\nPrint how many distinct binary strings of length exactly 2 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=4, L=2)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-f81742ba54f3ae61-t000", "input": {"accepting": [false, true, false, false], "length": 2, "start": 1, "transitions": [[0, 2], [1, 2], [1, 3], [3, 3]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-f81742ba54f3ae61"}], "verification": {"duration_ms": 0.9337919764220715, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 1)\n (= count_0_1 0)\n (= count_0_2 0)\n (= count_1_0 0)\n (= count_1_1 (+ count_0_0 count_0_1 count_0_1))\n (= count_1_2 (+ count_0_0 count_0_2 count_0_2))\n (= count_2_0 0)\n (= count_2_1 (+ count_1_0 count_1_1 count_1_1))\n (= count_2_2 (+ count_1_0 count_1_2 count_1_2))\n (= count_3_0 0)\n (= count_3_1 (+ count_2_0 count_2_1 count_2_1))\n (= count_3_2 (+ count_2_0 count_2_2 count_2_2))\n (= accepted_count (+ count_3_1 count_3_2)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "automata", "id": "automata-dfa_string_acceptance-4b4de150a22d116c", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [false, true, true], "answer": 8, "difficulty": "medium", "generator": "state_machine", "length": 3, "num_states": 3, "seed": 20261067, "start": 0, "transitions": [[2, 1], [1, 1], [2, 2]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 3 states numbered 0 to 2. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 0.\n\nPrint how many distinct binary strings of length exactly 3 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=3, L=3)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 8, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-4b4de150a22d116c-t000", "input": {"accepting": [false, true, true], "length": 3, "start": 0, "transitions": [[2, 1], [1, 1], [2, 2]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-4b4de150a22d116c"}], "verification": {"duration_ms": 0.885999936144799, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_1_0 (= depth_0_1 (+ depth_1_0 1)))\n (and pair_2_0 (= depth_0_1 (+ depth_2_0 1)))))\n (a!2 (or (and pair_1_3 (= depth_0_2 (+ depth_1_3 1)))\n (and pair_2_3 (= depth_0_2 (+ depth_2_3 1)))))\n (a!3 (or (and pair_1_1 (= depth_0_3 (+ depth_1_1 1)))\n (and pair_1_2 (= depth_0_3 (+ depth_1_2 1)))\n (and pair_2_1 (= depth_0_3 (+ depth_2_1 1)))\n (and pair_2_2 (= depth_0_3 (+ depth_2_2 1)))))\n (a!4 (or (and pair_1_4 (= depth_0_4 (+ depth_1_4 1)))\n (and pair_2_4 (= depth_0_4 (+ depth_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "automata", "id": "automata-dfa_product_reachability-0e7aa1c8f38451f2", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 4, "difficulty": "medium", "equivalent_by_construction": true, "first": {"accepting": [false, true, true, false], "start": 3, "transitions": [[2, 2], [2, 0], [2, 0], [1, 1]]}, "generator": "state_machine", "num_states": 4, "second": {"accepting": [false, true, true, false, false], "start": 0, "transitions": [[1, 1], [2, 3], [2, 3], [2, 2], [4, 4]]}, "seed": 20261068, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(3, 0) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (4x5 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-0e7aa1c8f38451f2-t000", "input": {"first": {"accepting": [false, true, true, false], "start": 3, "transitions": [[2, 2], [2, 0], [2, 0], [1, 1]]}, "second": {"accepting": [false, true, true, false, false], "start": 0, "transitions": [[1, 1], [2, 3], [2, 3], [2, 2], [4, 4]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-0e7aa1c8f38451f2"}], "verification": {"duration_ms": 2.148625033441931, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 1)\n (= count_0_1 0)\n (= count_0_2 0)\n (= count_1_0 (+ count_0_2))\n (= count_1_1 (+ count_0_0 count_0_0 count_0_1 count_0_2))\n (= count_1_2 (+ count_0_1))\n (= count_2_0 (+ count_1_2))\n (= count_2_1 (+ count_1_0 count_1_0 count_1_1 count_1_2))\n (= count_2_2 (+ count_1_1))\n (= count_3_0 (+ count_2_2))\n (= count_3_1 (+ count_2_0 count_2_0 count_2_1 count_2_2))\n (= count_3_2 (+ count_2_1))\n (= count_4_0 (+ count_3_2))\n (= count_4_1 (+ count_3_0 count_3_0 count_3_1 count_3_2))\n (= count_4_2 (+ count_3_1))\n (= accepted_c...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "automata", "id": "automata-dfa_string_acceptance-908a6af2d7262d32", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, true, true], "answer": 16, "difficulty": "medium", "generator": "state_machine", "length": 4, "num_states": 3, "seed": 20261069, "start": 0, "transitions": [[1, 1], [2, 1], [1, 0]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 3 states numbered 0 to 2. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 0.\n\nPrint how many distinct binary strings of length exactly 4 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=3, L=4)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 16, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-908a6af2d7262d32-t000", "input": {"accepting": [true, true, true], "length": 4, "start": 0, "transitions": [[1, 1], [2, 1], [1, 0]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-908a6af2d7262d32"}], "verification": {"duration_ms": 1.013917033560574, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 0)\n (= count_0_2 0)\n (= count_0_3 0)\n (= count_0_4 1)\n (= count_1_0 (+ count_0_0 count_0_1 count_0_2 count_0_4))\n (= count_1_1 (+ count_0_3 count_0_4))\n (= count_1_2 (+ count_0_0))\n (= count_1_3 (+ count_0_3))\n (= count_1_4 (+ count_0_1 count_0_2))\n (= count_2_0 (+ count_1_0 count_1_1 count_1_2 count_1_4))\n (= count_2_1 (+ count_1_3 count_1_4))\n (= count_2_2 (+ count_1_0))\n (= count_2_3 (+ count_1_3))\n (= count_2_4 (+ count_1_1 count_1_2))\n (= accepted_count (+ count_2_0)))", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "automata", "id": "automata-dfa_string_acceptance-903f80d954e0aa6a", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, false, false, false, false], "answer": 2, "difficulty": "medium", "generator": "state_machine", "length": 2, "num_states": 5, "seed": 20261070, "start": 4, "transitions": [[0, 2], [4, 0], [0, 4], [1, 3], [0, 1]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 5 states numbered 0 to 4. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 4.\n\nPrint how many distinct binary strings of length exactly 2 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=5, L=2)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-903f80d954e0aa6a-t000", "input": {"accepting": [true, false, false, false, false], "length": 2, "start": 4, "transitions": [[0, 2], [4, 0], [0, 4], [1, 3], [0, 1]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-903f80d954e0aa6a"}], "verification": {"duration_ms": 0.8542079594917595, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_4_3 (= depth_0_1 (+ depth_4_3 1)))\n (and pair_4_4 (= depth_0_1 (+ depth_4_4 1)))))\n (a!2 (or (and pair_4_0 (= depth_0_2 (+ depth_4_0 1)))))\n (a!3 (or (and pair_4_1 (= depth_0_3 (+ depth_4_1 1)))\n (and pair_4_2 (= depth_0_3 (+ depth_4_2 1)))))\n (a!4 (or (and pair_4_1 (= depth_2_0 (+ depth_4_1 1)))\n (and pair_4_4 (= depth_2_0 (+ depth_4_4 1)))))\n (a!5 (or (and pair_1_3 (= depth_2_1 (+ depth_1_3 1)))\n (and pair_1_4 (= depth_2_1 (+ depth_1_4 1)))\n (and pair_3_3 (= depth_2_1 (+ dept...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "automata", "id": "automata-dfa_product_reachability-1124d9d34a904208", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 5, "difficulty": "medium", "equivalent_by_construction": true, "first": {"accepting": [true, false, false, true, false], "start": 1, "transitions": [[3, 3], [2, 4], [3, 4], [2, 3], [0, 2]]}, "generator": "state_machine", "num_states": 5, "second": {"accepting": [false, false, true, true, false], "start": 4, "transitions": [[2, 1], [3, 0], [3, 3], [1, 3], [1, 0]]}, "seed": 20261071, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(1, 4) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (5x5 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-1124d9d34a904208-t000", "input": {"first": {"accepting": [true, false, false, true, false], "start": 1, "transitions": [[3, 3], [2, 4], [3, 4], [2, 3], [0, 2]]}, "second": {"accepting": [false, false, true, true, false], "start": 4, "transitions": [[2, 1], [3, 0], [3, 3], [1, 3], [1, 0]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-1124d9d34a904208"}], "verification": {"duration_ms": 3.177125006914139, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 0)\n (= count_0_2 0)\n (= count_0_3 1)\n (= count_0_4 0)\n (= count_1_0 (+ count_0_2 count_0_2 count_0_4))\n (= count_1_1 (+ count_0_0 count_0_3 count_0_4))\n (= count_1_2 (+ count_0_0))\n (= count_1_3 (+ count_0_1 count_0_3))\n (= count_1_4 (+ count_0_1))\n (= count_2_0 (+ count_1_2 count_1_2 count_1_4))\n (= count_2_1 (+ count_1_0 count_1_3 count_1_4))\n (= count_2_2 (+ count_1_0))\n (= count_2_3 (+ count_1_1 count_1_3))\n (= count_2_4 (+ count_1_1))\n (= count_3_0 (+ count_2_2 count_2_2 count_2_4))\n (= c...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "automata", "id": "automata-dfa_string_acceptance-5bf1ed9d8a8df569", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, true, false, true, false], "answer": 7, "difficulty": "medium", "generator": "state_machine", "length": 3, "num_states": 5, "seed": 20261072, "start": 3, "transitions": [[1, 2], [3, 4], [0, 0], [3, 1], [1, 0]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 5 states numbered 0 to 4. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 3.\n\nPrint how many distinct binary strings of length exactly 3 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=5, L=3)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 7, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-5bf1ed9d8a8df569-t000", "input": {"accepting": [true, true, false, true, false], "length": 3, "start": 3, "transitions": [[1, 2], [3, 4], [0, 0], [3, 1], [1, 0]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-5bf1ed9d8a8df569"}], "verification": {"duration_ms": 1.057624991517514, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 1)\n (= count_0_2 0)\n (= count_0_3 0)\n (= count_1_0 (+ count_0_1 count_0_3))\n (= count_1_1 (+ count_0_2))\n (= count_1_2 (+ count_0_0 count_0_0 count_0_1 count_0_3))\n (= count_1_3 (+ count_0_2))\n (= count_2_0 (+ count_1_1 count_1_3))\n (= count_2_1 (+ count_1_2))\n (= count_2_2 (+ count_1_0 count_1_0 count_1_1 count_1_3))\n (= count_2_3 (+ count_1_2))\n (= count_3_0 (+ count_2_1 count_2_3))\n (= count_3_1 (+ count_2_2))\n (= count_3_2 (+ count_2_0 count_2_0 count_2_1 count_2_3))\n (= count_3_3 (+ count_2_2...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "medium", "domain": "automata", "id": "automata-dfa_string_acceptance-24ebb861c8cca632", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, false, false, false], "answer": 2, "difficulty": "medium", "generator": "state_machine", "length": 3, "num_states": 4, "seed": 20261073, "start": 1, "transitions": [[2, 2], [2, 0], [1, 3], [2, 0]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 4 states numbered 0 to 3. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 1.\n\nPrint how many distinct binary strings of length exactly 3 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=4, L=3)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-24ebb861c8cca632-t000", "input": {"accepting": [true, false, false, false], "length": 3, "start": 1, "transitions": [[2, 2], [2, 0], [1, 3], [2, 0]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-24ebb861c8cca632"}], "verification": {"duration_ms": 0.8277079905383289, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 1)\n (= count_0_2 0)\n (= count_0_3 0)\n (= count_0_4 0)\n (= count_0_5 0)\n (= count_1_0 (+ count_0_2 count_0_4))\n (= count_1_1 (+ count_0_0 count_0_2 count_0_3 count_0_4 count_0_5))\n (= count_1_2 0)\n (= count_1_3 (+ count_0_1))\n (= count_1_4 (+ count_0_0 count_0_1))\n (= count_1_5 (+ count_0_3 count_0_5))\n (= count_2_0 (+ count_1_2 count_1_4))\n (= count_2_1 (+ count_1_0 count_1_2 count_1_3 count_1_4 count_1_5))\n (= count_2_2 0)\n (= count_2_3 (+ count_1_1))\n (= count_2_4 (+ count_1_0 count_1_1))\n ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "automata", "id": "automata-dfa_string_acceptance-f4e0434e45b1ab97", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [false, true, false, false, true, false], "answer": 5, "difficulty": "hard", "generator": "state_machine", "length": 3, "num_states": 6, "seed": 20261074, "start": 1, "transitions": [[1, 4], [3, 4], [0, 1], [5, 1], [0, 1], [5, 1]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 6 states numbered 0 to 5. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 1.\n\nPrint how many distinct binary strings of length exactly 3 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=6, L=3)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-f4e0434e45b1ab97-t000", "input": {"accepting": [false, true, false, false, true, false], "length": 3, "start": 1, "transitions": [[1, 4], [3, 4], [0, 1], [5, 1], [0, 1], [5, 1]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-f4e0434e45b1ab97"}], "verification": {"duration_ms": 1.000124029815197, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 1)\n (= count_0_1 0)\n (= count_0_2 0)\n (= count_0_3 0)\n (= count_0_4 0)\n (= count_1_0 (+ count_0_0 count_0_3))\n (= count_1_1 (+ count_0_1 count_0_2 count_0_3 count_0_4))\n (= count_1_2 0)\n (= count_1_3 (+ count_0_1))\n (= count_1_4 (+ count_0_0 count_0_2 count_0_4))\n (= count_2_0 (+ count_1_0 count_1_3))\n (= count_2_1 (+ count_1_1 count_1_2 count_1_3 count_1_4))\n (= count_2_2 0)\n (= count_2_3 (+ count_1_1))\n (= count_2_4 (+ count_1_0 count_1_2 count_1_4))\n (= count_3_0 (+ count_2_0 count_2_3))\n (= count_3_1 (+ co...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "automata", "id": "automata-dfa_string_acceptance-cf3fb2c5b6d957b0", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [false, true, true, false, false], "answer": 27, "difficulty": "hard", "generator": "state_machine", "length": 6, "num_states": 5, "seed": 20261075, "start": 0, "transitions": [[4, 0], [1, 3], [1, 4], [0, 1], [1, 4]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 5 states numbered 0 to 4. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 0.\n\nPrint how many distinct binary strings of length exactly 6 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=5, L=6)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 27, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-cf3fb2c5b6d957b0-t000", "input": {"accepting": [false, true, true, false, false], "length": 6, "start": 0, "transitions": [[4, 0], [1, 3], [1, 4], [0, 1], [1, 4]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-cf3fb2c5b6d957b0"}], "verification": {"duration_ms": 1.3902499922551215, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_0_2 (= depth_0_0 (+ depth_0_2 1)))\n (and pair_1_2 (= depth_0_0 (+ depth_1_2 1)))\n (and pair_2_1 (= depth_0_0 (+ depth_2_1 1)))\n (and pair_2_3 (= depth_0_0 (+ depth_2_3 1)))))\n (a!2 (or (and pair_2_0 (= depth_0_2 (+ depth_2_0 1)))))\n (a!3 (or (and pair_0_0 (= depth_0_3 (+ depth_0_0 1)))\n (and pair_1_0 (= depth_0_3 (+ depth_1_0 1)))))\n (a!4 (or (and pair_0_4 (= depth_0_4 (+ depth_0_4 1)))\n (and pair_1_4 (= depth_0_4 (+ depth_1_4 1)))\n (and pair_2_4 (= depth_0_4 (+ depth_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "automata", "id": "automata-dfa_product_reachability-2f2ac6ddd6ceeeb8", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 4, "difficulty": "hard", "equivalent_by_construction": true, "first": {"accepting": [true, true, false, false], "start": 0, "transitions": [[3, 0], [3, 0], [0, 3], [2, 1]]}, "generator": "state_machine", "num_states": 4, "second": {"accepting": [false, true, false, true, false], "start": 1, "transitions": [[2, 3], [0, 1], [1, 0], [0, 1], [4, 4]]}, "seed": 20261076, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(0, 1) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (4x5 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-2f2ac6ddd6ceeeb8-t000", "input": {"first": {"accepting": [true, true, false, false], "start": 0, "transitions": [[3, 0], [3, 0], [0, 3], [2, 1]]}, "second": {"accepting": [false, true, false, true, false], "start": 1, "transitions": [[2, 3], [0, 1], [1, 0], [0, 1], [4, 4]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-2f2ac6ddd6ceeeb8"}], "verification": {"duration_ms": 3.275667957495898, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 1)\n (= count_0_1 0)\n (= count_0_2 0)\n (= count_0_3 0)\n (= count_0_4 0)\n (= count_1_0 (+ count_0_1))\n (= count_1_1 (+ count_0_0 count_0_2 count_0_3))\n (= count_1_2 0)\n (= count_1_3 (+ count_0_0 count_0_2))\n (= count_1_4 (+ count_0_1 count_0_3 count_0_4 count_0_4))\n (= count_2_0 (+ count_1_1))\n (= count_2_1 (+ count_1_0 count_1_2 count_1_3))\n (= count_2_2 0)\n (= count_2_3 (+ count_1_0 count_1_2))\n (= count_2_4 (+ count_1_1 count_1_3 count_1_4 count_1_4))\n (= count_3_0 (+ count_2_1))\n (= count_3_1 (+ count_2_0 co...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "automata", "id": "automata-dfa_string_acceptance-1d8b38c16e390569", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, false, false, false, true], "answer": 13, "difficulty": "hard", "generator": "state_machine", "length": 4, "num_states": 5, "seed": 20261077, "start": 0, "transitions": [[1, 3], [0, 4], [3, 1], [1, 4], [4, 4]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 5 states numbered 0 to 4. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 0.\n\nPrint how many distinct binary strings of length exactly 4 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=5, L=4)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 13, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-1d8b38c16e390569-t000", "input": {"accepting": [true, false, false, false, true], "length": 4, "start": 0, "transitions": [[1, 3], [0, 4], [3, 1], [1, 4], [4, 4]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-1d8b38c16e390569"}], "verification": {"duration_ms": 1.2449590140022337, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_4_3 (= depth_0_1 (+ depth_4_3 1)))\n (and pair_4_4 (= depth_0_1 (+ depth_4_4 1)))))\n (a!2 (or (and pair_4_1 (= depth_0_2 (+ depth_4_1 1)))))\n (a!3 (or (and pair_4_2 (= depth_0_3 (+ depth_4_2 1)))))\n (a!4 (or (and pair_4_0 (= depth_0_4 (+ depth_4_0 1)))))\n (a!5 (or (and pair_1_2 (= depth_1_0 (+ depth_1_2 1)))\n (and pair_1_3 (= depth_1_0 (+ depth_1_3 1)))))\n (a!6 (or (and pair_1_1 (= depth_1_1 (+ depth_1_1 1)))\n (and pair_3_3 (= depth_1_1 (+ depth_3_3 1)))\n (and pair_3_4 (= depth_1_1 (+ de...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "automata", "id": "automata-dfa_product_reachability-d76ec71e588d775f", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 5, "difficulty": "hard", "equivalent_by_construction": true, "first": {"accepting": [false, false, false, true, false], "start": 2, "transitions": [[3, 2], [1, 4], [3, 4], [2, 1], [4, 0]]}, "generator": "state_machine", "num_states": 5, "second": {"accepting": [true, false, false, false, false], "start": 3, "transitions": [[3, 4], [1, 2], [0, 3], [0, 1], [4, 1]]}, "seed": 20261078, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(2, 3) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (5x5 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-d76ec71e588d775f-t000", "input": {"first": {"accepting": [false, false, false, true, false], "start": 2, "transitions": [[3, 2], [1, 4], [3, 4], [2, 1], [4, 0]]}, "second": {"accepting": [true, false, false, false, false], "start": 3, "transitions": [[3, 4], [1, 2], [0, 3], [0, 1], [4, 1]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-d76ec71e588d775f"}], "verification": {"duration_ms": 4.883791960310191, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_0_0 (= depth_0_0 (+ depth_0_0 1)))\n (and pair_0_0 (= depth_0_0 (+ depth_0_0 1)))\n (and pair_0_2 (= depth_0_0 (+ depth_0_2 1)))\n (and pair_1_0 (= depth_0_0 (+ depth_1_0 1)))\n (and pair_1_2 (= depth_0_0 (+ depth_1_2 1)))))\n (a!2 (or (and pair_0_1 (= depth_0_1 (+ depth_0_1 1)))\n (and pair_0_3 (= depth_0_1 (+ depth_0_3 1)))))\n (a!3 (or (and pair_0_1 (= depth_0_2 (+ depth_0_1 1)))\n (and pair_1_1 (= depth_0_2 (+ depth_1_1 1)))))\n (a!4 (or (and pair_0_2 (= depth_0_3 (+ depth_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "automata", "id": "automata-dfa_product_reachability-ebc3d1eaf9b0b15b", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 4, "difficulty": "hard", "equivalent_by_construction": true, "first": {"accepting": [true, true, false, false], "start": 1, "transitions": [[0, 0], [0, 2], [2, 3], [1, 3]]}, "generator": "state_machine", "num_states": 4, "second": {"accepting": [true, false, true, false], "start": 2, "transitions": [[0, 0], [2, 1], [0, 3], [3, 1]]}, "seed": 20261079, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(1, 2) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (4x4 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-ebc3d1eaf9b0b15b-t000", "input": {"first": {"accepting": [true, true, false, false], "start": 1, "transitions": [[0, 0], [0, 2], [2, 3], [1, 3]]}, "second": {"accepting": [true, false, true, false], "start": 2, "transitions": [[0, 0], [2, 1], [0, 3], [3, 1]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-ebc3d1eaf9b0b15b"}], "verification": {"duration_ms": 2.7804580167867243, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 0)\n (= count_0_2 0)\n (= count_0_3 1)\n (= count_1_0 (+ count_0_0 count_0_3 count_0_3))\n (= count_1_1 0)\n (= count_1_2 (+ count_0_1))\n (= count_1_3 (+ count_0_0 count_0_1 count_0_2 count_0_2))\n (= count_2_0 (+ count_1_0 count_1_3 count_1_3))\n (= count_2_1 0)\n (= count_2_2 (+ count_1_1))\n (= count_2_3 (+ count_1_0 count_1_1 count_1_2 count_1_2))\n (= count_3_0 (+ count_2_0 count_2_3 count_2_3))\n (= count_3_1 0)\n (= count_3_2 (+ count_2_1))\n (= count_3_3 (+ count_2_0 count_2_1 count_2_2 count_2_2))\n ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "automata", "id": "automata-dfa_string_acceptance-3713cd920dde9601", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, false, true, false], "answer": 22, "difficulty": "hard", "generator": "state_machine", "length": 5, "num_states": 4, "seed": 20261080, "start": 3, "transitions": [[3, 0], [3, 2], [3, 3], [0, 0]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 4 states numbered 0 to 3. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 3.\n\nPrint how many distinct binary strings of length exactly 5 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=4, L=5)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 22, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-3713cd920dde9601-t000", "input": {"accepting": [true, false, true, false], "length": 5, "start": 3, "transitions": [[3, 0], [3, 2], [3, 3], [0, 0]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-3713cd920dde9601"}], "verification": {"duration_ms": 1.1010000598616898, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_0_1 (= depth_0_1 (+ depth_0_1 1)))\n (and pair_0_2 (= depth_0_1 (+ depth_0_2 1)))\n (and pair_1_1 (= depth_0_1 (+ depth_1_1 1)))\n (and pair_1_2 (= depth_0_1 (+ depth_1_2 1)))\n (and pair_2_3 (= depth_0_1 (+ depth_2_3 1)))))\n (a!2 (or (and pair_0_0 (= depth_0_2 (+ depth_0_0 1)))\n (and pair_0_3 (= depth_0_2 (+ depth_0_3 1)))\n (and pair_1_0 (= depth_0_2 (+ depth_1_0 1)))\n (and pair_1_3 (= depth_0_2 (+ depth_1_3 1)))\n (and pair_2_1 (= depth_0_2 (+ depth_2_1 ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "automata", "id": "automata-dfa_product_reachability-0df76a98bc2b3976", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 3, "difficulty": "hard", "equivalent_by_construction": true, "first": {"accepting": [false, false, true, false], "start": 2, "transitions": [[1, 0], [1, 0], [0, 1], [2, 1]]}, "generator": "state_machine", "num_states": 4, "second": {"accepting": [false, false, false, true, false, false], "start": 3, "transitions": [[3, 2], [2, 1], [2, 1], [1, 2], [4, 4], [5, 5]]}, "seed": 20261081, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(2, 3) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (4x6 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 3, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-0df76a98bc2b3976-t000", "input": {"first": {"accepting": [false, false, true, false], "start": 2, "transitions": [[1, 0], [1, 0], [0, 1], [2, 1]]}, "second": {"accepting": [false, false, false, true, false, false], "start": 3, "transitions": [[3, 2], [2, 1], [2, 1], [1, 2], [4, 4], [5, 5]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-0df76a98bc2b3976"}], "verification": {"duration_ms": 2.2793749812990427, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 0)\n (= count_0_2 0)\n (= count_0_3 0)\n (= count_0_4 1)\n (= count_1_0 (+ count_0_3))\n (= count_1_1 (+ count_0_1 count_0_2 count_0_2))\n (= count_1_2 (+ count_0_4))\n (= count_1_3 (+ count_0_0 count_0_0 count_0_1 count_0_4))\n (= count_1_4 (+ count_0_3))\n (= count_2_0 (+ count_1_3))\n (= count_2_1 (+ count_1_1 count_1_2 count_1_2))\n (= count_2_2 (+ count_1_4))\n (= count_2_3 (+ count_1_0 count_1_0 count_1_1 count_1_4))\n (= count_2_4 (+ count_1_3))\n (= count_3_0 (+ count_2_3))\n (= count_3_1 (+ count_2_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "automata", "id": "automata-dfa_string_acceptance-121e4e8e09e2aeef", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, false, true, true, true], "answer": 6, "difficulty": "hard", "generator": "state_machine", "length": 3, "num_states": 5, "seed": 20261082, "start": 4, "transitions": [[3, 3], [1, 3], [1, 1], [0, 4], [2, 3]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 5 states numbered 0 to 4. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 4.\n\nPrint how many distinct binary strings of length exactly 3 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=5, L=3)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 6, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-121e4e8e09e2aeef-t000", "input": {"accepting": [true, false, true, true, true], "length": 3, "start": 4, "transitions": [[3, 3], [1, 3], [1, 1], [0, 4], [2, 3]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-121e4e8e09e2aeef"}], "verification": {"duration_ms": 0.8901239489205182, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_0_1 (= depth_0_0 (+ depth_0_1 1)))))\n (a!2 (or (and pair_3_0 (= depth_0_1 (+ depth_3_0 1)))))\n (a!3 (or (and pair_0_0 (= depth_0_2 (+ depth_0_0 1)))\n (and pair_0_2 (= depth_0_2 (+ depth_0_2 1)))))\n (a!4 (or (and pair_0_3 (= depth_0_3 (+ depth_0_3 1)))\n (and pair_3_1 (= depth_0_3 (+ depth_3_1 1)))\n (and pair_3_2 (= depth_0_3 (+ depth_3_2 1)))\n (and pair_3_4 (= depth_0_3 (+ depth_3_4 1)))))\n (a!5 (or (and pair_0_5 (= depth_0_5 (+ depth_0_5 1)))\n (and pair_3_5 (= depth_0_5 (+ dept...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "automata", "id": "automata-dfa_product_reachability-71bb58b78e46b48b", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 2, "difficulty": "hard", "equivalent_by_construction": true, "first": {"accepting": [false, false, false, false, true], "start": 0, "transitions": [[3, 0], [3, 2], [1, 4], [0, 3], [3, 4]]}, "generator": "state_machine", "num_states": 5, "second": {"accepting": [false, false, true, false, false, false], "start": 4, "transitions": [[1, 2], [3, 0], [3, 2], [4, 3], [3, 4], [5, 5]]}, "seed": 20261083, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(0, 4) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (5x6 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-71bb58b78e46b48b-t000", "input": {"first": {"accepting": [false, false, false, false, true], "start": 0, "transitions": [[3, 0], [3, 2], [1, 4], [0, 3], [3, 4]]}, "second": {"accepting": [false, false, true, false, false, false], "start": 4, "transitions": [[1, 2], [3, 0], [3, 2], [4, 3], [3, 4], [5, 5]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-71bb58b78e46b48b"}], "verification": {"duration_ms": 3.748999035451561, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 1)\n (= count_0_1 0)\n (= count_0_2 0)\n (= count_0_3 0)\n (= count_1_0 (+ count_0_0 count_0_0 count_0_2 count_0_3 count_0_3))\n (= count_1_1 (+ count_0_2))\n (= count_1_2 0)\n (= count_1_3 (+ count_0_1 count_0_1))\n (= count_2_0 (+ count_1_0 count_1_0 count_1_2 count_1_3 count_1_3))\n (= count_2_1 (+ count_1_2))\n (= count_2_2 0)\n (= count_2_3 (+ count_1_1 count_1_1))\n (= count_3_0 (+ count_2_0 count_2_0 count_2_2 count_2_3 count_2_3))\n (= count_3_1 (+ count_2_2))\n (= count_3_2 0)\n (= count_3_3 (+ count_2_1 count_2_1))\n ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "automata", "id": "automata-dfa_string_acceptance-94d6109632f3c557", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, true, false, false], "answer": 8, "difficulty": "hard", "generator": "state_machine", "length": 3, "num_states": 4, "seed": 20261084, "start": 0, "transitions": [[0, 0], [3, 3], [1, 0], [0, 0]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 4 states numbered 0 to 3. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 0.\n\nPrint how many distinct binary strings of length exactly 3 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=4, L=3)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 8, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-94d6109632f3c557-t000", "input": {"accepting": [true, true, false, false], "length": 3, "start": 0, "transitions": [[0, 0], [3, 3], [1, 0], [0, 0]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-94d6109632f3c557"}], "verification": {"duration_ms": 0.8479169919155538, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 0)\n (= count_0_2 1)\n (= count_0_3 0)\n (= count_1_0 (+ count_0_3))\n (= count_1_1 (+ count_0_1))\n (= count_1_2 (+ count_0_0 count_0_2))\n (= count_1_3 (+ count_0_0 count_0_1 count_0_2 count_0_3))\n (= count_2_0 (+ count_1_3))\n (= count_2_1 (+ count_1_1))\n (= count_2_2 (+ count_1_0 count_1_2))\n (= count_2_3 (+ count_1_0 count_1_1 count_1_2 count_1_3))\n (= count_3_0 (+ count_2_3))\n (= count_3_1 (+ count_2_1))\n (= count_3_2 (+ count_2_0 count_2_2))\n (= count_3_3 (+ count_2_0 count_2_1 count_2_2 count_2_3...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "automata", "id": "automata-dfa_string_acceptance-73e0395c8ea69ced", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [false, true, true, true], "answer": 24, "difficulty": "hard", "generator": "state_machine", "length": 5, "num_states": 4, "seed": 20261085, "start": 2, "transitions": [[2, 3], [3, 1], [3, 2], [3, 0]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 4 states numbered 0 to 3. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 2.\n\nPrint how many distinct binary strings of length exactly 5 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=4, L=5)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 24, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-73e0395c8ea69ced-t000", "input": {"accepting": [false, true, true, true], "length": 5, "start": 2, "transitions": [[2, 3], [3, 1], [3, 2], [3, 0]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-73e0395c8ea69ced"}], "verification": {"duration_ms": 1.029832987114787, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_2_2 (= depth_0_0 (+ depth_2_2 1)))\n (and pair_2_3 (= depth_0_0 (+ depth_2_3 1)))\n (and pair_3_2 (= depth_0_0 (+ depth_3_2 1)))\n (and pair_3_3 (= depth_0_0 (+ depth_3_3 1)))))\n (a!2 (or (and pair_2_1 (= depth_0_1 (+ depth_2_1 1)))\n (and pair_3_1 (= depth_0_1 (+ depth_3_1 1)))))\n (a!3 (or (and pair_2_0 (= depth_0_2 (+ depth_2_0 1)))\n (and pair_3_0 (= depth_0_2 (+ depth_3_0 1)))))\n (a!4 (or (and pair_2_4 (= depth_0_4 (+ depth_2_4 1)))\n (and pair_3_4 (= depth_0_4 (+ depth_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "automata", "id": "automata-dfa_product_reachability-846271b5747c10e3", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 3, "difficulty": "hard", "equivalent_by_construction": true, "first": {"accepting": [false, false, true, false], "start": 1, "transitions": [[1, 3], [3, 1], [1, 0], [3, 0]]}, "generator": "state_machine", "num_states": 4, "second": {"accepting": [false, false, false, true, false], "start": 1, "transitions": [[1, 2], [2, 1], [2, 0], [1, 0], [4, 4]]}, "seed": 20261086, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(1, 1) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (4x5 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 3, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-846271b5747c10e3-t000", "input": {"first": {"accepting": [false, false, true, false], "start": 1, "transitions": [[1, 3], [3, 1], [1, 0], [3, 0]]}, "second": {"accepting": [false, false, false, true, false], "start": 1, "transitions": [[1, 2], [2, 1], [2, 0], [1, 0], [4, 4]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-846271b5747c10e3"}], "verification": {"duration_ms": 2.617040940094739, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_0_0 (= depth_0_0 (+ depth_0_0 1)))\n (and pair_0_2 (= depth_0_0 (+ depth_0_2 1)))))\n (a!2 (or (and pair_0_1 (= depth_0_1 (+ depth_0_1 1)))))\n (a!3 (or (and pair_0_3 (= depth_0_3 (+ depth_0_3 1)))))\n (a!4 (or (and pair_0_4 (= depth_0_4 (+ depth_0_4 1)))))\n (a!5 (or (and pair_0_5 (= depth_0_5 (+ depth_0_5 1)))))\n (a!6 (or (and pair_0_1 (= depth_1_1 (+ depth_0_1 1)))\n (and pair_0_2 (= depth_1_1 (+ depth_0_2 1)))\n (and pair_3_1 (= depth_1_1 (+ depth_3_1 1)))\n (and pair_3_2 (= depth_1_1 (+ de...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "automata", "id": "automata-dfa_product_reachability-84ca6948c59ae7df", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 1, "difficulty": "hard", "equivalent_by_construction": true, "first": {"accepting": [false, false, true, true], "start": 2, "transitions": [[0, 1], [3, 2], [2, 2], [3, 1]]}, "generator": "state_machine", "num_states": 4, "second": {"accepting": [true, true, false, false, false, false], "start": 1, "transitions": [[0, 2], [1, 1], [0, 1], [3, 2], [4, 4], [5, 5]]}, "seed": 20261087, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(2, 1) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (4x6 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 1, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-84ca6948c59ae7df-t000", "input": {"first": {"accepting": [false, false, true, true], "start": 2, "transitions": [[0, 1], [3, 2], [2, 2], [3, 1]]}, "second": {"accepting": [true, true, false, false, false, false], "start": 1, "transitions": [[0, 2], [1, 1], [0, 1], [3, 2], [4, 4], [5, 5]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-84ca6948c59ae7df"}], "verification": {"duration_ms": 2.3023750400170684, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 0)\n (= count_0_2 1)\n (= count_0_3 0)\n (= count_0_4 0)\n (= count_1_0 (+ count_0_2 count_0_3))\n (= count_1_1 (+ count_0_0 count_0_2))\n (= count_1_2 (+ count_0_1))\n (= count_1_3 (+ count_0_1 count_0_4))\n (= count_1_4 (+ count_0_0 count_0_3 count_0_4))\n (= count_2_0 (+ count_1_2 count_1_3))\n (= count_2_1 (+ count_1_0 count_1_2))\n (= count_2_2 (+ count_1_1))\n (= count_2_3 (+ count_1_1 count_1_4))\n (= count_2_4 (+ count_1_0 count_1_3 count_1_4))\n (= count_3_0 (+ count_2_2 count_2_3))\n (= count_3_1 (...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "automata", "id": "automata-dfa_string_acceptance-1f7d19a0be9ec736", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [false, false, false, false, true], "answer": 25, "difficulty": "hard", "generator": "state_machine", "length": 6, "num_states": 5, "seed": 20261088, "start": 2, "transitions": [[1, 4], [2, 3], [1, 0], [4, 0], [3, 4]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 5 states numbered 0 to 4. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 2.\n\nPrint how many distinct binary strings of length exactly 6 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=5, L=6)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 25, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-1f7d19a0be9ec736-t000", "input": {"accepting": [false, false, false, false, true], "length": 6, "start": 2, "transitions": [[1, 4], [2, 3], [1, 0], [4, 0], [3, 4]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-1f7d19a0be9ec736"}], "verification": {"duration_ms": 1.1885410640388727, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 1)\n (= count_0_1 0)\n (= count_0_2 0)\n (= count_0_3 0)\n (= count_0_4 0)\n (= count_0_5 0)\n (= count_1_0 (+ count_0_0 count_0_5))\n (= count_1_1 (+ count_0_1 count_0_2))\n (= count_1_2 (+ count_0_4 count_0_4))\n (= count_1_3 (+ count_0_0))\n (= count_1_4 (+ count_0_2 count_0_3 count_0_5))\n (= count_1_5 (+ count_0_1 count_0_3))\n (= count_2_0 (+ count_1_0 count_1_5))\n (= count_2_1 (+ count_1_1 count_1_2))\n (= count_2_2 (+ count_1_4 count_1_4))\n (= count_2_3 (+ count_1_0))\n (= count_2_4 (+ count_1_2 count_1_3 count_1_5)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "automata", "id": "automata-dfa_string_acceptance-7e5c7b5221b963df", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, false, false, true, false, true], "answer": 4, "difficulty": "hard", "generator": "state_machine", "length": 3, "num_states": 6, "seed": 20261089, "start": 0, "transitions": [[0, 3], [1, 5], [4, 1], [4, 5], [2, 2], [4, 0]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 6 states numbered 0 to 5. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 0.\n\nPrint how many distinct binary strings of length exactly 3 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=6, L=3)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-7e5c7b5221b963df-t000", "input": {"accepting": [true, false, false, true, false, true], "length": 3, "start": 0, "transitions": [[0, 3], [1, 5], [4, 1], [4, 5], [2, 2], [4, 0]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-7e5c7b5221b963df"}], "verification": {"duration_ms": 0.8675840217620134, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_0_0 (= depth_0_0 (+ depth_0_0 1)))\n (and pair_0_2 (= depth_0_0 (+ depth_0_2 1)))\n (and pair_1_0 (= depth_0_0 (+ depth_1_0 1)))\n (and pair_1_2 (= depth_0_0 (+ depth_1_2 1)))))\n (a!2 (or (and pair_0_1 (= depth_0_2 (+ depth_0_1 1)))\n (and pair_0_3 (= depth_0_2 (+ depth_0_3 1)))\n (and pair_1_1 (= depth_0_2 (+ depth_1_1 1)))\n (and pair_1_3 (= depth_0_2 (+ depth_1_3 1)))))\n (a!3 (or (and pair_2_0 (= depth_1_0 (+ depth_2_0 1)))\n (and pair_2_2 (= depth_1_0 (+ depth_2_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "automata", "id": "automata-dfa_product_reachability-d61e9927bdb2fd4b", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 4, "difficulty": "hard", "equivalent_by_construction": true, "first": {"accepting": [false, false, false, false], "start": 1, "transitions": [[3, 0], [1, 0], [3, 1], [2, 1]]}, "generator": "state_machine", "num_states": 4, "second": {"accepting": [false, false, false, false], "start": 2, "transitions": [[3, 0], [3, 2], [2, 0], [1, 2]]}, "seed": 20261090, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(1, 2) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (4x4 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-d61e9927bdb2fd4b-t000", "input": {"first": {"accepting": [false, false, false, false], "start": 1, "transitions": [[3, 0], [1, 0], [3, 1], [2, 1]]}, "second": {"accepting": [false, false, false, false], "start": 2, "transitions": [[3, 0], [3, 2], [2, 0], [1, 2]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-d61e9927bdb2fd4b"}], "verification": {"duration_ms": 2.5684169959276915, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_0_4 (= depth_0_0 (+ depth_0_4 1)))\n (and pair_1_4 (= depth_0_0 (+ depth_1_4 1)))))\n (a!2 (or (and pair_0_1 (= depth_0_1 (+ depth_0_1 1)))\n (and pair_1_1 (= depth_0_1 (+ depth_1_1 1)))))\n (a!3 (or (and pair_0_2 (= depth_0_2 (+ depth_0_2 1)))\n (and pair_0_3 (= depth_0_2 (+ depth_0_3 1)))\n (and pair_1_2 (= depth_0_2 (+ depth_1_2 1)))\n (and pair_1_3 (= depth_0_2 (+ depth_1_3 1)))))\n (a!4 (or (and pair_0_0 (= depth_0_4 (+ depth_0_0 1)))\n (and pair_1_0 (= depth_0_4 (+ depth_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "automata", "id": "automata-dfa_product_reachability-cbad17d0832e0827", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 2, "difficulty": "hard", "equivalent_by_construction": true, "first": {"accepting": [true, false, true, true, false], "start": 3, "transitions": [[0, 1], [0, 1], [2, 3], [4, 3], [3, 3]]}, "generator": "state_machine", "num_states": 5, "second": {"accepting": [true, true, true, false, false, false], "start": 0, "transitions": [[4, 0], [1, 0], [2, 3], [2, 3], [0, 0], [5, 5]]}, "seed": 20261091, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(3, 0) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (5x6 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-cbad17d0832e0827-t000", "input": {"first": {"accepting": [true, false, true, true, false], "start": 3, "transitions": [[0, 1], [0, 1], [2, 3], [4, 3], [3, 3]]}, "second": {"accepting": [true, true, true, false, false, false], "start": 0, "transitions": [[4, 0], [1, 0], [2, 3], [2, 3], [0, 0], [5, 5]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-cbad17d0832e0827"}], "verification": {"duration_ms": 3.4383750171400607, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 0)\n (= count_0_2 1)\n (= count_0_3 0)\n (= count_0_4 0)\n (= count_0_5 0)\n (= count_1_0 (+ count_0_0 count_0_2 count_0_3))\n (= count_1_1 (+ count_0_2 count_0_3 count_0_5))\n (= count_1_2 (+ count_0_4 count_0_5))\n (= count_1_3 (+ count_0_0))\n (= count_1_4 0)\n (= count_1_5 (+ count_0_1 count_0_1 count_0_4))\n (= count_2_0 (+ count_1_0 count_1_2 count_1_3))\n (= count_2_1 (+ count_1_2 count_1_3 count_1_5))\n (= count_2_2 (+ count_1_4 count_1_5))\n (= count_2_3 (+ count_1_0))\n (= count_2_4 0)\n (= coun...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "automata", "id": "automata-dfa_string_acceptance-083ba62032bbd6f3", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, false, true, false, true, false], "answer": 24, "difficulty": "hard", "generator": "state_machine", "length": 6, "num_states": 6, "seed": 20261092, "start": 2, "transitions": [[0, 3], [5, 5], [0, 1], [1, 0], [5, 2], [2, 1]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 6 states numbered 0 to 5. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 2.\n\nPrint how many distinct binary strings of length exactly 6 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=6, L=6)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 24, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-083ba62032bbd6f3-t000", "input": {"accepting": [true, false, true, false, true, false], "length": 6, "start": 2, "transitions": [[0, 3], [5, 5], [0, 1], [1, 0], [5, 2], [2, 1]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-083ba62032bbd6f3"}], "verification": {"duration_ms": 1.3322079321369529, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_0_0 (= depth_0_2 (+ depth_0_0 1)))\n (and pair_0_3 (= depth_0_2 (+ depth_0_3 1)))\n (and pair_2_0 (= depth_0_2 (+ depth_2_0 1)))\n (and pair_2_3 (= depth_0_2 (+ depth_2_3 1)))))\n (a!2 (or (and pair_0_1 (= depth_0_3 (+ depth_0_1 1)))\n (and pair_2_1 (= depth_0_3 (+ depth_2_1 1)))))\n (a!3 (or (and pair_0_2 (= depth_0_4 (+ depth_0_2 1)))\n (and pair_0_4 (= depth_0_4 (+ depth_0_4 1)))\n (and pair_2_2 (= depth_0_4 (+ depth_2_2 1)))\n (and pair_2_4 (= depth_0_4 (+ depth_2_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "automata", "id": "automata-dfa_product_reachability-3f5ba40f5e755025", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 5, "difficulty": "hard", "equivalent_by_construction": true, "first": {"accepting": [true, false, false, false, true], "start": 3, "transitions": [[3, 0], [4, 2], [4, 0], [1, 2], [3, 3]]}, "generator": "state_machine", "num_states": 5, "second": {"accepting": [false, true, false, false, true, false, false], "start": 3, "transitions": [[1, 2], [3, 3], [1, 4], [0, 2], [3, 4], [5, 5], [6, 6]]}, "seed": 20261093, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(3, 3) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (5x7 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-3f5ba40f5e755025-t000", "input": {"first": {"accepting": [true, false, false, false, true], "start": 3, "transitions": [[3, 0], [4, 2], [4, 0], [1, 2], [3, 3]]}, "second": {"accepting": [false, true, false, false, true, false, false], "start": 3, "transitions": [[1, 2], [3, 3], [1, 4], [0, 2], [3, 4], [5, 5], [6, 6]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-3f5ba40f5e755025"}], "verification": {"duration_ms": 5.795083066914231, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 1)\n (= count_0_2 0)\n (= count_0_3 0)\n (= count_0_4 0)\n (= count_1_0 (+ count_0_2 count_0_3 count_0_4 count_0_4))\n (= count_1_1 (+ count_0_1 count_0_2 count_0_3))\n (= count_1_2 (+ count_0_1))\n (= count_1_3 (+ count_0_0))\n (= count_1_4 (+ count_0_0))\n (= count_2_0 (+ count_1_2 count_1_3 count_1_4 count_1_4))\n (= count_2_1 (+ count_1_1 count_1_2 count_1_3))\n (= count_2_2 (+ count_1_1))\n (= count_2_3 (+ count_1_0))\n (= count_2_4 (+ count_1_0))\n (= count_3_0 (+ count_2_2 count_2_3 count_2_4 count_2_4))...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "automata", "id": "automata-dfa_string_acceptance-09f622be6e91356d", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [false, true, true, false, true], "answer": 10, "difficulty": "hard", "generator": "state_machine", "length": 4, "num_states": 5, "seed": 20261094, "start": 1, "transitions": [[4, 3], [2, 1], [0, 1], [1, 0], [0, 0]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 5 states numbered 0 to 4. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 1.\n\nPrint how many distinct binary strings of length exactly 4 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=5, L=4)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 10, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-09f622be6e91356d-t000", "input": {"accepting": [false, true, true, false, true], "length": 4, "start": 1, "transitions": [[4, 3], [2, 1], [0, 1], [1, 0], [0, 0]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-09f622be6e91356d"}], "verification": {"duration_ms": 1.09420798253268, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_0_0 (= depth_0_0 (+ depth_0_0 1)))\n (and pair_3_1 (= depth_0_0 (+ depth_3_1 1)))))\n (a!2 (or (and pair_0_2 (= depth_0_1 (+ depth_0_2 1)))\n (and pair_3_2 (= depth_0_1 (+ depth_3_2 1)))))\n (a!3 (or (and pair_3_0 (= depth_0_2 (+ depth_3_0 1)))\n (and pair_3_3 (= depth_0_2 (+ depth_3_3 1)))))\n (a!4 (or (and pair_0_1 (= depth_0_3 (+ depth_0_1 1)))\n (and pair_0_3 (= depth_0_3 (+ depth_0_3 1)))))\n (a!5 (or (and pair_0_4 (= depth_0_4 (+ depth_0_4 1)))\n (and pair_3_4 (= depth_0_4 (+ dept...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "automata", "id": "automata-dfa_product_reachability-7355beb278ad3916", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 4, "difficulty": "hard", "equivalent_by_construction": true, "first": {"accepting": [false, false, true, false], "start": 3, "transitions": [[1, 0], [3, 3], [1, 2], [0, 2]]}, "generator": "state_machine", "num_states": 4, "second": {"accepting": [false, false, false, true, false], "start": 1, "transitions": [[2, 0], [0, 3], [1, 1], [2, 3], [4, 4]]}, "seed": 20261095, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(3, 1) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (4x5 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-7355beb278ad3916-t000", "input": {"first": {"accepting": [false, false, true, false], "start": 3, "transitions": [[1, 0], [3, 3], [1, 2], [0, 2]]}, "second": {"accepting": [false, false, false, true, false], "start": 1, "transitions": [[2, 0], [0, 3], [1, 1], [2, 3], [4, 4]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-7355beb278ad3916"}], "verification": {"duration_ms": 3.4363330341875553, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 0)\n (= count_0_2 0)\n (= count_0_3 0)\n (= count_0_4 1)\n (= count_1_0 (+ count_0_0 count_0_0 count_0_1))\n (= count_1_1 (+ count_0_2 count_0_4))\n (= count_1_2 (+ count_0_1 count_0_2))\n (= count_1_3 (+ count_0_3 count_0_3))\n (= count_1_4 (+ count_0_4))\n (= count_2_0 (+ count_1_0 count_1_0 count_1_1))\n (= count_2_1 (+ count_1_2 count_1_4))\n (= count_2_2 (+ count_1_1 count_1_2))\n (= count_2_3 (+ count_1_3 count_1_3))\n (= count_2_4 (+ count_1_4))\n (= count_3_0 (+ count_2_0 count_2_0 count_2_1))\n (= c...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "automata", "id": "automata-dfa_string_acceptance-de9b106adf8770bb", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, false, false, false, false], "answer": 3, "difficulty": "hard", "generator": "state_machine", "length": 3, "num_states": 5, "seed": 20261096, "start": 4, "transitions": [[0, 0], [0, 2], [1, 2], [3, 3], [1, 4]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 5 states numbered 0 to 4. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 4.\n\nPrint how many distinct binary strings of length exactly 3 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=5, L=3)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 3, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-de9b106adf8770bb-t000", "input": {"accepting": [true, false, false, false, false], "length": 3, "start": 4, "transitions": [[0, 0], [0, 2], [1, 2], [3, 3], [1, 4]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-de9b106adf8770bb"}], "verification": {"duration_ms": 1.0013749706558883, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 1)\n (= count_0_1 0)\n (= count_0_2 0)\n (= count_0_3 0)\n (= count_1_0 (+ count_0_0 count_0_1 count_0_2 count_0_3))\n (= count_1_1 (+ count_0_1 count_0_3))\n (= count_1_2 (+ count_0_0 count_0_2))\n (= count_1_3 0)\n (= count_2_0 (+ count_1_0 count_1_1 count_1_2 count_1_3))\n (= count_2_1 (+ count_1_1 count_1_3))\n (= count_2_2 (+ count_1_0 count_1_2))\n (= count_2_3 0)\n (= count_3_0 (+ count_2_0 count_2_1 count_2_2 count_2_3))\n (= count_3_1 (+ count_2_1 count_2_3))\n (= count_3_2 (+ count_2_0 count_2_2))\n (= count_3_3 0)\n ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "automata", "id": "automata-dfa_string_acceptance-591dae2a9dc74906", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, true, false, true], "answer": 16, "difficulty": "hard", "generator": "state_machine", "length": 5, "num_states": 4, "seed": 20261097, "start": 0, "transitions": [[2, 0], [0, 1], [0, 2], [0, 1]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 4 states numbered 0 to 3. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 0.\n\nPrint how many distinct binary strings of length exactly 5 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=4, L=5)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 16, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-591dae2a9dc74906-t000", "input": {"accepting": [true, true, false, true], "length": 5, "start": 0, "transitions": [[2, 0], [0, 1], [0, 2], [0, 1]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-591dae2a9dc74906"}], "verification": {"duration_ms": 1.0632920311763883, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_0_0 (= depth_0_0 (+ depth_0_0 1)))))\n (a!2 (or (and pair_0_1 (= depth_0_1 (+ depth_0_1 1)))\n (and pair_0_2 (= depth_0_1 (+ depth_0_2 1)))))\n (a!3 (or (and pair_0_3 (= depth_0_2 (+ depth_0_3 1)))))\n (a!4 (or (and pair_0_4 (= depth_0_4 (+ depth_0_4 1)))))\n (a!5 (or (and pair_3_0 (= depth_1_0 (+ depth_3_0 1)))))\n (a!6 (or (and pair_0_2 (= depth_1_1 (+ depth_0_2 1)))\n (and pair_3_1 (= depth_1_1 (+ depth_3_1 1)))\n (and pair_3_2 (= depth_1_1 (+ depth_3_2 1)))\n (and pair_3_2 (= depth_1_1 (+ de...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "hard", "domain": "automata", "id": "automata-dfa_product_reachability-49ca0157360e9f26", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 3, "difficulty": "hard", "equivalent_by_construction": true, "first": {"accepting": [true, false, false, false], "start": 2, "transitions": [[1, 0], [2, 2], [3, 2], [1, 1]]}, "generator": "state_machine", "num_states": 4, "second": {"accepting": [true, false, false, false, false], "start": 1, "transitions": [[2, 0], [3, 1], [1, 1], [2, 2], [4, 4]]}, "seed": 20261098, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(2, 1) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (4x5 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 3, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-49ca0157360e9f26-t000", "input": {"first": {"accepting": [true, false, false, false], "start": 2, "transitions": [[1, 0], [2, 2], [3, 2], [1, 1]]}, "second": {"accepting": [true, false, false, false, false], "start": 1, "transitions": [[2, 0], [3, 1], [1, 1], [2, 2], [4, 4]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-49ca0157360e9f26"}], "verification": {"duration_ms": 2.3952919873408973, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 0)\n (= count_0_2 0)\n (= count_0_3 0)\n (= count_0_4 1)\n (= count_1_0 (+ count_0_3 count_0_4))\n (= count_1_1 (+ count_0_0 count_0_0 count_0_2 count_0_3))\n (= count_1_2 0)\n (= count_1_3 (+ count_0_1 count_0_2))\n (= count_1_4 (+ count_0_1 count_0_4))\n (= count_2_0 (+ count_1_3 count_1_4))\n (= count_2_1 (+ count_1_0 count_1_0 count_1_2 count_1_3))\n (= count_2_2 0)\n (= count_2_3 (+ count_1_1 count_1_2))\n (= count_2_4 (+ count_1_1 count_1_4))\n (= count_3_0 (+ count_2_3 count_2_4))\n (= count_3_1 (+ co...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "automata", "id": "automata-dfa_string_acceptance-47bbe695899c5850", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [false, false, false, true, true], "answer": 33, "difficulty": "extreme", "generator": "state_machine", "length": 6, "num_states": 5, "seed": 20261099, "start": 4, "transitions": [[1, 1], [3, 4], [3, 1], [0, 1], [0, 4]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 5 states numbered 0 to 4. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 4.\n\nPrint how many distinct binary strings of length exactly 6 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=5, L=6)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 33, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-47bbe695899c5850-t000", "input": {"accepting": [false, false, false, true, true], "length": 6, "start": 4, "transitions": [[1, 1], [3, 4], [3, 1], [0, 1], [0, 4]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-47bbe695899c5850"}], "verification": {"duration_ms": 1.2667089467868209, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_2_1 (= depth_0_0 (+ depth_2_1 1)))\n (and pair_4_1 (= depth_0_0 (+ depth_4_1 1)))))\n (a!2 (or (and pair_2_3 (= depth_0_2 (+ depth_2_3 1)))\n (and pair_2_4 (= depth_0_2 (+ depth_2_4 1)))\n (and pair_4_3 (= depth_0_2 (+ depth_4_3 1)))\n (and pair_4_4 (= depth_0_2 (+ depth_4_4 1)))))\n (a!3 (or (and pair_2_0 (= depth_0_3 (+ depth_2_0 1)))\n (and pair_2_2 (= depth_0_3 (+ depth_2_2 1)))\n (and pair_4_0 (= depth_0_3 (+ depth_4_0 1)))\n (and pair_4_2 (= depth_0_3 (+ depth_4_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "automata", "id": "automata-dfa_product_reachability-ea459b086c553ed9", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 5, "difficulty": "extreme", "equivalent_by_construction": true, "first": {"accepting": [true, true, false, false, false], "start": 4, "transitions": [[2, 4], [1, 2], [3, 0], [1, 4], [3, 0]]}, "generator": "state_machine", "num_states": 5, "second": {"accepting": [false, true, false, true, false, false, false], "start": 2, "transitions": [[4, 3], [1, 0], [4, 3], [0, 2], [1, 2], [5, 5], [6, 6]]}, "seed": 20261100, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(4, 2) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (5x7 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-ea459b086c553ed9-t000", "input": {"first": {"accepting": [true, true, false, false, false], "start": 4, "transitions": [[2, 4], [1, 2], [3, 0], [1, 4], [3, 0]]}, "second": {"accepting": [false, true, false, true, false, false, false], "start": 2, "transitions": [[4, 3], [1, 0], [4, 3], [0, 2], [1, 2], [5, 5], [6, 6]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-ea459b086c553ed9"}], "verification": {"duration_ms": 4.447332990821451, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 0)\n (= count_0_2 0)\n (= count_0_3 0)\n (= count_0_4 1)\n (= count_1_0 (+ count_0_0 count_0_1 count_0_1))\n (= count_1_1 (+ count_0_2 count_0_3 count_0_4))\n (= count_1_2 (+ count_0_2 count_0_3))\n (= count_1_3 (+ count_0_0 count_0_4))\n (= count_1_4 0)\n (= count_2_0 (+ count_1_0 count_1_1 count_1_1))\n (= count_2_1 (+ count_1_2 count_1_3 count_1_4))\n (= count_2_2 (+ count_1_2 count_1_3))\n (= count_2_3 (+ count_1_0 count_1_4))\n (= count_2_4 0)\n (= count_3_0 (+ count_2_0 count_2_1 count_2_1))\n (= count...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "automata", "id": "automata-dfa_string_acceptance-27eb740f8e54ce0d", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, false, true, true, false], "answer": 205, "difficulty": "extreme", "generator": "state_machine", "length": 8, "num_states": 5, "seed": 20261101, "start": 4, "transitions": [[3, 0], [0, 0], [1, 2], [2, 1], [1, 3]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 5 states numbered 0 to 4. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 4.\n\nPrint how many distinct binary strings of length exactly 8 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=5, L=8)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 205, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-27eb740f8e54ce0d-t000", "input": {"accepting": [true, false, true, true, false], "length": 8, "start": 4, "transitions": [[3, 0], [0, 0], [1, 2], [2, 1], [1, 3]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-27eb740f8e54ce0d"}], "verification": {"duration_ms": 1.4204579638317227, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 1)\n (= count_0_2 0)\n (= count_0_3 0)\n (= count_0_4 0)\n (= count_0_5 0)\n (= count_1_0 (+ count_0_1 count_0_2))\n (= count_1_1 (+ count_0_1 count_0_3))\n (= count_1_2 0)\n (= count_1_3 (+ count_0_2 count_0_4 count_0_5))\n (= count_1_4 (+ count_0_4 count_0_5))\n (= count_1_5 (+ count_0_0 count_0_0 count_0_3))\n (= count_2_0 (+ count_1_1 count_1_2))\n (= count_2_1 (+ count_1_1 count_1_3))\n (= count_2_2 0)\n (= count_2_3 (+ count_1_2 count_1_4 count_1_5))\n (= count_2_4 (+ count_1_4 count_1_5))\n (= coun...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "automata", "id": "automata-dfa_string_acceptance-4afa20d7452c6b03", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, false, false, true, true, true], "answer": 399, "difficulty": "extreme", "generator": "state_machine", "length": 9, "num_states": 6, "seed": 20261102, "start": 1, "transitions": [[5, 5], [1, 0], [3, 0], [5, 1], [3, 4], [4, 3]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 6 states numbered 0 to 5. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 1.\n\nPrint how many distinct binary strings of length exactly 9 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=6, L=9)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 399, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-4afa20d7452c6b03-t000", "input": {"accepting": [true, false, false, true, true, true], "length": 9, "start": 1, "transitions": [[5, 5], [1, 0], [3, 0], [5, 1], [3, 4], [4, 3]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-4afa20d7452c6b03"}], "verification": {"duration_ms": 1.741749991197139, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 0)\n (= count_0_2 0)\n (= count_0_3 0)\n (= count_0_4 1)\n (= count_1_0 (+ count_0_3))\n (= count_1_1 (+ count_0_0 count_0_0 count_0_1 count_0_2))\n (= count_1_2 (+ count_0_4))\n (= count_1_3 0)\n (= count_1_4 (+ count_0_1 count_0_2 count_0_3 count_0_4))\n (= count_2_0 (+ count_1_3))\n (= count_2_1 (+ count_1_0 count_1_0 count_1_1 count_1_2))\n (= count_2_2 (+ count_1_4))\n (= count_2_3 0)\n (= count_2_4 (+ count_1_1 count_1_2 count_1_3 count_1_4))\n (= count_3_0 (+ count_2_3))\n (= count_3_1 (+ count_2_0 co...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "automata", "id": "automata-dfa_string_acceptance-3073f27314312234", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, false, true, false, true], "answer": 24, "difficulty": "extreme", "generator": "state_machine", "length": 5, "num_states": 5, "seed": 20261103, "start": 4, "transitions": [[1, 1], [1, 4], [4, 1], [0, 4], [2, 4]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 5 states numbered 0 to 4. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 4.\n\nPrint how many distinct binary strings of length exactly 5 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=5, L=5)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 24, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-3073f27314312234-t000", "input": {"accepting": [true, false, true, false, true], "length": 5, "start": 4, "transitions": [[1, 1], [1, 4], [4, 1], [0, 4], [2, 4]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-3073f27314312234"}], "verification": {"duration_ms": 1.0984589462168515, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 1)\n (= count_0_1 0)\n (= count_0_2 0)\n (= count_0_3 0)\n (= count_0_4 0)\n (= count_0_5 0)\n (= count_1_0 (+ count_0_1 count_0_3))\n (= count_1_1 (+ count_0_2))\n (= count_1_2 (+ count_0_0))\n (= count_1_3 (+ count_0_0 count_0_2))\n (= count_1_4 (+ count_0_1 count_0_4 count_0_4 count_0_5))\n (= count_1_5 (+ count_0_3 count_0_5))\n (= count_2_0 (+ count_1_1 count_1_3))\n (= count_2_1 (+ count_1_2))\n (= count_2_2 (+ count_1_0))\n (= count_2_3 (+ count_1_0 count_1_2))\n (= count_2_4 (+ count_1_1 count_1_4 count_1_4 count_1_5)...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "automata", "id": "automata-dfa_string_acceptance-49a61363f7509557", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [false, true, false, false, true, false], "answer": 203, "difficulty": "extreme", "generator": "state_machine", "length": 8, "num_states": 6, "seed": 20261104, "start": 0, "transitions": [[2, 3], [4, 0], [1, 3], [0, 5], [4, 4], [4, 5]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 6 states numbered 0 to 5. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 0.\n\nPrint how many distinct binary strings of length exactly 8 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=6, L=8)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 203, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-49a61363f7509557-t000", "input": {"accepting": [false, true, false, false, true, false], "length": 8, "start": 0, "transitions": [[2, 3], [4, 0], [1, 3], [0, 5], [4, 4], [4, 5]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-49a61363f7509557"}], "verification": {"duration_ms": 1.447082031518221, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 0)\n (= count_0_2 1)\n (= count_0_3 0)\n (= count_0_4 0)\n (= count_1_0 (+ count_0_0))\n (= count_1_1 (+ count_0_2 count_0_3))\n (= count_1_2 (+ count_0_3))\n (= count_1_3 (+ count_0_1 count_0_2 count_0_4))\n (= count_1_4 (+ count_0_0 count_0_1 count_0_4))\n (= count_2_0 (+ count_1_0))\n (= count_2_1 (+ count_1_2 count_1_3))\n (= count_2_2 (+ count_1_3))\n (= count_2_3 (+ count_1_1 count_1_2 count_1_4))\n (= count_2_4 (+ count_1_0 count_1_1 count_1_4))\n (= count_3_0 (+ count_2_0))\n (= count_3_1 (+ count_2_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "automata", "id": "automata-dfa_string_acceptance-9108f3e91f19e3e3", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, true, false, true, true], "answer": 53, "difficulty": "extreme", "generator": "state_machine", "length": 6, "num_states": 5, "seed": 20261105, "start": 2, "transitions": [[4, 0], [3, 4], [1, 3], [2, 1], [3, 4]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 5 states numbered 0 to 4. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 2.\n\nPrint how many distinct binary strings of length exactly 6 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=5, L=6)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 53, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-9108f3e91f19e3e3-t000", "input": {"accepting": [true, true, false, true, true], "length": 6, "start": 2, "transitions": [[4, 0], [3, 4], [1, 3], [2, 1], [3, 4]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-9108f3e91f19e3e3"}], "verification": {"duration_ms": 1.2102510081604123, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 1)\n (= count_0_2 0)\n (= count_0_3 0)\n (= count_0_4 0)\n (= count_1_0 (+ count_0_1 count_0_3 count_0_4 count_0_4))\n (= count_1_1 (+ count_0_0 count_0_0))\n (= count_1_2 (+ count_0_2 count_0_3))\n (= count_1_3 (+ count_0_2))\n (= count_1_4 (+ count_0_1))\n (= count_2_0 (+ count_1_1 count_1_3 count_1_4 count_1_4))\n (= count_2_1 (+ count_1_0 count_1_0))\n (= count_2_2 (+ count_1_2 count_1_3))\n (= count_2_3 (+ count_1_2))\n (= count_2_4 (+ count_1_1))\n (= count_3_0 (+ count_2_1 count_2_3 count_2_4 count_2_4))...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "automata", "id": "automata-dfa_string_acceptance-af7f82fef7c3d9ad", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, true, false, true, false], "answer": 104, "difficulty": "extreme", "generator": "state_machine", "length": 7, "num_states": 5, "seed": 20261106, "start": 1, "transitions": [[1, 1], [0, 4], [2, 3], [2, 0], [0, 0]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 5 states numbered 0 to 4. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 1.\n\nPrint how many distinct binary strings of length exactly 7 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=5, L=7)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 104, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-af7f82fef7c3d9ad-t000", "input": {"accepting": [true, true, false, true, false], "length": 7, "start": 1, "transitions": [[1, 1], [0, 4], [2, 3], [2, 0], [0, 0]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-af7f82fef7c3d9ad"}], "verification": {"duration_ms": 1.169875031337142, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_0_3 (= depth_0_0 (+ depth_0_3 1)))\n (and pair_4_3 (= depth_0_0 (+ depth_4_3 1)))\n (and pair_5_4 (= depth_0_0 (+ depth_5_4 1)))\n (and pair_5_5 (= depth_0_0 (+ depth_5_5 1)))))\n (a!2 (or (and pair_5_0 (= depth_0_1 (+ depth_5_0 1)))\n (and pair_5_1 (= depth_0_1 (+ depth_5_1 1)))))\n (a!3 (or (and pair_0_0 (= depth_0_2 (+ depth_0_0 1)))\n (and pair_4_0 (= depth_0_2 (+ depth_4_0 1)))\n (and pair_5_3 (= depth_0_2 (+ depth_5_3 1)))))\n (a!4 (or (and pair_0_1 (= depth_0_4 (+ depth_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "automata", "id": "automata-dfa_product_reachability-135933edf6ea16a9", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 5, "difficulty": "extreme", "equivalent_by_construction": true, "first": {"accepting": [true, true, false, true, true, true], "start": 1, "transitions": [[0, 1], [5, 4], [2, 1], [1, 5], [0, 4], [2, 0]]}, "generator": "state_machine", "num_states": 6, "second": {"accepting": [true, true, true, true, true, false], "start": 0, "transitions": [[2, 1], [4, 1], [5, 4], [0, 2], [4, 0], [5, 0]]}, "seed": 20261107, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(1, 0) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (6x6 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-135933edf6ea16a9-t000", "input": {"first": {"accepting": [true, true, false, true, true, true], "start": 1, "transitions": [[0, 1], [5, 4], [2, 1], [1, 5], [0, 4], [2, 0]]}, "second": {"accepting": [true, true, true, true, true, false], "start": 0, "transitions": [[2, 1], [4, 1], [5, 4], [0, 2], [4, 0], [5, 0]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-135933edf6ea16a9"}], "verification": {"duration_ms": 3.085416043177247, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_2_2 (= depth_0_0 (+ depth_2_2 1)))\n (and pair_4_2 (= depth_0_0 (+ depth_4_2 1)))\n (and pair_5_0 (= depth_0_0 (+ depth_5_0 1)))\n (and pair_5_3 (= depth_0_0 (+ depth_5_3 1)))))\n (a!2 (or (and pair_5_4 (= depth_0_1 (+ depth_5_4 1)))))\n (a!3 (or (and pair_2_4 (= depth_0_2 (+ depth_2_4 1)))\n (and pair_4_4 (= depth_0_2 (+ depth_4_4 1)))))\n (a!4 (or (and pair_5_1 (= depth_0_3 (+ depth_5_1 1)))\n (and pair_5_2 (= depth_0_3 (+ depth_5_2 1)))))\n (a!5 (or (and pair_2_0 (= depth_0_5 (+ dept...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "automata", "id": "automata-dfa_product_reachability-990ff5d004f2d474", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 6, "difficulty": "extreme", "equivalent_by_construction": true, "first": {"accepting": [true, false, false, false, true, true], "start": 0, "transitions": [[1, 2], [3, 4], [0, 4], [5, 3], [0, 3], [5, 0]]}, "generator": "state_machine", "num_states": 6, "second": {"accepting": [false, false, false, true, true, true], "start": 4, "transitions": [[5, 0], [4, 3], [0, 3], [4, 0], [2, 1], [5, 4]]}, "seed": 20261108, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(0, 4) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (6x6 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 6, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-990ff5d004f2d474-t000", "input": {"first": {"accepting": [true, false, false, false, true, true], "start": 0, "transitions": [[1, 2], [3, 4], [0, 4], [5, 3], [0, 3], [5, 0]]}, "second": {"accepting": [false, false, false, true, true, true], "start": 4, "transitions": [[5, 0], [4, 3], [0, 3], [4, 0], [2, 1], [5, 4]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-990ff5d004f2d474"}], "verification": {"duration_ms": 15.338208992034197, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 1)\n (= count_0_2 0)\n (= count_0_3 0)\n (= count_0_4 0)\n (= count_1_0 (+ count_0_3 count_0_4))\n (= count_1_1 (+ count_0_1 count_0_3))\n (= count_1_2 (+ count_0_0 count_0_2))\n (= count_1_3 (+ count_0_0 count_0_2 count_0_4))\n (= count_1_4 (+ count_0_1))\n (= count_2_0 (+ count_1_3 count_1_4))\n (= count_2_1 (+ count_1_1 count_1_3))\n (= count_2_2 (+ count_1_0 count_1_2))\n (= count_2_3 (+ count_1_0 count_1_2 count_1_4))\n (= count_2_4 (+ count_1_1))\n (= count_3_0 (+ count_2_3 count_2_4))\n (= count_3_1 (...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "automata", "id": "automata-dfa_string_acceptance-c81e9bc4d9d06c8b", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, true, true, false, false], "answer": 160, "difficulty": "extreme", "generator": "state_machine", "length": 8, "num_states": 5, "seed": 20261109, "start": 1, "transitions": [[3, 2], [4, 1], [3, 2], [1, 0], [3, 0]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 5 states numbered 0 to 4. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 1.\n\nPrint how many distinct binary strings of length exactly 8 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=5, L=8)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 160, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-c81e9bc4d9d06c8b-t000", "input": {"accepting": [true, true, true, false, false], "length": 8, "start": 1, "transitions": [[3, 2], [4, 1], [3, 2], [1, 0], [3, 0]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-c81e9bc4d9d06c8b"}], "verification": {"duration_ms": 1.606001053005457, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_2_1 (= depth_1_0 (+ depth_2_1 1)))\n (and pair_2_3 (= depth_1_0 (+ depth_2_3 1)))\n (and pair_2_4 (= depth_1_0 (+ depth_2_4 1)))\n (and pair_3_4 (= depth_1_0 (+ depth_3_4 1)))))\n (a!2 (or (and pair_3_2 (= depth_1_1 (+ depth_3_2 1)))))\n (a!3 (or (and pair_2_2 (= depth_1_3 (+ depth_2_2 1)))\n (and pair_3_0 (= depth_1_3 (+ depth_3_0 1)))\n (and pair_3_3 (= depth_1_3 (+ depth_3_3 1)))))\n (a!4 (or (and pair_2_0 (= depth_1_4 (+ depth_2_0 1)))\n (and pair_3_1 (= depth_1_4 (+ depth_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "automata", "id": "automata-dfa_product_reachability-a498b0b50c54dd57", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 3, "difficulty": "extreme", "equivalent_by_construction": true, "first": {"accepting": [false, true, false, false, false], "start": 4, "transitions": [[4, 3], [2, 2], [1, 4], [2, 1], [2, 4]]}, "generator": "state_machine", "num_states": 5, "second": {"accepting": [false, false, false, false, true, false, false], "start": 3, "transitions": [[4, 3], [0, 4], [3, 1], [0, 3], [0, 0], [5, 5], [6, 6]]}, "seed": 20261110, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(4, 3) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (5x7 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 3, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-a498b0b50c54dd57-t000", "input": {"first": {"accepting": [false, true, false, false, false], "start": 4, "transitions": [[4, 3], [2, 2], [1, 4], [2, 1], [2, 4]]}, "second": {"accepting": [false, false, false, false, true, false, false], "start": 3, "transitions": [[4, 3], [0, 4], [3, 1], [0, 3], [0, 0], [5, 5], [6, 6]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-a498b0b50c54dd57"}], "verification": {"duration_ms": 3.2745409407652915, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_4_5 (= depth_0_0 (+ depth_4_5 1)))))\n (a!2 (or (and pair_3_5 (= depth_0_1 (+ depth_3_5 1)))\n (and pair_4_5 (= depth_0_1 (+ depth_4_5 1)))))\n (a!3 (or (and pair_4_3 (= depth_0_2 (+ depth_4_3 1)))\n (and pair_4_4 (= depth_0_2 (+ depth_4_4 1)))))\n (a!4 (or (and pair_3_0 (= depth_0_3 (+ depth_3_0 1)))\n (and pair_3_4 (= depth_0_3 (+ depth_3_4 1)))\n (and pair_4_0 (= depth_0_3 (+ depth_4_0 1)))\n (and pair_4_0 (= depth_0_3 (+ depth_4_0 1)))\n (and pair_4_4 (= depth_0_3 (+ depth_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "automata", "id": "automata-dfa_product_reachability-6988e1014ba9ccf0", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 6, "difficulty": "extreme", "equivalent_by_construction": true, "first": {"accepting": [true, true, true, false, false, true], "start": 2, "transitions": [[2, 1], [3, 2], [5, 4], [0, 1], [0, 0], [3, 3]]}, "generator": "state_machine", "num_states": 6, "second": {"accepting": [false, true, true, true, false, true], "start": 5, "transitions": [[3, 3], [4, 4], [4, 5], [5, 2], [3, 2], [1, 0]]}, "seed": 20261111, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(2, 5) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (6x6 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 6, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-6988e1014ba9ccf0-t000", "input": {"first": {"accepting": [true, true, true, false, false, true], "start": 2, "transitions": [[2, 1], [3, 2], [5, 4], [0, 1], [0, 0], [3, 3]]}, "second": {"accepting": [false, true, true, true, false, true], "start": 5, "transitions": [[3, 3], [4, 4], [4, 5], [5, 2], [3, 2], [1, 0]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-6988e1014ba9ccf0"}], "verification": {"duration_ms": 43.00066694850102, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 1)\n (= count_0_2 0)\n (= count_0_3 0)\n (= count_0_4 0)\n (= count_1_0 (+ count_0_1))\n (= count_1_1 (+ count_0_0 count_0_2 count_0_3))\n (= count_1_2 (+ count_0_3))\n (= count_1_3 (+ count_0_0 count_0_2 count_0_4))\n (= count_1_4 (+ count_0_1 count_0_4))\n (= count_2_0 (+ count_1_1))\n (= count_2_1 (+ count_1_0 count_1_2 count_1_3))\n (= count_2_2 (+ count_1_3))\n (= count_2_3 (+ count_1_0 count_1_2 count_1_4))\n (= count_2_4 (+ count_1_1 count_1_4))\n (= count_3_0 (+ count_2_1))\n (= count_3_1 (+ count_2_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "automata", "id": "automata-dfa_string_acceptance-1effaa088a6c575d", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [false, false, false, true, false], "answer": 17, "difficulty": "extreme", "generator": "state_machine", "length": 6, "num_states": 5, "seed": 20261112, "start": 1, "transitions": [[3, 1], [4, 0], [1, 3], [2, 1], [3, 4]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 5 states numbered 0 to 4. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 1.\n\nPrint how many distinct binary strings of length exactly 6 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=5, L=6)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 17, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-1effaa088a6c575d-t000", "input": {"accepting": [false, false, false, true, false], "length": 6, "start": 1, "transitions": [[3, 1], [4, 0], [1, 3], [2, 1], [3, 4]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-1effaa088a6c575d"}], "verification": {"duration_ms": 1.2083739857189357, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 1)\n (= count_0_1 0)\n (= count_0_2 0)\n (= count_0_3 0)\n (= count_0_4 0)\n (= count_1_0 (+ count_0_1 count_0_2 count_0_3 count_0_4))\n (= count_1_1 (+ count_0_0 count_0_3))\n (= count_1_2 (+ count_0_0))\n (= count_1_3 (+ count_0_1 count_0_4))\n (= count_1_4 (+ count_0_2))\n (= count_2_0 (+ count_1_1 count_1_2 count_1_3 count_1_4))\n (= count_2_1 (+ count_1_0 count_1_3))\n (= count_2_2 (+ count_1_0))\n (= count_2_3 (+ count_1_1 count_1_4))\n (= count_2_4 (+ count_1_2))\n (= count_3_0 (+ count_2_1 count_2_2 count_2_3 count_2_4))...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "automata", "id": "automata-dfa_string_acceptance-3ab6dd2bf290674b", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, true, true, false, false], "answer": 385, "difficulty": "extreme", "generator": "state_machine", "length": 9, "num_states": 5, "seed": 20261113, "start": 0, "transitions": [[1, 2], [0, 3], [0, 4], [1, 0], [0, 3]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 5 states numbered 0 to 4. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 0.\n\nPrint how many distinct binary strings of length exactly 9 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=5, L=9)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 385, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-3ab6dd2bf290674b-t000", "input": {"accepting": [true, true, true, false, false], "length": 9, "start": 0, "transitions": [[1, 2], [0, 3], [0, 4], [1, 0], [0, 3]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-3ab6dd2bf290674b"}], "verification": {"duration_ms": 1.4393750461749732, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 1)\n (= count_0_2 0)\n (= count_0_3 0)\n (= count_0_4 0)\n (= count_1_0 (+ count_0_3))\n (= count_1_1 (+ count_0_2 count_0_4))\n (= count_1_2 0)\n (= count_1_3 (+ count_0_1 count_0_2 count_0_3 count_0_4))\n (= count_1_4 (+ count_0_0 count_0_0 count_0_1))\n (= count_2_0 (+ count_1_3))\n (= count_2_1 (+ count_1_2 count_1_4))\n (= count_2_2 0)\n (= count_2_3 (+ count_1_1 count_1_2 count_1_3 count_1_4))\n (= count_2_4 (+ count_1_0 count_1_0 count_1_1))\n (= count_3_0 (+ count_2_3))\n (= count_3_1 (+ count_2_2 co...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "automata", "id": "automata-dfa_string_acceptance-134993a273afb118", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [false, false, false, true, false], "answer": 13, "difficulty": "extreme", "generator": "state_machine", "length": 5, "num_states": 5, "seed": 20261114, "start": 1, "transitions": [[4, 4], [3, 4], [1, 3], [0, 3], [1, 3]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 5 states numbered 0 to 4. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 1.\n\nPrint how many distinct binary strings of length exactly 5 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=5, L=5)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 13, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-134993a273afb118-t000", "input": {"accepting": [false, false, false, true, false], "length": 5, "start": 1, "transitions": [[4, 4], [3, 4], [1, 3], [0, 3], [1, 3]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-134993a273afb118"}], "verification": {"duration_ms": 1.0474579758010805, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_3_1 (= depth_0_0 (+ depth_3_1 1)))))\n (a!2 (or (and pair_3_2 (= depth_0_1 (+ depth_3_2 1)))\n (and pair_4_0 (= depth_0_1 (+ depth_4_0 1)))\n (and pair_4_1 (= depth_0_1 (+ depth_4_1 1)))\n (and pair_4_4 (= depth_0_1 (+ depth_4_4 1)))))\n (a!3 (or (and pair_3_3 (= depth_0_2 (+ depth_3_3 1)))))\n (a!4 (or (and pair_3_4 (= depth_0_3 (+ depth_3_4 1)))\n (and pair_4_2 (= depth_0_3 (+ depth_4_2 1)))))\n (a!5 (or (and pair_3_0 (= depth_0_4 (+ depth_3_0 1)))\n (and pair_4_3 (= depth_0_4 (+ dept...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "automata", "id": "automata-dfa_product_reachability-57d64cbe9275d563", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 5, "difficulty": "extreme", "equivalent_by_construction": true, "first": {"accepting": [false, true, false, false, false], "start": 4, "transitions": [[3, 4], [1, 2], [1, 3], [1, 0], [0, 1]]}, "generator": "state_machine", "num_states": 5, "second": {"accepting": [false, true, false, false, false, false], "start": 2, "transitions": [[1, 4], [1, 0], [3, 1], [4, 2], [1, 3], [5, 5]]}, "seed": 20261115, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(4, 2) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (5x6 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-57d64cbe9275d563-t000", "input": {"first": {"accepting": [false, true, false, false, false], "start": 4, "transitions": [[3, 4], [1, 2], [1, 3], [1, 0], [0, 1]]}, "second": {"accepting": [false, true, false, false, false, false], "start": 2, "transitions": [[1, 4], [1, 0], [3, 1], [4, 2], [1, 3], [5, 5]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-57d64cbe9275d563"}], "verification": {"duration_ms": 5.504583066795021, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 0)\n (= count_0_2 0)\n (= count_0_3 0)\n (= count_0_4 1)\n (= count_1_0 (+ count_0_2 count_0_4))\n (= count_1_1 (+ count_0_3))\n (= count_1_2 (+ count_0_0 count_0_1))\n (= count_1_3 (+ count_0_0))\n (= count_1_4 (+ count_0_1 count_0_2 count_0_3 count_0_4))\n (= count_2_0 (+ count_1_2 count_1_4))\n (= count_2_1 (+ count_1_3))\n (= count_2_2 (+ count_1_0 count_1_1))\n (= count_2_3 (+ count_1_0))\n (= count_2_4 (+ count_1_1 count_1_2 count_1_3 count_1_4))\n (= count_3_0 (+ count_2_2 count_2_4))\n (= count_3_1 (...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "automata", "id": "automata-dfa_string_acceptance-7469fa36dda5d418", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [false, false, true, false, true], "answer": 273, "difficulty": "extreme", "generator": "state_machine", "length": 9, "num_states": 5, "seed": 20261116, "start": 4, "transitions": [[3, 2], [4, 2], [4, 0], [4, 1], [0, 4]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 5 states numbered 0 to 4. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 4.\n\nPrint how many distinct binary strings of length exactly 9 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=5, L=9)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 273, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-7469fa36dda5d418-t000", "input": {"accepting": [false, false, true, false, true], "length": 9, "start": 4, "transitions": [[3, 2], [4, 2], [4, 0], [4, 1], [0, 4]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-7469fa36dda5d418"}], "verification": {"duration_ms": 1.4900840469636023, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 0)\n (= count_0_2 1)\n (= count_0_3 0)\n (= count_0_4 0)\n (= count_1_0 (+ count_0_4 count_0_4))\n (= count_1_1 (+ count_0_0 count_0_0 count_0_1 count_0_2))\n (= count_1_2 (+ count_0_1 count_0_2 count_0_3))\n (= count_1_3 (+ count_0_3))\n (= count_1_4 0)\n (= count_2_0 (+ count_1_4 count_1_4))\n (= count_2_1 (+ count_1_0 count_1_0 count_1_1 count_1_2))\n (= count_2_2 (+ count_1_1 count_1_2 count_1_3))\n (= count_2_3 (+ count_1_3))\n (= count_2_4 0)\n (= count_3_0 (+ count_2_4 count_2_4))\n (= count_3_1 (+ co...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "automata", "id": "automata-dfa_string_acceptance-235285aa45d93a25", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, true, false, false, false], "answer": 64, "difficulty": "extreme", "generator": "state_machine", "length": 7, "num_states": 5, "seed": 20261117, "start": 2, "transitions": [[1, 1], [1, 2], [2, 1], [2, 3], [0, 0]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 5 states numbered 0 to 4. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 2.\n\nPrint how many distinct binary strings of length exactly 7 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=5, L=7)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 64, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-235285aa45d93a25-t000", "input": {"accepting": [true, true, false, false, false], "length": 7, "start": 2, "transitions": [[1, 1], [1, 2], [2, 1], [2, 3], [0, 0]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-235285aa45d93a25"}], "verification": {"duration_ms": 1.2566249934025109, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 0)\n (= count_0_2 0)\n (= count_0_3 0)\n (= count_0_4 0)\n (= count_0_5 1)\n (= count_1_0 (+ count_0_0 count_0_0 count_0_1))\n (= count_1_1 (+ count_0_1 count_0_5 count_0_5))\n (= count_1_2 (+ count_0_2))\n (= count_1_3 (+ count_0_2 count_0_3 count_0_4))\n (= count_1_4 0)\n (= count_1_5 (+ count_0_3 count_0_4))\n (= count_2_0 (+ count_1_0 count_1_0 count_1_1))\n (= count_2_1 (+ count_1_1 count_1_5 count_1_5))\n (= count_2_2 (+ count_1_2))\n (= count_2_3 (+ count_1_2 count_1_3 count_1_4))\n (= count_2_4 0)\n ...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "automata", "id": "automata-dfa_string_acceptance-7dd11f26212d8418", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, false, false, true, false, true], "answer": 30, "difficulty": "extreme", "generator": "state_machine", "length": 5, "num_states": 6, "seed": 20261118, "start": 5, "transitions": [[0, 0], [0, 1], [3, 2], [3, 5], [3, 5], [1, 1]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 6 states numbered 0 to 5. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 5.\n\nPrint how many distinct binary strings of length exactly 5 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=6, L=5)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 30, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-7dd11f26212d8418-t000", "input": {"accepting": [true, false, false, true, false, true], "length": 5, "start": 5, "transitions": [[0, 0], [0, 1], [3, 2], [3, 5], [3, 5], [1, 1]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-7dd11f26212d8418"}], "verification": {"duration_ms": 1.2253329623490572, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_0_2 (= depth_0_0 (+ depth_0_2 1)))\n (and pair_2_4 (= depth_0_0 (+ depth_2_4 1)))\n (and pair_4_2 (= depth_0_0 (+ depth_4_2 1)))))\n (a!2 (or (and pair_0_0 (= depth_0_1 (+ depth_0_0 1)))\n (and pair_0_1 (= depth_0_1 (+ depth_0_1 1)))\n (and pair_2_3 (= depth_0_1 (+ depth_2_3 1)))\n (and pair_4_0 (= depth_0_1 (+ depth_4_0 1)))\n (and pair_4_1 (= depth_0_1 (+ depth_4_1 1)))))\n (a!3 (or (and pair_2_0 (= depth_0_2 (+ depth_2_0 1)))))\n (a!4 (or (and pair_0_3 (= depth_0_3 (+ depth_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "automata", "id": "automata-dfa_product_reachability-0c33bfb052deb9ca", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 2, "difficulty": "extreme", "equivalent_by_construction": true, "first": {"accepting": [true, false, true, false, false], "start": 2, "transitions": [[0, 2], [4, 3], [2, 0], [2, 4], [0, 1]]}, "generator": "state_machine", "num_states": 5, "second": {"accepting": [false, true, false, true, false, false], "start": 3, "transitions": [[1, 2], [1, 3], [0, 4], [3, 1], [3, 0], [5, 5]]}, "seed": 20261119, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(2, 3) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (5x6 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 2, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-0c33bfb052deb9ca-t000", "input": {"first": {"accepting": [true, false, true, false, false], "start": 2, "transitions": [[0, 2], [4, 3], [2, 0], [2, 4], [0, 1]]}, "second": {"accepting": [false, true, false, true, false, false], "start": 3, "transitions": [[1, 2], [1, 3], [0, 4], [3, 1], [3, 0], [5, 5]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-0c33bfb052deb9ca"}], "verification": {"duration_ms": 5.113833001814783, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_3_3 (= depth_0_0 (+ depth_3_3 1)))))\n (a!2 (or (and pair_3_0 (= depth_0_1 (+ depth_3_0 1)))\n (and pair_3_5 (= depth_0_1 (+ depth_3_5 1)))))\n (a!3 (or (and pair_3_1 (= depth_0_4 (+ depth_3_1 1)))\n (and pair_3_2 (= depth_0_4 (+ depth_3_2 1)))\n (and pair_3_4 (= depth_0_4 (+ depth_3_4 1)))))\n (a!4 (or (and pair_3_6 (= depth_0_6 (+ depth_3_6 1)))))\n (a!5 (or (and pair_1_3 (= depth_1_0 (+ depth_1_3 1)))\n (and pair_2_3 (= depth_1_0 (+ depth_2_3 1)))\n (and pair_4_3 (= depth_1_0 (+ dept...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "automata", "id": "automata-dfa_product_reachability-d564d7013115325d", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 5, "difficulty": "extreme", "equivalent_by_construction": true, "first": {"accepting": [true, true, false, false, true, true], "start": 3, "transitions": [[2, 5], [1, 5], [1, 3], [0, 3], [1, 1], [2, 1]]}, "generator": "state_machine", "num_states": 6, "second": {"accepting": [true, false, true, false, true, true, false], "start": 3, "transitions": [[1, 5], [4, 3], [4, 4], [0, 3], [4, 5], [1, 4], [6, 6]]}, "seed": 20261120, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(3, 3) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (6x7 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 5, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-d564d7013115325d-t000", "input": {"first": {"accepting": [true, true, false, false, true, true], "start": 3, "transitions": [[2, 5], [1, 5], [1, 3], [0, 3], [1, 1], [2, 1]]}, "second": {"accepting": [true, false, true, false, true, true, false], "start": 3, "transitions": [[1, 5], [4, 3], [4, 4], [0, 3], [4, 5], [1, 4], [6, 6]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-d564d7013115325d"}], "verification": {"duration_ms": 5.13445696560666, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 0)\n (= count_0_2 0)\n (= count_0_3 0)\n (= count_0_4 1)\n (= count_0_5 0)\n (= count_1_0 (+ count_0_1 count_0_2 count_0_2))\n (= count_1_1 (+ count_0_3 count_0_3 count_0_5 count_0_5))\n (= count_1_2 (+ count_0_0 count_0_1 count_0_4 count_0_4))\n (= count_1_3 0)\n (= count_1_4 (+ count_0_0))\n (= count_1_5 0)\n (= count_2_0 (+ count_1_1 count_1_2 count_1_2))\n (= count_2_1 (+ count_1_3 count_1_3 count_1_5 count_1_5))\n (= count_2_2 (+ count_1_0 count_1_1 count_1_4 count_1_4))\n (= count_2_3 0)\n (= count_2_4...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "automata", "id": "automata-dfa_string_acceptance-af42c189465ccb56", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, true, false, true, false, false], "answer": 64, "difficulty": "extreme", "generator": "state_machine", "length": 7, "num_states": 6, "seed": 20261121, "start": 4, "transitions": [[2, 4], [0, 2], [0, 0], [1, 1], [2, 2], [1, 1]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 6 states numbered 0 to 5. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 4.\n\nPrint how many distinct binary strings of length exactly 7 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=6, L=7)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 64, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-af42c189465ccb56-t000", "input": {"accepting": [true, true, false, true, false, false], "length": 7, "start": 4, "transitions": [[2, 4], [0, 2], [0, 0], [1, 1], [2, 2], [1, 1]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-af42c189465ccb56"}], "verification": {"duration_ms": 1.4982919674366713, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(and (= count_0_0 0)\n (= count_0_1 1)\n (= count_0_2 0)\n (= count_0_3 0)\n (= count_0_4 0)\n (= count_0_5 0)\n (= count_1_0 (+ count_0_1 count_0_1 count_0_3))\n (= count_1_1 (+ count_0_3))\n (= count_1_2 0)\n (= count_1_3 (+ count_0_2 count_0_4 count_0_4 count_0_5))\n (= count_1_4 (+ count_0_0))\n (= count_1_5 (+ count_0_0 count_0_2 count_0_5))\n (= count_2_0 (+ count_1_1 count_1_1 count_1_3))\n (= count_2_1 (+ count_1_3))\n (= count_2_2 0)\n (= count_2_3 (+ count_1_2 count_1_4 count_1_4 count_1_5))\n (= count_2_4 (+ count_1_0))\n (= coun...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "automata", "id": "automata-dfa_string_acceptance-395db6e8e7f22f45", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"accepting": [true, false, false, true, false, false], "answer": 74, "difficulty": "extreme", "generator": "state_machine", "length": 7, "num_states": 6, "seed": 20261122, "start": 1, "transitions": [[5, 4], [0, 0], [5, 3], [0, 1], [3, 3], [5, 3]], "variant": "dfa_string_acceptance"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\ntransitions = data[\"transitions\"]\nstart = data[\"start\"]\naccepting = data[\"accepting\"]\nlength = data[\"length\"]\n\nsize = len(accepting)\ncounts = [0] * size\ncounts[start] = 1\nsteps = 0\nwhile steps < length:\n nxt = [0] * size\n for state in range(size):\n amount = counts[state]\n if amount != 0:\n for nxt_state in transitions[state]:\n nxt[nxt_state] += amount\n counts = nxt\n steps += 1\ntotal = 0\nfor state in range(size):\n if accepting[state]:\n total += counts[state]\nprint(total)\n", "statement": "You are given a deterministic finite automaton over the binary alphabet\n{0, 1} with 6 states numbered 0 to 5. The row transitions[s]\ncontains exactly two entries: the successor state after reading a 0 and the\nsuccessor state after reading a 1, respectively. The automaton starts in\nstate 1.\n\nPrint how many distinct binary strings of length exactly 7 move the\nautomaton from its start state to an accepting state. The count is unique;\nit has been formally proven for this instance with the Z3 SMT solver.\n", "title": "DFA String Acceptance Count (states=6, L=7)", "variant": "dfa_string_acceptance"}, "test_cases": [{"expected_output": 74, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_string_acceptance-395db6e8e7f22f45-t000", "input": {"accepting": [true, false, false, true, false, false], "length": 7, "start": 1, "transitions": [[5, 4], [0, 0], [5, 3], [0, 1], [3, 3], [5, 3]]}, "is_public": true, "problem_id": "automata-dfa_string_acceptance-395db6e8e7f22f45"}], "verification": {"duration_ms": 1.376416941639036, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}} {"batch_id": "Giavy-proceval-v1", "created_at": "2026-08-24T20:26:38.105007+00:00", "generator_name": "proceval-dataset", "spec": {"constraints": [{"description": "Transition rows fully define the next-state function.", "expression": "the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states", "name": "machine_shape"}, {"description": "The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.", "expression": "(let ((a!1 (or (and pair_1_3 (= depth_0_0 (+ depth_1_3 1)))\n (and pair_3_3 (= depth_0_0 (+ depth_3_3 1)))\n (and pair_4_3 (= depth_0_0 (+ depth_4_3 1)))\n (and pair_4_3 (= depth_0_0 (+ depth_4_3 1)))))\n (a!2 (or (and pair_1_0 (= depth_0_1 (+ depth_1_0 1)))\n (and pair_1_4 (= depth_0_1 (+ depth_1_4 1)))\n (and pair_4_0 (= depth_0_1 (+ depth_4_0 1)))\n (and pair_4_4 (= depth_0_1 (+ depth_4_4 1)))))\n (a!3 (or (and pair_3_1 (= depth_0_2 (+ depth_3_1 1)))\n (and pair_3_4 (= depth_0_2 (+ depth_3_...", "name": "solution_existence"}, {"description": "Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.", "expression": "the defining fixpoint admits exactly one value", "name": "solution_uniqueness"}], "difficulty": "extreme", "domain": "automata", "id": "automata-dfa_product_reachability-b8e46dbf0585e94a", "input_schema": {"additionalProperties": true, "properties": {}, "required": [], "type": "object"}, "metadata": {"answer": 4, "difficulty": "extreme", "equivalent_by_construction": true, "first": {"accepting": [false, false, false, true, true], "start": 3, "transitions": [[1, 4], [0, 4], [3, 3], [1, 0], [0, 0]]}, "generator": "state_machine", "num_states": 5, "second": {"accepting": [true, false, true, false, false], "start": 0, "transitions": [[1, 4], [4, 2], [4, 4], [0, 0], [1, 2]]}, "seed": 20261123, "variant": "dfa_product_reachability"}, "output_schema": {"description": "the unique verified answer", "minimum": 0, "type": "integer"}, "reference_solution": "import json\nimport sys\n\ndata = json.load(sys.stdin)\nfirst = data[\"first\"]\nsecond = data[\"second\"]\n\nrows = len(first[\"accepting\"])\ncols = len(second[\"accepting\"])\nseen = [[0] * cols for _ in range(rows)]\nstart_left = first[\"start\"]\nstart_right = second[\"start\"]\nseen[start_left][start_right] = 1\nstack = [[start_left, start_right]]\ntotal = 0\nwhile stack:\n left, right = stack.pop()\n total += 1\n for symbol in range(2):\n nxt_left = first[\"transitions\"][left][symbol]\n nxt_right = second[\"transitions\"][right][symbol]\n if not seen[nxt_left][nxt_right]:\n seen[nxt_left][nxt_right] = 1\n stack.append([nxt_left, nxt_right])\nprint(total)\n", "statement": "You are given two deterministic finite automata over the binary alphabet\n{0, 1} that recognize the same language (equivalence holds by construction).\nConsider their synchronous product: execution starts from the state pair\n(3, 0) and every input symbol advances both machines\nsimultaneously.\n\nPrint how many distinct state pairs are reachable in this product. The count\nis unique; it has been formally proven for this instance with the Z3 SMT\nsolver.\n", "title": "DFA Product Reachability (5x5 states)", "variant": "dfa_product_reachability"}, "test_cases": [{"expected_output": 4, "explanation": "Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.", "id": "automata-dfa_product_reachability-b8e46dbf0585e94a-t000", "input": {"first": {"accepting": [false, false, false, true, true], "start": 3, "transitions": [[1, 4], [0, 4], [3, 3], [1, 0], [0, 0]]}, "second": {"accepting": [true, false, true, false, false], "start": 0, "transitions": [[1, 4], [4, 2], [4, 4], [0, 0], [1, 2]]}}, "is_public": true, "problem_id": "automata-dfa_product_reachability-b8e46dbf0585e94a"}], "verification": {"duration_ms": 2.4594999849796295, "message": null, "models_blocked": 1, "solver": "z3", "status": "sat", "unique": true}}