TSTP Solution File: SCT079-1 by Prover9---1109a
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- Process Solution
%------------------------------------------------------------------------------
% File : Prover9---1109a
% Problem : SCT079-1 : TPTP v8.1.0. Released v4.1.0.
% Transfm : none
% Format : tptp:raw
% Command : tptp2X_and_run_prover9 %d %s
% Computer : n007.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 22:03:46 EDT 2022
% Result : Unsatisfiable 7.94s 8.24s
% Output : Refutation 7.94s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.12 % Problem : SCT079-1 : TPTP v8.1.0. Released v4.1.0.
% 0.12/0.13 % Command : tptp2X_and_run_prover9 %d %s
% 0.12/0.34 % Computer : n007.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 600
% 0.12/0.34 % DateTime : Sat Jul 2 07:36:55 EDT 2022
% 0.12/0.35 % CPUTime :
% 1.05/1.42 ============================== Prover9 ===============================
% 1.05/1.42 Prover9 (32) version 2009-11A, November 2009.
% 1.05/1.42 Process 3710 was started by sandbox on n007.cluster.edu,
% 1.05/1.42 Sat Jul 2 07:36:56 2022
% 1.05/1.42 The command was "/export/starexec/sandbox/solver/bin/prover9 -t 300 -f /tmp/Prover9_3557_n007.cluster.edu".
% 1.05/1.42 ============================== end of head ===========================
% 1.05/1.42
% 1.05/1.42 ============================== INPUT =================================
% 1.05/1.42
% 1.05/1.42 % Reading from file /tmp/Prover9_3557_n007.cluster.edu
% 1.05/1.42
% 1.05/1.42 set(prolog_style_variables).
% 1.05/1.42 set(auto2).
% 1.05/1.42 % set(auto2) -> set(auto).
% 1.05/1.42 % set(auto) -> set(auto_inference).
% 1.05/1.42 % set(auto) -> set(auto_setup).
% 1.05/1.42 % set(auto_setup) -> set(predicate_elim).
% 1.05/1.42 % set(auto_setup) -> assign(eq_defs, unfold).
% 1.05/1.42 % set(auto) -> set(auto_limits).
% 1.05/1.42 % set(auto_limits) -> assign(max_weight, "100.000").
% 1.05/1.42 % set(auto_limits) -> assign(sos_limit, 20000).
% 1.05/1.42 % set(auto) -> set(auto_denials).
% 1.05/1.42 % set(auto) -> set(auto_process).
% 1.05/1.42 % set(auto2) -> assign(new_constants, 1).
% 1.05/1.42 % set(auto2) -> assign(fold_denial_max, 3).
% 1.05/1.42 % set(auto2) -> assign(max_weight, "200.000").
% 1.05/1.42 % set(auto2) -> assign(max_hours, 1).
% 1.05/1.42 % assign(max_hours, 1) -> assign(max_seconds, 3600).
% 1.05/1.42 % set(auto2) -> assign(max_seconds, 0).
% 1.05/1.42 % set(auto2) -> assign(max_minutes, 5).
% 1.05/1.42 % assign(max_minutes, 5) -> assign(max_seconds, 300).
% 1.05/1.42 % set(auto2) -> set(sort_initial_sos).
% 1.05/1.42 % set(auto2) -> assign(sos_limit, -1).
% 1.05/1.42 % set(auto2) -> assign(lrs_ticks, 3000).
% 1.05/1.42 % set(auto2) -> assign(max_megs, 400).
% 1.05/1.42 % set(auto2) -> assign(stats, some).
% 1.05/1.42 % set(auto2) -> clear(echo_input).
% 1.05/1.42 % set(auto2) -> set(quiet).
% 1.05/1.42 % set(auto2) -> clear(print_initial_clauses).
% 1.05/1.42 % set(auto2) -> clear(print_given).
% 1.05/1.42 assign(lrs_ticks,-1).
% 1.05/1.42 assign(sos_limit,10000).
% 1.05/1.42 assign(order,kbo).
% 1.05/1.42 set(lex_order_vars).
% 1.05/1.42 clear(print_given).
% 1.05/1.42
% 1.05/1.42 % formulas(sos). % not echoed (770 formulas)
% 1.05/1.42
% 1.05/1.42 ============================== end of input ==========================
% 1.05/1.42
% 1.05/1.42 % From the command line: assign(max_seconds, 300).
% 1.05/1.42
% 1.05/1.42 ============================== PROCESS NON-CLAUSAL FORMULAS ==========
% 1.05/1.43
% 1.05/1.43 % Formulas that are not ordinary clauses:
% 1.05/1.43
% 1.05/1.43 ============================== end of process non-clausal formulas ===
% 1.05/1.43
% 1.05/1.43 ============================== PROCESS INITIAL CLAUSES ===============
% 1.05/1.43
% 1.05/1.43 ============================== PREDICATE ELIMINATION =================
% 1.05/1.43 1 class_OrderedGroup_Ocomm__monoid__mult(tc_nat) # label(clsarity_nat__OrderedGroup_Ocomm__monoid__mult) # label(axiom). [assumption].
% 1.05/1.43 2 -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].
% 1.05/1.43 3 -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].
% 1.05/1.43 4 -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].
% 1.05/1.43 5 -class_OrderedGroup_Ocomm__monoid__mult(A) | c_Finite__Set_Osetprod(hAPP(c_COMBC(c_COMBS(hAPP(hAPP(c_COMBB(tc_bool,tc_fun(A,tc_fun(A,A)),B),c_HOL_OIf(A)),hAPP(c_COMBC(c_fequal(B),B,B,tc_bool),C)),D,B,A,tc_fun(A,A)),B,A,A),E),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].
% 1.05/1.43 6 -class_OrderedGroup_Ocomm__monoid__mult(A) | c_Finite__Set_Osetprod(hAPP(c_COMBC(c_COMBS(hAPP(hAPP(c_COMBB(tc_bool,tc_fun(A,tc_fun(A,A)),B),c_HOL_OIf(A)),hAPP(c_fequal(B),C)),D,B,A,tc_fun(A,A)),B,A,A),c_HOL_Oone__class_Oone(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].
% 1.05/1.43 7 -class_OrderedGroup_Ocomm__monoid__mult(A) | c_Finite__Set_Osetprod(hAPP(c_COMBC(c_COMBS(hAPP(hAPP(c_COMBB(tc_bool,tc_fun(A,tc_fun(A,A)),B),c_HOL_OIf(A)),hAPP(c_fequal(B),C)),D,B,A,tc_fun(A,A)),B,A,A),c_HOL_Oone__class_Oone(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].
% 1.05/1.46 8 -class_OrderedGroup_Ocomm__monoid__mult(A) | c_Finite__Set_Osetprod(hAPP(c_COMBC(c_COMBS(hAPP(hAPP(c_COMBB(tc_bool,tc_fun(A,tc_fun(A,A)),B),c_HOL_OIf(A)),hAPP(c_COMBC(c_fequal(B),B,B,tc_bool),C)),D,B,A,tc_fun(A,A)),B,A,A),c_HOL_Oone__class_Oone(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].
% 1.05/1.46 9 -class_OrderedGroup_Ocomm__monoid__mult(A) | c_Finite__Set_Osetprod(hAPP(c_COMBC(c_COMBS(hAPP(hAPP(c_COMBB(tc_bool,tc_fun(A,tc_fun(A,A)),B),c_HOL_OIf(A)),hAPP(c_COMBC(c_fequal(B),B,B,tc_bool),C)),D,B,A,tc_fun(A,A)),B,A,A),c_HOL_Oone__class_Oone(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].
% 1.05/1.46 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(1,a,2,a)].
% 1.05/1.46 Derived: c_Finite__Set_Osetprod(A,B,C,tc_nat) = c_HOL_Oone__class_Oone(tc_nat) | c_Finite__Set_Ofinite(B,C). [resolve(1,a,3,a)].
% 1.05/1.46 Derived: c_Finite__Set_Osetprod(hAPP(c_COMBC(c_COMBS(hAPP(hAPP(c_COMBB(tc_bool,tc_fun(tc_nat,tc_fun(tc_nat,tc_nat)),A),c_HOL_OIf(tc_nat)),hAPP(c_COMBC(c_fequal(A),A,A,tc_bool),B)),C,A,tc_nat,tc_fun(tc_nat,tc_nat)),A,tc_nat,tc_nat),D),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(1,a,5,a)].
% 1.05/1.46 Derived: c_Finite__Set_Osetprod(hAPP(c_COMBC(c_COMBS(hAPP(hAPP(c_COMBB(tc_bool,tc_fun(tc_nat,tc_fun(tc_nat,tc_nat)),A),c_HOL_OIf(tc_nat)),hAPP(c_fequal(A),B)),C,A,tc_nat,tc_fun(tc_nat,tc_nat)),A,tc_nat,tc_nat),c_HOL_Oone__class_Oone(tc_nat)),D,A,tc_nat) = hAPP(C,B) | -hBOOL(hAPP(hAPP(c_in(A),B),D)) | -c_Finite__Set_Ofinite(D,A). [resolve(1,a,6,a)].
% 1.05/1.46 Derived: c_Finite__Set_Osetprod(hAPP(c_COMBC(c_COMBS(hAPP(hAPP(c_COMBB(tc_bool,tc_fun(tc_nat,tc_fun(tc_nat,tc_nat)),A),c_HOL_OIf(tc_nat)),hAPP(c_fequal(A),B)),C,A,tc_nat,tc_fun(tc_nat,tc_nat)),A,tc_nat,tc_nat),c_HOL_Oone__class_Oone(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(1,a,7,a)].
% 1.05/1.46 Derived: c_Finite__Set_Osetprod(hAPP(c_COMBC(c_COMBS(hAPP(hAPP(c_COMBB(tc_bool,tc_fun(tc_nat,tc_fun(tc_nat,tc_nat)),A),c_HOL_OIf(tc_nat)),hAPP(c_COMBC(c_fequal(A),A,A,tc_bool),B)),C,A,tc_nat,tc_fun(tc_nat,tc_nat)),A,tc_nat,tc_nat),c_HOL_Oone__class_Oone(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(1,a,8,a)].
% 1.05/1.46 Derived: c_Finite__Set_Osetprod(hAPP(c_COMBC(c_COMBS(hAPP(hAPP(c_COMBB(tc_bool,tc_fun(tc_nat,tc_fun(tc_nat,tc_nat)),A),c_HOL_OIf(tc_nat)),hAPP(c_COMBC(c_fequal(A),A,A,tc_bool),B)),C,A,tc_nat,tc_fun(tc_nat,tc_nat)),A,tc_nat,tc_nat),c_HOL_Oone__class_Oone(tc_nat)),D,A,tc_nat) = hAPP(C,B) | -hBOOL(hAPP(hAPP(c_in(A),B),D)) | -c_Finite__Set_Ofinite(D,A). [resolve(1,a,9,a)].
% 1.05/1.46 10 class_OrderedGroup_Opordered__comm__monoid__add(tc_nat) # label(clsarity_nat__OrderedGroup_Opordered__comm__monoid__add) # label(axiom). [assumption].
% 1.05/1.46 11 -class_OrderedGroup_Opordered__comm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_lessequals(A),c_HOL_Ozero__class_Ozero(A)),c_HOL_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_lessequals(A),c_HOL_Ozero__class_Ozero(A)),c_HOL_Ozero__class_Ozero(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].
% 1.05/1.46 12 -class_OrderedGroup_Opordered__comm__monoid__add(A) | hBOOL(hAPP(hAPP(c_lessequals(A),c_HOL_Ozero__class_Ozero(A)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C))) | -hBOOL(hAPP(hAPP(c_lessequals(A),c_HOL_Ozero__class_Ozero(A)),C)) | -hBOOL(hAPP(hAPP(c_lessequals(A),c_HOL_Ozero__class_Ozero(A)),B)) # label(cls_add__nonneg__nonneg_0) # label(axiom). [assumption].
% 1.05/1.46 13 -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].
% 1.05/1.46 14 -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_lessequals(A),c_HOL_Ozero__class_Ozero(A)),B)) # label(cls_add__nonneg__pos_0) # label(axiom). [assumption].
% 1.05/1.46 15 -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_lessequals(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__nonneg_0) # label(axiom). [assumption].
% 1.05/1.46 16 -class_OrderedGroup_Opordered__comm__monoid__add(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C) != c_HOL_Ozero__class_Ozero(A) | -hBOOL(hAPP(hAPP(c_lessequals(A),c_HOL_Ozero__class_Ozero(A)),C)) | -hBOOL(hAPP(hAPP(c_lessequals(A),c_HOL_Ozero__class_Ozero(A)),B)) | C = c_HOL_Ozero__class_Ozero(A) # label(cls_add__nonneg__eq__0__iff_1) # label(axiom). [assumption].
% 1.05/1.46 17 -class_OrderedGroup_Opordered__comm__monoid__add(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C) != c_HOL_Ozero__class_Ozero(A) | -hBOOL(hAPP(hAPP(c_lessequals(A),c_HOL_Ozero__class_Ozero(A)),C)) | -hBOOL(hAPP(hAPP(c_lessequals(A),c_HOL_Ozero__class_Ozero(A)),B)) | B = c_HOL_Ozero__class_Ozero(A) # label(cls_add__nonneg__eq__0__iff_0) # label(axiom). [assumption].
% 1.05/1.46 18 -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_lessequals(A),B),c_HOL_Ozero__class_Ozero(A))) # label(cls_add__nonpos__neg_0) # label(axiom). [assumption].
% 1.05/1.46 19 -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_lessequals(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__nonpos_0) # label(axiom). [assumption].
% 1.05/1.46 20 -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].
% 1.05/1.46 21 -class_OrderedGroup_Opordered__comm__monoid__add(A) | hBOOL(hAPP(hAPP(c_lessequals(A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C)),c_HOL_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_lessequals(A),C),c_HOL_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_lessequals(A),B),c_HOL_Ozero__class_Ozero(A))) # label(cls_add__nonpos__nonpos_0) # label(axiom). [assumption].
% 1.05/1.46 Derived: -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),c_HOL_Ozero__class_Ozero(tc_nat))) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),c_HOL_Ozero__class_Ozero(tc_nat))) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),c_HOL_Ozero__class_Ozero(tc_nat)) = c_HOL_Ozero__class_Ozero(tc_nat). [resolve(10,a,11,a)].
% 1.05/1.46 Derived: hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B))) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),B)) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),A)). [resolve(10,a,12,a)].
% 1.05/1.51 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(10,a,13,a)].
% 1.05/1.51 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)) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),A)). [resolve(10,a,14,a)].
% 1.05/1.51 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_lessequals(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),B)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),A)). [resolve(10,a,15,a)].
% 1.05/1.51 Derived: hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B) != c_HOL_Ozero__class_Ozero(tc_nat) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),B)) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),A)) | B = c_HOL_Ozero__class_Ozero(tc_nat). [resolve(10,a,16,a)].
% 1.05/1.51 Derived: hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B) != c_HOL_Ozero__class_Ozero(tc_nat) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),B)) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),A)) | A = c_HOL_Ozero__class_Ozero(tc_nat). [resolve(10,a,17,a)].
% 1.05/1.51 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_lessequals(tc_nat),A),c_HOL_Ozero__class_Ozero(tc_nat))). [resolve(10,a,18,a)].
% 1.05/1.51 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_lessequals(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(10,a,19,a)].
% 1.05/1.51 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)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),A)). [resolve(10,a,20,a)].
% 1.05/1.51 Derived: hBOOL(hAPP(hAPP(c_lessequals(tc_nat),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B)),c_HOL_Ozero__class_Ozero(tc_nat))) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),B),c_HOL_Ozero__class_Ozero(tc_nat))) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),A),c_HOL_Ozero__class_Ozero(tc_nat))). [resolve(10,a,21,a)].
% 1.05/1.51 22 class_OrderedGroup_Opordered__ab__semigroup__add(tc_nat) # label(clsarity_nat__OrderedGroup_Opordered__ab__semigroup__add) # label(axiom). [assumption].
% 1.05/1.51 23 -class_OrderedGroup_Opordered__ab__semigroup__add(A) | hBOOL(hAPP(hAPP(c_lessequals(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_lessequals(A),C),E)) | -hBOOL(hAPP(hAPP(c_lessequals(A),B),D)) # label(cls_add__mono_0) # label(axiom). [assumption].
% 1.05/1.51 24 -class_OrderedGroup_Opordered__ab__semigroup__add(A) | hBOOL(hAPP(hAPP(c_lessequals(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_lessequals(A),B),D)) # label(cls_add__right__mono_0) # label(axiom). [assumption].
% 1.05/1.51 25 -class_OrderedGroup_Opordered__ab__semigroup__add(A) | hBOOL(hAPP(hAPP(c_lessequals(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_lessequals(A),C),D)) # label(cls_add__left__mono_0) # label(axiom). [assumption].
% 1.05/1.51 Derived: hBOOL(hAPP(hAPP(c_lessequals(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_lessequals(tc_nat),B),D)) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),A),C)). [resolve(22,a,23,a)].
% 1.05/1.60 Derived: hBOOL(hAPP(hAPP(c_lessequals(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_lessequals(tc_nat),A),C)). [resolve(22,a,24,a)].
% 1.05/1.60 Derived: hBOOL(hAPP(hAPP(c_lessequals(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_lessequals(tc_nat),B),C)). [resolve(22,a,25,a)].
% 1.05/1.60 26 class_Ring__and__Field_Osemiring__0(tc_nat) # label(clsarity_nat__Ring__and__Field_Osemiring__0) # label(axiom). [assumption].
% 1.05/1.60 27 -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].
% 1.05/1.60 28 -class_Ring__and__Field_Osemiring__0(A) | -class_Power_Opower(A) | c_Power_Opower__class_Opower(c_HOL_Ozero__class_Ozero(A),c_Suc(B),A) = c_HOL_Ozero__class_Ozero(A) # label(cls_power__0__Suc_0) # label(axiom). [assumption].
% 1.05/1.60 29 -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].
% 1.05/1.60 Derived: -class_Power_Opower(tc_nat) | c_Power_Opower__class_Opower(c_HOL_Ozero__class_Ozero(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat),tc_nat) = c_HOL_Oone__class_Oone(tc_nat). [resolve(26,a,27,a)].
% 1.05/1.60 Derived: -class_Power_Opower(tc_nat) | c_Power_Opower__class_Opower(c_HOL_Ozero__class_Ozero(tc_nat),c_Suc(A),tc_nat) = c_HOL_Ozero__class_Ozero(tc_nat). [resolve(26,a,28,a)].
% 1.05/1.60 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(26,a,29,a)].
% 1.05/1.60 30 class_Ring__and__Field_Oordered__semidom(tc_nat) # label(clsarity_nat__Ring__and__Field_Oordered__semidom) # label(axiom). [assumption].
% 1.05/1.60 31 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_Power_Opower__class_Opower(B,c_Suc(C),A)),c_HOL_Oone__class_Oone(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)) # label(cls_power__Suc__less__one_0) # label(axiom). [assumption].
% 1.05/1.60 32 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_lessequals(A),c_Power_Opower__class_Opower(B,C,A)),c_Power_Opower__class_Opower(B,D,A))) | -hBOOL(hAPP(hAPP(c_lessequals(A),c_HOL_Oone__class_Oone(A)),B)) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),C),D)) # label(cls_power__increasing_0) # label(axiom). [assumption].
% 1.05/1.60 33 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_lessequals(A),c_Power_Opower__class_Opower(B,C,A)),c_Power_Opower__class_Opower(D,C,A))) | -hBOOL(hAPP(hAPP(c_lessequals(A),c_HOL_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_lessequals(A),B),D)) # label(cls_power__mono_0) # label(axiom). [assumption].
% 1.05/1.60 34 -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].
% 1.05/1.60 35 -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)) | -hBOOL(hAPP(hAPP(c_lessequals(A),c_HOL_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),D)) # label(cls_power__strict__mono_0) # label(axiom). [assumption].
% 1.05/1.60 36 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_lessequals(A),c_Nat_Osemiring__1__class_Oof__nat(B,A)),c_Nat_Osemiring__1__class_Oof__nat(C,A))) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),B),C)) # label(cls_of__nat__le__iff_1) # label(axiom). [assumption].
% 1.05/1.60 37 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_lessequals(tc_nat),B),C)) | -hBOOL(hAPP(hAPP(c_lessequals(A),c_Nat_Osemiring__1__class_Oof__nat(B,A)),c_Nat_Osemiring__1__class_Oof__nat(C,A))) # label(cls_of__nat__le__iff_0) # label(axiom). [assumption].
% 1.05/1.60 38 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_lessequals(A),c_HOL_Ozero__class_Ozero(A)),c_Power_Opower__class_Opower(B,C,A))) | -hBOOL(hAPP(hAPP(c_lessequals(A),c_HOL_Ozero__class_Ozero(A)),B)) # label(cls_zero__le__power_0) # label(axiom). [assumption].
% 1.05/1.60 39 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_lessequals(A),c_HOL_Oone__class_Oone(A)),c_Power_Opower__class_Opower(B,C,A))) | -hBOOL(hAPP(hAPP(c_lessequals(A),c_HOL_Oone__class_Oone(A)),B)) # label(cls_one__le__power_0) # label(axiom). [assumption].
% 1.05/1.60 40 -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].
% 1.05/1.60 41 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_lessequals(A),c_Power_Opower__class_Opower(B,C,A)),c_Power_Opower__class_Opower(B,D,A))) | -hBOOL(hAPP(hAPP(c_lessequals(A),B),c_HOL_Oone__class_Oone(A))) | -hBOOL(hAPP(hAPP(c_lessequals(A),c_HOL_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),D),C)) # label(cls_power__decreasing_0) # label(axiom). [assumption].
% 1.05/1.60 42 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_lessequals(A),c_HOL_Ozero__class_Ozero(A)),C)) | -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].
% 1.05/1.60 43 -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].
% 1.05/1.60 44 -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].
% 1.05/1.60 45 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_lessequals(A),c_HOL_Ozero__class_Ozero(A)),c_HOL_Oone__class_Oone(A))) # label(cls_zero__le__one_0) # label(axiom). [assumption].
% 1.05/1.60 46 -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].
% 1.05/1.60 47 -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)) | -hBOOL(hAPP(hAPP(c_lessequals(A),c_HOL_Ozero__class_Ozero(A)),D)) | -hBOOL(hAPP(hAPP(c_lessequals(A),c_HOL_Ozero__class_Ozero(A)),B)) # label(cls_power__eq__imp__eq__base_0) # label(axiom). [assumption].
% 1.05/1.60 48 -class_Ring__and__Field_Oordered__semidom(A) | c_Power_Opower__class_Opower(B,c_Suc(C),A) != c_Power_Opower__class_Opower(D,c_Suc(C),A) | B = D | -hBOOL(hAPP(hAPP(c_lessequals(A),c_HOL_Ozero__class_Ozero(A)),D)) | -hBOOL(hAPP(hAPP(c_lessequals(A),c_HOL_Ozero__class_Ozero(A)),B)) # label(cls_power__inject__base_0) # label(axiom). [assumption].
% 1.05/1.60 49 -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].
% 1.05/1.60 50 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_lessequals(A),c_HOL_Ozero__class_Ozero(A)),c_Nat_Osemiring__1__class_Oof__nat(B,A))) # label(cls_of__nat__0__le__iff_0) # label(axiom). [assumption].
% 1.05/1.60 51 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_lessequals(A),c_HOL_Ozero__class_Ozero(A)),c_Nat_Osemiring__1__class_Oof__nat(B,A))) # label(cls_zero__le__imp__of__nat_0) # label(axiom). [assumption].
% 1.05/1.60 52 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_lessequals(tc_nat),B),C)) | -hBOOL(hAPP(hAPP(c_lessequals(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__le__imp__le__exp_0) # label(axiom). [assumption].
% 1.05/1.60 53 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_lessequals(tc_nat),B),C)) | -hBOOL(hAPP(hAPP(c_lessequals(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__increasing__iff_0) # label(axiom). [assumption].
% 1.05/1.60 54 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_lessequals(A),c_Power_Opower__class_Opower(B,C,A)),c_Power_Opower__class_Opower(B,D,A))) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),C),D)) | -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].
% 1.05/1.60 55 -class_Ring__and__Field_Oordered__semidom(A) | -hBOOL(hAPP(hAPP(c_lessequals(A),c_HOL_Oone__class_Oone(A)),c_HOL_Ozero__class_Ozero(A))) # label(cls_not__one__le__zero_0) # label(axiom). [assumption].
% 1.05/1.60 56 -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_Suc(C),A))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Oone__class_Oone(A)),B)) # label(cls_power__gt1_0) # label(axiom). [assumption].
% 1.05/1.60 57 -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].
% 1.05/1.60 58 -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].
% 1.05/1.60 59 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_lessequals(A),B),C)) | -hBOOL(hAPP(hAPP(c_lessequals(A),c_HOL_Ozero__class_Ozero(A)),C)) | -hBOOL(hAPP(hAPP(c_lessequals(A),c_Power_Opower__class_Opower(B,c_Suc(D),A)),c_Power_Opower__class_Opower(C,c_Suc(D),A))) # label(cls_power__le__imp__le__base_0) # label(axiom). [assumption].
% 1.05/1.60 60 -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].
% 1.05/1.60 61 -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].
% 1.05/1.60 62 -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].
% 1.05/1.60 63 -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].
% 1.05/1.60 64 -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].
% 1.05/1.60 65 -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].
% 1.05/1.60 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(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].
% 1.05/1.60 67 -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].
% 1.05/1.60 68 -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].
% 1.05/1.60 69 -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].
% 1.05/1.60 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_Power_Opower__class_Opower(A,c_Suc(B),tc_nat)),c_HOL_Oone__class_Oone(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)). [resolve(30,a,31,a)].
% 1.05/1.60 Derived: hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_Power_Opower__class_Opower(A,B,tc_nat)),c_Power_Opower__class_Opower(A,C,tc_nat))) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_HOL_Oone__class_Oone(tc_nat)),A)) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),B),C)). [resolve(30,a,32,a)].
% 1.05/1.60 Derived: hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_Power_Opower__class_Opower(A,B,tc_nat)),c_Power_Opower__class_Opower(C,B,tc_nat))) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),A)) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),A),C)). [resolve(30,a,33,a)].
% 1.05/1.60 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(30,a,34,a)].
% 1.05/1.60 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)) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),A)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),C)). [resolve(30,a,35,a)].
% 1.05/1.60 Derived: hBOOL(hAPP(hAPP(c_lessequals(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_lessequals(tc_nat),A),B)). [resolve(30,a,36,a)].
% 1.05/1.60 Derived: hBOOL(hAPP(hAPP(c_lessequals(tc_nat),A),B)) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_Nat_Osemiring__1__class_Oof__nat(A,tc_nat)),c_Nat_Osemiring__1__class_Oof__nat(B,tc_nat))). [resolve(30,a,37,a)].
% 1.05/1.60 Derived: hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),c_Power_Opower__class_Opower(A,B,tc_nat))) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),A)). [resolve(30,a,38,a)].
% 1.05/1.60 Derived: hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_HOL_Oone__class_Oone(tc_nat)),c_Power_Opower__class_Opower(A,B,tc_nat))) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_HOL_Oone__class_Oone(tc_nat)),A)). [resolve(30,a,39,a)].
% 1.05/1.60 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(30,a,40,a)].
% 1.05/1.60 Derived: hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_Power_Opower__class_Opower(A,B,tc_nat)),c_Power_Opower__class_Opower(A,C,tc_nat))) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),A),c_HOL_Oone__class_Oone(tc_nat))) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),A)) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),C),B)). [resolve(30,a,41,a)].
% 1.05/1.60 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),B)) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),B)) | -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(30,a,42,a)].
% 1.05/1.60 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(30,a,43,a)].
% 1.05/1.60 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(30,a,44,a)].
% 1.05/1.60 Derived: hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),c_HOL_Oone__class_Oone(tc_nat))). [resolve(30,a,45,a)].
% 1.05/1.60 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(30,a,46,a)].
% 1.05/1.60 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)) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),C)) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),A)). [resolve(30,a,47,a)].
% 1.05/1.60 Derived: c_Power_Opower__class_Opower(A,c_Suc(B),tc_nat) != c_Power_Opower__class_Opower(C,c_Suc(B),tc_nat) | A = C | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),C)) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),A)). [resolve(30,a,48,a)].
% 1.05/1.60 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(30,a,49,a)].
% 1.05/1.60 Derived: hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),c_Nat_Osemiring__1__class_Oof__nat(A,tc_nat))). [resolve(30,a,50,a)].
% 1.05/1.60 Derived: hBOOL(hAPP(hAPP(c_lessequals(tc_nat),A),B)) | -hBOOL(hAPP(hAPP(c_lessequals(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(30,a,52,a)].
% 1.05/1.60 Derived: hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_Power_Opower__class_Opower(A,B,tc_nat)),c_Power_Opower__class_Opower(A,C,tc_nat))) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),B),C)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Oone__class_Oone(tc_nat)),A)). [resolve(30,a,54,a)].
% 1.05/1.62 Derived: -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_HOL_Oone__class_Oone(tc_nat)),c_HOL_Ozero__class_Ozero(tc_nat))). [resolve(30,a,55,a)].
% 1.05/1.62 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Oone__class_Oone(tc_nat)),c_Power_Opower__class_Opower(A,c_Suc(B),tc_nat))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Oone__class_Oone(tc_nat)),A)). [resolve(30,a,56,a)].
% 1.05/1.62 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(30,a,57,a)].
% 1.05/1.62 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(30,a,58,a)].
% 1.05/1.62 Derived: hBOOL(hAPP(hAPP(c_lessequals(tc_nat),A),B)) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),B)) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_Power_Opower__class_Opower(A,c_Suc(C),tc_nat)),c_Power_Opower__class_Opower(B,c_Suc(C),tc_nat))). [resolve(30,a,59,a)].
% 1.05/1.62 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(30,a,60,a)].
% 1.05/1.62 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(30,a,61,a)].
% 1.05/1.62 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(30,a,62,a)].
% 1.05/1.62 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(30,a,63,a)].
% 1.05/1.62 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(30,a,64,a)].
% 1.05/1.62 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(30,a,67,a)].
% 1.05/1.62 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(30,a,68,a)].
% 1.05/1.62 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(30,a,69,a)].
% 1.05/1.62 70 class_Ring__and__Field_Ocomm__semiring__1(tc_nat) # label(clsarity_nat__Ring__and__Field_Ocomm__semiring__1) # label(axiom). [assumption].
% 1.05/1.62 71 -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].
% 1.05/1.62 72 -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].
% 1.05/1.62 73 -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].
% 1.05/1.62 74 -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].
% 1.05/1.62 75 -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].
% 1.05/1.62 76 -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].
% 1.05/1.62 77 -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].
% 1.05/1.62 78 -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].
% 1.05/1.62 79 -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].
% 1.05/1.62 80 -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].
% 1.05/1.62 81 -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].
% 1.05/1.62 82 -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].
% 1.05/1.62 83 -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].
% 1.05/1.62 84 -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].
% 1.05/1.62 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(70,a,71,a)].
% 1.05/1.62 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(70,a,73,a)].
% 1.05/1.62 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(70,a,74,a)].
% 1.05/1.62 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(70,a,75,a)].
% 1.05/1.62 Derived: hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),B),A). [resolve(70,a,76,a)].
% 1.05/1.62 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(70,a,78,a)].
% 1.05/1.66 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(70,a,79,b)].
% 1.05/1.66 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(70,a,80,a)].
% 1.05/1.66 Derived: hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),A) = A. [resolve(70,a,81,a)].
% 1.05/1.66 Derived: hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),c_HOL_Ozero__class_Ozero(tc_nat)) = A. [resolve(70,a,82,a)].
% 1.05/1.66 Derived: c_Power_Opower__class_Opower(A,c_HOL_Ozero__class_Ozero(tc_nat),tc_nat) = c_HOL_Oone__class_Oone(tc_nat). [resolve(70,a,84,a)].
% 1.05/1.66 85 class_OrderedGroup_Ocomm__monoid__add(tc_nat) # label(clsarity_nat__OrderedGroup_Ocomm__monoid__add) # label(axiom). [assumption].
% 1.05/1.66 86 -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].
% 1.05/1.66 87 -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].
% 1.05/1.66 88 -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)) | -hBOOL(hAPP(hAPP(c_lessequals(A),c_HOL_Ozero__class_Ozero(A)),C)) # label(cls_add__strict__increasing2_0) # label(axiom). [assumption].
% 1.05/1.66 89 -class_OrderedGroup_Ocomm__monoid__add(A) | c_Finite__Set_Osetsum(c_COMBS(hAPP(hAPP(c_COMBB(A,tc_fun(A,A),B),c_HOL_Oplus__class_Oplus(A)),C),D,B,A,A),E,B,A) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),c_Finite__Set_Osetsum(C,E,B,A)),c_Finite__Set_Osetsum(D,E,B,A)) # label(cls_setsum__addf_0) # label(axiom). [assumption].
% 1.05/1.66 90 -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_lessequals(A),B),D)) | -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].
% 1.05/1.66 91 -class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(A) | -class_OrderedGroup_Ocomm__monoid__add(A) | hBOOL(hAPP(hAPP(c_lessequals(A),B),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),C),D))) | -hBOOL(hAPP(hAPP(c_lessequals(A),B),D)) | -hBOOL(hAPP(hAPP(c_lessequals(A),c_HOL_Ozero__class_Ozero(A)),C)) # label(cls_add__increasing_0) # label(axiom). [assumption].
% 1.05/1.66 92 -class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(A) | -class_OrderedGroup_Ocomm__monoid__add(A) | hBOOL(hAPP(hAPP(c_lessequals(A),B),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),C),D))) | -hBOOL(hAPP(hAPP(c_lessequals(A),B),C)) | -hBOOL(hAPP(hAPP(c_lessequals(A),c_HOL_Ozero__class_Ozero(A)),D)) # label(cls_add__increasing2_0) # label(axiom). [assumption].
% 1.05/1.66 93 -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].
% 1.05/1.66 94 -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].
% 1.05/1.66 95 -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].
% 1.05/1.66 96 -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].
% 1.05/1.66 97 -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].
% 1.05/1.66 98 -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].
% 1.05/1.66 99 -class_OrderedGroup_Ocomm__monoid__add(A) | c_Finite__Set_Osetsum(hAPP(c_COMBC(c_COMBS(hAPP(hAPP(c_COMBB(tc_bool,tc_fun(A,tc_fun(A,A)),B),c_HOL_OIf(A)),hAPP(c_fequal(B),C)),D,B,A,tc_fun(A,A)),B,A,A),c_HOL_Ozero__class_Ozero(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].
% 1.05/1.66 100 -class_OrderedGroup_Ocomm__monoid__add(A) | c_Finite__Set_Osetsum(hAPP(c_COMBC(c_COMBS(hAPP(hAPP(c_COMBB(tc_bool,tc_fun(A,tc_fun(A,A)),B),c_HOL_OIf(A)),hAPP(c_fequal(B),C)),D,B,A,tc_fun(A,A)),B,A,A),c_HOL_Ozero__class_Ozero(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].
% 1.05/1.66 101 -class_OrderedGroup_Ocomm__monoid__add(A) | c_Finite__Set_Osetsum(hAPP(c_COMBC(c_COMBS(hAPP(hAPP(c_COMBB(tc_bool,tc_fun(A,tc_fun(A,A)),B),c_HOL_OIf(A)),hAPP(c_COMBC(c_fequal(B),B,B,tc_bool),C)),D,B,A,tc_fun(A,A)),B,A,A),c_HOL_Ozero__class_Ozero(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].
% 1.05/1.66 102 -class_OrderedGroup_Ocomm__monoid__add(A) | c_Finite__Set_Osetsum(hAPP(c_COMBC(c_COMBS(hAPP(hAPP(c_COMBB(tc_bool,tc_fun(A,tc_fun(A,A)),B),c_HOL_OIf(A)),hAPP(c_COMBC(c_fequal(B),B,B,tc_bool),C)),D,B,A,tc_fun(A,A)),B,A,A),c_HOL_Ozero__class_Ozero(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].
% 1.05/1.66 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)) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),B)). [resolve(85,a,88,b)].
% 1.05/1.66 Derived: c_Finite__Set_Osetsum(c_COMBS(hAPP(hAPP(c_COMBB(tc_nat,tc_fun(tc_nat,tc_nat),A),c_HOL_Oplus__class_Oplus(tc_nat)),B),C,A,tc_nat,tc_nat),D,A,tc_nat) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),c_Finite__Set_Osetsum(B,D,A,tc_nat)),c_Finite__Set_Osetsum(C,D,A,tc_nat)). [resolve(85,a,89,a)].
% 1.05/1.66 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_lessequals(tc_nat),A),C)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),B)). [resolve(85,a,90,b)].
% 1.05/1.66 Derived: -class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(tc_nat) | hBOOL(hAPP(hAPP(c_lessequals(tc_nat),A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),B),C))) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),A),C)) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),B)). [resolve(85,a,91,b)].
% 1.05/1.66 Derived: -class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(tc_nat) | hBOOL(hAPP(hAPP(c_lessequals(tc_nat),A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),B),C))) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),A),B)) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),C)). [resolve(85,a,92,b)].
% 1.05/1.66 Derived: c_Finite__Set_Osetsum(A,B,C,tc_nat) = c_HOL_Ozero__class_Ozero(tc_nat) | c_Finite__Set_Ofinite(B,C). [resolve(85,a,94,a)].
% 1.05/1.70 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(85,a,98,a)].
% 1.05/1.70 Derived: c_Finite__Set_Osetsum(hAPP(c_COMBC(c_COMBS(hAPP(hAPP(c_COMBB(tc_bool,tc_fun(tc_nat,tc_fun(tc_nat,tc_nat)),A),c_HOL_OIf(tc_nat)),hAPP(c_fequal(A),B)),C,A,tc_nat,tc_fun(tc_nat,tc_nat)),A,tc_nat,tc_nat),c_HOL_Ozero__class_Ozero(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(85,a,99,a)].
% 1.05/1.70 Derived: c_Finite__Set_Osetsum(hAPP(c_COMBC(c_COMBS(hAPP(hAPP(c_COMBB(tc_bool,tc_fun(tc_nat,tc_fun(tc_nat,tc_nat)),A),c_HOL_OIf(tc_nat)),hAPP(c_fequal(A),B)),C,A,tc_nat,tc_fun(tc_nat,tc_nat)),A,tc_nat,tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),D,A,tc_nat) = hAPP(C,B) | -hBOOL(hAPP(hAPP(c_in(A),B),D)) | -c_Finite__Set_Ofinite(D,A). [resolve(85,a,100,a)].
% 1.05/1.70 Derived: c_Finite__Set_Osetsum(hAPP(c_COMBC(c_COMBS(hAPP(hAPP(c_COMBB(tc_bool,tc_fun(tc_nat,tc_fun(tc_nat,tc_nat)),A),c_HOL_OIf(tc_nat)),hAPP(c_COMBC(c_fequal(A),A,A,tc_bool),B)),C,A,tc_nat,tc_fun(tc_nat,tc_nat)),A,tc_nat,tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),D,A,tc_nat) = hAPP(C,B) | -hBOOL(hAPP(hAPP(c_in(A),B),D)) | -c_Finite__Set_Ofinite(D,A). [resolve(85,a,101,a)].
% 1.05/1.70 Derived: c_Finite__Set_Osetsum(hAPP(c_COMBC(c_COMBS(hAPP(hAPP(c_COMBB(tc_bool,tc_fun(tc_nat,tc_fun(tc_nat,tc_nat)),A),c_HOL_OIf(tc_nat)),hAPP(c_COMBC(c_fequal(A),A,A,tc_bool),B)),C,A,tc_nat,tc_fun(tc_nat,tc_nat)),A,tc_nat,tc_nat),c_HOL_Ozero__class_Ozero(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(85,a,102,a)].
% 1.05/1.70 103 class_OrderedGroup_Oab__semigroup__add(tc_nat) # label(clsarity_nat__OrderedGroup_Oab__semigroup__add) # label(axiom). [assumption].
% 1.05/1.70 104 -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].
% 1.05/1.70 105 class_Ring__and__Field_Ozero__neq__one(tc_nat) # label(clsarity_nat__Ring__and__Field_Ozero__neq__one) # label(axiom). [assumption].
% 1.05/1.70 106 -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].
% 1.05/1.70 107 -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].
% 1.05/1.70 108 -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].
% 1.05/1.70 109 -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].
% 1.05/1.70 110 -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].
% 1.05/1.70 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(105,a,106,a)].
% 1.05/1.70 Derived: c_HOL_Ozero__class_Ozero(tc_nat) != c_HOL_Oone__class_Oone(tc_nat). [resolve(105,a,107,a)].
% 1.05/1.70 Derived: c_HOL_Oone__class_Oone(tc_nat) != c_HOL_Ozero__class_Ozero(tc_nat). [resolve(105,a,108,a)].
% 1.05/1.70 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(105,a,109,a)].
% 1.05/1.75 111 class_OrderedGroup_Omonoid__mult(tc_nat) # label(clsarity_nat__OrderedGroup_Omonoid__mult) # label(axiom). [assumption].
% 1.05/1.75 112 -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].
% 1.05/1.75 Derived: c_Power_Opower__class_Opower(c_HOL_Oone__class_Oone(tc_nat),A,tc_nat) = c_HOL_Oone__class_Oone(tc_nat). [resolve(111,a,112,a)].
% 1.05/1.75 113 class_Ring__and__Field_Osemiring__1(tc_nat) # label(clsarity_nat__Ring__and__Field_Osemiring__1) # label(axiom). [assumption].
% 1.05/1.75 114 -class_Ring__and__Field_Osemiring__1(A) | c_Nat_Osemiring__1__class_Oof__nat(c_Suc(B),A) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),c_HOL_Oone__class_Oone(A)),c_Nat_Osemiring__1__class_Oof__nat(B,A)) # label(cls_of__nat__Suc_0) # label(axiom). [assumption].
% 1.05/1.75 115 -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].
% 1.05/1.75 116 -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].
% 1.05/1.75 Derived: c_Nat_Osemiring__1__class_Oof__nat(c_Suc(A),tc_nat) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),c_HOL_Oone__class_Oone(tc_nat)),c_Nat_Osemiring__1__class_Oof__nat(A,tc_nat)). [resolve(113,a,114,a)].
% 1.05/1.75 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(113,a,115,a)].
% 1.05/1.75 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(113,a,116,a)].
% 1.05/1.75 117 class_Orderings_Olinorder(tc_nat) # label(clsarity_nat__Orderings_Olinorder) # label(axiom). [assumption].
% 1.05/1.75 118 -class_Orderings_Olinorder(A) | c_SetInterval_Oord__class_OlessThan(B,A) != c_SetInterval_Oord__class_OlessThan(C,A) | B = C # label(cls_lessThan__eq__iff_0) # label(axiom). [assumption].
% 1.05/1.75 119 -class_Orderings_Olinorder(A) | c_SetInterval_Oord__class_OgreaterThan(B,A) != c_SetInterval_Oord__class_OgreaterThan(C,A) | B = C # label(cls_greaterThan__eq__iff_0) # label(axiom). [assumption].
% 1.05/1.75 120 -class_Orderings_Olinorder(A) | B = C | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_lessequals(A),B),C)) # label(cls_linorder__antisym__conv2_0) # label(axiom). [assumption].
% 1.05/1.75 121 -class_Orderings_Olinorder(A) | B = C | -hBOOL(hAPP(hAPP(c_lessequals(A),B),C)) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),C)) # label(cls_linorder__antisym__conv1_0) # label(axiom). [assumption].
% 1.05/1.75 122 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_lessequals(A),C),B)) # label(cls_not__leE_0) # label(axiom). [assumption].
% 1.05/1.75 123 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_lessequals(A),B),B)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),B)) # label(cls_linorder__antisym__conv2_1) # label(axiom). [assumption].
% 1.05/1.75 124 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),B)) | hBOOL(hAPP(hAPP(c_lessequals(A),B),B)) # label(cls_linorder__antisym__conv1_1) # label(axiom). [assumption].
% 1.05/1.75 125 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_lessequals(A),C),B)) # label(cls_linorder__not__less_1) # label(axiom). [assumption].
% 1.05/1.75 126 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_lessequals(A),B),C)) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),C),B)) # label(cls_linorder__not__less_0) # label(axiom). [assumption].
% 1.05/1.80 127 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_lessequals(A),B),C)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),C),B)) # label(cls_linorder__not__le_1) # label(axiom). [assumption].
% 1.05/1.80 128 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_lessequals(A),C),B)) # label(cls_linorder__not__le_0) # label(axiom). [assumption].
% 1.05/1.80 129 -class_Orderings_Olinorder(A) | c_Orderings_Olinorder(c_COMBC(c_lessequals(A),A,A,tc_bool),c_COMBC(c_HOL_Oord__class_Oless(A),A,A,tc_bool),A) # label(cls_dual__linorder_0) # label(axiom). [assumption].
% 1.05/1.80 130 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_lessequals(A),B),C)) | hBOOL(hAPP(hAPP(c_lessequals(A),C),B)) # label(cls_linorder__linear_0) # label(axiom). [assumption].
% 1.05/1.80 131 -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].
% 1.05/1.80 132 -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].
% 1.05/1.80 133 -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].
% 1.05/1.80 134 -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].
% 1.05/1.80 135 -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].
% 1.05/1.80 136 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),B)) # label(cls_linorder__neq__iff_1) # label(axiom). [assumption].
% 1.05/1.80 Derived: c_SetInterval_Oord__class_OlessThan(A,tc_nat) != c_SetInterval_Oord__class_OlessThan(B,tc_nat) | A = B. [resolve(117,a,118,a)].
% 1.05/1.80 Derived: c_SetInterval_Oord__class_OgreaterThan(A,tc_nat) != c_SetInterval_Oord__class_OgreaterThan(B,tc_nat) | A = B. [resolve(117,a,119,a)].
% 1.05/1.80 Derived: A = B | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),B)) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),A),B)). [resolve(117,a,120,a)].
% 1.05/1.80 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),B)) | hBOOL(hAPP(hAPP(c_lessequals(tc_nat),B),A)). [resolve(117,a,122,a)].
% 1.05/1.80 Derived: -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),A),A)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),A)). [resolve(117,a,123,a)].
% 1.05/1.80 Derived: -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),B)) | -hBOOL(hAPP(hAPP(c_lessequals(tc_nat),B),A)). [resolve(117,a,125,a)].
% 1.05/1.80 Derived: c_Orderings_Olinorder(c_COMBC(c_lessequals(tc_nat),tc_nat,tc_nat,tc_bool),c_COMBC(c_HOL_Oord__class_Oless(tc_nat),tc_nat,tc_nat,tc_bool),tc_nat). [resolve(117,a,129,a)].
% 1.05/1.80 Derived: hBOOL(hAPP(hAPP(c_lessequals(tc_nat),A),B)) | hBOOL(hAPP(hAPP(c_lessequals(tc_nat),B),A)). [resolve(117,a,130,a)].
% 1.05/1.80 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(117,a,131,a)].
% 1.05/1.80 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(117,a,132,a)].
% 1.05/1.80 137 class_OrderedGroup_Ocancel__ab__semigroup__add(tc_nat) # label(clsarity_nat__OrderedGroup_Ocancel__ab__semigroup__add) # label(axiom). [assumption].
% 1.05/1.80 138 -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].
% 1.05/1.80 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(137,a,138,a)].
% 1.05/1.80 139 class_OrderedGroup_Ocancel__semigroup__add(tc_nat) # label(clsarity_nat__OrderedGroup_Ocancel__semigroup__add) # label(axiom). [assumption].
% 1.05/1.88 140 -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].
% 1.05/1.88 141 -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].
% 1.05/1.88 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(139,a,141,a)].
% 1.05/1.88 142 class_Nat_Osemiring__char__0(tc_nat) # label(clsarity_nat__Nat_Osemiring__char__0) # label(axiom). [assumption].
% 1.05/1.88 143 -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].
% 1.05/1.88 Derived: c_Nat_Osemiring__1__class_Oof__nat(A,tc_nat) != c_Nat_Osemiring__1__class_Oof__nat(B,tc_nat) | A = B. [resolve(142,a,143,a)].
% 1.05/1.88 144 class_OrderedGroup_Omonoid__add(tc_nat) # label(clsarity_nat__OrderedGroup_Omonoid__add) # label(axiom). [assumption].
% 1.05/1.88 145 -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].
% 1.05/1.88 146 -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].
% 1.05/1.88 147 class_OrderedGroup_Opordered__cancel__ab__semigroup__add(tc_nat) # label(clsarity_nat__OrderedGroup_Opordered__cancel__ab__semigroup__add) # label(axiom). [assumption].
% 1.05/1.88 148 -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_lessequals(A),B),D)) # label(cls_add__le__less__mono_0) # label(axiom). [assumption].
% 1.05/1.88 149 -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_lessequals(A),C),E)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),D)) # label(cls_add__less__le__mono_0) # label(axiom). [assumption].
% 1.05/1.88 150 -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].
% 1.05/1.88 151 -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].
% 1.05/1.88 152 -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].
% 1.05/1.88 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_lessequals(tc_nat),A),C)). [resolve(147,a,148,a)].
% 1.05/1.88 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_lessequals(tc_nat),B),D)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),C)). [resolve(147,a,149,a)].
% 1.05/1.88 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(147,a,150,a)].
% 2.86/3.13 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(147,a,151,a)].
% 2.86/3.13 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(147,a,152,a)].
% 2.86/3.13
% 2.86/3.13 ============================== end predicate elimination =============
% 2.86/3.13
% 2.86/3.13 Auto_denials: (non-Horn, no changes).
% 2.86/3.13
% 2.86/3.13 Term ordering decisions:
% 2.86/3.13 Function symbol KB weights: tc_nat=1. tc_Arrow__Order__Mirabelle_Oalt=1. tc_bool=1. c_Arrow__Order__Mirabelle_OLin=1. tc_Arrow__Order__Mirabelle_Oindi=1. v_a=1. c_Arrow__Order__Mirabelle_OProf=1. t_a=1. v_Lab____=1. v_Lba____=1. v_h____=1. v_n=1. v_F=1. v_sko__Arrow__Order__Mirabelle__Xlinear__alt__1=1. hAPP=1. tc_fun=1. tc_prod=1. c_Nat_Osemiring__1__class_Oof__nat=1. c_Finite__Set_Ocard=1. c_Hilbert__Choice_OEps=1. c_Arrow__Order__Mirabelle_Omkbot=1. c_Arrow__Order__Mirabelle_Omktop=1. c_Complete__Lattice_OInf__class_OInf=1. c_Complete__Lattice_OSup__class_OSup=1. c_SetInterval_Oord__class_OgreaterThan=1. c_SetInterval_Oord__class_OlessThan=1. c_Set_OPow=1. c_List_Osko__Divides__Xmod__induct__1__1=1. c_The=1. v_sko__Arrow__Order__Mirabelle__Xcomplete__Lin__1=1. c_List_Osko__List__Xall__nat__less__eq__1__2=1. c_List_Osko__List__Xex__nat__less__eq__1__1=1. c_HOL_Oord__class_Oless=1. c_lessequals=1. c_in=1. c_HOL_Oplus__class_Oplus=1. c_HOL_Ozero__class_Ozero=1. c_Suc=1. c_HOL_Ominus__class_Ominus=1. c_HOL_Oone__class_Oone=1. c_Orderings_Otop__class_Otop=1. c_Orderings_Obot__class_Obot=1. c_HOL_Ouminus__class_Ouminus=1. c_HOL_OIf=1. c_fequal=1. c_ATP__Linkup_Osko__Nat__Xinfinite__descent__1__1=1. c_ATP__Linkup_Osko__Nat__Xnat__less__induct__1__1=1. c_Power_Opower__class_Opower=1. c_COMBB=1. c_Arrow__Order__Mirabelle_Oabove=1. c_Arrow__Order__Mirabelle_Obelow=1. c_Relation_Oconverse=1. c_ATP__Linkup_Osko__Nat__Xinfinite__descent__measure__1__1=1. c_ATP__Linkup_Osko__Nat__Xmeasure__induct__1__1=1. c_ATP__Linkup_Osko__Nat__Xmeasure__induct__rule__1__1=1. c_COMBK=1. c_Pair=1. c_COMBC=1. c_Set_Ovimage=1. c_Finite__Set_Osetprod=1. c_Finite__Set_Osetsum=1. c_FuncSet_OPi=1. c_COMBS=1. c_Fun_Ooverride__on=1.
% 2.86/3.13
% 2.86/3.13 ============================== end of process initial clauses ========
% 2.86/3.13
% 2.86/3.13 ============================== CLAUSES FOR SEARCH ====================
% 2.86/3.13
% 2.86/3.13 ============================== end of clauses for search =============
% 2.86/3.13
% 2.86/3.13 ============================== SEARCH ================================
% 2.86/3.13
% 2.86/3.13 % Starting search at 0.68 seconds.
% 2.86/3.13
% 2.86/3.13 NOTE: Back_subsumption disabled, ratio of kept to back_subsumed is 34 (0.00 of 1.23 sec).
% 2.86/3.13
% 2.86/3.13 Low Water (keep): wt=66.000, iters=3345
% 2.86/3.13
% 2.86/3.13 Low Water (keep): wt=64.000, iters=3412
% 2.86/3.13
% 2.86/3.13 Low Water (keep): wt=56.000, iters=3400
% 2.86/3.13
% 2.86/3.13 Low Water (keep): wt=49.000, iters=3414
% 2.86/3.13
% 2.86/3.13 Low Water (keep): wt=45.000, iters=3344
% 2.86/3.13
% 2.86/3.13 Low Water (keep): wt=43.000, iters=3366
% 2.86/3.13
% 2.86/3.13 Low Water (keep): wt=42.000, iters=3346
% 2.86/3.13
% 2.86/3.13 Low Water (keep): wt=41.000, iters=3420
% 2.86/3.13
% 2.86/3.13 Low Water (keep): wt=40.000, iters=3363
% 2.86/3.13
% 2.86/3.13 Low Water (keep): wt=39.000, iters=3416
% 2.86/3.13
% 2.86/3.13 Low Water (keep): wt=38.000, iters=3463
% 2.86/3.13
% 2.86/3.13 Low Water (keep): wt=37.000, iters=3373
% 2.86/3.13
% 2.86/3.13 Low Water (keep): wt=36.000, iters=3353
% 2.86/3.13
% 2.86/3.13 Low Water (keep): wt=33.000, iters=3395
% 2.86/3.13
% 2.86/3.13 Low Water (keep): wt=32.000, iters=3606
% 2.86/3.13
% 2.86/3.13 Low Water (keep): wt=31.000, iters=3529
% 2.86/3.13
% 2.86/3.13 Low Water (keep): wt=30.000, iters=3390
% 2.86/3.13
% 2.86/3.13 Low Water (keep): wt=29.000, iters=3382
% 2.86/3.13
% 2.86/3.13 Low Water (keep): wt=28.000, iters=3334
% 2.86/3.13
% 2.86/3.13 Low Water (keep): wt=27.000, iters=3346
% 2.86/3.13
% 2.86/3.13 Low Water (keep): wt=26.000, iters=3342
% 2.86/3.13
% 2.86/3.13 Low Water (keep): wt=25.000, iters=3403
% 2.86/3.13
% 2.86/3.13 Low Water (keep): wt=24.000, iters=3576
% 2.86/3.13
% 2.86/3.13 Low Water (keep): wt=22.000, iters=3720
% 2.86/3.13
% 2.86/3.13 Low Water (keep): wt=21.000, iters=3844
% 2.86/3.13
% 2.86/3.13 Low Water (keep): wt=19.000, iters=3434
% 7.94/8.24
% 7.94/8.24 Low Water (keep): wt=16.000, iters=3344
% 7.94/8.24
% 7.94/8.24 Low Water (keep): wt=14.000, iters=3345
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=2901, wt=198.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=2900, wt=196.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=3738, wt=194.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=2896, wt=193.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=2898, wt=192.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=3844, wt=191.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=2902, wt=187.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=2751, wt=186.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=3843, wt=185.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=3842, wt=181.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=3154, wt=176.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=2904, wt=174.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=2710, wt=137.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=3819, wt=132.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=2715, wt=127.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=2670, wt=125.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=2718, wt=124.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=1574, wt=123.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=3824, wt=122.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=3827, wt=119.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=2711, wt=118.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=2721, wt=117.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=2897, wt=116.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=2693, wt=115.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=3779, wt=114.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=2873, wt=113.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=3812, wt=112.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=2723, wt=111.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=2866, wt=110.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=3830, wt=109.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=2689, wt=108.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=3820, wt=107.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=3832, wt=106.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=2878, wt=105.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=3802, wt=104.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=3828, wt=103.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=3766, wt=102.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=2884, wt=101.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=2829, wt=100.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=3777, wt=99.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=8087, wt=76.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=6896, wt=73.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=10604, wt=64.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=8452, wt=59.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=8481, wt=57.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=11966, wt=56.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=11897, wt=55.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=12068, wt=54.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=2576, wt=52.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=12037, wt=51.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=11859, wt=50.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=8088, wt=49.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=12167, wt=48.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=12023, wt=47.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=8477, wt=46.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=12131, wt=45.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=12048, wt=44.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=12448, wt=43.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=12371, wt=42.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=12635, wt=41.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=12718, wt=40.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=12733, wt=39.000
% 7.94/8.24
% 7.94/8.24 Low Water (displace): id=12863, wt=12.000
% 7.94/8.24
% 7.94/8.24 ============================== PROOF =================================
% 7.94/8.24 % SZS status Unsatisfiable
% 7.94/8.24 % SZS output start Refutation
% 7.94/8.24
% 7.94/8.24 % Proof 1 at 6.77 (+ 0.18) seconds.
% 7.94/8.24 % Length of proof is 42.
% 7.94/8.24 % Level of proof is 10.
% 7.94/8.24 % Maximum clause weight is 29.000.
% 7.94/8.24 % Given clauses 4172.
% 7.94/8.24
% 7.94/8.24 30 class_Ring__and__Field_Oordered__semidom(tc_nat) # label(clsarity_nat__Ring__and__Field_Oordered__semidom) # label(axiom). [assumption].
% 7.94/8.24 58 -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].
% 7.94/8.24 70 class_Ring__and__Field_Ocomm__semiring__1(tc_nat) # label(clsarity_nat__Ring__and__Field_Ocomm__semiring__1) # label(axiom). [assumption].
% 7.94/8.24 84 -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].
% 7.94/8.24 105 class_Ring__and__Field_Ozero__neq__one(tc_nat) # label(clsarity_nat__Ring__and__Field_Ozero__neq__one) # label(axiom). [assumption].
% 7.94/8.24 106 -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].
% 7.94/8.24 166 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)) # label(cls_nat__add__left__commute_0) # label(axiom). [assumption].
% 7.94/8.24 167 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)) # label(cls_nat__add__assoc_0) # label(axiom). [assumption].
% 7.94/8.24 175 hAPP(c_COMBS(A,B,C,D,E),F) = hAPP(hAPP(A,F),hAPP(B,F)) # label(cls_COMBS__def_0) # label(axiom). [assumption].
% 7.94/8.24 193 hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),c_Suc(A)),B) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),c_Suc(B)) # label(cls_add__Suc__shift_0) # label(axiom). [assumption].
% 7.94/8.24 229 hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),c_Suc(B)) = c_Suc(hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B)) # label(cls_add__Suc__right_0) # label(axiom). [assumption].
% 7.94/8.24 230 c_Suc(hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),c_Suc(B)). [copy(229),flip(a)].
% 7.94/8.24 232 c_HOL_Ozero__class_Ozero(tc_nat) != c_Suc(A) # label(cls_nat_Osimps_I2_J_0) # label(axiom). [assumption].
% 7.94/8.24 246 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].
% 7.94/8.24 248 hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B) != A | B = c_HOL_Ozero__class_Ozero(tc_nat) # label(cls_add__eq__self__zero_0) # label(axiom). [assumption].
% 7.94/8.24 249 hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B) != A | c_HOL_Ozero__class_Ozero(tc_nat) = B. [copy(248),flip(b)].
% 7.94/8.24 314 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].
% 7.94/8.24 317 hAPP(hAPP(c_HOL_Ominus__class_Ominus(tc_nat),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B)),B) = A # label(cls_diff__add__inverse2_0) # label(axiom). [assumption].
% 7.94/8.24 527 c_Suc(hAPP(hAPP(c_HOL_Ominus__class_Ominus(tc_nat),A),c_Suc(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)),A)) # label(cls_Suc__pred_0) # label(axiom). [assumption].
% 7.94/8.24 610 hAPP(hAPP(hAPP(c_COMBB(A,B,C),D),E),F) = hAPP(D,hAPP(E,F)) # label(cls_COMBB__def_0) # label(axiom). [assumption].
% 7.94/8.24 624 class_Ring__and__Field_Ono__zero__divisors(tc_nat) # label(clsarity_nat__Ring__and__Field_Ono__zero__divisors) # label(axiom). [assumption].
% 7.94/8.24 625 class_Ring__and__Field_Omult__zero(tc_nat) # label(clsarity_nat__Ring__and__Field_Omult__zero) # label(axiom). [assumption].
% 7.94/8.24 629 class_Power_Opower(tc_nat) # label(clsarity_nat__Power_Opower) # label(axiom). [assumption].
% 7.94/8.24 693 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(30,a,58,a)].
% 7.94/8.24 713 c_Power_Opower__class_Opower(A,c_HOL_Ozero__class_Ozero(tc_nat),tc_nat) = c_HOL_Oone__class_Oone(tc_nat). [resolve(70,a,84,a)].
% 7.94/8.24 726 -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(105,a,106,a)].
% 7.94/8.24 727 c_HOL_Oone__class_Oone(tc_nat) != c_HOL_Ozero__class_Ozero(tc_nat). [copy(726),rewrite([713(10)]),unit_del(a,624),unit_del(b,625),unit_del(c,629)].
% 7.94/8.24 779 c_HOL_Oone__class_Oone(tc_nat) = c_0. [new_symbol(727)].
% 7.94/8.24 792 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(693),rewrite([779(8)])].
% 7.94/8.24 824 hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),hAPP(c_COMBS(c_HOL_Oplus__class_Oplus(tc_nat),B,C,D,E),F)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),F),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),hAPP(B,F))). [para(175(a,2),166(a,1,2))].
% 7.94/8.24 880 hAPP(c_COMBS(c_HOL_Oplus__class_Oplus(tc_nat),A,B,C,D),c_Suc(E)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),E),c_Suc(hAPP(A,c_Suc(E)))). [para(193(a,1),175(a,2))].
% 7.94/8.24 1260 hAPP(c_COMBS(A,hAPP(c_HOL_Oplus__class_Oplus(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),B,C,D),E) = hAPP(hAPP(A,E),E). [para(314(a,1),175(a,2,2))].
% 7.94/8.24 4010 c_Suc(hAPP(hAPP(c_HOL_Ominus__class_Ominus(tc_nat),c_0),c_Suc(c_HOL_Ozero__class_Ozero(tc_nat)))) = c_0. [resolve(792,a,527,b),rewrite([314(9),314(16)])].
% 7.94/8.24 10744 hAPP(c_COMBS(c_HOL_Oplus__class_Oplus(tc_nat),A,B,C,D),c_Suc(E)) != E. [para(880(a,2),249(a,1)),flip(b),unit_del(b(flip),232)].
% 7.94/8.24 10806 hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),c_Suc(hAPP(B,c_Suc(A)))) != A. [para(175(a,1),10744(a,1)),rewrite([193(7)])].
% 7.94/8.24 10819 hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),c_Suc(B)) != A. [para(317(a,1),10806(a,1,2,1))].
% 7.94/8.24 10898 hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),B),c_Suc(C))) != A. [para(230(a,1),10819(a,1,2))].
% 7.94/8.24 14329 hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),B),c_0)) != A. [para(4010(a,1),10898(a,1,2,2))].
% 7.94/8.24 14730 hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),c_0),B)) != B. [para(246(a,1),14329(a,1)),rewrite([167(9)])].
% 7.94/8.24 14901 hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),c_0),hAPP(c_COMBS(c_HOL_Oplus__class_Oplus(tc_nat),A,B,C,D),E)) != hAPP(A,E). [para(824(a,2),14730(a,1))].
% 7.94/8.24 27152 hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),c_0),hAPP(c_COMBS(c_HOL_Oplus__class_Oplus(tc_nat),c_COMBS(A,hAPP(c_HOL_Oplus__class_Oplus(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),B,C,D),E,F,V6),V7)) != hAPP(hAPP(A,V7),V7). [para(1260(a,1),14901(a,2))].
% 7.94/8.24 27153 $F. [resolve(27152,a,610,a(flip))].
% 7.94/8.24
% 7.94/8.24 % SZS output end Refutation
% 7.94/8.24 ============================== end of proof ==========================
% 7.94/8.24
% 7.94/8.24 ============================== STATISTICS ============================
% 7.94/8.24
% 7.94/8.24 Given=4172. Generated=312177. Kept=26895. proofs=1.
% 7.94/8.24 Usable=4156. Sos=9995. Demods=418. Limbo=281, Disabled=13339. Hints=0.
% 7.94/8.24 Megabytes=37.02.
% 7.94/8.24 User_CPU=6.77, System_CPU=0.18, Wall_clock=7.
% 7.94/8.24
% 7.94/8.24 ============================== end of statistics =====================
% 7.94/8.24
% 7.94/8.24 ============================== end of search =========================
% 7.94/8.24
% 7.94/8.24 THEOREM PROVED
% 7.94/8.24 % SZS status Unsatisfiable
% 7.94/8.24
% 7.94/8.24 Exiting with 1 proof.
% 7.94/8.24
% 7.94/8.24 Process 3710 exit (max_proofs) Sat Jul 2 07:37:03 2022
% 7.94/8.24 Prover9 interrupted
%------------------------------------------------------------------------------