%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SEV637+1 : TPTP v9.3.0. Released v9.3.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n022.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue May 5 06:56:02 PM UTC 2026
% Result : Theorem 5.14s 5.34s
% Output : Proof 5.34s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEV637+1 : TPTP v9.3.0. Released v9.3.0.
% 0.00/0.12 % Command : run_E %s %d THM
% 0.16/0.33 % Computer : n022.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue May 5 06:49:07 EDT 2026
% 0.16/0.33 % CPUTime :
% 5.14/5.34 % SZS status Theorem
% 5.14/5.34 % SZS output start Proof
% 5.14/5.34 tff(p_type, type, (
% 5.14/5.34 p: $i > $o)).
% 5.14/5.34 tff(cCoq_2eInit_2ePeano_2ele___24a2_type, type, (
% 5.14/5.34 cCoq_2eInit_2ePeano_2ele___24a2: ( $i * $i ) > $i)).
% 5.14/5.34 tff(tptp_fun_V1__m_1438_type, type, (
% 5.14/5.34 tptp_fun_V1__m_1438: $i)).
% 5.14/5.34 tff(tptp_fun_V0__n_1437_type, type, (
% 5.14/5.34 tptp_fun_V0__n_1437: $i)).
% 5.14/5.34 tff(cCoq_2eInit_2eDatatypes_2eS___24a1_type, type, (
% 5.14/5.34 cCoq_2eInit_2eDatatypes_2eS___24a1: $i > $i)).
% 5.14/5.34 tff(cIndProp_2elt___24a2_type, type, (
% 5.14/5.34 cIndProp_2elt___24a2: ( $i * $i ) > $o)).
% 5.14/5.34 tff(cCoq_2eInit_2eDatatypes_2enat___24t_type, type, (
% 5.14/5.34 cCoq_2eInit_2eDatatypes_2enat___24t: $i > $o)).
% 5.14/5.34 tff(1,plain,
% 5.14/5.34 (^[V0__n: $i, V1__m: $i] : refl((cIndProp_2elt___24a2(V0__n, V1__m) <=> p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m))) <=> (cIndProp_2elt___24a2(V0__n, V1__m) <=> p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m))))),
% 5.14/5.34 inference(bind,[status(th)],[])).
% 5.14/5.34 tff(2,plain,
% 5.14/5.34 (![V0__n: $i, V1__m: $i] : (cIndProp_2elt___24a2(V0__n, V1__m) <=> p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m))) <=> ![V0__n: $i, V1__m: $i] : (cIndProp_2elt___24a2(V0__n, V1__m) <=> p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)))),
% 5.14/5.34 inference(quant_intro,[status(thm)],[1])).
% 5.14/5.34 tff(3,plain,
% 5.14/5.34 (![V0__n: $i, V1__m: $i] : (cIndProp_2elt___24a2(V0__n, V1__m) <=> p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m))) <=> ![V0__n: $i, V1__m: $i] : (cIndProp_2elt___24a2(V0__n, V1__m) <=> p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)))),
% 5.14/5.34 inference(rewrite,[status(thm)],[])).
% 5.14/5.34 tff(4,axiom,(![V0__n: $i, V1__m: $i] : (cIndProp_2elt___24a2(V0__n, V1__m) <=> p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p',''$_def_IndProp.lt'')).
% 5.14/5.34 tff(5,plain,
% 5.14/5.34 (![V0__n: $i, V1__m: $i] : (cIndProp_2elt___24a2(V0__n, V1__m) <=> p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)))),
% 5.14/5.34 inference(modus_ponens,[status(thm)],[4, 3])).
% 5.14/5.34 tff(6,plain,(
% 5.14/5.34 ![V0__n: $i, V1__m: $i] : (cIndProp_2elt___24a2(V0__n, V1__m) <=> p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)))),
% 5.14/5.34 inference(skolemize,[status(sab)],[5])).
% 5.14/5.34 tff(7,plain,
% 5.14/5.34 (![V0__n: $i, V1__m: $i] : (cIndProp_2elt___24a2(V0__n, V1__m) <=> p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)))),
% 5.14/5.34 inference(modus_ponens,[status(thm)],[6, 2])).
% 5.14/5.34 tff(8,plain,
% 5.14/5.34 ((~![V0__n: $i, V1__m: $i] : (cIndProp_2elt___24a2(V0__n, V1__m) <=> p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)))) | (cIndProp_2elt___24a2(V0__n!1437, V1__m!1438) <=> p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), V1__m!1438)))),
% 5.14/5.34 inference(quant_inst,[status(thm)],[])).
% 5.14/5.34 tff(9,plain,
% 5.14/5.34 (cIndProp_2elt___24a2(V0__n!1437, V1__m!1438) <=> p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), V1__m!1438))),
% 5.14/5.34 inference(unit_resolution,[status(thm)],[8, 7])).
% 5.14/5.34 tff(10,plain,
% 5.14/5.34 ((~(cIndProp_2elt___24a2(V0__n!1437, cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m!1438)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m!1438)) | (~cIndProp_2elt___24a2(V0__n!1437, V1__m!1438)))) <=> (~(cIndProp_2elt___24a2(V0__n!1437, cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m!1438)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m!1438)) | (~cIndProp_2elt___24a2(V0__n!1437, V1__m!1438))))),
% 5.14/5.34 inference(rewrite,[status(thm)],[])).
% 5.14/5.34 tff(11,plain,
% 5.14/5.34 ((cCoq_2eInit_2eDatatypes_2enat___24t(V0__n!1437) & (~(cIndProp_2elt___24a2(V0__n!1437, cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m!1438)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m!1438)) | (~cIndProp_2elt___24a2(V0__n!1437, V1__m!1438))))) <=> (cCoq_2eInit_2eDatatypes_2enat___24t(V0__n!1437) & (~(cIndProp_2elt___24a2(V0__n!1437, cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m!1438)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m!1438)) | (~cIndProp_2elt___24a2(V0__n!1437, V1__m!1438)))))),
% 5.14/5.34 inference(monotonicity,[status(thm)],[10])).
% 5.14/5.34 tff(12,plain,
% 5.14/5.34 ((~![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (cIndProp_2elt___24a2(V0__n, cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~cIndProp_2elt___24a2(V0__n, V1__m))))) <=> (~![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (cIndProp_2elt___24a2(V0__n, cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~cIndProp_2elt___24a2(V0__n, V1__m)))))),
% 5.14/5.34 inference(rewrite,[status(thm)],[])).
% 5.14/5.34 tff(13,plain,
% 5.14/5.34 ((~![V0__n: $i] : (cCoq_2eInit_2eDatatypes_2enat___24t(V0__n) => ![V1__m: $i] : (cCoq_2eInit_2eDatatypes_2enat___24t(V1__m) => (cIndProp_2elt___24a2(V0__n, V1__m) => cIndProp_2elt___24a2(V0__n, cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m)))))) <=> (~![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (cIndProp_2elt___24a2(V0__n, cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~cIndProp_2elt___24a2(V0__n, V1__m)))))),
% 5.14/5.34 inference(rewrite,[status(thm)],[])).
% 5.14/5.34 tff(14,axiom,(~![V0__n: $i] : (cCoq_2eInit_2eDatatypes_2enat___24t(V0__n) => ![V1__m: $i] : (cCoq_2eInit_2eDatatypes_2enat___24t(V1__m) => (cIndProp_2elt___24a2(V0__n, V1__m) => cIndProp_2elt___24a2(V0__n, cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m)))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p',''_HAMMER_GOAL'')).
% 5.14/5.34 tff(15,plain,
% 5.14/5.34 (~![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (cIndProp_2elt___24a2(V0__n, cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~cIndProp_2elt___24a2(V0__n, V1__m))))),
% 5.14/5.34 inference(modus_ponens,[status(thm)],[14, 13])).
% 5.14/5.34 tff(16,plain,
% 5.14/5.34 (~![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (cIndProp_2elt___24a2(V0__n, cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~cIndProp_2elt___24a2(V0__n, V1__m))))),
% 5.14/5.34 inference(modus_ponens,[status(thm)],[15, 12])).
% 5.14/5.34 tff(17,plain,
% 5.14/5.34 (~![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (cIndProp_2elt___24a2(V0__n, cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~cIndProp_2elt___24a2(V0__n, V1__m))))),
% 5.14/5.34 inference(modus_ponens,[status(thm)],[16, 12])).
% 5.14/5.34 tff(18,plain,
% 5.14/5.34 (~![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (cIndProp_2elt___24a2(V0__n, cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~cIndProp_2elt___24a2(V0__n, V1__m))))),
% 5.14/5.34 inference(modus_ponens,[status(thm)],[17, 12])).
% 5.14/5.34 tff(19,plain,
% 5.14/5.34 (cCoq_2eInit_2eDatatypes_2enat___24t(V0__n!1437) & (~(cIndProp_2elt___24a2(V0__n!1437, cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m!1438)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m!1438)) | (~cIndProp_2elt___24a2(V0__n!1437, V1__m!1438))))),
% 5.14/5.34 inference(modus_ponens,[status(thm)],[18, 11])).
% 5.14/5.34 tff(20,plain,
% 5.14/5.34 (~(cIndProp_2elt___24a2(V0__n!1437, cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m!1438)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m!1438)) | (~cIndProp_2elt___24a2(V0__n!1437, V1__m!1438)))),
% 5.14/5.34 inference(and_elim,[status(thm)],[19])).
% 5.14/5.34 tff(21,plain,
% 5.14/5.34 (cIndProp_2elt___24a2(V0__n!1437, V1__m!1438)),
% 5.14/5.34 inference(or_elim,[status(thm)],[20])).
% 5.14/5.34 tff(22,plain,
% 5.14/5.34 ((~(cIndProp_2elt___24a2(V0__n!1437, V1__m!1438) <=> p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), V1__m!1438)))) | (~cIndProp_2elt___24a2(V0__n!1437, V1__m!1438)) | p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), V1__m!1438))),
% 5.14/5.34 inference(tautology,[status(thm)],[])).
% 5.14/5.34 tff(23,plain,
% 5.14/5.34 ((~(cIndProp_2elt___24a2(V0__n!1437, V1__m!1438) <=> p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), V1__m!1438)))) | p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), V1__m!1438))),
% 5.14/5.34 inference(unit_resolution,[status(thm)],[22, 21])).
% 5.14/5.34 tff(24,plain,
% 5.14/5.34 (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), V1__m!1438))),
% 5.14/5.34 inference(unit_resolution,[status(thm)],[23, 9])).
% 5.14/5.34 tff(25,plain,
% 5.14/5.34 (cCoq_2eInit_2eDatatypes_2enat___24t(V0__n!1437)),
% 5.14/5.34 inference(and_elim,[status(thm)],[19])).
% 5.14/5.34 tff(26,plain,
% 5.14/5.34 (^[V0__n: $i] : refl(((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m))))) <=> ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m))))))),
% 5.14/5.34 inference(bind,[status(th)],[])).
% 5.14/5.34 tff(27,plain,
% 5.14/5.34 (![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m))))) <=> ![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)))))),
% 5.14/5.34 inference(quant_intro,[status(thm)],[26])).
% 5.14/5.34 tff(28,plain,
% 5.14/5.34 (^[V0__n: $i] : rewrite(((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m))))) <=> ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m))))))),
% 5.14/5.34 inference(bind,[status(th)],[])).
% 5.14/5.34 tff(29,plain,
% 5.14/5.34 (![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m))))) <=> ![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)))))),
% 5.14/5.34 inference(quant_intro,[status(thm)],[28])).
% 5.14/5.34 tff(30,plain,
% 5.14/5.34 (![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m))))) <=> ![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)))))),
% 5.14/5.34 inference(transitivity,[status(thm)],[29, 27])).
% 5.14/5.34 tff(31,plain,
% 5.14/5.34 (![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m))))) <=> ![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)))))),
% 5.14/5.34 inference(rewrite,[status(thm)],[])).
% 5.14/5.34 tff(32,plain,
% 5.14/5.34 (^[V0__n: $i] : trans(monotonicity(quant_intro(proof_bind(^[V1__m: $i] : trans(monotonicity(rewrite((p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)) => p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m))) <=> (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m))))), ((cCoq_2eInit_2eDatatypes_2enat___24t(V1__m) => (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)) => p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)))) <=> (cCoq_2eInit_2eDatatypes_2enat___24t(V1__m) => (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m))))))), rewrite((cCoq_2eInit_2eDatatypes_2enat___24t(V1__m) => (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m))))) <=> (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m))))), ((cCoq_2eInit_2eDatatypes_2enat___24t(V1__m) => (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)) => p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)))) <=> (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m))))))), (![V1__m: $i] : (cCoq_2eInit_2eDatatypes_2enat___24t(V1__m) => (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)) => p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)))) <=> ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)))))), ((cCoq_2eInit_2eDatatypes_2enat___24t(V0__n) => ![V1__m: $i] : (cCoq_2eInit_2eDatatypes_2enat___24t(V1__m) => (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)) => p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m))))) <=> (cCoq_2eInit_2eDatatypes_2enat___24t(V0__n) => ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m))))))), rewrite((cCoq_2eInit_2eDatatypes_2enat___24t(V0__n) => ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m))))) <=> ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)))))), ((cCoq_2eInit_2eDatatypes_2enat___24t(V0__n) => ![V1__m: $i] : (cCoq_2eInit_2eDatatypes_2enat___24t(V1__m) => (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)) => p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m))))) <=> ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)))))))),
% 5.14/5.34 inference(bind,[status(th)],[])).
% 5.14/5.34 tff(33,plain,
% 5.14/5.34 (![V0__n: $i] : (cCoq_2eInit_2eDatatypes_2enat___24t(V0__n) => ![V1__m: $i] : (cCoq_2eInit_2eDatatypes_2enat___24t(V1__m) => (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)) => p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m))))) <=> ![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)))))),
% 5.14/5.34 inference(quant_intro,[status(thm)],[32])).
% 5.14/5.34 tff(34,axiom,(![V0__n: $i] : (cCoq_2eInit_2eDatatypes_2enat___24t(V0__n) => ![V1__m: $i] : (cCoq_2eInit_2eDatatypes_2enat___24t(V1__m) => (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)) => p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p',''Coq.Arith.Le.le_Sn_le'')).
% 5.14/5.34 tff(35,plain,
% 5.14/5.34 (![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)))))),
% 5.14/5.34 inference(modus_ponens,[status(thm)],[34, 33])).
% 5.14/5.34 tff(36,plain,
% 5.14/5.34 (![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)))))),
% 5.14/5.34 inference(modus_ponens,[status(thm)],[35, 31])).
% 5.14/5.34 tff(37,plain,(
% 5.14/5.34 ![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)))))),
% 5.14/5.34 inference(skolemize,[status(sab)],[36])).
% 5.14/5.34 tff(38,plain,
% 5.14/5.34 (![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)))))),
% 5.14/5.34 inference(modus_ponens,[status(thm)],[37, 30])).
% 5.14/5.34 tff(39,plain,
% 5.14/5.34 (((~![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)))))) | ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n!1437)) | ![V1__m: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), V1__m)))))) <=> ((~![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)))))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n!1437)) | ![V1__m: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), V1__m)))))),
% 5.14/5.34 inference(rewrite,[status(thm)],[])).
% 5.14/5.34 tff(40,plain,
% 5.14/5.34 (((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n!1437)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), V1__m))))) <=> ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n!1437)) | ![V1__m: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), V1__m)))))),
% 5.14/5.34 inference(rewrite,[status(thm)],[])).
% 5.14/5.34 tff(41,plain,
% 5.14/5.34 (((~![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)))))) | ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n!1437)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), V1__m)))))) <=> ((~![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)))))) | ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n!1437)) | ![V1__m: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), V1__m))))))),
% 5.14/5.34 inference(monotonicity,[status(thm)],[40])).
% 5.14/5.34 tff(42,plain,
% 5.14/5.34 (((~![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)))))) | ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n!1437)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), V1__m)))))) <=> ((~![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)))))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n!1437)) | ![V1__m: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), V1__m)))))),
% 5.14/5.34 inference(transitivity,[status(thm)],[41, 39])).
% 5.14/5.34 tff(43,plain,
% 5.14/5.34 ((~![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)))))) | ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n!1437)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), V1__m)))))),
% 5.14/5.34 inference(quant_inst,[status(thm)],[])).
% 5.14/5.34 tff(44,plain,
% 5.14/5.34 ((~![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)))))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n!1437)) | ![V1__m: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), V1__m))))),
% 5.14/5.34 inference(modus_ponens,[status(thm)],[43, 42])).
% 5.14/5.34 tff(45,plain,
% 5.14/5.34 (![V1__m: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), V1__m))))),
% 5.14/5.34 inference(unit_resolution,[status(thm)],[44, 38, 25])).
% 5.14/5.34 tff(46,plain,
% 5.14/5.34 (cCoq_2eInit_2eDatatypes_2enat___24t(V1__m!1438)),
% 5.14/5.34 inference(or_elim,[status(thm)],[20])).
% 5.14/5.34 tff(47,plain,
% 5.14/5.34 (((~![V1__m: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), V1__m))))) | ((~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m!1438)) | p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m!1438)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), V1__m!1438))))) <=> ((~![V1__m: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), V1__m))))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m!1438)) | p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m!1438)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), V1__m!1438))))),
% 5.14/5.34 inference(rewrite,[status(thm)],[])).
% 5.14/5.34 tff(48,plain,
% 5.14/5.34 ((~![V1__m: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), V1__m))))) | ((~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m!1438)) | p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m!1438)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), V1__m!1438))))),
% 5.14/5.34 inference(quant_inst,[status(thm)],[])).
% 5.14/5.34 tff(49,plain,
% 5.14/5.34 ((~![V1__m: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), V1__m))))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m!1438)) | p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m!1438)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), V1__m!1438)))),
% 5.14/5.34 inference(modus_ponens,[status(thm)],[48, 47])).
% 5.14/5.34 tff(50,plain,
% 5.14/5.34 (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m!1438))),
% 5.14/5.34 inference(unit_resolution,[status(thm)],[49, 46, 45, 24])).
% 5.14/5.34 tff(51,plain,
% 5.14/5.34 ((~![V0__n: $i, V1__m: $i] : (cIndProp_2elt___24a2(V0__n, V1__m) <=> p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), V1__m)))) | (cIndProp_2elt___24a2(V0__n!1437, cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m!1438)) <=> p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m!1438))))),
% 5.14/5.34 inference(quant_inst,[status(thm)],[])).
% 5.14/5.34 tff(52,plain,
% 5.14/5.34 (cIndProp_2elt___24a2(V0__n!1437, cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m!1438)) <=> p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m!1438)))),
% 5.14/5.34 inference(unit_resolution,[status(thm)],[51, 7])).
% 5.14/5.34 tff(53,plain,
% 5.14/5.34 (~cIndProp_2elt___24a2(V0__n!1437, cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m!1438))),
% 5.14/5.34 inference(or_elim,[status(thm)],[20])).
% 5.14/5.34 tff(54,plain,
% 5.14/5.34 ((~(cIndProp_2elt___24a2(V0__n!1437, cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m!1438)) <=> p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m!1438))))) | cIndProp_2elt___24a2(V0__n!1437, cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m!1438)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m!1438))))),
% 5.14/5.34 inference(tautology,[status(thm)],[])).
% 5.14/5.34 tff(55,plain,
% 5.14/5.34 ((~(cIndProp_2elt___24a2(V0__n!1437, cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m!1438)) <=> p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m!1438))))) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m!1438))))),
% 5.14/5.34 inference(unit_resolution,[status(thm)],[54, 53])).
% 5.14/5.34 tff(56,plain,
% 5.14/5.34 (~p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m!1438)))),
% 5.14/5.34 inference(unit_resolution,[status(thm)],[55, 52])).
% 5.14/5.34 tff(57,plain,
% 5.14/5.34 (^[V0__n: $i] : refl(((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m))))) <=> ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m))))))),
% 5.14/5.34 inference(bind,[status(th)],[])).
% 5.14/5.34 tff(58,plain,
% 5.14/5.34 (![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m))))) <=> ![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)))))),
% 5.14/5.34 inference(quant_intro,[status(thm)],[57])).
% 5.14/5.34 tff(59,plain,
% 5.14/5.34 (^[V0__n: $i] : rewrite(((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m))))) <=> ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m))))))),
% 5.14/5.34 inference(bind,[status(th)],[])).
% 5.14/5.34 tff(60,plain,
% 5.14/5.34 (![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m))))) <=> ![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)))))),
% 5.14/5.34 inference(quant_intro,[status(thm)],[59])).
% 5.14/5.34 tff(61,plain,
% 5.14/5.34 (![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m))))) <=> ![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)))))),
% 5.14/5.34 inference(transitivity,[status(thm)],[60, 58])).
% 5.14/5.34 tff(62,plain,
% 5.14/5.34 (![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m))))) <=> ![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)))))),
% 5.14/5.34 inference(rewrite,[status(thm)],[])).
% 5.14/5.34 tff(63,plain,
% 5.14/5.34 (^[V0__n: $i] : trans(monotonicity(quant_intro(proof_bind(^[V1__m: $i] : trans(monotonicity(rewrite((p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) => p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m)))) <=> ((~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m))) | p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))))), ((cCoq_2eInit_2eDatatypes_2enat___24t(V1__m) => (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) => p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))))) <=> (cCoq_2eInit_2eDatatypes_2enat___24t(V1__m) => ((~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m))) | p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))))))), rewrite((cCoq_2eInit_2eDatatypes_2enat___24t(V1__m) => ((~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m))) | p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))))) <=> (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m))))), ((cCoq_2eInit_2eDatatypes_2enat___24t(V1__m) => (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) => p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))))) <=> (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m))))))), (![V1__m: $i] : (cCoq_2eInit_2eDatatypes_2enat___24t(V1__m) => (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) => p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))))) <=> ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)))))), ((cCoq_2eInit_2eDatatypes_2enat___24t(V0__n) => ![V1__m: $i] : (cCoq_2eInit_2eDatatypes_2enat___24t(V1__m) => (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) => p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m)))))) <=> (cCoq_2eInit_2eDatatypes_2enat___24t(V0__n) => ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m))))))), rewrite((cCoq_2eInit_2eDatatypes_2enat___24t(V0__n) => ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m))))) <=> ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)))))), ((cCoq_2eInit_2eDatatypes_2enat___24t(V0__n) => ![V1__m: $i] : (cCoq_2eInit_2eDatatypes_2enat___24t(V1__m) => (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) => p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m)))))) <=> ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)))))))),
% 5.14/5.34 inference(bind,[status(th)],[])).
% 5.14/5.34 tff(64,plain,
% 5.14/5.34 (![V0__n: $i] : (cCoq_2eInit_2eDatatypes_2enat___24t(V0__n) => ![V1__m: $i] : (cCoq_2eInit_2eDatatypes_2enat___24t(V1__m) => (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) => p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m)))))) <=> ![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)))))),
% 5.14/5.34 inference(quant_intro,[status(thm)],[63])).
% 5.14/5.34 tff(65,axiom,(![V0__n: $i] : (cCoq_2eInit_2eDatatypes_2enat___24t(V0__n) => ![V1__m: $i] : (cCoq_2eInit_2eDatatypes_2enat___24t(V1__m) => (p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)) => p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p',''Coq.Arith.Le.le_n_S'')).
% 5.14/5.34 tff(66,plain,
% 5.14/5.34 (![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)))))),
% 5.14/5.34 inference(modus_ponens,[status(thm)],[65, 64])).
% 5.14/5.34 tff(67,plain,
% 5.14/5.34 (![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)))))),
% 5.14/5.34 inference(modus_ponens,[status(thm)],[66, 62])).
% 5.14/5.34 tff(68,plain,(
% 5.14/5.34 ![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)))))),
% 5.14/5.34 inference(skolemize,[status(sab)],[67])).
% 5.14/5.34 tff(69,plain,
% 5.14/5.34 (![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)))))),
% 5.14/5.34 inference(modus_ponens,[status(thm)],[68, 61])).
% 5.14/5.34 tff(70,plain,
% 5.14/5.34 (((~![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)))))) | ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n!1437)) | ![V1__m: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m))) | p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m)))))) <=> ((~![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)))))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n!1437)) | ![V1__m: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m))) | p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m)))))),
% 5.14/5.34 inference(rewrite,[status(thm)],[])).
% 5.14/5.34 tff(71,plain,
% 5.14/5.34 (((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n!1437)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m))))) <=> ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n!1437)) | ![V1__m: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m))) | p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m)))))),
% 5.14/5.34 inference(rewrite,[status(thm)],[])).
% 5.14/5.34 tff(72,plain,
% 5.14/5.34 (((~![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)))))) | ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n!1437)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m)))))) <=> ((~![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)))))) | ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n!1437)) | ![V1__m: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m))) | p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))))))),
% 5.14/5.34 inference(monotonicity,[status(thm)],[71])).
% 5.14/5.34 tff(73,plain,
% 5.14/5.34 (((~![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)))))) | ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n!1437)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m)))))) <=> ((~![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)))))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n!1437)) | ![V1__m: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m))) | p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m)))))),
% 5.14/5.34 inference(transitivity,[status(thm)],[72, 70])).
% 5.14/5.34 tff(74,plain,
% 5.14/5.34 ((~![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)))))) | ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n!1437)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m)))))),
% 5.14/5.34 inference(quant_inst,[status(thm)],[])).
% 5.14/5.34 tff(75,plain,
% 5.14/5.34 ((~![V0__n: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n)) | ![V1__m: $i] : (p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n, V1__m)))))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V0__n!1437)) | ![V1__m: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m))) | p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))))),
% 5.14/5.34 inference(modus_ponens,[status(thm)],[74, 73])).
% 5.14/5.34 tff(76,plain,
% 5.14/5.34 (![V1__m: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m))) | p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))))),
% 5.14/5.34 inference(unit_resolution,[status(thm)],[75, 69, 25])).
% 5.14/5.34 tff(77,plain,
% 5.14/5.34 (((~![V1__m: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m))) | p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))))) | ((~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m!1438)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m!1438))) | p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m!1438))))) <=> ((~![V1__m: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m))) | p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m!1438)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m!1438))) | p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m!1438))))),
% 5.34/5.58 inference(rewrite,[status(thm)],[])).
% 5.34/5.58 tff(78,plain,
% 5.34/5.58 ((~![V1__m: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m))) | p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))))) | ((~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m!1438)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m!1438))) | p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m!1438))))),
% 5.34/5.58 inference(quant_inst,[status(thm)],[])).
% 5.34/5.58 tff(79,plain,
% 5.34/5.58 ((~![V1__m: $i] : ((~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m))) | p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m))))) | (~cCoq_2eInit_2eDatatypes_2enat___24t(V1__m!1438)) | (~p(cCoq_2eInit_2ePeano_2ele___24a2(V0__n!1437, V1__m!1438))) | p(cCoq_2eInit_2ePeano_2ele___24a2(cCoq_2eInit_2eDatatypes_2eS___24a1(V0__n!1437), cCoq_2eInit_2eDatatypes_2eS___24a1(V1__m!1438)))),
% 5.34/5.58 inference(modus_ponens,[status(thm)],[78, 77])).
% 5.34/5.58 tff(80,plain,
% 5.34/5.58 ($false),
% 5.34/5.58 inference(unit_resolution,[status(thm)],[79, 46, 76, 56, 50])).
% 5.34/5.58 % SZS output end Proof
% 0.20/5.60 % E exiting
%------------------------------------------------------------------------------