%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SWX034+1 : TPTP v9.1.0. Released v9.1.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n008.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 : Sat Jun 21 05:37:49 AM UTC 2025
% Result : Theorem 0.48s 0.63s
% Output : Assurance 0s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----No solution output by system
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : SWX034+1 : TPTP v9.1.0. Released v9.1.0.
% 0.07/0.13 % Command : run_E %s %d THM
% 0.14/0.34 % Computer : n008.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Fri Jun 20 12:19:33 EDT 2025
% 0.14/0.35 % CPUTime :
% 0.48/0.63 % SZS status Theorem
% 0.48/0.63 % SZS output start Proof
% 0.48/0.63 tff(times_terminates_type, type, (
% 0.48/0.63 times_terminates: ( $i * $i * $i ) > $o)).
% 0.48/0.63 tff(tptp_fun_Xx5_4_type, type, (
% 0.48/0.63 tptp_fun_Xx5_4: ( $i * $i * $i ) > $i)).
% 0.48/0.63 tff(tptp_fun_Xx_30_type, type, (
% 0.48/0.63 tptp_fun_Xx_30: $i)).
% 0.48/0.63 tff(tptp_fun_Xy_33_type, type, (
% 0.48/0.63 tptp_fun_Xy_33: $i)).
% 0.48/0.63 tff(tptp_fun_Xz_32_type, type, (
% 0.48/0.63 tptp_fun_Xz_32: $i)).
% 0.48/0.63 tff(tptp_fun_Xx2_31_type, type, (
% 0.48/0.63 tptp_fun_Xx2_31: $i)).
% 0.48/0.63 tff(tptp_fun_Xx4_5_type, type, (
% 0.48/0.63 tptp_fun_Xx4_5: ( $i * $i * $i ) > $i)).
% 0.48/0.63 tff(inj_37_type, type, (
% 0.48/0.63 inj_37: $i > $i)).
% 0.48/0.63 tff(s_type, type, (
% 0.48/0.63 s: $i > $i)).
% 0.48/0.63 tff(nat_succeeds_type, type, (
% 0.48/0.63 nat_succeeds: $i > $o)).
% 0.48/0.63 tff(tptp_fun__0__type, type, (
% 0.48/0.63 tptp_fun__0_: $i)).
% 0.48/0.63 tff(tptp_fun_Xx_36_type, type, (
% 0.48/0.63 tptp_fun_Xx_36: $i)).
% 0.48/0.63 tff(tptp_fun_Xz_34_type, type, (
% 0.48/0.63 tptp_fun_Xz_34: $i)).
% 0.48/0.63 tff(tptp_fun_Xy_35_type, type, (
% 0.48/0.63 tptp_fun_Xy_35: $i)).
% 0.48/0.63 tff(times_fails_type, type, (
% 0.48/0.63 times_fails: ( $i * $i * $i ) > $o)).
% 0.48/0.63 tff(plus_terminates_type, type, (
% 0.48/0.63 plus_terminates: ( $i * $i * $i ) > $o)).
% 0.48/0.63 tff(1,plain,
% 0.48/0.63 (![Xk!0: $i] : (inj_37(s(Xk!0)) = Xk!0) <=> ![Xk!0: $i] : (inj_37(s(Xk!0)) = Xk!0)),
% 0.48/0.63 inference(rewrite,[status(thm)],[])).
% 0.48/0.63 tff(2,plain,
% 0.48/0.63 (![Xx4: $i, Xx5: $i] : ((~(s(Xx4) = s(Xx5))) | (Xx4 = Xx5)) <=> ![Xk!0: $i] : (inj_37(s(Xk!0)) = Xk!0)),
% 0.48/0.63 inference(rewrite,[status(thm)],[])).
% 0.48/0.63 tff(3,plain,
% 0.48/0.63 (![Xx4: $i, Xx5: $i] : ((~(s(Xx4) = s(Xx5))) | (Xx4 = Xx5)) <=> ![Xx4: $i, Xx5: $i] : ((~(s(Xx4) = s(Xx5))) | (Xx4 = Xx5))),
% 0.48/0.63 inference(rewrite,[status(thm)],[])).
% 0.48/0.63 tff(4,plain,
% 0.48/0.63 (^[Xx4: $i, Xx5: $i] : rewrite(((s(Xx4) = s(Xx5)) => (Xx4 = Xx5)) <=> ((~(s(Xx4) = s(Xx5))) | (Xx4 = Xx5)))),
% 0.48/0.63 inference(bind,[status(th)],[])).
% 0.48/0.63 tff(5,plain,
% 0.48/0.63 (![Xx4: $i, Xx5: $i] : ((s(Xx4) = s(Xx5)) => (Xx4 = Xx5)) <=> ![Xx4: $i, Xx5: $i] : ((~(s(Xx4) = s(Xx5))) | (Xx4 = Xx5))),
% 0.48/0.63 inference(quant_intro,[status(thm)],[4])).
% 0.48/0.63 tff(6,axiom,(![Xx4: $i, Xx5: $i] : ((s(Xx4) = s(Xx5)) => (Xx4 = Xx5))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','id2')).
% 0.48/0.63 tff(7,plain,
% 0.48/0.63 (![Xx4: $i, Xx5: $i] : ((~(s(Xx4) = s(Xx5))) | (Xx4 = Xx5))),
% 0.48/0.63 inference(modus_ponens,[status(thm)],[6, 5])).
% 0.48/0.63 tff(8,plain,
% 0.48/0.63 (![Xx4: $i, Xx5: $i] : ((~(s(Xx4) = s(Xx5))) | (Xx4 = Xx5))),
% 0.48/0.63 inference(modus_ponens,[status(thm)],[7, 3])).
% 0.48/0.63 tff(9,plain,(
% 0.48/0.63 ![Xx4: $i, Xx5: $i] : ((~(s(Xx4) = s(Xx5))) | (Xx4 = Xx5))),
% 0.48/0.63 inference(skolemize,[status(sab)],[8])).
% 0.48/0.63 tff(10,plain,
% 0.48/0.63 (![Xk!0: $i] : (inj_37(s(Xk!0)) = Xk!0)),
% 0.48/0.63 inference(modus_ponens,[status(thm)],[9, 2])).
% 0.48/0.63 tff(11,plain,
% 0.48/0.63 (![Xk!0: $i] : (inj_37(s(Xk!0)) = Xk!0)),
% 0.48/0.63 inference(modus_ponens,[status(thm)],[10, 1])).
% 0.48/0.63 tff(12,plain,
% 0.48/0.63 ((~![Xk!0: $i] : (inj_37(s(Xk!0)) = Xk!0)) | (inj_37(s(tptp_fun_Xx4_5(Xz!32, Xy!33, Xx!30))) = tptp_fun_Xx4_5(Xz!32, Xy!33, Xx!30))),
% 0.48/0.63 inference(quant_inst,[status(thm)],[])).
% 0.48/0.63 tff(13,plain,
% 0.48/0.63 (inj_37(s(tptp_fun_Xx4_5(Xz!32, Xy!33, Xx!30))) = tptp_fun_Xx4_5(Xz!32, Xy!33, Xx!30)),
% 0.48/0.63 inference(unit_resolution,[status(thm)],[12, 11])).
% 0.48/0.63 tff(14,assumption,(![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx!36, Xy, Xz))), introduced(assumption)).
% 0.48/0.63 tff(15,plain,
% 0.48/0.63 ((~![Xx: $i, Xy: $i, Xz: $i] : ((~(nat_succeeds(Xx) & nat_succeeds(Xy))) | times_terminates(Xx, Xy, Xz))) <=> (~![Xx: $i, Xy: $i, Xz: $i] : ((~(nat_succeeds(Xx) & nat_succeeds(Xy))) | times_terminates(Xx, Xy, Xz)))),
% 0.48/0.63 inference(rewrite,[status(thm)],[])).
% 0.48/0.63 tff(16,plain,
% 0.48/0.63 ((~![Xx: $i, Xy: $i, Xz: $i] : ((nat_succeeds(Xx) & nat_succeeds(Xy)) => times_terminates(Xx, Xy, Xz))) <=> (~![Xx: $i, Xy: $i, Xz: $i] : ((~(nat_succeeds(Xx) & nat_succeeds(Xy))) | times_terminates(Xx, Xy, Xz)))),
% 0.48/0.63 inference(rewrite,[status(thm)],[])).
% 0.48/0.63 tff(17,axiom,(~![Xx: $i, Xy: $i, Xz: $i] : ((nat_succeeds(Xx) & nat_succeeds(Xy)) => times_terminates(Xx, Xy, Xz))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p',''lemma-(times:termination)'')).
% 0.48/0.63 tff(18,plain,
% 0.48/0.63 (~![Xx: $i, Xy: $i, Xz: $i] : ((~(nat_succeeds(Xx) & nat_succeeds(Xy))) | times_terminates(Xx, Xy, Xz))),
% 0.48/0.63 inference(modus_ponens,[status(thm)],[17, 16])).
% 0.48/0.63 tff(19,plain,
% 0.48/0.63 (~![Xx: $i, Xy: $i, Xz: $i] : ((~(nat_succeeds(Xx) & nat_succeeds(Xy))) | times_terminates(Xx, Xy, Xz))),
% 0.48/0.63 inference(modus_ponens,[status(thm)],[18, 15])).
% 0.48/0.63 tff(20,plain,
% 0.48/0.63 (~![Xx: $i, Xy: $i, Xz: $i] : ((~(nat_succeeds(Xx) & nat_succeeds(Xy))) | times_terminates(Xx, Xy, Xz))),
% 0.48/0.63 inference(modus_ponens,[status(thm)],[19, 15])).
% 0.48/0.63 tff(21,plain,
% 0.48/0.63 (~![Xx: $i, Xy: $i, Xz: $i] : ((~(nat_succeeds(Xx) & nat_succeeds(Xy))) | times_terminates(Xx, Xy, Xz))),
% 0.48/0.63 inference(modus_ponens,[status(thm)],[20, 15])).
% 0.48/0.63 tff(22,plain,(
% 0.48/0.63 ~((~(nat_succeeds(Xx!36) & nat_succeeds(Xy!35))) | times_terminates(Xx!36, Xy!35, Xz!34))),
% 0.48/0.63 inference(skolemize,[status(sab)],[21])).
% 0.48/0.63 tff(23,plain,
% 0.48/0.63 (~times_terminates(Xx!36, Xy!35, Xz!34)),
% 0.48/0.63 inference(or_elim,[status(thm)],[22])).
% 0.48/0.63 tff(24,plain,
% 0.48/0.63 (nat_succeeds(Xx!36) & nat_succeeds(Xy!35)),
% 0.48/0.63 inference(or_elim,[status(thm)],[22])).
% 0.48/0.63 tff(25,plain,
% 0.48/0.63 (nat_succeeds(Xy!35)),
% 0.48/0.63 inference(and_elim,[status(thm)],[24])).
% 0.48/0.63 tff(26,plain,
% 0.48/0.63 (((~![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx!36, Xy, Xz))) | ((~nat_succeeds(Xy!35)) | times_terminates(Xx!36, Xy!35, Xz!34))) <=> ((~![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx!36, Xy, Xz))) | (~nat_succeeds(Xy!35)) | times_terminates(Xx!36, Xy!35, Xz!34))),
% 0.48/0.63 inference(rewrite,[status(thm)],[])).
% 0.48/0.63 tff(27,plain,
% 0.48/0.63 ((~![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx!36, Xy, Xz))) | ((~nat_succeeds(Xy!35)) | times_terminates(Xx!36, Xy!35, Xz!34))),
% 0.48/0.63 inference(quant_inst,[status(thm)],[])).
% 0.48/0.63 tff(28,plain,
% 0.48/0.63 ((~![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx!36, Xy, Xz))) | (~nat_succeeds(Xy!35)) | times_terminates(Xx!36, Xy!35, Xz!34)),
% 0.48/0.63 inference(modus_ponens,[status(thm)],[27, 26])).
% 0.48/0.63 tff(29,plain,
% 0.48/0.63 ($false),
% 0.48/0.63 inference(unit_resolution,[status(thm)],[28, 25, 23, 14])).
% 0.48/0.63 tff(30,plain,(~![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx!36, Xy, Xz))), inference(lemma,lemma(discharge,[]))).
% 0.48/0.63 tff(31,plain,
% 0.48/0.63 (nat_succeeds(Xx!36)),
% 0.48/0.63 inference(and_elim,[status(thm)],[24])).
% 0.48/0.63 tff(32,assumption,(![Xx: $i] : ((~nat_succeeds(Xx)) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))), introduced(assumption)).
% 0.48/0.63 tff(33,plain,
% 0.48/0.63 (((~![Xx: $i] : ((~nat_succeeds(Xx)) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))) | ((~nat_succeeds(Xx!36)) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx!36, Xy, Xz)))) <=> ((~![Xx: $i] : ((~nat_succeeds(Xx)) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))) | (~nat_succeeds(Xx!36)) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx!36, Xy, Xz)))),
% 0.48/0.63 inference(rewrite,[status(thm)],[])).
% 0.48/0.63 tff(34,plain,
% 0.48/0.63 ((~![Xx: $i] : ((~nat_succeeds(Xx)) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))) | ((~nat_succeeds(Xx!36)) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx!36, Xy, Xz)))),
% 0.48/0.63 inference(quant_inst,[status(thm)],[])).
% 0.48/0.63 tff(35,plain,
% 0.48/0.63 ((~![Xx: $i] : ((~nat_succeeds(Xx)) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))) | (~nat_succeeds(Xx!36)) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx!36, Xy, Xz))),
% 0.48/0.63 inference(modus_ponens,[status(thm)],[34, 33])).
% 0.48/0.63 tff(36,plain,
% 0.48/0.63 ($false),
% 0.48/0.63 inference(unit_resolution,[status(thm)],[35, 32, 31, 30])).
% 0.48/0.63 tff(37,plain,(~![Xx: $i] : ((~nat_succeeds(Xx)) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))), inference(lemma,lemma(discharge,[]))).
% 0.48/0.63 tff(38,plain,
% 0.48/0.63 (((~((~nat_succeeds(Xy!33)) | times_terminates(Xx!30, Xy!33, Xz!32) | (~((Xx!30 = |'0'|) | (~((~(Xx!30 = s(Xx2!31))) | (~nat_succeeds(Xx2!31)) | (~![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2!31, Xy, Xz))))))))) | ![Xx: $i] : ((~nat_succeeds(Xx)) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))) <=> ((~((~nat_succeeds(Xy!33)) | times_terminates(Xx!30, Xy!33, Xz!32) | (~((Xx!30 = |'0'|) | (~((~(Xx!30 = s(Xx2!31))) | (~nat_succeeds(Xx2!31)) | (~![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2!31, Xy, Xz))))))))) | ![Xx: $i] : ((~nat_succeeds(Xx)) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz))))),
% 0.48/0.63 inference(rewrite,[status(thm)],[])).
% 0.48/0.63 tff(39,plain,
% 0.48/0.63 (((((Xx!30 = |'0'|) | ((Xx!30 = s(Xx2!31)) & nat_succeeds(Xx2!31) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2!31, Xy, Xz)))) & (~((~nat_succeeds(Xy!33)) | times_terminates(Xx!30, Xy!33, Xz!32)))) | ![Xx: $i] : ((~nat_succeeds(Xx)) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))) <=> ((~((~nat_succeeds(Xy!33)) | times_terminates(Xx!30, Xy!33, Xz!32) | (~((Xx!30 = |'0'|) | (~((~(Xx!30 = s(Xx2!31))) | (~nat_succeeds(Xx2!31)) | (~![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2!31, Xy, Xz))))))))) | ![Xx: $i] : ((~nat_succeeds(Xx)) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz))))),
% 0.48/0.63 inference(rewrite,[status(thm)],[])).
% 0.48/0.63 tff(40,plain,
% 0.48/0.63 (((Xx!30 = s(Xx2!31)) & nat_succeeds(Xx2!31) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2!31, Xy, Xz))) <=> ((Xx!30 = s(Xx2!31)) & nat_succeeds(Xx2!31) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2!31, Xy, Xz)))),
% 0.48/0.63 inference(rewrite,[status(thm)],[])).
% 0.48/0.63 tff(41,plain,
% 0.48/0.63 (((Xx!30 = |'0'|) | ((Xx!30 = s(Xx2!31)) & nat_succeeds(Xx2!31) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2!31, Xy, Xz)))) <=> ((Xx!30 = |'0'|) | ((Xx!30 = s(Xx2!31)) & nat_succeeds(Xx2!31) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2!31, Xy, Xz))))),
% 0.48/0.63 inference(monotonicity,[status(thm)],[40])).
% 0.48/0.63 tff(42,plain,
% 0.48/0.63 ((((Xx!30 = |'0'|) | ((Xx!30 = s(Xx2!31)) & nat_succeeds(Xx2!31) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2!31, Xy, Xz)))) & (~((~nat_succeeds(Xy!33)) | times_terminates(Xx!30, Xy!33, Xz!32)))) <=> (((Xx!30 = |'0'|) | ((Xx!30 = s(Xx2!31)) & nat_succeeds(Xx2!31) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2!31, Xy, Xz)))) & (~((~nat_succeeds(Xy!33)) | times_terminates(Xx!30, Xy!33, Xz!32))))),
% 0.48/0.63 inference(monotonicity,[status(thm)],[41])).
% 0.48/0.63 tff(43,plain,
% 0.48/0.63 (((((Xx!30 = |'0'|) | ((Xx!30 = s(Xx2!31)) & nat_succeeds(Xx2!31) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2!31, Xy, Xz)))) & (~((~nat_succeeds(Xy!33)) | times_terminates(Xx!30, Xy!33, Xz!32)))) | ![Xx: $i] : ((~nat_succeeds(Xx)) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))) <=> ((((Xx!30 = |'0'|) | ((Xx!30 = s(Xx2!31)) & nat_succeeds(Xx2!31) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2!31, Xy, Xz)))) & (~((~nat_succeeds(Xy!33)) | times_terminates(Xx!30, Xy!33, Xz!32)))) | ![Xx: $i] : ((~nat_succeeds(Xx)) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz))))),
% 0.48/0.63 inference(monotonicity,[status(thm)],[42])).
% 0.48/0.63 tff(44,plain,
% 0.48/0.63 (((~![Xx: $i] : ((~((Xx = |'0'|) | ?[Xx2: $i] : ((Xx = s(Xx2)) & nat_succeeds(Xx2) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2, Xy, Xz))))) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))) | ![Xx: $i] : ((~nat_succeeds(Xx)) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))) <=> ((~![Xx: $i] : ((~((Xx = |'0'|) | ?[Xx2: $i] : ((Xx = s(Xx2)) & nat_succeeds(Xx2) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2, Xy, Xz))))) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))) | ![Xx: $i] : ((~nat_succeeds(Xx)) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz))))),
% 0.48/0.63 inference(rewrite,[status(thm)],[])).
% 0.48/0.63 tff(45,plain,
% 0.48/0.63 ((![Xx: $i] : ((~((Xx = |'0'|) | ?[Xx2: $i] : ((Xx = s(Xx2)) & nat_succeeds(Xx2) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2, Xy, Xz))))) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz))) => ![Xx: $i] : ((~nat_succeeds(Xx)) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))) <=> ((~![Xx: $i] : ((~((Xx = |'0'|) | ?[Xx2: $i] : ((Xx = s(Xx2)) & nat_succeeds(Xx2) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2, Xy, Xz))))) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))) | ![Xx: $i] : ((~nat_succeeds(Xx)) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz))))),
% 0.48/0.64 inference(rewrite,[status(thm)],[])).
% 0.48/0.64 tff(46,plain,
% 0.48/0.64 (^[Xx: $i] : trans(monotonicity(quant_intro(proof_bind(^[Xy: $i, Xz: $i] : rewrite((nat_succeeds(Xy) => times_terminates(Xx, Xy, Xz)) <=> ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))), (![Xy: $i, Xz: $i] : (nat_succeeds(Xy) => times_terminates(Xx, Xy, Xz)) <=> ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))), ((nat_succeeds(Xx) => ![Xy: $i, Xz: $i] : (nat_succeeds(Xy) => times_terminates(Xx, Xy, Xz))) <=> (nat_succeeds(Xx) => ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz))))), rewrite((nat_succeeds(Xx) => ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz))) <=> ((~nat_succeeds(Xx)) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))), ((nat_succeeds(Xx) => ![Xy: $i, Xz: $i] : (nat_succeeds(Xy) => times_terminates(Xx, Xy, Xz))) <=> ((~nat_succeeds(Xx)) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))))),
% 0.48/0.64 inference(bind,[status(th)],[])).
% 0.48/0.64 tff(47,plain,
% 0.48/0.64 (![Xx: $i] : (nat_succeeds(Xx) => ![Xy: $i, Xz: $i] : (nat_succeeds(Xy) => times_terminates(Xx, Xy, Xz))) <=> ![Xx: $i] : ((~nat_succeeds(Xx)) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))),
% 0.48/0.64 inference(quant_intro,[status(thm)],[46])).
% 0.48/0.64 tff(48,plain,
% 0.48/0.64 (^[Xx: $i] : trans(monotonicity(trans(monotonicity(quant_intro(proof_bind(^[Xx2: $i] : trans(monotonicity(quant_intro(proof_bind(^[Xy: $i, Xz: $i] : rewrite((nat_succeeds(Xy) => times_terminates(Xx2, Xy, Xz)) <=> ((~nat_succeeds(Xy)) | times_terminates(Xx2, Xy, Xz)))), (![Xy: $i, Xz: $i] : (nat_succeeds(Xy) => times_terminates(Xx2, Xy, Xz)) <=> ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2, Xy, Xz)))), ((((Xx = s(Xx2)) & nat_succeeds(Xx2)) & ![Xy: $i, Xz: $i] : (nat_succeeds(Xy) => times_terminates(Xx2, Xy, Xz))) <=> (((Xx = s(Xx2)) & nat_succeeds(Xx2)) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2, Xy, Xz))))), rewrite((((Xx = s(Xx2)) & nat_succeeds(Xx2)) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2, Xy, Xz))) <=> ((Xx = s(Xx2)) & nat_succeeds(Xx2) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2, Xy, Xz)))), ((((Xx = s(Xx2)) & nat_succeeds(Xx2)) & ![Xy: $i, Xz: $i] : (nat_succeeds(Xy) => times_terminates(Xx2, Xy, Xz))) <=> ((Xx = s(Xx2)) & nat_succeeds(Xx2) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2, Xy, Xz)))))), (?[Xx2: $i] : (((Xx = s(Xx2)) & nat_succeeds(Xx2)) & ![Xy: $i, Xz: $i] : (nat_succeeds(Xy) => times_terminates(Xx2, Xy, Xz))) <=> ?[Xx2: $i] : ((Xx = s(Xx2)) & nat_succeeds(Xx2) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2, Xy, Xz))))), ((?[Xx2: $i] : (((Xx = s(Xx2)) & nat_succeeds(Xx2)) & ![Xy: $i, Xz: $i] : (nat_succeeds(Xy) => times_terminates(Xx2, Xy, Xz))) | (Xx = |'0'|)) <=> (?[Xx2: $i] : ((Xx = s(Xx2)) & nat_succeeds(Xx2) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2, Xy, Xz))) | (Xx = |'0'|)))), rewrite((?[Xx2: $i] : ((Xx = s(Xx2)) & nat_succeeds(Xx2) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2, Xy, Xz))) | (Xx = |'0'|)) <=> ((Xx = |'0'|) | ?[Xx2: $i] : ((Xx = s(Xx2)) & nat_succeeds(Xx2) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2, Xy, Xz))))), ((?[Xx2: $i] : (((Xx = s(Xx2)) & nat_succeeds(Xx2)) & ![Xy: $i, Xz: $i] : (nat_succeeds(Xy) => times_terminates(Xx2, Xy, Xz))) | (Xx = |'0'|)) <=> ((Xx = |'0'|) | ?[Xx2: $i] : ((Xx = s(Xx2)) & nat_succeeds(Xx2) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2, Xy, Xz)))))), quant_intro(proof_bind(^[Xy: $i, Xz: $i] : rewrite((nat_succeeds(Xy) => times_terminates(Xx, Xy, Xz)) <=> ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))), (![Xy: $i, Xz: $i] : (nat_succeeds(Xy) => times_terminates(Xx, Xy, Xz)) <=> ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))), (((?[Xx2: $i] : (((Xx = s(Xx2)) & nat_succeeds(Xx2)) & ![Xy: $i, Xz: $i] : (nat_succeeds(Xy) => times_terminates(Xx2, Xy, Xz))) | (Xx = |'0'|)) => ![Xy: $i, Xz: $i] : (nat_succeeds(Xy) => times_terminates(Xx, Xy, Xz))) <=> (((Xx = |'0'|) | ?[Xx2: $i] : ((Xx = s(Xx2)) & nat_succeeds(Xx2) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2, Xy, Xz)))) => ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz))))), rewrite((((Xx = |'0'|) | ?[Xx2: $i] : ((Xx = s(Xx2)) & nat_succeeds(Xx2) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2, Xy, Xz)))) => ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz))) <=> ((~((Xx = |'0'|) | ?[Xx2: $i] : ((Xx = s(Xx2)) & nat_succeeds(Xx2) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2, Xy, Xz))))) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))), (((?[Xx2: $i] : (((Xx = s(Xx2)) & nat_succeeds(Xx2)) & ![Xy: $i, Xz: $i] : (nat_succeeds(Xy) => times_terminates(Xx2, Xy, Xz))) | (Xx = |'0'|)) => ![Xy: $i, Xz: $i] : (nat_succeeds(Xy) => times_terminates(Xx, Xy, Xz))) <=> ((~((Xx = |'0'|) | ?[Xx2: $i] : ((Xx = s(Xx2)) & nat_succeeds(Xx2) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2, Xy, Xz))))) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))))),
% 0.48/0.64 inference(bind,[status(th)],[])).
% 0.48/0.64 tff(49,plain,
% 0.48/0.64 (![Xx: $i] : ((?[Xx2: $i] : (((Xx = s(Xx2)) & nat_succeeds(Xx2)) & ![Xy: $i, Xz: $i] : (nat_succeeds(Xy) => times_terminates(Xx2, Xy, Xz))) | (Xx = |'0'|)) => ![Xy: $i, Xz: $i] : (nat_succeeds(Xy) => times_terminates(Xx, Xy, Xz))) <=> ![Xx: $i] : ((~((Xx = |'0'|) | ?[Xx2: $i] : ((Xx = s(Xx2)) & nat_succeeds(Xx2) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2, Xy, Xz))))) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))),
% 0.48/0.64 inference(quant_intro,[status(thm)],[48])).
% 0.48/0.64 tff(50,plain,
% 0.48/0.64 ((![Xx: $i] : ((?[Xx2: $i] : (((Xx = s(Xx2)) & nat_succeeds(Xx2)) & ![Xy: $i, Xz: $i] : (nat_succeeds(Xy) => times_terminates(Xx2, Xy, Xz))) | (Xx = |'0'|)) => ![Xy: $i, Xz: $i] : (nat_succeeds(Xy) => times_terminates(Xx, Xy, Xz))) => ![Xx: $i] : (nat_succeeds(Xx) => ![Xy: $i, Xz: $i] : (nat_succeeds(Xy) => times_terminates(Xx, Xy, Xz)))) <=> (![Xx: $i] : ((~((Xx = |'0'|) | ?[Xx2: $i] : ((Xx = s(Xx2)) & nat_succeeds(Xx2) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2, Xy, Xz))))) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz))) => ![Xx: $i] : ((~nat_succeeds(Xx)) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz))))),
% 0.48/0.64 inference(monotonicity,[status(thm)],[49, 47])).
% 0.48/0.64 tff(51,plain,
% 0.48/0.64 ((![Xx: $i] : ((?[Xx2: $i] : (((Xx = s(Xx2)) & nat_succeeds(Xx2)) & ![Xy: $i, Xz: $i] : (nat_succeeds(Xy) => times_terminates(Xx2, Xy, Xz))) | (Xx = |'0'|)) => ![Xy: $i, Xz: $i] : (nat_succeeds(Xy) => times_terminates(Xx, Xy, Xz))) => ![Xx: $i] : (nat_succeeds(Xx) => ![Xy: $i, Xz: $i] : (nat_succeeds(Xy) => times_terminates(Xx, Xy, Xz)))) <=> ((~![Xx: $i] : ((~((Xx = |'0'|) | ?[Xx2: $i] : ((Xx = s(Xx2)) & nat_succeeds(Xx2) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2, Xy, Xz))))) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))) | ![Xx: $i] : ((~nat_succeeds(Xx)) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz))))),
% 0.48/0.64 inference(transitivity,[status(thm)],[50, 45])).
% 0.48/0.64 tff(52,axiom,(![Xx: $i] : ((?[Xx2: $i] : (((Xx = s(Xx2)) & nat_succeeds(Xx2)) & ![Xy: $i, Xz: $i] : (nat_succeeds(Xy) => times_terminates(Xx2, Xy, Xz))) | (Xx = |'0'|)) => ![Xy: $i, Xz: $i] : (nat_succeeds(Xy) => times_terminates(Xx, Xy, Xz))) => ![Xx: $i] : (nat_succeeds(Xx) => ![Xy: $i, Xz: $i] : (nat_succeeds(Xy) => times_terminates(Xx, Xy, Xz)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','induction')).
% 0.48/0.64 tff(53,plain,
% 0.48/0.64 ((~![Xx: $i] : ((~((Xx = |'0'|) | ?[Xx2: $i] : ((Xx = s(Xx2)) & nat_succeeds(Xx2) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2, Xy, Xz))))) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))) | ![Xx: $i] : ((~nat_succeeds(Xx)) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))),
% 0.48/0.64 inference(modus_ponens,[status(thm)],[52, 51])).
% 0.48/0.64 tff(54,plain,
% 0.48/0.64 ((~![Xx: $i] : ((~((Xx = |'0'|) | ?[Xx2: $i] : ((Xx = s(Xx2)) & nat_succeeds(Xx2) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2, Xy, Xz))))) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))) | ![Xx: $i] : ((~nat_succeeds(Xx)) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))),
% 0.48/0.64 inference(modus_ponens,[status(thm)],[53, 44])).
% 0.48/0.64 tff(55,plain,
% 0.48/0.64 ((~![Xx: $i] : ((~((Xx = |'0'|) | ?[Xx2: $i] : ((Xx = s(Xx2)) & nat_succeeds(Xx2) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2, Xy, Xz))))) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))) | ![Xx: $i] : ((~nat_succeeds(Xx)) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))),
% 0.48/0.64 inference(modus_ponens,[status(thm)],[54, 44])).
% 0.48/0.64 tff(56,plain,
% 0.48/0.64 ((~![Xx: $i] : ((~((Xx = |'0'|) | ?[Xx2: $i] : ((Xx = s(Xx2)) & nat_succeeds(Xx2) & ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx2, Xy, Xz))))) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))) | ![Xx: $i] : ((~nat_succeeds(Xx)) | ![Xy: $i, Xz: $i] : ((~nat_succeeds(Xy)) | times_terminates(Xx, Xy, Xz)))),
% 0.48/0.64 inference(modus_ponens,[status(thm)],[55, 44])).
% 0.48/0.64 Proof display could not be completed: monotonicity rule is not handled
% 0.48/0.64 % E exiting
%------------------------------------------------------------------------------