TSTP Solution File: SCT098-1 by Prover9---1109a
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Prover9---1109a
% Problem : SCT098-1 : TPTP v8.1.0. Released v4.1.0.
% Transfm : none
% Format : tptp:raw
% Command : tptp2X_and_run_prover9 %d %s
% Computer : n026.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 22:03:53 EDT 2022
% Result : Timeout 300.05s 300.34s
% Output : None
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----No solution output by system
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : SCT098-1 : TPTP v8.1.0. Released v4.1.0.
% 0.11/0.13 % Command : tptp2X_and_run_prover9 %d %s
% 0.13/0.34 % Computer : n026.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 600
% 0.13/0.34 % DateTime : Sat Jul 2 03:49:50 EDT 2022
% 0.13/0.34 % CPUTime :
% 1.57/1.81 ============================== Prover9 ===============================
% 1.57/1.81 Prover9 (32) version 2009-11A, November 2009.
% 1.57/1.81 Process 12409 was started by sandbox on n026.cluster.edu,
% 1.57/1.81 Sat Jul 2 03:49:51 2022
% 1.57/1.81 The command was "/export/starexec/sandbox/solver/bin/prover9 -t 300 -f /tmp/Prover9_12254_n026.cluster.edu".
% 1.57/1.81 ============================== end of head ===========================
% 1.57/1.81
% 1.57/1.81 ============================== INPUT =================================
% 1.57/1.81
% 1.57/1.81 % Reading from file /tmp/Prover9_12254_n026.cluster.edu
% 1.57/1.81
% 1.57/1.81 set(prolog_style_variables).
% 1.57/1.81 set(auto2).
% 1.57/1.81 % set(auto2) -> set(auto).
% 1.57/1.81 % set(auto) -> set(auto_inference).
% 1.57/1.81 % set(auto) -> set(auto_setup).
% 1.57/1.81 % set(auto_setup) -> set(predicate_elim).
% 1.57/1.81 % set(auto_setup) -> assign(eq_defs, unfold).
% 1.57/1.81 % set(auto) -> set(auto_limits).
% 1.57/1.81 % set(auto_limits) -> assign(max_weight, "100.000").
% 1.57/1.81 % set(auto_limits) -> assign(sos_limit, 20000).
% 1.57/1.81 % set(auto) -> set(auto_denials).
% 1.57/1.81 % set(auto) -> set(auto_process).
% 1.57/1.81 % set(auto2) -> assign(new_constants, 1).
% 1.57/1.81 % set(auto2) -> assign(fold_denial_max, 3).
% 1.57/1.81 % set(auto2) -> assign(max_weight, "200.000").
% 1.57/1.81 % set(auto2) -> assign(max_hours, 1).
% 1.57/1.81 % assign(max_hours, 1) -> assign(max_seconds, 3600).
% 1.57/1.81 % set(auto2) -> assign(max_seconds, 0).
% 1.57/1.81 % set(auto2) -> assign(max_minutes, 5).
% 1.57/1.81 % assign(max_minutes, 5) -> assign(max_seconds, 300).
% 1.57/1.81 % set(auto2) -> set(sort_initial_sos).
% 1.57/1.81 % set(auto2) -> assign(sos_limit, -1).
% 1.57/1.81 % set(auto2) -> assign(lrs_ticks, 3000).
% 1.57/1.81 % set(auto2) -> assign(max_megs, 400).
% 1.57/1.81 % set(auto2) -> assign(stats, some).
% 1.57/1.81 % set(auto2) -> clear(echo_input).
% 1.57/1.81 % set(auto2) -> set(quiet).
% 1.57/1.81 % set(auto2) -> clear(print_initial_clauses).
% 1.57/1.81 % set(auto2) -> clear(print_given).
% 1.57/1.81 assign(lrs_ticks,-1).
% 1.57/1.81 assign(sos_limit,10000).
% 1.57/1.81 assign(order,kbo).
% 1.57/1.81 set(lex_order_vars).
% 1.57/1.81 clear(print_given).
% 1.57/1.81
% 1.57/1.81 % formulas(sos). % not echoed (1345 formulas)
% 1.57/1.81
% 1.57/1.81 ============================== end of input ==========================
% 1.57/1.81
% 1.57/1.81 % From the command line: assign(max_seconds, 300).
% 1.57/1.81
% 1.57/1.81 ============================== PROCESS NON-CLAUSAL FORMULAS ==========
% 1.57/1.81
% 1.57/1.81 % Formulas that are not ordinary clauses:
% 1.57/1.81
% 1.57/1.81 ============================== end of process non-clausal formulas ===
% 1.57/1.81
% 1.57/1.81 ============================== PROCESS INITIAL CLAUSES ===============
% 1.57/1.81
% 1.57/1.81 ============================== PREDICATE ELIMINATION =================
% 1.57/1.81 1 class_Ring__and__Field_Ocomm__semiring__1(tc_nat) # label(clsarity_nat__Ring__and__Field_Ocomm__semiring__1) # label(axiom). [assumption].
% 1.57/1.81 2 -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].
% 1.57/1.81 3 -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].
% 1.57/1.81 4 -class_Ring__and__Field_Ocomm__semiring__1(A) | c_Ring__and__Field_Odvd__class_Odvd(c_Power_Opower__class_Opower(B,C,A),c_Power_Opower__class_Opower(B,D,A),A) | -c_lessequals(C,D,tc_nat) # label(cls_le__imp__power__dvd_0) # label(axiom). [assumption].
% 1.57/1.81 5 -class_Ring__and__Field_Ocomm__semiring__1(A) | c_Ring__and__Field_Odvd__class_Odvd(c_HOL_Oone__class_Oone(A),c_Power_Opower__class_Opower(c_HOL_Oone__class_Oone(A),B,A),A) # label(cls_dvd__power_1) # label(axiom). [assumption].
% 1.57/1.81 6 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),B) = hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),c_HOL_Oone__class_Oone(A)),c_HOL_Oone__class_Oone(A))),B) # label(cls_class__semiring_Osemiring__rules_I4_J_0) # label(axiom). [assumption].
% 1.57/1.81 7 -class_Ring__and__Field_Ocomm__semiring__1(A) | c_Ring__and__Field_Odvd__class_Odvd(B,c_HOL_Ozero__class_Ozero(A),A) # label(cls_dvd__0__right_0) # label(axiom). [assumption].
% 1.57/1.81 8 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),D),E)) = hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),C),E)) # label(cls_class__semiring_Osemiring__rules_I13_J_0) # label(axiom). [assumption].
% 1.57/1.81 9 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),D),E)) = hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),C),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),D),E))) # label(cls_class__semiring_Osemiring__rules_I14_J_0) # label(axiom). [assumption].
% 1.57/1.81 10 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),D),E)) = hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),D),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C)),E)) # label(cls_class__semiring_Osemiring__rules_I15_J_0) # label(axiom). [assumption].
% 1.57/1.81 11 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),c_HOL_Ozero__class_Ozero(A)),B) = c_HOL_Ozero__class_Ozero(A) # label(cls_class__semiring_Osemiring__rules_I9_J_0) # label(axiom). [assumption].
% 1.57/1.81 12 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),c_HOL_Ozero__class_Ozero(A)) = c_HOL_Ozero__class_Ozero(A) # label(cls_class__semiring_Osemiring__rules_I10_J_0) # label(axiom). [assumption].
% 1.57/1.81 13 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),c_HOL_Oone__class_Oone(A)),B) = B # label(cls_class__semiring_Omul__1_0) # label(axiom). [assumption].
% 1.57/1.81 14 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),c_HOL_Oone__class_Oone(A)) = B # label(cls_class__semiring_Osemiring__rules_I12_J_0) # label(axiom). [assumption].
% 1.57/1.81 15 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),c_HOL_Oone__class_Oone(A)),B) = B # label(cls_class__semiring_Osemiring__rules_I11_J_0) # label(axiom). [assumption].
% 1.57/1.81 16 -class_Ring__and__Field_Ocomm__semiring__1(A) | c_Ring__and__Field_Odvd__class_Odvd(B,hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C),A) # label(cls_dvd__triv__left_0) # label(axiom). [assumption].
% 1.57/1.81 17 -class_Ring__and__Field_Ocomm__semiring__1(A) | c_Ring__and__Field_Odvd__class_Odvd(B,hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),C),B),A) # label(cls_dvd__triv__right_0) # label(axiom). [assumption].
% 1.57/1.81 18 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C) = hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),C),B) # label(cls_class__semiring_Omul__c_0) # label(axiom). [assumption].
% 1.57/1.81 19 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C) = hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),C),B) # label(cls_class__semiring_Osemiring__rules_I7_J_0) # label(axiom). [assumption].
% 1.57/1.81 20 -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].
% 1.57/1.81 21 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),C),D)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C)),D) # label(cls_class__semiring_Osemiring__rules_I25_J_0) # label(axiom). [assumption].
% 1.57/1.81 22 -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].
% 1.57/1.81 23 -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].
% 1.57/1.81 24 -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].
% 1.57/1.81 25 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),c_HOL_Ozero__class_Ozero(A)),B) = c_HOL_Ozero__class_Ozero(A) # label(cls_class__semiring_Omul__0_0) # label(axiom). [assumption].
% 1.57/1.81 26 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C)),C) = hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),c_HOL_Oone__class_Oone(A))),C) # label(cls_class__semiring_Osemiring__rules_I2_J_0) # label(axiom). [assumption].
% 1.57/1.81 27 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),C),B)) = hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),C),c_HOL_Oone__class_Oone(A))),B) # label(cls_class__semiring_Osemiring__rules_I3_J_0) # label(axiom). [assumption].
% 1.57/1.81 28 -class_Ring__and__Field_Ocomm__semiring__1(A) | c_Ring__and__Field_Odvd__class_Odvd(B,C,A) | -c_Ring__and__Field_Odvd__class_Odvd(D,C,A) | -c_Ring__and__Field_Odvd__class_Odvd(B,D,A) # label(cls_dvd__trans_0) # label(axiom). [assumption].
% 1.57/1.81 29 -class_Ring__and__Field_Ocomm__semiring__1(A) | c_Ring__and__Field_Odvd__class_Odvd(B,B,A) # label(cls_dvd__refl_0) # label(axiom). [assumption].
% 1.57/1.81 30 -class_Ring__and__Field_Ocomm__semiring__1(A) | c_Ring__and__Field_Odvd__class_Odvd(c_HOL_Oone__class_Oone(A),B,A) # label(cls_one__dvd_0) # label(axiom). [assumption].
% 1.57/1.81 31 -class_Ring__and__Field_Ocomm__semiring__1(A) | B = c_HOL_Ozero__class_Ozero(A) | -c_Ring__and__Field_Odvd__class_Odvd(c_HOL_Ozero__class_Ozero(A),B,A) # label(cls_dvd__0__left_0) # label(axiom). [assumption].
% 1.57/1.81 32 -class_Ring__and__Field_Ocomm__semiring__1(A) | c_Ring__and__Field_Odvd__class_Odvd(c_Power_Opower__class_Opower(B,C,A),c_Power_Opower__class_Opower(D,C,A),A) | -c_Ring__and__Field_Odvd__class_Odvd(B,D,A) # label(cls_dvd__power__same_0) # label(axiom). [assumption].
% 1.57/1.81 33 -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].
% 1.57/1.81 34 -class_Ring__and__Field_Ocomm__semiring__1(A) | c_Ring__and__Field_Odvd__class_Odvd(c_Power_Opower__class_Opower(B,C,A),c_Power_Opower__class_Opower(D,E,A),A) | -c_lessequals(C,E,tc_nat) | -c_Ring__and__Field_Odvd__class_Odvd(B,D,A) # label(cls_dvd__power__le_0) # label(axiom). [assumption].
% 1.57/1.81 35 -class_Ring__and__Field_Ocomm__semiring__1(A) | c_Ring__and__Field_Odvd__class_Odvd(B,C,A) | -c_Ring__and__Field_Odvd__class_Odvd(hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),D),B),C,A) # label(cls_dvd__mult__right_0) # label(axiom). [assumption].
% 1.57/1.81 36 -class_Ring__and__Field_Ocomm__semiring__1(A) | c_Ring__and__Field_Odvd__class_Odvd(B,C,A) | -c_Ring__and__Field_Odvd__class_Odvd(hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),D),C,A) # label(cls_dvd__mult__left_0) # label(axiom). [assumption].
% 1.57/1.81 37 -class_Ring__and__Field_Ocomm__semiring__1(A) | c_Ring__and__Field_Odvd__class_Odvd(B,hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),C),D),A) | -c_Ring__and__Field_Odvd__class_Odvd(B,C,A) # label(cls_dvd__mult2_0) # label(axiom). [assumption].
% 1.57/1.81 38 -class_Ring__and__Field_Ocomm__semiring__1(A) | c_Ring__and__Field_Odvd__class_Odvd(B,hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),C),D),A) | -c_Ring__and__Field_Odvd__class_Odvd(B,D,A) # label(cls_dvd__mult_0) # label(axiom). [assumption].
% 1.57/1.81 39 -class_Ring__and__Field_Ocomm__semiring__1(A) | c_Ring__and__Field_Odvd__class_Odvd(hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),D),E),A) | -c_Ring__and__Field_Odvd__class_Odvd(C,E,A) | -c_Ring__and__Field_Odvd__class_Odvd(B,D,A) # label(cls_mult__dvd__mono_0) # label(axiom). [assumption].
% 1.57/1.81 40 -class_Ring__and__Field_Ocomm__semiring__1(A) | c_Ring__and__Field_Odvd__class_Odvd(B,c_Power_Opower__class_Opower(B,C,A),A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),C)) # label(cls_dvd__power_0) # label(axiom). [assumption].
% 1.57/1.81 41 -class_Ring__and__Field_Ocomm__semiring__1(A) | c_Ring__and__Field_Odvd__class_Odvd(c_Power_Opower__class_Opower(B,C,A),D,A) | -c_lessequals(C,E,tc_nat) | -c_Ring__and__Field_Odvd__class_Odvd(c_Power_Opower__class_Opower(B,E,A),D,A) # label(cls_power__le__dvd_0) # label(axiom). [assumption].
% 1.57/1.81 42 -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].
% 1.57/1.81 43 -class_Ring__and__Field_Ocomm__semiring__1(A) | c_Ring__and__Field_Odvd__class_Odvd(B,hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),C),D),A) | -c_Ring__and__Field_Odvd__class_Odvd(B,D,A) | -c_Ring__and__Field_Odvd__class_Odvd(B,C,A) # label(cls_dvd__add_0) # label(axiom). [assumption].
% 1.57/1.81 44 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),D),C)) = hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),D)),C) # label(cls_class__semiring_Osemiring__rules_I1_J_0) # label(axiom). [assumption].
% 1.57/1.81 45 -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].
% 1.57/1.81 46 -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].
% 1.57/1.81 47 -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].
% 1.57/1.81 48 -class_Ring__and__Field_Ocomm__semiring__1(A) | c_Power_Opower__class_Opower(hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C),D,A) = hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),c_Power_Opower__class_Opower(B,D,A)),c_Power_Opower__class_Opower(C,D,A)) # label(cls_class__semiring_Opwr__mul_0) # label(axiom). [assumption].
% 1.57/1.81 49 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C)),D) = hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),D)),C) # label(cls_class__semiring_Osemiring__rules_I16_J_0) # label(axiom). [assumption].
% 1.57/1.81 50 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C)),D) = hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),C),D)) # label(cls_class__semiring_Osemiring__rules_I17_J_0) # label(axiom). [assumption].
% 1.57/1.81 51 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),C),D)) = hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C)),D) # label(cls_class__semiring_Osemiring__rules_I18_J_0) # label(axiom). [assumption].
% 1.57/1.81 52 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),C),D)) = hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),C),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),D)) # label(cls_class__semiring_Osemiring__rules_I19_J_0) # label(axiom). [assumption].
% 1.57/1.81 53 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),C),D)) = hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C)),D) # label(cls_class__semiring_Omul__a_0) # label(axiom). [assumption].
% 1.57/1.81 54 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C)),D) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),C),D)) # label(cls_class__semiring_Osemiring__rules_I8_J_0) # label(axiom). [assumption].
% 1.57/1.81 55 -class_Ring__and__Field_Ocomm__semiring__1(A) | hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),C),D)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),D)) # label(cls_class__semiring_Omul__d_0) # label(axiom). [assumption].
% 1.57/1.81 56 -class_Ring__and__Field_Ocomm__semiring__1(A) | c_Ring__and__Field_Odvd__class_Odvd(hAPP(B,C),c_Finite__Set_Osetprod(B,D,E,A),A) | -hBOOL(hAPP(hAPP(c_in(E),C),D)) | -c_Finite__Set_Ofinite(D,E) # label(cls_dvd__setprod_0) # label(axiom). [assumption].
% 1.57/1.81 57 -class_Ring__and__Field_Ono__zero__divisors(A) | -class_Ring__and__Field_Ocomm__semiring__1(A) | c_Finite__Set_Osetprod(B,C,D,A) != c_HOL_Ozero__class_Ozero(A) | -c_Finite__Set_Ofinite(C,D) | hAPP(B,c_ATP__Linkup_Osko__Finite__Set__Xsetprod__zero__iff__1__1(C,B,D,A)) = c_HOL_Ozero__class_Ozero(A) # label(cls_setprod__zero__iff_1) # label(axiom). [assumption].
% 1.57/1.81 58 -class_Ring__and__Field_Ono__zero__divisors(A) | -class_Ring__and__Field_Ocomm__semiring__1(A) | c_Finite__Set_Osetprod(B,C,D,A) != c_HOL_Ozero__class_Ozero(A) | -c_Finite__Set_Ofinite(C,D) | hBOOL(hAPP(hAPP(c_in(D),c_ATP__Linkup_Osko__Finite__Set__Xsetprod__zero__iff__1__1(C,B,D,A)),C)) # label(cls_setprod__zero__iff_0) # label(axiom). [assumption].
% 1.57/1.81 59 -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].
% 1.57/1.81 60 -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].
% 1.57/1.81 Derived: hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),B),A). [resolve(1,a,2,a)].
% 1.57/1.81 Derived: c_Ring__and__Field_Odvd__class_Odvd(c_Power_Opower__class_Opower(A,B,tc_nat),c_Power_Opower__class_Opower(A,C,tc_nat),tc_nat) | -c_lessequals(B,C,tc_nat). [resolve(1,a,4,a)].
% 1.57/1.81 Derived: c_Ring__and__Field_Odvd__class_Odvd(c_HOL_Oone__class_Oone(tc_nat),c_Power_Opower__class_Opower(c_HOL_Oone__class_Oone(tc_nat),A,tc_nat),tc_nat). [resolve(1,a,5,a)].
% 1.57/1.81 Derived: hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),A) = hAPP(hAPP(c_HOL_Otimes__class_Otimes(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))),A). [resolve(1,a,6,a)].
% 1.57/1.81 Derived: c_Ring__and__Field_Odvd__class_Odvd(A,c_HOL_Ozero__class_Ozero(tc_nat),tc_nat). [resolve(1,a,7,a)].
% 1.57/1.81 Derived: hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),B)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),C),D)) = hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),C)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),B),D)). [resolve(1,a,8,a)].
% 1.57/1.81 Derived: hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),B)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),C),D)) = hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),B),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),C),D))). [resolve(1,a,9,a)].
% 1.57/1.81 Derived: hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),B)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),C),D)) = hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),C),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),B)),D)). [resolve(1,a,10,a)].
% 1.58/1.81 Derived: hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),A) = c_HOL_Ozero__class_Ozero(tc_nat). [resolve(1,a,11,a)].
% 1.58/1.81 Derived: hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),c_HOL_Ozero__class_Ozero(tc_nat)) = c_HOL_Ozero__class_Ozero(tc_nat). [resolve(1,a,12,a)].
% 1.58/1.81 Derived: hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),c_HOL_Oone__class_Oone(tc_nat)),A) = A. [resolve(1,a,13,a)].
% 1.58/1.81 Derived: hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),c_HOL_Oone__class_Oone(tc_nat)) = A. [resolve(1,a,14,a)].
% 1.58/1.81 Derived: c_Ring__and__Field_Odvd__class_Odvd(A,hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),B),tc_nat). [resolve(1,a,16,a)].
% 1.58/1.81 Derived: c_Ring__and__Field_Odvd__class_Odvd(A,hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),B),A),tc_nat). [resolve(1,a,17,a)].
% 1.58/1.81 Derived: hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),B) = hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),B),A). [resolve(1,a,18,a)].
% 1.58/1.81 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(1,a,20,a)].
% 1.58/1.81 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(1,a,22,a)].
% 1.58/1.81 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(1,a,23,a)].
% 1.58/1.81 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(1,a,24,a)].
% 1.58/1.81 Derived: hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),B)),B) = hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),c_HOL_Oone__class_Oone(tc_nat))),B). [resolve(1,a,26,a)].
% 1.58/1.81 Derived: hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),B),A)) = hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),B),c_HOL_Oone__class_Oone(tc_nat))),A). [resolve(1,a,27,a)].
% 1.58/1.81 Derived: c_Ring__and__Field_Odvd__class_Odvd(A,B,tc_nat) | -c_Ring__and__Field_Odvd__class_Odvd(C,B,tc_nat) | -c_Ring__and__Field_Odvd__class_Odvd(A,C,tc_nat). [resolve(1,a,28,a)].
% 1.58/1.81 Derived: c_Ring__and__Field_Odvd__class_Odvd(A,A,tc_nat). [resolve(1,a,29,a)].
% 1.58/1.81 Derived: c_Ring__and__Field_Odvd__class_Odvd(c_HOL_Oone__class_Oone(tc_nat),A,tc_nat). [resolve(1,a,30,a)].
% 1.58/1.81 Derived: A = c_HOL_Ozero__class_Ozero(tc_nat) | -c_Ring__and__Field_Odvd__class_Odvd(c_HOL_Ozero__class_Ozero(tc_nat),A,tc_nat). [resolve(1,a,31,a)].
% 1.58/1.81 Derived: c_Ring__and__Field_Odvd__class_Odvd(c_Power_Opower__class_Opower(A,B,tc_nat),c_Power_Opower__class_Opower(C,B,tc_nat),tc_nat) | -c_Ring__and__Field_Odvd__class_Odvd(A,C,tc_nat). [resolve(1,a,32,a)].
% 1.58/1.81 Derived: c_Power_Opower__class_Opower(A,c_HOL_Ozero__class_Ozero(tc_nat),tc_nat) = c_HOL_Oone__class_Oone(tc_nat). [resolve(1,a,33,a)].
% 1.58/1.81 Derived: c_Ring__and__Field_Odvd__class_Odvd(c_Power_Opower__class_Opower(A,B,tc_nat),c_Power_Opower__class_Opower(C,D,tc_nat),tc_nat) | -c_lessequals(B,D,tc_nat) | -c_Ring__and__Field_Odvd__class_Odvd(A,C,tc_nat). [resolve(1,a,34,a)].
% 1.58/1.81 Derived: c_Ring__and__Field_Odvd__class_Odvd(A,B,tc_nat) | -c_Ring__and__Field_Odvd__class_Odvd(hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),C),A),B,tc_nat). [resolve(1,a,35,a)].
% 1.58/1.81 Derived: c_Ring__and__Field_Odvd__class_Odvd(A,B,tc_nat) | -c_Ring__and__Field_Odvd__class_Odvd(hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),C),B,tc_nat). [resolve(1,a,36,a)].
% 1.58/1.81 Derived: c_Ring__and__Field_Odvd__class_Odvd(A,hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),B),C),tc_nat) | -c_Ring__and__Field_Odvd__class_Odvd(A,B,tc_nat). [resolve(1,a,37,a)].
% 1.58/1.81 Derived: c_Ring__and__Field_Odvd__class_Odvd(A,hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),B),C),tc_nat) | -c_Ring__and__Field_Odvd__class_Odvd(A,C,tc_nat). [resolve(1,a,38,a)].
% 1.58/1.81 Derived: c_Ring__and__Field_Odvd__class_Odvd(hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),B),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),C),D),tc_nat) | -c_Ring__and__Field_Odvd__class_Odvd(B,D,tc_nat) | -c_Ring__and__Field_Odvd__class_Odvd(A,C,tc_nat). [resolve(1,a,39,a)].
% 1.58/1.81 Derived: c_Ring__and__Field_Odvd__class_Odvd(A,c_Power_Opower__class_Opower(A,B,tc_nat),tc_nat) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),B)). [resolve(1,a,40,a)].
% 1.58/1.81 Derived: c_Ring__and__Field_Odvd__class_Odvd(c_Power_Opower__class_Opower(A,B,tc_nat),C,tc_nat) | -c_lessequals(B,D,tc_nat) | -c_Ring__and__Field_Odvd__class_Odvd(c_Power_Opower__class_Opower(A,D,tc_nat),C,tc_nat). [resolve(1,a,41,a)].
% 1.58/1.81 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(1,a,42,a)].
% 1.58/1.81 Derived: c_Ring__and__Field_Odvd__class_Odvd(A,hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),B),C),tc_nat) | -c_Ring__and__Field_Odvd__class_Odvd(A,C,tc_nat) | -c_Ring__and__Field_Odvd__class_Odvd(A,B,tc_nat). [resolve(1,a,43,a)].
% 1.58/1.81 Derived: hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),B)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),C),B)) = hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),C)),B). [resolve(1,a,44,a)].
% 1.58/1.81 Derived: hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),A) = A. [resolve(1,a,45,a)].
% 1.58/1.81 Derived: hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),c_HOL_Ozero__class_Ozero(tc_nat)) = A. [resolve(1,a,46,a)].
% 1.58/1.81 Derived: c_Power_Opower__class_Opower(hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),B),C,tc_nat) = hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),c_Power_Opower__class_Opower(A,C,tc_nat)),c_Power_Opower__class_Opower(B,C,tc_nat)). [resolve(1,a,48,a)].
% 1.58/1.82 Derived: hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),B)),C) = hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),C)),B). [resolve(1,a,49,a)].
% 1.58/1.82 Derived: hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),B)),C) = hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),B),C)). [resolve(1,a,50,a)].
% 1.58/1.82 Derived: hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),B),C)) = hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),B)),C). [resolve(1,a,51,a)].
% 1.58/1.82 Derived: hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),B),C)) = hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),B),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),C)). [resolve(1,a,52,a)].
% 1.58/1.82 Derived: hAPP(hAPP(c_HOL_Otimes__class_Otimes(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_Otimes__class_Otimes(tc_nat),A),C)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),B),C)). [resolve(1,a,54,a)].
% 1.58/1.82 Derived: hAPP(hAPP(c_HOL_Otimes__class_Otimes(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_Otimes__class_Otimes(tc_nat),A),B)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),C)). [resolve(1,a,55,a)].
% 1.58/1.82 Derived: c_Ring__and__Field_Odvd__class_Odvd(hAPP(A,B),c_Finite__Set_Osetprod(A,C,D,tc_nat),tc_nat) | -hBOOL(hAPP(hAPP(c_in(D),B),C)) | -c_Finite__Set_Ofinite(C,D). [resolve(1,a,56,a)].
% 1.58/1.82 Derived: -class_Ring__and__Field_Ono__zero__divisors(tc_nat) | c_Finite__Set_Osetprod(A,B,C,tc_nat) != c_HOL_Ozero__class_Ozero(tc_nat) | -c_Finite__Set_Ofinite(B,C) | hAPP(A,c_ATP__Linkup_Osko__Finite__Set__Xsetprod__zero__iff__1__1(B,A,C,tc_nat)) = c_HOL_Ozero__class_Ozero(tc_nat). [resolve(1,a,57,b)].
% 1.58/1.88 Derived: -class_Ring__and__Field_Ono__zero__divisors(tc_nat) | c_Finite__Set_Osetprod(A,B,C,tc_nat) != c_HOL_Ozero__class_Ozero(tc_nat) | -c_Finite__Set_Ofinite(B,C) | hBOOL(hAPP(hAPP(c_in(C),c_ATP__Linkup_Osko__Finite__Set__Xsetprod__zero__iff__1__1(B,A,C,tc_nat)),B)). [resolve(1,a,58,b)].
% 1.58/1.88 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(1,a,59,a)].
% 1.58/1.88 61 class_OrderedGroup_Ocomm__monoid__add(tc_nat) # label(clsarity_nat__OrderedGroup_Ocomm__monoid__add) # label(axiom). [assumption].
% 1.58/1.88 62 -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].
% 1.58/1.88 63 -class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(A) | -class_OrderedGroup_Ocomm__monoid__add(A) | c_lessequals(B,hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),C),D),A) | -c_lessequals(B,C,A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),D,A) # label(cls_add__increasing2_0) # label(axiom). [assumption].
% 1.58/1.88 64 -class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(A) | -class_OrderedGroup_Ocomm__monoid__add(A) | c_lessequals(B,hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),C),D),A) | -c_lessequals(B,D,A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),C,A) # label(cls_add__increasing_0) # label(axiom). [assumption].
% 1.58/1.88 65 -class_OrderedGroup_Ocomm__monoid__add(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),hAPP(hAPP(c_Finite__Set_Osetsum(B,A),C),c_Lattices_Oupper__semilattice__class_Osup(D,E,tc_fun(B,tc_bool)))),hAPP(hAPP(c_Finite__Set_Osetsum(B,A),C),c_Lattices_Olower__semilattice__class_Oinf(D,E,tc_fun(B,tc_bool)))) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),hAPP(hAPP(c_Finite__Set_Osetsum(B,A),C),D)),hAPP(hAPP(c_Finite__Set_Osetsum(B,A),C),E)) | -c_Finite__Set_Ofinite(E,B) | -c_Finite__Set_Ofinite(D,B) # label(cls_setsum__Un__Int_0) # label(axiom). [assumption].
% 1.58/1.88 66 -class_OrderedGroup_Ocomm__monoid__add(A) | c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_fun(D,tc_bool)) != c_Orderings_Obot__class_Obot(tc_fun(D,tc_bool)) | -c_Finite__Set_Ofinite(C,D) | -c_Finite__Set_Ofinite(B,D) | hAPP(hAPP(c_Finite__Set_Osetsum(D,A),E),c_Lattices_Oupper__semilattice__class_Osup(B,C,tc_fun(D,tc_bool))) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),hAPP(hAPP(c_Finite__Set_Osetsum(D,A),E),B)),hAPP(hAPP(c_Finite__Set_Osetsum(D,A),E),C)) # label(cls_setsum__Un__disjoint_0) # label(axiom). [assumption].
% 1.58/1.88 67 -class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(A) | -class_OrderedGroup_Ocomm__monoid__add(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),C),D))) | -c_lessequals(B,D,A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),C)) # label(cls_add__strict__increasing_0) # label(axiom). [assumption].
% 1.58/1.88 68 -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].
% 1.58/1.88 69 -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].
% 1.58/1.88 70 -class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(A) | -class_OrderedGroup_Ocomm__monoid__add(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),C),D))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),D)) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),C,A) # label(cls_add__strict__increasing2_0) # label(axiom). [assumption].
% 1.58/1.88 71 -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].
% 1.58/1.88 72 -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].
% 1.58/1.88 73 -class_OrderedGroup_Ocomm__monoid__add(A) | hAPP(hAPP(c_Finite__Set_Osetsum(B,A),C),c_Set_Oinsert(D,E,B)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),hAPP(C,D)),hAPP(hAPP(c_Finite__Set_Osetsum(B,A),C),E)) | hBOOL(hAPP(hAPP(c_in(B),D),E)) | -c_Finite__Set_Ofinite(E,B) # label(cls_setsum__insert_0) # label(axiom). [assumption].
% 1.58/1.88 74 -class_OrderedGroup_Ocomm__monoid__add(A) | hAPP(hAPP(c_Finite__Set_Osetsum(B,A),c_COMBS(c_COMBS(c_COMBB(c_HOL_OIf(A),hAPP(c_COMBC(c_in(B),B,tc_fun(B,tc_bool),tc_bool),C),tc_bool,tc_fun(A,tc_fun(A,A)),B),D,B,A,tc_fun(A,A)),E,B,A,A)),F) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),hAPP(hAPP(c_Finite__Set_Osetsum(B,A),D),c_Lattices_Olower__semilattice__class_Oinf(F,C,tc_fun(B,tc_bool)))),hAPP(hAPP(c_Finite__Set_Osetsum(B,A),E),c_Lattices_Olower__semilattice__class_Oinf(F,hAPP(c_HOL_Ouminus__class_Ouminus(tc_fun(B,tc_bool)),C),tc_fun(B,tc_bool)))) | -c_Finite__Set_Ofinite(F,B) # label(cls_setsum__cases_0) # label(axiom). [assumption].
% 1.58/1.88 75 -class_OrderedGroup_Ocomm__monoid__add(A) | hAPP(B,c_ATP__Linkup_Osko__Finite__Set__Xsetsum__0_H__1__1(C,B,D,A)) != c_HOL_Ozero__class_Ozero(A) | hAPP(hAPP(c_Finite__Set_Osetsum(D,A),B),C) = c_HOL_Ozero__class_Ozero(A) # label(cls_setsum__0_H_1) # label(axiom). [assumption].
% 1.58/1.88 76 -class_OrderedGroup_Ocomm__monoid__add(A) | hAPP(hAPP(c_Finite__Set_Osetsum(B,A),C),c_Orderings_Obot__class_Obot(tc_fun(B,tc_bool))) = c_HOL_Ozero__class_Ozero(A) # label(cls_setsum__empty_0) # label(axiom). [assumption].
% 1.58/1.88 77 -class_OrderedGroup_Ocomm__monoid__add(A) | hAPP(hAPP(c_Finite__Set_Osetsum(B,A),C),D) = c_HOL_Ozero__class_Ozero(A) | hBOOL(hAPP(hAPP(c_in(B),c_ATP__Linkup_Osko__Finite__Set__Xsetsum__0_H__1__1(D,C,B,A)),D)) # label(cls_setsum__0_H_0) # label(axiom). [assumption].
% 1.58/1.88 78 -class_OrderedGroup_Ocomm__monoid__add(A) | hAPP(hAPP(c_Finite__Set_Osetsum(B,A),c_COMBS(c_COMBB(c_HOL_Oplus__class_Oplus(A),C,A,tc_fun(A,A),B),D,B,A,A)),E) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),hAPP(hAPP(c_Finite__Set_Osetsum(B,A),C),E)),hAPP(hAPP(c_Finite__Set_Osetsum(B,A),D),E)) # label(cls_setsum__addf_0) # label(axiom). [assumption].
% 1.58/1.88 79 -class_OrderedGroup_Ocomm__monoid__add(A) | hAPP(hAPP(c_Finite__Set_Osetsum(B,A),hAPP(c_COMBC(c_COMBB(c_Finite__Set_Osetsum(C,A),D,tc_fun(C,A),tc_fun(tc_fun(C,tc_bool),A),B),B,tc_fun(C,tc_bool),A),E)),F) = hAPP(hAPP(c_Finite__Set_Osetsum(C,A),hAPP(c_COMBC(c_COMBB(c_Finite__Set_Osetsum(B,A),c_COMBC(D,B,C,A),tc_fun(B,A),tc_fun(tc_fun(B,tc_bool),A),C),C,tc_fun(B,tc_bool),A),F)),E) # label(cls_setsum__commute_0) # label(axiom). [assumption].
% 1.58/1.88 80 -class_OrderedGroup_Ocomm__monoid__add(A) | hAPP(hAPP(c_Finite__Set_Osetsum(B,A),C),c_Lattices_Olower__semilattice__class_Oinf(D,E,tc_fun(B,tc_bool))) = hAPP(hAPP(c_Finite__Set_Osetsum(B,A),hAPP(c_COMBC(c_COMBS(c_COMBB(c_HOL_OIf(A),hAPP(c_COMBC(c_in(B),B,tc_fun(B,tc_bool),tc_bool),E),tc_bool,tc_fun(A,tc_fun(A,A)),B),C,B,A,tc_fun(A,A)),B,A,A),c_HOL_Ozero__class_Ozero(A))),D) | -c_Finite__Set_Ofinite(D,B) # label(cls_setsum__restrict__set_0) # label(axiom). [assumption].
% 1.58/1.88 81 -class_OrderedGroup_Ocomm__monoid__add(A) | hAPP(hAPP(c_Finite__Set_Osetsum(B,A),hAPP(c_COMBC(c_COMBS(c_COMBB(c_HOL_OIf(A),hAPP(c_fequal(B),C),tc_bool,tc_fun(A,tc_fun(A,A)),B),D,B,A,tc_fun(A,A)),B,A,A),c_HOL_Ozero__class_Ozero(A))),E) = 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].
% 1.58/1.88 82 -class_OrderedGroup_Ocomm__monoid__add(A) | hAPP(hAPP(c_Finite__Set_Osetsum(B,A),hAPP(c_COMBC(c_COMBS(c_COMBB(c_HOL_OIf(A),hAPP(c_fequal(B),C),tc_bool,tc_fun(A,tc_fun(A,A)),B),D,B,A,tc_fun(A,A)),B,A,A),c_HOL_Ozero__class_Ozero(A))),E) = 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].
% 1.58/1.88 83 -class_OrderedGroup_Ocomm__monoid__add(A) | hAPP(hAPP(c_Finite__Set_Osetsum(B,A),C),D) = c_HOL_Ozero__class_Ozero(A) | c_Finite__Set_Ofinite(D,B) # label(cls_setsum__def_1) # label(axiom). [assumption].
% 1.58/1.88 84 -class_OrderedGroup_Ocomm__monoid__add(A) | hAPP(hAPP(c_Finite__Set_Osetsum(B,A),C),D) = c_HOL_Ozero__class_Ozero(A) | c_Finite__Set_Ofinite(D,B) # label(cls_setsum__infinite_0) # label(axiom). [assumption].
% 1.58/1.88 85 -class_OrderedGroup_Ocomm__monoid__add(A) | hAPP(hAPP(c_Finite__Set_Osetsum(B,A),hAPP(c_COMBC(c_COMBS(c_COMBB(c_HOL_OIf(A),hAPP(c_COMBC(c_fequal(B),B,B,tc_bool),C),tc_bool,tc_fun(A,tc_fun(A,A)),B),D,B,A,tc_fun(A,A)),B,A,A),c_HOL_Ozero__class_Ozero(A))),E) = 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].
% 1.58/1.88 86 -class_OrderedGroup_Ocomm__monoid__add(A) | hAPP(hAPP(c_Finite__Set_Osetsum(B,A),hAPP(c_COMBC(c_COMBS(c_COMBB(c_HOL_OIf(A),hAPP(c_COMBC(c_fequal(B),B,B,tc_bool),C),tc_bool,tc_fun(A,tc_fun(A,A)),B),D,B,A,tc_fun(A,A)),B,A,A),c_HOL_Ozero__class_Ozero(A))),E) = 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].
% 1.58/1.88 Derived: -class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(tc_nat) | c_lessequals(A,hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),B),C),tc_nat) | -c_lessequals(A,B,tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),C,tc_nat). [resolve(61,a,63,b)].
% 1.58/1.88 Derived: -class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(tc_nat) | c_lessequals(A,hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),B),C),tc_nat) | -c_lessequals(A,C,tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),B,tc_nat). [resolve(61,a,64,b)].
% 1.58/1.88 Derived: hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),hAPP(hAPP(c_Finite__Set_Osetsum(A,tc_nat),B),c_Lattices_Oupper__semilattice__class_Osup(C,D,tc_fun(A,tc_bool)))),hAPP(hAPP(c_Finite__Set_Osetsum(A,tc_nat),B),c_Lattices_Olower__semilattice__class_Oinf(C,D,tc_fun(A,tc_bool)))) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),hAPP(hAPP(c_Finite__Set_Osetsum(A,tc_nat),B),C)),hAPP(hAPP(c_Finite__Set_Osetsum(A,tc_nat),B),D)) | -c_Finite__Set_Ofinite(D,A) | -c_Finite__Set_Ofinite(C,A). [resolve(61,a,65,a)].
% 1.58/1.88 Derived: c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_fun(C,tc_bool)) != c_Orderings_Obot__class_Obot(tc_fun(C,tc_bool)) | -c_Finite__Set_Ofinite(B,C) | -c_Finite__Set_Ofinite(A,C) | hAPP(hAPP(c_Finite__Set_Osetsum(C,tc_nat),D),c_Lattices_Oupper__semilattice__class_Osup(A,B,tc_fun(C,tc_bool))) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),hAPP(hAPP(c_Finite__Set_Osetsum(C,tc_nat),D),A)),hAPP(hAPP(c_Finite__Set_Osetsum(C,tc_nat),D),B)). [resolve(61,a,66,a)].
% 1.58/1.88 Derived: -class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(tc_nat) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),B),C))) | -c_lessequals(A,C,tc_nat) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),B)). [resolve(61,a,67,b)].
% 1.58/1.88 Derived: -class_OrderedGroup_Opordered__ab__semigroup__add__imp__le(tc_nat) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),B),C))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),C)) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),B,tc_nat). [resolve(61,a,70,b)].
% 1.58/1.88 Derived: hAPP(hAPP(c_Finite__Set_Osetsum(A,tc_nat),B),c_Set_Oinsert(C,D,A)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),hAPP(B,C)),hAPP(hAPP(c_Finite__Set_Osetsum(A,tc_nat),B),D)) | hBOOL(hAPP(hAPP(c_in(A),C),D)) | -c_Finite__Set_Ofinite(D,A). [resolve(61,a,73,a)].
% 1.58/1.88 Derived: hAPP(hAPP(c_Finite__Set_Osetsum(A,tc_nat),c_COMBS(c_COMBS(c_COMBB(c_HOL_OIf(tc_nat),hAPP(c_COMBC(c_in(A),A,tc_fun(A,tc_bool),tc_bool),B),tc_bool,tc_fun(tc_nat,tc_fun(tc_nat,tc_nat)),A),C,A,tc_nat,tc_fun(tc_nat,tc_nat)),D,A,tc_nat,tc_nat)),E) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),hAPP(hAPP(c_Finite__Set_Osetsum(A,tc_nat),C),c_Lattices_Olower__semilattice__class_Oinf(E,B,tc_fun(A,tc_bool)))),hAPP(hAPP(c_Finite__Set_Osetsum(A,tc_nat),D),c_Lattices_Olower__semilattice__class_Oinf(E,hAPP(c_HOL_Ouminus__class_Ouminus(tc_fun(A,tc_bool)),B),tc_fun(A,tc_bool)))) | -c_Finite__Set_Ofinite(E,A). [resolve(61,a,74,a)].
% 1.76/2.03 Derived: hAPP(A,c_ATP__Linkup_Osko__Finite__Set__Xsetsum__0_H__1__1(B,A,C,tc_nat)) != c_HOL_Ozero__class_Ozero(tc_nat) | hAPP(hAPP(c_Finite__Set_Osetsum(C,tc_nat),A),B) = c_HOL_Ozero__class_Ozero(tc_nat). [resolve(61,a,75,a)].
% 1.76/2.03 Derived: hAPP(hAPP(c_Finite__Set_Osetsum(A,tc_nat),B),c_Orderings_Obot__class_Obot(tc_fun(A,tc_bool))) = c_HOL_Ozero__class_Ozero(tc_nat). [resolve(61,a,76,a)].
% 1.76/2.03 Derived: hAPP(hAPP(c_Finite__Set_Osetsum(A,tc_nat),B),C) = c_HOL_Ozero__class_Ozero(tc_nat) | hBOOL(hAPP(hAPP(c_in(A),c_ATP__Linkup_Osko__Finite__Set__Xsetsum__0_H__1__1(C,B,A,tc_nat)),C)). [resolve(61,a,77,a)].
% 1.76/2.03 Derived: hAPP(hAPP(c_Finite__Set_Osetsum(A,tc_nat),c_COMBS(c_COMBB(c_HOL_Oplus__class_Oplus(tc_nat),B,tc_nat,tc_fun(tc_nat,tc_nat),A),C,A,tc_nat,tc_nat)),D) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),hAPP(hAPP(c_Finite__Set_Osetsum(A,tc_nat),B),D)),hAPP(hAPP(c_Finite__Set_Osetsum(A,tc_nat),C),D)). [resolve(61,a,78,a)].
% 1.76/2.03 Derived: hAPP(hAPP(c_Finite__Set_Osetsum(A,tc_nat),hAPP(c_COMBC(c_COMBB(c_Finite__Set_Osetsum(B,tc_nat),C,tc_fun(B,tc_nat),tc_fun(tc_fun(B,tc_bool),tc_nat),A),A,tc_fun(B,tc_bool),tc_nat),D)),E) = hAPP(hAPP(c_Finite__Set_Osetsum(B,tc_nat),hAPP(c_COMBC(c_COMBB(c_Finite__Set_Osetsum(A,tc_nat),c_COMBC(C,A,B,tc_nat),tc_fun(A,tc_nat),tc_fun(tc_fun(A,tc_bool),tc_nat),B),B,tc_fun(A,tc_bool),tc_nat),E)),D). [resolve(61,a,79,a)].
% 1.76/2.03 Derived: hAPP(hAPP(c_Finite__Set_Osetsum(A,tc_nat),B),c_Lattices_Olower__semilattice__class_Oinf(C,D,tc_fun(A,tc_bool))) = hAPP(hAPP(c_Finite__Set_Osetsum(A,tc_nat),hAPP(c_COMBC(c_COMBS(c_COMBB(c_HOL_OIf(tc_nat),hAPP(c_COMBC(c_in(A),A,tc_fun(A,tc_bool),tc_bool),D),tc_bool,tc_fun(tc_nat,tc_fun(tc_nat,tc_nat)),A),B,A,tc_nat,tc_fun(tc_nat,tc_nat)),A,tc_nat,tc_nat),c_HOL_Ozero__class_Ozero(tc_nat))),C) | -c_Finite__Set_Ofinite(C,A). [resolve(61,a,80,a)].
% 1.76/2.03 Derived: hAPP(hAPP(c_Finite__Set_Osetsum(A,tc_nat),hAPP(c_COMBC(c_COMBS(c_COMBB(c_HOL_OIf(tc_nat),hAPP(c_fequal(A),B),tc_bool,tc_fun(tc_nat,tc_fun(tc_nat,tc_nat)),A),C,A,tc_nat,tc_fun(tc_nat,tc_nat)),A,tc_nat,tc_nat),c_HOL_Ozero__class_Ozero(tc_nat))),D) = hAPP(C,B) | -hBOOL(hAPP(hAPP(c_in(A),B),D)) | -c_Finite__Set_Ofinite(D,A). [resolve(61,a,81,a)].
% 1.76/2.03 Derived: hAPP(hAPP(c_Finite__Set_Osetsum(A,tc_nat),hAPP(c_COMBC(c_COMBS(c_COMBB(c_HOL_OIf(tc_nat),hAPP(c_fequal(A),B),tc_bool,tc_fun(tc_nat,tc_fun(tc_nat,tc_nat)),A),C,A,tc_nat,tc_fun(tc_nat,tc_nat)),A,tc_nat,tc_nat),c_HOL_Ozero__class_Ozero(tc_nat))),D) = c_HOL_Ozero__class_Ozero(tc_nat) | hBOOL(hAPP(hAPP(c_in(A),B),D)) | -c_Finite__Set_Ofinite(D,A). [resolve(61,a,82,a)].
% 1.76/2.03 Derived: hAPP(hAPP(c_Finite__Set_Osetsum(A,tc_nat),B),C) = c_HOL_Ozero__class_Ozero(tc_nat) | c_Finite__Set_Ofinite(C,A). [resolve(61,a,83,a)].
% 1.76/2.03 Derived: hAPP(hAPP(c_Finite__Set_Osetsum(A,tc_nat),hAPP(c_COMBC(c_COMBS(c_COMBB(c_HOL_OIf(tc_nat),hAPP(c_COMBC(c_fequal(A),A,A,tc_bool),B),tc_bool,tc_fun(tc_nat,tc_fun(tc_nat,tc_nat)),A),C,A,tc_nat,tc_fun(tc_nat,tc_nat)),A,tc_nat,tc_nat),c_HOL_Ozero__class_Ozero(tc_nat))),D) = hAPP(C,B) | -hBOOL(hAPP(hAPP(c_in(A),B),D)) | -c_Finite__Set_Ofinite(D,A). [resolve(61,a,85,a)].
% 1.76/2.03 Derived: hAPP(hAPP(c_Finite__Set_Osetsum(A,tc_nat),hAPP(c_COMBC(c_COMBS(c_COMBB(c_HOL_OIf(tc_nat),hAPP(c_COMBC(c_fequal(A),A,A,tc_bool),B),tc_bool,tc_fun(tc_nat,tc_fun(tc_nat,tc_nat)),A),C,A,tc_nat,tc_fun(tc_nat,tc_nat)),A,tc_nat,tc_nat),c_HOL_Ozero__class_Ozero(tc_nat))),D) = c_HOL_Ozero__class_Ozero(tc_nat) | hBOOL(hAPP(hAPP(c_in(A),B),D)) | -c_Finite__Set_Ofinite(D,A). [resolve(61,a,86,a)].
% 1.76/2.03 87 class_Ring__and__Field_Oordered__semidom(tc_nat) # label(clsarity_nat__Ring__and__Field_Oordered__semidom) # label(axiom). [assumption].
% 1.76/2.03 88 -class_Ring__and__Field_Oordered__semidom(A) | c_lessequals(c_Power_Opower__class_Opower(B,C,A),c_Power_Opower__class_Opower(B,D,A),A) | -c_lessequals(B,c_HOL_Oone__class_Oone(A),A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),B,A) | -c_lessequals(D,C,tc_nat) # label(cls_power__decreasing_0) # label(axiom). [assumption].
% 1.76/2.03 89 -class_Ring__and__Field_Oordered__semidom(A) | c_lessequals(c_HOL_Oone__class_Oone(A),c_Power_Opower__class_Opower(B,C,A),A) | -c_lessequals(c_HOL_Oone__class_Oone(A),B,A) # label(cls_one__le__power_0) # label(axiom). [assumption].
% 1.76/2.03 90 -class_Ring__and__Field_Oordered__semidom(A) | c_lessequals(c_HOL_Ozero__class_Ozero(A),c_Power_Opower__class_Opower(B,C,A),A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),B,A) # label(cls_zero__le__power_0) # label(axiom). [assumption].
% 1.76/2.03 91 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),C)) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),C,A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_Power_Opower__class_Opower(B,D,A)),c_Power_Opower__class_Opower(C,D,A))) # label(cls_power__less__imp__less__base_0) # label(axiom). [assumption].
% 1.76/2.03 92 -class_Ring__and__Field_Oordered__semidom(A) | c_lessequals(B,C,tc_nat) | -c_lessequals(hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),C),A) # label(cls_of__nat__le__iff_0) # label(axiom). [assumption].
% 1.76/2.03 93 -class_Ring__and__Field_Oordered__semidom(A) | c_lessequals(hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),C),A) | -c_lessequals(B,C,tc_nat) # label(cls_of__nat__le__iff_1) # label(axiom). [assumption].
% 1.76/2.03 94 -class_Ring__and__Field_Oordered__semidom(A) | c_lessequals(c_Power_Opower__class_Opower(B,C,A),c_Power_Opower__class_Opower(B,D,A),A) | -c_lessequals(c_HOL_Oone__class_Oone(A),B,A) | -c_lessequals(C,D,tc_nat) # label(cls_power__increasing_0) # label(axiom). [assumption].
% 1.76/2.03 95 -class_Ring__and__Field_Oordered__semidom(A) | c_lessequals(c_Power_Opower__class_Opower(B,C,A),c_Power_Opower__class_Opower(D,C,A),A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),B,A) | -c_lessequals(B,D,A) # label(cls_power__mono_0) # label(axiom). [assumption].
% 1.76/2.03 96 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Oone__class_Oone(A)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),c_Power_Opower__class_Opower(B,C,A)))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Oone__class_Oone(A)),B)) # label(cls_power__gt1__lemma_0) # label(axiom). [assumption].
% 1.76/2.03 97 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_Power_Opower__class_Opower(B,C,A)),c_Power_Opower__class_Opower(D,C,A))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),C)) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),B,A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),D)) # label(cls_power__strict__mono_0) # label(axiom). [assumption].
% 1.76/2.03 98 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_Power_Opower__class_Opower(B,C,A)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),c_Power_Opower__class_Opower(B,C,A)))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Oone__class_Oone(A)),B)) # label(cls_power__less__power__Suc_0) # label(axiom). [assumption].
% 1.76/2.03 99 -class_Ring__and__Field_Oordered__semidom(A) | c_Power_Opower__class_Opower(B,C,A) != c_Power_Opower__class_Opower(D,C,A) | B = D | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),C)) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),D,A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),B,A) # label(cls_power__eq__imp__eq__base_0) # label(axiom). [assumption].
% 1.76/2.03 100 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),c_Power_Opower__class_Opower(B,C,A))),c_Power_Opower__class_Opower(B,C,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_0) # label(axiom). [assumption].
% 1.76/2.03 101 -class_Ring__and__Field_Oordered__semidom(A) | c_lessequals(B,C,tc_nat) | -c_lessequals(c_Power_Opower__class_Opower(D,B,A),c_Power_Opower__class_Opower(D,C,A),A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Oone__class_Oone(A)),D)) # label(cls_power__le__imp__le__exp_0) # label(axiom). [assumption].
% 1.76/2.03 102 -class_Ring__and__Field_Oordered__semidom(A) | c_lessequals(B,C,tc_nat) | -c_lessequals(c_Power_Opower__class_Opower(D,B,A),c_Power_Opower__class_Opower(D,C,A),A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Oone__class_Oone(A)),D)) # label(cls_power__increasing__iff_0) # label(axiom). [assumption].
% 1.76/2.03 103 -class_Ring__and__Field_Oordered__semidom(A) | c_lessequals(c_Power_Opower__class_Opower(B,C,A),c_Power_Opower__class_Opower(B,D,A),A) | -c_lessequals(C,D,tc_nat) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Oone__class_Oone(A)),B)) # label(cls_power__increasing__iff_1) # label(axiom). [assumption].
% 1.76/2.03 104 -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].
% 1.76/2.03 105 -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].
% 1.76/2.03 106 -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].
% 1.76/2.03 107 -class_Ring__and__Field_Oordered__semidom(A) | c_lessequals(c_HOL_Ozero__class_Ozero(A),c_HOL_Oone__class_Oone(A),A) # label(cls_zero__le__one_0) # label(axiom). [assumption].
% 1.76/2.03 108 -class_Ring__and__Field_Oordered__semidom(A) | -c_lessequals(c_HOL_Oone__class_Oone(A),c_HOL_Ozero__class_Ozero(A),A) # label(cls_not__one__le__zero_0) # label(axiom). [assumption].
% 1.76/2.03 109 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Oone__class_Oone(A)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Oone__class_Oone(A)),C)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Oone__class_Oone(A)),B)) # label(cls_less__1__mult_0) # label(axiom). [assumption].
% 1.76/2.03 110 -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].
% 1.76/2.03 111 -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].
% 1.76/2.03 112 -class_Ring__and__Field_Oordered__semidom(A) | c_lessequals(c_HOL_Ozero__class_Ozero(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B),A) # label(cls_zero__le__imp__of__nat_0) # label(axiom). [assumption].
% 1.76/2.03 113 -class_Ring__and__Field_Oordered__semidom(A) | c_lessequals(c_HOL_Ozero__class_Ozero(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B),A) # label(cls_of__nat__0__le__iff_0) # label(axiom). [assumption].
% 1.76/2.03 114 -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].
% 1.76/2.03 115 -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].
% 1.76/2.03 116 -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].
% 1.76/2.03 117 -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].
% 1.76/2.03 118 -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].
% 1.76/2.03 119 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),c_Finite__Set_Osetprod(B,C,D,A))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),hAPP(B,c_ATP__Linkup_Osko__Finite__Set__Xsetprod__pos__1__1(C,B,D,A)))) # label(cls_setprod__pos_1) # label(axiom). [assumption].
% 1.76/2.03 120 -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].
% 1.76/2.03 121 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),c_Finite__Set_Osetprod(B,C,D,A))) | hBOOL(hAPP(hAPP(c_in(D),c_ATP__Linkup_Osko__Finite__Set__Xsetprod__pos__1__1(C,B,D,A)),C)) # label(cls_setprod__pos_0) # label(axiom). [assumption].
% 1.76/2.03 122 -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)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B))) # label(cls_of__nat__0__less__iff_0) # label(axiom). [assumption].
% 1.76/2.03 123 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B))) | -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].
% 1.76/2.03 124 -class_Ring__and__Field_Oordered__semidom(A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)),c_HOL_Ozero__class_Ozero(A))) # label(cls_of__nat__less__0__iff_0) # label(axiom). [assumption].
% 1.76/2.03 125 -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].
% 1.76/2.03 126 -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].
% 1.76/2.03 127 -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),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),C))) # label(cls_of__nat__less__iff_0) # label(axiom). [assumption].
% 1.76/2.03 128 -class_Ring__and__Field_Oordered__semidom(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),C))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),B),C)) # label(cls_of__nat__less__iff_1) # label(axiom). [assumption].
% 1.76/2.03 Derived: c_lessequals(c_Power_Opower__class_Opower(A,B,tc_nat),c_Power_Opower__class_Opower(A,C,tc_nat),tc_nat) | -c_lessequals(A,c_HOL_Oone__class_Oone(tc_nat),tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),A,tc_nat) | -c_lessequals(C,B,tc_nat). [resolve(87,a,88,a)].
% 1.76/2.03 Derived: c_lessequals(c_HOL_Oone__class_Oone(tc_nat),c_Power_Opower__class_Opower(A,B,tc_nat),tc_nat) | -c_lessequals(c_HOL_Oone__class_Oone(tc_nat),A,tc_nat). [resolve(87,a,89,a)].
% 1.76/2.03 Derived: c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),c_Power_Opower__class_Opower(A,B,tc_nat),tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),A,tc_nat). [resolve(87,a,90,a)].
% 1.76/2.03 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),B)) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),B,tc_nat) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_Power_Opower__class_Opower(A,C,tc_nat)),c_Power_Opower__class_Opower(B,C,tc_nat))). [resolve(87,a,91,a)].
% 1.76/2.03 Derived: c_lessequals(A,B,tc_nat) | -c_lessequals(hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_nat),A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_nat),B),tc_nat). [resolve(87,a,92,a)].
% 1.76/2.03 Derived: c_lessequals(hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_nat),A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_nat),B),tc_nat) | -c_lessequals(A,B,tc_nat). [resolve(87,a,93,a)].
% 1.76/2.03 Derived: c_lessequals(c_Power_Opower__class_Opower(A,B,tc_nat),c_Power_Opower__class_Opower(A,C,tc_nat),tc_nat) | -c_lessequals(c_HOL_Oone__class_Oone(tc_nat),A,tc_nat) | -c_lessequals(B,C,tc_nat). [resolve(87,a,94,a)].
% 1.76/2.03 Derived: c_lessequals(c_Power_Opower__class_Opower(A,B,tc_nat),c_Power_Opower__class_Opower(C,B,tc_nat),tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),A,tc_nat) | -c_lessequals(A,C,tc_nat). [resolve(87,a,95,a)].
% 1.76/2.03 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Oone__class_Oone(tc_nat)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),c_Power_Opower__class_Opower(A,B,tc_nat)))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Oone__class_Oone(tc_nat)),A)). [resolve(87,a,96,a)].
% 1.76/2.03 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_Power_Opower__class_Opower(A,B,tc_nat)),c_Power_Opower__class_Opower(C,B,tc_nat))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),B)) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),A,tc_nat) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),C)). [resolve(87,a,97,a)].
% 1.76/2.03 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_Power_Opower__class_Opower(A,B,tc_nat)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),c_Power_Opower__class_Opower(A,B,tc_nat)))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Oone__class_Oone(tc_nat)),A)). [resolve(87,a,98,a)].
% 1.76/2.03 Derived: c_Power_Opower__class_Opower(A,B,tc_nat) != c_Power_Opower__class_Opower(C,B,tc_nat) | A = C | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),B)) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),C,tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),A,tc_nat). [resolve(87,a,99,a)].
% 1.76/2.03 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),c_Power_Opower__class_Opower(A,B,tc_nat))),c_Power_Opower__class_Opower(A,B,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(87,a,100,a)].
% 1.76/2.03 Derived: c_lessequals(A,B,tc_nat) | -c_lessequals(c_Power_Opower__class_Opower(C,A,tc_nat),c_Power_Opower__class_Opower(C,B,tc_nat),tc_nat) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Oone__class_Oone(tc_nat)),C)). [resolve(87,a,101,a)].
% 1.76/2.03 Derived: c_lessequals(c_Power_Opower__class_Opower(A,B,tc_nat),c_Power_Opower__class_Opower(A,C,tc_nat),tc_nat) | -c_lessequals(B,C,tc_nat) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Oone__class_Oone(tc_nat)),A)). [resolve(87,a,103,a)].
% 1.76/2.03 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(87,a,104,a)].
% 1.76/2.03 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(87,a,105,a)].
% 1.76/2.03 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(87,a,106,a)].
% 1.76/2.03 Derived: c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),c_HOL_Oone__class_Oone(tc_nat),tc_nat). [resolve(87,a,107,a)].
% 1.76/2.03 Derived: -c_lessequals(c_HOL_Oone__class_Oone(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat),tc_nat). [resolve(87,a,108,a)].
% 1.76/2.03 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Oone__class_Oone(tc_nat)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),B))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Oone__class_Oone(tc_nat)),B)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Oone__class_Oone(tc_nat)),A)). [resolve(87,a,109,a)].
% 1.76/2.03 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(87,a,110,a)].
% 1.76/2.03 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(87,a,111,a)].
% 1.76/2.03 Derived: c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_nat),A),tc_nat). [resolve(87,a,112,a)].
% 1.76/2.03 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(87,a,114,a)].
% 1.76/2.03 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(87,a,115,a)].
% 1.76/2.03 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(87,a,116,a)].
% 1.76/2.03 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),c_Finite__Set_Osetprod(A,B,C,tc_nat))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),hAPP(A,c_ATP__Linkup_Osko__Finite__Set__Xsetprod__pos__1__1(B,A,C,tc_nat)))). [resolve(87,a,119,a)].
% 1.76/2.03 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(87,a,120,a)].
% 1.76/2.03 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),c_Finite__Set_Osetprod(A,B,C,tc_nat))) | hBOOL(hAPP(hAPP(c_in(C),c_ATP__Linkup_Osko__Finite__Set__Xsetprod__pos__1__1(B,A,C,tc_nat)),B)). [resolve(87,a,121,a)].
% 1.76/2.03 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)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_nat),A))). [resolve(87,a,122,a)].
% 1.76/2.03 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_nat),A))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),A)). [resolve(87,a,123,a)].
% 1.88/2.18 Derived: -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_nat),A)),c_HOL_Ozero__class_Ozero(tc_nat))). [resolve(87,a,124,a)].
% 1.88/2.18 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(87,a,125,a)].
% 1.88/2.18 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(87,a,126,a)].
% 1.88/2.18 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),B)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_nat),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_nat),B))). [resolve(87,a,127,a)].
% 1.88/2.18 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_nat),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_nat),B))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),B)). [resolve(87,a,128,a)].
% 1.88/2.18 129 class_Ring__and__Field_Oordered__semiring__strict(tc_nat) # label(clsarity_nat__Ring__and__Field_Oordered__semiring__strict) # label(axiom). [assumption].
% 1.88/2.18 130 -class_Ring__and__Field_Oordered__semiring__strict(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),D),E))) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),C,A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),D)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),C),E)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),D)) # label(cls_mult__strict__mono_0) # label(axiom). [assumption].
% 1.88/2.18 131 -class_Ring__and__Field_Oordered__semiring__strict(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),C)) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),D,A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),C),D))) # label(cls_mult__less__imp__less__right_0) # label(axiom). [assumption].
% 1.88/2.18 132 -class_Ring__and__Field_Oordered__semiring__strict(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),C)) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),D,A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),D),B)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),D),C))) # label(cls_mult__less__imp__less__left_0) # label(axiom). [assumption].
% 1.88/2.18 133 -class_Ring__and__Field_Oordered__semiring__strict(A) | c_lessequals(B,C,A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),D)) | -c_lessequals(hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),D),B),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),D),C),A) # label(cls_mult__left__le__imp__le_0) # label(axiom). [assumption].
% 1.88/2.18 134 -class_Ring__and__Field_Oordered__semiring__strict(A) | c_lessequals(B,C,A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),D)) | -c_lessequals(hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),D),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),C),D),A) # label(cls_mult__right__le__imp__le_0) # label(axiom). [assumption].
% 1.88/2.18 135 -class_Ring__and__Field_Oordered__semiring__strict(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),D),E))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),C)) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),B,A) | -c_lessequals(C,E,A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),D)) # label(cls_mult__less__le__imp__less_0) # label(axiom). [assumption].
% 1.88/2.18 136 -class_Ring__and__Field_Oordered__semiring__strict(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),D),E))) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),C,A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),C),E)) | -c_lessequals(B,D,A) # label(cls_mult__le__less__imp__less_0) # label(axiom). [assumption].
% 1.88/2.18 137 -class_Ring__and__Field_Oordered__semiring__strict(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),D),E))) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),C,A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),B,A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),C),E)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),D)) # label(cls_mult__strict__mono_H_0) # label(axiom). [assumption].
% 1.88/2.18 138 -class_Ring__and__Field_Oordered__semiring__strict(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),D),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),B),D)) # label(cls_mult__strict__right__mono_0) # label(axiom). [assumption].
% 1.88/2.18 139 -class_Ring__and__Field_Oordered__semiring__strict(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),D))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),C),D)) # label(cls_mult__strict__left__mono_0) # label(axiom). [assumption].
% 1.88/2.18 140 -class_Ring__and__Field_Oordered__semiring__strict(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(A),c_HOL_Ozero__class_Ozero(A)),C)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C))) # label(cls_zero__less__mult__pos2_0) # label(axiom). [assumption].
% 1.88/2.18 141 -class_Ring__and__Field_Oordered__semiring__strict(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(A),c_HOL_Ozero__class_Ozero(A)),C)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),C),B))) # label(cls_zero__less__mult__pos_0) # label(axiom). [assumption].
% 1.88/2.18 142 -class_Ring__and__Field_Oordered__semiring__strict(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(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_mult__pos__pos_0) # label(axiom). [assumption].
% 1.88/2.18 143 -class_Ring__and__Field_Oordered__semiring__strict(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C)),c_HOL_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),c_HOL_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),C)) # label(cls_mult__pos__neg2_0) # label(axiom). [assumption].
% 1.88/2.18 144 -class_Ring__and__Field_Oordered__semiring__strict(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(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),c_HOL_Ozero__class_Ozero(A)),B)) # label(cls_mult__pos__neg_0) # label(axiom). [assumption].
% 1.88/2.18 145 -class_Ring__and__Field_Oordered__semiring__strict(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C)),c_HOL_Ozero__class_Ozero(A))) | -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),B),c_HOL_Ozero__class_Ozero(A))) # label(cls_mult__neg__pos_0) # label(axiom). [assumption].
% 1.88/2.18 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),B)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),C),D))) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),B,tc_nat) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),C)) | -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(129,a,130,a)].
% 1.88/2.18 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),B)) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),C,tc_nat) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),C)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),B),C))). [resolve(129,a,131,a)].
% 1.88/2.18 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),B)) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),C,tc_nat) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),C),A)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),C),B))). [resolve(129,a,132,a)].
% 1.88/2.18 Derived: c_lessequals(A,B,tc_nat) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),C)) | -c_lessequals(hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),C),A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),C),B),tc_nat). [resolve(129,a,133,a)].
% 1.88/2.18 Derived: c_lessequals(A,B,tc_nat) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),C)) | -c_lessequals(hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),C),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),B),C),tc_nat). [resolve(129,a,134,a)].
% 1.88/2.19 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),B)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),C),D))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),B)) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),A,tc_nat) | -c_lessequals(B,D,tc_nat) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),C)). [resolve(129,a,135,a)].
% 1.88/2.19 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),B)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),C),D))) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),B,tc_nat) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),A)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),B),D)) | -c_lessequals(A,C,tc_nat). [resolve(129,a,136,a)].
% 1.88/2.19 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),B)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),C),D))) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),B,tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),A,tc_nat) | -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(129,a,137,a)].
% 1.88/2.19 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),B)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),C),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),A),C)). [resolve(129,a,138,a)].
% 1.88/2.19 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),B)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),C))) | -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),B),C)). [resolve(129,a,139,a)].
% 1.88/2.19 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)),B)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),B))). [resolve(129,a,140,a)].
% 1.88/2.19 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)),B)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),B),A))). [resolve(129,a,141,a)].
% 1.88/2.19 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(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(129,a,142,a)].
% 2.16/2.48 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),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))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),B)). [resolve(129,a,143,a)].
% 2.16/2.48 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(hAPP(c_HOL_Otimes__class_Otimes(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),c_HOL_Ozero__class_Ozero(tc_nat)),A)). [resolve(129,a,144,a)].
% 2.16/2.48 146 class_Ring__and__Field_Oordered__semiring(tc_nat) # label(clsarity_nat__Ring__and__Field_Oordered__semiring) # label(axiom). [assumption].
% 2.16/2.48 147 -class_Ring__and__Field_Oordered__semiring(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),C)) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),D,A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),C),D))) # label(cls_mult__right__less__imp__less_0) # label(axiom). [assumption].
% 2.16/2.48 148 -class_Ring__and__Field_Oordered__semiring(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),C)) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),D,A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),D),B)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),D),C))) # label(cls_mult__left__less__imp__less_0) # label(axiom). [assumption].
% 2.16/2.48 149 class_OrderedGroup_Omonoid__mult(tc_nat) # label(clsarity_nat__OrderedGroup_Omonoid__mult) # label(axiom). [assumption].
% 2.16/2.48 150 -class_OrderedGroup_Omonoid__mult(A) | hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),c_Power_Opower__class_Opower(B,C,A)),B) = hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),c_Power_Opower__class_Opower(B,C,A)) # label(cls_power__commutes_0) # label(axiom). [assumption].
% 2.16/2.48 151 -class_OrderedGroup_Omonoid__mult(A) | hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),c_HOL_Oone__class_Oone(A)),B) = B # label(cls_mult__1__left_0) # label(axiom). [assumption].
% 2.16/2.48 152 -class_OrderedGroup_Omonoid__mult(A) | hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),c_HOL_Oone__class_Oone(A)) = B # label(cls_mult__1__right_0) # label(axiom). [assumption].
% 2.16/2.48 153 -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].
% 2.16/2.48 Derived: c_Power_Opower__class_Opower(c_HOL_Oone__class_Oone(tc_nat),A,tc_nat) = c_HOL_Oone__class_Oone(tc_nat). [resolve(149,a,153,a)].
% 2.16/2.48 154 class_Ring__and__Field_Odvd(tc_nat) # label(clsarity_nat__Ring__and__Field_Odvd) # label(axiom). [assumption].
% 2.16/2.48 155 -class_Ring__and__Field_Odvd(A) | -class_OrderedGroup_Ocomm__monoid__mult(A) | c_Ring__and__Field_Odvd__class_Odvd(c_Finite__Set_Osetprod(B,C,D,A),c_Finite__Set_Osetprod(B,E,D,A),A) | -c_lessequals(C,E,tc_fun(D,tc_bool)) | -c_Finite__Set_Ofinite(E,D) # label(cls_setprod__dvd__setprod__subset_0) # label(axiom). [assumption].
% 2.16/2.48 156 -class_Ring__and__Field_Odvd(A) | c_Ring__and__Field_Odvd__class_Odvd(B,hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C),A) # label(cls_dvdI_0) # label(axiom). [assumption].
% 2.16/2.48 157 -class_Ring__and__Field_Odvd(A) | -class_Ring__and__Field_Ocomm__ring(A) | c_Ring__and__Field_Odvd__class_Odvd(B,hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),C),D),A) | -c_Ring__and__Field_Odvd__class_Odvd(B,hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),hAPP(hAPP(c_HOL_Ominus__class_Ominus(A),C),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),E),F))),D),A) | -c_Ring__and__Field_Odvd__class_Odvd(B,F,A) # label(cls_inf__period_I4_J_1) # label(axiom). [assumption].
% 2.16/2.48 158 -class_Ring__and__Field_Odvd(A) | -class_Ring__and__Field_Ocomm__ring(A) | c_Ring__and__Field_Odvd__class_Odvd(B,hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),hAPP(hAPP(c_HOL_Ominus__class_Ominus(A),C),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),D),E))),F),A) | -c_Ring__and__Field_Odvd__class_Odvd(B,hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),C),F),A) | -c_Ring__and__Field_Odvd__class_Odvd(B,E,A) # label(cls_inf__period_I4_J_0) # label(axiom). [assumption].
% 2.27/2.51 Derived: -class_OrderedGroup_Ocomm__monoid__mult(tc_nat) | c_Ring__and__Field_Odvd__class_Odvd(c_Finite__Set_Osetprod(A,B,C,tc_nat),c_Finite__Set_Osetprod(A,D,C,tc_nat),tc_nat) | -c_lessequals(B,D,tc_fun(C,tc_bool)) | -c_Finite__Set_Ofinite(D,C). [resolve(154,a,155,a)].
% 2.27/2.51 Derived: -class_Ring__and__Field_Ocomm__ring(tc_nat) | c_Ring__and__Field_Odvd__class_Odvd(A,hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),B),C),tc_nat) | -c_Ring__and__Field_Odvd__class_Odvd(A,hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),hAPP(hAPP(c_HOL_Ominus__class_Ominus(tc_nat),B),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),D),E))),C),tc_nat) | -c_Ring__and__Field_Odvd__class_Odvd(A,E,tc_nat). [resolve(154,a,157,a)].
% 2.27/2.51 Derived: -class_Ring__and__Field_Ocomm__ring(tc_nat) | c_Ring__and__Field_Odvd__class_Odvd(A,hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),hAPP(hAPP(c_HOL_Ominus__class_Ominus(tc_nat),B),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),C),D))),E),tc_nat) | -c_Ring__and__Field_Odvd__class_Odvd(A,hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),B),E),tc_nat) | -c_Ring__and__Field_Odvd__class_Odvd(A,D,tc_nat). [resolve(154,a,158,a)].
% 2.27/2.51 159 class_OrderedGroup_Opordered__comm__monoid__add(tc_nat) # label(clsarity_nat__OrderedGroup_Opordered__comm__monoid__add) # label(axiom). [assumption].
% 2.27/2.51 160 -class_OrderedGroup_Opordered__comm__monoid__add(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C) != c_HOL_Ozero__class_Ozero(A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),C,A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),B,A) | C = c_HOL_Ozero__class_Ozero(A) # label(cls_add__nonneg__eq__0__iff_1) # label(axiom). [assumption].
% 2.27/2.51 161 -class_OrderedGroup_Opordered__comm__monoid__add(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C) != c_HOL_Ozero__class_Ozero(A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),C,A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),B,A) | B = c_HOL_Ozero__class_Ozero(A) # label(cls_add__nonneg__eq__0__iff_0) # label(axiom). [assumption].
% 2.27/2.51 162 -class_OrderedGroup_Opordered__comm__monoid__add(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C)),c_HOL_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),C),c_HOL_Ozero__class_Ozero(A))) | -c_lessequals(B,c_HOL_Ozero__class_Ozero(A),A) # label(cls_add__nonpos__neg_0) # label(axiom). [assumption].
% 2.27/2.51 163 -class_OrderedGroup_Opordered__comm__monoid__add(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C)),c_HOL_Ozero__class_Ozero(A))) | -c_lessequals(C,c_HOL_Ozero__class_Ozero(A),A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),c_HOL_Ozero__class_Ozero(A))) # label(cls_add__neg__nonpos_0) # label(axiom). [assumption].
% 2.27/2.51 164 -class_OrderedGroup_Opordered__comm__monoid__add(A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),c_HOL_Ozero__class_Ozero(A),A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),c_HOL_Ozero__class_Ozero(A),A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),c_HOL_Ozero__class_Ozero(A)),c_HOL_Ozero__class_Ozero(A)) = c_HOL_Ozero__class_Ozero(A) # label(cls_add__nonneg__eq__0__iff_2) # label(axiom). [assumption].
% 2.27/2.51 165 -class_OrderedGroup_Opordered__comm__monoid__add(A) | c_lessequals(c_HOL_Ozero__class_Ozero(A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C),A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),C,A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),B,A) # label(cls_add__nonneg__nonneg_0) # label(axiom). [assumption].
% 2.27/2.51 166 -class_OrderedGroup_Opordered__comm__monoid__add(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),C)) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),B,A) # label(cls_add__nonneg__pos_0) # label(axiom). [assumption].
% 2.27/2.51 167 -class_OrderedGroup_Opordered__comm__monoid__add(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C))) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),C,A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),B)) # label(cls_add__pos__nonneg_0) # label(axiom). [assumption].
% 2.27/2.59 168 -class_OrderedGroup_Opordered__comm__monoid__add(A) | c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C),c_HOL_Ozero__class_Ozero(A),A) | -c_lessequals(C,c_HOL_Ozero__class_Ozero(A),A) | -c_lessequals(B,c_HOL_Ozero__class_Ozero(A),A) # label(cls_add__nonpos__nonpos_0) # label(axiom). [assumption].
% 2.27/2.59 169 -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].
% 2.27/2.59 170 -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].
% 2.27/2.59 Derived: hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B) != c_HOL_Ozero__class_Ozero(tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),B,tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),A,tc_nat) | B = c_HOL_Ozero__class_Ozero(tc_nat). [resolve(159,a,160,a)].
% 2.27/2.59 Derived: hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B) != c_HOL_Ozero__class_Ozero(tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),B,tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),A,tc_nat) | A = c_HOL_Ozero__class_Ozero(tc_nat). [resolve(159,a,161,a)].
% 2.27/2.59 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B)),c_HOL_Ozero__class_Ozero(tc_nat))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),B),c_HOL_Ozero__class_Ozero(tc_nat))) | -c_lessequals(A,c_HOL_Ozero__class_Ozero(tc_nat),tc_nat). [resolve(159,a,162,a)].
% 2.27/2.59 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B)),c_HOL_Ozero__class_Ozero(tc_nat))) | -c_lessequals(B,c_HOL_Ozero__class_Ozero(tc_nat),tc_nat) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),c_HOL_Ozero__class_Ozero(tc_nat))). [resolve(159,a,163,a)].
% 2.27/2.59 Derived: c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B),tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),B,tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),A,tc_nat). [resolve(159,a,165,a)].
% 2.27/2.59 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),B)) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),A,tc_nat). [resolve(159,a,166,a)].
% 2.27/2.59 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B))) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),B,tc_nat) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)),A)). [resolve(159,a,167,a)].
% 2.27/2.59 Derived: c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B),c_HOL_Ozero__class_Ozero(tc_nat),tc_nat) | -c_lessequals(B,c_HOL_Ozero__class_Ozero(tc_nat),tc_nat) | -c_lessequals(A,c_HOL_Ozero__class_Ozero(tc_nat),tc_nat). [resolve(159,a,168,a)].
% 2.27/2.59 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(159,a,169,a)].
% 2.27/2.59 171 class_Lattices_Olower__semilattice(tc_fun(A,B)) | -class_Lattices_Olattice(B) # label(clsarity_fun__Lattices_Olower__semilattice) # label(axiom). [assumption].
% 2.27/2.59 172 -class_Lattices_Olower__semilattice(A) | c_Lattices_Olower__semilattice__class_Oinf(B,c_Lattices_Olower__semilattice__class_Oinf(C,D,A),A) = c_Lattices_Olower__semilattice__class_Oinf(C,c_Lattices_Olower__semilattice__class_Oinf(B,D,A),A) # label(cls_inf__left__commute_0) # label(axiom). [assumption].
% 2.27/2.59 173 -class_Lattices_Olower__semilattice(A) | c_Lattices_Olower__semilattice__class_Oinf(c_Lattices_Olower__semilattice__class_Oinf(B,C,A),D,A) = c_Lattices_Olower__semilattice__class_Oinf(B,c_Lattices_Olower__semilattice__class_Oinf(C,D,A),A) # label(cls_inf__assoc_0) # label(axiom). [assumption].
% 2.27/2.59 174 -class_Lattices_Olower__semilattice(A) | c_Lattices_Olower__semilattice__class_Oinf(B,C,A) = B | -c_lessequals(B,C,A) # label(cls_le__iff__inf_0) # label(axiom). [assumption].
% 2.27/2.59 175 -class_Lattices_Olower__semilattice(A) | c_Lattices_Olower__semilattice__class_Oinf(B,C,A) != B | c_lessequals(B,C,A) # label(cls_le__iff__inf_1) # label(axiom). [assumption].
% 2.27/2.59 176 -class_Lattices_Olower__semilattice(A) | c_Lattices_Olower__semilattice__class_Oinf(B,C,A) = C | -c_lessequals(C,B,A) # label(cls_inf__absorb2_0) # label(axiom). [assumption].
% 2.27/2.59 177 -class_Lattices_Olower__semilattice(A) | c_Lattices_Olower__semilattice__class_Oinf(B,C,A) = c_Lattices_Olower__semilattice__class_Oinf(C,B,A) # label(cls_inf__commute_0) # label(axiom). [assumption].
% 2.27/2.59 178 -class_Lattices_Olower__semilattice(A) | c_lessequals(B,C,A) | -c_lessequals(B,c_Lattices_Olower__semilattice__class_Oinf(C,D,A),A) # label(cls_le__infE_0) # label(axiom). [assumption].
% 2.27/2.59 179 -class_Lattices_Olower__semilattice(A) | c_lessequals(B,C,A) | -c_lessequals(B,c_Lattices_Olower__semilattice__class_Oinf(D,C,A),A) # label(cls_le__infE_1) # label(axiom). [assumption].
% 2.27/2.59 180 -class_Lattices_Olower__semilattice(A) | c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(B,C,A),D,A) | -c_lessequals(B,D,A) # label(cls_le__infI1_0) # label(axiom). [assumption].
% 2.27/2.59 181 -class_Lattices_Olower__semilattice(A) | c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(B,C,A),D,A) | -c_lessequals(C,D,A) # label(cls_le__infI2_0) # label(axiom). [assumption].
% 2.27/2.59 182 -class_Lattices_Olower__semilattice(A) | c_lessequals(B,C,A) | -c_lessequals(B,c_Lattices_Olower__semilattice__class_Oinf(C,D,A),A) # label(cls_le__inf__iff_0) # label(axiom). [assumption].
% 2.27/2.59 183 -class_Lattices_Olower__semilattice(A) | c_lessequals(B,C,A) | -c_lessequals(B,c_Lattices_Olower__semilattice__class_Oinf(D,C,A),A) # label(cls_le__inf__iff_1) # label(axiom). [assumption].
% 2.27/2.59 184 -class_Lattices_Olower__semilattice(A) | c_lessequals(B,c_Lattices_Olower__semilattice__class_Oinf(C,D,A),A) | -c_lessequals(B,D,A) | -c_lessequals(B,C,A) # label(cls_inf__greatest_0) # label(axiom). [assumption].
% 2.27/2.59 185 -class_Lattices_Olower__semilattice(A) | c_lessequals(B,c_Lattices_Olower__semilattice__class_Oinf(C,D,A),A) | -c_lessequals(B,D,A) | -c_lessequals(B,C,A) # label(cls_le__inf__iff_2) # label(axiom). [assumption].
% 2.27/2.59 186 -class_Lattices_Olower__semilattice(A) | c_lessequals(B,c_Lattices_Olower__semilattice__class_Oinf(C,D,A),A) | -c_lessequals(B,D,A) | -c_lessequals(B,C,A) # label(cls_le__infI_0) # label(axiom). [assumption].
% 2.27/2.59 187 -class_Lattices_Olower__semilattice(A) | c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(B,C,A),C,A) # label(cls_inf__le2_0) # label(axiom). [assumption].
% 2.27/2.59 188 -class_Lattices_Olower__semilattice(A) | c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(B,C,A),B,A) # label(cls_inf__le1_0) # label(axiom). [assumption].
% 2.27/2.59 189 -class_Lattices_Olower__semilattice(A) | c_Lattices_Olower__semilattice__class_Oinf(B,c_Lattices_Olower__semilattice__class_Oinf(B,C,A),A) = c_Lattices_Olower__semilattice__class_Oinf(B,C,A) # label(cls_inf__left__idem_0) # label(axiom). [assumption].
% 2.27/2.59 190 -class_Lattices_Olower__semilattice(A) | c_Lattices_Olower__semilattice__class_Oinf(B,B,A) = B # label(cls_inf__idem_0) # label(axiom). [assumption].
% 2.27/2.59 191 -class_Lattices_Olower__semilattice(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_Lattices_Olower__semilattice__class_Oinf(B,C,A)),D)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),D)) # label(cls_less__infI1_0) # label(axiom). [assumption].
% 2.27/2.59 192 -class_Lattices_Olower__semilattice(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_Lattices_Olower__semilattice__class_Oinf(B,C,A)),D)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),C),D)) # label(cls_less__infI2_0) # label(axiom). [assumption].
% 2.27/2.59 Derived: -class_Lattices_Olattice(A) | c_Lattices_Olower__semilattice__class_Oinf(B,c_Lattices_Olower__semilattice__class_Oinf(C,D,tc_fun(E,A)),tc_fun(E,A)) = c_Lattices_Olower__semilattice__class_Oinf(C,c_Lattices_Olower__semilattice__class_Oinf(B,D,tc_fun(E,A)),tc_fun(E,A)). [resolve(171,a,172,a)].
% 2.27/2.59 Derived: -class_Lattices_Olattice(A) | c_Lattices_Olower__semilattice__class_Oinf(c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_fun(D,A)),E,tc_fun(D,A)) = c_Lattices_Olower__semilattice__class_Oinf(B,c_Lattices_Olower__semilattice__class_Oinf(C,E,tc_fun(D,A)),tc_fun(D,A)). [resolve(171,a,173,a)].
% 2.27/2.59 Derived: -class_Lattices_Olattice(A) | c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_fun(D,A)) = B | -c_lessequals(B,C,tc_fun(D,A)). [resolve(171,a,174,a)].
% 2.27/2.59 Derived: -class_Lattices_Olattice(A) | c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_fun(D,A)) != B | c_lessequals(B,C,tc_fun(D,A)). [resolve(171,a,175,a)].
% 2.27/2.59 Derived: -class_Lattices_Olattice(A) | c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_fun(D,A)) = C | -c_lessequals(C,B,tc_fun(D,A)). [resolve(171,a,176,a)].
% 2.27/2.59 Derived: -class_Lattices_Olattice(A) | c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_fun(D,A)) = c_Lattices_Olower__semilattice__class_Oinf(C,B,tc_fun(D,A)). [resolve(171,a,177,a)].
% 2.27/2.59 Derived: -class_Lattices_Olattice(A) | c_lessequals(B,C,tc_fun(D,A)) | -c_lessequals(B,c_Lattices_Olower__semilattice__class_Oinf(C,E,tc_fun(D,A)),tc_fun(D,A)). [resolve(171,a,178,a)].
% 2.27/2.59 Derived: -class_Lattices_Olattice(A) | c_lessequals(B,C,tc_fun(D,A)) | -c_lessequals(B,c_Lattices_Olower__semilattice__class_Oinf(E,C,tc_fun(D,A)),tc_fun(D,A)). [resolve(171,a,179,a)].
% 2.27/2.59 Derived: -class_Lattices_Olattice(A) | c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_fun(D,A)),E,tc_fun(D,A)) | -c_lessequals(B,E,tc_fun(D,A)). [resolve(171,a,180,a)].
% 2.27/2.59 Derived: -class_Lattices_Olattice(A) | c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_fun(D,A)),E,tc_fun(D,A)) | -c_lessequals(C,E,tc_fun(D,A)). [resolve(171,a,181,a)].
% 2.27/2.59 Derived: -class_Lattices_Olattice(A) | c_lessequals(B,c_Lattices_Olower__semilattice__class_Oinf(C,D,tc_fun(E,A)),tc_fun(E,A)) | -c_lessequals(B,D,tc_fun(E,A)) | -c_lessequals(B,C,tc_fun(E,A)). [resolve(171,a,184,a)].
% 2.27/2.59 Derived: -class_Lattices_Olattice(A) | c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_fun(D,A)),C,tc_fun(D,A)). [resolve(171,a,187,a)].
% 2.27/2.59 Derived: -class_Lattices_Olattice(A) | c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_fun(D,A)),B,tc_fun(D,A)). [resolve(171,a,188,a)].
% 2.27/2.59 Derived: -class_Lattices_Olattice(A) | c_Lattices_Olower__semilattice__class_Oinf(B,c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_fun(D,A)),tc_fun(D,A)) = c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_fun(D,A)). [resolve(171,a,189,a)].
% 2.27/2.59 Derived: -class_Lattices_Olattice(A) | c_Lattices_Olower__semilattice__class_Oinf(B,B,tc_fun(C,A)) = B. [resolve(171,a,190,a)].
% 2.27/2.59 Derived: -class_Lattices_Olattice(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_fun(B,A)),c_Lattices_Olower__semilattice__class_Oinf(C,D,tc_fun(B,A))),E)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_fun(B,A)),C),E)). [resolve(171,a,191,a)].
% 2.27/2.59 Derived: -class_Lattices_Olattice(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_fun(B,A)),c_Lattices_Olower__semilattice__class_Oinf(C,D,tc_fun(B,A))),E)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_fun(B,A)),D),E)). [resolve(171,a,192,a)].
% 2.27/2.59 193 class_Lattices_Olower__semilattice(tc_nat) # label(clsarity_nat__Lattices_Olower__semilattice) # label(axiom). [assumption].
% 2.27/2.59 Derived: c_Lattices_Olower__semilattice__class_Oinf(A,c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_nat),tc_nat) = c_Lattices_Olower__semilattice__class_Oinf(B,c_Lattices_Olower__semilattice__class_Oinf(A,C,tc_nat),tc_nat). [resolve(193,a,172,a)].
% 2.27/2.59 Derived: c_Lattices_Olower__semilattice__class_Oinf(c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_nat),C,tc_nat) = c_Lattices_Olower__semilattice__class_Oinf(A,c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_nat),tc_nat). [resolve(193,a,173,a)].
% 2.27/2.59 Derived: c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_nat) = A | -c_lessequals(A,B,tc_nat). [resolve(193,a,174,a)].
% 2.27/2.59 Derived: c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_nat) != A | c_lessequals(A,B,tc_nat). [resolve(193,a,175,a)].
% 2.27/2.59 Derived: c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_nat) = B | -c_lessequals(B,A,tc_nat). [resolve(193,a,176,a)].
% 2.27/2.59 Derived: c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_nat) = c_Lattices_Olower__semilattice__class_Oinf(B,A,tc_nat). [resolve(193,a,177,a)].
% 2.27/2.59 Derived: c_lessequals(A,B,tc_nat) | -c_lessequals(A,c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_nat),tc_nat). [resolve(193,a,178,a)].
% 2.27/2.59 Derived: c_lessequals(A,B,tc_nat) | -c_lessequals(A,c_Lattices_Olower__semilattice__class_Oinf(C,B,tc_nat),tc_nat). [resolve(193,a,179,a)].
% 2.27/2.59 Derived: c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_nat),C,tc_nat) | -c_lessequals(A,C,tc_nat). [resolve(193,a,180,a)].
% 2.27/2.59 Derived: c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_nat),C,tc_nat) | -c_lessequals(B,C,tc_nat). [resolve(193,a,181,a)].
% 2.27/2.59 Derived: c_lessequals(A,c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_nat),tc_nat) | -c_lessequals(A,C,tc_nat) | -c_lessequals(A,B,tc_nat). [resolve(193,a,184,a)].
% 2.27/2.59 Derived: c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_nat),B,tc_nat). [resolve(193,a,187,a)].
% 2.27/2.59 Derived: c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_nat),A,tc_nat). [resolve(193,a,188,a)].
% 2.27/2.59 Derived: c_Lattices_Olower__semilattice__class_Oinf(A,c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_nat),tc_nat) = c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_nat). [resolve(193,a,189,a)].
% 2.27/2.59 Derived: c_Lattices_Olower__semilattice__class_Oinf(A,A,tc_nat) = A. [resolve(193,a,190,a)].
% 2.27/2.59 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_nat)),C)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),C)). [resolve(193,a,191,a)].
% 2.27/2.59 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_nat)),C)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),B),C)). [resolve(193,a,192,a)].
% 2.27/2.59 194 class_Lattices_Olower__semilattice(tc_bool) # label(clsarity_bool__Lattices_Olower__semilattice) # label(axiom). [assumption].
% 2.27/2.59 Derived: c_Lattices_Olower__semilattice__class_Oinf(A,c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_bool),tc_bool) = c_Lattices_Olower__semilattice__class_Oinf(B,c_Lattices_Olower__semilattice__class_Oinf(A,C,tc_bool),tc_bool). [resolve(194,a,172,a)].
% 2.27/2.59 Derived: c_Lattices_Olower__semilattice__class_Oinf(c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_bool),C,tc_bool) = c_Lattices_Olower__semilattice__class_Oinf(A,c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_bool),tc_bool). [resolve(194,a,173,a)].
% 2.27/2.59 Derived: c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_bool) = A | -c_lessequals(A,B,tc_bool). [resolve(194,a,174,a)].
% 2.27/2.59 Derived: c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_bool) != A | c_lessequals(A,B,tc_bool). [resolve(194,a,175,a)].
% 2.27/2.59 Derived: c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_bool) = B | -c_lessequals(B,A,tc_bool). [resolve(194,a,176,a)].
% 2.27/2.59 Derived: c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_bool) = c_Lattices_Olower__semilattice__class_Oinf(B,A,tc_bool). [resolve(194,a,177,a)].
% 2.27/2.59 Derived: c_lessequals(A,B,tc_bool) | -c_lessequals(A,c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_bool),tc_bool). [resolve(194,a,178,a)].
% 2.27/2.59 Derived: c_lessequals(A,B,tc_bool) | -c_lessequals(A,c_Lattices_Olower__semilattice__class_Oinf(C,B,tc_bool),tc_bool). [resolve(194,a,179,a)].
% 2.27/2.59 Derived: c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_bool),C,tc_bool) | -c_lessequals(A,C,tc_bool). [resolve(194,a,180,a)].
% 2.27/2.59 Derived: c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_bool),C,tc_bool) | -c_lessequals(B,C,tc_bool). [resolve(194,a,181,a)].
% 2.37/2.65 Derived: c_lessequals(A,c_Lattices_Olower__semilattice__class_Oinf(B,C,tc_bool),tc_bool) | -c_lessequals(A,C,tc_bool) | -c_lessequals(A,B,tc_bool). [resolve(194,a,184,a)].
% 2.37/2.65 Derived: c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_bool),B,tc_bool). [resolve(194,a,187,a)].
% 2.37/2.65 Derived: c_lessequals(c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_bool),A,tc_bool). [resolve(194,a,188,a)].
% 2.37/2.65 Derived: c_Lattices_Olower__semilattice__class_Oinf(A,c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_bool),tc_bool) = c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_bool). [resolve(194,a,189,a)].
% 2.37/2.65 Derived: c_Lattices_Olower__semilattice__class_Oinf(A,A,tc_bool) = A. [resolve(194,a,190,a)].
% 2.37/2.65 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_bool),c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_bool)),C)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_bool),A),C)). [resolve(194,a,191,a)].
% 2.37/2.65 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_bool),c_Lattices_Olower__semilattice__class_Oinf(A,B,tc_bool)),C)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_bool),B),C)). [resolve(194,a,192,a)].
% 2.37/2.65 195 class_Ring__and__Field_Osemiring__0(tc_nat) # label(clsarity_nat__Ring__and__Field_Osemiring__0) # label(axiom). [assumption].
% 2.37/2.65 196 -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].
% 2.37/2.65 197 -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].
% 2.37/2.65 198 -class_Ring__and__Field_Osemiring__0(A) | hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Finite__Set_Osetsum(C,A),D),E)) = hAPP(hAPP(c_Finite__Set_Osetsum(C,A),c_COMBB(hAPP(c_HOL_Otimes__class_Otimes(A),B),D,A,A,C)),E) # label(cls_setsum__right__distrib_0) # label(axiom). [assumption].
% 2.37/2.65 199 -class_Ring__and__Field_Osemiring__0(A) | hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),hAPP(hAPP(c_Finite__Set_Osetsum(B,A),C),D)),E) = hAPP(hAPP(c_Finite__Set_Osetsum(B,A),hAPP(c_COMBC(c_COMBB(c_HOL_Otimes__class_Otimes(A),C,A,tc_fun(A,A),B),B,A,A),E)),D) # label(cls_setsum__left__distrib_0) # label(axiom). [assumption].
% 2.37/2.65 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(195,a,197,a)].
% 2.37/2.65 Derived: hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),hAPP(hAPP(c_Finite__Set_Osetsum(B,tc_nat),C),D)) = hAPP(hAPP(c_Finite__Set_Osetsum(B,tc_nat),c_COMBB(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),C,tc_nat,tc_nat,B)),D). [resolve(195,a,198,a)].
% 2.37/2.65 Derived: hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),hAPP(hAPP(c_Finite__Set_Osetsum(A,tc_nat),B),C)),D) = hAPP(hAPP(c_Finite__Set_Osetsum(A,tc_nat),hAPP(c_COMBC(c_COMBB(c_HOL_Otimes__class_Otimes(tc_nat),B,tc_nat,tc_fun(tc_nat,tc_nat),A),A,tc_nat,tc_nat),D)),C). [resolve(195,a,199,a)].
% 2.37/2.65 200 class_OrderedGroup_Opordered__ab__semigroup__add(tc_nat) # label(clsarity_nat__OrderedGroup_Opordered__ab__semigroup__add) # label(axiom). [assumption].
% 2.37/2.65 201 -class_OrderedGroup_Opordered__ab__semigroup__add(A) | c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),D),E),A) | -c_lessequals(C,E,A) | -c_lessequals(B,D,A) # label(cls_add__mono_0) # label(axiom). [assumption].
% 2.37/2.65 202 -class_OrderedGroup_Opordered__ab__semigroup__add(A) | c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),D),C),A) | -c_lessequals(B,D,A) # label(cls_add__right__mono_0) # label(axiom). [assumption].
% 2.37/2.65 203 -class_OrderedGroup_Opordered__ab__semigroup__add(A) | c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),D),A) | -c_lessequals(C,D,A) # label(cls_add__left__mono_0) # label(axiom). [assumption].
% 2.46/2.70 Derived: c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),C),D),tc_nat) | -c_lessequals(B,D,tc_nat) | -c_lessequals(A,C,tc_nat). [resolve(200,a,201,a)].
% 2.46/2.70 Derived: c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),C),B),tc_nat) | -c_lessequals(A,C,tc_nat). [resolve(200,a,202,a)].
% 2.46/2.70 Derived: c_lessequals(hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),C),tc_nat) | -c_lessequals(B,C,tc_nat). [resolve(200,a,203,a)].
% 2.46/2.70 204 class_Lattices_Oupper__semilattice(tc_fun(A,B)) | -class_Lattices_Olattice(B) # label(clsarity_fun__Lattices_Oupper__semilattice) # label(axiom). [assumption].
% 2.46/2.70 205 -class_Lattices_Oupper__semilattice(A) | c_Lattices_Oupper__semilattice__class_Osup(B,B,A) = B # label(cls_sup__idem_0) # label(axiom). [assumption].
% 2.46/2.70 206 -class_Lattices_Oupper__semilattice(A) | c_Lattices_Oupper__semilattice__class_Osup(c_Lattices_Oupper__semilattice__class_Osup(B,C,A),D,A) = c_Lattices_Oupper__semilattice__class_Osup(B,c_Lattices_Oupper__semilattice__class_Osup(C,D,A),A) # label(cls_sup__assoc_0) # label(axiom). [assumption].
% 2.46/2.70 207 -class_Lattices_Oupper__semilattice(A) | c_Lattices_Oupper__semilattice__class_Osup(B,C,A) = B | -c_lessequals(C,B,A) # label(cls_sup__absorb1_0) # label(axiom). [assumption].
% 2.46/2.70 208 -class_Lattices_Oupper__semilattice(A) | c_Lattices_Oupper__semilattice__class_Osup(B,C,A) != C | c_lessequals(B,C,A) # label(cls_le__iff__sup_1) # label(axiom). [assumption].
% 2.46/2.70 209 -class_Lattices_Oupper__semilattice(A) | c_Lattices_Oupper__semilattice__class_Osup(B,C,A) = C | -c_lessequals(B,C,A) # label(cls_le__iff__sup_0) # label(axiom). [assumption].
% 2.46/2.70 210 -class_Lattices_Oupper__semilattice(A) | c_Lattices_Oupper__semilattice__class_Osup(B,c_Lattices_Oupper__semilattice__class_Osup(C,D,A),A) = c_Lattices_Oupper__semilattice__class_Osup(C,c_Lattices_Oupper__semilattice__class_Osup(B,D,A),A) # label(cls_sup__left__commute_0) # label(axiom). [assumption].
% 2.46/2.70 211 -class_Lattices_Oupper__semilattice(A) | c_lessequals(c_Lattices_Oupper__semilattice__class_Osup(B,C,A),D,A) | -c_lessequals(C,D,A) | -c_lessequals(B,D,A) # label(cls_le__supI_0) # label(axiom). [assumption].
% 2.46/2.70 212 -class_Lattices_Oupper__semilattice(A) | c_lessequals(B,c_Lattices_Oupper__semilattice__class_Osup(B,C,A),A) # label(cls_sup__ge1_0) # label(axiom). [assumption].
% 2.46/2.70 213 -class_Lattices_Oupper__semilattice(A) | c_lessequals(B,c_Lattices_Oupper__semilattice__class_Osup(C,B,A),A) # label(cls_sup__ge2_0) # label(axiom). [assumption].
% 2.46/2.70 214 -class_Lattices_Oupper__semilattice(A) | c_lessequals(c_Lattices_Oupper__semilattice__class_Osup(B,C,A),D,A) | -c_lessequals(C,D,A) | -c_lessequals(B,D,A) # label(cls_sup__least_0) # label(axiom). [assumption].
% 2.46/2.70 215 -class_Lattices_Oupper__semilattice(A) | c_lessequals(c_Lattices_Oupper__semilattice__class_Osup(B,C,A),D,A) | -c_lessequals(C,D,A) | -c_lessequals(B,D,A) # label(cls_le__sup__iff_2) # label(axiom). [assumption].
% 2.46/2.70 216 -class_Lattices_Oupper__semilattice(A) | c_lessequals(B,C,A) | -c_lessequals(c_Lattices_Oupper__semilattice__class_Osup(D,B,A),C,A) # label(cls_le__sup__iff_1) # label(axiom). [assumption].
% 2.46/2.70 217 -class_Lattices_Oupper__semilattice(A) | c_lessequals(B,C,A) | -c_lessequals(c_Lattices_Oupper__semilattice__class_Osup(B,D,A),C,A) # label(cls_le__sup__iff_0) # label(axiom). [assumption].
% 2.46/2.70 218 -class_Lattices_Oupper__semilattice(A) | c_lessequals(B,c_Lattices_Oupper__semilattice__class_Osup(C,D,A),A) | -c_lessequals(B,D,A) # label(cls_le__supI2_0) # label(axiom). [assumption].
% 2.46/2.70 219 -class_Lattices_Oupper__semilattice(A) | c_lessequals(B,c_Lattices_Oupper__semilattice__class_Osup(C,D,A),A) | -c_lessequals(B,C,A) # label(cls_le__supI1_0) # label(axiom). [assumption].
% 2.46/2.70 220 -class_Lattices_Oupper__semilattice(A) | c_lessequals(B,C,A) | -c_lessequals(c_Lattices_Oupper__semilattice__class_Osup(D,B,A),C,A) # label(cls_le__supE_1) # label(axiom). [assumption].
% 2.46/2.70 221 -class_Lattices_Oupper__semilattice(A) | c_lessequals(B,C,A) | -c_lessequals(c_Lattices_Oupper__semilattice__class_Osup(B,D,A),C,A) # label(cls_le__supE_0) # label(axiom). [assumption].
% 2.46/2.70 222 -class_Lattices_Oupper__semilattice(A) | c_Lattices_Oupper__semilattice__class_Osup(B,C,A) = c_Lattices_Oupper__semilattice__class_Osup(C,B,A) # label(cls_sup__commute_0) # label(axiom). [assumption].
% 2.46/2.70 223 -class_Lattices_Oupper__semilattice(A) | c_Lattices_Oupper__semilattice__class_Osup(B,c_Lattices_Oupper__semilattice__class_Osup(B,C,A),A) = c_Lattices_Oupper__semilattice__class_Osup(B,C,A) # label(cls_sup__left__idem_0) # label(axiom). [assumption].
% 2.46/2.70 224 -class_Lattices_Oupper__semilattice(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),c_Lattices_Oupper__semilattice__class_Osup(C,D,A))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),C)) # label(cls_less__supI1_0) # label(axiom). [assumption].
% 2.46/2.70 225 -class_Lattices_Oupper__semilattice(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),c_Lattices_Oupper__semilattice__class_Osup(C,D,A))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),D)) # label(cls_less__supI2_0) # label(axiom). [assumption].
% 2.46/2.70 Derived: -class_Lattices_Olattice(A) | c_Lattices_Oupper__semilattice__class_Osup(B,B,tc_fun(C,A)) = B. [resolve(204,a,205,a)].
% 2.46/2.70 Derived: -class_Lattices_Olattice(A) | c_Lattices_Oupper__semilattice__class_Osup(c_Lattices_Oupper__semilattice__class_Osup(B,C,tc_fun(D,A)),E,tc_fun(D,A)) = c_Lattices_Oupper__semilattice__class_Osup(B,c_Lattices_Oupper__semilattice__class_Osup(C,E,tc_fun(D,A)),tc_fun(D,A)). [resolve(204,a,206,a)].
% 2.46/2.70 Derived: -class_Lattices_Olattice(A) | c_Lattices_Oupper__semilattice__class_Osup(B,C,tc_fun(D,A)) = B | -c_lessequals(C,B,tc_fun(D,A)). [resolve(204,a,207,a)].
% 2.46/2.70 Derived: -class_Lattices_Olattice(A) | c_Lattices_Oupper__semilattice__class_Osup(B,C,tc_fun(D,A)) != C | c_lessequals(B,C,tc_fun(D,A)). [resolve(204,a,208,a)].
% 2.46/2.70 Derived: -class_Lattices_Olattice(A) | c_Lattices_Oupper__semilattice__class_Osup(B,C,tc_fun(D,A)) = C | -c_lessequals(B,C,tc_fun(D,A)). [resolve(204,a,209,a)].
% 2.46/2.70 Derived: -class_Lattices_Olattice(A) | c_Lattices_Oupper__semilattice__class_Osup(B,c_Lattices_Oupper__semilattice__class_Osup(C,D,tc_fun(E,A)),tc_fun(E,A)) = c_Lattices_Oupper__semilattice__class_Osup(C,c_Lattices_Oupper__semilattice__class_Osup(B,D,tc_fun(E,A)),tc_fun(E,A)). [resolve(204,a,210,a)].
% 2.46/2.70 Derived: -class_Lattices_Olattice(A) | c_lessequals(c_Lattices_Oupper__semilattice__class_Osup(B,C,tc_fun(D,A)),E,tc_fun(D,A)) | -c_lessequals(C,E,tc_fun(D,A)) | -c_lessequals(B,E,tc_fun(D,A)). [resolve(204,a,211,a)].
% 2.46/2.70 Derived: -class_Lattices_Olattice(A) | c_lessequals(B,c_Lattices_Oupper__semilattice__class_Osup(B,C,tc_fun(D,A)),tc_fun(D,A)). [resolve(204,a,212,a)].
% 2.46/2.70 Derived: -class_Lattices_Olattice(A) | c_lessequals(B,c_Lattices_Oupper__semilattice__class_Osup(C,B,tc_fun(D,A)),tc_fun(D,A)). [resolve(204,a,213,a)].
% 2.46/2.70 Derived: -class_Lattices_Olattice(A) | c_lessequals(B,C,tc_fun(D,A)) | -c_lessequals(c_Lattices_Oupper__semilattice__class_Osup(E,B,tc_fun(D,A)),C,tc_fun(D,A)). [resolve(204,a,216,a)].
% 2.46/2.70 Derived: -class_Lattices_Olattice(A) | c_lessequals(B,C,tc_fun(D,A)) | -c_lessequals(c_Lattices_Oupper__semilattice__class_Osup(B,E,tc_fun(D,A)),C,tc_fun(D,A)). [resolve(204,a,217,a)].
% 2.46/2.70 Derived: -class_Lattices_Olattice(A) | c_lessequals(B,c_Lattices_Oupper__semilattice__class_Osup(C,D,tc_fun(E,A)),tc_fun(E,A)) | -c_lessequals(B,D,tc_fun(E,A)). [resolve(204,a,218,a)].
% 2.46/2.70 Derived: -class_Lattices_Olattice(A) | c_lessequals(B,c_Lattices_Oupper__semilattice__class_Osup(C,D,tc_fun(E,A)),tc_fun(E,A)) | -c_lessequals(B,C,tc_fun(E,A)). [resolve(204,a,219,a)].
% 2.46/2.70 Derived: -class_Lattices_Olattice(A) | c_Lattices_Oupper__semilattice__class_Osup(B,C,tc_fun(D,A)) = c_Lattices_Oupper__semilattice__class_Osup(C,B,tc_fun(D,A)). [resolve(204,a,222,a)].
% 2.46/2.70 Derived: -class_Lattices_Olattice(A) | c_Lattices_Oupper__semilattice__class_Osup(B,c_Lattices_Oupper__semilattice__class_Osup(B,C,tc_fun(D,A)),tc_fun(D,A)) = c_Lattices_Oupper__semilattice__class_Osup(B,C,tc_fun(D,A)). [resolve(204,a,223,a)].
% 2.46/2.70 Derived: -class_Lattices_Olattice(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_fun(B,A)),C),c_Lattices_Oupper__semilattice__class_Osup(D,E,tc_fun(B,A)))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_fun(B,A)),C),D)). [resolve(204,a,224,a)].
% 2.46/2.71 Derived: -class_Lattices_Olattice(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_fun(B,A)),C),c_Lattices_Oupper__semilattice__class_Osup(D,E,tc_fun(B,A)))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_fun(B,A)),C),E)). [resolve(204,a,225,a)].
% 2.46/2.71 226 class_Lattices_Oupper__semilattice(tc_nat) # label(clsarity_nat__Lattices_Oupper__semilattice) # label(axiom). [assumption].
% 2.46/2.71 Derived: c_Lattices_Oupper__semilattice__class_Osup(A,A,tc_nat) = A. [resolve(226,a,205,a)].
% 2.46/2.71 Derived: c_Lattices_Oupper__semilattice__class_Osup(c_Lattices_Oupper__semilattice__class_Osup(A,B,tc_nat),C,tc_nat) = c_Lattices_Oupper__semilattice__class_Osup(A,c_Lattices_Oupper__semilattice__class_Osup(B,C,tc_nat),tc_nat). [resolve(226,a,206,a)].
% 2.46/2.71 Derived: c_Lattices_Oupper__semilattice__class_Osup(A,B,tc_nat) = A | -c_lessequals(B,A,tc_nat). [resolve(226,a,207,a)].
% 2.46/2.71 Derived: c_Lattices_Oupper__semilattice__class_Osup(A,B,tc_nat) != B | c_lessequals(A,B,tc_nat). [resolve(226,a,208,a)].
% 2.46/2.71 Derived: c_Lattices_Oupper__semilattice__class_Osup(A,B,tc_nat) = B | -c_lessequals(A,B,tc_nat). [resolve(226,a,209,a)].
% 2.46/2.71 Derived: c_Lattices_Oupper__semilattice__class_Osup(A,c_Lattices_Oupper__semilattice__class_Osup(B,C,tc_nat),tc_nat) = c_Lattices_Oupper__semilattice__class_Osup(B,c_Lattices_Oupper__semilattice__class_Osup(A,C,tc_nat),tc_nat). [resolve(226,a,210,a)].
% 2.46/2.71 Derived: c_lessequals(c_Lattices_Oupper__semilattice__class_Osup(A,B,tc_nat),C,tc_nat) | -c_lessequals(B,C,tc_nat) | -c_lessequals(A,C,tc_nat). [resolve(226,a,211,a)].
% 2.46/2.71 Derived: c_lessequals(A,c_Lattices_Oupper__semilattice__class_Osup(A,B,tc_nat),tc_nat). [resolve(226,a,212,a)].
% 2.46/2.71 Derived: c_lessequals(A,c_Lattices_Oupper__semilattice__class_Osup(B,A,tc_nat),tc_nat). [resolve(226,a,213,a)].
% 2.46/2.71 Derived: c_lessequals(A,B,tc_nat) | -c_lessequals(c_Lattices_Oupper__semilattice__class_Osup(C,A,tc_nat),B,tc_nat). [resolve(226,a,216,a)].
% 2.46/2.71 Derived: c_lessequals(A,B,tc_nat) | -c_lessequals(c_Lattices_Oupper__semilattice__class_Osup(A,C,tc_nat),B,tc_nat). [resolve(226,a,217,a)].
% 2.46/2.71 Derived: c_lessequals(A,c_Lattices_Oupper__semilattice__class_Osup(B,C,tc_nat),tc_nat) | -c_lessequals(A,C,tc_nat). [resolve(226,a,218,a)].
% 2.46/2.71 Derived: c_lessequals(A,c_Lattices_Oupper__semilattice__class_Osup(B,C,tc_nat),tc_nat) | -c_lessequals(A,B,tc_nat). [resolve(226,a,219,a)].
% 2.46/2.71 Derived: c_Lattices_Oupper__semilattice__class_Osup(A,B,tc_nat) = c_Lattices_Oupper__semilattice__class_Osup(B,A,tc_nat). [resolve(226,a,222,a)].
% 2.46/2.71 Derived: c_Lattices_Oupper__semilattice__class_Osup(A,c_Lattices_Oupper__semilattice__class_Osup(A,B,tc_nat),tc_nat) = c_Lattices_Oupper__semilattice__class_Osup(A,B,tc_nat). [resolve(226,a,223,a)].
% 2.46/2.71 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),c_Lattices_Oupper__semilattice__class_Osup(B,C,tc_nat))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),B)). [resolve(226,a,224,a)].
% 2.46/2.71 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),c_Lattices_Oupper__semilattice__class_Osup(B,C,tc_nat))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),C)). [resolve(226,a,225,a)].
% 2.46/2.71 227 class_Lattices_Oupper__semilattice(tc_bool) # label(clsarity_bool__Lattices_Oupper__semilattice) # label(axiom). [assumption].
% 2.46/2.71 Derived: c_Lattices_Oupper__semilattice__class_Osup(A,A,tc_bool) = A. [resolve(227,a,205,a)].
% 2.46/2.71 Derived: c_Lattices_Oupper__semilattice__class_Osup(c_Lattices_Oupper__semilattice__class_Osup(A,B,tc_bool),C,tc_bool) = c_Lattices_Oupper__semilattice__class_Osup(A,c_Lattices_Oupper__semilattice__class_Osup(B,C,tc_bool),tc_bool). [resolve(227,a,206,a)].
% 2.46/2.71 Derived: c_Lattices_Oupper__semilattice__class_Osup(A,B,tc_bool) = A | -c_lessequals(B,A,tc_bool). [resolve(227,a,207,a)].
% 2.46/2.71 Derived: c_Lattices_Oupper__semilattice__class_Osup(A,B,tc_bool) != B | c_lessequals(A,B,tc_bool). [resolve(227,a,208,a)].
% 2.46/2.71 Derived: c_Lattices_Oupper__semilattice__class_Osup(A,B,tc_bool) = B | -c_lessequals(A,B,tc_bool). [resolve(227,a,209,a)].
% 2.55/2.79 Derived: c_Lattices_Oupper__semilattice__class_Osup(A,c_Lattices_Oupper__semilattice__class_Osup(B,C,tc_bool),tc_bool) = c_Lattices_Oupper__semilattice__class_Osup(B,c_Lattices_Oupper__semilattice__class_Osup(A,C,tc_bool),tc_bool). [resolve(227,a,210,a)].
% 2.55/2.79 Derived: c_lessequals(c_Lattices_Oupper__semilattice__class_Osup(A,B,tc_bool),C,tc_bool) | -c_lessequals(B,C,tc_bool) | -c_lessequals(A,C,tc_bool). [resolve(227,a,211,a)].
% 2.55/2.79 Derived: c_lessequals(A,c_Lattices_Oupper__semilattice__class_Osup(A,B,tc_bool),tc_bool). [resolve(227,a,212,a)].
% 2.55/2.79 Derived: c_lessequals(A,c_Lattices_Oupper__semilattice__class_Osup(B,A,tc_bool),tc_bool). [resolve(227,a,213,a)].
% 2.55/2.79 Derived: c_lessequals(A,B,tc_bool) | -c_lessequals(c_Lattices_Oupper__semilattice__class_Osup(C,A,tc_bool),B,tc_bool). [resolve(227,a,216,a)].
% 2.55/2.79 Derived: c_lessequals(A,B,tc_bool) | -c_lessequals(c_Lattices_Oupper__semilattice__class_Osup(A,C,tc_bool),B,tc_bool). [resolve(227,a,217,a)].
% 2.55/2.79 Derived: c_lessequals(A,c_Lattices_Oupper__semilattice__class_Osup(B,C,tc_bool),tc_bool) | -c_lessequals(A,C,tc_bool). [resolve(227,a,218,a)].
% 2.56/2.79 Derived: c_lessequals(A,c_Lattices_Oupper__semilattice__class_Osup(B,C,tc_bool),tc_bool) | -c_lessequals(A,B,tc_bool). [resolve(227,a,219,a)].
% 2.56/2.79 Derived: c_Lattices_Oupper__semilattice__class_Osup(A,B,tc_bool) = c_Lattices_Oupper__semilattice__class_Osup(B,A,tc_bool). [resolve(227,a,222,a)].
% 2.56/2.79 Derived: c_Lattices_Oupper__semilattice__class_Osup(A,c_Lattices_Oupper__semilattice__class_Osup(A,B,tc_bool),tc_bool) = c_Lattices_Oupper__semilattice__class_Osup(A,B,tc_bool). [resolve(227,a,223,a)].
% 2.56/2.79 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_bool),A),c_Lattices_Oupper__semilattice__class_Osup(B,C,tc_bool))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_bool),A),B)). [resolve(227,a,224,a)].
% 2.56/2.79 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_bool),A),c_Lattices_Oupper__semilattice__class_Osup(B,C,tc_bool))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_bool),A),C)). [resolve(227,a,225,a)].
% 2.56/2.79 228 class_OrderedGroup_Oab__semigroup__add(tc_nat) # label(clsarity_nat__OrderedGroup_Oab__semigroup__add) # label(axiom). [assumption].
% 2.56/2.79 229 -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].
% 2.56/2.79 230 class_Ring__and__Field_Ozero__neq__one(tc_nat) # label(clsarity_nat__Ring__and__Field_Ozero__neq__one) # label(axiom). [assumption].
% 2.56/2.79 231 -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].
% 2.56/2.79 232 -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].
% 2.56/2.79 233 -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].
% 2.56/2.79 234 -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].
% 2.56/2.79 235 -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].
% 2.56/2.79 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(230,a,231,a)].
% 2.68/2.92 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(230,a,232,a)].
% 2.68/2.92 Derived: c_HOL_Ozero__class_Ozero(tc_nat) != c_HOL_Oone__class_Oone(tc_nat). [resolve(230,a,234,a)].
% 2.68/2.92 Derived: c_HOL_Oone__class_Oone(tc_nat) != c_HOL_Ozero__class_Ozero(tc_nat). [resolve(230,a,235,a)].
% 2.68/2.92 236 c_Relation_Orel__comp(c_Relation_Oconverse(A,B,B),A,B,B,B) != A | c_Equiv__Relations_Oequiv(c_Relation_ODomain(A,B,B),A,B) # label(cls_comp__equivI_0) # label(axiom). [assumption].
% 2.68/2.92 237 c_lessequals(c_Relation_OImage(A,c_Set_Oinsert(B,c_Orderings_Obot__class_Obot(tc_fun(C,tc_bool)),C),C,C),c_Relation_OImage(A,c_Set_Oinsert(D,c_Orderings_Obot__class_Obot(tc_fun(C,tc_bool)),C),C,C),tc_fun(C,tc_bool)) | -hBOOL(hAPP(hAPP(c_in(tc_prod(C,C)),c_Pair(B,D,C,C)),A)) | -c_Equiv__Relations_Oequiv(E,A,C) # label(cls_equiv__class__subset_0) # label(axiom). [assumption].
% 2.68/2.92 238 hBOOL(hAPP(hAPP(c_in(A),B),c_Relation_OImage(C,c_Set_Oinsert(B,c_Orderings_Obot__class_Obot(tc_fun(A,tc_bool)),A),A,A))) | -hBOOL(hAPP(hAPP(c_in(A),B),D)) | -c_Equiv__Relations_Oequiv(D,C,A) # label(cls_equiv__class__self_0) # label(axiom). [assumption].
% 2.68/2.92 239 hBOOL(hAPP(hAPP(c_in(tc_prod(A,A)),c_Pair(B,C,A,A)),D)) | -hBOOL(hAPP(hAPP(c_in(A),E),c_Lattices_Olower__semilattice__class_Oinf(c_Relation_OImage(D,c_Set_Oinsert(B,c_Orderings_Obot__class_Obot(tc_fun(A,tc_bool)),A),A,A),c_Relation_OImage(D,c_Set_Oinsert(C,c_Orderings_Obot__class_Obot(tc_fun(A,tc_bool)),A),A,A),tc_fun(A,tc_bool)))) | -c_Equiv__Relations_Oequiv(F,D,A) # label(cls_equiv__class__nondisjoint_0) # label(axiom). [assumption].
% 2.68/2.92 240 hBOOL(hAPP(hAPP(c_in(tc_prod(A,A)),c_Pair(B,C,A,A)),D)) | -hBOOL(hAPP(hAPP(c_in(A),C),E)) | -c_lessequals(c_Relation_OImage(D,c_Set_Oinsert(C,c_Orderings_Obot__class_Obot(tc_fun(A,tc_bool)),A),A,A),c_Relation_OImage(D,c_Set_Oinsert(B,c_Orderings_Obot__class_Obot(tc_fun(A,tc_bool)),A),A,A),tc_fun(A,tc_bool)) | -c_Equiv__Relations_Oequiv(E,D,A) # label(cls_subset__equiv__class_0) # label(axiom). [assumption].
% 2.68/2.92 241 c_Relation_OImage(A,c_Set_Oinsert(B,c_Orderings_Obot__class_Obot(tc_fun(C,tc_bool)),C),C,C) != c_Relation_OImage(A,c_Set_Oinsert(D,c_Orderings_Obot__class_Obot(tc_fun(C,tc_bool)),C),C,C) | hBOOL(hAPP(hAPP(c_in(tc_prod(C,C)),c_Pair(B,D,C,C)),A)) | -hBOOL(hAPP(hAPP(c_in(C),D),E)) | -c_Equiv__Relations_Oequiv(E,A,C) # label(cls_eq__equiv__class_0) # label(axiom). [assumption].
% 2.68/2.92 242 c_Relation_OImage(A,c_Set_Oinsert(B,c_Orderings_Obot__class_Obot(tc_fun(C,tc_bool)),C),C,C) != c_Relation_OImage(A,c_Set_Oinsert(D,c_Orderings_Obot__class_Obot(tc_fun(C,tc_bool)),C),C,C) | -c_Equiv__Relations_Oequiv(E,A,C) | hBOOL(hAPP(hAPP(c_in(tc_prod(C,C)),c_Pair(B,D,C,C)),A)) | -hBOOL(hAPP(hAPP(c_in(C),D),E)) | -hBOOL(hAPP(hAPP(c_in(C),B),E)) # label(cls_equiv__class__eq__iff_3) # label(axiom). [assumption].
% 2.68/2.92 243 c_Relation_OImage(A,c_Set_Oinsert(B,c_Orderings_Obot__class_Obot(tc_fun(C,tc_bool)),C),C,C) = c_Relation_OImage(A,c_Set_Oinsert(D,c_Orderings_Obot__class_Obot(tc_fun(C,tc_bool)),C),C,C) | -hBOOL(hAPP(hAPP(c_in(tc_prod(C,C)),c_Pair(B,D,C,C)),A)) | -hBOOL(hAPP(hAPP(c_in(C),D),E)) | -hBOOL(hAPP(hAPP(c_in(C),B),E)) | -c_Equiv__Relations_Oequiv(E,A,C) # label(cls_eq__equiv__class__iff_1) # label(axiom). [assumption].
% 2.68/2.92 244 c_Relation_OImage(A,c_Set_Oinsert(B,c_Orderings_Obot__class_Obot(tc_fun(C,tc_bool)),C),C,C) != c_Relation_OImage(A,c_Set_Oinsert(D,c_Orderings_Obot__class_Obot(tc_fun(C,tc_bool)),C),C,C) | -hBOOL(hAPP(hAPP(c_in(C),D),E)) | -hBOOL(hAPP(hAPP(c_in(C),B),E)) | -c_Equiv__Relations_Oequiv(E,A,C) | hBOOL(hAPP(hAPP(c_in(tc_prod(C,C)),c_Pair(B,D,C,C)),A)) # label(cls_eq__equiv__class__iff_0) # label(axiom). [assumption].
% 2.68/2.92 245 c_Relation_OImage(A,c_Set_Oinsert(B,c_Orderings_Obot__class_Obot(tc_fun(C,tc_bool)),C),C,C) = c_Relation_OImage(A,c_Set_Oinsert(D,c_Orderings_Obot__class_Obot(tc_fun(C,tc_bool)),C),C,C) | -hBOOL(hAPP(hAPP(c_in(tc_prod(C,C)),c_Pair(B,D,C,C)),A)) | -c_Equiv__Relations_Oequiv(E,A,C) # label(cls_equiv__class__eq_0) # label(axiom). [assumption].
% 2.68/2.92 246 c_Relation_OImage(A,c_Set_Oinsert(B,c_Orderings_Obot__class_Obot(tc_fun(C,tc_bool)),C),C,C) = c_Relation_OImage(A,c_Set_Oinsert(D,c_Orderings_Obot__class_Obot(tc_fun(C,tc_bool)),C),C,C) | -hBOOL(hAPP(hAPP(c_in(tc_prod(C,C)),c_Pair(B,D,C,C)),A)) | -c_Equiv__Relations_Oequiv(E,A,C) # label(cls_equiv__class__eq__iff_0) # label(axiom). [assumption].
% 2.68/2.92 247 c_Equiv__Relations_Ocongruent(A,hAPP(B,C),D,E) | -hBOOL(hAPP(hAPP(c_in(F),C),V6)) | -c_Equiv__Relations_Ocongruent2(V7,A,B,F,D,E) | -c_Equiv__Relations_Oequiv(V6,V7,F) # label(cls_congruent2__implies__congruent_0) # label(axiom). [assumption].
% 2.68/2.92 Derived: c_Relation_Orel__comp(c_Relation_Oconverse(A,B,B),A,B,B,B) != A | c_lessequals(c_Relation_OImage(A,c_Set_Oinsert(C,c_Orderings_Obot__class_Obot(tc_fun(B,tc_bool)),B),B,B),c_Relation_OImage(A,c_Set_Oinsert(D,c_Orderings_Obot__class_Obot(tc_fun(B,tc_bool)),B),B,B),tc_fun(B,tc_bool)) | -hBOOL(hAPP(hAPP(c_in(tc_prod(B,B)),c_Pair(C,D,B,B)),A)). [resolve(236,b,237,c)].
% 2.68/2.92 Derived: c_Relation_Orel__comp(c_Relation_Oconverse(A,B,B),A,B,B,B) != A | hBOOL(hAPP(hAPP(c_in(B),C),c_Relation_OImage(A,c_Set_Oinsert(C,c_Orderings_Obot__class_Obot(tc_fun(B,tc_bool)),B),B,B))) | -hBOOL(hAPP(hAPP(c_in(B),C),c_Relation_ODomain(A,B,B))). [resolve(236,b,238,c)].
% 2.68/2.92 Derived: c_Relation_Orel__comp(c_Relation_Oconverse(A,B,B),A,B,B,B) != A | hBOOL(hAPP(hAPP(c_in(tc_prod(B,B)),c_Pair(C,D,B,B)),A)) | -hBOOL(hAPP(hAPP(c_in(B),E),c_Lattices_Olower__semilattice__class_Oinf(c_Relation_OImage(A,c_Set_Oinsert(C,c_Orderings_Obot__class_Obot(tc_fun(B,tc_bool)),B),B,B),c_Relation_OImage(A,c_Set_Oinsert(D,c_Orderings_Obot__class_Obot(tc_fun(B,tc_bool)),B),B,B),tc_fun(B,tc_bool)))). [resolve(236,b,239,c)].
% 2.68/2.92 Derived: c_Relation_Orel__comp(c_Relation_Oconverse(A,B,B),A,B,B,B) != A | hBOOL(hAPP(hAPP(c_in(tc_prod(B,B)),c_Pair(C,D,B,B)),A)) | -hBOOL(hAPP(hAPP(c_in(B),D),c_Relation_ODomain(A,B,B))) | -c_lessequals(c_Relation_OImage(A,c_Set_Oinsert(D,c_Orderings_Obot__class_Obot(tc_fun(B,tc_bool)),B),B,B),c_Relation_OImage(A,c_Set_Oinsert(C,c_Orderings_Obot__class_Obot(tc_fun(B,tc_bool)),B),B,B),tc_fun(B,tc_bool)). [resolve(236,b,240,d)].
% 2.68/2.92 Derived: c_Relation_Orel__comp(c_Relation_Oconverse(A,B,B),A,B,B,B) != A | c_Relation_OImage(A,c_Set_Oinsert(C,c_Orderings_Obot__class_Obot(tc_fun(B,tc_bool)),B),B,B) != c_Relation_OImage(A,c_Set_Oinsert(D,c_Orderings_Obot__class_Obot(tc_fun(B,tc_bool)),B),B,B) | hBOOL(hAPP(hAPP(c_in(tc_prod(B,B)),c_Pair(C,D,B,B)),A)) | -hBOOL(hAPP(hAPP(c_in(B),D),c_Relation_ODomain(A,B,B))). [resolve(236,b,241,d)].
% 2.68/2.92 Derived: c_Relation_Orel__comp(c_Relation_Oconverse(A,B,B),A,B,B,B) != A | c_Relation_OImage(A,c_Set_Oinsert(C,c_Orderings_Obot__class_Obot(tc_fun(B,tc_bool)),B),B,B) = c_Relation_OImage(A,c_Set_Oinsert(D,c_Orderings_Obot__class_Obot(tc_fun(B,tc_bool)),B),B,B) | -hBOOL(hAPP(hAPP(c_in(tc_prod(B,B)),c_Pair(C,D,B,B)),A)) | -hBOOL(hAPP(hAPP(c_in(B),D),c_Relation_ODomain(A,B,B))) | -hBOOL(hAPP(hAPP(c_in(B),C),c_Relation_ODomain(A,B,B))). [resolve(236,b,243,e)].
% 2.68/2.92 Derived: c_Relation_Orel__comp(c_Relation_Oconverse(A,B,B),A,B,B,B) != A | c_Relation_OImage(A,c_Set_Oinsert(C,c_Orderings_Obot__class_Obot(tc_fun(B,tc_bool)),B),B,B) = c_Relation_OImage(A,c_Set_Oinsert(D,c_Orderings_Obot__class_Obot(tc_fun(B,tc_bool)),B),B,B) | -hBOOL(hAPP(hAPP(c_in(tc_prod(B,B)),c_Pair(C,D,B,B)),A)). [resolve(236,b,245,c)].
% 2.68/2.92 Derived: c_Relation_Orel__comp(c_Relation_Oconverse(A,B,B),A,B,B,B) != A | c_Equiv__Relations_Ocongruent(C,hAPP(D,E),F,V6) | -hBOOL(hAPP(hAPP(c_in(B),E),c_Relation_ODomain(A,B,B))) | -c_Equiv__Relations_Ocongruent2(A,C,D,B,F,V6). [resolve(236,b,247,d)].
% 2.68/2.92 248 c_Relation_Orel__comp(c_Relation_Oconverse(A,B,B),A,B,B,B) = A | -c_Equiv__Relations_Oequiv(C,A,B) # label(cls_equiv__comp__eq_0) # label(axiom). [assumption].
% 2.68/2.92 249 c_Relation_Orefl__on(A,B,C) | -c_Equiv__Relations_Oequiv(A,B,C) # label(cls_equiv_Orefl__on_0) # label(axiom). [assumption].
% 2.68/2.92 Derived: c_Relation_Orefl__on(c_Relation_ODomain(A,B,B),A,B) | c_Relation_Orel__comp(c_Relation_Oconverse(A,B,B),A,B,B,B) != A. [resolve(249,b,236,b)].
% 2.68/2.98 250 c_Relation_Osym(A,B) | -c_Equiv__Relations_Oequiv(C,A,B) # label(cls_equiv_Osym_0) # label(axiom). [assumption].
% 2.68/2.98 Derived: c_Relation_Osym(A,B) | c_Relation_Orel__comp(c_Relation_Oconverse(A,B,B),A,B,B,B) != A. [resolve(250,b,236,b)].
% 2.68/2.98 251 hBOOL(hAPP(hAPP(c_in(A),B),C)) | -hBOOL(hAPP(hAPP(c_in(tc_prod(A,A)),c_Pair(D,B,A,A)),E)) | -c_Equiv__Relations_Oequiv(C,E,A) # label(cls_equiv__class__eq__iff_2) # label(axiom). [assumption].
% 2.68/2.98 Derived: hBOOL(hAPP(hAPP(c_in(A),B),c_Relation_ODomain(C,A,A))) | -hBOOL(hAPP(hAPP(c_in(tc_prod(A,A)),c_Pair(D,B,A,A)),C)) | c_Relation_Orel__comp(c_Relation_Oconverse(C,A,A),C,A,A,A) != C. [resolve(251,c,236,b)].
% 2.68/2.98 252 hBOOL(hAPP(hAPP(c_in(A),B),C)) | -hBOOL(hAPP(hAPP(c_in(tc_prod(A,A)),c_Pair(B,D,A,A)),E)) | -c_Equiv__Relations_Oequiv(C,E,A) # label(cls_equiv__class__eq__iff_1) # label(axiom). [assumption].
% 2.68/2.98 253 class_Ring__and__Field_Osemiring(tc_nat) # label(clsarity_nat__Ring__and__Field_Osemiring) # label(axiom). [assumption].
% 2.68/2.98 254 -class_Ring__and__Field_Osemiring(A) | hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),D),C)),E)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),D)),C)),E) # label(cls_combine__common__factor_0) # label(axiom). [assumption].
% 2.68/2.98 Derived: hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),B)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),C),B)),D)) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),C)),B)),D). [resolve(253,a,254,a)].
% 2.68/2.98 255 class_Ring__and__Field_Opordered__cancel__semiring(tc_nat) # label(clsarity_nat__Ring__and__Field_Opordered__cancel__semiring) # label(axiom). [assumption].
% 2.68/2.98 256 -class_Ring__and__Field_Opordered__cancel__semiring(A) | c_lessequals(hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C),c_HOL_Ozero__class_Ozero(A),A) | -c_lessequals(C,c_HOL_Ozero__class_Ozero(A),A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),B,A) # label(cls_split__mult__neg__le_0) # label(axiom). [assumption].
% 2.68/2.98 257 -class_Ring__and__Field_Opordered__cancel__semiring(A) | c_lessequals(hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C),c_HOL_Ozero__class_Ozero(A),A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),C,A) | -c_lessequals(B,c_HOL_Ozero__class_Ozero(A),A) # label(cls_split__mult__neg__le_1) # label(axiom). [assumption].
% 2.68/2.98 258 -class_Ring__and__Field_Opordered__cancel__semiring(A) | c_lessequals(hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C),c_HOL_Ozero__class_Ozero(A),A) | -c_lessequals(C,c_HOL_Ozero__class_Ozero(A),A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),B,A) # label(cls_mult__nonneg__nonpos_0) # label(axiom). [assumption].
% 2.68/2.98 259 -class_Ring__and__Field_Opordered__cancel__semiring(A) | c_lessequals(hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C),c_HOL_Ozero__class_Ozero(A),A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),C,A) | -c_lessequals(B,c_HOL_Ozero__class_Ozero(A),A) # label(cls_mult__nonpos__nonneg_0) # label(axiom). [assumption].
% 2.68/2.98 260 -class_Ring__and__Field_Opordered__cancel__semiring(A) | c_lessequals(hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C),c_HOL_Ozero__class_Ozero(A),A) | -c_lessequals(B,c_HOL_Ozero__class_Ozero(A),A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),C,A) # label(cls_mult__nonneg__nonpos2_0) # label(axiom). [assumption].
% 2.68/2.98 261 -class_Ring__and__Field_Opordered__cancel__semiring(A) | c_lessequals(c_HOL_Ozero__class_Ozero(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C),A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),C,A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),B,A) # label(cls_mult__nonneg__nonneg_0) # label(axiom). [assumption].
% 2.68/2.98 Derived: c_lessequals(hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),B),c_HOL_Ozero__class_Ozero(tc_nat),tc_nat) | -c_lessequals(B,c_HOL_Ozero__class_Ozero(tc_nat),tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),A,tc_nat). [resolve(255,a,256,a)].
% 2.89/3.12 Derived: c_lessequals(hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),B),c_HOL_Ozero__class_Ozero(tc_nat),tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),B,tc_nat) | -c_lessequals(A,c_HOL_Ozero__class_Ozero(tc_nat),tc_nat). [resolve(255,a,257,a)].
% 2.89/3.12 Derived: c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),B),tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),B,tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),A,tc_nat). [resolve(255,a,261,a)].
% 2.89/3.12 262 class_OrderedGroup_Ocancel__ab__semigroup__add(tc_nat) # label(clsarity_nat__OrderedGroup_Ocancel__ab__semigroup__add) # label(axiom). [assumption].
% 2.89/3.12 263 -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].
% 2.89/3.12 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(262,a,263,a)].
% 2.89/3.12 264 class_OrderedGroup_Ocancel__semigroup__add(tc_nat) # label(clsarity_nat__OrderedGroup_Ocancel__semigroup__add) # label(axiom). [assumption].
% 2.89/3.12 265 -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].
% 2.89/3.12 266 -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].
% 2.89/3.12 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(264,a,266,a)].
% 2.89/3.12 267 class_Ring__and__Field_Omult__mono(tc_nat) # label(clsarity_nat__Ring__and__Field_Omult__mono) # label(axiom). [assumption].
% 2.89/3.12 268 -class_Ring__and__Field_Omult__mono(A) | c_lessequals(hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),D),C),A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),C,A) | -c_lessequals(B,D,A) # label(cls_mult__right__mono_0) # label(axiom). [assumption].
% 2.89/3.12 269 -class_Ring__and__Field_Omult__mono(A) | c_lessequals(hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),D),A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),B,A) | -c_lessequals(C,D,A) # label(cls_mult__left__mono_0) # label(axiom). [assumption].
% 2.89/3.12 Derived: c_lessequals(hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),B),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),C),B),tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),B,tc_nat) | -c_lessequals(A,C,tc_nat). [resolve(267,a,268,a)].
% 2.89/3.12 Derived: c_lessequals(hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),B),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),C),tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),A,tc_nat) | -c_lessequals(B,C,tc_nat). [resolve(267,a,269,a)].
% 2.89/3.12 270 class_Ring__and__Field_Omult__mono1(tc_nat) # label(clsarity_nat__Ring__and__Field_Omult__mono1) # label(axiom). [assumption].
% 2.89/3.12 271 -class_Ring__and__Field_Omult__mono1(A) | c_lessequals(hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),D),A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),B,A) | -c_lessequals(C,D,A) # label(cls_mult__mono1_0) # label(axiom). [assumption].
% 2.89/3.12 272 class_Ring__and__Field_Opordered__semiring(tc_nat) # label(clsarity_nat__Ring__and__Field_Opordered__semiring) # label(axiom). [assumption].
% 2.89/3.12 273 -class_Ring__and__Field_Opordered__semiring(A) | c_lessequals(hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),D),E),A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),C,A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),D,A) | -c_lessequals(C,E,A) | -c_lessequals(B,D,A) # label(cls_mult__mono_0) # label(axiom). [assumption].
% 2.89/3.12 274 -class_Ring__and__Field_Opordered__semiring(A) | c_lessequals(hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),D),E),A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),C,A) | -c_lessequals(c_HOL_Ozero__class_Ozero(A),B,A) | -c_lessequals(C,E,A) | -c_lessequals(B,D,A) # label(cls_mult__mono_H_0) # label(axiom). [assumption].
% 2.98/3.22 Derived: c_lessequals(hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),B),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),C),D),tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),B,tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),C,tc_nat) | -c_lessequals(B,D,tc_nat) | -c_lessequals(A,C,tc_nat). [resolve(272,a,273,a)].
% 2.98/3.22 Derived: c_lessequals(hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),A),B),hAPP(hAPP(c_HOL_Otimes__class_Otimes(tc_nat),C),D),tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),B,tc_nat) | -c_lessequals(c_HOL_Ozero__class_Ozero(tc_nat),A,tc_nat) | -c_lessequals(B,D,tc_nat) | -c_lessequals(A,C,tc_nat). [resolve(272,a,274,a)].
% 2.98/3.22 275 class_OrderedGroup_Omonoid__add(tc_nat) # label(clsarity_nat__OrderedGroup_Omonoid__add) # label(axiom). [assumption].
% 2.98/3.22 276 -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].
% 2.98/3.22 277 -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].
% 2.98/3.22 278 class_OrderedGroup_Oab__semigroup__mult(tc_nat) # label(clsarity_nat__OrderedGroup_Oab__semigroup__mult) # label(axiom). [assumption].
% 2.98/3.22 279 -class_OrderedGroup_Oab__semigroup__mult(A) | hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C)),D) = hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),C),D)) # label(cls_ab__semigroup__mult__class_Omult__ac_I1_J_0) # label(axiom). [assumption].
% 2.98/3.22 280 class_Ring__and__Field_Ocomm__semiring(tc_nat) # label(clsarity_nat__Ring__and__Field_Ocomm__semiring) # label(axiom). [assumption].
% 2.98/3.22 281 -class_Ring__and__Field_Ocomm__semiring(A) | hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C)),D) = hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),C),D)) # label(cls_comm__semiring__class_Odistrib_0) # label(axiom). [assumption].
% 2.98/3.22 282 class_Orderings_Olinorder(tc_nat) # label(clsarity_nat__Orderings_Olinorder) # label(axiom). [assumption].
% 2.98/3.22 283 -class_Orderings_Olinorder(A) | c_lessequals(B,C,A) | c_lessequals(C,B,A) # label(cls_linorder__linear_0) # label(axiom). [assumption].
% 2.98/3.22 284 -class_Orderings_Olinorder(A) | B = C | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),C)) | -c_lessequals(B,C,A) # label(cls_linorder__antisym__conv2_0) # label(axiom). [assumption].
% 2.98/3.22 285 -class_Orderings_Olinorder(A) | B = C | -c_lessequals(B,C,A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),C)) # label(cls_linorder__antisym__conv1_0) # label(axiom). [assumption].
% 2.98/3.22 286 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),C)) | c_lessequals(C,B,A) # label(cls_not__leE_0) # label(axiom). [assumption].
% 2.98/3.22 287 -class_Orderings_Olinorder(A) | -c_lessequals(B,B,A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),B)) # label(cls_linorder__antisym__conv2_1) # label(axiom). [assumption].
% 2.98/3.22 288 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),B)) | c_lessequals(B,B,A) # label(cls_linorder__antisym__conv1_1) # label(axiom). [assumption].
% 2.98/3.22 289 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),C)) | -c_lessequals(C,B,A) # label(cls_linorder__not__less_1) # label(axiom). [assumption].
% 2.98/3.22 290 -class_Orderings_Olinorder(A) | c_lessequals(B,C,A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),C),B)) # label(cls_linorder__not__less_0) # label(axiom). [assumption].
% 2.98/3.22 291 -class_Orderings_Olinorder(A) | -c_lessequals(B,C,A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),C),B)) # label(cls_linorder__not__le_1) # label(axiom). [assumption].
% 2.98/3.22 292 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),C)) | c_lessequals(C,B,A) # label(cls_linorder__not__le_0) # label(axiom). [assumption].
% 2.98/3.27 293 -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].
% 2.98/3.27 294 -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].
% 2.98/3.27 295 -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].
% 2.98/3.27 296 -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].
% 2.98/3.27 297 -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].
% 2.98/3.27 298 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),B)) # label(cls_linorder__neq__iff_1) # label(axiom). [assumption].
% 2.98/3.27 Derived: c_lessequals(A,B,tc_nat) | c_lessequals(B,A,tc_nat). [resolve(282,a,283,a)].
% 2.98/3.27 Derived: A = B | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),B)) | -c_lessequals(A,B,tc_nat). [resolve(282,a,284,a)].
% 2.98/3.27 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),B)) | c_lessequals(B,A,tc_nat). [resolve(282,a,286,a)].
% 2.98/3.27 Derived: -c_lessequals(A,A,tc_nat) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),A)). [resolve(282,a,287,a)].
% 2.98/3.27 Derived: -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),B)) | -c_lessequals(B,A,tc_nat). [resolve(282,a,289,a)].
% 2.98/3.27 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(282,a,293,a)].
% 2.98/3.27 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(282,a,294,a)].
% 2.98/3.27 299 class_Ring__and__Field_Oordered__comm__semiring__strict(tc_nat) # label(clsarity_nat__Ring__and__Field_Oordered__comm__semiring__strict) # label(axiom). [assumption].
% 2.98/3.27 300 -class_Ring__and__Field_Oordered__comm__semiring__strict(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_HOL_Otimes__class_Otimes(A),B),D))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),c_HOL_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),C),D)) # label(cls_mult__strict__left__mono__comm_0) # label(axiom). [assumption].
% 2.98/3.27 301 class_OrderedGroup_Opordered__cancel__ab__semigroup__add(tc_nat) # label(clsarity_nat__OrderedGroup_Opordered__cancel__ab__semigroup__add) # label(axiom). [assumption].
% 2.98/3.27 302 -class_OrderedGroup_Opordered__cancel__ab__semigroup__add(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),D),E))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),C),E)) | -c_lessequals(B,D,A) # label(cls_add__le__less__mono_0) # label(axiom). [assumption].
% 2.98/3.27 303 -class_OrderedGroup_Opordered__cancel__ab__semigroup__add(A) | hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(A),D),E))) | -c_lessequals(C,E,A) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(A),B),D)) # label(cls_add__less__le__mono_0) # label(axiom). [assumption].
% 2.98/3.27 304 -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].
% 2.98/3.27 305 -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].
% 3.06/3.33 306 -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].
% 3.06/3.33 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),C),D))) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),B),D)) | -c_lessequals(A,C,tc_nat). [resolve(301,a,302,a)].
% 3.06/3.33 Derived: hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),A),B)),hAPP(hAPP(c_HOL_Oplus__class_Oplus(tc_nat),C),D))) | -c_lessequals(B,D,tc_nat) | -hBOOL(hAPP(hAPP(c_HOL_Oord__class_Oless(tc_nat),A),C)). [resolve(301,a,303,a)].
% 3.06/3.33 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(301,a,304,a)].
% 3.06/3.33 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(301,a,305,a)].
% 3.06/3.33 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(301,a,306,a)].
% 3.06/3.33 307 class_Ring__and__Field_Osemiring__1(tc_nat) # label(clsarity_nat__Ring__and__Field_Osemiring__1) # label(axiom). [assumption].
% 3.06/3.33 308 -class_Ring__and__Field_Osemiring__1(A) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),c_HOL_Ozero__class_Ozero(tc_nat)) = c_HOL_Ozero__class_Ozero(A) # label(cls_of__nat__0_0) # label(axiom). [assumption].
% 3.06/3.33 309 -class_Ring__and__Field_Osemiring__1(A) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(hAPP(c_Finite__Set_Osetsum(B,tc_nat),C),D)) = hAPP(hAPP(c_Finite__Set_Osetsum(B,A),c_COMBB(c_Nat_Osemiring__1__class_Oof__nat(A),C,tc_nat,A,B)),D) # label(cls_of__nat__setsum_0) # label(axiom). [assumption].
% 3.06/3.33 310 -class_Ring__and__Field_Osemiring__1(A) | -class_OrderedGroup_Ocomm__monoid__mult(A) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),c_Finite__Set_Osetprod(B,C,D,tc_nat)) = c_Finite__Set_Osetprod(c_COMBB(c_Nat_Osemiring__1__class_Oof__nat(A),B,tc_nat,A,D),C,D,A) # label(cls_of__nat__setprod_0) # label(axiom). [assumption].
% 3.06/3.33 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_nat),c_HOL_Ozero__class_Ozero(tc_nat)) = c_HOL_Ozero__class_Ozero(tc_nat). [resolve(307,a,308,a)].
% 3.06/3.33 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_nat),hAPP(hAPP(c_Finite__Set_Osetsum(A,tc_nat),B),C)) = hAPP(hAPP(c_Finite__Set_Osetsum(A,tc_nat),c_COMBB(c_Nat_Osemiring__1__class_Oof__nat(tc_nat),B,tc_nat,tc_nat,A)),C). [resolve(307,a,309,a)].
% 3.06/3.33 Derived: -class_OrderedGroup_Ocomm__monoid__mult(tc_nat) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_nat),c_Finite__Set_Osetprod(A,B,C,tc_nat)) = c_Finite__Set_Osetprod(c_COMBB(c_Nat_Osemiring__1__class_Oof__nat(tc_nat),A,tc_nat,tc_nat,C),B,C,tc_nat). [resolve(307,a,310,a)].
% 3.06/3.33 311 -hBOOL(hAPP(hAPP(c_in(tc_prod(A,A)),c_Pair(B,B,A,A)),C)) | -c_Relation_Oirrefl(C,A) # label(cls_irrefl__def_0) # label(axiom). [assumption].
% 3.06/3.33 312 c_Relation_Oirrefl(A,B) | hBOOL(hAPP(hAPP(c_in(tc_prod(B,B)),c_Pair(c_ATP__Linkup_Osko__Relation__Xirrefl__def__1__1(A,B),c_ATP__Linkup_Osko__Relation__Xirrefl__def__1__1(A,B),B,B)),A)) # label(cls_irrefl__def_1) # label(axiom). [assumption].
% 3.06/3.33 Derived: -hBOOL(hAPP(hAPP(c_in(tc_prod(A,A)),c_Pair(B,B,A,A)),C)) | hBOOL(hAPP(hAPP(c_in(tc_prod(A,A)),c_Pair(c_ATP__Linkup_Osko__Relation__Xirrefl__def__1__1(C,A),c_ATP__Linkup_Osko__Relation__Xirrefl__def__1__1(C,A),A,A)),C)). [resolve(311,b,312,a)].
% 3.28/3.52 313 hBOOL(hAPP(hAPP(c_in(tc_prod(A,A)),c_Pair(B,B,A,A)),C)) | -c_Nitpick_Orefl_H(C,A) # label(cls_Nitpick_Orefl_H__def_0) # label(axiom). [assumption].
% 3.28/3.52 314 c_Nitpick_Orefl_H(A,B) | -hBOOL(hAPP(hAPP(c_in(tc_prod(B,B)),c_Pair(c_Nitpick_Osko__Nitpick__Xrefl_H__def__1__1(A,B),c_Nitpick_Osko__Nitpick__Xrefl_H__def__1__1(A,B),B,B)),A)) # label(cls_Nitpick_Orefl_H__def_1) # label(axiom). [assumption].
% 3.28/3.52 Derived: hBOOL(hAPP(hAPP(c_in(tc_prod(A,A)),c_Pair(B,B,A,A)),C)) | -hBOOL(hAPP(hAPP(c_in(tc_prod(A,A)),c_Pair(c_Nitpick_Osko__Nitpick__Xrefl_H__def__1__1(C,A),c_Nitpick_Osko__Nitpick__Xrefl_H__def__1__1(C,A),A,A)),C)). [resolve(313,b,314,a)].
% 3.28/3.52 315 class_Nat_Osemiring__char__0(tc_nat) # label(clsarity_nat__Nat_Osemiring__char__0) # label(axiom). [assumption].
% 3.28/3.52 316 -class_Nat_Osemiring__char__0(A) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B) != hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),C) | B = C # label(cls_of__nat__eq__iff_0) # label(axiom). [assumption].
% 3.28/3.52 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_nat),A) != hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_nat),B) | A = B. [resolve(315,a,316,a)].
% 3.28/3.52
% 3.28/3.52 ============================== end predicate elimination =============
% 3.28/3.52
% 3.28/3.52 Auto_denials: (non-Horn, no changes).
% 3.28/3.52
% 3.28/3.52 Term ordering decisions:
% 3.28/3.52 Function symbol KB weights: tc_nat=1. tc_bool=1. tc_Arrow__Order__Mirabelle_Oalt=1. tc_Arrow__Order__Mirabelle_Oindi=1. t_a=1. v_P____=1. v_e____=1. v_c____=1. v_h____=1. v_n____=1. c_Arrow__Order__Mirabelle_Omkbot=1. c_Arrow__Order__Mirabelle_Omktop=1. c_Arrow__Order__Mirabelle_Oabove=1. v_d____=1. t_b=1. v_r=1. c_Arrow__Order__Mirabelle_OProf=1. hAPP=1. tc_fun=1. tc_prod=1. c_Transitive__Closure_Ortrancl=1. c_Transitive__Closure_Otrancl=1. c_Finite__Set_Osetsum=1. c_Wellfounded_Oacc=1. c_Finite__Set_Ocard=1. c_Relation_OId__on=1. c_Wellfounded_Omeasure=1. v_sko__Wellfounded__Xacc__Xinduct__1=1. v_sko__Wellfounded__Xacc__Xinducts__1=1. c_ATP__Linkup_Osko__Relation__XIdE__1__1=1. c_ATP__Linkup_Osko__Relation__Xirrefl__def__1__1=1. c_ATP__Linkup_Osko__Transitive__Closure__Xirrefl__trancl__rD__1__1=1. c_Nitpick_Osko__Nitpick__Xrefl_H__def__1__1=1. c_ATP__Linkup_Osko__Wellfounded__Xacc__wfI__1__1=1. c_ATP__Linkup_Osko__Wellfounded__Xwf__acc__iff__1__1=1. v_sko__Arrow__Order__Mirabelle__Xcomplete__Lin__1=1. c_in=1. c_HOL_Oord__class_Oless=1. c_HOL_Ozero__class_Ozero=1. c_HOL_Otimes__class_Otimes=1. c_HOL_Oplus__class_Oplus=1. c_Orderings_Obot__class_Obot=1. c_HOL_Ouminus__class_Ouminus=1. c_HOL_Oone__class_Oone=1. c_Orderings_Otop__class_Otop=1. c_HOL_OIf=1. c_Relation_OId=1. c_Nat_Osemiring__1__class_Oof__nat=1. c_fequal=1. v_F=1. c_HOL_Ominus__class_Ominus=1. c_Lattices_Oupper__semilattice__class_Osup=1. c_Lattices_Olower__semilattice__class_Oinf=1. c_Set_Oinsert=1. c_Relation_Oconverse=1. c_Power_Opower__class_Opower=1. c_Relation_ODomain=1. c_Relation_ORange=1. c_ATP__Linkup_Osko__Relation__XId__onE__1__1=1. c_ATP__Linkup_Osko__Wellfounded__Xacc__induct__1__1=1. c_ATP__Linkup_Osko__Wellfounded__Xacc__induct__rule__1__1=1. c_Wellfounded_Omlex__prod=1. c_ATP__Linkup_Osko__Wellfounded__Xacc__Xintros__1__1=1. c_ATP__Linkup_Osko__Wellfounded__Xnot__acc__down__1__1=1. c_ATP__Linkup_Osko__Wellfounded__Xwf__def__1__1=1. c_ATP__Linkup_Osko__Wellfounded__Xwf__induct__1__1=1. c_ATP__Linkup_Osko__Wellfounded__Xwf__induct__rule__1__1=1. c_List_Osko__Recdef__Xtfl__wf__induct__1__1=1. v_sko__Transitive__Closure__Xrtrancl__Xcases__1=1. v_sko__Transitive__Closure__Xtrancl__Xcases__1=1. c_ATP__Linkup_Osko__Wellfounded__XwfE__min__1__1=1. c_ATP__Linkup_Osko__Wellfounded__Xwf__eq__minimal__1__1=1. c_Pair=1. c_COMBC=1. c_Relation_OImage=1. c_Finite__Set_Osetprod=1. c_Set_Ovimage=1. c_Product__Type_OSigma=1. c_Relation_Oinv__image=1. c_ATP__Linkup_Osko__Transitive__Closure__Xconverse__rtrancl__induct__1__2=1. c_ATP__Linkup_Osko__Transitive__Closure__Xrtrancl__induct__1__1=1. c_ATP__Linkup_Osko__Finite__Set__Xsetprod__1_H__1__1=1. c_ATP__Linkup_Osko__Finite__Set__Xsetprod__pos__1__1=1. c_ATP__Linkup_Osko__Finite__Set__Xsetprod__zero__iff__1__1=1. c_ATP__Linkup_Osko__Finite__Set__Xsetsum__0_H__1__1=1. c_ATP__Linkup_Osko__Transitive__Closure__Xconverse__rtranclE__1__1=1. c_ATP__Linkup_Osko__Transitive__Closure__Xconverse__rtrancl__induct__1__1=1. c_ATP__Linkup_Osko__Transitive__Closure__Xconverse__tranclE__1__1=1. c_ATP__Linkup_Osko__Transitive__Closure__XrtranclE__1__1=1. c_ATP__Linkup_Osko__Transitive__Closure__Xrtrancl__induct__1__2=1. c_ATP__Linkup_Osko__Transitive__Closure__XtranclD2__1__1=1. c_ATP__Linkup_Osko__Transitive__Closure__XtranclD__1__1=1. c_ATP__Linkup_Osko__Transitive__Closure__XtranclE__1__1=1. c_ATP__Linkup_Osko__Relation__XDomainE__1__1=1. c_ATP__Linkup_Osko__Relation__XDomain__iff__1__1=1. c_ATP__Linkup_Osko__Relation__XRangeE__1__1=1. c_ATP__Linkup_Osko__Relation__XRange__iff__1__1=1. c_COMBB=1. c_Relation_Orel__comp=1. c_COMBS=1. c_Recdef_Ocut=1. c_ATP__Linkup_Osko__Relation__XImageE__1__1=1. c_ATP__Linkup_Osko__Relation__XImage__iff__1__1=1. c_ATP__Linkup_Osko__Product__Type__XSigmaE__1__1=1. c_List_Osko__Recdef__Xcuts__eq__1__1=1. c_ATP__Linkup_Osko__Relation__Xrel__compEpair__1__1=1.
% 4.87/5.15
% 4.87/5.15 ============================== end of process initial clauses ========
% 4.87/5.15
% 4.87/5.15 ============================== CLAUSES FOR SEARCH ====================
% 4.87/5.15
% 4.87/5.15 ============================== end of clauses for search =============
% 4.87/5.15
% 4.87/5.15 ============================== SEARCH ================================
% 4.87/5.15
% 4.87/5.15 % Starting search at 1.99 seconds.
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% 4.87/5.15 NOTE: Back_subsumption disabled, ratio of kept to back_subsumed is 86 (0.00 of 2.54 sec).
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% 4.87/5.15 Low Water (displace): id=4336, wt=101Cputime limit exceeded (core dumped)
%------------------------------------------------------------------------------