YES proof of prog.inttrs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given IRSwT could be proven: (0) IRSwT (1) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (2) IRSwT (3) IRSwTTerminationDigraphProof [EQUIVALENT, 480 ms] (4) IRSwT (5) IntTRSCompressionProof [EQUIVALENT, 0 ms] (6) IRSwT (7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (8) IRSwT (9) TempFilterProof [SOUND, 37 ms] (10) IntTRS (11) PolynomialOrderProcessor [EQUIVALENT, 9 ms] (12) IntTRS (13) PolynomialOrderProcessor [EQUIVALENT, 0 ms] (14) YES ---------------------------------------- (0) Obligation: Rules: l0(__const_5000HAT0, x_13HAT0, x_27HAT0, x_32HAT0, y_16HAT0, y_28HAT0, y_33HAT0) -> l1(__const_5000HATpost, x_13HATpost, x_27HATpost, x_32HATpost, y_16HATpost, y_28HATpost, y_33HATpost) :|: y_33HAT0 = y_33HATpost && y_28HAT0 = y_28HATpost && x_32HAT0 = x_32HATpost && x_27HAT0 = x_27HATpost && __const_5000HAT0 = __const_5000HATpost && y_16HATpost = y_16HATpost && x_13HATpost = x_13HATpost l1(x, x1, x2, x3, x4, x5, x6) -> l2(x7, x8, x9, x10, x11, x12, x13) :|: x6 = x13 && x5 = x12 && x3 = x10 && x2 = x9 && x1 = x8 && x = x7 && 1 <= x1 && 1 <= x11 && x11 = x && 1 <= x1 l3(x14, x15, x16, x17, x18, x19, x20) -> l2(x21, x22, x23, x24, x25, x26, x27) :|: x20 = x27 && x17 = x24 && x14 = x21 && 1 <= x26 && 1 <= x23 && -1 + x26 <= x25 && x25 <= -1 + x26 && -1 + x23 <= x22 && x22 <= -1 + x23 && x25 = -1 + x18 && x22 = -1 + x15 && 1 <= x18 && x26 = x26 && x23 = x23 l2(x28, x29, x30, x31, x32, x33, x34) -> l1(x35, x36, x37, x38, x39, x40, x41) :|: x34 = x41 && x33 = x40 && x32 = x39 && x31 = x38 && x30 = x37 && x29 = x36 && x28 = x35 && x32 <= 0 && x32 <= 0 l2(x42, x43, x44, x45, x46, x47, x48) -> l4(x49, x50, x51, x52, x53, x54, x55) :|: x47 = x54 && x44 = x51 && x42 = x49 && 1 <= x55 && -1 + x55 <= x53 && x53 <= -1 + x55 && -1 + x52 <= x50 && x50 <= -1 + x52 && x53 = -1 + x46 && x50 = -1 + x43 && 1 <= x46 && x55 = x55 && x52 = x52 l4(x56, x57, x58, x59, x60, x61, x62) -> l2(x63, x64, x65, x66, x67, x68, x69) :|: x62 = x69 && x61 = x68 && x60 = x67 && x59 = x66 && x58 = x65 && x57 = x64 && x56 = x63 l5(x70, x71, x72, x73, x74, x75, x76) -> l0(x77, x78, x79, x80, x81, x82, x83) :|: x76 = x83 && x75 = x82 && x74 = x81 && x73 = x80 && x72 = x79 && x71 = x78 && x70 = x77 Start term: l5(__const_5000HAT0, x_13HAT0, x_27HAT0, x_32HAT0, y_16HAT0, y_28HAT0, y_33HAT0) ---------------------------------------- (1) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (2) Obligation: Rules: l0(__const_5000HAT0, x_13HAT0, x_27HAT0, x_32HAT0, y_16HAT0, y_28HAT0, y_33HAT0) -> l1(__const_5000HATpost, x_13HATpost, x_27HATpost, x_32HATpost, y_16HATpost, y_28HATpost, y_33HATpost) :|: y_33HAT0 = y_33HATpost && y_28HAT0 = y_28HATpost && x_32HAT0 = x_32HATpost && x_27HAT0 = x_27HATpost && __const_5000HAT0 = __const_5000HATpost && y_16HATpost = y_16HATpost && x_13HATpost = x_13HATpost l1(x, x1, x2, x3, x4, x5, x6) -> l2(x7, x8, x9, x10, x11, x12, x13) :|: x6 = x13 && x5 = x12 && x3 = x10 && x2 = x9 && x1 = x8 && x = x7 && 1 <= x1 && 1 <= x11 && x11 = x && 1 <= x1 l3(x14, x15, x16, x17, x18, x19, x20) -> l2(x21, x22, x23, x24, x25, x26, x27) :|: x20 = x27 && x17 = x24 && x14 = x21 && 1 <= x26 && 1 <= x23 && -1 + x26 <= x25 && x25 <= -1 + x26 && -1 + x23 <= x22 && x22 <= -1 + x23 && x25 = -1 + x18 && x22 = -1 + x15 && 1 <= x18 && x26 = x26 && x23 = x23 l2(x28, x29, x30, x31, x32, x33, x34) -> l1(x35, x36, x37, x38, x39, x40, x41) :|: x34 = x41 && x33 = x40 && x32 = x39 && x31 = x38 && x30 = x37 && x29 = x36 && x28 = x35 && x32 <= 0 && x32 <= 0 l2(x42, x43, x44, x45, x46, x47, x48) -> l4(x49, x50, x51, x52, x53, x54, x55) :|: x47 = x54 && x44 = x51 && x42 = x49 && 1 <= x55 && -1 + x55 <= x53 && x53 <= -1 + x55 && -1 + x52 <= x50 && x50 <= -1 + x52 && x53 = -1 + x46 && x50 = -1 + x43 && 1 <= x46 && x55 = x55 && x52 = x52 l4(x56, x57, x58, x59, x60, x61, x62) -> l2(x63, x64, x65, x66, x67, x68, x69) :|: x62 = x69 && x61 = x68 && x60 = x67 && x59 = x66 && x58 = x65 && x57 = x64 && x56 = x63 l5(x70, x71, x72, x73, x74, x75, x76) -> l0(x77, x78, x79, x80, x81, x82, x83) :|: x76 = x83 && x75 = x82 && x74 = x81 && x73 = x80 && x72 = x79 && x71 = x78 && x70 = x77 Start term: l5(__const_5000HAT0, x_13HAT0, x_27HAT0, x_32HAT0, y_16HAT0, y_28HAT0, y_33HAT0) ---------------------------------------- (3) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) l0(__const_5000HAT0, x_13HAT0, x_27HAT0, x_32HAT0, y_16HAT0, y_28HAT0, y_33HAT0) -> l1(__const_5000HATpost, x_13HATpost, x_27HATpost, x_32HATpost, y_16HATpost, y_28HATpost, y_33HATpost) :|: y_33HAT0 = y_33HATpost && y_28HAT0 = y_28HATpost && x_32HAT0 = x_32HATpost && x_27HAT0 = x_27HATpost && __const_5000HAT0 = __const_5000HATpost && y_16HATpost = y_16HATpost && x_13HATpost = x_13HATpost (2) l1(x, x1, x2, x3, x4, x5, x6) -> l2(x7, x8, x9, x10, x11, x12, x13) :|: x6 = x13 && x5 = x12 && x3 = x10 && x2 = x9 && x1 = x8 && x = x7 && 1 <= x1 && 1 <= x11 && x11 = x && 1 <= x1 (3) l3(x14, x15, x16, x17, x18, x19, x20) -> l2(x21, x22, x23, x24, x25, x26, x27) :|: x20 = x27 && x17 = x24 && x14 = x21 && 1 <= x26 && 1 <= x23 && -1 + x26 <= x25 && x25 <= -1 + x26 && -1 + x23 <= x22 && x22 <= -1 + x23 && x25 = -1 + x18 && x22 = -1 + x15 && 1 <= x18 && x26 = x26 && x23 = x23 (4) l2(x28, x29, x30, x31, x32, x33, x34) -> l1(x35, x36, x37, x38, x39, x40, x41) :|: x34 = x41 && x33 = x40 && x32 = x39 && x31 = x38 && x30 = x37 && x29 = x36 && x28 = x35 && x32 <= 0 && x32 <= 0 (5) l2(x42, x43, x44, x45, x46, x47, x48) -> l4(x49, x50, x51, x52, x53, x54, x55) :|: x47 = x54 && x44 = x51 && x42 = x49 && 1 <= x55 && -1 + x55 <= x53 && x53 <= -1 + x55 && -1 + x52 <= x50 && x50 <= -1 + x52 && x53 = -1 + x46 && x50 = -1 + x43 && 1 <= x46 && x55 = x55 && x52 = x52 (6) l4(x56, x57, x58, x59, x60, x61, x62) -> l2(x63, x64, x65, x66, x67, x68, x69) :|: x62 = x69 && x61 = x68 && x60 = x67 && x59 = x66 && x58 = x65 && x57 = x64 && x56 = x63 (7) l5(x70, x71, x72, x73, x74, x75, x76) -> l0(x77, x78, x79, x80, x81, x82, x83) :|: x76 = x83 && x75 = x82 && x74 = x81 && x73 = x80 && x72 = x79 && x71 = x78 && x70 = x77 Arcs: (1) -> (2) (2) -> (5) (3) -> (4), (5) (4) -> (2) (5) -> (6) (6) -> (4), (5) (7) -> (1) This digraph is fully evaluated! ---------------------------------------- (4) Obligation: Termination digraph: Nodes: (1) l1(x, x1, x2, x3, x4, x5, x6) -> l2(x7, x8, x9, x10, x11, x12, x13) :|: x6 = x13 && x5 = x12 && x3 = x10 && x2 = x9 && x1 = x8 && x = x7 && 1 <= x1 && 1 <= x11 && x11 = x && 1 <= x1 (2) l2(x28, x29, x30, x31, x32, x33, x34) -> l1(x35, x36, x37, x38, x39, x40, x41) :|: x34 = x41 && x33 = x40 && x32 = x39 && x31 = x38 && x30 = x37 && x29 = x36 && x28 = x35 && x32 <= 0 && x32 <= 0 (3) l4(x56, x57, x58, x59, x60, x61, x62) -> l2(x63, x64, x65, x66, x67, x68, x69) :|: x62 = x69 && x61 = x68 && x60 = x67 && x59 = x66 && x58 = x65 && x57 = x64 && x56 = x63 (4) l2(x42, x43, x44, x45, x46, x47, x48) -> l4(x49, x50, x51, x52, x53, x54, x55) :|: x47 = x54 && x44 = x51 && x42 = x49 && 1 <= x55 && -1 + x55 <= x53 && x53 <= -1 + x55 && -1 + x52 <= x50 && x50 <= -1 + x52 && x53 = -1 + x46 && x50 = -1 + x43 && 1 <= x46 && x55 = x55 && x52 = x52 Arcs: (1) -> (4) (2) -> (1) (3) -> (2), (4) (4) -> (3) This digraph is fully evaluated! ---------------------------------------- (5) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (6) Obligation: Rules: l2(x42:0, x43:0, x44:0, x45:0, x46:0, x47:0, x48:0) -> l2(x42:0, -1 + x43:0, x44:0, x52:0, -1 + x46:0, x47:0, x55:0) :|: x55:0 > 0 && x46:0 > 0 && -1 + x55:0 = -1 + x46:0 && -1 + x52:0 = -1 + x43:0 l2(x11:0, x29:0, x30:0, x10:0, x32:0, x12:0, x13:0) -> l2(x11:0, x29:0, x30:0, x10:0, x11:0, x12:0, x13:0) :|: x32:0 < 1 && x29:0 > 0 && x11:0 > 0 ---------------------------------------- (7) IntTRSUnneededArgumentFilterProof (EQUIVALENT) Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: l2(x1, x2, x3, x4, x5, x6, x7) -> l2(x1, x2, x4, x5, x7) ---------------------------------------- (8) Obligation: Rules: l2(x42:0, x43:0, x45:0, x46:0, x48:0) -> l2(x42:0, -1 + x43:0, x52:0, -1 + x46:0, x55:0) :|: x55:0 > 0 && x46:0 > 0 && -1 + x55:0 = -1 + x46:0 && -1 + x52:0 = -1 + x43:0 l2(x11:0, x29:0, x10:0, x32:0, x13:0) -> l2(x11:0, x29:0, x10:0, x11:0, x13:0) :|: x32:0 < 1 && x29:0 > 0 && x11:0 > 0 ---------------------------------------- (9) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l2(VARIABLE, INTEGER, VARIABLE, INTEGER, VARIABLE) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (10) Obligation: Rules: l2(x42:0, x43:0, x45:0, x46:0, x48:0) -> l2(x42:0, c, x52:0, c1, x55:0) :|: c1 = -1 + x46:0 && c = -1 + x43:0 && (x55:0 > 0 && x46:0 > 0 && -1 + x55:0 = -1 + x46:0 && -1 + x52:0 = -1 + x43:0) l2(x11:0, x29:0, x10:0, x32:0, x13:0) -> l2(x11:0, x29:0, x10:0, x11:0, x13:0) :|: x32:0 < 1 && x29:0 > 0 && x11:0 > 0 ---------------------------------------- (11) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l2(x, x1, x2, x3, x4)] = x + x1 - x3 The following rules are decreasing: l2(x11:0, x29:0, x10:0, x32:0, x13:0) -> l2(x11:0, x29:0, x10:0, x11:0, x13:0) :|: x32:0 < 1 && x29:0 > 0 && x11:0 > 0 The following rules are bounded: l2(x11:0, x29:0, x10:0, x32:0, x13:0) -> l2(x11:0, x29:0, x10:0, x11:0, x13:0) :|: x32:0 < 1 && x29:0 > 0 && x11:0 > 0 ---------------------------------------- (12) Obligation: Rules: l2(x42:0, x43:0, x45:0, x46:0, x48:0) -> l2(x42:0, c, x52:0, c1, x55:0) :|: c1 = -1 + x46:0 && c = -1 + x43:0 && (x55:0 > 0 && x46:0 > 0 && -1 + x55:0 = -1 + x46:0 && -1 + x52:0 = -1 + x43:0) ---------------------------------------- (13) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l2(x, x1, x2, x3, x4)] = x3 The following rules are decreasing: l2(x42:0, x43:0, x45:0, x46:0, x48:0) -> l2(x42:0, c, x52:0, c1, x55:0) :|: c1 = -1 + x46:0 && c = -1 + x43:0 && (x55:0 > 0 && x46:0 > 0 && -1 + x55:0 = -1 + x46:0 && -1 + x52:0 = -1 + x43:0) The following rules are bounded: l2(x42:0, x43:0, x45:0, x46:0, x48:0) -> l2(x42:0, c, x52:0, c1, x55:0) :|: c1 = -1 + x46:0 && c = -1 + x43:0 && (x55:0 > 0 && x46:0 > 0 && -1 + x55:0 = -1 + x46:0 && -1 + x52:0 = -1 + x43:0) ---------------------------------------- (14) YES