/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern gcd(a,a,g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToIRSwTTransformerProof [SOUND, 99 ms] (2) AND (3) IRSwT (4) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 2 ms] (5) IRSwT (6) IntTRSCompressionProof [EQUIVALENT, 13 ms] (7) IRSwT (8) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (9) IRSwT (10) IRSwTTerminationDigraphProof [EQUIVALENT, 15 ms] (11) TRUE (12) IRSwT (13) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (14) IRSwT (15) IntTRSCompressionProof [EQUIVALENT, 0 ms] (16) IRSwT (17) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (18) IRSwT (19) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms] (20) TRUE ---------------------------------------- (0) Obligation: Clauses: gcd(X, 0, X) :- !. gcd(0, X, X) :- !. gcd(X, Y, Divisor) :- ','(>=(X, Y), ','(!, ','(>(Y, 0), ','(is(Remainder, mod(X, Y)), gcd(Y, Remainder, Divisor))))). gcd(X, Y, Divisor) :- ','(<(X, Y), ','(!, ','(>(X, 0), gcd(Y, X, Divisor)))). Query: gcd(a,a,g) ---------------------------------------- (1) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 5, "program": { "directives": [], "clauses": [ [ "(gcd X (0) X)", "(!)" ], [ "(gcd (0) X X)", "(!)" ], [ "(gcd X Y Divisor)", "(',' (>= X Y) (',' (!) (',' (> Y (0)) (',' (is Remainder (mod X Y)) (gcd Y Remainder Divisor)))))" ], [ "(gcd X Y Divisor)", "(',' (< X Y) (',' (!) 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T28)", "(gcd X28 (0) X28)" ], [ "(gcd T30 T31 T28)", "(gcd (0) X31 X31)" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [ { "lhs": { "name": "T31", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T29", "type": "PlainIntegerVariable" }, { "name": "T30", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "mod" }, "operation": "=" }, { "lhs": { "type": "PlainIntegerConstant", "value": "0" }, "type": "PlainIntegerRelation", "rhs": { "name": "T30", "type": "PlainIntegerVariable" }, "operation": "<" }, { "lhs": { "name": "T30", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "name": "T29", "type": "PlainIntegerVariable" }, "operation": "<=" } ] }, "ground": [ "T31", "T30", "T28" ], "free": [ "X3", "X6", "X28", "X31" ], "exprvars": [ "T31", "T30", "T29" ] } }, "4518": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (is X49 (mod T51 T52)) (gcd T52 X49 T53))" }], "kb": { "nonunifying": [ [ "(gcd T1 T2 T53)", "(gcd X3 (0) X3)" ], [ "(gcd T1 T2 T53)", "(gcd (0) X6 X6)" ], [ "(gcd T51 T52 T53)", "(gcd X28 (0) X28)" ], [ "(gcd T51 T52 T53)", "(gcd (0) X31 X31)" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [ { "lhs": { "name": "T31", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T29", "type": "PlainIntegerVariable" }, { "name": "T30", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "mod" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T29", "type": "PlainIntegerVariable" }, { "name": "T51", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "mod" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T29", "type": "PlainIntegerVariable" }, { "name": "T30", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "mod" }, "operation": "=" }, { "lhs": { "type": "PlainIntegerConstant", "value": "0" }, "type": "PlainIntegerRelation", "rhs": { "name": "T30", "type": "PlainIntegerVariable" }, "operation": "<" }, { "lhs": { "name": "T51", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "name": "T30", "type": "PlainIntegerVariable" }, "operation": "<" }, { "lhs": { "type": "PlainIntegerConstant", "value": "0" }, "type": "PlainIntegerRelation", "rhs": { "name": "T52", "type": "PlainIntegerVariable" }, "operation": "<" }, { "lhs": { "name": "T30", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "name": "T29", "type": "PlainIntegerVariable" }, "operation": "<=" }, { "lhs": { "name": "T29", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "name": "T29", "type": "PlainIntegerVariable" }, "operation": "<=" }, { "lhs": { "name": "T52", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "name": "T51", "type": "PlainIntegerVariable" }, "operation": "<=" } ] }, "ground": [ "T53", "T52", "T51" ], "free": [ "X3", "X6", "X28", "X31", "X49" ], "exprvars": [ "T31", "T30", "T52", "T29", "T51" ] } }, "4519": { "goal": [], "kb": { "nonunifying": [ [ "(gcd T1 T2 T53)", "(gcd X3 (0) X3)" ], [ "(gcd T1 T2 T53)", "(gcd (0) X6 X6)" ], [ "(gcd T51 T52 T53)", "(gcd X28 (0) X28)" ], [ "(gcd T51 T52 T53)", "(gcd (0) X31 X31)" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [ { "lhs": { "name": "T31", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T29", "type": "PlainIntegerVariable" }, { "name": "T30", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "mod" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T29", "type": "PlainIntegerVariable" }, { "name": "T51", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "mod" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T29", "type": "PlainIntegerVariable" }, { "name": "T30", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "mod" }, "operation": "=" }, { "lhs": { "type": "PlainIntegerConstant", "value": "0" }, "type": "PlainIntegerRelation", "rhs": { "name": "T30", "type": "PlainIntegerVariable" }, "operation": "<" }, { "lhs": { "name": "T51", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "name": "T30", "type": "PlainIntegerVariable" }, "operation": "<" }, { "lhs": { "name": "T30", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "name": "T29", "type": "PlainIntegerVariable" }, "operation": "<=" }, { "lhs": { "name": "T29", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "name": "T29", "type": "PlainIntegerVariable" }, "operation": "<=" }, { "lhs": { "name": "T52", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "name": "T51", "type": "PlainIntegerVariable" }, "operation": "<=" }, { "lhs": { "type": "PlainIntegerConstant", "value": "0" }, "type": "PlainIntegerRelation", "rhs": { "name": "T52", "type": "PlainIntegerVariable" }, "operation": ">=" } ] }, "ground": [ "T53", "T52", "T51" ], "free": [ "X3", "X6", "X28", "X31", "X49" ], "exprvars": [ "T31", "T30", "T52", "T29", "T51" ] } }, "5206": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (is X99 (mod T102 T103)) (gcd T103 X99 T104))" }], "kb": { "nonunifying": [ [ "(gcd T1 T2 T104)", "(gcd X3 (0) X3)" ], [ "(gcd T1 T2 T104)", "(gcd (0) X6 X6)" ], [ "(gcd T102 T103 T104)", "(gcd X78 (0) X78)" ], [ "(gcd T102 T103 T104)", "(gcd (0) X81 X81)" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [ { "lhs": { "name": "T81", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "name": "T82", "type": "PlainIntegerVariable" }, "operation": "<" }, { "lhs": { "type": "PlainIntegerConstant", "value": "0" }, "type": "PlainIntegerRelation", "rhs": { "name": "T81", "type": "PlainIntegerVariable" }, "operation": "<" }, { "lhs": { "name": "T102", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "name": "T82", "type": "PlainIntegerVariable" }, "operation": "<" }, { "lhs": { "name": "T103", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "name": "T81", "type": "PlainIntegerVariable" }, "operation": "<" }, { "lhs": { "type": "PlainIntegerConstant", "value": "0" }, "type": "PlainIntegerRelation", "rhs": { "name": "T103", "type": "PlainIntegerVariable" }, "operation": "<" }, { "lhs": { "name": "T103", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "name": "T102", "type": "PlainIntegerVariable" }, "operation": "<=" } ] }, "ground": [ "T104", "T103", "T102" ], "free": [ "X3", "X6", "X78", "X81", "X99" ], "exprvars": [ "T81", "T103", "T82", "T102" ] } }, "5207": { "goal": [], "kb": { "nonunifying": [ [ "(gcd T1 T2 T104)", "(gcd X3 (0) X3)" ], [ "(gcd T1 T2 T104)", "(gcd (0) X6 X6)" ], [ "(gcd T102 T103 T104)", "(gcd X78 (0) X78)" ], [ "(gcd T102 T103 T104)", "(gcd (0) X81 X81)" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [ { "lhs": { "name": "T81", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "name": "T82", "type": "PlainIntegerVariable" }, "operation": "<" }, { "lhs": { "type": "PlainIntegerConstant", "value": "0" }, "type": "PlainIntegerRelation", "rhs": { "name": "T81", "type": "PlainIntegerVariable" }, "operation": "<" }, { "lhs": { "name": "T102", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "name": "T82", "type": "PlainIntegerVariable" }, "operation": "<" }, { "lhs": { "name": "T103", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "name": "T81", "type": "PlainIntegerVariable" }, "operation": "<" }, { "lhs": { "name": "T103", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "name": "T102", "type": "PlainIntegerVariable" }, "operation": "<=" }, { "lhs": { "type": "PlainIntegerConstant", "value": "0" }, "type": "PlainIntegerRelation", "rhs": { "name": "T103", "type": "PlainIntegerVariable" }, "operation": ">=" } ] }, "ground": [ "T104", "T103", "T102" ], "free": [ "X3", "X6", "X78", "X81", "X99" ], "exprvars": [ "T81", "T103", "T82", "T102" ] } }, "5208": { "goal": [{ "clause": -1, "scope": -1, "term": "(gcd T103 T105 T104)" }], "kb": { "nonunifying": [ [ "(gcd T1 T2 T104)", "(gcd X3 (0) X3)" ], [ "(gcd T1 T2 T104)", "(gcd (0) X6 X6)" ], [ "(gcd T102 T103 T104)", "(gcd X78 (0) X78)" ], [ "(gcd T102 T103 T104)", "(gcd (0) X81 X81)" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [ { "lhs": { "name": "T105", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T102", "type": "PlainIntegerVariable" }, { "name": "T103", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "mod" }, "operation": "=" }, { "lhs": { "name": "T81", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "name": "T82", "type": "PlainIntegerVariable" }, "operation": "<" }, { "lhs": { "type": "PlainIntegerConstant", "value": "0" }, "type": "PlainIntegerRelation", "rhs": { "name": "T81", "type": "PlainIntegerVariable" }, "operation": "<" }, { "lhs": { "name": "T102", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "name": "T82", "type": "PlainIntegerVariable" }, "operation": "<" }, { "lhs": { "name": "T103", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "name": "T81", "type": "PlainIntegerVariable" }, "operation": "<" }, { "lhs": { "type": "PlainIntegerConstant", "value": "0" }, "type": "PlainIntegerRelation", "rhs": { "name": "T103", "type": "PlainIntegerVariable" }, "operation": "<" }, { "lhs": { "name": "T103", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "name": "T102", "type": "PlainIntegerVariable" }, "operation": "<=" } ] }, "ground": [ "T104", "T103", "T105", "T102" ], "free": [ "X3", "X6", "X78", "X81", "X99" ], "exprvars": [ "T81", "T103", "T105", "T82", "T102" ] } } }, "edges": [ { "from": 5, "to": 6, "label": "CASE" }, { "from": 6, "to": 8, "label": "EVAL with clause\ngcd(X3, 0, X3) :- !_1.\nand substitutionT1 -> T6,\nX3 -> T6,\nT2 -> 0,\nT3 -> T6" }, { "from": 6, "to": 9, "label": "EVAL-BACKTRACK" }, { "from": 8, "to": 13, "label": "CUT" }, { "from": 9, "to": 18, "label": "EVAL with clause\ngcd(0, X6, X6) :- !_1.\nand substitutionT1 -> 0,\nT2 -> T9,\nX6 -> T9,\nT3 -> T9" }, { "from": 9, "to": 19, "label": "EVAL-BACKTRACK" }, { "from": 13, "to": 15, "label": "SUCCESS" }, { "from": 18, "to": 21, "label": "CUT" }, { "from": 19, "to": 26, "label": "PARALLEL" }, { "from": 19, "to": 28, "label": "PARALLEL" }, { "from": 21, "to": 23, "label": "SUCCESS" }, { "from": 26, "to": 29, "label": "ONLY EVAL with clause\ngcd(X21, X22, X23) :- ','(>=(X21, X22), ','(!_1, ','(>(X22, 0), ','(is(X24, mod(X21, X22)), gcd(X22, X24, X23))))).\nand substitutionT1 -> T29,\nX21 -> T29,\nT2 -> T30,\nX22 -> T30,\nT3 -> T28,\nX23 -> T28,\nT26 -> T29,\nT27 -> T30" }, { "from": 28, "to": 5155, "label": "ONLY EVAL with clause\ngcd(X72, X73, X74) :- ','(<(X72, X73), ','(!_1, ','(>(X72, 0), gcd(X73, X72, X74)))).\nand substitutionT1 -> T81,\nX72 -> T81,\nT2 -> T82,\nX73 -> T82,\nT3 -> T80,\nX74 -> T80,\nT78 -> T81,\nT79 -> T82" }, { "from": 29, "to": 30, "label": "IS ERROR" }, { "from": 29, "to": 1665, "label": "ARITHCOMP SUCCESS" }, { "from": 29, "to": 1670, "label": "ARITHCOMP FAIL" }, { "from": 1665, "to": 1676, "label": "CUT" }, { "from": 1676, "to": 3051, "label": "ARITHCOMP SUCCESS" }, { "from": 1676, "to": 3052, "label": "ARITHCOMP FAIL" }, { "from": 3051, "to": 3053, "label": "\nX24 -> T31" }, { "from": 3053, "to": 3054, "label": "CASE" }, { "from": 3054, "to": 3055, "label": "EVAL with clause\ngcd(X28, 0, X28) :- !_2.\nand substitutionT30 -> T35,\nX28 -> T35,\nT31 -> 0,\nT28 -> T35" }, { "from": 3054, "to": 3056, "label": "EVAL-BACKTRACK" }, { "from": 3055, "to": 3057, "label": "CUT" }, { "from": 3056, "to": 3059, "label": "EVAL with clause\ngcd(0, X31, X31) :- !_2.\nand substitutionT30 -> 0,\nT31 -> T38,\nX31 -> T38,\nT28 -> T38" }, { "from": 3056, "to": 3060, "label": "EVAL-BACKTRACK" }, { "from": 3057, "to": 3058, "label": "SUCCESS" }, { "from": 3059, "to": 3061, "label": "CUT" }, { "from": 3060, "to": 3076, "label": "PARALLEL" }, { "from": 3060, "to": 3085, "label": "PARALLEL" }, { "from": 3061, "to": 3062, "label": "SUCCESS" }, { "from": 3076, "to": 3119, "label": "ONLY EVAL with clause\ngcd(X46, X47, X48) :- ','(>=(X46, X47), ','(!_2, ','(>(X47, 0), ','(is(X49, mod(X46, X47)), gcd(X47, X49, X48))))).\nand substitutionT30 -> T51,\nX46 -> T51,\nT31 -> T52,\nX47 -> T52,\nT28 -> T53,\nX48 -> T53" }, { "from": 3085, "to": 5014, "label": "ONLY EVAL with clause\ngcd(X57, X58, X59) :- ','(<(X57, X58), ','(!_2, ','(>(X57, 0), gcd(X58, X57, X59)))).\nand substitutionT30 -> T62,\nX57 -> T62,\nT31 -> T63,\nX58 -> T63,\nT28 -> T64,\nX59 -> T64" }, { "from": 3119, "to": 3753, "label": "ARITHCOMP SUCCESS" }, { "from": 3119, "to": 3754, "label": "ARITHCOMP FAIL" }, { "from": 3753, "to": 3755, "label": "CUT" }, { "from": 3755, "to": 4518, "label": "ARITHCOMP SUCCESS" }, { "from": 3755, "to": 4519, "label": "ARITHCOMP FAIL" }, { "from": 4518, "to": 4520, "label": "\nX49 -> T54" }, { "from": 4520, "to": 3053, "label": "INSTANCE with matching:\nT30 -> T52\nT31 -> T54\nT28 -> T53\nX24 -> X28" }, { "from": 5014, "to": 5143, "label": "ARITHCOMP SUCCESS" }, { "from": 5014, "to": 5144, "label": "ARITHCOMP FAIL" }, { "from": 5143, "to": 5145, "label": "CUT" }, { "from": 5145, "to": 5146, "label": "ARITHCOMP SUCCESS" }, { "from": 5145, "to": 5147, "label": "ARITHCOMP FAIL" }, { "from": 5146, "to": 3053, "label": "INSTANCE with matching:\nT30 -> T63\nT31 -> T62\nT28 -> T64\nX24 -> X28" }, { "from": 5155, "to": 5156, "label": "IS ERROR" }, { "from": 5155, "to": 5166, "label": "ARITHCOMP SUCCESS" }, { "from": 5155, "to": 5167, "label": "ARITHCOMP FAIL" }, { "from": 5166, "to": 5168, "label": "CUT" }, { "from": 5168, "to": 5185, "label": "ARITHCOMP SUCCESS" }, { "from": 5168, "to": 5186, "label": "ARITHCOMP FAIL" }, { "from": 5185, "to": 5188, "label": "CASE" }, { "from": 5188, "to": 5189, "label": "EVAL with clause\ngcd(X78, 0, X78) :- !_3.\nand substitutionT82 -> T86,\nX78 -> T86,\nT81 -> 0,\nT80 -> T86" }, { "from": 5188, "to": 5190, "label": "EVAL-BACKTRACK" }, { "from": 5189, "to": 5191, "label": "CUT" }, { "from": 5190, "to": 5193, "label": "EVAL with clause\ngcd(0, X81, X81) :- !_3.\nand substitutionT82 -> 0,\nT81 -> T89,\nX81 -> T89,\nT80 -> T89" }, { "from": 5190, "to": 5194, "label": "EVAL-BACKTRACK" }, { "from": 5191, "to": 5192, "label": "SUCCESS" }, { "from": 5193, "to": 5195, "label": "CUT" }, { "from": 5194, "to": 5197, "label": "PARALLEL" }, { "from": 5194, "to": 5198, "label": "PARALLEL" }, { "from": 5195, "to": 5196, "label": "SUCCESS" }, { "from": 5197, "to": 5199, "label": "ONLY EVAL with clause\ngcd(X96, X97, X98) :- ','(>=(X96, X97), ','(!_3, ','(>(X97, 0), ','(is(X99, mod(X96, X97)), gcd(X97, X99, X98))))).\nand substitutionT82 -> T102,\nX96 -> T102,\nT81 -> T103,\nX97 -> T103,\nT80 -> T104,\nX98 -> T104" }, { "from": 5198, "to": 5233, "label": "ONLY EVAL with clause\ngcd(X107, X108, X109) :- ','(<(X107, X108), ','(!_3, ','(>(X107, 0), gcd(X108, X107, X109)))).\nand substitutionT82 -> T113,\nX107 -> T113,\nT81 -> T114,\nX108 -> T114,\nT80 -> T115,\nX109 -> T115" }, { "from": 5199, "to": 5200, "label": "ARITHCOMP SUCCESS" }, { "from": 5199, "to": 5201, "label": "ARITHCOMP FAIL" }, { "from": 5200, "to": 5202, "label": "CUT" }, { "from": 5202, "to": 5206, "label": "ARITHCOMP SUCCESS" }, { "from": 5202, "to": 5207, "label": "ARITHCOMP FAIL" }, { "from": 5206, "to": 5208, "label": "\nX99 -> T105" }, { "from": 5208, "to": 5185, "label": "INSTANCE with matching:\nT82 -> T103\nT81 -> T105\nT80 -> T104" }, { "from": 5233, "to": 5243, "label": "ARITHCOMP SUCCESS" }, { "from": 5233, "to": 5244, "label": "ARITHCOMP FAIL" }, { "from": 5243, "to": 5245, "label": "CUT" }, { "from": 5245, "to": 5266, "label": "ARITHCOMP SUCCESS" }, { "from": 5245, "to": 5267, "label": "ARITHCOMP FAIL" }, { "from": 5266, "to": 5185, "label": "INSTANCE with matching:\nT82 -> T114\nT81 -> T113\nT80 -> T115" } ], "type": "Graph" } } ---------------------------------------- (2) Complex Obligation (AND) ---------------------------------------- (3) Obligation: Rules: f5185_in(T82, T81, T80) -> f5188_in(T82, T81, T80) :|: TRUE f5188_out(x, x1, x2) -> f5185_out(x, x1, x2) :|: TRUE f5198_in(T113, T114, T115) -> f5233_in(T113, T114, T115) :|: TRUE f5233_out(x3, x4, x5) -> f5198_out(x3, x4, x5) :|: TRUE f5199_in(T102, T103, T104) -> f5201_in(T104, T103, T102) :|: T102 < T103 f5199_in(x6, x7, x8) -> f5200_in(x7, x6, x8) :|: x6 >= x7 f5200_out(x9, x10, x11) -> f5199_out(x10, x9, x11) :|: x10 >= x9 f5201_out(x12, x13, x14) -> f5199_out(x14, x13, x12) :|: x14 < x13 f5243_in(x15, x16, x17) -> f5245_in(x15, x16, x17) :|: TRUE f5245_out(x18, x19, x20) -> f5243_out(x18, x19, x20) :|: TRUE f5197_out(x21, x22, x23) -> f5194_out(x21, x22, x23) :|: TRUE f5194_in(x24, x25, x26) -> f5198_in(x24, x25, x26) :|: TRUE f5194_in(x27, x28, x29) -> f5197_in(x27, x28, x29) :|: TRUE f5198_out(x30, x31, x32) -> f5194_out(x30, x31, x32) :|: TRUE f5245_in(x33, x34, x35) -> f5266_in(x34, x33, x35) :|: x33 > 0 f5266_out(x36, x37, x38) -> f5245_out(x37, x36, x38) :|: x37 > 0 f5245_in(x39, x40, x41) -> f5267_in(x41, x40, x39) :|: x39 <= 0 f5267_out(x42, x43, x44) -> f5245_out(x44, x43, x42) :|: x44 <= 0 f5197_in(x45, x46, x47) -> f5199_in(x45, x46, x47) :|: TRUE f5199_out(x48, x49, x50) -> f5197_out(x48, x49, x50) :|: TRUE f5206_in(x51, x52, x53) -> f5208_in(x52, x54, x53, x51) :|: x54 = mod(x51, x52) f5208_out(x55, x56, x57, x58) -> f5206_out(x58, x55, x57) :|: TRUE f5208_in(x59, x60, x61, x62) -> f5185_in(x59, x60, x61) :|: TRUE f5185_out(x63, x64, x65) -> f5208_out(x63, x64, x65, x66) :|: TRUE f5207_out(x67, x68, x69) -> f5202_out(x68, x69, x67) :|: x68 <= 0 f5202_in(x70, x71, x72) -> f5207_in(x72, x70, x71) :|: x70 <= 0 f5202_in(x73, x74, x75) -> f5206_in(x74, x73, x75) :|: x73 > 0 f5206_out(x76, x77, x78) -> f5202_out(x77, x76, x78) :|: x77 > 0 f5266_in(x79, x80, x81) -> f5185_in(x79, x80, x81) :|: TRUE f5185_out(x82, x83, x84) -> f5266_out(x82, x83, x84) :|: TRUE f5194_out(x85, x86, x87) -> f5190_out(x85, x86, x87) :|: TRUE f5193_out(T89) -> f5190_out(0, T89, T89) :|: TRUE f5190_in(0, x88, x88) -> f5193_in(x88) :|: TRUE f5190_in(x89, x90, x91) -> f5194_in(x89, x90, x91) :|: TRUE f5233_in(x92, x93, x94) -> f5244_in(x94, x93, x92) :|: x92 >= x93 f5243_out(x95, x96, x97) -> f5233_out(x95, x96, x97) :|: x95 < x96 f5233_in(x98, x99, x100) -> f5243_in(x98, x99, x100) :|: x98 < x99 f5244_out(x101, x102, x103) -> f5233_out(x103, x102, x101) :|: x103 >= x102 f5202_out(x104, x105, x106) -> f5200_out(x104, x105, x106) :|: TRUE f5200_in(x107, x108, x109) -> f5202_in(x107, x108, x109) :|: TRUE f5188_in(T86, 0, T86) -> f5189_in(T86) :|: TRUE f5190_out(x110, x111, x112) -> f5188_out(x110, x111, x112) :|: TRUE f5189_out(x113) -> f5188_out(x113, 0, x113) :|: TRUE f5188_in(x114, x115, x116) -> f5190_in(x114, x115, x116) :|: TRUE f5_in(T3) -> f6_in(T3) :|: TRUE f6_out(x117) -> f5_out(x117) :|: TRUE f6_in(T6) -> f8_in(T6) :|: TRUE f6_in(x118) -> f9_in(x118) :|: TRUE f8_out(x119) -> f6_out(x119) :|: TRUE f9_out(x120) -> f6_out(x120) :|: TRUE f9_in(x121) -> f19_in(x121) :|: TRUE f19_out(x122) -> f9_out(x122) :|: TRUE f9_in(T9) -> f18_in(T9) :|: TRUE f18_out(x123) -> f9_out(x123) :|: TRUE f26_out(x124) -> f19_out(x124) :|: TRUE f19_in(x125) -> f28_in(x125) :|: TRUE f19_in(x126) -> f26_in(x126) :|: TRUE f28_out(x127) -> f19_out(x127) :|: TRUE f5155_out(x128) -> f28_out(x128) :|: TRUE f28_in(x129) -> f5155_in(x129) :|: TRUE f5155_in(x130) -> f5167_in(x131, x130, x132) :|: x131 >= x132 f5166_out(x133, x134, x135) -> f5155_out(x135) :|: x133 < x134 f5155_in(x136) -> f5166_in(x137, x138, x136) :|: x137 < x138 f5156_out -> f5155_out(x139) :|: TRUE f5167_out(x140, x141, x142) -> f5155_out(x141) :|: x140 >= x142 f5155_in(x143) -> f5156_in :|: TRUE f5168_out(x144, x145, x146) -> f5166_out(x144, x145, x146) :|: TRUE f5166_in(x147, x148, x149) -> f5168_in(x147, x148, x149) :|: TRUE f5185_out(x150, x151, x152) -> f5168_out(x151, x150, x152) :|: x151 > 0 f5168_in(x153, x154, x155) -> f5185_in(x154, x153, x155) :|: x153 > 0 f5186_out(x156, x157, x158) -> f5168_out(x156, x158, x157) :|: x156 <= 0 f5168_in(x159, x160, x161) -> f5186_in(x159, x161, x160) :|: x159 <= 0 Start term: f5_in(T3) ---------------------------------------- (4) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f5185_in(T82, T81, T80) -> f5188_in(T82, T81, T80) :|: TRUE f5198_in(T113, T114, T115) -> f5233_in(T113, T114, T115) :|: TRUE f5199_in(x6, x7, x8) -> f5200_in(x7, x6, x8) :|: x6 >= x7 f5243_in(x15, x16, x17) -> f5245_in(x15, x16, x17) :|: TRUE f5194_in(x24, x25, x26) -> f5198_in(x24, x25, x26) :|: TRUE f5194_in(x27, x28, x29) -> f5197_in(x27, x28, x29) :|: TRUE f5245_in(x33, x34, x35) -> f5266_in(x34, x33, x35) :|: x33 > 0 f5197_in(x45, x46, x47) -> f5199_in(x45, x46, x47) :|: TRUE f5206_in(x51, x52, x53) -> f5208_in(x52, x54, x53, x51) :|: x54 = mod(x51, x52) f5208_in(x59, x60, x61, x62) -> f5185_in(x59, x60, x61) :|: TRUE f5202_in(x73, x74, x75) -> f5206_in(x74, x73, x75) :|: x73 > 0 f5266_in(x79, x80, x81) -> f5185_in(x79, x80, x81) :|: TRUE f5190_in(x89, x90, x91) -> f5194_in(x89, x90, x91) :|: TRUE f5233_in(x98, x99, x100) -> f5243_in(x98, x99, x100) :|: x98 < x99 f5200_in(x107, x108, x109) -> f5202_in(x107, x108, x109) :|: TRUE f5188_in(x114, x115, x116) -> f5190_in(x114, x115, x116) :|: TRUE ---------------------------------------- (5) Obligation: Rules: f5185_in(T82, T81, T80) -> f5188_in(T82, T81, T80) :|: TRUE f5198_in(T113, T114, T115) -> f5233_in(T113, T114, T115) :|: TRUE f5199_in(x6, x7, x8) -> f5200_in(x7, x6, x8) :|: x6 >= x7 f5243_in(x15, x16, x17) -> f5245_in(x15, x16, x17) :|: TRUE f5194_in(x24, x25, x26) -> f5198_in(x24, x25, x26) :|: TRUE f5194_in(x27, x28, x29) -> f5197_in(x27, x28, x29) :|: TRUE f5245_in(x33, x34, x35) -> f5266_in(x34, x33, x35) :|: x33 > 0 f5197_in(x45, x46, x47) -> f5199_in(x45, x46, x47) :|: TRUE f5206_in(x51, x52, x53) -> f5208_in(x52, x54, x53, x51) :|: x54 = mod(x51, x52) f5208_in(x59, x60, x61, x62) -> f5185_in(x59, x60, x61) :|: TRUE f5202_in(x73, x74, x75) -> f5206_in(x74, x73, x75) :|: x73 > 0 f5266_in(x79, x80, x81) -> f5185_in(x79, x80, x81) :|: TRUE f5190_in(x89, x90, x91) -> f5194_in(x89, x90, x91) :|: TRUE f5233_in(x98, x99, x100) -> f5243_in(x98, x99, x100) :|: x98 < x99 f5200_in(x107, x108, x109) -> f5202_in(x107, x108, x109) :|: TRUE f5188_in(x114, x115, x116) -> f5190_in(x114, x115, x116) :|: TRUE ---------------------------------------- (6) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (7) Obligation: Rules: f5190_in(x89:0, x90:0, x91:0) -> f5190_in(x90:0, x89:0, x91:0) :|: x90:0 > x89:0 && x89:0 > 0 f5190_in(x, x1, x2) -> f5190_in(x1, x3, x2) :|: x3 = mod(x, x1) && x >= x1 && TRUE && x1 > 0 ---------------------------------------- (8) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (9) Obligation: Rules: f5190_in(x89:0, x90:0, x91:0) -> f5190_in(x90:0, x89:0, x91:0) :|: x90:0 > x89:0 && x89:0 > 0 f5190_in(x, x1, x2) -> f5190_in(x1, x3, x2) :|: x3 = mod(x, x1) && x >= x1 && TRUE && x1 > 0 ---------------------------------------- (10) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f5190_in(x89:0, x90:0, x91:0) -> f5190_in(x90:0, x89:0, x91:0) :|: x90:0 > x89:0 && x89:0 > 0 (2) f5190_in(x, x1, x2) -> f5190_in(x1, x3, x2) :|: x3 = mod(x, x1) && x >= x1 && TRUE && x1 > 0 No arcs! This digraph is fully evaluated! ---------------------------------------- (11) TRUE ---------------------------------------- (12) Obligation: Rules: f3054_in(T35, 0, T35) -> f3055_in(T35) :|: TRUE f3056_out(T30, T31, T28) -> f3054_out(T30, T31, T28) :|: TRUE f3055_out(x) -> f3054_out(x, 0, x) :|: TRUE f3054_in(x1, x2, x3) -> f3056_in(x1, x2, x3) :|: TRUE f5146_out(T63, T62, T64) -> f5145_out(T62, T63, T64) :|: T62 > 0 f5145_in(x4, x5, x6) -> f5147_in(x6, x5, x4) :|: x4 <= 0 f5147_out(x7, x8, x9) -> f5145_out(x9, x8, x7) :|: x9 <= 0 f5145_in(x10, x11, x12) -> f5146_in(x11, x10, x12) :|: x10 > 0 f3076_in(T51, T52, T53) -> f3119_in(T51, T52, T53) :|: TRUE f3119_out(x13, x14, x15) -> f3076_out(x13, x14, x15) :|: TRUE f5143_in(x16, x17, x18) -> f5145_in(x16, x17, x18) :|: TRUE f5145_out(x19, x20, x21) -> f5143_out(x19, x20, x21) :|: TRUE f3053_out(x22, x23, x24, x25) -> f5146_out(x22, x23, x24) :|: TRUE f5146_in(x26, x27, x28) -> f3053_in(x26, x27, x28, x29) :|: TRUE f3753_in(x30, x31, x32) -> f3755_in(x30, x31, x32) :|: TRUE f3755_out(x33, x34, x35) -> f3753_out(x33, x34, x35) :|: TRUE f3059_out(T38) -> f3056_out(0, T38, T38) :|: TRUE f3056_in(x36, x37, x38) -> f3060_in(x36, x37, x38) :|: TRUE f3060_out(x39, x40, x41) -> f3056_out(x39, x40, x41) :|: TRUE f3056_in(0, x42, x42) -> f3059_in(x42) :|: TRUE f4519_out(x43, x44, x45) -> f3755_out(x44, x45, x43) :|: x44 <= 0 f4518_out(x46, x47, x48) -> f3755_out(x47, x46, x48) :|: x47 > 0 f3755_in(x49, x50, x51) -> f4518_in(x50, x49, x51) :|: x49 > 0 f3755_in(x52, x53, x54) -> f4519_in(x54, x52, x53) :|: x52 <= 0 f3053_in(x55, x56, x57, x58) -> f3054_in(x55, x56, x57) :|: TRUE f3054_out(x59, x60, x61) -> f3053_out(x59, x60, x61, x62) :|: TRUE f5014_out(x63, x64, x65) -> f3085_out(x63, x64, x65) :|: TRUE f3085_in(x66, x67, x68) -> f5014_in(x66, x67, x68) :|: TRUE f3060_in(x69, x70, x71) -> f3076_in(x69, x70, x71) :|: TRUE f3060_in(x72, x73, x74) -> f3085_in(x72, x73, x74) :|: TRUE f3076_out(x75, x76, x77) -> f3060_out(x75, x76, x77) :|: TRUE f3085_out(x78, x79, x80) -> f3060_out(x78, x79, x80) :|: TRUE f5144_out(x81, x82, x83) -> f5014_out(x83, x82, x81) :|: x83 >= x82 f5014_in(x84, x85, x86) -> f5143_in(x84, x85, x86) :|: x84 < x85 f5143_out(x87, x88, x89) -> f5014_out(x87, x88, x89) :|: x87 < x88 f5014_in(x90, x91, x92) -> f5144_in(x92, x91, x90) :|: x90 >= x91 f3753_out(x93, x94, x95) -> f3119_out(x94, x93, x95) :|: x94 >= x93 f3754_out(x96, x97, x98) -> f3119_out(x98, x97, x96) :|: x98 < x97 f3119_in(x99, x100, x101) -> f3754_in(x101, x100, x99) :|: x99 < x100 f3119_in(x102, x103, x104) -> f3753_in(x103, x102, x104) :|: x102 >= x103 f4518_in(x105, x106, x107) -> f4520_in(x106, x108, x107, x105) :|: x108 = mod(x105, x106) f4520_out(x109, x110, x111, x112) -> f4518_out(x112, x109, x111) :|: TRUE f4520_in(x113, x114, x115, x116) -> f3053_in(x113, x114, x115, x117) :|: TRUE f3053_out(x118, x119, x120, x121) -> f4520_out(x118, x119, x120, x122) :|: TRUE f5_in(T3) -> f6_in(T3) :|: TRUE f6_out(x123) -> f5_out(x123) :|: TRUE f6_in(T6) -> f8_in(T6) :|: TRUE f6_in(x124) -> f9_in(x124) :|: TRUE f8_out(x125) -> f6_out(x125) :|: TRUE f9_out(x126) -> f6_out(x126) :|: TRUE f9_in(x127) -> f19_in(x127) :|: TRUE f19_out(x128) -> f9_out(x128) :|: TRUE f9_in(T9) -> f18_in(T9) :|: TRUE f18_out(x129) -> f9_out(x129) :|: TRUE f26_out(x130) -> f19_out(x130) :|: TRUE f19_in(x131) -> f28_in(x131) :|: TRUE f19_in(x132) -> f26_in(x132) :|: TRUE f28_out(x133) -> f19_out(x133) :|: TRUE f29_out(x134) -> f26_out(x134) :|: TRUE f26_in(x135) -> f29_in(x135) :|: TRUE f30_out -> f29_out(x136) :|: TRUE f29_in(x137) -> f1665_in(x138, x139, x137) :|: x139 >= x138 f29_in(x140) -> f30_in :|: TRUE f1665_out(x141, x142, x143) -> f29_out(x143) :|: x142 >= x141 f1670_out(x144, x145, x146) -> f29_out(x146) :|: x145 < x144 f29_in(x147) -> f1670_in(x148, x149, x147) :|: x149 < x148 f1676_out(x150, x151, x152) -> f1665_out(x150, x151, x152) :|: TRUE f1665_in(x153, x154, x155) -> f1676_in(x153, x154, x155) :|: TRUE f3051_out(x156, x157, x158) -> f1676_out(x157, x156, x158) :|: x157 > 0 f1676_in(x159, x160, x161) -> f3051_in(x160, x159, x161) :|: x159 > 0 f1676_in(x162, x163, x164) -> f3052_in(x162, x163, x164) :|: x162 <= 0 f3052_out(x165, x166, x167) -> f1676_out(x165, x166, x167) :|: x165 <= 0 f3053_out(x168, x169, x170, x171) -> f3051_out(x171, x168, x170) :|: TRUE f3051_in(x172, x173, x174) -> f3053_in(x173, x175, x174, x172) :|: x175 = mod(x172, x173) Start term: f5_in(T3) ---------------------------------------- (13) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f3054_in(x1, x2, x3) -> f3056_in(x1, x2, x3) :|: TRUE f5145_in(x10, x11, x12) -> f5146_in(x11, x10, x12) :|: x10 > 0 f3076_in(T51, T52, T53) -> f3119_in(T51, T52, T53) :|: TRUE f5143_in(x16, x17, x18) -> f5145_in(x16, x17, x18) :|: TRUE f5146_in(x26, x27, x28) -> f3053_in(x26, x27, x28, x29) :|: TRUE f3753_in(x30, x31, x32) -> f3755_in(x30, x31, x32) :|: TRUE f3056_in(x36, x37, x38) -> f3060_in(x36, x37, x38) :|: TRUE f3755_in(x49, x50, x51) -> f4518_in(x50, x49, x51) :|: x49 > 0 f3053_in(x55, x56, x57, x58) -> f3054_in(x55, x56, x57) :|: TRUE f3085_in(x66, x67, x68) -> f5014_in(x66, x67, x68) :|: TRUE f3060_in(x69, x70, x71) -> f3076_in(x69, x70, x71) :|: TRUE f3060_in(x72, x73, x74) -> f3085_in(x72, x73, x74) :|: TRUE f5014_in(x84, x85, x86) -> f5143_in(x84, x85, x86) :|: x84 < x85 f3119_in(x102, x103, x104) -> f3753_in(x103, x102, x104) :|: x102 >= x103 f4518_in(x105, x106, x107) -> f4520_in(x106, x108, x107, x105) :|: x108 = mod(x105, x106) f4520_in(x113, x114, x115, x116) -> f3053_in(x113, x114, x115, x117) :|: TRUE ---------------------------------------- (14) Obligation: Rules: f3054_in(x1, x2, x3) -> f3056_in(x1, x2, x3) :|: TRUE f5145_in(x10, x11, x12) -> f5146_in(x11, x10, x12) :|: x10 > 0 f3076_in(T51, T52, T53) -> f3119_in(T51, T52, T53) :|: TRUE f5143_in(x16, x17, x18) -> f5145_in(x16, x17, x18) :|: TRUE f5146_in(x26, x27, x28) -> f3053_in(x26, x27, x28, x29) :|: TRUE f3753_in(x30, x31, x32) -> f3755_in(x30, x31, x32) :|: TRUE f3056_in(x36, x37, x38) -> f3060_in(x36, x37, x38) :|: TRUE f3755_in(x49, x50, x51) -> f4518_in(x50, x49, x51) :|: x49 > 0 f3053_in(x55, x56, x57, x58) -> f3054_in(x55, x56, x57) :|: TRUE f3085_in(x66, x67, x68) -> f5014_in(x66, x67, x68) :|: TRUE f3060_in(x69, x70, x71) -> f3076_in(x69, x70, x71) :|: TRUE f3060_in(x72, x73, x74) -> f3085_in(x72, x73, x74) :|: TRUE f5014_in(x84, x85, x86) -> f5143_in(x84, x85, x86) :|: x84 < x85 f3119_in(x102, x103, x104) -> f3753_in(x103, x102, x104) :|: x102 >= x103 f4518_in(x105, x106, x107) -> f4520_in(x106, x108, x107, x105) :|: x108 = mod(x105, x106) f4520_in(x113, x114, x115, x116) -> f3053_in(x113, x114, x115, x117) :|: TRUE ---------------------------------------- (15) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (16) Obligation: Rules: f3054_in(x1:0, x2:0, x3:0) -> f3054_in(x2:0, x1:0, x3:0) :|: x2:0 > x1:0 && x1:0 > 0 f3054_in(x, x1, x2) -> f3054_in(x1, x3, x2) :|: x1 > 0 && x >= x1 && TRUE && x3 = mod(x, x1) ---------------------------------------- (17) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (18) Obligation: Rules: f3054_in(x1:0, x2:0, x3:0) -> f3054_in(x2:0, x1:0, x3:0) :|: x2:0 > x1:0 && x1:0 > 0 f3054_in(x, x1, x2) -> f3054_in(x1, x3, x2) :|: x1 > 0 && x >= x1 && TRUE && x3 = mod(x, x1) ---------------------------------------- (19) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f3054_in(x1:0, x2:0, x3:0) -> f3054_in(x2:0, x1:0, x3:0) :|: x2:0 > x1:0 && x1:0 > 0 (2) f3054_in(x, x1, x2) -> f3054_in(x1, x3, x2) :|: x1 > 0 && x >= x1 && TRUE && x3 = mod(x, x1) No arcs! This digraph is fully evaluated! ---------------------------------------- (20) TRUE