YES proof of prog.inttrs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given IRSwT could be proven: (0) IRSwT (1) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (2) IRSwT (3) IRSwTTerminationDigraphProof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, 382 ms] (4) IRSwT (5) IntTRSCompressionProof [EQUIVALENT, 8 ms] (6) IRSwT (7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (8) IRSwT (9) TempFilterProof [SOUND, 16 ms] (10) IntTRS (11) PolynomialOrderProcessor [EQUIVALENT, 0 ms] (12) YES ---------------------------------------- (0) Obligation: Rules: l0(__const_10HAT0, a_20HAT0, i_13HAT0, i_21HAT0, rt_11HAT0, st_14HAT0) -> l1(__const_10HATpost, a_20HATpost, i_13HATpost, i_21HATpost, rt_11HATpost, st_14HATpost) :|: st_14HAT0 = st_14HATpost && rt_11HAT0 = rt_11HATpost && i_21HAT0 = i_21HATpost && a_20HAT0 = a_20HATpost && __const_10HAT0 = __const_10HATpost && i_13HATpost <= 0 && 0 <= i_13HATpost && i_13HATpost = 0 l2(x, x1, x2, x3, x4, x5) -> l1(x6, x7, x8, x9, x10, x11) :|: 1 + x2 <= x && x12 = 1 + x2 && x8 = x8 && 2 <= x8 && x8 <= 2 && x7 = x7 && x7 <= x8 && x8 <= x7 && x = x6 && x3 = x9 && x4 = x10 && x5 = x11 l3(x13, x14, x15, x16, x17, x18) -> l1(x19, x20, x21, x22, x23, x24) :|: x18 = x24 && x17 = x23 && x16 = x22 && x14 = x20 && x13 = x19 && x21 <= 1 && 1 <= x21 && x21 = 1 + x15 && 1 + x15 <= x13 l1(x25, x26, x27, x28, x29, x30) -> l4(x31, x32, x33, x34, x35, x36) :|: x30 = x36 && x28 = x34 && x27 = x33 && x26 = x32 && x25 = x31 && x35 = x30 && x25 <= x27 l1(x37, x38, x39, x40, x41, x42) -> l5(x43, x44, x45, x46, x47, x48) :|: x42 = x48 && x41 = x47 && x38 = x44 && x37 = x43 && 1 + x46 <= x37 && 1 + x46 <= x45 && x45 <= 1 + x46 && x45 = 1 + x39 && 1 + x39 <= x37 && x46 = x46 l5(x49, x50, x51, x52, x53, x54) -> l1(x55, x56, x57, x58, x59, x60) :|: x54 = x60 && x53 = x59 && x52 = x58 && x51 = x57 && x50 = x56 && x49 = x55 l6(x61, x62, x63, x64, x65, x66) -> l0(x67, x68, x69, x70, x71, x72) :|: x66 = x72 && x65 = x71 && x64 = x70 && x63 = x69 && x62 = x68 && x61 = x67 Start term: l6(__const_10HAT0, a_20HAT0, i_13HAT0, i_21HAT0, rt_11HAT0, st_14HAT0) ---------------------------------------- (1) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (2) Obligation: Rules: l0(__const_10HAT0, a_20HAT0, i_13HAT0, i_21HAT0, rt_11HAT0, st_14HAT0) -> l1(__const_10HATpost, a_20HATpost, i_13HATpost, i_21HATpost, rt_11HATpost, st_14HATpost) :|: st_14HAT0 = st_14HATpost && rt_11HAT0 = rt_11HATpost && i_21HAT0 = i_21HATpost && a_20HAT0 = a_20HATpost && __const_10HAT0 = __const_10HATpost && i_13HATpost <= 0 && 0 <= i_13HATpost && i_13HATpost = 0 l2(x, x1, x2, x3, x4, x5) -> l1(x6, x7, x8, x9, x10, x11) :|: 1 + x2 <= x && x12 = 1 + x2 && x8 = x8 && 2 <= x8 && x8 <= 2 && x7 = x7 && x7 <= x8 && x8 <= x7 && x = x6 && x3 = x9 && x4 = x10 && x5 = x11 l3(x13, x14, x15, x16, x17, x18) -> l1(x19, x20, x21, x22, x23, x24) :|: x18 = x24 && x17 = x23 && x16 = x22 && x14 = x20 && x13 = x19 && x21 <= 1 && 1 <= x21 && x21 = 1 + x15 && 1 + x15 <= x13 l1(x25, x26, x27, x28, x29, x30) -> l4(x31, x32, x33, x34, x35, x36) :|: x30 = x36 && x28 = x34 && x27 = x33 && x26 = x32 && x25 = x31 && x35 = x30 && x25 <= x27 l1(x37, x38, x39, x40, x41, x42) -> l5(x43, x44, x45, x46, x47, x48) :|: x42 = x48 && x41 = x47 && x38 = x44 && x37 = x43 && 1 + x46 <= x37 && 1 + x46 <= x45 && x45 <= 1 + x46 && x45 = 1 + x39 && 1 + x39 <= x37 && x46 = x46 l5(x49, x50, x51, x52, x53, x54) -> l1(x55, x56, x57, x58, x59, x60) :|: x54 = x60 && x53 = x59 && x52 = x58 && x51 = x57 && x50 = x56 && x49 = x55 l6(x61, x62, x63, x64, x65, x66) -> l0(x67, x68, x69, x70, x71, x72) :|: x66 = x72 && x65 = x71 && x64 = x70 && x63 = x69 && x62 = x68 && x61 = x67 Start term: l6(__const_10HAT0, a_20HAT0, i_13HAT0, i_21HAT0, rt_11HAT0, st_14HAT0) ---------------------------------------- (3) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) l0(__const_10HAT0, a_20HAT0, i_13HAT0, i_21HAT0, rt_11HAT0, st_14HAT0) -> l1(__const_10HATpost, a_20HATpost, i_13HATpost, i_21HATpost, rt_11HATpost, st_14HATpost) :|: st_14HAT0 = st_14HATpost && rt_11HAT0 = rt_11HATpost && i_21HAT0 = i_21HATpost && a_20HAT0 = a_20HATpost && __const_10HAT0 = __const_10HATpost && i_13HATpost <= 0 && 0 <= i_13HATpost && i_13HATpost = 0 (2) l2(x, x1, x2, x3, x4, x5) -> l1(x6, x7, x8, x9, x10, x11) :|: 1 + x2 <= x && x12 = 1 + x2 && x8 = x8 && 2 <= x8 && x8 <= 2 && x7 = x7 && x7 <= x8 && x8 <= x7 && x = x6 && x3 = x9 && x4 = x10 && x5 = x11 (3) l3(x13, x14, x15, x16, x17, x18) -> l1(x19, x20, x21, x22, x23, x24) :|: x18 = x24 && x17 = x23 && x16 = x22 && x14 = x20 && x13 = x19 && x21 <= 1 && 1 <= x21 && x21 = 1 + x15 && 1 + x15 <= x13 (4) l1(x25, x26, x27, x28, x29, x30) -> l4(x31, x32, x33, x34, x35, x36) :|: x30 = x36 && x28 = x34 && x27 = x33 && x26 = x32 && x25 = x31 && x35 = x30 && x25 <= x27 (5) l1(x37, x38, x39, x40, x41, x42) -> l5(x43, x44, x45, x46, x47, x48) :|: x42 = x48 && x41 = x47 && x38 = x44 && x37 = x43 && 1 + x46 <= x37 && 1 + x46 <= x45 && x45 <= 1 + x46 && x45 = 1 + x39 && 1 + x39 <= x37 && x46 = x46 (6) l5(x49, x50, x51, x52, x53, x54) -> l1(x55, x56, x57, x58, x59, x60) :|: x54 = x60 && x53 = x59 && x52 = x58 && x51 = x57 && x50 = x56 && x49 = x55 (7) l6(x61, x62, x63, x64, x65, x66) -> l0(x67, x68, x69, x70, x71, x72) :|: x66 = x72 && x65 = x71 && x64 = x70 && x63 = x69 && x62 = x68 && x61 = x67 Arcs: (1) -> (4), (5) (2) -> (4), (5) (3) -> (4), (5) (5) -> (6) (6) -> (4), (5) (7) -> (1) This digraph is fully evaluated! ---------------------------------------- (4) Obligation: Termination digraph: Nodes: (1) l1(x37, x38, x39, x40, x41, x42) -> l5(x43, x44, x45, x46, x47, x48) :|: x42 = x48 && x41 = x47 && x38 = x44 && x37 = x43 && 1 + x46 <= x37 && 1 + x46 <= x45 && x45 <= 1 + x46 && x45 = 1 + x39 && 1 + x39 <= x37 && x46 = x46 (2) l5(x49, x50, x51, x52, x53, x54) -> l1(x55, x56, x57, x58, x59, x60) :|: x54 = x60 && x53 = x59 && x52 = x58 && x51 = x57 && x50 = x56 && x49 = x55 Arcs: (1) -> (2) (2) -> (1) This digraph is fully evaluated! ---------------------------------------- (5) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (6) Obligation: Rules: l1(x37:0, x38:0, x39:0, x40:0, x41:0, x42:0) -> l1(x37:0, x38:0, 1 + x39:0, x46:0, x41:0, x42:0) :|: x37:0 >= 1 + x46:0 && 1 + x46:0 = 1 + x39:0 && x37:0 >= 1 + x39:0 ---------------------------------------- (7) IntTRSUnneededArgumentFilterProof (EQUIVALENT) Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: l1(x1, x2, x3, x4, x5, x6) -> l1(x1, x3, x4) ---------------------------------------- (8) Obligation: Rules: l1(x37:0, x39:0, x40:0) -> l1(x37:0, 1 + x39:0, x46:0) :|: x37:0 >= 1 + x46:0 && 1 + x46:0 = 1 + x39:0 && x37:0 >= 1 + x39:0 ---------------------------------------- (9) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l1(INTEGER, INTEGER, VARIABLE) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (10) Obligation: Rules: l1(x37:0, x39:0, x40:0) -> l1(x37:0, c, x46:0) :|: c = 1 + x39:0 && (x37:0 >= 1 + x46:0 && 1 + x46:0 = 1 + x39:0 && x37:0 >= 1 + x39:0) ---------------------------------------- (11) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l1(x, x1, x2)] = x - x1 The following rules are decreasing: l1(x37:0, x39:0, x40:0) -> l1(x37:0, c, x46:0) :|: c = 1 + x39:0 && (x37:0 >= 1 + x46:0 && 1 + x46:0 = 1 + x39:0 && x37:0 >= 1 + x39:0) The following rules are bounded: l1(x37:0, x39:0, x40:0) -> l1(x37:0, c, x46:0) :|: c = 1 + x39:0 && (x37:0 >= 1 + x46:0 && 1 + x46:0 = 1 + x39:0 && x37:0 >= 1 + x39:0) ---------------------------------------- (12) YES