TSTP Solution File: SCT096-1 by Prover9---1109a
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- Process Solution
%------------------------------------------------------------------------------
% File : Prover9---1109a
% Problem : SCT096-1 : TPTP v8.1.0. Released v4.1.0.
% Transfm : none
% Format : tptp:raw
% Command : tptp2X_and_run_prover9 %d %s
% Computer : n020.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 : 600s
% DateTime : Mon Jul 18 22:03:52 EDT 2022
% Result : Unsatisfiable 5.73s 6.00s
% Output : Refutation 5.73s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : SCT096-1 : TPTP v8.1.0. Released v4.1.0.
% 0.07/0.14 % Command : tptp2X_and_run_prover9 %d %s
% 0.15/0.35 % Computer : n020.cluster.edu
% 0.15/0.35 % Model : x86_64 x86_64
% 0.15/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.35 % Memory : 8042.1875MB
% 0.15/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.35 % CPULimit : 300
% 0.15/0.35 % WCLimit : 600
% 0.15/0.35 % DateTime : Sat Jul 2 07:34:15 EDT 2022
% 0.15/0.35 % CPUTime :
% 1.18/1.68 ============================== Prover9 ===============================
% 1.18/1.68 Prover9 (32) version 2009-11A, November 2009.
% 1.18/1.68 Process 11440 was started by sandbox on n020.cluster.edu,
% 1.18/1.68 Sat Jul 2 07:34:16 2022
% 1.18/1.68 The command was "/export/starexec/sandbox/solver/bin/prover9 -t 300 -f /tmp/Prover9_11287_n020.cluster.edu".
% 1.18/1.68 ============================== end of head ===========================
% 1.18/1.68
% 1.18/1.68 ============================== INPUT =================================
% 1.18/1.68
% 1.18/1.68 % Reading from file /tmp/Prover9_11287_n020.cluster.edu
% 1.18/1.68
% 1.18/1.68 set(prolog_style_variables).
% 1.18/1.68 set(auto2).
% 1.18/1.68 % set(auto2) -> set(auto).
% 1.18/1.68 % set(auto) -> set(auto_inference).
% 1.18/1.68 % set(auto) -> set(auto_setup).
% 1.18/1.68 % set(auto_setup) -> set(predicate_elim).
% 1.18/1.68 % set(auto_setup) -> assign(eq_defs, unfold).
% 1.18/1.68 % set(auto) -> set(auto_limits).
% 1.18/1.68 % set(auto_limits) -> assign(max_weight, "100.000").
% 1.18/1.68 % set(auto_limits) -> assign(sos_limit, 20000).
% 1.18/1.68 % set(auto) -> set(auto_denials).
% 1.18/1.68 % set(auto) -> set(auto_process).
% 1.18/1.68 % set(auto2) -> assign(new_constants, 1).
% 1.18/1.68 % set(auto2) -> assign(fold_denial_max, 3).
% 1.18/1.68 % set(auto2) -> assign(max_weight, "200.000").
% 1.18/1.68 % set(auto2) -> assign(max_hours, 1).
% 1.18/1.68 % assign(max_hours, 1) -> assign(max_seconds, 3600).
% 1.18/1.68 % set(auto2) -> assign(max_seconds, 0).
% 1.18/1.68 % set(auto2) -> assign(max_minutes, 5).
% 1.18/1.68 % assign(max_minutes, 5) -> assign(max_seconds, 300).
% 1.18/1.68 % set(auto2) -> set(sort_initial_sos).
% 1.18/1.68 % set(auto2) -> assign(sos_limit, -1).
% 1.18/1.68 % set(auto2) -> assign(lrs_ticks, 3000).
% 1.18/1.68 % set(auto2) -> assign(max_megs, 400).
% 1.18/1.68 % set(auto2) -> assign(stats, some).
% 1.18/1.68 % set(auto2) -> clear(echo_input).
% 1.18/1.68 % set(auto2) -> set(quiet).
% 1.18/1.68 % set(auto2) -> clear(print_initial_clauses).
% 1.18/1.68 % set(auto2) -> clear(print_given).
% 1.18/1.68 assign(lrs_ticks,-1).
% 1.18/1.68 assign(sos_limit,10000).
% 1.18/1.68 assign(order,kbo).
% 1.18/1.68 set(lex_order_vars).
% 1.18/1.68 clear(print_given).
% 1.18/1.68
% 1.18/1.68 % formulas(sos). % not echoed (820 formulas)
% 1.18/1.68
% 1.18/1.68 ============================== end of input ==========================
% 1.18/1.68
% 1.18/1.68 % From the command line: assign(max_seconds, 300).
% 1.18/1.68
% 1.18/1.68 ============================== PROCESS NON-CLAUSAL FORMULAS ==========
% 1.18/1.68
% 1.18/1.68 % Formulas that are not ordinary clauses:
% 1.18/1.68
% 1.18/1.68 ============================== end of process non-clausal formulas ===
% 1.18/1.68
% 1.18/1.68 ============================== PROCESS INITIAL CLAUSES ===============
% 1.18/1.68
% 1.18/1.68 ============================== PREDICATE ELIMINATION =================
% 1.18/1.68 1 class_Ring__and__Field_Ozero__neq__one(tc_nat) # label(clsarity_nat__Ring__and__Field_Ozero__neq__one) # label(axiom). [assumption].
% 1.18/1.68 2 -class_Ring__and__Field_Ozero__neq__one(A) | -class_Ring__and__Field_Ono__zero__divisors(A) | -class_Ring__and__Field_Omult__zero(A) | -class_Power_Opower(A) | c_Power_Opower__class_Opower(B,C,A) != c_HOL_Ozero__class_Ozero(A) | B = c_HOL_Ozero__class_Ozero(A) # label(cls_power__eq__0__iff_0) # label(axiom). [assumption].
% 1.18/1.68 3 -class_Ring__and__Field_Ozero__neq__one(A) | -class_Ring__and__Field_Ono__zero__divisors(A) | -class_Ring__and__Field_Omult__zero(A) | -class_Power_Opower(A) | c_Power_Opower__class_Opower(c_HOL_Ozero__class_Ozero(A),B,A) = c_HOL_Ozero__class_Ozero(A) | B = c_HOL_Ozero__class_Ozero(tc_nat) # label(cls_power__eq__0__iff_2) # label(axiom). [assumption].
% 1.18/1.68 4 -class_Ring__and__Field_Ozero__neq__one(A) | -class_Ring__and__Field_Ono__zero__divisors(A) | -class_Ring__and__Field_Omult__zero(A) | -class_Power_Opower(A) | c_Power_Opower__class_Opower(B,c_HOL_Ozero__class_Ozero(tc_nat),A) != c_HOL_Ozero__class_Ozero(A) # label(cls_power__eq__0__iff_1) # label(axiom). [assumption].
% 1.18/1.68 5 -class_Ring__and__Field_Ozero__neq__one(A) | c_HOL_Ozero__class_Ozero(A) != c_HOL_Oone__class_Oone(A) # label(cls_zero__neq__one_0) # label(axiom). [assumption].
% 1.18/1.68 6 -class_Ring__and__Field_Ozero__neq__one(A) | c_HOL_Oone__class_Oone(A) != c_HOL_Ozero__class_Ozero(A) # label(cls_one__neq__zero_0) # label(axiom). [assumption].
% 1.18/1.68 Derived: -class_Ring__and__Field_Ono__zero__divisors(tc_nat) | -class_Ring__and__Field_Omult__zero(tc_nat) | -class_Power_Opower(tc_nat) | c_Power_Opower__class_Opower(A,B,tc_nat) != c_HOL_Ozero__class_Ozero(tc_nat) | A = c_HOL_Ozero__class_Ozero(tc_nat). [resolve(1,a,2,a)].
% 1.18/1.71 Derived: -class_Ring__and__Field_Ono__zero__divisors(tc_nat) | -class_Ring__and__Field_Omult__zero(tc_nat) | -class_Power_Opower(tc_nat) | c_Power_Opower__class_Opower(c_HOL_Ozero__class_Ozero(tc_nat),A,tc_nat) = c_HOL_Ozero__class_Ozero(tc_nat) | A = c_HOL_Ozero__class_Ozero(tc_nat). [resolve(1,a,3,a)].
% 1.18/1.71 Derived: -class_Ring__and__Field_Ono__zero__divisors(tc_nat) | -class_Ring__and__Field_Omult__zero(tc_nat) | -class_Power_Opower(tc_nat) | c_Power_Opower__class_Opower(A,c_HOL_Ozero__class_Ozero(tc_nat),tc_nat) != c_HOL_Ozero__class_Ozero(tc_nat). [resolve(1,a,4,a)].
% 1.18/1.71 Derived: c_HOL_Ozero__class_Ozero(tc_nat) != c_HOL_Oone__class_Oone(tc_nat). [resolve(1,a,5,a)].
% 1.18/1.71 Derived: c_HOL_Oone__class_Oone(tc_nat) != c_HOL_Ozero__class_Ozero(tc_nat). [resolve(1,a,6,a)].
% 1.18/1.71 7 class_Ring__and__Field_Ocomm__semiring__1(tc_nat) # label(clsarity_nat__Ring__and__Field_Ocomm__semiring__1) # label(axiom). [assumption].
% 1.18/1.71 8 -class_Ring__and__Field_Ocomm__semiring__1(A) | c_Power_Opower__class_Opower(B,c_HOL_Ozero__class_Ozero(tc_nat),A) = c_HOL_Oone__class_Oone(A) # label(cls_class__semiring_Opwr__0_0) # label(axiom). [assumption].
% 1.18/1.71 9 -class_Ring__and__Field_Ocomm__semiring__1(A) | c_Power_Opower__class_Opower(B,c_HOL_Oone__class_Oone(tc_nat),A) = B # label(cls_class__semiring_Osemiring__rules_I33_J_0) # label(axiom). [assumption].
% 1.18/1.71 10 -class_Ring__and__Field_Ono__zero__divisors(A) | -class_Ring__and__Field_Ocomm__semiring__1(A) | c_Finite__Set_Osetprod(B,C,D,A) != c_HOL_Ozero__class_Ozero(A) | -c_Finite__Set_Ofinite(C,D) | hAPP(B,c_ATP__Linkup_Osko__Finite__Set__Xsetprod__zero__iff__1__1(C,B,D,A)) = c_HOL_Ozero__class_Ozero(A) # label(cls_setprod__zero__iff_1) # label(axiom). [assumption].
% 1.18/1.71 11 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C)),D) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),C),D)) # label(cls_class__semiring_Osemiring__rules_I21_J_0) # label(axiom). [assumption].
% 1.18/1.71 12 -class_Ring__and__Field_Ono__zero__divisors(A) | -class_Ring__and__Field_Ocomm__semiring__1(A) | c_Finite__Set_Osetprod(B,C,D,A) != c_HOL_Ozero__class_Ozero(A) | -c_Finite__Set_Ofinite(C,D) | hBOOL(hAPP(hAPP(c_in(D),c_ATP__Linkup_Osko__Finite__Set__Xsetprod__zero__iff__1__1(C,B,D,A)),C)) # label(cls_setprod__zero__iff_0) # label(axiom). [assumption].
% 1.18/1.71 13 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),D),E)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),D)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),C),E)) # label(cls_class__semiring_Osemiring__rules_I20_J_0) # label(axiom). [assumption].
% 1.18/1.71 14 -class_Ring__and__Field_Ono__zero__divisors(A) | -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(B,C) != c_HOL_Ozero__class_Ozero(A) | -hBOOL(hAPP(hAPP(c_in(D),C),E)) | -c_Finite__Set_Ofinite(E,D) | c_Finite__Set_Osetprod(B,E,D,A) = c_HOL_Ozero__class_Ozero(A) # label(cls_setprod__zero__iff_2) # label(axiom). [assumption].
% 1.18/1.71 15 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(B,C) != c_HOL_Ozero__class_Ozero(A) | -hBOOL(hAPP(hAPP(c_in(D),C),E)) | -c_Finite__Set_Ofinite(E,D) | c_Finite__Set_Osetprod(B,E,D,A) = c_HOL_Ozero__class_Ozero(A) # label(cls_setprod__zero_0) # label(axiom). [assumption].
% 1.18/1.71 16 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),c_HOL_Ozero__class_Ozero(A)),B) = B # label(cls_class__semiring_Oadd__0_0) # label(axiom). [assumption].
% 1.18/1.71 17 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),c_HOL_Ozero__class_Ozero(A)) = B # label(cls_class__semiring_Osemiring__rules_I6_J_0) # label(axiom). [assumption].
% 1.18/1.71 18 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),c_HOL_Ozero__class_Ozero(A)),B) = B # label(cls_class__semiring_Osemiring__rules_I5_J_0) # label(axiom). [assumption].
% 1.18/1.71 19 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),C),D)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),C),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),D)) # label(cls_class__semiring_Osemiring__rules_I22_J_0) # label(axiom). [assumption].
% 1.18/1.71 20 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C)),D) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),D)),C) # label(cls_class__semiring_Osemiring__rules_I23_J_0) # label(axiom). [assumption].
% 1.18/1.71 21 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),C),D)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C)),D) # label(cls_class__semiring_Osemiring__rules_I25_J_0) # label(axiom). [assumption].
% 1.18/1.71 22 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),C),D)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C)),D) # label(cls_class__semiring_Oadd__a_0) # label(axiom). [assumption].
% 1.18/1.71 23 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),C),B) # label(cls_class__semiring_Osemiring__rules_I24_J_0) # label(axiom). [assumption].
% 1.18/1.71 24 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),C),B) # label(cls_class__semiring_Oadd__c_0) # label(axiom). [assumption].
% 1.18/1.71 Derived: c_Power_Opower__class_Opower(A,c_HOL_Ozero__class_Ozero(tc_nat),tc_nat) = c_HOL_Oone__class_Oone(tc_nat). [resolve(7,a,8,a)].
% 1.18/1.71 Derived: c_Power_Opower__class_Opower(A,c_HOL_Oone__class_Oone(tc_nat),tc_nat) = A. [resolve(7,a,9,a)].
% 1.18/1.71 Derived: -class_Ring__and__Field_Ono__zero__divisors(tc_nat) | c_Finite__Set_Osetprod(A,B,C,tc_nat) != c_HOL_Ozero__class_Ozero(tc_nat) | -c_Finite__Set_Ofinite(B,C) | hAPP(A,c_ATP__Linkup_Osko__Finite__Set__Xsetprod__zero__iff__1__1(B,A,C,tc_nat)) = c_HOL_Ozero__class_Ozero(tc_nat). [resolve(7,a,10,b)].
% 1.18/1.71 Derived: hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B)),C) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),B),C)). [resolve(7,a,11,a)].
% 1.18/1.71 Derived: -class_Ring__and__Field_Ono__zero__divisors(tc_nat) | c_Finite__Set_Osetprod(A,B,C,tc_nat) != c_HOL_Ozero__class_Ozero(tc_nat) | -c_Finite__Set_Ofinite(B,C) | hBOOL(hAPP(hAPP(c_in(C),c_ATP__Linkup_Osko__Finite__Set__Xsetprod__zero__iff__1__1(B,A,C,tc_nat)),B)). [resolve(7,a,12,b)].
% 1.18/1.71 Derived: hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),C),D)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),C)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),B),D)). [resolve(7,a,13,a)].
% 1.18/1.71 Derived: -class_Ring__and__Field_Ono__zero__divisors(tc_nat) | hAPP(A,B) != c_HOL_Ozero__class_Ozero(tc_nat) | -hBOOL(hAPP(hAPP(c_in(C),B),D)) | -c_Finite__Set_Ofinite(D,C) | c_Finite__Set_Osetprod(A,D,C,tc_nat) = c_HOL_Ozero__class_Ozero(tc_nat). [resolve(7,a,14,b)].
% 1.18/1.71 Derived: hAPP(A,B) != c_HOL_Ozero__class_Ozero(tc_nat) | -hBOOL(hAPP(hAPP(c_in(C),B),D)) | -c_Finite__Set_Ofinite(D,C) | c_Finite__Set_Osetprod(A,D,C,tc_nat) = c_HOL_Ozero__class_Ozero(tc_nat). [resolve(7,a,15,a)].
% 1.18/1.71 Derived: hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),A) = A. [resolve(7,a,16,a)].
% 1.18/1.71 Derived: hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),c_HOL_Ozero__class_Ozero(tc_nat)) = A. [resolve(7,a,17,a)].
% 1.18/1.71 Derived: hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),B),C)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),B),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),C)). [resolve(7,a,19,a)].
% 1.18/1.71 Derived: hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B)),C) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),C)),B). [resolve(7,a,20,a)].
% 1.54/1.79 Derived: hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),B),C)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B)),C). [resolve(7,a,21,a)].
% 1.54/1.79 Derived: hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),B),A). [resolve(7,a,23,a)].
% 1.54/1.79 25 hBOOL(hAPP(hAPP(c_in(tc_prod(A,A)),c_Pair(B,C,A,A)),D)) | hBOOL(hAPP(hAPP(c_in(tc_prod(A,A)),c_Pair(C,B,A,A)),D)) | C = B | -hBOOL(hAPP(hAPP(c_in(A),B),E)) | -hBOOL(hAPP(hAPP(c_in(A),C),E)) | -c_Relation_Ototal__on(E,D,A) # label(cls_total__on__def_0) # label(axiom). [assumption].
% 1.54/1.79 26 c_Relation_Ototal__on(c_Orderings_Obot__class_Obot(tc_fun(A,tc_bool)),B,A) # label(cls_total__on__empty_0) # label(axiom). [assumption].
% 1.54/1.79 Derived: hBOOL(hAPP(hAPP(c_in(tc_prod(A,A)),c_Pair(B,C,A,A)),D)) | hBOOL(hAPP(hAPP(c_in(tc_prod(A,A)),c_Pair(C,B,A,A)),D)) | C = B | -hBOOL(hAPP(hAPP(c_in(A),B),c_Orderings_Obot__class_Obot(tc_fun(A,tc_bool)))) | -hBOOL(hAPP(hAPP(c_in(A),C),c_Orderings_Obot__class_Obot(tc_fun(A,tc_bool)))). [resolve(25,f,26,a)].
% 1.54/1.79 27 class_Ring__and__Field_Osemiring__0(tc_nat) # label(clsarity_nat__Ring__and__Field_Osemiring__0) # label(axiom). [assumption].
% 1.54/1.79 28 -class_Ring__and__Field_Osemiring__0(A) | -class_Power_Opower(A) | c_Power_Opower__class_Opower(c_HOL_Ozero__class_Ozero(A),B,A) = c_HOL_Ozero__class_Ozero(A) | B = c_HOL_Ozero__class_Ozero(tc_nat) # label(cls_power__0__left_1) # label(axiom). [assumption].
% 1.54/1.79 29 -class_Ring__and__Field_Osemiring__0(A) | -class_Power_Opower(A) | c_Power_Opower__class_Opower(c_HOL_Ozero__class_Ozero(A),c_HOL_Ozero__class_Ozero(tc_nat),A) = c_HOL_Oone__class_Oone(A) # label(cls_power__0__left_0) # label(axiom). [assumption].
% 1.54/1.79 Derived: -class_Power_Opower(tc_nat) | c_Power_Opower__class_Opower(c_HOL_Ozero__class_Ozero(tc_nat),A,tc_nat) = c_HOL_Ozero__class_Ozero(tc_nat) | A = c_HOL_Ozero__class_Ozero(tc_nat). [resolve(27,a,28,a)].
% 1.54/1.79 30 class_Lattices_Olower__semilattice(tc_fun(A,B)) | -class_Lattices_Olattice(B) # label(clsarity_fun__Lattices_Olower__semilattice) # label(axiom). [assumption].
% 1.54/1.79 31 -class_Lattices_Olower__semilattice(A) | c_lessequals(B,c_Lattices_Olower__semilattice__class_Oinf(C,D,A),A) | -c_lessequals(B,D,A) | -c_lessequals(B,C,A) # label(cls_inf__greatest_0) # label(axiom). [assumption].
% 1.54/1.79 32 -class_Lattices_Olower__semilattice(A) | c_lessequals(B,c_Lattices_Olower__semilattice__class_Oinf(C,D,A),A) | -c_lessequals(B,D,A) | -c_lessequals(B,C,A) # label(cls_le__inf__iff_2) # label(axiom). [assumption].
% 1.54/1.79 33 -class_Lattices_Olower__semilattice(A) | c_lessequals(B,c_Lattices_Olower__semilattice__class_Oinf(C,D,A),A) | -c_lessequals(B,D,A) | -c_lessequals(B,C,A) # label(cls_le__infI_0) # label(axiom). [assumption].
% 1.54/1.79 34 -class_Lattices_Olower__semilattice(A) | c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(B,C,A),C,A) # label(cls_inf__le2_0) # label(axiom). [assumption].
% 1.54/1.79 35 -class_Lattices_Olower__semilattice(A) | c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(B,C,A),B,A) # label(cls_inf__le1_0) # label(axiom). [assumption].
% 1.54/1.79 36 -class_Lattices_Olower__semilattice(A) | c_Lattices_Olower__semilattice__class_Oinf(B,c_Lattices_Olower__semilattice__class_Oinf(B,C,A),A) = c_Lattices_Olower__semilattice__class_Oinf(B,C,A) # label(cls_inf__left__idem_0) # label(axiom). [assumption].
% 1.54/1.79 37 -class_Lattices_Olower__semilattice(A) | c_Lattices_Olower__semilattice__class_Oinf(B,B,A) = B # label(cls_inf__idem_0) # label(axiom). [assumption].
% 1.54/1.79 38 -class_Lattices_Olower__semilattice(A) | c_Lattices_Olower__semilattice__class_Oinf(c_Lattices_Olower__semilattice__class_Oinf(B,C,A),D,A) = c_Lattices_Olower__semilattice__class_Oinf(B,c_Lattices_Olower__semilattice__class_Oinf(C,D,A),A) # label(cls_inf__assoc_0) # label(axiom). [assumption].
% 1.54/1.79 39 -class_Lattices_Olower__semilattice(A) | c_Lattices_Olower__semilattice__class_Oinf(B,c_Lattices_Olower__semilattice__class_Oinf(C,D,A),A) = c_Lattices_Olower__semilattice__class_Oinf(C,c_Lattices_Olower__semilattice__class_Oinf(B,D,A),A) # label(cls_inf__left__commute_0) # label(axiom). [assumption].
% 1.54/1.79 40 -class_Lattices_Olower__semilattice(A) | c_Lattices_Olower__semilattice__class_Oinf(B,C,A) = B | -c_lessequals(B,C,A) # label(cls_le__iff__inf_0) # label(axiom). [assumption].
% 1.54/1.79 41 -class_Lattices_Olower__semilattice(A) | c_Lattices_Olower__semilattice__class_Oinf(B,C,A) != B | c_lessequals(B,C,A) # label(cls_le__iff__inf_1) # label(axiom). [assumption].
% 1.54/1.79 42 -class_Lattices_Olower__semilattice(A) | c_Lattices_Olower__semilattice__class_Oinf(B,C,A) = C | -c_lessequals(C,B,A) # label(cls_inf__absorb2_0) # label(axiom). [assumption].
% 1.54/1.79 43 -class_Lattices_Olower__semilattice(A) | c_Lattices_Olower__semilattice__class_Oinf(B,C,A) = c_Lattices_Olower__semilattice__class_Oinf(C,B,A) # label(cls_inf__commute_0) # label(axiom). [assumption].
% 1.54/1.79 44 -class_Lattices_Olower__semilattice(A) | c_lessequals(B,C,A) | -c_lessequals(B,c_Lattices_Olower__semilattice__class_Oinf(C,D,A),A) # label(cls_le__infE_0) # label(axiom). [assumption].
% 1.54/1.79 45 -class_Lattices_Olower__semilattice(A) | c_lessequals(B,C,A) | -c_lessequals(B,c_Lattices_Olower__semilattice__class_Oinf(D,C,A),A) # label(cls_le__infE_1) # label(axiom). [assumption].
% 1.54/1.79 46 -class_Lattices_Olower__semilattice(A) | c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(B,C,A),D,A) | -c_lessequals(B,D,A) # label(cls_le__infI1_0) # label(axiom). [assumption].
% 1.54/1.79 47 -class_Lattices_Olower__semilattice(A) | c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(B,C,A),D,A) | -c_lessequals(C,D,A) # label(cls_le__infI2_0) # label(axiom). [assumption].
% 1.54/1.79 48 -class_Lattices_Olower__semilattice(A) | c_lessequals(B,C,A) | -c_lessequals(B,c_Lattices_Olower__semilattice__class_Oinf(C,D,A),A) # label(cls_le__inf__iff_0) # label(axiom). [assumption].
% 1.54/1.79 49 -class_Lattices_Olower__semilattice(A) | c_lessequals(B,C,A) | -c_lessequals(B,c_Lattices_Olower__semilattice__class_Oinf(D,C,A),A) # label(cls_le__inf__iff_1) # label(axiom). [assumption].
% 1.54/1.79 50 -class_Lattices_Olower__semilattice(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_Lattices_Olower__semilattice__class_Oinf(B,C,A)),D)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),D)) # label(cls_less__infI1_0) # label(axiom). [assumption].
% 1.54/1.79 51 -class_Lattices_Olower__semilattice(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_Lattices_Olower__semilattice__class_Oinf(B,C,A)),D)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),C),D)) # label(cls_less__infI2_0) # label(axiom). [assumption].
% 1.54/1.79 Derived: -class_Lattices_Olattice(A) | c_lessequals(B,c_Lattices_Olower__semilattice__class_Oinf(C,D,tc_fun(E,A)),tc_fun(E,A)) | -c_lessequals(B,D,tc_fun(E,A)) | -c_lessequals(B,C,tc_fun(E,A)). [resolve(30,a,31,a)].
% 1.54/1.79 Derived: -class_Lattices_Olattice(A) | c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_fun(D,A)),C,tc_fun(D,A)). [resolve(30,a,34,a)].
% 1.54/1.79 Derived: -class_Lattices_Olattice(A) | c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_fun(D,A)),B,tc_fun(D,A)). [resolve(30,a,35,a)].
% 1.54/1.79 Derived: -class_Lattices_Olattice(A) | c_Lattices_Olower__semilattice__class_Oinf(B,c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_fun(D,A)),tc_fun(D,A)) = c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_fun(D,A)). [resolve(30,a,36,a)].
% 1.54/1.79 Derived: -class_Lattices_Olattice(A) | c_Lattices_Olower__semilattice__class_Oinf(B,B,tc_fun(C,A)) = B. [resolve(30,a,37,a)].
% 1.54/1.79 Derived: -class_Lattices_Olattice(A) | c_Lattices_Olower__semilattice__class_Oinf(c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_fun(D,A)),E,tc_fun(D,A)) = c_Lattices_Olower__semilattice__class_Oinf(B,c_Lattices_Olower__semilattice__class_Oinf(C,E,tc_fun(D,A)),tc_fun(D,A)). [resolve(30,a,38,a)].
% 1.54/1.79 Derived: -class_Lattices_Olattice(A) | c_Lattices_Olower__semilattice__class_Oinf(B,c_Lattices_Olower__semilattice__class_Oinf(C,D,tc_fun(E,A)),tc_fun(E,A)) = c_Lattices_Olower__semilattice__class_Oinf(C,c_Lattices_Olower__semilattice__class_Oinf(B,D,tc_fun(E,A)),tc_fun(E,A)). [resolve(30,a,39,a)].
% 1.54/1.79 Derived: -class_Lattices_Olattice(A) | c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_fun(D,A)) = B | -c_lessequals(B,C,tc_fun(D,A)). [resolve(30,a,40,a)].
% 1.54/1.79 Derived: -class_Lattices_Olattice(A) | c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_fun(D,A)) != B | c_lessequals(B,C,tc_fun(D,A)). [resolve(30,a,41,a)].
% 1.54/1.80 Derived: -class_Lattices_Olattice(A) | c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_fun(D,A)) = C | -c_lessequals(C,B,tc_fun(D,A)). [resolve(30,a,42,a)].
% 1.54/1.80 Derived: -class_Lattices_Olattice(A) | c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_fun(D,A)) = c_Lattices_Olower__semilattice__class_Oinf(C,B,tc_fun(D,A)). [resolve(30,a,43,a)].
% 1.54/1.80 Derived: -class_Lattices_Olattice(A) | c_lessequals(B,C,tc_fun(D,A)) | -c_lessequals(B,c_Lattices_Olower__semilattice__class_Oinf(C,E,tc_fun(D,A)),tc_fun(D,A)). [resolve(30,a,44,a)].
% 1.54/1.80 Derived: -class_Lattices_Olattice(A) | c_lessequals(B,C,tc_fun(D,A)) | -c_lessequals(B,c_Lattices_Olower__semilattice__class_Oinf(E,C,tc_fun(D,A)),tc_fun(D,A)). [resolve(30,a,45,a)].
% 1.54/1.80 Derived: -class_Lattices_Olattice(A) | c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_fun(D,A)),E,tc_fun(D,A)) | -c_lessequals(B,E,tc_fun(D,A)). [resolve(30,a,46,a)].
% 1.54/1.80 Derived: -class_Lattices_Olattice(A) | c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_fun(D,A)),E,tc_fun(D,A)) | -c_lessequals(C,E,tc_fun(D,A)). [resolve(30,a,47,a)].
% 1.54/1.80 Derived: -class_Lattices_Olattice(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_fun(B,A)),c_Lattices_Olower__semilattice__class_Oinf(C,D,tc_fun(B,A))),E)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_fun(B,A)),C),E)). [resolve(30,a,50,a)].
% 1.54/1.80 Derived: -class_Lattices_Olattice(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_fun(B,A)),c_Lattices_Olower__semilattice__class_Oinf(C,D,tc_fun(B,A))),E)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_fun(B,A)),D),E)). [resolve(30,a,51,a)].
% 1.54/1.80 52 class_Lattices_Olower__semilattice(tc_nat) # label(clsarity_nat__Lattices_Olower__semilattice) # label(axiom). [assumption].
% 1.54/1.80 Derived: c_lessequals(A,c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_nat),tc_nat) | -c_lessequals(A,C,tc_nat) | -c_lessequals(A,B,tc_nat). [resolve(52,a,31,a)].
% 1.54/1.80 Derived: c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_nat),B,tc_nat). [resolve(52,a,34,a)].
% 1.54/1.80 Derived: c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_nat),A,tc_nat). [resolve(52,a,35,a)].
% 1.54/1.80 Derived: c_Lattices_Olower__semilattice__class_Oinf(A,c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_nat),tc_nat) = c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_nat). [resolve(52,a,36,a)].
% 1.54/1.80 Derived: c_Lattices_Olower__semilattice__class_Oinf(A,A,tc_nat) = A. [resolve(52,a,37,a)].
% 1.54/1.80 Derived: c_Lattices_Olower__semilattice__class_Oinf(c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_nat),C,tc_nat) = c_Lattices_Olower__semilattice__class_Oinf(A,c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_nat),tc_nat). [resolve(52,a,38,a)].
% 1.54/1.80 Derived: c_Lattices_Olower__semilattice__class_Oinf(A,c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_nat),tc_nat) = c_Lattices_Olower__semilattice__class_Oinf(B,c_Lattices_Olower__semilattice__class_Oinf(A,C,tc_nat),tc_nat). [resolve(52,a,39,a)].
% 1.54/1.80 Derived: c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_nat) = A | -c_lessequals(A,B,tc_nat). [resolve(52,a,40,a)].
% 1.54/1.80 Derived: c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_nat) != A | c_lessequals(A,B,tc_nat). [resolve(52,a,41,a)].
% 1.54/1.80 Derived: c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_nat) = B | -c_lessequals(B,A,tc_nat). [resolve(52,a,42,a)].
% 1.54/1.80 Derived: c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_nat) = c_Lattices_Olower__semilattice__class_Oinf(B,A,tc_nat). [resolve(52,a,43,a)].
% 1.54/1.80 Derived: c_lessequals(A,B,tc_nat) | -c_lessequals(A,c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_nat),tc_nat). [resolve(52,a,44,a)].
% 1.54/1.80 Derived: c_lessequals(A,B,tc_nat) | -c_lessequals(A,c_Lattices_Olower__semilattice__class_Oinf(C,B,tc_nat),tc_nat). [resolve(52,a,45,a)].
% 1.54/1.80 Derived: c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_nat),C,tc_nat) | -c_lessequals(A,C,tc_nat). [resolve(52,a,46,a)].
% 1.54/1.80 Derived: c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_nat),C,tc_nat) | -c_lessequals(B,C,tc_nat). [resolve(52,a,47,a)].
% 1.59/1.86 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_nat)),C)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),C)). [resolve(52,a,50,a)].
% 1.59/1.86 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_nat)),C)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),B),C)). [resolve(52,a,51,a)].
% 1.59/1.86 53 class_Lattices_Olower__semilattice(tc_bool) # label(clsarity_bool__Lattices_Olower__semilattice) # label(axiom). [assumption].
% 1.59/1.86 Derived: c_lessequals(A,c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_bool),tc_bool) | -c_lessequals(A,C,tc_bool) | -c_lessequals(A,B,tc_bool). [resolve(53,a,31,a)].
% 1.59/1.86 Derived: c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_bool),B,tc_bool). [resolve(53,a,34,a)].
% 1.59/1.86 Derived: c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_bool),A,tc_bool). [resolve(53,a,35,a)].
% 1.59/1.86 Derived: c_Lattices_Olower__semilattice__class_Oinf(A,c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_bool),tc_bool) = c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_bool). [resolve(53,a,36,a)].
% 1.59/1.86 Derived: c_Lattices_Olower__semilattice__class_Oinf(A,A,tc_bool) = A. [resolve(53,a,37,a)].
% 1.59/1.86 Derived: c_Lattices_Olower__semilattice__class_Oinf(c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_bool),C,tc_bool) = c_Lattices_Olower__semilattice__class_Oinf(A,c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_bool),tc_bool). [resolve(53,a,38,a)].
% 1.59/1.86 Derived: c_Lattices_Olower__semilattice__class_Oinf(A,c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_bool),tc_bool) = c_Lattices_Olower__semilattice__class_Oinf(B,c_Lattices_Olower__semilattice__class_Oinf(A,C,tc_bool),tc_bool). [resolve(53,a,39,a)].
% 1.59/1.86 Derived: c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_bool) = A | -c_lessequals(A,B,tc_bool). [resolve(53,a,40,a)].
% 1.59/1.86 Derived: c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_bool) != A | c_lessequals(A,B,tc_bool). [resolve(53,a,41,a)].
% 1.59/1.86 Derived: c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_bool) = B | -c_lessequals(B,A,tc_bool). [resolve(53,a,42,a)].
% 1.59/1.86 Derived: c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_bool) = c_Lattices_Olower__semilattice__class_Oinf(B,A,tc_bool). [resolve(53,a,43,a)].
% 1.59/1.86 Derived: c_lessequals(A,B,tc_bool) | -c_lessequals(A,c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_bool),tc_bool). [resolve(53,a,44,a)].
% 1.59/1.86 Derived: c_lessequals(A,B,tc_bool) | -c_lessequals(A,c_Lattices_Olower__semilattice__class_Oinf(C,B,tc_bool),tc_bool). [resolve(53,a,45,a)].
% 1.59/1.86 Derived: c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_bool),C,tc_bool) | -c_lessequals(A,C,tc_bool). [resolve(53,a,46,a)].
% 1.59/1.86 Derived: c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_bool),C,tc_bool) | -c_lessequals(B,C,tc_bool). [resolve(53,a,47,a)].
% 1.59/1.86 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_bool),c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_bool)),C)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_bool),A),C)). [resolve(53,a,50,a)].
% 1.59/1.86 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_bool),c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_bool)),C)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_bool),B),C)). [resolve(53,a,51,a)].
% 1.59/1.86 54 class_Ring__and__Field_Osemiring__1(tc_nat) # label(clsarity_nat__Ring__and__Field_Osemiring__1) # label(axiom). [assumption].
% 1.59/1.86 55 -class_Ring__and__Field_Osemiring__1(A) | c_Nat_Osemiring__1__class_Oof__nat__aux(B,c_HOL_Ozero__class_Ozero(tc_nat),C,A) = C # label(cls_of__nat__aux_Osimps_I1_J_0) # label(axiom). [assumption].
% 1.59/1.86 56 -class_Ring__and__Field_Osemiring__1(A) | c_Nat_Osemiring__1__class_Oof__nat(c_HOL_Ozero__class_Ozero(tc_nat),A) = c_HOL_Ozero__class_Ozero(A) # label(cls_of__nat__0_0) # label(axiom). [assumption].
% 1.59/1.86 57 -class_Ring__and__Field_Osemiring__1(A) | c_Nat_Osemiring__1__class_Oof__nat(c_HOL_Oone__class_Oone(tc_nat),A) = c_HOL_Oone__class_Oone(A) # label(cls_of__nat__1_0) # label(axiom). [assumption].
% 1.59/1.86 58 -class_Ring__and__Field_Osemiring__1(A) | c_Nat_Osemiring__1__class_Oof__nat(B,A) = c_Nat_Osemiring__1__class_Oof__nat__aux(c_COMBC(c_HOL_Oplus__class_Oplus(A),c_HOL_Oone__class_Oone(A),A,A,A),B,c_HOL_Ozero__class_Ozero(A),A) # label(cls_semiring__1__class_Oof__nat__code_0) # label(axiom). [assumption].
% 1.59/1.88 59 -class_Ring__and__Field_Osemiring__1(A) | c_Nat_Osemiring__1__class_Oof__nat(hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),B),C),A) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),c_Nat_Osemiring__1__class_Oof__nat(B,A)),c_Nat_Osemiring__1__class_Oof__nat(C,A)) # label(cls_of__nat__add_0) # label(axiom). [assumption].
% 1.59/1.88 Derived: c_Nat_Osemiring__1__class_Oof__nat__aux(A,c_HOL_Ozero__class_Ozero(tc_nat),B,tc_nat) = B. [resolve(54,a,55,a)].
% 1.59/1.88 Derived: c_Nat_Osemiring__1__class_Oof__nat(c_HOL_Ozero__class_Ozero(tc_nat),tc_nat) = c_HOL_Ozero__class_Ozero(tc_nat). [resolve(54,a,56,a)].
% 1.59/1.88 Derived: c_Nat_Osemiring__1__class_Oof__nat(c_HOL_Oone__class_Oone(tc_nat),tc_nat) = c_HOL_Oone__class_Oone(tc_nat). [resolve(54,a,57,a)].
% 1.59/1.88 Derived: c_Nat_Osemiring__1__class_Oof__nat(A,tc_nat) = c_Nat_Osemiring__1__class_Oof__nat__aux(c_COMBC(c_HOL_Oplus__class_Oplus(tc_nat),c_HOL_Oone__class_Oone(tc_nat),tc_nat,tc_nat,tc_nat),A,c_HOL_Ozero__class_Ozero(tc_nat),tc_nat). [resolve(54,a,58,a)].
% 1.59/1.88 Derived: c_Nat_Osemiring__1__class_Oof__nat(hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B),tc_nat) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),c_Nat_Osemiring__1__class_Oof__nat(A,tc_nat)),c_Nat_Osemiring__1__class_Oof__nat(B,tc_nat)). [resolve(54,a,59,a)].
% 1.59/1.88 60 class_Ring__and__Field_Oordered__semidom(tc_nat) # label(clsarity_nat__Ring__and__Field_Oordered__semidom) # label(axiom). [assumption].
% 1.59/1.88 61 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Oone__class_Oone(A)),c_Power_Opower__class_Opower(B,C,A))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),C)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Oone__class_Oone(A)),B)) # label(cls_one__less__power_0) # label(axiom). [assumption].
% 1.59/1.88 62 -class_Ring__and__Field_Oordered__semidom(A) | c_lessequals(c_Power_Opower__class_Opower(B,C,A),c_Power_Opower__class_Opower(D,C,A),A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),B,A) | -c_lessequals(B,D,A) # label(cls_power__mono_0) # label(axiom). [assumption].
% 1.59/1.88 63 -class_Ring__and__Field_Oordered__semidom(A) | c_Power_Opower__class_Opower(B,C,A) != c_Power_Opower__class_Opower(D,C,A) | B = D | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),C)) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),D,A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),B,A) # label(cls_power__eq__imp__eq__base_0) # label(axiom). [assumption].
% 1.59/1.88 64 -class_Ring__and__Field_Oordered__semidom(A) | c_lessequals(c_HOL_Ozero__class_Ozero(A),c_Power_Opower__class_Opower(B,C,A),A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),B,A) # label(cls_zero__le__power_0) # label(axiom). [assumption].
% 1.59/1.88 65 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),c_Power_Opower__class_Opower(B,C,A))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),B)) # label(cls_zero__less__power_0) # label(axiom). [assumption].
% 1.59/1.88 66 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_Power_Opower__class_Opower(B,C,A)),c_Power_Opower__class_Opower(D,C,A))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),C)) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),B,A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),D)) # label(cls_power__strict__mono_0) # label(axiom). [assumption].
% 1.59/1.88 67 -class_Ring__and__Field_Oordered__semidom(A) | c_lessequals(c_HOL_Oone__class_Oone(A),c_Power_Opower__class_Opower(B,C,A),A) | -c_lessequals(c_HOL_Oone__class_Oone(A),B,A) # label(cls_one__le__power_0) # label(axiom). [assumption].
% 1.59/1.88 68 -class_Ring__and__Field_Oordered__semidom(A) | c_Power_Opower__class_Opower(B,C,A) != c_Power_Opower__class_Opower(B,D,A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Oone__class_Oone(A)),B)) | C = D # label(cls_power__inject__exp_0) # label(axiom). [assumption].
% 1.59/1.88 69 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),C)) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),C,A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_Power_Opower__class_Opower(B,D,A)),c_Power_Opower__class_Opower(C,D,A))) # label(cls_power__less__imp__less__base_0) # label(axiom). [assumption].
% 1.59/1.88 70 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),B),C)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_Power_Opower__class_Opower(D,B,A)),c_Power_Opower__class_Opower(D,C,A))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Oone__class_Oone(A)),D)) # label(cls_power__less__imp__less__exp_0) # label(axiom). [assumption].
% 1.59/1.88 71 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_Power_Opower__class_Opower(B,C,A)),c_Power_Opower__class_Opower(B,D,A))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Oone__class_Oone(A)),B)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),C),D)) # label(cls_power__strict__increasing_0) # label(axiom). [assumption].
% 1.59/1.88 72 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),B),C)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_Power_Opower__class_Opower(D,B,A)),c_Power_Opower__class_Opower(D,C,A))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Oone__class_Oone(A)),D)) # label(cls_power__strict__increasing__iff_0) # label(axiom). [assumption].
% 1.59/1.88 73 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_Power_Opower__class_Opower(B,C,A)),c_Power_Opower__class_Opower(B,D,A))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),C),D)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Oone__class_Oone(A)),B)) # label(cls_power__strict__increasing__iff_1) # label(axiom). [assumption].
% 1.59/1.88 74 -class_Ring__and__Field_Oordered__semidom(A) | c_lessequals(c_Power_Opower__class_Opower(B,C,A),c_Power_Opower__class_Opower(B,D,A),A) | -c_lessequals(c_HOL_Oone__class_Oone(A),B,A) | -c_lessequals(C,D,tc_nat) # label(cls_power__increasing_0) # label(axiom). [assumption].
% 1.59/1.88 75 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_Power_Opower__class_Opower(B,C,A)),c_Power_Opower__class_Opower(B,D,A))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),c_HOL_Oone__class_Oone(A))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),D),C)) # label(cls_power__strict__decreasing_0) # label(axiom). [assumption].
% 1.59/1.88 76 -class_Ring__and__Field_Oordered__semidom(A) | c_lessequals(c_Power_Opower__class_Opower(B,C,A),c_Power_Opower__class_Opower(B,D,A),A) | -c_lessequals(B,c_HOL_Oone__class_Oone(A),A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),B,A) | -c_lessequals(D,C,tc_nat) # label(cls_power__decreasing_0) # label(axiom). [assumption].
% 1.59/1.88 77 -class_Ring__and__Field_Oordered__semidom(A) | c_lessequals(B,C,tc_nat) | -c_lessequals(c_Power_Opower__class_Opower(D,B,A),c_Power_Opower__class_Opower(D,C,A),A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Oone__class_Oone(A)),D)) # label(cls_power__le__imp__le__exp_0) # label(axiom). [assumption].
% 1.59/1.88 78 -class_Ring__and__Field_Oordered__semidom(A) | c_lessequals(B,C,tc_nat) | -c_lessequals(c_Power_Opower__class_Opower(D,B,A),c_Power_Opower__class_Opower(D,C,A),A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Oone__class_Oone(A)),D)) # label(cls_power__increasing__iff_0) # label(axiom). [assumption].
% 1.59/1.88 79 -class_Ring__and__Field_Oordered__semidom(A) | c_lessequals(c_Power_Opower__class_Opower(B,C,A),c_Power_Opower__class_Opower(B,D,A),A) | -c_lessequals(C,D,tc_nat) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Oone__class_Oone(A)),B)) # label(cls_power__increasing__iff_1) # label(axiom). [assumption].
% 1.59/1.88 80 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),c_Nat_Osemiring__1__class_Oof__nat(B,A))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),B)) # label(cls_of__nat__0__less__iff_1) # label(axiom). [assumption].
% 1.59/1.88 81 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),B)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),c_Nat_Osemiring__1__class_Oof__nat(B,A))) # label(cls_of__nat__0__less__iff_0) # label(axiom). [assumption].
% 1.59/1.88 82 -class_Ring__and__Field_Oordered__semidom(A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_Nat_Osemiring__1__class_Oof__nat(B,A)),c_HOL_Ozero__class_Ozero(A))) # label(cls_of__nat__less__0__iff_0) # label(axiom). [assumption].
% 1.59/1.88 83 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),c_HOL_Oone__class_Oone(A)),c_HOL_Oone__class_Oone(A)))) # label(cls_zero__less__two_0) # label(axiom). [assumption].
% 1.59/1.88 84 -class_Ring__and__Field_Oordered__semidom(A) | c_lessequals(c_HOL_Ozero__class_Ozero(A),c_Nat_Osemiring__1__class_Oof__nat(B,A),A) # label(cls_of__nat__0__le__iff_0) # label(axiom). [assumption].
% 1.59/1.88 85 -class_Ring__and__Field_Oordered__semidom(A) | c_lessequals(c_HOL_Ozero__class_Ozero(A),c_Nat_Osemiring__1__class_Oof__nat(B,A),A) # label(cls_zero__le__imp__of__nat_0) # label(axiom). [assumption].
% 1.59/1.88 86 -class_Ring__and__Field_Oordered__semidom(A) | c_lessequals(c_Nat_Osemiring__1__class_Oof__nat(B,A),c_Nat_Osemiring__1__class_Oof__nat(C,A),A) | -c_lessequals(B,C,tc_nat) # label(cls_of__nat__le__iff_1) # label(axiom). [assumption].
% 1.59/1.88 87 -class_Ring__and__Field_Oordered__semidom(A) | c_lessequals(B,C,tc_nat) | -c_lessequals(c_Nat_Osemiring__1__class_Oof__nat(B,A),c_Nat_Osemiring__1__class_Oof__nat(C,A),A) # label(cls_of__nat__le__iff_0) # label(axiom). [assumption].
% 1.59/1.88 88 -class_Ring__and__Field_Oordered__semidom(A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Oone__class_Oone(A)),c_HOL_Ozero__class_Ozero(A))) # label(cls_not__one__less__zero_0) # label(axiom). [assumption].
% 1.59/1.88 89 -class_Ring__and__Field_Oordered__semidom(A) | c_lessequals(c_HOL_Ozero__class_Ozero(A),c_HOL_Oone__class_Oone(A),A) # label(cls_zero__le__one_0) # label(axiom). [assumption].
% 1.59/1.88 90 -class_Ring__and__Field_Oordered__semidom(A) | -c_lessequals(c_HOL_Oone__class_Oone(A),c_HOL_Ozero__class_Ozero(A),A) # label(cls_not__one__le__zero_0) # label(axiom). [assumption].
% 1.59/1.88 91 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),c_HOL_Oone__class_Oone(A))) # label(cls_zero__less__one_0) # label(axiom). [assumption].
% 1.59/1.88 92 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),c_HOL_Oone__class_Oone(A)))) # label(cls_less__add__one_0) # label(axiom). [assumption].
% 1.59/1.88 93 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),C),D))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),D)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),C)) # label(cls_pos__add__strict_0) # label(axiom). [assumption].
% 1.59/1.88 94 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),B),C)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_Nat_Osemiring__1__class_Oof__nat(B,A)),c_Nat_Osemiring__1__class_Oof__nat(C,A))) # label(cls_of__nat__less__iff_0) # label(axiom). [assumption].
% 1.59/1.88 95 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_Nat_Osemiring__1__class_Oof__nat(B,A)),c_Nat_Osemiring__1__class_Oof__nat(C,A))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),B),C)) # label(cls_of__nat__less__iff_1) # label(axiom). [assumption].
% 1.59/1.88 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Oone__class_Oone(tc_nat)),c_Power_Opower__class_Opower(A,B,tc_nat))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),B)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Oone__class_Oone(tc_nat)),A)). [resolve(60,a,61,a)].
% 1.59/1.88 Derived: c_lessequals(c_Power_Opower__class_Opower(A,B,tc_nat),c_Power_Opower__class_Opower(C,B,tc_nat),tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),A,tc_nat) | -c_lessequals(A,C,tc_nat). [resolve(60,a,62,a)].
% 1.59/1.89 Derived: c_Power_Opower__class_Opower(A,B,tc_nat) != c_Power_Opower__class_Opower(C,B,tc_nat) | A = C | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),B)) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),C,tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),A,tc_nat). [resolve(60,a,63,a)].
% 1.59/1.89 Derived: c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),c_Power_Opower__class_Opower(A,B,tc_nat),tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),A,tc_nat). [resolve(60,a,64,a)].
% 1.59/1.89 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),c_Power_Opower__class_Opower(A,B,tc_nat))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),A)). [resolve(60,a,65,a)].
% 1.59/1.89 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_Power_Opower__class_Opower(A,B,tc_nat)),c_Power_Opower__class_Opower(C,B,tc_nat))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),B)) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),A,tc_nat) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),C)). [resolve(60,a,66,a)].
% 1.59/1.89 Derived: c_lessequals(c_HOL_Oone__class_Oone(tc_nat),c_Power_Opower__class_Opower(A,B,tc_nat),tc_nat) | -c_lessequals(c_HOL_Oone__class_Oone(tc_nat),A,tc_nat). [resolve(60,a,67,a)].
% 1.59/1.89 Derived: c_Power_Opower__class_Opower(A,B,tc_nat) != c_Power_Opower__class_Opower(A,C,tc_nat) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Oone__class_Oone(tc_nat)),A)) | B = C. [resolve(60,a,68,a)].
% 1.59/1.89 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),B)) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),B,tc_nat) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_Power_Opower__class_Opower(A,C,tc_nat)),c_Power_Opower__class_Opower(B,C,tc_nat))). [resolve(60,a,69,a)].
% 1.59/1.89 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),B)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_Power_Opower__class_Opower(C,A,tc_nat)),c_Power_Opower__class_Opower(C,B,tc_nat))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Oone__class_Oone(tc_nat)),C)). [resolve(60,a,70,a)].
% 1.59/1.89 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_Power_Opower__class_Opower(A,B,tc_nat)),c_Power_Opower__class_Opower(A,C,tc_nat))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Oone__class_Oone(tc_nat)),A)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),B),C)). [resolve(60,a,71,a)].
% 1.59/1.89 Derived: c_lessequals(c_Power_Opower__class_Opower(A,B,tc_nat),c_Power_Opower__class_Opower(A,C,tc_nat),tc_nat) | -c_lessequals(c_HOL_Oone__class_Oone(tc_nat),A,tc_nat) | -c_lessequals(B,C,tc_nat). [resolve(60,a,74,a)].
% 1.59/1.89 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_Power_Opower__class_Opower(A,B,tc_nat)),c_Power_Opower__class_Opower(A,C,tc_nat))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),c_HOL_Oone__class_Oone(tc_nat))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),A)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),C),B)). [resolve(60,a,75,a)].
% 1.59/1.89 Derived: c_lessequals(c_Power_Opower__class_Opower(A,B,tc_nat),c_Power_Opower__class_Opower(A,C,tc_nat),tc_nat) | -c_lessequals(A,c_HOL_Oone__class_Oone(tc_nat),tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),A,tc_nat) | -c_lessequals(C,B,tc_nat). [resolve(60,a,76,a)].
% 1.59/1.89 Derived: c_lessequals(A,B,tc_nat) | -c_lessequals(c_Power_Opower__class_Opower(C,A,tc_nat),c_Power_Opower__class_Opower(C,B,tc_nat),tc_nat) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Oone__class_Oone(tc_nat)),C)). [resolve(60,a,77,a)].
% 1.59/1.89 Derived: c_lessequals(c_Power_Opower__class_Opower(A,B,tc_nat),c_Power_Opower__class_Opower(A,C,tc_nat),tc_nat) | -c_lessequals(B,C,tc_nat) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Oone__class_Oone(tc_nat)),A)). [resolve(60,a,79,a)].
% 1.59/1.89 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),c_Nat_Osemiring__1__class_Oof__nat(A,tc_nat))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),A)). [resolve(60,a,80,a)].
% 1.74/2.04 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),A)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),c_Nat_Osemiring__1__class_Oof__nat(A,tc_nat))). [resolve(60,a,81,a)].
% 1.74/2.04 Derived: -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_Nat_Osemiring__1__class_Oof__nat(A,tc_nat)),c_HOL_Ozero__class_Ozero(tc_nat))). [resolve(60,a,82,a)].
% 1.74/2.04 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),c_HOL_Oone__class_Oone(tc_nat)),c_HOL_Oone__class_Oone(tc_nat)))). [resolve(60,a,83,a)].
% 1.74/2.04 Derived: c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),c_Nat_Osemiring__1__class_Oof__nat(A,tc_nat),tc_nat). [resolve(60,a,84,a)].
% 1.74/2.04 Derived: c_lessequals(c_Nat_Osemiring__1__class_Oof__nat(A,tc_nat),c_Nat_Osemiring__1__class_Oof__nat(B,tc_nat),tc_nat) | -c_lessequals(A,B,tc_nat). [resolve(60,a,86,a)].
% 1.74/2.04 Derived: c_lessequals(A,B,tc_nat) | -c_lessequals(c_Nat_Osemiring__1__class_Oof__nat(A,tc_nat),c_Nat_Osemiring__1__class_Oof__nat(B,tc_nat),tc_nat). [resolve(60,a,87,a)].
% 1.74/2.04 Derived: -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Oone__class_Oone(tc_nat)),c_HOL_Ozero__class_Ozero(tc_nat))). [resolve(60,a,88,a)].
% 1.74/2.04 Derived: c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),c_HOL_Oone__class_Oone(tc_nat),tc_nat). [resolve(60,a,89,a)].
% 1.74/2.04 Derived: -c_lessequals(c_HOL_Oone__class_Oone(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat),tc_nat). [resolve(60,a,90,a)].
% 1.74/2.04 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),c_HOL_Oone__class_Oone(tc_nat))). [resolve(60,a,91,a)].
% 1.74/2.04 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),c_HOL_Oone__class_Oone(tc_nat)))). [resolve(60,a,92,a)].
% 1.74/2.04 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),B),C))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),C)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),B)). [resolve(60,a,93,a)].
% 1.74/2.04 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),B)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_Nat_Osemiring__1__class_Oof__nat(A,tc_nat)),c_Nat_Osemiring__1__class_Oof__nat(B,tc_nat))). [resolve(60,a,94,a)].
% 1.74/2.04 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_Nat_Osemiring__1__class_Oof__nat(A,tc_nat)),c_Nat_Osemiring__1__class_Oof__nat(B,tc_nat))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),B)). [resolve(60,a,95,a)].
% 1.74/2.04 96 class_OrderedGroup_Omonoid__mult(tc_nat) # label(clsarity_nat__OrderedGroup_Omonoid__mult) # label(axiom). [assumption].
% 1.74/2.04 97 -class_OrderedGroup_Omonoid__mult(A) | c_Power_Opower__class_Opower(B,c_HOL_Oone__class_Oone(tc_nat),A) = B # label(cls_power__one__right_0) # label(axiom). [assumption].
% 1.74/2.04 98 -class_OrderedGroup_Omonoid__mult(A) | c_Power_Opower__class_Opower(c_HOL_Oone__class_Oone(A),B,A) = c_HOL_Oone__class_Oone(A) # label(cls_power__one_0) # label(axiom). [assumption].
% 1.74/2.04 Derived: c_Power_Opower__class_Opower(c_HOL_Oone__class_Oone(tc_nat),A,tc_nat) = c_HOL_Oone__class_Oone(tc_nat). [resolve(96,a,98,a)].
% 1.74/2.04 99 class_OrderedGroup_Ocomm__monoid__add(tc_nat) # label(clsarity_nat__OrderedGroup_Ocomm__monoid__add) # label(axiom). [assumption].
% 1.74/2.04 100 -class_OrderedGroup_Ocomm__monoid__add(A) | c_Finite__Set_Ofold__image(c_HOL_Oplus__class_Oplus(A),B,C,c_Set_Oinsert(D,E,F),A,F) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),hAPP(B,D)),c_Finite__Set_Ofold__image(c_HOL_Oplus__class_Oplus(A),B,C,E,A,F)) | hBOOL(hAPP(hAPP(c_in(F),D),E)) | -c_Finite__Set_Ofinite(E,F) # label(cls_comm__monoid__add_Ofold__image__insert_0) # label(axiom). [assumption].
% 1.74/2.04 101 -class_OrderedGroup_Ocomm__monoid__add(A) | c_Finite__Set_Osetsum(B,c_Orderings_Obot__class_Obot(tc_fun(C,tc_bool)),C,A) = c_HOL_Ozero__class_Ozero(A) # label(cls_setsum__empty_0) # label(axiom). [assumption].
% 1.74/2.04 102 -class_OrderedGroup_Ocomm__monoid__add(A) | c_Finite__Set_Osetsum(B,c_Set_Oinsert(C,D,E),E,A) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),hAPP(B,C)),c_Finite__Set_Osetsum(B,D,E,A)) | hBOOL(hAPP(hAPP(c_in(E),C),D)) | -c_Finite__Set_Ofinite(D,E) # label(cls_setsum__insert_0) # label(axiom). [assumption].
% 1.74/2.04 103 -class_OrderedGroup_Ocomm__monoid__add(A) | c_Finite__Set_Osetsum(B,C,D,A) = c_Finite__Set_Ofold__image(c_HOL_Oplus__class_Oplus(A),B,c_HOL_Ozero__class_Ozero(A),C,A,D) | -c_Finite__Set_Ofinite(C,D) # label(cls_setsum__def_0) # label(axiom). [assumption].
% 1.74/2.04 104 -class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(A) | -class_OrderedGroup_Ocomm__monoid__add(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),C),D))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),D)) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),C,A) # label(cls_add__strict__increasing2_0) # label(axiom). [assumption].
% 1.74/2.04 105 -class_OrderedGroup_Ocomm__monoid__add(A) | c_Finite__Set_Osetsum(B,c_Lattices_Olower__semilattice__class_Oinf(C,D,tc_fun(E,tc_bool)),E,A) = c_Finite__Set_Osetsum(c_COMBC(c_COMBS(c_COMBB(c_HOL_OIf(A),c_COMBC(c_in(E),D,E,tc_fun(E,tc_bool),tc_bool),tc_bool,tc_fun(A,tc_fun(A,A)),E),B,E,A,tc_fun(A,A)),c_HOL_Ozero__class_Ozero(A),E,A,A),C,E,A) | -c_Finite__Set_Ofinite(C,E) # label(cls_setsum__restrict__set_0) # label(axiom). [assumption].
% 1.74/2.04 106 -class_OrderedGroup_Ocomm__monoid__add(A) | c_Finite__Set_Ofold__image(c_HOL_Oplus__class_Oplus(A),c_COMBS(c_COMBB(c_HOL_Oplus__class_Oplus(A),B,A,tc_fun(A,A),C),D,C,A,A),c_HOL_Ozero__class_Ozero(A),E,A,C) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),c_Finite__Set_Ofold__image(c_HOL_Oplus__class_Oplus(A),B,c_HOL_Ozero__class_Ozero(A),E,A,C)),c_Finite__Set_Ofold__image(c_HOL_Oplus__class_Oplus(A),D,c_HOL_Ozero__class_Ozero(A),E,A,C)) | -c_Finite__Set_Ofinite(E,C) # label(cls_comm__monoid__add_Ofold__image__distrib_0) # label(axiom). [assumption].
% 1.74/2.04 107 -class_OrderedGroup_Ocomm__monoid__add(A) | c_Finite__Set_Osetsum(c_COMBS(c_COMBS(c_COMBB(c_HOL_OIf(A),c_COMBC(c_in(B),C,B,tc_fun(B,tc_bool),tc_bool),tc_bool,tc_fun(A,tc_fun(A,A)),B),D,B,A,tc_fun(A,A)),E,B,A,A),F,B,A) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),c_Finite__Set_Osetsum(D,c_Lattices_Olower__semilattice__class_Oinf(F,C,tc_fun(B,tc_bool)),B,A)),c_Finite__Set_Osetsum(E,c_Lattices_Olower__semilattice__class_Oinf(F,hAPP(c_HOL_Ouminus__class_Ouminus(tc_fun(B,tc_bool)),C),tc_fun(B,tc_bool)),B,A)) | -c_Finite__Set_Ofinite(F,B) # label(cls_setsum__cases_0) # label(axiom). [assumption].
% 1.74/2.04 108 -class_OrderedGroup_Ocomm__monoid__add(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),c_HOL_Ozero__class_Ozero(A)),B) = B # label(cls_comm__monoid__add_Omult__1_0) # label(axiom). [assumption].
% 1.74/2.04 109 -class_OrderedGroup_Ocomm__monoid__add(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),c_HOL_Ozero__class_Ozero(A)) = B # label(cls_comm__monoid__add_Omult__1__right_0) # label(axiom). [assumption].
% 1.74/2.04 110 -class_OrderedGroup_Ocomm__monoid__add(A) | c_Finite__Set_Osetsum(c_COMBS(c_COMBB(c_HOL_Oplus__class_Oplus(A),B,A,tc_fun(A,A),C),D,C,A,A),E,C,A) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),c_Finite__Set_Osetsum(B,E,C,A)),c_Finite__Set_Osetsum(D,E,C,A)) # label(cls_setsum__addf_0) # label(axiom). [assumption].
% 1.74/2.04 111 -class_OrderedGroup_Ocomm__monoid__add(A) | c_Finite__Set_Osetsum(c_COMBC(c_COMBS(c_COMBB(c_HOL_OIf(A),hAPP(c_fequal(B),C),tc_bool,tc_fun(A,tc_fun(A,A)),B),D,B,A,tc_fun(A,A)),c_HOL_Ozero__class_Ozero(A),B,A,A),E,B,A) = c_HOL_Ozero__class_Ozero(A) | hBOOL(hAPP(hAPP(c_in(B),C),E)) | -c_Finite__Set_Ofinite(E,B) # label(cls_setsum__delta_H_1) # label(axiom). [assumption].
% 1.74/2.04 112 -class_OrderedGroup_Ocomm__monoid__add(A) | c_Finite__Set_Osetsum(c_COMBC(c_COMBS(c_COMBB(c_HOL_OIf(A),hAPP(c_fequal(B),C),tc_bool,tc_fun(A,tc_fun(A,A)),B),D,B,A,tc_fun(A,A)),c_HOL_Ozero__class_Ozero(A),B,A,A),E,B,A) = hAPP(D,C) | -hBOOL(hAPP(hAPP(c_in(B),C),E)) | -c_Finite__Set_Ofinite(E,B) # label(cls_setsum__delta_H_0) # label(axiom). [assumption].
% 1.74/2.04 113 -class_OrderedGroup_Ocomm__monoid__add(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),C),D)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),C),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),D)) # label(cls_comm__monoid__add_Omult__left__commute_0) # label(axiom). [assumption].
% 1.74/2.04 114 -class_OrderedGroup_Ocomm__monoid__add(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C)),D) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),C),D)) # label(cls_comm__monoid__add_Omult__assoc_0) # label(axiom). [assumption].
% 1.74/2.04 115 -class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(A) | -class_OrderedGroup_Ocomm__monoid__add(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),C),D))) | -c_lessequals(B,D,A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),C)) # label(cls_add__strict__increasing_0) # label(axiom). [assumption].
% 1.74/2.04 116 -class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(A) | -class_OrderedGroup_Ocomm__monoid__add(A) | c_lessequals(B,hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),C),D),A) | -c_lessequals(B,D,A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),C,A) # label(cls_add__increasing_0) # label(axiom). [assumption].
% 1.74/2.04 117 -class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(A) | -class_OrderedGroup_Ocomm__monoid__add(A) | c_lessequals(B,hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),C),D),A) | -c_lessequals(B,C,A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),D,A) # label(cls_add__increasing2_0) # label(axiom). [assumption].
% 1.74/2.04 118 -class_OrderedGroup_Ocomm__monoid__add(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),C),B) # label(cls_comm__monoid__add_Omult__commute_0) # label(axiom). [assumption].
% 1.74/2.04 119 -class_OrderedGroup_Ocomm__monoid__add(A) | c_Finite__Set_Osetsum(B,C,D,A) = c_HOL_Ozero__class_Ozero(A) | c_Finite__Set_Ofinite(C,D) # label(cls_setsum__def_1) # label(axiom). [assumption].
% 1.74/2.04 120 -class_OrderedGroup_Ocomm__monoid__add(A) | c_Finite__Set_Osetsum(B,C,D,A) = c_HOL_Ozero__class_Ozero(A) | c_Finite__Set_Ofinite(C,D) # label(cls_setsum__infinite_0) # label(axiom). [assumption].
% 1.74/2.04 121 -class_OrderedGroup_Ocomm__monoid__add(A) | c_Finite__Set_Osetsum(c_COMBC(c_COMBS(c_COMBB(c_HOL_OIf(A),c_COMBC(c_fequal(B),C,B,B,tc_bool),tc_bool,tc_fun(A,tc_fun(A,A)),B),D,B,A,tc_fun(A,A)),c_HOL_Ozero__class_Ozero(A),B,A,A),E,B,A) = hAPP(D,C) | -hBOOL(hAPP(hAPP(c_in(B),C),E)) | -c_Finite__Set_Ofinite(E,B) # label(cls_setsum__delta_0) # label(axiom). [assumption].
% 1.74/2.04 122 -class_OrderedGroup_Ocomm__monoid__add(A) | c_Finite__Set_Osetsum(c_COMBC(c_COMBS(c_COMBB(c_HOL_OIf(A),c_COMBC(c_fequal(B),C,B,B,tc_bool),tc_bool,tc_fun(A,tc_fun(A,A)),B),D,B,A,tc_fun(A,A)),c_HOL_Ozero__class_Ozero(A),B,A,A),E,B,A) = c_HOL_Ozero__class_Ozero(A) | hBOOL(hAPP(hAPP(c_in(B),C),E)) | -c_Finite__Set_Ofinite(E,B) # label(cls_setsum__delta_1) # label(axiom). [assumption].
% 1.74/2.04 Derived: c_Finite__Set_Ofold__image(c_HOL_Oplus__class_Oplus(tc_nat),A,B,c_Set_Oinsert(C,D,E),tc_nat,E) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),hAPP(A,C)),c_Finite__Set_Ofold__image(c_HOL_Oplus__class_Oplus(tc_nat),A,B,D,tc_nat,E)) | hBOOL(hAPP(hAPP(c_in(E),C),D)) | -c_Finite__Set_Ofinite(D,E). [resolve(99,a,100,a)].
% 1.74/2.04 Derived: c_Finite__Set_Osetsum(A,c_Orderings_Obot__class_Obot(tc_fun(B,tc_bool)),B,tc_nat) = c_HOL_Ozero__class_Ozero(tc_nat). [resolve(99,a,101,a)].
% 1.74/2.04 Derived: c_Finite__Set_Osetsum(A,c_Set_Oinsert(B,C,D),D,tc_nat) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),hAPP(A,B)),c_Finite__Set_Osetsum(A,C,D,tc_nat)) | hBOOL(hAPP(hAPP(c_in(D),B),C)) | -c_Finite__Set_Ofinite(C,D). [resolve(99,a,102,a)].
% 1.74/2.04 Derived: c_Finite__Set_Osetsum(A,B,C,tc_nat) = c_Finite__Set_Ofold__image(c_HOL_Oplus__class_Oplus(tc_nat),A,c_HOL_Ozero__class_Ozero(tc_nat),B,tc_nat,C) | -c_Finite__Set_Ofinite(B,C). [resolve(99,a,103,a)].
% 1.74/2.04 Derived: -class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(tc_nat) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),B),C))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),C)) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),B,tc_nat). [resolve(99,a,104,b)].
% 1.74/2.05 Derived: c_Finite__Set_Osetsum(A,c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_fun(D,tc_bool)),D,tc_nat) = c_Finite__Set_Osetsum(c_COMBC(c_COMBS(c_COMBB(c_HOL_OIf(tc_nat),c_COMBC(c_in(D),C,D,tc_fun(D,tc_bool),tc_bool),tc_bool,tc_fun(tc_nat,tc_fun(tc_nat,tc_nat)),D),A,D,tc_nat,tc_fun(tc_nat,tc_nat)),c_HOL_Ozero__class_Ozero(tc_nat),D,tc_nat,tc_nat),B,D,tc_nat) | -c_Finite__Set_Ofinite(B,D). [resolve(99,a,105,a)].
% 1.74/2.05 Derived: c_Finite__Set_Ofold__image(c_HOL_Oplus__class_Oplus(tc_nat),c_COMBS(c_COMBB(c_HOL_Oplus__class_Oplus(tc_nat),A,tc_nat,tc_fun(tc_nat,tc_nat),B),C,B,tc_nat,tc_nat),c_HOL_Ozero__class_Ozero(tc_nat),D,tc_nat,B) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),c_Finite__Set_Ofold__image(c_HOL_Oplus__class_Oplus(tc_nat),A,c_HOL_Ozero__class_Ozero(tc_nat),D,tc_nat,B)),c_Finite__Set_Ofold__image(c_HOL_Oplus__class_Oplus(tc_nat),C,c_HOL_Ozero__class_Ozero(tc_nat),D,tc_nat,B)) | -c_Finite__Set_Ofinite(D,B). [resolve(99,a,106,a)].
% 1.74/2.05 Derived: c_Finite__Set_Osetsum(c_COMBS(c_COMBS(c_COMBB(c_HOL_OIf(tc_nat),c_COMBC(c_in(A),B,A,tc_fun(A,tc_bool),tc_bool),tc_bool,tc_fun(tc_nat,tc_fun(tc_nat,tc_nat)),A),C,A,tc_nat,tc_fun(tc_nat,tc_nat)),D,A,tc_nat,tc_nat),E,A,tc_nat) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),c_Finite__Set_Osetsum(C,c_Lattices_Olower__semilattice__class_Oinf(E,B,tc_fun(A,tc_bool)),A,tc_nat)),c_Finite__Set_Osetsum(D,c_Lattices_Olower__semilattice__class_Oinf(E,hAPP(c_HOL_Ouminus__class_Ouminus(tc_fun(A,tc_bool)),B),tc_fun(A,tc_bool)),A,tc_nat)) | -c_Finite__Set_Ofinite(E,A). [resolve(99,a,107,a)].
% 1.74/2.05 Derived: c_Finite__Set_Osetsum(c_COMBS(c_COMBB(c_HOL_Oplus__class_Oplus(tc_nat),A,tc_nat,tc_fun(tc_nat,tc_nat),B),C,B,tc_nat,tc_nat),D,B,tc_nat) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),c_Finite__Set_Osetsum(A,D,B,tc_nat)),c_Finite__Set_Osetsum(C,D,B,tc_nat)). [resolve(99,a,110,a)].
% 1.74/2.05 Derived: c_Finite__Set_Osetsum(c_COMBC(c_COMBS(c_COMBB(c_HOL_OIf(tc_nat),hAPP(c_fequal(A),B),tc_bool,tc_fun(tc_nat,tc_fun(tc_nat,tc_nat)),A),C,A,tc_nat,tc_fun(tc_nat,tc_nat)),c_HOL_Ozero__class_Ozero(tc_nat),A,tc_nat,tc_nat),D,A,tc_nat) = c_HOL_Ozero__class_Ozero(tc_nat) | hBOOL(hAPP(hAPP(c_in(A),B),D)) | -c_Finite__Set_Ofinite(D,A). [resolve(99,a,111,a)].
% 1.74/2.05 Derived: c_Finite__Set_Osetsum(c_COMBC(c_COMBS(c_COMBB(c_HOL_OIf(tc_nat),hAPP(c_fequal(A),B),tc_bool,tc_fun(tc_nat,tc_fun(tc_nat,tc_nat)),A),C,A,tc_nat,tc_fun(tc_nat,tc_nat)),c_HOL_Ozero__class_Ozero(tc_nat),A,tc_nat,tc_nat),D,A,tc_nat) = hAPP(C,B) | -hBOOL(hAPP(hAPP(c_in(A),B),D)) | -c_Finite__Set_Ofinite(D,A). [resolve(99,a,112,a)].
% 1.74/2.05 Derived: -class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(tc_nat) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),B),C))) | -c_lessequals(A,C,tc_nat) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),B)). [resolve(99,a,115,b)].
% 1.74/2.05 Derived: -class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(tc_nat) | c_lessequals(A,hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),B),C),tc_nat) | -c_lessequals(A,C,tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),B,tc_nat). [resolve(99,a,116,b)].
% 1.74/2.05 Derived: -class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(tc_nat) | c_lessequals(A,hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),B),C),tc_nat) | -c_lessequals(A,B,tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),C,tc_nat). [resolve(99,a,117,b)].
% 1.74/2.05 Derived: c_Finite__Set_Osetsum(A,B,C,tc_nat) = c_HOL_Ozero__class_Ozero(tc_nat) | c_Finite__Set_Ofinite(B,C). [resolve(99,a,119,a)].
% 1.74/2.05 Derived: c_Finite__Set_Osetsum(c_COMBC(c_COMBS(c_COMBB(c_HOL_OIf(tc_nat),c_COMBC(c_fequal(A),B,A,A,tc_bool),tc_bool,tc_fun(tc_nat,tc_fun(tc_nat,tc_nat)),A),C,A,tc_nat,tc_fun(tc_nat,tc_nat)),c_HOL_Ozero__class_Ozero(tc_nat),A,tc_nat,tc_nat),D,A,tc_nat) = hAPP(C,B) | -hBOOL(hAPP(hAPP(c_in(A),B),D)) | -c_Finite__Set_Ofinite(D,A). [resolve(99,a,121,a)].
% 1.74/2.05 Derived: c_Finite__Set_Osetsum(c_COMBC(c_COMBS(c_COMBB(c_HOL_OIf(tc_nat),c_COMBC(c_fequal(A),B,A,A,tc_bool),tc_bool,tc_fun(tc_nat,tc_fun(tc_nat,tc_nat)),A),C,A,tc_nat,tc_fun(tc_nat,tc_nat)),c_HOL_Ozero__class_Ozero(tc_nat),A,tc_nat,tc_nat),D,A,tc_nat) = c_HOL_Ozero__class_Ozero(tc_nat) | hBOOL(hAPP(hAPP(c_in(A),B),D)) | -c_Finite__Set_Ofinite(D,A). [resolve(99,a,122,a)].
% 1.82/2.07 123 class_OrderedGroup_Ocomm__monoid__mult(tc_nat) # label(clsarity_nat__OrderedGroup_Ocomm__monoid__mult) # label(axiom). [assumption].
% 1.82/2.07 124 -class_OrderedGroup_Ocomm__monoid__mult(A) | c_Finite__Set_Osetprod(B,c_Orderings_Obot__class_Obot(tc_fun(C,tc_bool)),C,A) = c_HOL_Oone__class_Oone(A) # label(cls_setprod__empty_0) # label(axiom). [assumption].
% 1.82/2.07 125 -class_OrderedGroup_Ocomm__monoid__mult(A) | c_Finite__Set_Osetprod(c_COMBC(c_COMBS(c_COMBB(c_HOL_OIf(A),c_COMBC(c_fequal(B),C,B,B,tc_bool),tc_bool,tc_fun(A,tc_fun(A,A)),B),D,B,A,tc_fun(A,A)),E,B,A,A),F,B,A) = c_Power_Opower__class_Opower(E,c_Finite__Set_Ocard(F,B),A) | hBOOL(hAPP(hAPP(c_in(B),C),F)) | -c_Finite__Set_Ofinite(F,B) # label(cls_setprod__gen__delta_1) # label(axiom). [assumption].
% 1.82/2.07 126 -class_OrderedGroup_Ocomm__monoid__mult(A) | c_Finite__Set_Osetprod(c_COMBC(c_COMBS(c_COMBB(c_HOL_OIf(A),hAPP(c_fequal(B),C),tc_bool,tc_fun(A,tc_fun(A,A)),B),D,B,A,tc_fun(A,A)),c_HOL_Oone__class_Oone(A),B,A,A),E,B,A) = hAPP(D,C) | -hBOOL(hAPP(hAPP(c_in(B),C),E)) | -c_Finite__Set_Ofinite(E,B) # label(cls_setprod__delta_H_0) # label(axiom). [assumption].
% 1.82/2.07 127 -class_OrderedGroup_Ocomm__monoid__mult(A) | c_Finite__Set_Osetprod(c_COMBC(c_COMBS(c_COMBB(c_HOL_OIf(A),hAPP(c_fequal(B),C),tc_bool,tc_fun(A,tc_fun(A,A)),B),D,B,A,tc_fun(A,A)),c_HOL_Oone__class_Oone(A),B,A,A),E,B,A) = c_HOL_Oone__class_Oone(A) | hBOOL(hAPP(hAPP(c_in(B),C),E)) | -c_Finite__Set_Ofinite(E,B) # label(cls_setprod__delta_H_1) # label(axiom). [assumption].
% 1.82/2.07 128 -class_OrderedGroup_Ocomm__monoid__mult(A) | c_Finite__Set_Osetprod(B,C,D,A) = c_HOL_Oone__class_Oone(A) | c_Finite__Set_Ofinite(C,D) # label(cls_setprod__def_1) # label(axiom). [assumption].
% 1.82/2.07 129 -class_OrderedGroup_Ocomm__monoid__mult(A) | c_Finite__Set_Osetprod(B,C,D,A) = c_HOL_Oone__class_Oone(A) | c_Finite__Set_Ofinite(C,D) # label(cls_setprod__infinite_0) # label(axiom). [assumption].
% 1.82/2.07 130 -class_OrderedGroup_Ocomm__monoid__mult(A) | c_Finite__Set_Osetprod(c_COMBC(c_COMBS(c_COMBB(c_HOL_OIf(A),c_COMBC(c_fequal(B),C,B,B,tc_bool),tc_bool,tc_fun(A,tc_fun(A,A)),B),D,B,A,tc_fun(A,A)),c_HOL_Oone__class_Oone(A),B,A,A),E,B,A) = c_HOL_Oone__class_Oone(A) | hBOOL(hAPP(hAPP(c_in(B),C),E)) | -c_Finite__Set_Ofinite(E,B) # label(cls_setprod__delta_1) # label(axiom). [assumption].
% 1.82/2.07 131 -class_OrderedGroup_Ocomm__monoid__mult(A) | c_Finite__Set_Osetprod(c_COMBC(c_COMBS(c_COMBB(c_HOL_OIf(A),c_COMBC(c_fequal(B),C,B,B,tc_bool),tc_bool,tc_fun(A,tc_fun(A,A)),B),D,B,A,tc_fun(A,A)),c_HOL_Oone__class_Oone(A),B,A,A),E,B,A) = hAPP(D,C) | -hBOOL(hAPP(hAPP(c_in(B),C),E)) | -c_Finite__Set_Ofinite(E,B) # label(cls_setprod__delta_0) # label(axiom). [assumption].
% 1.82/2.07 Derived: c_Finite__Set_Osetprod(A,c_Orderings_Obot__class_Obot(tc_fun(B,tc_bool)),B,tc_nat) = c_HOL_Oone__class_Oone(tc_nat). [resolve(123,a,124,a)].
% 1.82/2.07 Derived: c_Finite__Set_Osetprod(c_COMBC(c_COMBS(c_COMBB(c_HOL_OIf(tc_nat),c_COMBC(c_fequal(A),B,A,A,tc_bool),tc_bool,tc_fun(tc_nat,tc_fun(tc_nat,tc_nat)),A),C,A,tc_nat,tc_fun(tc_nat,tc_nat)),D,A,tc_nat,tc_nat),E,A,tc_nat) = c_Power_Opower__class_Opower(D,c_Finite__Set_Ocard(E,A),tc_nat) | hBOOL(hAPP(hAPP(c_in(A),B),E)) | -c_Finite__Set_Ofinite(E,A). [resolve(123,a,125,a)].
% 1.82/2.07 Derived: c_Finite__Set_Osetprod(c_COMBC(c_COMBS(c_COMBB(c_HOL_OIf(tc_nat),hAPP(c_fequal(A),B),tc_bool,tc_fun(tc_nat,tc_fun(tc_nat,tc_nat)),A),C,A,tc_nat,tc_fun(tc_nat,tc_nat)),c_HOL_Oone__class_Oone(tc_nat),A,tc_nat,tc_nat),D,A,tc_nat) = hAPP(C,B) | -hBOOL(hAPP(hAPP(c_in(A),B),D)) | -c_Finite__Set_Ofinite(D,A). [resolve(123,a,126,a)].
% 1.82/2.07 Derived: c_Finite__Set_Osetprod(c_COMBC(c_COMBS(c_COMBB(c_HOL_OIf(tc_nat),hAPP(c_fequal(A),B),tc_bool,tc_fun(tc_nat,tc_fun(tc_nat,tc_nat)),A),C,A,tc_nat,tc_fun(tc_nat,tc_nat)),c_HOL_Oone__class_Oone(tc_nat),A,tc_nat,tc_nat),D,A,tc_nat) = c_HOL_Oone__class_Oone(tc_nat) | hBOOL(hAPP(hAPP(c_in(A),B),D)) | -c_Finite__Set_Ofinite(D,A). [resolve(123,a,127,a)].
% 1.82/2.07 Derived: c_Finite__Set_Osetprod(A,B,C,tc_nat) = c_HOL_Oone__class_Oone(tc_nat) | c_Finite__Set_Ofinite(B,C). [resolve(123,a,128,a)].
% 1.84/2.11 Derived: c_Finite__Set_Osetprod(c_COMBC(c_COMBS(c_COMBB(c_HOL_OIf(tc_nat),c_COMBC(c_fequal(A),B,A,A,tc_bool),tc_bool,tc_fun(tc_nat,tc_fun(tc_nat,tc_nat)),A),C,A,tc_nat,tc_fun(tc_nat,tc_nat)),c_HOL_Oone__class_Oone(tc_nat),A,tc_nat,tc_nat),D,A,tc_nat) = c_HOL_Oone__class_Oone(tc_nat) | hBOOL(hAPP(hAPP(c_in(A),B),D)) | -c_Finite__Set_Ofinite(D,A). [resolve(123,a,130,a)].
% 1.84/2.11 Derived: c_Finite__Set_Osetprod(c_COMBC(c_COMBS(c_COMBB(c_HOL_OIf(tc_nat),c_COMBC(c_fequal(A),B,A,A,tc_bool),tc_bool,tc_fun(tc_nat,tc_fun(tc_nat,tc_nat)),A),C,A,tc_nat,tc_fun(tc_nat,tc_nat)),c_HOL_Oone__class_Oone(tc_nat),A,tc_nat,tc_nat),D,A,tc_nat) = hAPP(C,B) | -hBOOL(hAPP(hAPP(c_in(A),B),D)) | -c_Finite__Set_Ofinite(D,A). [resolve(123,a,131,a)].
% 1.84/2.11 132 hBOOL(hAPP(hAPP(c_in(tc_prod(A,A)),c_Pair(B,B,A,A)),C)) | -hBOOL(hAPP(hAPP(c_in(A),B),D)) | -c_Relation_Orefl__on(D,C,A) # label(cls_refl__on__def_1) # label(axiom). [assumption].
% 1.84/2.11 133 c_Relation_Orefl__on(c_Orderings_Otop__class_Otop(tc_fun(A,tc_bool)),c_Transitive__Closure_Ortrancl(B,A),A) # label(cls_refl__rtrancl_0) # label(axiom). [assumption].
% 1.84/2.11 134 c_Relation_Orefl__on(c_Orderings_Otop__class_Otop(tc_fun(A,tc_bool)),c_Relation_OId(A),A) # label(cls_refl__Id_0) # label(axiom). [assumption].
% 1.84/2.11 135 c_Relation_Orefl__on(A,c_Relation_OId__on(A,B),B) # label(cls_refl__on__Id__on_0) # label(axiom). [assumption].
% 1.84/2.11 Derived: hBOOL(hAPP(hAPP(c_in(tc_prod(A,A)),c_Pair(B,B,A,A)),c_Transitive__Closure_Ortrancl(C,A))) | -hBOOL(hAPP(hAPP(c_in(A),B),c_Orderings_Otop__class_Otop(tc_fun(A,tc_bool)))). [resolve(132,c,133,a)].
% 1.84/2.11 Derived: hBOOL(hAPP(hAPP(c_in(tc_prod(A,A)),c_Pair(B,B,A,A)),c_Relation_OId(A))) | -hBOOL(hAPP(hAPP(c_in(A),B),c_Orderings_Otop__class_Otop(tc_fun(A,tc_bool)))). [resolve(132,c,134,a)].
% 1.84/2.11 Derived: hBOOL(hAPP(hAPP(c_in(tc_prod(A,A)),c_Pair(B,B,A,A)),c_Relation_OId__on(C,A))) | -hBOOL(hAPP(hAPP(c_in(A),B),C)). [resolve(132,c,135,a)].
% 1.84/2.11 136 hBOOL(hAPP(hAPP(c_in(A),B),C)) | -hBOOL(hAPP(hAPP(c_in(tc_prod(A,A)),c_Pair(D,B,A,A)),E)) | -c_Relation_Orefl__on(C,E,A) # label(cls_refl__onD2_0) # label(axiom). [assumption].
% 1.84/2.11 Derived: hBOOL(hAPP(hAPP(c_in(A),B),c_Orderings_Otop__class_Otop(tc_fun(A,tc_bool)))) | -hBOOL(hAPP(hAPP(c_in(tc_prod(A,A)),c_Pair(C,B,A,A)),c_Transitive__Closure_Ortrancl(D,A))). [resolve(136,c,133,a)].
% 1.84/2.11 Derived: hBOOL(hAPP(hAPP(c_in(A),B),c_Orderings_Otop__class_Otop(tc_fun(A,tc_bool)))) | -hBOOL(hAPP(hAPP(c_in(tc_prod(A,A)),c_Pair(C,B,A,A)),c_Relation_OId(A))). [resolve(136,c,134,a)].
% 1.84/2.11 Derived: hBOOL(hAPP(hAPP(c_in(A),B),C)) | -hBOOL(hAPP(hAPP(c_in(tc_prod(A,A)),c_Pair(D,B,A,A)),c_Relation_OId__on(C,A))). [resolve(136,c,135,a)].
% 1.84/2.11 137 hBOOL(hAPP(hAPP(c_in(A),B),C)) | -hBOOL(hAPP(hAPP(c_in(tc_prod(A,A)),c_Pair(B,D,A,A)),E)) | -c_Relation_Orefl__on(C,E,A) # label(cls_refl__onD1_0) # label(axiom). [assumption].
% 1.84/2.11 Derived: hBOOL(hAPP(hAPP(c_in(A),B),c_Orderings_Otop__class_Otop(tc_fun(A,tc_bool)))) | -hBOOL(hAPP(hAPP(c_in(tc_prod(A,A)),c_Pair(B,C,A,A)),c_Transitive__Closure_Ortrancl(D,A))). [resolve(137,c,133,a)].
% 1.84/2.11 Derived: hBOOL(hAPP(hAPP(c_in(A),B),c_Orderings_Otop__class_Otop(tc_fun(A,tc_bool)))) | -hBOOL(hAPP(hAPP(c_in(tc_prod(A,A)),c_Pair(B,C,A,A)),c_Relation_OId(A))). [resolve(137,c,134,a)].
% 1.84/2.11 Derived: hBOOL(hAPP(hAPP(c_in(A),B),C)) | -hBOOL(hAPP(hAPP(c_in(tc_prod(A,A)),c_Pair(B,D,A,A)),c_Relation_OId__on(C,A))). [resolve(137,c,135,a)].
% 1.84/2.11 138 hBOOL(hAPP(hAPP(c_in(tc_prod(A,A)),c_Pair(B,B,A,A)),C)) | -hBOOL(hAPP(hAPP(c_in(A),B),D)) | -c_Relation_Orefl__on(D,C,A) # label(cls_refl__onD_0) # label(axiom). [assumption].
% 1.84/2.11 139 A = B | -hBOOL(hAPP(hAPP(c_in(tc_prod(C,C)),c_Pair(B,A,C,C)),D)) | -hBOOL(hAPP(hAPP(c_in(tc_prod(C,C)),c_Pair(A,B,C,C)),D)) | -c_Relation_Oantisym(D,C) # label(cls_antisym__def_0) # label(axiom). [assumption].
% 1.84/2.11 140 c_Relation_Oantisym(c_Relation_OId__on(A,B),B) # label(cls_antisym__Id__on_0) # label(axiom). [assumption].
% 1.84/2.11 141 c_Relation_Oantisym(c_Relation_OId(A),A) # label(cls_antisym__Id_0) # label(axiom). [assumption].
% 1.84/2.11 142 A = B | -hBOOL(hAPP(hAPP(c_in(tc_prod(C,C)),c_Pair(B,A,C,C)),D)) | -hBOOL(hAPP(hAPP(c_in(tc_prod(C,C)),c_Pair(A,B,C,C)),D)) | -c_Relation_Oantisym(D,C) # label(cls_antisymD_0) # label(axiom). [assumption].
% 1.89/2.17 143 class_OrderedGroup_Ocancel__semigroup__add(tc_nat) # label(clsarity_nat__OrderedGroup_Ocancel__semigroup__add) # label(axiom). [assumption].
% 1.89/2.17 144 -class_OrderedGroup_Ocancel__semigroup__add(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C) != hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),D),C) | B = D # label(cls_add__right__cancel_0) # label(axiom). [assumption].
% 1.89/2.17 145 -class_OrderedGroup_Ocancel__semigroup__add(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C) != hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),D) | C = D # label(cls_add__left__cancel_0) # label(axiom). [assumption].
% 1.89/2.17 Derived: hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B) != hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),C),B) | A = C. [resolve(143,a,144,a)].
% 1.89/2.17 Derived: hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B) != hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),C) | B = C. [resolve(143,a,145,a)].
% 1.89/2.17 146 class_OrderedGroup_Ocancel__ab__semigroup__add(tc_nat) # label(clsarity_nat__OrderedGroup_Ocancel__ab__semigroup__add) # label(axiom). [assumption].
% 1.89/2.17 147 -class_OrderedGroup_Ocancel__ab__semigroup__add(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C) != hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),D) | C = D # label(cls_add__imp__eq_0) # label(axiom). [assumption].
% 1.89/2.17 148 class_OrderedGroup_Opordered__comm__monoid__add(tc_nat) # label(clsarity_nat__OrderedGroup_Opordered__comm__monoid__add) # label(axiom). [assumption].
% 1.89/2.17 149 -class_OrderedGroup_Opordered__comm__monoid__add(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C))) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),C,A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),B)) # label(cls_add__pos__nonneg_0) # label(axiom). [assumption].
% 1.89/2.17 150 -class_OrderedGroup_Opordered__comm__monoid__add(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),C)) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),B,A) # label(cls_add__nonneg__pos_0) # label(axiom). [assumption].
% 1.89/2.17 151 -class_OrderedGroup_Opordered__comm__monoid__add(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C)),c_HOL_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),C),c_HOL_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),c_HOL_Ozero__class_Ozero(A))) # label(cls_add__neg__neg_0) # label(axiom). [assumption].
% 1.89/2.17 152 -class_OrderedGroup_Opordered__comm__monoid__add(A) | c_lessequals(c_HOL_Ozero__class_Ozero(A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C),A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),C,A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),B,A) # label(cls_add__nonneg__nonneg_0) # label(axiom). [assumption].
% 1.89/2.17 153 -class_OrderedGroup_Opordered__comm__monoid__add(A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),c_HOL_Ozero__class_Ozero(A),A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),c_HOL_Ozero__class_Ozero(A),A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),c_HOL_Ozero__class_Ozero(A)),c_HOL_Ozero__class_Ozero(A)) = c_HOL_Ozero__class_Ozero(A) # label(cls_add__nonneg__eq__0__iff_2) # label(axiom). [assumption].
% 1.89/2.17 154 -class_OrderedGroup_Opordered__comm__monoid__add(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C) != c_HOL_Ozero__class_Ozero(A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),C,A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),B,A) | C = c_HOL_Ozero__class_Ozero(A) # label(cls_add__nonneg__eq__0__iff_1) # label(axiom). [assumption].
% 1.89/2.17 155 -class_OrderedGroup_Opordered__comm__monoid__add(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C) != c_HOL_Ozero__class_Ozero(A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),C,A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),B,A) | B = c_HOL_Ozero__class_Ozero(A) # label(cls_add__nonneg__eq__0__iff_0) # label(axiom). [assumption].
% 1.89/2.17 156 -class_OrderedGroup_Opordered__comm__monoid__add(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C)),c_HOL_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),C),c_HOL_Ozero__class_Ozero(A))) | -c_lessequals(B,c_HOL_Ozero__class_Ozero(A),A) # label(cls_add__nonpos__neg_0) # label(axiom). [assumption].
% 1.89/2.18 157 -class_OrderedGroup_Opordered__comm__monoid__add(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C)),c_HOL_Ozero__class_Ozero(A))) | -c_lessequals(C,c_HOL_Ozero__class_Ozero(A),A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),c_HOL_Ozero__class_Ozero(A))) # label(cls_add__neg__nonpos_0) # label(axiom). [assumption].
% 1.89/2.18 158 -class_OrderedGroup_Opordered__comm__monoid__add(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),C)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),B)) # label(cls_add__pos__pos_0) # label(axiom). [assumption].
% 1.89/2.18 159 -class_OrderedGroup_Opordered__comm__monoid__add(A) | c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C),c_HOL_Ozero__class_Ozero(A),A) | -c_lessequals(C,c_HOL_Ozero__class_Ozero(A),A) | -c_lessequals(B,c_HOL_Ozero__class_Ozero(A),A) # label(cls_add__nonpos__nonpos_0) # label(axiom). [assumption].
% 1.89/2.18 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B))) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),B,tc_nat) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),A)). [resolve(148,a,149,a)].
% 1.89/2.18 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),B)) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),A,tc_nat). [resolve(148,a,150,a)].
% 1.89/2.18 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B)),c_HOL_Ozero__class_Ozero(tc_nat))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),B),c_HOL_Ozero__class_Ozero(tc_nat))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),c_HOL_Ozero__class_Ozero(tc_nat))). [resolve(148,a,151,a)].
% 1.89/2.18 Derived: c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B),tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),B,tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),A,tc_nat). [resolve(148,a,152,a)].
% 1.89/2.18 Derived: hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B) != c_HOL_Ozero__class_Ozero(tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),B,tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),A,tc_nat) | B = c_HOL_Ozero__class_Ozero(tc_nat). [resolve(148,a,154,a)].
% 1.89/2.18 Derived: hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B) != c_HOL_Ozero__class_Ozero(tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),B,tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),A,tc_nat) | A = c_HOL_Ozero__class_Ozero(tc_nat). [resolve(148,a,155,a)].
% 1.89/2.18 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B)),c_HOL_Ozero__class_Ozero(tc_nat))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),B),c_HOL_Ozero__class_Ozero(tc_nat))) | -c_lessequals(A,c_HOL_Ozero__class_Ozero(tc_nat),tc_nat). [resolve(148,a,156,a)].
% 1.89/2.18 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B)),c_HOL_Ozero__class_Ozero(tc_nat))) | -c_lessequals(B,c_HOL_Ozero__class_Ozero(tc_nat),tc_nat) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),c_HOL_Ozero__class_Ozero(tc_nat))). [resolve(148,a,157,a)].
% 1.89/2.18 Derived: c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B),c_HOL_Ozero__class_Ozero(tc_nat),tc_nat) | -c_lessequals(B,c_HOL_Ozero__class_Ozero(tc_nat),tc_nat) | -c_lessequals(A,c_HOL_Ozero__class_Ozero(tc_nat),tc_nat). [resolve(148,a,159,a)].
% 1.89/2.18 160 class_OrderedGroup_Omonoid__add(tc_nat) # label(clsarity_nat__OrderedGroup_Omonoid__add) # label(axiom). [assumption].
% 1.89/2.21 161 -class_OrderedGroup_Omonoid__add(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),c_HOL_Ozero__class_Ozero(A)),B) = B # label(cls_add__0__left_0) # label(axiom). [assumption].
% 1.89/2.21 162 -class_OrderedGroup_Omonoid__add(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),c_HOL_Ozero__class_Ozero(A)) = B # label(cls_monoid__add__class_Oadd__0__right_0) # label(axiom). [assumption].
% 1.89/2.21 163 class_OrderedGroup_Opordered__cancel__ab__semigroup__add(tc_nat) # label(clsarity_nat__OrderedGroup_Opordered__cancel__ab__semigroup__add) # label(axiom). [assumption].
% 1.89/2.21 164 -class_OrderedGroup_Opordered__cancel__ab__semigroup__add(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),D),E))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),C),E)) | -c_lessequals(B,D,A) # label(cls_add__le__less__mono_0) # label(axiom). [assumption].
% 1.89/2.21 165 -class_OrderedGroup_Opordered__cancel__ab__semigroup__add(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),D),E))) | -c_lessequals(C,E,A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),D)) # label(cls_add__less__le__mono_0) # label(axiom). [assumption].
% 1.89/2.21 166 -class_OrderedGroup_Opordered__cancel__ab__semigroup__add(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),D))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),C),D)) # label(cls_add__strict__left__mono_0) # label(axiom). [assumption].
% 1.89/2.21 167 -class_OrderedGroup_Opordered__cancel__ab__semigroup__add(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),D),C))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),D)) # label(cls_add__strict__right__mono_0) # label(axiom). [assumption].
% 1.89/2.21 168 -class_OrderedGroup_Opordered__cancel__ab__semigroup__add(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),D),E))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),C),E)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),D)) # label(cls_add__strict__mono_0) # label(axiom). [assumption].
% 1.89/2.21 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),C),D))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),B),D)) | -c_lessequals(A,C,tc_nat). [resolve(163,a,164,a)].
% 1.89/2.21 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),C),D))) | -c_lessequals(B,D,tc_nat) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),C)). [resolve(163,a,165,a)].
% 1.89/2.21 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),C))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),B),C)). [resolve(163,a,166,a)].
% 1.89/2.21 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),C),B))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),C)). [resolve(163,a,167,a)].
% 1.89/2.21 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),C),D))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),B),D)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),C)). [resolve(163,a,168,a)].
% 1.89/2.21 169 class_Orderings_Olinorder(tc_nat) # label(clsarity_nat__Orderings_Olinorder) # label(axiom). [assumption].
% 1.89/2.21 170 -class_Orderings_Olinorder(A) | B = C | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),C)) | -c_lessequals(B,C,A) # label(cls_linorder__antisym__conv2_0) # label(axiom). [assumption].
% 1.89/2.21 171 -class_Orderings_Olinorder(A) | B = C | -c_lessequals(B,C,A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),C)) # label(cls_linorder__antisym__conv1_0) # label(axiom). [assumption].
% 1.97/2.23 172 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),C)) | c_lessequals(C,B,A) # label(cls_not__leE_0) # label(axiom). [assumption].
% 1.97/2.23 173 -class_Orderings_Olinorder(A) | -c_lessequals(B,B,A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),B)) # label(cls_linorder__antisym__conv2_1) # label(axiom). [assumption].
% 1.97/2.23 174 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),B)) | c_lessequals(B,B,A) # label(cls_linorder__antisym__conv1_1) # label(axiom). [assumption].
% 1.97/2.23 175 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),C)) | -c_lessequals(C,B,A) # label(cls_linorder__not__less_1) # label(axiom). [assumption].
% 1.97/2.23 176 -class_Orderings_Olinorder(A) | c_lessequals(B,C,A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),C),B)) # label(cls_linorder__not__less_0) # label(axiom). [assumption].
% 1.97/2.23 177 -class_Orderings_Olinorder(A) | -c_lessequals(B,C,A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),C),B)) # label(cls_linorder__not__le_1) # label(axiom). [assumption].
% 1.97/2.23 178 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),C)) | c_lessequals(C,B,A) # label(cls_linorder__not__le_0) # label(axiom). [assumption].
% 1.97/2.23 179 -class_Orderings_Olinorder(A) | c_lessequals(B,C,A) | c_lessequals(C,B,A) # label(cls_linorder__linear_0) # label(axiom). [assumption].
% 1.97/2.23 180 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),C),B)) # label(cls_not__less__iff__gr__or__eq_1) # label(axiom). [assumption].
% 1.97/2.23 181 -class_Orderings_Olinorder(A) | B = C | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),C),B)) # label(cls_linorder__antisym__conv3_0) # label(axiom). [assumption].
% 1.97/2.23 182 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),C)) | C = B | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),C),B)) # label(cls_linorder__less__linear_0) # label(axiom). [assumption].
% 1.97/2.23 183 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),C),B)) | C = B # label(cls_linorder__neqE_0) # label(axiom). [assumption].
% 1.97/2.23 184 -class_Orderings_Olinorder(A) | B = C | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),C),B)) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),C)) # label(cls_not__less__iff__gr__or__eq_0) # label(axiom). [assumption].
% 1.97/2.23 185 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),B)) # label(cls_linorder__neq__iff_1) # label(axiom). [assumption].
% 1.97/2.23 Derived: A = B | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),B)) | -c_lessequals(A,B,tc_nat). [resolve(169,a,170,a)].
% 1.97/2.23 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),B)) | c_lessequals(B,A,tc_nat). [resolve(169,a,172,a)].
% 1.97/2.23 Derived: -c_lessequals(A,A,tc_nat) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),A)). [resolve(169,a,173,a)].
% 1.97/2.23 Derived: -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),B)) | -c_lessequals(B,A,tc_nat). [resolve(169,a,175,a)].
% 1.97/2.23 Derived: c_lessequals(A,B,tc_nat) | c_lessequals(B,A,tc_nat). [resolve(169,a,179,a)].
% 1.97/2.23 Derived: -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),B)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),B),A)). [resolve(169,a,180,a)].
% 1.97/2.23 Derived: A = B | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),B)) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),B),A)). [resolve(169,a,181,a)].
% 1.97/2.23 186 class_OrderedGroup_Oab__semigroup__add(tc_nat) # label(clsarity_nat__OrderedGroup_Oab__semigroup__add) # label(axiom). [assumption].
% 1.97/2.23 187 -class_OrderedGroup_Oab__semigroup__add(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C)),D) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),C),D)) # label(cls_ab__semigroup__add__class_Oadd__ac_I1_J_0) # label(axiom). [assumption].
% 1.97/2.23 188 class_OrderedGroup_Opordered__ab__semigroup__add(tc_nat) # label(clsarity_nat__OrderedGroup_Opordered__ab__semigroup__add) # label(axiom). [assumption].
% 2.05/2.37 189 -class_OrderedGroup_Opordered__ab__semigroup__add(A) | c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),D),E),A) | -c_lessequals(C,E,A) | -c_lessequals(B,D,A) # label(cls_add__mono_0) # label(axiom). [assumption].
% 2.05/2.37 190 -class_OrderedGroup_Opordered__ab__semigroup__add(A) | c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),D),C),A) | -c_lessequals(B,D,A) # label(cls_add__right__mono_0) # label(axiom). [assumption].
% 2.05/2.37 191 -class_OrderedGroup_Opordered__ab__semigroup__add(A) | c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),D),A) | -c_lessequals(C,D,A) # label(cls_add__left__mono_0) # label(axiom). [assumption].
% 2.05/2.37 Derived: c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),C),D),tc_nat) | -c_lessequals(B,D,tc_nat) | -c_lessequals(A,C,tc_nat). [resolve(188,a,189,a)].
% 2.05/2.37 Derived: c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),C),B),tc_nat) | -c_lessequals(A,C,tc_nat). [resolve(188,a,190,a)].
% 2.05/2.37 Derived: c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),C),tc_nat) | -c_lessequals(B,C,tc_nat). [resolve(188,a,191,a)].
% 2.05/2.37 192 class_Nat_Osemiring__char__0(tc_nat) # label(clsarity_nat__Nat_Osemiring__char__0) # label(axiom). [assumption].
% 2.05/2.37 193 -class_Nat_Osemiring__char__0(A) | c_Nat_Osemiring__1__class_Oof__nat(B,A) != c_Nat_Osemiring__1__class_Oof__nat(C,A) | B = C # label(cls_of__nat__eq__iff_0) # label(axiom). [assumption].
% 2.05/2.37 Derived: c_Nat_Osemiring__1__class_Oof__nat(A,tc_nat) != c_Nat_Osemiring__1__class_Oof__nat(B,tc_nat) | A = B. [resolve(192,a,193,a)].
% 2.05/2.37
% 2.05/2.37 ============================== end predicate elimination =============
% 2.05/2.37
% 2.05/2.37 Auto_denials: (non-Horn, no changes).
% 2.05/2.37
% 2.05/2.37 Term ordering decisions:
% 2.05/2.37 Function symbol KB weights: tc_Arrow__Order__Mirabelle_Oalt=1. tc_nat=1. tc_bool=1. tc_Arrow__Order__Mirabelle_Oindi=1. v_e____=1. v_h____=1. v_n____=1. c_Arrow__Order__Mirabelle_OProf=1. v_P____=1. t_a=1. v_d____=1. v_a____=1. v_b____=1. c_Arrow__Order__Mirabelle_Omktop=1. c_Arrow__Order__Mirabelle_Omkbot=1. v_Lba____=1. v_c____=1. v_Lab____=1. c_Arrow__Order__Mirabelle_Oabove=1. hAPP=1. tc_fun=1. tc_prod=1. c_Transitive__Closure_Ortrancl=1. c_Transitive__Closure_Otrancl=1. c_Wellfounded_Oacc=1. c_Nat_Osemiring__1__class_Oof__nat=1. c_Finite__Set_Ocard=1. c_Relation_OId__on=1. c_Wellfounded_Omeasure=1. v_sko__Wellfounded__Xacc__Xinduct__1=1. v_sko__Wellfounded__Xacc__Xinducts__1=1. v_sko__Arrow__Order__Mirabelle__Xcomplete__Lin__1=1. c_in=1. c_HOL_Oord__class_Oless=1. c_HOL_Oplus__class_Oplus=1. c_HOL_Ozero__class_Ozero=1. c_Orderings_Obot__class_Obot=1. v_F=1. c_Orderings_Otop__class_Otop=1. c_HOL_Oone__class_Oone=1. c_HOL_OIf=1. c_HOL_Ouminus__class_Ouminus=1. c_fequal=1. c_Relation_OId=1. c_Lattices_Olower__semilattice__class_Oinf=1. c_Set_Oinsert=1. c_Power_Opower__class_Opower=1. c_Relation_ODomain=1. c_Relation_ORange=1. c_ATP__Linkup_Osko__Wellfounded__Xacc__induct__1__1=1. c_ATP__Linkup_Osko__Wellfounded__Xacc__induct__rule__1__1=1. c_Wellfounded_Omlex__prod=1. c_ATP__Linkup_Osko__Wellfounded__Xacc__Xintros__1__1=1. c_ATP__Linkup_Osko__Wellfounded__Xnot__acc__down__1__1=1. c_ATP__Linkup_Osko__Wellfounded__XwfE__min__1__1=1. c_ATP__Linkup_Osko__Wellfounded__Xwf__eq__minimal__1__1=1. c_Pair=1. c_Set_Ovimage=1. c_Finite__Set_Osetsum=1. c_Finite__Set_Osetprod=1. c_Relation_OImage=1. v_sko__local__X3__1=1. c_Product__Type_OSigma=1. c_Relation_Oinv__image=1. c_ATP__Linkup_Osko__Finite__Set__Xsetprod__zero__iff__1__1=1. c_ATP__Linkup_Osko__Transitive__Closure__XtranclD2__1__1=1. c_ATP__Linkup_Osko__Transitive__Closure__XtranclD__1__1=1. c_Nat_Osemiring__1__class_Oof__nat__aux=1. c_ATP__Linkup_Osko__Relation__XDomainE__1__1=1. c_ATP__Linkup_Osko__Relation__XDomain__iff__1__1=1. c_ATP__Linkup_Osko__Relation__XRangeE__1__1=1. c_ATP__Linkup_Osko__Relation__XRange__iff__1__1=1. c_COMBC=1. c_COMBB=1. c_COMBS=1. c_Relation_Orel__comp=1. v_sko__local__X4__1=1. c_Recdef_Ocut=1. c_ATP__Linkup_Osko__Relation__XImageE__1__1=1. c_ATP__Linkup_Osko__Relation__XImage__iff__1__1=1. v_sko__local__X2__1=1. v_sko__local__Xpairwise__neutrality__1=1. c_Finite__Set_Ofold__image=1. v_sko__local__X1__1=1.
% 5.73/6.00
% 5.73/6.00 ============================== end of process initial clauses ========
% 5.73/6.00
% 5.73/6.00 ============================== CLAUSES FOR SEARCH ====================
% 5.73/6.00
% 5.73/6.00 ============================== end of clauses for search =============
% 5.73/6.00
% 5.73/6.00 ============================== SEARCH ================================
% 5.73/6.00
% 5.73/6.00 % Starting search at 0.95 seconds.
% 5.73/6.00
% 5.73/6.00 Low Water (keep): wt=144.000, iters=3336
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% 5.73/6.00 Low Water (keep): wt=140.000, iters=3374
% 5.73/6.00
% 5.73/6.00 Low Water (keep): wt=137.000, iters=3349
% 5.73/6.00
% 5.73/6.00 NOTE: Back_subsumption disabled, ratio of kept to back_subsumed is 90 (0.00 of 1.93 sec).
% 5.73/6.00
% 5.73/6.00 Low Water (keep): wt=130.000, iters=3438
% 5.73/6.00
% 5.73/6.00 Low Water (keep): wt=113.000, iters=3618
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% 5.73/6.00 Low Water (keep): wt=83.000, iters=3364
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% 5.73/6.00 Low Water (keep): wt=65.000, iters=3345
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% 5.73/6.00 Low Water (keep): wt=57.000, iters=3363
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% 5.73/6.00 Low Water (keep): wt=56.000, iters=3335
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% 5.73/6.00 Low Water (keep): wt=47.000, iters=3364
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% 5.73/6.00 Low Water (keep): wt=45.000, iters=3373
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% 5.73/6.00 Low Water (keep): wt=43.000, iters=3369
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% 5.73/6.00 Low Water (keep): wt=42.000, iters=3378
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% 5.73/6.00 Low Water (keep): wt=40.000, iters=3346
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% 5.73/6.00 Low Water (keep): wt=37.000, iters=3336
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% 5.73/6.00 Low Water (keep): wt=36.000, iters=3367
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% 5.73/6.00 Low Water (keep): wt=35.000, iters=3339
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% 5.73/6.00 Low Water (keep): wt=34.000, iters=3370
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% 5.73/6.00 Low Water (keep): wt=33.000, iters=3344
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% 5.73/6.00 Low Water (keep): wt=32.000, iters=3368
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% 5.73/6.00 Low Water (keep): wt=31.000, iters=3349
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% 5.73/6.00 Low Water (keep): wt=29.000, iters=3395
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% 5.73/6.00 Low Water (keep): wt=28.000, iters=3418
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% 5.73/6.00 Low Water (keep): wt=27.000, iters=3493
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% 5.73/6.00 Low Water (keep): wt=25.000, iters=3370
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% 5.73/6.00 Low Water (keep): wt=24.000, iters=3751
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% 5.73/6.00 Low Water (keep): wt=20.000, iters=3370
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% 5.73/6.00 Low Water (keep): wt=18.000, iters=3541
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% 5.73/6.00 Low Water (keep): wt=16.000, iters=3375
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% 5.73/6.00 Low Water (keep): wt=14.000, iters=3350
% 5.73/6.00
% 5.73/6.00 Low Water (displace): id=1656, wt=200.000
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% 5.73/6.00 Low Water (displace): id=4168, wt=195.000
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% 5.73/6.00 Low Water (displace): id=4140, wt=190.000
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% 5.73/6.00 Low Water (displace): id=4156, wt=189.000
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% 5.73/6.00 Low Water (displace): id=3011, wt=187.000
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% 5.73/6.00 Low Water (displace): id=3015, wt=186.000
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% 5.73/6.00 Low Water (displace): id=3028, wt=185.000
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% 5.73/6.00 Low Water (displace): id=3017, wt=184.000
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% 5.73/6.00 Low Water (displace): id=4148, wt=182.000
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% 5.73/6.00 Low Water (displace): id=3014, wt=181.000
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% 5.73/6.00 Low Water (displace): id=3021, wt=179.000
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% 5.73/6.00 Low Water (displace): id=1542, wt=176.000
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% 5.73/6.00 Low Water (displace): id=6951, wt=112.000
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% 5.73/6.00 Low Water (displace): id=6949, wt=110.000
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% 5.73/6.00 Low Water (displace): id=3024, wt=107.000
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% 5.73/6.00 Low Water (displace): id=5329, wt=103.000
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% 5.73/6.00 Low Water (displace): id=5325, wt=101.000
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% 5.73/6.00 Low Water (displace): id=9250, wt=99.000
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% 5.73/6.00 Low Water (displace): id=9251, wt=97.000
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% 5.73/6.00 Low Water (displace): id=9236, wt=95.000
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% 5.73/6.00 Low Water (displace): id=4611, wt=85.000
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% 5.73/6.00 Low Water (displace): id=11432, wt=63.000
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% 5.73/6.00 Low Water (displace): id=11428, wt=59.000
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% 5.73/6.00 Low Water (displace): id=6387, wt=55.000
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% 5.73/6.00 Low Water (displace): id=10718, wt=54.000
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% 5.73/6.00 Low Water (displace): id=11750, wt=52.000
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% 5.73/6.00 Low Water (displace): id=11596, wt=51.000
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% 5.73/6.00 Low Water (displace): id=7865, wt=50.000
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% 5.73/6.00 Low Water (displace): id=9643, wt=49.000
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% 5.73/6.00 Low Water (displace): id=11752, wt=48.000
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% 5.73/6.00 Low Water (displace): id=11859, wt=47.000
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% 5.73/6.00 Low Water (displace): id=11909, wt=46.000
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% 5.73/6.00 Low Water (displace): id=11947, wt=45.000
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% 5.73/6.00 Low Water (displace): id=11854, wt=44.000
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% 5.73/6.00 Low Water (displace): id=12026, wt=12.000
% 5.73/6.00
% 5.73/6.00 ============================== PROOF =================================
% 5.73/6.00 % SZS status Unsatisfiable
% 5.73/6.00 % SZS output start Refutation
% 5.73/6.00
% 5.73/6.00 % Proof 1 at 4.52 (+ 0.06) seconds.
% 5.73/6.00 % Length of proof is 42.
% 5.73/6.00 % Level of proof is 16.
% 5.73/6.00 % Maximum clause weight is 36.000.
% 5.73/6.00 % Given clauses 2709.
% 5.73/6.00
% 5.73/6.00 1 class_Ring__and__Field_Ozero__neq__one(tc_nat) # label(clsarity_nat__Ring__and__Field_Ozero__neq__one) # label(axiom). [assumption].
% 5.73/6.00 5 -class_Ring__and__Field_Ozero__neq__one(A) | c_HOL_Ozero__class_Ozero(A) != c_HOL_Oone__class_Oone(A) # label(cls_zero__neq__one_0) # label(axiom). [assumption].
% 5.73/6.00 34 -class_Lattices_Olower__semilattice(A) | c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(B,C,A),C,A) # label(cls_inf__le2_0) # label(axiom). [assumption].
% 5.73/6.00 37 -class_Lattices_Olower__semilattice(A) | c_Lattices_Olower__semilattice__class_Oinf(B,B,A) = B # label(cls_inf__idem_0) # label(axiom). [assumption].
% 5.73/6.00 43 -class_Lattices_Olower__semilattice(A) | c_Lattices_Olower__semilattice__class_Oinf(B,C,A) = c_Lattices_Olower__semilattice__class_Oinf(C,B,A) # label(cls_inf__commute_0) # label(axiom). [assumption].
% 5.73/6.00 52 class_Lattices_Olower__semilattice(tc_nat) # label(clsarity_nat__Lattices_Olower__semilattice) # label(axiom). [assumption].
% 5.73/6.00 60 class_Ring__and__Field_Oordered__semidom(tc_nat) # label(clsarity_nat__Ring__and__Field_Oordered__semidom) # label(axiom). [assumption].
% 5.73/6.00 92 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),c_HOL_Oone__class_Oone(A)))) # label(cls_less__add__one_0) # label(axiom). [assumption].
% 5.73/6.00 169 class_Orderings_Olinorder(tc_nat) # label(clsarity_nat__Orderings_Olinorder) # label(axiom). [assumption].
% 5.73/6.00 175 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),C)) | -c_lessequals(C,B,A) # label(cls_linorder__not__less_1) # label(axiom). [assumption].
% 5.73/6.00 321 A = hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),c_HOL_Ozero__class_Ozero(tc_nat)) # label(cls_natgb_Oadd__r0__iff_1) # label(axiom). [assumption].
% 5.73/6.00 322 hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),c_HOL_Ozero__class_Ozero(tc_nat)) = A. [copy(321),flip(a)].
% 5.73/6.00 323 hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),A) = A # label(cls_plus__nat_Oadd__0_0) # label(axiom). [assumption].
% 5.73/6.00 359 c_lessequals(A,hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),B),C),tc_nat) | -c_lessequals(A,C,tc_nat) # label(cls_termination__basic__simps_I4_J_0) # label(axiom). [assumption].
% 5.73/6.00 463 hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),B),A) # label(cls_nat__add__commute_0) # label(axiom). [assumption].
% 5.73/6.00 557 hAPP(c_COMBB(A,B,C,D,E),F) = hAPP(A,hAPP(B,F)) # label(cls_COMBB__def_0) # label(axiom). [assumption].
% 5.73/6.00 559 hAPP(c_COMBS(A,B,C,D,E),F) = hAPP(hAPP(A,F),hAPP(B,F)) # label(cls_COMBS__def_0) # label(axiom). [assumption].
% 5.73/6.00 620 c_HOL_Ozero__class_Ozero(tc_nat) != c_HOL_Oone__class_Oone(tc_nat). [resolve(1,a,5,a)].
% 5.73/6.00 621 c_HOL_Oone__class_Oone(tc_nat) != c_HOL_Ozero__class_Ozero(tc_nat). [copy(620),flip(a)].
% 5.73/6.00 657 c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_nat),B,tc_nat). [resolve(52,a,34,a)].
% 5.73/6.00 660 c_Lattices_Olower__semilattice__class_Oinf(A,A,tc_nat) = A. [resolve(52,a,37,a)].
% 5.73/6.00 666 c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_nat) = c_Lattices_Olower__semilattice__class_Oinf(B,A,tc_nat). [resolve(52,a,43,a)].
% 5.73/6.00 725 hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),c_HOL_Oone__class_Oone(tc_nat)))). [resolve(60,a,92,a)].
% 5.73/6.00 772 -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),B)) | -c_lessequals(B,A,tc_nat). [resolve(169,a,175,a)].
% 5.73/6.00 854 c_HOL_Oone__class_Oone(tc_nat) = c_0. [new_symbol(621)].
% 5.73/6.00 885 hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),c_0))). [back_rewrite(725),rewrite([854(8)])].
% 5.73/6.00 3771 c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_nat),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),C),B),tc_nat). [resolve(657,a,359,b)].
% 5.73/6.00 4511 hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),c_0),A))). [para(463(a,1),885(a,1,2))].
% 5.73/6.00 15579 -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B)),c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_nat))). [resolve(3771,a,772,b),rewrite([666(9)])].
% 5.73/6.00 15598 -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(c_COMBS(c_HOL_Oplus__class_Oplus(tc_nat),A,B,C,D),E)),c_Lattices_Olower__semilattice__class_Oinf(F,hAPP(A,E),tc_nat))). [para(559(a,2),15579(a,1,1,2)),rewrite([666(10)])].
% 5.73/6.00 15657 -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),hAPP(B,A))),c_Lattices_Olower__semilattice__class_Oinf(C,hAPP(B,A),tc_nat))). [para(559(a,1),15598(a,1,1,2))].
% 5.73/6.00 15676 -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(A,c_HOL_Ozero__class_Ozero(tc_nat))),c_Lattices_Olower__semilattice__class_Oinf(B,hAPP(A,c_HOL_Ozero__class_Ozero(tc_nat)),tc_nat))). [para(323(a,1),15657(a,1,1,2))].
% 5.73/6.00 15695 -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(A,hAPP(B,c_HOL_Ozero__class_Ozero(tc_nat)))),c_Lattices_Olower__semilattice__class_Oinf(C,hAPP(A,hAPP(B,c_HOL_Ozero__class_Ozero(tc_nat))),tc_nat))). [para(557(a,1),15676(a,1,1,2)),rewrite([557(11)])].
% 5.73/6.00 15738 -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(A,B)),c_Lattices_Olower__semilattice__class_Oinf(C,hAPP(A,B),tc_nat))). [para(322(a,1),15695(a,1,1,2,2)),rewrite([322(10)])].
% 5.73/6.00 15751 -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(hAPP(A,B),hAPP(C,B))),c_Lattices_Olower__semilattice__class_Oinf(D,hAPP(c_COMBS(A,C,E,F,V6),B),tc_nat))). [para(559(a,1),15738(a,1,1,2))].
% 5.73/6.00 16464 -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(hAPP(A,B),hAPP(C,B))),hAPP(c_COMBS(A,C,D,E,F),B))). [para(660(a,1),15751(a,1,2))].
% 5.73/6.00 16496 -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(c_COMBS(A,B,C,D,E),F)),hAPP(c_COMBS(A,B,V6,V7,V8),F))). [para(559(a,2),16464(a,1,1,2))].
% 5.73/6.00 16507 -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(c_COMBS(A,B,C,D,E),F)),hAPP(hAPP(A,F),hAPP(B,F)))). [para(559(a,1),16496(a,1,2))].
% 5.73/6.00 16510 -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(c_COMBS(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),A,B,C,D),E)),hAPP(E,hAPP(A,E)))). [para(323(a,1),16507(a,1,2,1))].
% 5.73/6.00 16586 -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(c_COMBS(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),hAPP(c_HOL_Oplus__class_Oplus(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),A,B,C),D)),hAPP(D,D))). [para(323(a,1),16510(a,1,2,2))].
% 5.73/6.00 17087 -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(c_COMBS(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),hAPP(c_HOL_Oplus__class_Oplus(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),A,B,C),c_COMBB(D,E,F,V6,V7))),hAPP(D,hAPP(E,c_COMBB(D,E,F,V6,V7))))). [para(557(a,1),16586(a,1,2))].
% 5.73/6.00 17088 $F. [resolve(17087,a,4511,a)].
% 5.73/6.00
% 5.73/6.00 % SZS output end Refutation
% 5.73/6.00 ============================== end of proof ==========================
% 5.73/6.00
% 5.73/6.00 ============================== STATISTICS ============================
% 5.73/6.00
% 5.73/6.00 Given=2709. Generated=114667. Kept=16816. proofs=1.
% 5.73/6.00 Usable=2698. Sos=9999. Demods=350. Limbo=3, Disabled=5104. Hints=0.
% 5.73/6.00 Megabytes=37.95.
% 5.73/6.00 User_CPU=4.52, System_CPU=0.06, Wall_clock=5.
% 5.73/6.00
% 5.73/6.00 ============================== end of statistics =====================
% 5.73/6.00
% 5.73/6.00 ============================== end of search =========================
% 5.73/6.00
% 5.73/6.00 THEOREM PROVED
% 5.73/6.00 % SZS status Unsatisfiable
% 5.73/6.00
% 5.73/6.00 Exiting with 1 proof.
% 5.73/6.00
% 5.73/6.00 Process 11440 exit (max_proofs) Sat Jul 2 07:34:21 2022
% 5.73/6.00 Prover9 interrupted
%------------------------------------------------------------------------------