# Colon/Sipma: Practical Methods for Proving Program Termination
(VAR n i j k l t h m p q r up)
(RULES
eval_0(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_1(n, i, j, k, l, t, h, m, p, q, r, TRUE) :|: n > 0
eval_1(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_2(n, i, j, k, l, t, h, m, 1, q, r, up)
eval_2(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_3(n, i, j, k, l, t, h, m, p, q, r, up)
eval_3(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_4(n, i, j, k, l, t, 1, m, p, q, r, up)
eval_4(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_5(n, i, j, k, l, t, h, n, p, q, r, up)
eval_5(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_6(n, i, j, k, l, t, h, m, p, q, r, up) :|: up
eval_6(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_7(n, 1, j, k, l, t, h, m, p, q, r, up)
eval_7(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_8(n, i, n, k, l, t, h, m, p, q, r, up)
eval_8(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_9(n, i, j, n + 1, l, t, h, n, p, q, r, up)
eval_9(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_14(n, i, j, k, 2 * n, t, h, n, p, q, r, up)
eval_5(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_10(n, i, j, k, l, t, h, m, p, q, r, up) :|: !up
eval_10(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_11(n, i, j, 1, l, t, h, m, p, q, r, up)
eval_11(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_12(n, i, j, k, n, t, h, m, p, q, r, up)
eval_12(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_13(n, n + 1, j, k, l, t, h, m, p, q, r, up)
eval_13(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_14(n, i, 2 * n, k, l, t, h, m, p, q, r, up)
eval_14(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_15(n, i, j, k, l, t, h, m, p, q, r, up)
eval_15(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_16(n, i, j, k, l, t, h, m, p, q, r, up) :|: m >= p
eval_16(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_18(n, i, j, k, l, t, h, m, p, p, r, up)
eval_15(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_17(n, i, j, k, l, t, h, m, p, q, r, up) :|: m < p
eval_17(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_18(n, i, j, k, l, t, h, m, p, m, r, up)
eval_18(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_19(n, i, j, k, l, t, h, m - q, p, q, r, up)
eval_19(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_20(n, i, j, k, l, t, h, m, p, q, r, up) :|: m >= p
eval_20(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_22(n, i, j, k, l, t, h, m, p, q, p, up)
eval_19(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_21(n, i, j, k, l, t, h, m, p, q, r, up) :|: m < p
eval_21(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_22(n, i, j, k, l, t, h, m, p, q, m, up)
eval_22(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_23(n, i, j, k, l, t, h, m - r, p, q, r, up)
eval_23(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_24(n, i, j, k, l, t, h, m, p, q, r, up) :|: q > 0 && r > 0
eval_24(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_25(n, i, j, k, l, t, h, m, p, q, r, up)
eval_25(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_26(n, i, j, k, l, t, h, m, p, q, r, up)
eval_26(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_27(n, i, j, k + h, l, t, h, m, p, q, r, up)
eval_27(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_28(n, i + 1, j, k, l, t, h, m, p, q, r, up)
eval_28(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_23(n, i, j, k, l, t, h, m, p, q - 1, r, up)
eval_24(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_29(n, i, j, k, l, t, h, m, p, q, r, up)
eval_29(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_30(n, i, j, k, l, t, h, m, p, q, r, up)
eval_30(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_31(n, i, j, k + h, l, t, h, m, p, q, r, up)
eval_31(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_32(n, i, j - 1, k, l, t, h, m, p, q, r, up)
eval_32(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_23(n, i, j, k, l, t, h, m, p, q, r - 1, up)
eval_23(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_33(n, i, j, k, l, t, h, m, p, q, r, up) :|: q <= 0 || r <= 0
eval_33(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_34(n, i, j, k, l, t, h, m, p, q, r, up) :|: r > 0
eval_34(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_35(n, i, j, k, l, t, h, m, p, q, r, up)
eval_35(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_36(n, i, j, k + h, l, t, h, m, p, q, r, up)
eval_36(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_37(n, i, j - 1, k, l, t, h, m, p, q, r, up)
eval_37(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_33(n, i, j, k, l, t, h, m, p, q, r - 1, up)
eval_33(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_38(n, i, j, k, l, t, h, m, p, q, r, up) :|: r <= 0
eval_38(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_39(n, i, j, k, l, t, h, m, p, q, r, up) :|: q > 0
eval_39(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_40(n, i, j, k, l, t, h, m, p, q, r, up)
eval_40(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_41(n, i, j, k + h, l, t, h, m, p, q, r, up)
eval_41(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_42(n, i + 1, j, k, l, t, h, m, p, q, r, up)
eval_42(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_38(n, i, j, k, l, t, h, m, p, q - 1, r, up)
eval_38(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_43(n, i, j, k, l, t, h, m, p, q, r, up) :|: q <= 0
eval_43(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_44(n, i, j, k, l, t, -h, m, p, q, r, up)
eval_44(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_45(n, i, j, k, l, k, h, m, p, q, r, up)
eval_45(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_46(n, i, j, l, l, t, h, m, p, q, r, up)
eval_46(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_47(n, i, j, k, t, t, h, m, p, q, r, up) :|: m <= 0
eval_46(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_14(n, i, j, k, l, t, h, m, p, q, r, up) :|: m > 0
eval_47(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_48(n, i, j, k, l, t, h, m, p, q, r, !up)
eval_48(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_49(n, i, j, k, l, t, h, m, 2 * p, q, r, up) :|: p >= n
eval_48(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_2(n, i, j, k, l, t, h, m, p, q, r, up) :|: p < n
eval_49(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_50(n, i, j, k, l, t, h, m, p, q, r, up) :|: !up
eval_50(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_51(n, 1, j, k, l, t, h, m, p, q, r, up)
eval_51(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_52(n, i, j, k, l, t, h, m, p, q, r, up) :|: i <= n
eval_52(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_53(n, i, j, k, l, t, h, m, p, q, r, up)
eval_53(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_51(n, i + 1, j, k, l, t, h, m, p, q, r, up)
eval_51(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_end(n, i, j, k, l, t, h, m, p, q, r, up) :|: i > n
eval_49(n, i, j, k, l, t, h, m, p, q, r, up) -> eval_end(n, i, j, k, l, t, h, m, p, q, r, up) :|: up
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