0.02/0.10 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.02/0.11 % Command : tptp2X_and_run_prover9 %d %s 0.10/0.31 % Computer : n017.cluster.edu 0.10/0.31 % Model : x86_64 x86_64 0.10/0.31 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.10/0.31 % Memory : 8042.1875MB 0.10/0.31 % OS : Linux 3.10.0-693.el7.x86_64 0.10/0.31 % CPULimit : 1440 0.10/0.31 % WCLimit : 180 0.10/0.31 % DateTime : Mon Jul 3 08:51:08 EDT 2023 0.10/0.31 % CPUTime : 2.22/2.58 ============================== Prover9 =============================== 2.22/2.58 Prover9 (32) version 2009-11A, November 2009. 2.22/2.58 Process 20774 was started by sandbox2 on n017.cluster.edu, 2.22/2.58 Mon Jul 3 08:51:10 2023 2.22/2.58 The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 1440 -f /tmp/Prover9_20608_n017.cluster.edu". 2.22/2.58 ============================== end of head =========================== 2.22/2.58 2.22/2.58 ============================== INPUT ================================= 2.22/2.58 2.22/2.58 % Reading from file /tmp/Prover9_20608_n017.cluster.edu 2.22/2.58 2.22/2.58 set(prolog_style_variables). 2.22/2.58 set(auto2). 2.22/2.58 % set(auto2) -> set(auto). 2.22/2.58 % set(auto) -> set(auto_inference). 2.22/2.58 % set(auto) -> set(auto_setup). 2.22/2.58 % set(auto_setup) -> set(predicate_elim). 2.22/2.58 % set(auto_setup) -> assign(eq_defs, unfold). 2.22/2.58 % set(auto) -> set(auto_limits). 2.22/2.58 % set(auto_limits) -> assign(max_weight, "100.000"). 2.22/2.58 % set(auto_limits) -> assign(sos_limit, 20000). 2.22/2.58 % set(auto) -> set(auto_denials). 2.22/2.58 % set(auto) -> set(auto_process). 2.22/2.58 % set(auto2) -> assign(new_constants, 1). 2.22/2.58 % set(auto2) -> assign(fold_denial_max, 3). 2.22/2.58 % set(auto2) -> assign(max_weight, "200.000"). 2.22/2.58 % set(auto2) -> assign(max_hours, 1). 2.22/2.58 % assign(max_hours, 1) -> assign(max_seconds, 3600). 2.22/2.58 % set(auto2) -> assign(max_seconds, 0). 2.22/2.58 % set(auto2) -> assign(max_minutes, 5). 2.22/2.58 % assign(max_minutes, 5) -> assign(max_seconds, 300). 2.22/2.58 % set(auto2) -> set(sort_initial_sos). 2.22/2.58 % set(auto2) -> assign(sos_limit, -1). 2.22/2.58 % set(auto2) -> assign(lrs_ticks, 3000). 2.22/2.58 % set(auto2) -> assign(max_megs, 400). 2.22/2.58 % set(auto2) -> assign(stats, some). 2.22/2.58 % set(auto2) -> clear(echo_input). 2.22/2.58 % set(auto2) -> set(quiet). 2.22/2.58 % set(auto2) -> clear(print_initial_clauses). 2.22/2.58 % set(auto2) -> clear(print_given). 2.22/2.58 assign(lrs_ticks,-1). 2.22/2.58 assign(sos_limit,10000). 2.22/2.58 assign(order,kbo). 2.22/2.58 set(lex_order_vars). 2.22/2.58 clear(print_given). 2.22/2.58 2.22/2.58 % formulas(sos). % not echoed (623 formulas) 2.22/2.58 2.22/2.58 ============================== end of input ========================== 2.22/2.58 2.22/2.58 % From the command line: assign(max_seconds, 1440). 2.22/2.58 2.22/2.58 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 2.22/2.58 2.22/2.58 % Formulas that are not ordinary clauses: 2.22/2.58 1 (all X_b all A_3 all C all D (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),C),D)) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),C)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),D))))) # label(fact_256_insert__mono) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 2 (all X_b all A_1 all B (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),B)) -> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B) = ti(fun(X_b,bool),B))) # label(fact_328_Un__absorb1) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 3 (all Loc_2 all Loc_3 (hAPP(loc_1,vname,loc,Loc_2) = hAPP(loc_1,vname,loc,Loc_3) <-> ti(loc_1,Loc_3) = ti(loc_1,Loc_2))) # label(fact_89_vname_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 4 (all X_b all X_c all Z_1 all F all A_1 ((exists X_2 (hAPP(X_c,X_b,F,X_2) = ti(X_b,Z_1) & hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),X_2),A_1)))) <-> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Z_1),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),A_1))))) # label(fact_67_image__iff) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 5 (all Loc_3 all Fun all Com all Vname all Fun_1 hAPP(com,com,hAPP(fun(state,nat),fun(com,com),hAPP(loc_1,fun(fun(state,nat),fun(com,com)),local,Loc_3),Fun),Com) != hAPP(fun(state,nat),com,hAPP(vname,fun(fun(state,nat),com),ass,Vname),Fun_1)) # label(fact_93_com_Osimps_I23_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 6 (all X_b all Pa all Q_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Q_1))) & hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Pa))) <-> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fdisj),Pa)),Q_1)))))) # label(fact_141_finite__Collect__disjI) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 7 (all X_b all A_1 all B all C hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),B),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),C)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),B),C))) # label(fact_337_Un__left__commute) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 8 (all X_c all X_b all X_1 all F all Z_1 all G all F_1 (hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite908156982e_idem(X_b,X_c),F),Z_1),G),F_1)) -> ti(X_b,X_1) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,X_1),X_1))) # label(fact_163_folding__image__simple__idem_Oidem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 9 (all Ca hAPP(com,hoare_1656922687triple(state),hoare_Mirabelle_MGT,Ca) = hAPP(fun(state,fun(state,bool)),hoare_1656922687triple(state),hAPP(com,fun(fun(state,fun(state,bool)),hoare_1656922687triple(state)),hAPP(fun(state,fun(state,bool)),fun(com,fun(fun(state,fun(state,bool)),hoare_1656922687triple(state))),hoare_246368825triple(state),fequal(state)),Ca),hAPP(com,fun(state,fun(state,bool)),evalc,Ca))) # label(fact_128_MGT__def) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 10 (all X_b all A_3 all B hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),A_3))),B) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),B)) # label(fact_362_insert__def) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 11 (all X_a (order(X_a) -> (all Z all Y all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Z),Y)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Z),X))))))) # label(fact_304_xt1_I6_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 12 (all T_2 all T_1 (lattice(T_1) -> semilattice_inf(fun(T_2,T_1)))) # label(arity_fun___Lattices_Osemilattice__inf) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 13 (all X_b all A_1 all Ca all B ((-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),B)) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),A_1))) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B))))) # label(fact_239_UnCI) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 14 (all X_b all X_c all Z_1 all X_1 all A_1 all V all F (hBOOL(hAPP(fun(X_b,fun(X_c,X_c)),bool,finite100568337ommute(X_b,X_c),F)) -> (hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),finite_fold_graph(X_b,X_c),F),Z_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),A_1)),V)) -> (-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_1)) -> -(all Y_2 (hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_1),Y_2) = ti(X_c,V) -> -hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),finite_fold_graph(X_b,X_c),F),Z_1),A_1),Y_2)))))))) # label(fact_208_comp__fun__commute_Ofold__graph__insertE) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 15 (all X_b all B all D all A_1 all C (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),C)) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B),D)) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),C),D)))))) # label(fact_432_Int__mono) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 16 (all X_b all F all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (ti(fun(X_b,bool),A_1) != bot_bot(fun(X_b,bool)) -> (exists X1 hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),fun(X_b,bool)),finite_fold1Set(X_b),F),A_1),X1)))))) # label(fact_149_finite__nonempty__imp__fold1Set) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 17 (all X_b ti(fun(fun(X_b,bool),fun(X_b,bool)),collect(X_b)) = collect(X_b)) # label(tsy_c_Set_OCollect_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 18 (all X_b all X_c ti(fun(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)))),finite1357897459simple(X_b,X_c)) = finite1357897459simple(X_b,X_c)) # label(tsy_c_Finite__Set_Ofolding__image__simple_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 19 (all X_b all Q_1 all Ga all Pa all Ca all Q_3 (hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Pa),Ca),Q_3)),bot_bot(fun(hoare_1656922687triple(X_b),bool))))) -> ((all Z_2 all S_2 (hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),Q_3,Z_2),S_2)) -> hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),Q_1,Z_2),S_2)))) -> hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Pa),Ca),Q_1)),bot_bot(fun(hoare_1656922687triple(X_b),bool)))))))) # label(fact_6_conseq2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 20 (all X_3 all A_3 all S_4 hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,hAPP(fun(state,nat),com,hAPP(vname,fun(fun(state,nat),com),ass,X_3),A_3)),S_4),hAPP(nat,state,hAPP(vname,fun(nat,state),hAPP(state,fun(vname,fun(nat,state)),update,S_4),X_3),hAPP(state,nat,A_3,S_4))))) # label(fact_115_evalc_OAssign) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 21 (all X_b all Ba all A_1 all B (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),B)) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Ba),B))))) # label(fact_255_subset__insertI2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 22 (all X_a all X_c all X_b all P all Q all R hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),P,R),hAPP(X_a,X_b,Q,R)) = hAPP(X_a,X_c,hAPP(fun(X_a,X_b),fun(X_a,X_c),hAPP(fun(X_a,fun(X_b,X_c)),fun(fun(X_a,X_b),fun(X_a,X_c)),combs(X_a,X_b,X_c),P),Q),R)) # label(help_COMBS_1_1_U) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 23 (all X_b all Ca all A_1 all B (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B))) -> -(hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),A_1)) -> -hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),B))))) # label(fact_416_IntE) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 24 (all X_b all A_3 all A_1 all B hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),A_1)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),B)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B))) # label(fact_448_insert__inter__insert) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 25 (all X_b all Pa all A_1 all F_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),F_1)) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),F_1),A_1)) -> (hBOOL(hAPP(fun(X_b,bool),bool,Pa,bot_bot(fun(X_b,bool)))) -> ((all A_4 all F_2 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),F_2)) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_4),A_1)) -> (-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_4),F_2)) -> (hBOOL(hAPP(fun(X_b,bool),bool,Pa,F_2)) -> hBOOL(hAPP(fun(X_b,bool),bool,Pa,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_4),F_2)))))))) -> hBOOL(hAPP(fun(X_b,bool),bool,Pa,F_1))))))) # label(fact_375_finite__subset__induct) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 26 (all Q all P (hBOOL(P) | hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fimplies,P),Q)))) # label(help_fimplies_1_1_U) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 27 (all X_a (semilattice_inf(X_a) -> (all A_2 all B_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),B_1),A_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),A_2),B_1)))) # label(fact_473_inf_Ocommute) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 28 (all X_b all A_1 ti(fun(X_b,bool),A_1) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),A_1)) # label(fact_343_Un__absorb) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 29 (all X_b all X_a all P all Q ti(X_a,P) = hAPP(X_b,X_a,hAPP(X_a,fun(X_b,X_a),combk(X_a,X_b),P),Q)) # label(help_COMBK_1_1_U) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 30 (all X_b (ab_sem1668676832m_mult(X_b) -> (all A_3 all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),A_1)) = hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),X_b)),finite_fold(X_b,X_b),times_times(X_b)),A_3),A_1))))) # label(fact_214_fold1__eq__fold__idem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 31 (all X_b all A_1 all B hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),B)),B) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),B)) # label(fact_166_Diff__idemp) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 32 (all X_b (semilattice_inf(X_b) -> (all X_1 all Y_1 (hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),Y_1)) <-> ti(X_b,X_1) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_inf_inf(X_b),X_1),Y_1))))) # label(fact_427_le__iff__inf) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 33 (all T_2 all T_1 (lattice(T_1) -> lattice(fun(T_2,T_1)))) # label(arity_fun___Lattices_Olattice) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 34 (all X_a (semilattice_inf(X_a) -> (all A_2 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),A_2),A_2) = ti(X_a,A_2)))) # label(fact_470_inf_Oidem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 35 (all X_a (lattice(X_a) -> (all X all Y all Z hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X),Y)),Z) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),Y),Z))))) # label(fact_483_inf__sup__aci_I2_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 36 (all X_c all X_b all X_d all F all G all A_1 hAPP(fun(X_d,bool),fun(X_b,bool),hAPP(fun(X_d,X_b),fun(fun(X_d,bool),fun(X_b,bool)),image(X_d,X_b),hAPP(fun(X_d,X_c),fun(X_d,X_b),hAPP(fun(X_c,X_b),fun(fun(X_d,X_c),fun(X_d,X_b)),combb(X_c,X_b,X_d),F),G)),A_1) = hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),hAPP(fun(X_d,bool),fun(X_c,bool),hAPP(fun(X_d,X_c),fun(fun(X_d,bool),fun(X_c,bool)),image(X_d,X_c),G),A_1))) # label(fact_63_image__image) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 37 (all X_b all A_3 all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_3),A_1)) -> ti(fun(X_b,bool),A_1) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),bot_bot(fun(X_b,bool))))))) # label(fact_176_insert__Diff) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 38 (all X_b all A_3 all A_1 hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),A_1) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),bot_bot(fun(X_b,bool)))),A_1)) # label(fact_363_insert__is__Un) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 39 (all X_a (semilattice_sup(X_a) -> (all X all Y hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),Y),X) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X),Y)))) # label(fact_284_sup__commute) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 40 (all T all A ti(T,A) = ti(T,ti(T,A))) # label(help_ti_idem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 41 (all X_b all A_1 all B (ti(fun(X_b,bool),A_1) = ti(fun(X_b,bool),B) -> -(hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),B)) -> -hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B),A_1))))) # label(fact_320_equalityE) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 42 (all X_b all A_1 all B (hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B) = bot_bot(fun(X_b,bool)) <-> (all X_2 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_1)) -> (all Xa (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Xa),B)) -> ti(X_b,Xa) != ti(X_b,X_2))))))) # label(fact_465_disjoint__iff__not__equal) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 43 (all P all Q (-hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fdisj,P),Q)) | hBOOL(Q) | hBOOL(P))) # label(help_fdisj_3_1_U) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 44 (all X_a (semilattice_sup(X_a) -> (all X all Y hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X),Y)))))) # label(fact_292_sup__ge1) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 45 (all X_b ti(fun(X_b,fun(fun(X_b,bool),X_b)),partial_flat_lub(X_b)) = partial_flat_lub(X_b)) # label(tsy_c_Partial__Function_Oflat__lub_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 46 (all X_b all A_1 all B hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),B)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B)) = ti(fun(X_b,bool),A_1)) # label(fact_490_Un__Diff__Int) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 47 (all X_b (lattice(X_b) -> (all X_1 ti(X_b,X_1) = hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),bot_bot(fun(X_b,bool))))))) # label(fact_398_Sup__fin_Osingleton) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 48 (all X_b (ab_semigroup_mult(X_b) -> (all A_3 all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_3),A_1)) -> hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),A_1)) = hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),X_b)),finite_fold(X_b,X_b),times_times(X_b)),A_3),A_1)))))) # label(fact_215_fold1__eq__fold) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 49 (all X_a (semilattice_sup(X_a) -> (all X all Y hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X),Y) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X),Y))))) # label(fact_280_sup__left__idem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 50 (all X_b all B all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B),A_1)) -> ti(fun(X_b,bool),A_1) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B))) # label(fact_327_Un__absorb2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 51 (all P (fFalse = ti(bool,P) | ti(bool,P) = fTrue)) # label(help_fFalse_1_1_T) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 52 (all Vname_1 all Fun hAPP(fun(state,nat),com,hAPP(vname,fun(fun(state,nat),com),ass,Vname_1),Fun) != skip) # label(fact_70_com_Osimps_I8_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 53 (all X_b insert(X_b) = ti(fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),insert(X_b))) # label(tsy_c_Set_Oinsert_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 54 (all X_b all A_1 all B all C hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),B),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),C)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),B),C))) # label(fact_460_Int__left__commute) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 55 (all X_b all A_1 all B all X_1 ((-hBOOL(hAPP(X_b,bool,B,X_1)) -> hBOOL(hAPP(X_b,bool,A_1,X_1))) -> hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B),X_1)))) # label(fact_241_sup1CI) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 56 (all P (hBOOL(hAPP(bool,bool,fNot,P)) | hBOOL(P))) # label(help_fNot_2_1_U) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 57 (all X_a (linorder(X_a) -> (all X all Y (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X)))))) # label(fact_302_linorder__le__cases) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 58 (all X_b all B hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),bot_bot(fun(X_b,bool))),B) = bot_bot(fun(X_b,bool))) # label(fact_467_Int__empty__left) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 59 (all X_a (ab_sem1668676832m_mult(X_a) -> (all A_2 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_2),A_2) = ti(X_a,A_2)))) # label(fact_199_times_Oidem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 60 (all X_b all A_3 all G all F (hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),F) = G -> ti(X_b,A_3) = hAPP(fun(X_b,bool),X_b,G,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),bot_bot(fun(X_b,bool)))))) # label(fact_218_fold1__singleton__def) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 61 (all X_b all A_1 all B (hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B) = bot_bot(fun(X_b,bool)) -> ti(fun(X_b,bool),A_1) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),B))) # label(fact_493_Diff__triv) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 62 (all X_b (ab_semigroup_mult(X_b) -> ti(fun(X_b,fun(X_b,X_b)),times_times(X_b)) = times_times(X_b))) # label(tsy_c_Groups_Otimes__class_Otimes_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 63 (all X_c all X_b (lattice(X_b) -> (all F all G all X_1 hAPP(X_c,X_b,hAPP(fun(X_c,X_b),fun(X_c,X_b),hAPP(fun(X_c,X_b),fun(fun(X_c,X_b),fun(X_c,X_b)),semilattice_inf_inf(fun(X_c,X_b)),F),G),X_1) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_inf_inf(X_b),hAPP(X_c,X_b,F,X_1)),hAPP(X_c,X_b,G,X_1))))) # label(fact_485_inf__apply) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 64 (all T_2 all T_1 (order(T_1) -> order(fun(T_2,T_1)))) # label(arity_fun___Orderings_Oorder) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 65 (all X_c all X_b all F all Z_1 all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (exists X1 hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),finite_fold_graph(X_b,X_c),F),Z_1),A_1),X1))))) # label(fact_153_finite__imp__fold__graph) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 66 (all X_b all Ca all A_1 all B (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B))) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),B)))) # label(fact_464_IntD2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 67 (all P all Q (-hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,P),Q)) | hBOOL(Q))) # label(help_fconj_3_1_U) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 68 (all X_b all C all A_1 all B hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),C),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),B)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),C),A_1)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),C),B))) # label(fact_489_Diff__Int__distrib) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 69 (all X_a (bot(X_a) -> (all A_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),bot_bot(X_a))) -> bot_bot(X_a) = ti(X_a,A_2))))) # label(fact_355_le__bot) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 70 (all X_b all Y_1 all A_1 all X_1 (hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Y_1),A_1),X_1)) <-> hBOOL(hAPP(X_b,bool,A_1,X_1)) | ti(X_b,Y_1) = ti(X_b,X_1))) # label(fact_27_insert__code) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 71 (all X_a (semilattice_sup(X_a) -> (all A_2 all B_1 all C_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),A_2),B_1)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),A_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),B_1),C_1))))) # label(fact_276_sup_Oassoc) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 72 (all X_b all Pa all A_3 ((hBOOL(hAPP(X_b,bool,Pa,A_3)) -> hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(X_b,fun(X_b,bool),fequal(X_b),A_3))),Pa)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),bot_bot(fun(X_b,bool)))) & (-hBOOL(hAPP(X_b,bool,Pa,A_3)) -> bot_bot(fun(X_b,bool)) = hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(X_b,fun(X_b,bool),fequal(X_b),A_3))),Pa))))) # label(fact_14_Collect__conv__if2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 73 (all X_b (semilattice_sup(X_b) -> ti(fun(X_b,fun(X_b,X_b)),semilattice_sup_sup(X_b)) = semilattice_sup_sup(X_b))) # label(tsy_c_Lattices_Osemilattice__sup__class_Osup_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 74 (all X_b all Pa all F_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),F_1)) -> (hBOOL(hAPP(fun(X_b,bool),bool,Pa,bot_bot(fun(X_b,bool)))) -> ((all X_2 all F_2 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),F_2)) -> (-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),F_2)) -> (hBOOL(hAPP(fun(X_b,bool),bool,Pa,F_2)) -> hBOOL(hAPP(fun(X_b,bool),bool,Pa,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),F_2))))))) -> hBOOL(hAPP(fun(X_b,bool),bool,Pa,F_1)))))) # label(fact_150_finite__induct) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 75 (all X_b all Ca all A_1 all B (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),B)) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),A_1)) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),B))))) # label(fact_238_subsetD) # label(axiom) # label(non_clause). [assumption]. 2.22/2.58 76 (all X_b all X_c (lattice(X_c) -> (all F all G all X_2 hAPP(X_b,X_c,hAPP(fun(X_b,X_c),fun(X_b,X_c),hAPP(fun(X_b,X_c),fun(fun(X_b,X_c),fun(X_b,X_c)),semilattice_inf_inf(fun(X_b,X_c)),F),G),X_2) = hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),semilattice_inf_inf(X_c),hAPP(X_b,X_c,F,X_2)),hAPP(X_b,X_c,G,X_2))))) # label(fact_472_inf__fun__def) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 77 (all X_a (semilattice_sup(X_a) -> (all X ti(X_a,X) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X),X)))) # label(fact_288_sup__idem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 78 (all X_b (lattice(X_b) -> (all N all H ((all X_2 all Y_2 hAPP(X_b,X_b,H,hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),X_2),Y_2)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),hAPP(X_b,X_b,H,X_2)),hAPP(X_b,X_b,H,Y_2))) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),N)) -> (ti(fun(X_b,bool),N) != bot_bot(fun(X_b,bool)) -> hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,X_b),fun(fun(X_b,bool),fun(X_b,bool)),image(X_b,X_b),H),N)) = hAPP(X_b,X_b,H,hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),N)))))))) # label(fact_409_Sup__fin_Ohom__commute) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 79 (all X_b all R_1 all S all X_2 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),R_1),S))) <-> hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),R_1)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),S)),X_2)))) # label(fact_352_sup__Un__eq) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 80 (all C2 all S2 all N2 all T2 all C1 all S1 all N1 all T1 (hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C1),S1),N1),T1)) -> (hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C2),S2),N2),T2)) -> (exists N_1 (hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C1),S1),N_1),T1)) & hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C2),S2),N_1),T2))))))) # label(fact_132_evaln__max2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 81 (all X_c all X_b all X_1 all A_1 all F all Z_1 all G all F_1 (hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite1357897459simple(X_b,X_c),F),Z_1),G),F_1)) -> (hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_1)) -> (-hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),X_1),A_1)) -> hAPP(fun(X_c,bool),X_b,F_1,hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_c,bool),fun(X_c,bool)),insert(X_c),X_1),A_1)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(X_c,X_b,G,X_1)),hAPP(fun(X_c,bool),X_b,F_1,A_1)))))) # label(fact_191_folding__image__simple_Oinsert) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 82 (all X_b all X_c all Z_1 all X_1 all A_1 all F (hBOOL(hAPP(fun(X_b,fun(X_c,X_c)),bool,finite_comp_fun_idem(X_b,X_c),F)) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c)),finite_fold(X_b,X_c),F),Z_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),A_1)) = hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_1),hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c)),finite_fold(X_b,X_c),F),Z_1),A_1))))) # label(fact_225_comp__fun__idem_Ofold__insert__idem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 83 (all X_b hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(bool,fun(X_b,bool),combk(bool,X_b),fFalse)) = bot_bot(fun(X_b,bool))) # label(fact_23_empty__def) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 84 (all X_b all X_1 all A_1 all F all F_1 (hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite2073411215e_idem(X_b),F),F_1)) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (bot_bot(fun(X_b,bool)) != ti(fun(X_b,bool),A_1) -> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,X_1),hAPP(fun(X_b,bool),X_b,F_1,A_1)) = hAPP(fun(X_b,bool),X_b,F_1,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),A_1)))))) # label(fact_154_folding__one__idem_Oinsert__idem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 85 (all X_b ti(fun(fun(X_b,bool),X_b),the(X_b)) = the(X_b)) # label(tsy_c_HOL_OThe_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 86 (all X_a (lattice(X_a) -> (all X all Y hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X),Y) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X),Y))))) # label(fact_281_inf__sup__aci_I8_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 87 (all X_b all A_1 all B all C hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B)),C) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),B),C))) # label(fact_335_Un__assoc) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 88 (all X_b all X_c all Ba all F all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ba),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),A_1))) -> -(all X_2 (ti(X_b,Ba) = hAPP(X_c,X_b,F,X_2) -> -hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),X_2),A_1)))))) # label(fact_80_imageE) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 89 (all X_b ti(fun(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool)),finite2073411215e_idem(X_b)) = finite2073411215e_idem(X_b)) # label(tsy_c_Finite__Set_Ofolding__one__idem_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 90 (all X_b all A_3 all B all C hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),B),C)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),B)),C)) # label(fact_358_Un__insert__left) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 91 (all X_b hoare_246368825triple(X_b) = ti(fun(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)))),hoare_246368825triple(X_b))) # label(tsy_c_Hoare__Mirabelle__nrugjuseim_Otriple_Otriple_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 92 (all X_b all B all A_1 all X_1 (hBOOL(hAPP(X_b,bool,A_1,X_1)) -> (hBOOL(hAPP(X_b,bool,B,X_1)) -> hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B),X_1))))) # label(fact_413_inf1I) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 93 (all U_1 all C_1 all S_1 all T_4 (hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C_1),S_1),T_4)) -> (hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C_1),S_1),U_1)) -> U_1 = T_4))) # label(fact_118_com__det) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 94 (all X_b all X_1 hAPP(fun(X_b,bool),X_b,the(X_b),hAPP(X_b,fun(X_b,bool),fequal(X_b),X_1)) = ti(X_b,X_1)) # label(fact_81_the__sym__eq__trivial) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 95 (all X_b all A_1 all B hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fdisj),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),A_1))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),B))) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B)) # label(fact_342_Un__def) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 96 (all C1 all S2 all C0 all S0 all N_2 all S1 (hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C0),S0),N_2),S1)) -> (hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C1),S1),N_2),S2)) -> hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,hAPP(com,com,hAPP(com,fun(com,com),semi,C0),C1)),S0),N_2),S2))))) # label(fact_107_evaln_OSemi) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 97 (all X_a (semilattice_inf(X_a) -> (all X hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X),X) = ti(X_a,X)))) # label(fact_471_inf__idem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 98 (all X_c all X_b all F all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_c,bool),fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,bool)),fun(fun(X_c,bool),fun(fun(X_b,bool),fun(X_c,bool))),hAPP(fun(fun(X_c,bool),fun(fun(X_c,bool),fun(X_c,bool))),fun(fun(X_b,fun(X_c,bool)),fun(fun(X_c,bool),fun(fun(X_b,bool),fun(X_c,bool)))),finite_fold_image(fun(X_c,bool),X_b),semilattice_sup_sup(fun(X_c,bool))),hAPP(fun(X_c,bool),fun(X_b,fun(X_c,bool)),hAPP(fun(X_b,fun(fun(X_c,bool),fun(X_c,bool))),fun(fun(X_c,bool),fun(X_b,fun(X_c,bool))),combc(X_b,fun(X_c,bool),fun(X_c,bool)),hAPP(fun(X_b,X_c),fun(X_b,fun(fun(X_c,bool),fun(X_c,bool))),hAPP(fun(X_c,fun(fun(X_c,bool),fun(X_c,bool))),fun(fun(X_b,X_c),fun(X_b,fun(fun(X_c,bool),fun(X_c,bool)))),combb(X_c,fun(fun(X_c,bool),fun(X_c,bool)),X_b),insert(X_c)),F)),bot_bot(fun(X_c,bool)))),bot_bot(fun(X_c,bool))),A_1) = hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),A_1))) # label(fact_387_image__eq__fold__image) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 99 (all X_b (lattice(X_b) -> (all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),A_1) = hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),semilattice_sup_sup(X_b)),A_1))))) # label(fact_401_Sup__fin_OF__eq) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 100 (all X_b all Q_1 all Pa (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Pa))) | hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Q_1))) -> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),Pa)),Q_1)))))) # label(fact_137_finite__Collect__conjI) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 101 (all X_b all Pa all Q_1 hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Pa)),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Q_1)) = hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),Pa)),Q_1))) # label(fact_454_Collect__conj__eq) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 102 (all X_c all X_b all F all Z_1 all A1 all A2 (bot_bot(fun(X_b,bool)) = ti(fun(X_b,bool),A1) & ti(X_c,A2) = ti(X_c,Z_1) | (exists X_2 exists A_5 exists Y_2 (hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_5) = ti(fun(X_b,bool),A1) & -hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_5)) & hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),finite_fold_graph(X_b,X_c),F),Z_1),A_5),Y_2)) & ti(X_c,A2) = hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_2),Y_2))) <-> hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),finite_fold_graph(X_b,X_c),F),Z_1),A1),A2)))) # label(fact_131_fold__graph_Osimps) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 103 (all X_a (semilattice_inf(X_a) -> (all X all Y hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X),Y) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X),Y))))) # label(fact_478_inf__left__idem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 104 (all X_b ti(fun(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b)),finite_fold1(X_b)) = finite_fold1(X_b)) # label(tsy_c_Finite__Set_Ofold1_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 105 (all X_b all X_c all B all F all A_1 ((exists AA (ti(fun(X_b,bool),B) = hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),AA) & hBOOL(hAPP(fun(X_c,bool),bool,hAPP(fun(X_c,bool),fun(fun(X_c,bool),bool),ord_less_eq(fun(X_c,bool)),AA),A_1)))) <-> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),A_1))))) # label(fact_257_subset__image__iff) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 106 (all X_b all F_1 all Ga (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),Ga)) & hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),F_1)) <-> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),F_1),Ga))))) # label(fact_300_finite__Un) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 107 (all X_b all A_3 all Ba (hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Ba),bot_bot(fun(X_b,bool))) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),bot_bot(fun(X_b,bool))) -> ti(X_b,A_3) = ti(X_b,Ba))) # label(fact_35_singleton__inject) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 108 (all X_b (lattice(X_b) -> (all X_1 all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_1)) -> (bot_bot(fun(X_b,bool)) != ti(fun(X_b,bool),A_1) -> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),X_1),hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),A_1)) = hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),A_1)))))))) # label(fact_403_Sup__fin_Oinsert) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 109 (all X_b all X_1 all Pa all Q_1 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),Pa),Q_1)) -> (hBOOL(hAPP(X_b,bool,Pa,X_1)) -> hBOOL(hAPP(X_b,bool,Q_1,X_1))))) # label(fact_347_predicate1D) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 110 (all X_b all X_1 all A_1 all B (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),B)) & hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),B)) <-> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),A_1)),B)))) # label(fact_253_insert__subset) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 111 (all X_b all X_c (ab_semigroup_mult(X_c) -> (all G all Z_1 all A_3 all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_3),A_1)) -> hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),times_times(X_c),hAPP(X_b,X_c,G,A_3)),hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c)),hAPP(fun(X_c,fun(X_c,X_c)),fun(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c))),finite_fold_image(X_c,X_b),times_times(X_c)),G),Z_1),A_1)) = hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c)),hAPP(fun(X_c,fun(X_c,X_c)),fun(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c))),finite_fold_image(X_c,X_b),times_times(X_c)),G),Z_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),A_1))))))) # label(fact_391_fold__image__insert) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 112 (all X_a ti(X_a,undefined(X_a)) = undefined(X_a)) # label(tsy_c_HOL_Oundefined_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 113 (all X_b all Ba all A_3 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ba),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),bot_bot(fun(X_b,bool))))) -> ti(X_b,A_3) = ti(X_b,Ba))) # label(fact_36_singletonE) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 114 (all X_b all A_1 all B all C hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B)),C) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),B),C))) # label(fact_488_Int__Diff) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 115 (all X_a (bot(X_a) -> (all A_2 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),bot_bot(X_a)),A_2))))) # label(fact_353_bot__least) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 116 (all X_b all Ga all Pa hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Pa),skip),Pa)),bot_bot(fun(hoare_1656922687triple(X_b),bool)))))) # label(fact_45_hoare__derivs_OSkip) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 117 (all X_b all X_1 all A_1 all F all F_1 (hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite_folding_one(X_b),F),F_1)) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_1)) -> hAPP(fun(X_b,bool),X_b,F_1,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),A_1)) = hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),X_b)),finite_fold(X_b,X_b),F),X_1),A_1))))) # label(fact_226_folding__one_Oeq__fold_H) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 118 (all X_b all B all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B),A_1)) -> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B) = ti(fun(X_b,bool),B))) # label(fact_434_Int__absorb1) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 119 (all X_b all X_c all F all G ((all X_2 hAPP(X_b,X_c,F,X_2) = hAPP(X_b,X_c,G,X_2)) -> ti(fun(X_b,X_c),G) = ti(fun(X_b,X_c),F))) # label(fact_75_ext) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 120 (all X_b all Pa (bot_bot(fun(X_b,bool)) = hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Pa) <-> (all X_2 -hBOOL(hAPP(X_b,bool,Pa,X_2))))) # label(fact_18_Collect__empty__eq) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 121 (all X_b all X_c all F all A_1 all B hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(fun(X_c,bool),fun(fun(X_c,bool),fun(X_c,bool)),semilattice_inf_inf(fun(X_c,bool)),A_1),B))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),A_1)),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),B))))) # label(fact_440_image__Int__subset) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 122 (all X_b all B all A_3 all A_1 ((hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_3),A_1)) -> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),B)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B))) & (-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_3),A_1)) -> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),B)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B)))) # label(fact_450_Int__insert__right) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 123 (all X_b all B all A_1 ((all X_2 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_1)) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),B)))) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),B)))) # label(fact_376_subsetI) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 124 (all X_a (semilattice_sup(X_a) -> (all A_2 all B_1 all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),A_2),B_1)),X)) -> -(hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),X)) -> -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_1),X))))))) # label(fact_264_le__supE) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 125 (all Com1 all Com2 all Vname all Fun_1 hAPP(fun(state,nat),com,hAPP(vname,fun(fun(state,nat),com),ass,Vname),Fun_1) != hAPP(com,com,hAPP(com,fun(com,com),semi,Com1),Com2)) # label(fact_69_com_Osimps_I25_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 126 (all X_b all B all A_1 all X_1 (hBOOL(hAPP(X_b,bool,A_1,X_1)) -> hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B),X_1)))) # label(fact_350_sup1I1) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 127 (all X_c all X_b all B all A_1 all F all Z_1 all G all F_1 (hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite908156982e_idem(X_b,X_c),F),Z_1),G),F_1)) -> (hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_1)) -> (hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),B)) -> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(fun(X_c,bool),X_b,F_1,A_1)),hAPP(fun(X_c,bool),X_b,F_1,B)) = hAPP(fun(X_c,bool),X_b,F_1,hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(fun(X_c,bool),fun(fun(X_c,bool),fun(X_c,bool)),semilattice_sup_sup(fun(X_c,bool)),A_1),B)))))) # label(fact_369_folding__image__simple__idem_Ounion__idem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 128 (all X_c all X_b all F all A_1 all B hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(fun(X_c,bool),fun(fun(X_c,bool),fun(X_c,bool)),semilattice_sup_sup(fun(X_c,bool)),A_1),B)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),A_1)),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),B))) # label(fact_356_image__Un) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 129 (all X_c all X_b all F all Z_1 hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),fold_graph(X_b,X_c),F),Z_1),bot_bot(fun(X_b,bool))),Z_1))) # label(fact_204_fold__graph_H_Ointros_I1_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 130 (all X_b all B all A_3 all C ((-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_3),C)) -> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),B)),C) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),B),C)) & (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_3),C)) -> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),B)),C) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),B),C))))) # label(fact_449_Int__insert__left) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 131 (all X_a (lattice(X_a) -> (all X all Y all Z hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),Y),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X),Y)),Z)))) # label(fact_275_inf__sup__aci_I6_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 132 (all X_b all X_c ti(fun(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool))),image(X_b,X_c)) = image(X_b,X_c)) # label(tsy_c_Set_Oimage_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 133 (all X_b (lattice(X_b) -> (all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (ti(fun(X_b,bool),A_1) != bot_bot(fun(X_b,bool)) -> ((all X_2 all Y_2 hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),X_2),Y_2)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Y_2),bot_bot(fun(X_b,bool))))))) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),A_1)),A_1)))))))) # label(fact_410_Sup__fin_Oclosed) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 134 (all X_b ti(fun(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool)),finite_folding_one(X_b)) = finite_folding_one(X_b)) # label(tsy_c_Finite__Set_Ofolding__one_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 135 (all Y_4 all A_3 all Ca all S_4 all N_3 all T_5 (hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,hAPP(com,com,hAPP(fun(state,nat),fun(com,com),hAPP(loc_1,fun(fun(state,nat),fun(com,com)),local,Y_4),A_3),Ca)),S_4),N_3),T_5)) -> -(all S1_1 (T_5 = hAPP(nat,state,hAPP(vname,fun(nat,state),hAPP(state,fun(vname,fun(nat,state)),update,S1_1),hAPP(loc_1,vname,loc,Y_4)),hAPP(loc_1,nat,hAPP(state,fun(loc_1,nat),getlocs,S_4),Y_4)) -> -hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,Ca),hAPP(nat,state,hAPP(vname,fun(nat,state),hAPP(state,fun(vname,fun(nat,state)),update,S_4),hAPP(loc_1,vname,loc,Y_4)),hAPP(state,nat,A_3,S_4))),N_3),S1_1)))))) # label(fact_124_evaln__elim__cases_I3_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 136 (all X_b (lattice(X_b) -> (all X_1 all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_1)) -> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),X_1),hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),A_1)) = hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),A_1)))))) # label(fact_399_Sup__fin_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 137 (all X_a (semilattice_inf(X_a) -> (all B_1 all A_2 all C_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),A_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),B_1),C_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),B_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),A_2),C_1))))) # label(fact_479_inf_Oleft__commute) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 138 (all X_b (semilattice_inf(X_b) -> hBOOL(hAPP(fun(X_b,fun(X_b,X_b)),bool,finite_comp_fun_idem(X_b,X_b),semilattice_inf_inf(X_b))))) # label(fact_443_comp__fun__idem__inf) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 139 (all X_b all A_1 all A_3 all B (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),B)))) <-> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),B))))) # label(fact_181_finite__Diff__insert) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 140 (all X_a (semilattice_sup(X_a) -> (all A_2 all X all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),B_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),A_2),B_1))))))) # label(fact_270_le__supI2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 141 (all X_a (order(X_a) -> (all X all Y (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X)) -> ti(X_a,Y) = ti(X_a,X)))))) # label(fact_307_order__antisym) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 142 (all X_c all X_b all F all Z_1 all Y_1 all X_1 all A_1 (-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_1)) -> (hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),finite_fold_graph(X_b,X_c),F),Z_1),A_1),Y_1)) -> hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),finite_fold_graph(X_b,X_c),F),Z_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),A_1)),hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_1),Y_1)))))) # label(fact_122_fold__graph_OinsertI) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 143 (all X_a all X_b all X_c combc(X_a,X_b,X_c) = ti(fun(fun(X_a,fun(X_b,X_c)),fun(X_b,fun(X_a,X_c))),combc(X_a,X_b,X_c))) # label(tsy_c_COMBC_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 144 (all X_c all X_b all X_d (comm_monoid_mult(X_d) -> (all E all G all F all H all K all T_3 all S (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),S)) -> ((all Y_2 (hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),Y_2),T_3)) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(X_c,X_b,K,Y_2)),S)) & ti(X_c,Y_2) = hAPP(X_b,X_c,H,hAPP(X_c,X_b,K,Y_2)))) -> ((all X_2 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),S)) -> hAPP(X_c,X_d,G,hAPP(X_b,X_c,H,X_2)) = hAPP(X_b,X_d,F,X_2) & ti(X_b,X_2) = hAPP(X_c,X_b,K,hAPP(X_b,X_c,H,X_2)) & hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),hAPP(X_b,X_c,H,X_2)),T_3)))) -> hAPP(fun(X_b,bool),X_d,hAPP(X_d,fun(fun(X_b,bool),X_d),hAPP(fun(X_b,X_d),fun(X_d,fun(fun(X_b,bool),X_d)),hAPP(fun(X_d,fun(X_d,X_d)),fun(fun(X_b,X_d),fun(X_d,fun(fun(X_b,bool),X_d))),finite_fold_image(X_d,X_b),times_times(X_d)),F),E),S) = hAPP(fun(X_c,bool),X_d,hAPP(X_d,fun(fun(X_c,bool),X_d),hAPP(fun(X_c,X_d),fun(X_d,fun(fun(X_c,bool),X_d)),hAPP(fun(X_d,fun(X_d,X_d)),fun(fun(X_c,X_d),fun(X_d,fun(fun(X_c,bool),X_d))),finite_fold_image(X_d,X_c),times_times(X_d)),G),E),T_3))))))) # label(fact_394_fold__image__eq__general__inverses) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 145 (all T_2 all T_1 (bounded_lattice(T_1) -> bounded_lattice_bot(fun(T_2,T_1)))) # label(arity_fun___Lattices_Obounded__lattice__bot) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 146 (all X_c all X_b all F all A_3 all B hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_c,bool),fun(X_c,bool)),insert(X_c),A_3),B)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),hAPP(X_c,X_b,F,A_3)),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),B))) # label(fact_79_image__insert) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 147 (all X_a (lattice(X_a) -> (all Y all X hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X),Y)))))) # label(fact_291_inf__sup__ord_I4_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.59 148 (all X_c all X_b all X_1 all A_1 all F all Z_1 all G all F_1 (hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite1357897459simple(X_b,X_c),F),Z_1),G),F_1)) -> (hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_1)) -> (hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),X_1),A_1)) -> hAPP(fun(X_c,bool),X_b,F_1,A_1) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(X_c,X_b,G,X_1)),hAPP(fun(X_c,bool),X_b,F_1,hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(fun(X_c,bool),fun(fun(X_c,bool),fun(X_c,bool)),minus_minus(fun(X_c,bool)),A_1),hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_c,bool),fun(X_c,bool)),insert(X_c),X_1),bot_bot(fun(X_c,bool)))))))))) # label(fact_186_folding__image__simple_Oremove) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 149 (all C1 all C2 all S_1 all N_2 all T_4 (hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,hAPP(com,com,hAPP(com,fun(com,com),semi,C1),C2)),S_1),N_2),T_4)) -> -(all S1_1 (hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C1),S_1),N_2),S1_1)) -> -hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C2),S1_1),N_2),T_4)))))) # label(fact_126_evaln__elim__cases_I4_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 150 (all X_b all A_1 all B (ti(fun(X_b,bool),B) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B) <-> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),B)))) # label(fact_340_subset__Un__eq) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 151 (all X_b all Pa all A_1 all B ((exists X_2 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B))) & hBOOL(hAPP(X_b,bool,Pa,X_2)))) <-> (exists X_2 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),B)) & hBOOL(hAPP(X_b,bool,Pa,X_2)))) | (exists X_2 (hBOOL(hAPP(X_b,bool,Pa,X_2)) & hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_1)))))) # label(fact_334_bex__Un) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 152 (all X_a (semilattice_sup(X_a) -> (all B_1 all D_1 all A_2 all C_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),C_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_1),D_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),A_2),B_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),C_1),D_1)))))))) # label(fact_265_sup__mono) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 153 (all X_a (lattice(X_a) -> (all X all Y all Z hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),Y),Z))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X),Z))))))) # label(fact_438_distrib__sup__le) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 154 (all X_b all X_3 hAPP(fun(X_b,bool),X_b,the(X_b),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,bool),hAPP(fun(fun(X_b,bool),bool),fun(fun(X_b,fun(X_b,bool)),fun(X_b,bool)),combb(fun(X_b,bool),bool,X_b),hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),fequal(fun(X_b,bool)),X_3)),hAPP(fun(X_b,bool),fun(X_b,fun(X_b,bool)),hAPP(fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,bool),fun(X_b,fun(X_b,bool))),combc(X_b,fun(X_b,bool),fun(X_b,bool)),insert(X_b)),bot_bot(fun(X_b,bool))))) = hAPP(fun(X_b,bool),X_b,the_elem(X_b),X_3)) # label(fact_54_the__elem__def) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 155 (all X_b (ab_sem1668676832m_mult(X_b) -> (all N all H ((all X_2 all Y_2 hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(X_b,X_b,H,X_2)),hAPP(X_b,X_b,H,Y_2)) = hAPP(X_b,X_b,H,hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),X_2),Y_2))) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),N)) -> (bot_bot(fun(X_b,bool)) != ti(fun(X_b,bool),N) -> hAPP(X_b,X_b,H,hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),N)) = hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,X_b),fun(fun(X_b,bool),fun(X_b,bool)),image(X_b,X_b),H),N)))))))) # label(fact_231_hom__fold1__commute) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 156 (all X_b all A_1 all B all C hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),B),C)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B)),C)) # label(fact_462_Int__assoc) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 157 (all X_b all A_1 all B (ti(fun(X_b,bool),B) = ti(fun(X_b,bool),A_1) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B),A_1)))) # label(fact_331_equalityD2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 158 (all T_2 all T_1 (finite_finite(T_2) & finite_finite(T_1) -> finite_finite(fun(T_2,T_1)))) # label(arity_fun___Finite__Set_Ofinite) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 159 (all X_b (lattice(X_b) -> (all X_1 all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_1)) -> (hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),bot_bot(fun(X_b,bool)))) = bot_bot(fun(X_b,bool)) -> hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),A_1) = ti(X_b,X_1)) & (bot_bot(fun(X_b,bool)) != hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),bot_bot(fun(X_b,bool)))) -> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),X_1),hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),bot_bot(fun(X_b,bool)))))) = hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),A_1))))))) # label(fact_397_Sup__fin_Oremove) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 160 (all X_b all A_1 hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),bot_bot(fun(X_b,bool))),A_1) = bot_bot(fun(X_b,bool))) # label(fact_172_empty__Diff) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 161 (all X_b (semilattice_sup(X_b) -> (all X_1 all Y_1 all Z_1 (hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Y_1),Z_1)) & hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),Z_1)) <-> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),X_1),Y_1)),Z_1)))))) # label(fact_273_le__sup__iff) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 162 (all X_b all C all A_1 all B (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),B)) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B),C)) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),C))))) # label(fact_323_subset__trans) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 163 (all X_b ti(fun(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool)),hoare_279057269derivs(X_b)) = hoare_279057269derivs(X_b)) # label(tsy_c_Hoare__Mirabelle__nrugjuseim_Ohoare__derivs_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 164 (all X_b all X_c all A_1 all Ba all F all X_1 (hAPP(X_c,X_b,F,X_1) = ti(X_b,Ba) -> (hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),X_1),A_1)) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ba),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),A_1)))))) # label(fact_61_image__eqI) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 165 (all X_b all X_c (ord(X_c) -> (all F all G ((all X_2 hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less_eq(X_c),hAPP(X_b,X_c,F,X_2)),hAPP(X_b,X_c,G,X_2)))) <-> hBOOL(hAPP(fun(X_b,X_c),bool,hAPP(fun(X_b,X_c),fun(fun(X_b,X_c),bool),ord_less_eq(fun(X_b,X_c)),F),G)))))) # label(fact_317_le__fun__def) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 166 (all X_c all X_b all F all Z_1 all X_1 (hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),finite_fold_graph(X_b,X_c),F),Z_1),bot_bot(fun(X_b,bool))),X_1)) -> ti(X_c,X_1) = ti(X_c,Z_1))) # label(fact_120_empty__fold__graphE) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 167 (all X_b all A_3 hAPP(fun(X_b,bool),X_b,the(X_b),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),A_3)) = ti(X_b,A_3)) # label(fact_82_the__eq__trivial) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 168 (all X_3 all A_3 all S_4 all N_3 hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,hAPP(fun(state,nat),com,hAPP(vname,fun(fun(state,nat),com),ass,X_3),A_3)),S_4),N_3),hAPP(nat,state,hAPP(vname,fun(nat,state),hAPP(state,fun(vname,fun(nat,state)),update,S_4),X_3),hAPP(state,nat,A_3,S_4))))) # label(fact_113_evaln_OAssign) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 169 (all X_b all X_c all Y_1 all Z_1 all A_1 all X_1 all F (hBOOL(hAPP(fun(X_b,fun(X_c,X_c)),bool,finite100568337ommute(X_b,X_c),F)) -> (hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),finite_fold_graph(X_b,X_c),F),Z_1),A_1),X_1)) -> (hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),finite_fold_graph(X_b,X_c),F),Z_1),A_1),Y_1)) -> ti(X_c,X_1) = ti(X_c,Y_1))))) # label(fact_206_comp__fun__commute_Ofold__graph__determ) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 170 (all P (-hBOOL(hAPP(bool,bool,fNot,P)) | -hBOOL(P))) # label(help_fNot_1_1_U) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 171 (all X_b all A_1 all B hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),B),A_1)) # label(fact_341_Un__commute) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 172 (all X_b all Y_4 ti(fun(X_b,bool),Y_4) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,X_b),fun(fun(X_b,bool),fun(X_b,bool)),image(X_b,X_b),combi(X_b)),Y_4)) # label(fact_62_image__ident) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 173 (all X_a (preorder(X_a) -> (all Z all X all Y (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),Z)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Z))))))) # label(fact_306_order__trans) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 174 (all X_b (lattice(X_b) -> (all X_1 all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (bot_bot(fun(X_b,bool)) != hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),bot_bot(fun(X_b,bool)))) -> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),X_1),hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),bot_bot(fun(X_b,bool)))))) = hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),A_1))) & (hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),bot_bot(fun(X_b,bool)))) = bot_bot(fun(X_b,bool)) -> hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),A_1)) = ti(X_b,X_1)))))) # label(fact_408_Sup__fin_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 175 (all X_b all A_3 hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),bot_bot(fun(X_b,bool))) = hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(X_b,fun(X_b,bool),fequal(X_b),A_3))) # label(fact_12_singleton__conv2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 176 (all S_1 all T_4 (hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,skip),S_1),T_4)) -> S_1 = T_4)) # label(fact_112_evalc__elim__cases_I1_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 177 (all X_b all Ca all A_1 all B (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B))) -> (-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),A_1)) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),B))))) # label(fact_240_UnE) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 178 (all X_a (semilattice_inf(X_a) -> (all A_2 all B_1 all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_1),X)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),A_2),B_1)),X)))))) # label(fact_424_le__infI2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 179 (all X_c all X_b all Ca all X_1 all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_1)) -> hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),hAPP(X_c,fun(X_b,X_c),combk(X_c,X_b),Ca)),A_1) = hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_c,bool),fun(X_c,bool)),insert(X_c),Ca),bot_bot(fun(X_c,bool))))) # label(fact_60_image__constant) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 180 (all X_b all Pa all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (hBOOL(hAPP(fun(X_b,bool),bool,Pa,A_1)) -> ((all A_4 all A_5 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_5)) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_4),A_5)) -> (hBOOL(hAPP(fun(X_b,bool),bool,Pa,A_5)) -> hBOOL(hAPP(fun(X_b,bool),bool,Pa,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_5),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_4),bot_bot(fun(X_b,bool)))))))))) -> hBOOL(hAPP(fun(X_b,bool),bool,Pa,bot_bot(fun(X_b,bool)))))))) # label(fact_185_finite__empty__induct) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 181 (all X_b all A_1 all B all X_1 (hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B),X_1)) -> (-hBOOL(hAPP(X_b,bool,A_1,X_1)) -> hBOOL(hAPP(X_b,bool,B,X_1))))) # label(fact_242_sup1E) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 182 (all C1 all C2 all S_1 all T_4 (hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,hAPP(com,com,hAPP(com,fun(com,com),semi,C1),C2)),S_1),T_4)) -> -(all S1_1 (hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C1),S_1),S1_1)) -> -hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C2),S1_1),T_4)))))) # label(fact_125_evalc__elim__cases_I4_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 183 (all X_b all Pa (bot_bot(fun(X_b,bool)) = hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Pa) <-> (all X_2 -hBOOL(hAPP(X_b,bool,Pa,X_2))))) # label(fact_20_empty__Collect__eq) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 184 (all X_b all Ga all Pa all Ca all Q_1 all C ((hBOOL(C) -> hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Pa),Ca),Q_1)),bot_bot(fun(hoare_1656922687triple(X_b),bool)))))) -> hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),hAPP(bool,fun(X_b,fun(state,bool)),hAPP(fun(X_b,fun(bool,fun(state,bool))),fun(bool,fun(X_b,fun(state,bool))),combc(X_b,bool,fun(state,bool)),hAPP(fun(X_b,fun(state,fun(bool,bool))),fun(X_b,fun(bool,fun(state,bool))),hAPP(fun(fun(state,fun(bool,bool)),fun(bool,fun(state,bool))),fun(fun(X_b,fun(state,fun(bool,bool))),fun(X_b,fun(bool,fun(state,bool)))),combb(fun(state,fun(bool,bool)),fun(bool,fun(state,bool)),X_b),combc(state,bool,bool)),hAPP(fun(X_b,fun(state,bool)),fun(X_b,fun(state,fun(bool,bool))),hAPP(fun(fun(state,bool),fun(state,fun(bool,bool))),fun(fun(X_b,fun(state,bool)),fun(X_b,fun(state,fun(bool,bool)))),combb(fun(state,bool),fun(state,fun(bool,bool)),X_b),hAPP(fun(bool,fun(bool,bool)),fun(fun(state,bool),fun(state,fun(bool,bool))),combb(bool,fun(bool,bool),state),fconj)),Pa))),C)),Ca),Q_1)),bot_bot(fun(hoare_1656922687triple(X_b),bool))))))) # label(fact_4_constant) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 185 (all X_a (ord(X_a) -> (all C_1 all A_2 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),B_1)) -> (B_1 = C_1 -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),C_1))))))) # label(fact_309_ord__le__eq__trans) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 186 (all X_b all B all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool))),finite_fold(X_b,fun(X_b,bool)),insert(X_b)),B),A_1))) # label(fact_365_union__fold__insert) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 187 (all X_b all A_1 hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),bot_bot(fun(X_b,bool))) = bot_bot(fun(X_b,bool))) # label(fact_466_Int__empty__right) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 188 (all K hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less_eq(nat)),K))))) # label(fact_381_finite__Collect__le__nat) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 189 (all X_b all A_1 all B all X_1 (hBOOL(hAPP(X_b,bool,B,X_1)) -> hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B),X_1)))) # label(fact_349_sup1I2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 190 (all X_a all X_c all X_b all P all Q all R hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),P,R),Q) = hAPP(X_a,X_c,hAPP(X_b,fun(X_a,X_c),hAPP(fun(X_a,fun(X_b,X_c)),fun(X_b,fun(X_a,X_c)),combc(X_a,X_b,X_c),P),Q),R)) # label(help_COMBC_1_1_U) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 191 (all X_a (semilattice_inf(X_a) -> (all X all Y (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X),Y) = ti(X_a,X))))) # label(fact_423_inf__absorb1) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 192 (all X_b all X_c all F all A_1 all B hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),A_1)),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),B))),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(fun(X_c,bool),fun(fun(X_c,bool),fun(X_c,bool)),minus_minus(fun(X_c,bool)),A_1),B))))) # label(fact_368_image__diff__subset) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 193 (all S_1 all N_2 hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,skip),S_1),N_2),S_1))) # label(fact_108_evaln_OSkip) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 194 (all X_b all Pa ti(fun(X_b,bool),Pa) = hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Pa)) # label(fact_77_Collect__def) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 195 (all X_b all Y_1 -(all Fun1 all Com_2 all Fun2 hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Fun1),Com_2),Fun2) != Y_1)) # label(fact_47_triple_Oexhaust) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 196 (all Loc_1 all Glb_1 hAPP(glb_1,vname,glb,Glb_1) != hAPP(loc_1,vname,loc,Loc_1)) # label(fact_146_vname_Osimps_I4_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 197 (all X_b all Q_1 all Ga all Ca all Pa ((all Z_2 all S_2 (hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),Pa,Z_2),S_2)) -> (exists P_1 exists Q_2 (hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),P_1),Ca),Q_2)),bot_bot(fun(hoare_1656922687triple(X_b),bool))))) & (all S_3 ((all Z_3 (hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),P_1,Z_3),S_2)) -> hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),Q_2,Z_3),S_3)))) -> hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),Q_1,Z_2),S_3)))))))) -> hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Pa),Ca),Q_1)),bot_bot(fun(hoare_1656922687triple(X_b),bool))))))) # label(fact_51_conseq) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 198 (all X_b all Ga all G_1 all Ts (hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),G_1),Ts)) -> (hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),G_1)) -> hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),Ts))))) # label(fact_2_cut) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 199 (all X_b all X_c (ord(X_c) -> (all X_1 all F all G (hBOOL(hAPP(fun(X_b,X_c),bool,hAPP(fun(X_b,X_c),fun(fun(X_b,X_c),bool),ord_less_eq(fun(X_b,X_c)),F),G)) -> hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less_eq(X_c),hAPP(X_b,X_c,F,X_1)),hAPP(X_b,X_c,G,X_1))))))) # label(fact_313_le__funD) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 200 (all X_a (semilattice_sup(X_a) -> (all Y all X hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X),Y)))))) # label(fact_290_sup__ge2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 201 (all X_c all X_b all F all Z_1 hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),finite_fold_graph(X_b,X_c),F),Z_1),bot_bot(fun(X_b,bool))),Z_1))) # label(fact_121_fold__graph_OemptyI) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 202 (all X_a (semilattice_inf(X_a) -> (all A_2 all B_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),A_2),B_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),A_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),A_2),B_1))))) # label(fact_476_inf_Oleft__idem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 203 (all X_b all X_c all X_1 all Y_1 all Z_1 all F (hBOOL(hAPP(fun(X_b,fun(X_c,X_c)),bool,finite100568337ommute(X_b,X_c),F)) -> hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_1),hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,Y_1),Z_1)) = hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,Y_1),hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_1),Z_1)))) # label(fact_200_comp__fun__commute_Ofun__left__comm) # label(axiom) # label(non_clause). [assumption]. 2.22/2.60 204 (all X_b all A_1 all B hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),B) = hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),A_1))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),combb(bool,bool,X_b),fNot),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),B))))) # label(fact_168_set__diff__eq) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 205 (all X_b all X_c all A_3 all Z_1 all A_1 all Y_1 all F (hBOOL(hAPP(fun(X_b,fun(X_c,X_c)),bool,finite100568337ommute(X_b,X_c),F)) -> (hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),finite_fold_graph(X_b,X_c),F),Z_1),A_1),Y_1)) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_3),A_1)) -> (exists Y_3 (ti(X_c,Y_1) = hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,A_3),Y_3) & hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),finite_fold_graph(X_b,X_c),F),Z_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),bot_bot(fun(X_b,bool))))),Y_3)))))))) # label(fact_194_comp__fun__commute_Ofold__graph__insertE__aux) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 206 (all X_b all A_1 all B (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),B)) -> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),B),A_1)) = ti(fun(X_b,bool),B))) # label(fact_262_Diff__partition) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 207 (all X_b all X_c ti(fun(fun(X_b,fun(X_c,X_c)),bool),finite100568337ommute(X_b,X_c)) = finite100568337ommute(X_b,X_c)) # label(tsy_c_Finite__Set_Ocomp__fun__commute_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 208 (all X_b all A_1 (ti(fun(X_b,bool),A_1) = bot_bot(fun(X_b,bool)) <-> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),bot_bot(fun(X_b,bool)))))) # label(fact_246_subset__empty) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 209 (all X_b all B all C all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),C),A_1)) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),C),B)) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),C),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B)))))) # label(fact_433_Int__greatest) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 210 (all X_b all B hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),bot_bot(fun(X_b,bool))),B) = ti(fun(X_b,bool),B)) # label(fact_297_Un__empty__left) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 211 (all X_b all A_1 all B all C hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),B)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),C)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),B),C))) # label(fact_491_Diff__Un) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 212 (all X_b all B all A_3 hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),B)))) # label(fact_252_subset__insertI) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 213 (all X_a (bounded_lattice_bot(X_a) -> (all X hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),bot_bot(X_a)),X) = bot_bot(X_a)))) # label(fact_468_inf__bot__left) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 214 (all X_b (ab_semigroup_mult(X_b) -> (all Z_1 all Ba all A_1 all Y_1 (hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),finite_fold_graph(X_b,X_b),times_times(X_b)),Ba),A_1),Y_1)) -> (-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ba),A_1)) -> hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),finite_fold_graph(X_b,X_b),times_times(X_b)),Z_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Ba),A_1)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),Z_1),Y_1)))))))) # label(fact_207_fold__graph__insert__swap) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 215 (all X_a (lattice(X_a) -> (all X all Y all Z hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X),Z))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),Y),Z))))))) # label(fact_439_distrib__inf__le) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 216 (all X_b all F1 all F2 all Loc_2 hAPP(loc_1,X_b,F2,Loc_2) = hAPP(vname,X_b,hAPP(fun(loc_1,X_b),fun(vname,X_b),hAPP(fun(glb_1,X_b),fun(fun(loc_1,X_b),fun(vname,X_b)),vname_rec(X_b),F1),F2),hAPP(loc_1,vname,loc,Loc_2))) # label(fact_102_vname_Orecs_I2_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 217 (all X_b (ab_sem1668676832m_mult(X_b) -> hBOOL(hAPP(fun(X_b,fun(X_b,X_b)),bool,finite_comp_fun_idem(X_b,X_b),times_times(X_b))))) # label(fact_203_comp__fun__idem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 218 (all X_b all B all A_1 all F all F_1 (hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite2073411215e_idem(X_b),F),F_1)) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (ti(fun(X_b,bool),A_1) != bot_bot(fun(X_b,bool)) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B)) -> (bot_bot(fun(X_b,bool)) != ti(fun(X_b,bool),B) -> hAPP(fun(X_b,bool),X_b,F_1,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(fun(X_b,bool),X_b,F_1,A_1)),hAPP(fun(X_b,bool),X_b,F_1,B)))))))) # label(fact_373_folding__one__idem_Ounion__idem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 219 (all X_b all B all A_3 all C (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_3),C)) -> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),B),C)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),B)),C))) # label(fact_444_Int__insert__left__if1) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 220 (all X_b all A_1 all B all X_1 (hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B),X_1)) -> -(hBOOL(hAPP(X_b,bool,A_1,X_1)) -> -hBOOL(hAPP(X_b,bool,B,X_1))))) # label(fact_414_inf1E) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 221 (all X_b all X_1 all A_1 all Pa (hBOOL(hAPP(X_b,bool,Pa,X_1)) & hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_1)) <-> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Pa)))))) # label(fact_455_Int__Collect) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 222 (all X_b all Pa all A_3 ((-hBOOL(hAPP(X_b,bool,Pa,A_3)) -> bot_bot(fun(X_b,bool)) = hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),A_3))),Pa))) & (hBOOL(hAPP(X_b,bool,Pa,A_3)) -> hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),A_3))),Pa)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),bot_bot(fun(X_b,bool)))))) # label(fact_15_Collect__conv__if) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 223 (all X_c all X_b ti(fun(fun(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),X_b))),fun(hoare_1656922687triple(X_c),X_b)),hoare_1312322281e_case(X_c,X_b)) = hoare_1312322281e_case(X_c,X_b)) # label(tsy_c_Hoare__Mirabelle__nrugjuseim_Otriple_Otriple__case_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 224 (all X_b all F1 all F2 all Glb_3 hAPP(vname,X_b,hAPP(fun(loc_1,X_b),fun(vname,X_b),hAPP(fun(glb_1,X_b),fun(fun(loc_1,X_b),fun(vname,X_b)),vname_rec(X_b),F1),F2),hAPP(glb_1,vname,glb,Glb_3)) = hAPP(glb_1,X_b,F1,Glb_3)) # label(fact_133_vname_Orecs_I1_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 225 (all X_c all X_b (bot(X_b) -> (all X_1 hAPP(X_c,X_b,bot_bot(fun(X_c,X_b)),X_1) = bot_bot(X_b)))) # label(fact_43_bot__apply) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 226 (all X_a (ab_sem1668676832m_mult(X_a) -> (all X hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),X) = ti(X_a,X)))) # label(fact_198_mult__idem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 227 (all X_b all X_c ti(fun(fun(X_b,fun(X_c,X_c)),bool),finite_comp_fun_idem(X_b,X_c)) = finite_comp_fun_idem(X_b,X_c)) # label(tsy_c_Finite__Set_Ocomp__fun__idem_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 228 (all X_b hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),bot_bot(fun(X_b,bool))))) # label(fact_138_finite_OemptyI) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 229 (all X_b all A_3 all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),A_1))) <-> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)))) # label(fact_145_finite__insert) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 230 (all X_b all A_1 all B (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),B)) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B)) -> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1))))) # label(fact_248_finite__subset) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 231 (all X_b all X_1 all F all F_1 (hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite_folding_one(X_b),F),F_1)) -> ti(X_b,X_1) = hAPP(fun(X_b,bool),X_b,F_1,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),bot_bot(fun(X_b,bool)))))) # label(fact_87_folding__one_Osingleton) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 232 (all X_b all Q_1 all Pa all X_1 (hBOOL(hAPP(X_b,bool,Pa,X_1)) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),Pa),Q_1)) -> hBOOL(hAPP(X_b,bool,Q_1,X_1))))) # label(fact_348_rev__predicate1D) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 233 (all X_a (lattice(X_a) -> (all X all Y hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),Y),X) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X),Y)))) # label(fact_285_inf__sup__aci_I5_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 234 (all X_b all X_c all X_1 all Z_1 all F (hBOOL(hAPP(fun(X_b,fun(X_c,X_c)),bool,finite_comp_fun_idem(X_b,X_c),F)) -> hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_1),hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_1),Z_1)) = hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_1),Z_1))) # label(fact_201_comp__fun__idem_Ofun__left__idem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 235 (all X_b all A_1 all B all C hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),C)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),B),C)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),B)),C)) # label(fact_487_Diff__Int__distrib2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 236 (all X_a (lattice(X_a) -> (all X all Y hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X),Y) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X),Y))))) # label(fact_477_inf__sup__aci_I4_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 237 (all X_b all X_c all Z_1 all X_1 all A_1 all F (hBOOL(hAPP(fun(X_b,fun(X_c,X_c)),bool,finite100568337ommute(X_b,X_c),F)) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_1)) -> hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_1),hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c)),finite_fold(X_b,X_c),F),Z_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),bot_bot(fun(X_b,bool)))))) = hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c)),finite_fold(X_b,X_c),F),Z_1),A_1))))) # label(fact_209_comp__fun__commute_Ofold__rec) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 238 (all Glb_3 all Glb_2 (ti(glb_1,Glb_3) = ti(glb_1,Glb_2) <-> hAPP(glb_1,vname,glb,Glb_2) = hAPP(glb_1,vname,glb,Glb_3))) # label(fact_142_vname_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 239 (all X_a all X_c all X_b all P all Q all R hAPP(X_b,X_c,P,hAPP(X_a,X_b,Q,R)) = hAPP(X_a,X_c,hAPP(fun(X_a,X_b),fun(X_a,X_c),hAPP(fun(X_b,X_c),fun(fun(X_a,X_b),fun(X_a,X_c)),combb(X_b,X_c,X_a),P),Q),R)) # label(help_COMBB_1_1_U) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 240 (all X_b all X_1 all A_1 all B (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),B)) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_1)) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),B))))) # label(fact_326_in__mono) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 241 (all X_b all X_c fold_graph(X_b,X_c) = ti(fun(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool)))),fold_graph(X_b,X_c))) # label(tsy_c_Nitpick_Ofold__graph_H_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 242 (all S_1 hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,skip),S_1),S_1))) # label(fact_111_evalc_OSkip) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 243 (all X_b all X_c (ord(X_c) -> (all F all G ((all X_2 hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less_eq(X_c),hAPP(X_b,X_c,F,X_2)),hAPP(X_b,X_c,G,X_2)))) -> hBOOL(hAPP(fun(X_b,X_c),bool,hAPP(fun(X_b,X_c),fun(fun(X_b,X_c),bool),ord_less_eq(fun(X_b,X_c)),F),G)))))) # label(fact_380_le__funI) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 244 (all X_b all A_1 all B hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),B),A_1)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B)) # label(fact_249_Un__Diff__cancel) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 245 (all S_1 all N_2 all T_4 (hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,skip),S_1),N_2),T_4)) -> S_1 = T_4)) # label(fact_109_evaln__elim__cases_I1_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 246 (all X_b all X_1 all Y_1 all A_1 hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Y_1),A_1)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Y_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),A_1))) # label(fact_29_insert__commute) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 247 (all X_b all A_1 all F all F_1 (hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite_folding_one(X_b),F),F_1)) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),F),A_1) = hAPP(fun(X_b,bool),X_b,F_1,A_1)))) # label(fact_221_folding__one_Oeq__fold) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 248 (all X_b all X_1 all A_1 hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),A_1)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),A_1)) # label(fact_30_insert__absorb2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 249 (all X_b all B all A_1 all C (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),C)) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B),C)) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B)),C))))) # label(fact_322_Un__least) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 250 (all X_b all A_1 all A_3 all B hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),B))) # label(fact_357_Un__insert__right) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 251 (all X_b all Ca all A_1 all B (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B))) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),A_1)))) # label(fact_463_IntD1) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 252 (all X_b all F all A_3 all X_3 all X_1 (hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),fun(X_b,bool)),finite_fold1Set(X_b),F),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),X_3)),X_1)) -> -(all A_4 all A_5 (hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_4),A_5) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),X_3) -> (hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),finite_fold_graph(X_b,X_b),F),A_4),A_5),X_1)) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_4),A_5))))))) # label(fact_127_insert__fold1SetE) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 253 (all X_b all A_1 all B all C hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B)),C) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),C)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),B),C))) # label(fact_251_Un__Diff) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 254 (all X_b all A_3 all Pa hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fimplies),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),combb(bool,bool,X_b),fNot),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),A_3)))),Pa)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Pa))) # label(fact_31_insert__Collect) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 255 (all X_b all Ts all Ga all T_5 (hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),T_5),bot_bot(fun(hoare_1656922687triple(X_b),bool))))) -> (hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),Ts)) -> hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),T_5),Ts)))))) # label(fact_3_hoare__derivs_Oinsert) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 256 (all X_b all A_1 all B (ti(fun(X_b,bool),A_1) = bot_bot(fun(X_b,bool)) & ti(fun(X_b,bool),B) = bot_bot(fun(X_b,bool)) <-> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B) = bot_bot(fun(X_b,bool)))) # label(fact_299_Un__empty) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 257 (all X_a (semilattice_inf(X_a) -> (all A_2 all B_1 all C_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),A_2),B_1)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),A_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),B_1),C_1))))) # label(fact_482_inf_Oassoc) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 258 (all X_b all B all A_3 all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_3),A_1)) -> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),B)))) # label(fact_445_Int__insert__right__if1) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 259 (all X_a (lattice(X_a) -> (all X all Y all Z hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),Y),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),Y),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X),Z))))) # label(fact_480_inf__sup__aci_I3_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 260 (all X_a (bounded_lattice_bot(X_a) -> (all X bot_bot(X_a) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X),bot_bot(X_a))))) # label(fact_469_inf__bot__right) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 261 (all X_b all B all A_1 hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),B),A_1) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),B),A_1)),A_1)) # label(fact_250_Un__Diff__cancel2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 262 (all X_c all X_b all G all A_1 all F all Z_1 all F_1 (hBOOL(hAPP(fun(fun(X_c,X_b),fun(fun(X_c,bool),X_b)),bool,hAPP(X_b,fun(fun(fun(X_c,X_b),fun(fun(X_c,bool),X_b)),bool),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(fun(X_c,X_b),fun(fun(X_c,bool),X_b)),bool)),big_comm_monoid_big(X_b,X_c),F),Z_1),F_1)) -> (-hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_1)) -> ti(X_b,Z_1) = hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),F_1,G),A_1)))) # label(fact_400_comm__monoid__big_Oinfinite) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 263 (all X_b (ordered_ab_group_add(X_b) -> (all A_3 all Ba all Ca all D_2 (hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),Ca),D_2) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),A_3),Ba) -> (hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Ca),D_2)) <-> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),A_3),Ba))))))) # label(fact_382_diff__eq__diff__less__eq) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 264 (all X_b all Ca all A_1 all B (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B))) <-> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),A_1)) & hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),B)))) # label(fact_461_Int__iff) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 265 (all X_a all X_c all B_1_1 all B_2_1 hAPP(X_a,X_c,ti(fun(X_a,X_c),B_1_1),B_2_1) = hAPP(X_a,X_c,B_1_1,B_2_1)) # label(tsy_c_hAPP_arg1) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 266 (all X_b all A_1 all B all C hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),B)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),C)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),B),C))) # label(fact_492_Diff__Int) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 267 (all X_a all X_b combk(X_a,X_b) = ti(fun(X_a,fun(X_b,X_a)),combk(X_a,X_b))) # label(tsy_c_COMBK_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 268 (all X_b all X_1 all A_1 all F all F_1 (hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite_folding_one(X_b),F),F_1)) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (bot_bot(fun(X_b,bool)) != hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),bot_bot(fun(X_b,bool)))) -> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,X_1),hAPP(fun(X_b,bool),X_b,F_1,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),bot_bot(fun(X_b,bool)))))) = hAPP(fun(X_b,bool),X_b,F_1,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),A_1))) & (hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),bot_bot(fun(X_b,bool)))) = bot_bot(fun(X_b,bool)) -> ti(X_b,X_1) = hAPP(fun(X_b,bool),X_b,F_1,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),A_1)))))) # label(fact_159_folding__one_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 269 (all X_b (semilattice_inf(X_b) -> (all X_1 all Y_1 all Z_1 (hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),Z_1)) & hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),Y_1)) <-> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_inf_inf(X_b),Y_1),Z_1))))))) # label(fact_426_le__inf__iff) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 270 (all C_1 all S_1 all T_4 (hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C_1),S_1),T_4)) -> (exists N_1 hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C_1),S_1),N_1),T_4))))) # label(fact_129_evalc__evaln) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 271 (all X_b (order(X_b) -> (all X_1 all Y_1 (hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),Y_1)) & hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Y_1),X_1)) <-> ti(X_b,Y_1) = ti(X_b,X_1))))) # label(fact_315_order__eq__iff) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 272 (all X_b all Ca all A_1 all B (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),B))) -> -(hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),A_1)) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),B))))) # label(fact_160_DiffE) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 273 (all X_b all A_1 ((exists X_2 exists B_2 (-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),B_2)) & ti(fun(X_b,bool),A_1) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),B_2))) <-> ti(fun(X_b,bool),A_1) != bot_bot(fun(X_b,bool)))) # label(fact_56_nonempty__iff) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 274 (all X_c all X_b (minus(X_b) -> (all A_1 all B all X_1 hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(X_c,X_b,A_1,X_1)),hAPP(X_c,X_b,B,X_1)) = hAPP(X_c,X_b,hAPP(fun(X_c,X_b),fun(X_c,X_b),hAPP(fun(X_c,X_b),fun(fun(X_c,X_b),fun(X_c,X_b)),minus_minus(fun(X_c,X_b)),A_1),B),X_1)))) # label(fact_189_minus__apply) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 275 (all X_b all F all A_3 all A_1 all X_1 (hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),finite_fold_graph(X_b,X_b),F),A_3),A_1),X_1)) -> (-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_3),A_1)) -> hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),fun(X_b,bool)),finite_fold1Set(X_b),F),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),A_1)),X_1))))) # label(fact_106_fold1Set_Ointros) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 276 (all X_c all X_b all X_1 all A_1 all F all Z_1 all G all F_1 (hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite908156982e_idem(X_b,X_c),F),Z_1),G),F_1)) -> (hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_1)) -> (hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),X_1),A_1)) -> hAPP(fun(X_c,bool),X_b,F_1,A_1) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(X_c,X_b,G,X_1)),hAPP(fun(X_c,bool),X_b,F_1,A_1)))))) # label(fact_183_folding__image__simple__idem_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 277 (all Q all P (-hBOOL(P) | -hBOOL(Q) | hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,P),Q)))) # label(help_fconj_1_1_U) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 278 (all X_b all X_c all Z_1 all X_1 all A_1 all F (hBOOL(hAPP(fun(X_b,fun(X_c,X_c)),bool,finite100568337ommute(X_b,X_c),F)) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_1),hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c)),finite_fold(X_b,X_c),F),Z_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),bot_bot(fun(X_b,bool)))))) = hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c)),finite_fold(X_b,X_c),F),Z_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),A_1))))) # label(fact_210_comp__fun__commute_Ofold__insert__remove) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 279 (all X_b (order(X_b) -> (all Y_1 all X_1 (hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Y_1),X_1)) -> (ti(X_b,Y_1) = ti(X_b,X_1) <-> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),Y_1))))))) # label(fact_312_order__antisym__conv) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 280 (all X_a (lattice(X_a) -> (all X all Y hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X),Y)),X))))) # label(fact_431_inf__sup__ord_I1_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 281 (all X_b all A_3 all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_3),A_1)) -> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),A_1) = ti(fun(X_b,bool),A_1))) # label(fact_24_insert__absorb) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 282 (all X_b all A_3 all A_1 hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),bot_bot(fun(X_b,bool))))) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),A_1)) # label(fact_178_insert__Diff__single) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 283 (all X_c all X_b (comm_monoid_mult(X_b) -> (all H all G all S all R_1 all E (hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),R_1,E),E)) -> ((all X1 all Y1 all X2 all Y2 (hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),R_1,Y1),Y2)) & hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),R_1,X1),X2)) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),R_1,hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),X1),Y1)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),X2),Y2))))) -> (hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),S)) -> ((all X_2 (hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),X_2),S)) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),R_1,hAPP(X_c,X_b,H,X_2)),hAPP(X_c,X_b,G,X_2))))) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),R_1,hAPP(fun(X_c,bool),X_b,hAPP(X_b,fun(fun(X_c,bool),X_b),hAPP(fun(X_c,X_b),fun(X_b,fun(fun(X_c,bool),X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_c,X_b),fun(X_b,fun(fun(X_c,bool),X_b))),finite_fold_image(X_b,X_c),times_times(X_b)),H),E),S)),hAPP(fun(X_c,bool),X_b,hAPP(X_b,fun(fun(X_c,bool),X_b),hAPP(fun(X_c,X_b),fun(X_b,fun(fun(X_c,bool),X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_c,X_b),fun(X_b,fun(fun(X_c,bool),X_b))),finite_fold_image(X_b,X_c),times_times(X_b)),G),E),S)))))))))) # label(fact_395_fold__image__related) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 284 (all X_b all A_1 all B (ti(fun(X_b,bool),B) = ti(fun(X_b,bool),A_1) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),B)))) # label(fact_332_equalityD1) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 285 (all X_b all X_c finite908156982e_idem(X_b,X_c) = ti(fun(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)))),finite908156982e_idem(X_b,X_c))) # label(tsy_c_Finite__Set_Ofolding__image__simple__idem_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 286 (all Vname all Fun_1 all Loc_3 all Fun all Com hAPP(fun(state,nat),com,hAPP(vname,fun(fun(state,nat),com),ass,Vname),Fun_1) != hAPP(com,com,hAPP(fun(state,nat),fun(com,com),hAPP(loc_1,fun(fun(state,nat),fun(com,com)),local,Loc_3),Fun),Com)) # label(fact_94_com_Osimps_I22_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 287 (all X_c all X_b all F all G hAPP(fun(X_c,X_b),fun(X_b,fun(fun(X_c,bool),X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_c,X_b),fun(X_b,fun(fun(X_c,bool),X_b))),finite_fold_image(X_b,X_c),F),G) = hAPP(fun(X_c,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,bool),X_b)),finite_fold(X_c,X_b),hAPP(fun(X_c,X_b),fun(X_c,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_c,X_b),fun(X_c,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_c),F),G))) # label(fact_388_fold__image__def) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 288 (all X_b all X_c (lattice(X_c) -> (all F all G all X_2 hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),semilattice_sup_sup(X_c),hAPP(X_b,X_c,F,X_2)),hAPP(X_b,X_c,G,X_2)) = hAPP(X_b,X_c,hAPP(fun(X_b,X_c),fun(X_b,X_c),hAPP(fun(X_b,X_c),fun(fun(X_b,X_c),fun(X_b,X_c)),semilattice_sup_sup(fun(X_b,X_c)),F),G),X_2)))) # label(fact_287_sup__fun__def) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 289 (all X_b all R_1 all S (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),R_1)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),S))) <-> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),R_1),S)))) # label(fact_351_pred__subset__eq) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 290 (all X_a (preorder(X_a) -> (all X hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),X))))) # label(fact_236_order__refl) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 291 (all X_c all X_b all H all F_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),F_1)) -> hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),H),F_1))))) # label(fact_140_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 292 (all X_b all B all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),B))))) # label(fact_162_finite__Diff) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 293 (all X_c all X_b all B all F all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (hBOOL(hAPP(fun(X_c,bool),bool,hAPP(fun(X_c,bool),fun(fun(X_c,bool),bool),ord_less_eq(fun(X_c,bool)),B),hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),A_1))) -> hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),B))))) # label(fact_367_finite__surj) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 294 (all X_a (lattice(X_a) -> (all X hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X),X) = ti(X_a,X)))) # label(fact_442_Inf__fin_Oidem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 295 (all X_b (ab_semigroup_mult(X_b) -> (all X_1 all A_1 (bot_bot(fun(X_b,bool)) != ti(fun(X_b,bool),A_1) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_1)) -> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),X_1),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),A_1)) = hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),A_1)))))))) # label(fact_211_fold1__insert) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 296 (all X_b all X_c big_comm_monoid_big(X_b,X_c) = ti(fun(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(fun(X_c,X_b),fun(fun(X_c,bool),X_b)),bool))),big_comm_monoid_big(X_b,X_c))) # label(tsy_c_Big__Operators_Ocomm__monoid__big_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 297 (all X_c all X_b all B all A_1 all F all Z_1 all G all F_1 (hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite908156982e_idem(X_b,X_c),F),Z_1),G),F_1)) -> (hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_1)) -> (hBOOL(hAPP(fun(X_c,bool),bool,hAPP(fun(X_c,bool),fun(fun(X_c,bool),bool),ord_less_eq(fun(X_c,bool)),B),A_1)) -> hAPP(fun(X_c,bool),X_b,F_1,A_1) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(fun(X_c,bool),X_b,F_1,B)),hAPP(fun(X_c,bool),X_b,F_1,A_1)))))) # label(fact_370_folding__image__simple__idem_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 298 (all Vname_1 all Fun skip != hAPP(fun(state,nat),com,hAPP(vname,fun(fun(state,nat),com),ass,Vname_1),Fun)) # label(fact_71_com_Osimps_I9_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 299 (all Loc_3 all Fun all Com hAPP(com,com,hAPP(fun(state,nat),fun(com,com),hAPP(loc_1,fun(fun(state,nat),fun(com,com)),local,Loc_3),Fun),Com) != skip) # label(fact_96_com_Osimps_I10_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 300 (all X_b all A_1 all B hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),B)),A_1))) # label(fact_259_Diff__subset) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 301 (all X_c all X_b all F1 all Fun1_1 all Com_1 all Fun2_1 hAPP(hoare_1656922687triple(X_c),X_b,hAPP(fun(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),X_b))),fun(hoare_1656922687triple(X_c),X_b),hoare_1312322281e_case(X_c,X_b),F1),hAPP(fun(X_c,fun(state,bool)),hoare_1656922687triple(X_c),hAPP(com,fun(fun(X_c,fun(state,bool)),hoare_1656922687triple(X_c)),hAPP(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),hoare_1656922687triple(X_c))),hoare_246368825triple(X_c),Fun1_1),Com_1),Fun2_1)) = hAPP(fun(X_c,fun(state,bool)),X_b,hAPP(com,fun(fun(X_c,fun(state,bool)),X_b),hAPP(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),X_b)),F1,Fun1_1),Com_1),Fun2_1)) # label(fact_42_triple_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 302 (all X_c all X_b all X_1 all A_1 all F all Z_1 all G all F_1 (hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite1357897459simple(X_b,X_c),F),Z_1),G),F_1)) -> (hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_1)) -> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(X_c,X_b,G,X_1)),hAPP(fun(X_c,bool),X_b,F_1,hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(fun(X_c,bool),fun(fun(X_c,bool),fun(X_c,bool)),minus_minus(fun(X_c,bool)),A_1),hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_c,bool),fun(X_c,bool)),insert(X_c),X_1),bot_bot(fun(X_c,bool)))))) = hAPP(fun(X_c,bool),X_b,F_1,hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_c,bool),fun(X_c,bool)),insert(X_c),X_1),A_1))))) # label(fact_187_folding__image__simple_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 303 (all X_a (semilattice_sup(X_a) -> (all Z all Y all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Z),X)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),Y),Z)),X))))))) # label(fact_266_sup__least) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 304 (all X_c all X_b all G all A_1 all F all Z_1 all F_1 (hBOOL(hAPP(fun(fun(X_c,X_b),fun(fun(X_c,bool),X_b)),bool,hAPP(X_b,fun(fun(fun(X_c,X_b),fun(fun(X_c,bool),X_b)),bool),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(fun(X_c,X_b),fun(fun(X_c,bool),X_b)),bool)),big_comm_monoid_big(X_b,X_c),F),Z_1),F_1)) -> (-hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_1)) -> hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),F_1,G),A_1) = ti(X_b,Z_1)) & (hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_1)) -> hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),F_1,G),A_1) = hAPP(fun(X_c,bool),X_b,hAPP(X_b,fun(fun(X_c,bool),X_b),hAPP(fun(X_c,X_b),fun(X_b,fun(fun(X_c,bool),X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_c,X_b),fun(X_b,fun(fun(X_c,bool),X_b))),finite_fold_image(X_b,X_c),F),G),Z_1),A_1)))) # label(fact_396_comm__monoid__big_OF__eq) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 305 (all X_b all A_3 all Ba all Ca all D_2 (hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Ca),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),D_2),bot_bot(fun(X_b,bool)))) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Ba),bot_bot(fun(X_b,bool)))) <-> ti(X_b,Ba) = ti(X_b,D_2) & ti(X_b,A_3) = ti(X_b,Ca) | ti(X_b,D_2) = ti(X_b,A_3) & ti(X_b,Ca) = ti(X_b,Ba))) # label(fact_37_doubleton__eq__iff) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 306 (all X_b all A_1 all A_3 all B hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),B)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),bot_bot(fun(X_b,bool)))) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),B))) # label(fact_180_Diff__insert) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 307 (all X_b the_elem(X_b) = ti(fun(fun(X_b,bool),X_b),the_elem(X_b))) # label(tsy_c_Set_Othe__elem_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 308 (all X_c all X_b all F all G all Z_1 hAPP(fun(X_c,bool),X_b,hAPP(X_b,fun(fun(X_c,bool),X_b),hAPP(fun(X_c,X_b),fun(X_b,fun(fun(X_c,bool),X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_c,X_b),fun(X_b,fun(fun(X_c,bool),X_b))),finite_fold_image(X_b,X_c),F),G),Z_1),bot_bot(fun(X_c,bool))) = ti(X_b,Z_1)) # label(fact_389_fold__image__empty) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 309 (all X_c all X_b all Ba all F all X_1 all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_1)) -> (ti(X_c,Ba) = hAPP(X_b,X_c,F,X_1) -> hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),Ba),hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),A_1)))))) # label(fact_65_rev__image__eqI) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 310 (all X_b all F all A_3 all Ba (hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),fun(X_b,bool)),finite_fold1Set(X_b),F),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),bot_bot(fun(X_b,bool)))),Ba)) <-> ti(X_b,A_3) = ti(X_b,Ba))) # label(fact_85_fold1Set__sing) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 311 (all X_b (ab_semigroup_mult(X_b) -> hBOOL(hAPP(fun(X_b,fun(X_b,X_b)),bool,finite100568337ommute(X_b,X_b),times_times(X_b))))) # label(fact_202_comp__fun__commute) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 312 (all X_b (lattice(X_b) -> (all A_3 all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),A_1)) = hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),X_b)),finite_fold(X_b,X_b),semilattice_sup_sup(X_b)),A_3),A_1))))) # label(fact_406_Sup__fin_Oeq__fold__idem_H) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 313 (all Y_4 all A_3 all Ca all S_4 all T_5 (hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,hAPP(com,com,hAPP(fun(state,nat),fun(com,com),hAPP(loc_1,fun(fun(state,nat),fun(com,com)),local,Y_4),A_3),Ca)),S_4),T_5)) -> -(all S1_1 (hAPP(nat,state,hAPP(vname,fun(nat,state),hAPP(state,fun(vname,fun(nat,state)),update,S1_1),hAPP(loc_1,vname,loc,Y_4)),hAPP(loc_1,nat,hAPP(state,fun(loc_1,nat),getlocs,S_4),Y_4)) = T_5 -> -hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,Ca),hAPP(nat,state,hAPP(vname,fun(nat,state),hAPP(state,fun(vname,fun(nat,state)),update,S_4),hAPP(loc_1,vname,loc,Y_4)),hAPP(state,nat,A_3,S_4))),S1_1)))))) # label(fact_123_evalc__elim__cases_I3_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 314 (all X_b all A_1 all B (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B)) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),B)) -> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1))))) # label(fact_247_rev__finite__subset) # label(axiom) # label(non_clause). [assumption]. 2.22/2.61 315 (all X_a (semilattice_inf(X_a) -> (all X all Y all Z hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),Y),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X),Y)),Z)))) # label(fact_484_inf__assoc) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 316 (all X_a (order(X_a) -> (all C_1 all A_2 all B_1 (ti(X_a,B_1) = ti(X_a,A_2) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),B_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),A_2))))))) # label(fact_310_xt1_I3_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 317 (all X_b all Ga all Pa all X_3 all A_3 hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),hAPP(fun(state,state),fun(X_b,fun(state,bool)),hAPP(fun(X_b,fun(fun(state,state),fun(state,bool))),fun(fun(state,state),fun(X_b,fun(state,bool))),combc(X_b,fun(state,state),fun(state,bool)),hAPP(fun(X_b,fun(state,bool)),fun(X_b,fun(fun(state,state),fun(state,bool))),hAPP(fun(fun(state,bool),fun(fun(state,state),fun(state,bool))),fun(fun(X_b,fun(state,bool)),fun(X_b,fun(fun(state,state),fun(state,bool)))),combb(fun(state,bool),fun(fun(state,state),fun(state,bool)),X_b),combb(state,bool,state)),Pa)),hAPP(fun(state,nat),fun(state,state),hAPP(fun(state,fun(nat,state)),fun(fun(state,nat),fun(state,state)),combs(state,nat,state),hAPP(vname,fun(state,fun(nat,state)),hAPP(fun(state,fun(vname,fun(nat,state))),fun(vname,fun(state,fun(nat,state))),combc(state,vname,fun(nat,state)),update),X_3)),A_3))),hAPP(fun(state,nat),com,hAPP(vname,fun(fun(state,nat),com),ass,X_3),A_3)),Pa)),bot_bot(fun(hoare_1656922687triple(X_b),bool)))))) # label(fact_58_Ass) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 318 (all X_b all A_1 all B hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),A_1))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),B))) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B)) # label(fact_457_Int__def) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 319 (all X_b all X_c all Z_1 all X_1 all A_1 all F (hBOOL(hAPP(fun(X_b,fun(X_c,X_c)),bool,finite_comp_fun_idem(X_b,X_c),F)) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c)),finite_fold(X_b,X_c),F),Z_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),A_1)) = hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c)),finite_fold(X_b,X_c),F),hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_1),Z_1)),A_1)))) # label(fact_224_comp__fun__idem_Ofold__insert__idem2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 320 (all X_c all X_b all H all G all A_1 all B all F all Z_1 all F_1 (hBOOL(hAPP(fun(fun(X_c,X_b),fun(fun(X_c,bool),X_b)),bool,hAPP(X_b,fun(fun(fun(X_c,X_b),fun(fun(X_c,bool),X_b)),bool),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(fun(X_c,X_b),fun(fun(X_c,bool),X_b)),bool)),big_comm_monoid_big(X_b,X_c),F),Z_1),F_1)) -> (ti(fun(X_c,bool),B) = ti(fun(X_c,bool),A_1) -> ((all X_2 (hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),X_2),B)) -> hAPP(X_c,X_b,G,X_2) = hAPP(X_c,X_b,H,X_2))) -> hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),F_1,G),B) = hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),F_1,H),A_1))))) # label(fact_411_comm__monoid__big_OF__cong) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 321 (all X_b (semilattice_sup(X_b) -> (all Ca all Ba all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> ((all X_2 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_1)) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_2),Ba)))) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),X_b)),finite_fold(X_b,X_b),semilattice_sup_sup(X_b)),Ca),A_1)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),Ba),Ca)))))))) # label(fact_374_fold__sup__le__sup) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 322 (all X_c all X_b all F all X_1 all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_1)) -> hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),hAPP(X_b,X_c,F,X_1)),hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),A_1))))) # label(fact_66_imageI) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 323 (all Vname all Fun_1 all Com1 all Com2 hAPP(com,com,hAPP(com,fun(com,com),semi,Com1),Com2) != hAPP(fun(state,nat),com,hAPP(vname,fun(fun(state,nat),com),ass,Vname),Fun_1)) # label(fact_68_com_Osimps_I24_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 324 (all X_b all C all A_1 all B (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),B)) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B),C)) -> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),B),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),C),A_1)) = ti(fun(X_b,bool),A_1)))) # label(fact_261_double__diff) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 325 (all X_b all A_1 all B hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),B),A_1) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B)) # label(fact_458_Int__commute) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 326 (all X_b all X_1 all A_1 all F all F_1 (hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite_folding_one(X_b),F),F_1)) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_1)) -> (hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),bot_bot(fun(X_b,bool)))) != bot_bot(fun(X_b,bool)) -> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,X_1),hAPP(fun(X_b,bool),X_b,F_1,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),bot_bot(fun(X_b,bool)))))) = hAPP(fun(X_b,bool),X_b,F_1,A_1)) & (bot_bot(fun(X_b,bool)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),bot_bot(fun(X_b,bool)))) -> hAPP(fun(X_b,bool),X_b,F_1,A_1) = ti(X_b,X_1)))))) # label(fact_158_folding__one_Oremove) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 327 (all X_b all Ca all A_1 all B (-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),B)) & hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),A_1)) <-> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),B))))) # label(fact_167_Diff__iff) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 328 (all X_b all X_c ti(fun(fun(X_b,fun(X_b,X_b)),fun(fun(X_c,X_b),fun(X_b,fun(fun(X_c,bool),X_b)))),finite_fold_image(X_b,X_c)) = finite_fold_image(X_b,X_c)) # label(tsy_c_Finite__Set_Ofold__image_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 329 (all X_a (ab_semigroup_mult(X_a) -> (all A_2 all B_1 all C_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_2),B_1)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_1),C_1))))) # label(fact_384_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 330 (all X_b ti(fun(X_b,fun(fun(X_b,bool),bool)),member(X_b)) = member(X_b)) # label(tsy_c_member_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 331 (all X_b all F all X_1 -hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),fun(X_b,bool)),finite_fold1Set(X_b),F),bot_bot(fun(X_b,bool))),X_1))) # label(fact_97_empty__fold1SetE) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 332 (all M_1 all C_1 all S_1 all N_2 all T_4 (hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C_1),S_1),N_2),T_4)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_2),M_1)) -> hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C_1),S_1),M_1),T_4))))) # label(fact_378_evaln__nonstrict) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 333 (all X_a (semilattice_inf(X_a) -> (all Y all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X)) -> ti(X_a,Y) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X),Y))))) # label(fact_422_inf__absorb2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 334 (all X_a (semilattice_inf(X_a) -> (all X all A_2 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),A_2),B_1))) -> -(hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),A_2)) -> -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),B_1))))))) # label(fact_418_le__infE) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 335 (all X_b all A_1 all B hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B)),B))) # label(fact_436_Int__lower2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 336 (all X_b (lattice(X_b) -> (all B all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (ti(fun(X_b,bool),B) != bot_bot(fun(X_b,bool)) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B),A_1)) -> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),B)),hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),A_1)) = hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),A_1))))))) # label(fact_404_Sup__fin_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 337 (all X_b (bot(X_b) -> (all A_3 (ti(X_b,A_3) = bot_bot(X_b) <-> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),A_3),bot_bot(X_b))))))) # label(fact_354_bot__unique) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 338 (all X_c all X_b all F all B all A_1 ((all X_2 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_1)) -> hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),hAPP(X_b,X_c,F,X_2)),B)))) -> hBOOL(hAPP(fun(X_c,bool),bool,hAPP(fun(X_c,bool),fun(fun(X_c,bool),bool),ord_less_eq(fun(X_c,bool)),hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),A_1)),B)))) # label(fact_379_image__subsetI) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 339 (all X_b all A_1 ((all Y_2 -hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Y_2),A_1))) -> ti(fun(X_b,bool),A_1) = bot_bot(fun(X_b,bool)))) # label(fact_50_equals0I) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 340 (all X_b all D_2 all R_1 all Ga all Pa all Ca all Q_1 (hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Pa),Ca),Q_1)),bot_bot(fun(hoare_1656922687triple(X_b),bool))))) -> (hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Q_1),D_2),R_1)),bot_bot(fun(hoare_1656922687triple(X_b),bool))))) -> hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Pa),hAPP(com,com,hAPP(com,fun(com,com),semi,Ca),D_2)),R_1)),bot_bot(fun(hoare_1656922687triple(X_b),bool)))))))) # label(fact_46_Comp) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 341 (all X_a (order(X_a) -> (all Y all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)) -> ti(X_a,Y) = ti(X_a,X)))))) # label(fact_305_xt1_I5_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 342 (all X_b all B all A_1 hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B)))) # label(fact_344_Un__upper2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 343 (all X_b all A_3 all Ga all Pa all Ca all Q_1 all X_3 all S_5 (hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Pa),Ca),hAPP(fun(state,state),fun(X_b,fun(state,bool)),hAPP(fun(X_b,fun(fun(state,state),fun(state,bool))),fun(fun(state,state),fun(X_b,fun(state,bool))),combc(X_b,fun(state,state),fun(state,bool)),hAPP(fun(X_b,fun(state,bool)),fun(X_b,fun(fun(state,state),fun(state,bool))),hAPP(fun(fun(state,bool),fun(fun(state,state),fun(state,bool))),fun(fun(X_b,fun(state,bool)),fun(X_b,fun(fun(state,state),fun(state,bool)))),combb(fun(state,bool),fun(fun(state,state),fun(state,bool)),X_b),combb(state,bool,state)),Q_1)),hAPP(nat,fun(state,state),hAPP(fun(state,fun(nat,state)),fun(nat,fun(state,state)),combc(state,nat,state),hAPP(vname,fun(state,fun(nat,state)),hAPP(fun(state,fun(vname,fun(nat,state))),fun(vname,fun(state,fun(nat,state))),combc(state,vname,fun(nat,state)),update),hAPP(loc_1,vname,loc,X_3))),hAPP(loc_1,nat,hAPP(state,fun(loc_1,nat),getlocs,S_5),X_3))))),bot_bot(fun(hoare_1656922687triple(X_b),bool))))) -> hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),hAPP(fun(X_b,fun(state,bool)),fun(X_b,fun(state,bool)),hAPP(fun(fun(state,bool),fun(state,bool)),fun(fun(X_b,fun(state,bool)),fun(X_b,fun(state,bool))),combb(fun(state,bool),fun(state,bool),X_b),hAPP(fun(state,fun(bool,bool)),fun(fun(state,bool),fun(state,bool)),combs(state,bool,bool),hAPP(fun(state,bool),fun(state,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(state,bool),fun(state,fun(bool,bool))),combb(bool,fun(bool,bool),state),fconj),hAPP(state,fun(state,bool),fequal(state),S_5)))),hAPP(fun(state,state),fun(X_b,fun(state,bool)),hAPP(fun(X_b,fun(fun(state,state),fun(state,bool))),fun(fun(state,state),fun(X_b,fun(state,bool))),combc(X_b,fun(state,state),fun(state,bool)),hAPP(fun(X_b,fun(state,bool)),fun(X_b,fun(fun(state,state),fun(state,bool))),hAPP(fun(fun(state,bool),fun(fun(state,state),fun(state,bool))),fun(fun(X_b,fun(state,bool)),fun(X_b,fun(fun(state,state),fun(state,bool)))),combb(fun(state,bool),fun(fun(state,state),fun(state,bool)),X_b),combb(state,bool,state)),Pa)),hAPP(fun(state,nat),fun(state,state),hAPP(fun(state,fun(nat,state)),fun(fun(state,nat),fun(state,state)),combs(state,nat,state),hAPP(vname,fun(state,fun(nat,state)),hAPP(fun(state,fun(vname,fun(nat,state))),fun(vname,fun(state,fun(nat,state))),combc(state,vname,fun(nat,state)),update),hAPP(loc_1,vname,loc,X_3))),A_3)))),hAPP(com,com,hAPP(fun(state,nat),fun(com,com),hAPP(loc_1,fun(fun(state,nat),fun(com,com)),local,X_3),A_3),Ca)),Q_1)),bot_bot(fun(hoare_1656922687triple(X_b),bool))))))) # label(fact_88_hoare__derivs_OLocal) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 344 (all X_b all A_3 all B hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),B)))) # label(fact_33_insertI1) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 345 (all X_a (semilattice_sup(X_a) -> (all X all Y all Z hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),Y),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X),Y)),Z)))) # label(fact_274_sup__assoc) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 346 (all X_b all B all A_3 all C (-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_3),C)) -> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),B)),C) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),B),C))) # label(fact_446_Int__insert__left__if0) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 347 (all X_b all A_1 all B all X_1 (hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B),X_1)) -> hBOOL(hAPP(X_b,bool,B,X_1)))) # label(fact_453_inf1D2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 348 (all X_a (ord(X_a) -> (all C_1 all A_2 all B_1 (A_2 = B_1 -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_1),C_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),C_1))))))) # label(fact_311_ord__eq__le__trans) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 349 (all X_b all Ca all A_1 all B (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),B))) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),A_1)))) # label(fact_165_DiffD1) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 350 (all X_b all A_1 all X_1 all B (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),bot_bot(fun(X_b,bool))))),B)) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_1)) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),B)))))) # label(fact_371_diff__single__insert) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 351 (all X_a (semilattice_inf(X_a) -> (all B_1 all A_2 all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),X)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),A_2),B_1)),X)))))) # label(fact_425_le__infI1) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 352 (all X_b all B all X_1 all A_1 (-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_1)) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),B)) <-> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),B)))))) # label(fact_254_subset__insert) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 353 (all X_b all X_1 all A_1 (hBOOL(hAPP(X_b,bool,A_1,X_1)) <-> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_1)))) # label(fact_76_mem__def) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 354 (all X_b all X_c (bot(X_c) -> (all X_2 hAPP(X_b,X_c,bot_bot(fun(X_b,X_c)),X_2) = bot_bot(X_c)))) # label(fact_44_bot__fun__def) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 355 (all X_b hBOOL(hAPP(fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),bool,finite_comp_fun_idem(X_b,fun(X_b,bool)),insert(X_b)))) # label(fact_205_comp__fun__idem__insert) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 356 (all X_c all X_b (lattice(X_b) -> (all F all G all X_1 hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),hAPP(X_c,X_b,F,X_1)),hAPP(X_c,X_b,G,X_1)) = hAPP(X_c,X_b,hAPP(fun(X_c,X_b),fun(X_c,X_b),hAPP(fun(X_c,X_b),fun(fun(X_c,X_b),fun(X_c,X_b)),semilattice_sup_sup(fun(X_c,X_b)),F),G),X_1)))) # label(fact_272_sup__apply) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 357 (all X_b all A_1 all B hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B)))) # label(fact_345_Un__upper1) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 358 (all X_a (semilattice_inf(X_a) -> semilattice_inf_inf(X_a) = ti(fun(X_a,fun(X_a,X_a)),semilattice_inf_inf(X_a)))) # label(tsy_c_Lattices_Osemilattice__inf__class_Oinf_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 359 (all P all Q (hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fdisj,P),Q)) | -hBOOL(Q))) # label(help_fdisj_2_1_U) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 360 (all X_b all Ga all Ca all Q_1 all Pa ((all Z_2 all S_2 (hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),Pa,Z_2),S_2)) -> hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),hAPP(fun(state,bool),fun(X_b,fun(state,bool)),combk(fun(state,bool),X_b),hAPP(state,fun(state,bool),hAPP(fun(state,fun(state,bool)),fun(state,fun(state,bool)),combc(state,state,bool),fequal(state)),S_2))),Ca),hAPP(fun(state,bool),fun(X_b,fun(state,bool)),combk(fun(state,bool),X_b),hAPP(X_b,fun(state,bool),Q_1,Z_2)))),bot_bot(fun(hoare_1656922687triple(X_b),bool))))))) -> hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Pa),Ca),Q_1)),bot_bot(fun(hoare_1656922687triple(X_b),bool))))))) # label(fact_5_escape) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 361 (all X_b (ab_sem1668676832m_mult(X_b) -> (all X_1 all A_1 (ti(fun(X_b,bool),A_1) != bot_bot(fun(X_b,bool)) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),X_1),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),A_1)) = hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),A_1))))))) # label(fact_212_fold1__insert__idem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 362 (all X_b all Ga all F_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),F_1)) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),Ga)) -> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),F_1),Ga)))))) # label(fact_301_finite__UnI) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 363 (all X_c all X_b all F all A_1 all B (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),B)) -> hBOOL(hAPP(fun(X_c,bool),bool,hAPP(fun(X_c,bool),fun(fun(X_c,bool),bool),ord_less_eq(fun(X_c,bool)),hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),A_1)),hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),B))))) # label(fact_258_image__mono) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 364 (all B_1_1 (hBOOL(ti(bool,B_1_1)) <-> hBOOL(B_1_1))) # label(tsy_c_hBOOL_arg1) # label(hypothesis) # label(non_clause). [assumption]. 2.22/2.62 365 (all X_a (lattice(X_a) -> (all X all Y all Z hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),Y),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),Y),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X),Z))))) # label(fact_278_inf__sup__aci_I7_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 366 (all X_b all A_1 ti(fun(X_b,bool),A_1) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),bot_bot(fun(X_b,bool)))) # label(fact_298_Un__empty__right) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 367 (all X_b all A_1 hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),A_1) = bot_bot(fun(X_b,bool))) # label(fact_170_Diff__cancel) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 368 (all X_b all A_1 all B hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B)) # label(fact_338_Un__left__absorb) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 369 (all X_b all X_1 all A_1 all F all F_1 (hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite2073411215e_idem(X_b),F),F_1)) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_1)) -> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,X_1),hAPP(fun(X_b,bool),X_b,F_1,A_1)) = hAPP(fun(X_b,bool),X_b,F_1,A_1))))) # label(fact_182_folding__one__idem_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 370 (all X_b all X_c all Z_1 all A_1 all Y_1 all F (hBOOL(hAPP(fun(X_b,fun(X_c,X_c)),bool,finite100568337ommute(X_b,X_c),F)) -> (hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),finite_fold_graph(X_b,X_c),F),Z_1),A_1),Y_1)) -> ti(X_c,Y_1) = hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c)),finite_fold(X_b,X_c),F),Z_1),A_1)))) # label(fact_219_comp__fun__commute_Ofold__equality) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 371 (all X_b all A_1 hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),bot_bot(fun(X_b,bool))),A_1))) # label(fact_243_empty__subsetI) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 372 (all X_b all A_3 all B hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),B) = hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fdisj),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),A_3))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),B)))) # label(fact_32_insert__compr) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 373 (all Ca all S0_1 all Y_4 all A_3 all N_3 all S1_2 (hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,Ca),hAPP(nat,state,hAPP(vname,fun(nat,state),hAPP(state,fun(vname,fun(nat,state)),update,S0_1),hAPP(loc_1,vname,loc,Y_4)),hAPP(state,nat,A_3,S0_1))),N_3),S1_2)) -> hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,hAPP(com,com,hAPP(fun(state,nat),fun(com,com),hAPP(loc_1,fun(fun(state,nat),fun(com,com)),local,Y_4),A_3),Ca)),S0_1),N_3),hAPP(nat,state,hAPP(vname,fun(nat,state),hAPP(state,fun(vname,fun(nat,state)),update,S1_2),hAPP(loc_1,vname,loc,Y_4)),hAPP(loc_1,nat,hAPP(state,fun(loc_1,nat),getlocs,S0_1),Y_4)))))) # label(fact_105_evaln_OLocal) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 374 (all X_b all B all X_1 all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_1)) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),B)) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),B))))) # label(fact_325_set__rev__mp) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 375 (all X_b all Pa all F_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),F_1)) -> (bot_bot(fun(X_b,bool)) != ti(fun(X_b,bool),F_1) -> ((all X_2 hBOOL(hAPP(fun(X_b,bool),bool,Pa,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool)))))) -> ((all X_2 all F_2 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),F_2)) -> (ti(fun(X_b,bool),F_2) != bot_bot(fun(X_b,bool)) -> (-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),F_2)) -> (hBOOL(hAPP(fun(X_b,bool),bool,Pa,F_2)) -> hBOOL(hAPP(fun(X_b,bool),bool,Pa,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),F_2)))))))) -> hBOOL(hAPP(fun(X_b,bool),bool,Pa,F_1))))))) # label(fact_155_finite__ne__induct) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 376 (all X_b all F all A_1 hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),F),A_1) = hAPP(fun(X_b,bool),X_b,the(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),fun(X_b,bool)),finite_fold1Set(X_b),F),A_1))) # label(fact_229_fold1__def) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 377 (all T_2 all T_1 (bot(T_1) -> bot(fun(T_2,T_1)))) # label(arity_fun___Orderings_Obot) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 378 (all X_b all A_1 ((exists X_2 hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_1))) <-> ti(fun(X_b,bool),A_1) != bot_bot(fun(X_b,bool)))) # label(fact_21_ex__in__conv) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 379 (all X_b all A_1 all X_1 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),bot_bot(fun(X_b,bool))))) -> ti(fun(X_b,bool),A_1) = bot_bot(fun(X_b,bool)) | ti(fun(X_b,bool),A_1) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),bot_bot(fun(X_b,bool))))) # label(fact_366_subset__singletonD) # label(axiom) # label(non_clause). [assumption]. 2.22/2.62 380 (all X_c all X_b all F all Z_1 all Y_1 all X_1 all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_1)) -> (hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),fold_graph(X_b,X_c),F),Z_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),bot_bot(fun(X_b,bool))))),Y_1)) -> hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),fold_graph(X_b,X_c),F),Z_1),A_1),hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_1),Y_1)))))) # label(fact_195_fold__graph_H_Ointros_I2_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 381 (all X_b (lattice(X_b) -> big_lattice_Sup_fin(X_b) = ti(fun(fun(X_b,bool),X_b),big_lattice_Sup_fin(X_b)))) # label(tsy_c_Big__Operators_Olattice__class_OSup__fin_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 382 (all X_b all A_1 all A_3 all B hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),bot_bot(fun(X_b,bool))))),B) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),B))) # label(fact_179_Diff__insert2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 383 (all X_b all A_1 all C all B hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),C)),B) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),C)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),B),C))) # label(fact_486_Diff__Int2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 384 (all X_b all B all A_3 all A_1 (-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_3),A_1)) -> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),B)))) # label(fact_447_Int__insert__right__if0) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 385 (all X_b (semilattice_sup(X_b) -> (all X_1 all Y_1 (hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),Y_1)) <-> ti(X_b,Y_1) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),X_1),Y_1))))) # label(fact_283_le__iff__sup) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 386 (all X_b all X_2 (hBOOL(hAPP(X_b,bool,bot_bot(fun(X_b,bool)),X_2)) <-> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),bot_bot(fun(X_b,bool)))))) # label(fact_57_bot__empty__eq) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 387 (all X_a (linorder(X_a) -> (all X all Y (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X)) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)))))) # label(fact_316_linorder__linear) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 388 (all X_b all X_1 all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_1)) -> -(all B_2 (ti(fun(X_b,bool),A_1) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),B_2) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),B_2)))))) # label(fact_48_Set_Oset__insert) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 389 (all X_a (semilattice_inf(X_a) -> (all X all Y all Z hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),Y),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),Y),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X),Z))))) # label(fact_481_inf__left__commute) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 390 (all X_c all X_b all F all G all M_2 all N (ti(fun(X_b,bool),M_2) = ti(fun(X_b,bool),N) -> ((all X_2 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),N)) -> hAPP(X_b,X_c,G,X_2) = hAPP(X_b,X_c,F,X_2))) -> hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),M_2) = hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),G),N)))) # label(fact_84_image__cong) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 391 (all X_b all A_3 hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),A_3)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),bot_bot(fun(X_b,bool)))) # label(fact_13_singleton__conv) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 392 (all X_c all X_b all F hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),bot_bot(fun(X_c,bool))) = bot_bot(fun(X_b,bool))) # label(fact_73_image__empty) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 393 (all X_b all X_1 all A_1 all B (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),B)) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_1)) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),B))))) # label(fact_324_set__mp) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 394 (all X_b (bounded_lattice_bot(X_b) -> (all X_1 all Y_1 (hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),X_1),Y_1) = bot_bot(X_b) <-> ti(X_b,Y_1) = bot_bot(X_b) & ti(X_b,X_1) = bot_bot(X_b))))) # label(fact_294_sup__eq__bot__iff) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 395 (all X_a (semilattice_inf(X_a) -> (all X all Y hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X),Y)),Y))))) # label(fact_428_inf__le2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 396 (all X_b all A_3 all A_1 all F all F_1 (hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite2073411215e_idem(X_b),F),F_1)) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),X_b)),finite_fold(X_b,X_b),F),A_3),A_1) = hAPP(fun(X_b,bool),X_b,F_1,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),A_1))))) # label(fact_227_folding__one__idem_Oeq__fold__idem_H) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 397 (all X_b all B all Ca all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),A_1)) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B))))) # label(fact_330_UnI1) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 398 (all X_a (lattice(X_a) -> (all X all Y hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X),Y)),Y))))) # label(fact_429_inf__sup__ord_I2_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 399 (all Vname all Fun_1 all Vname_1 all Fun (ti(vname,Vname) = ti(vname,Vname_1) & Fun = Fun_1 <-> hAPP(fun(state,nat),com,hAPP(vname,fun(fun(state,nat),com),ass,Vname_1),Fun) = hAPP(fun(state,nat),com,hAPP(vname,fun(fun(state,nat),com),ass,Vname),Fun_1))) # label(fact_64_com_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 400 (all X_a (bounded_lattice_bot(X_a) -> (all X ti(X_a,X) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),bot_bot(X_a)),X)))) # label(fact_296_sup__bot__left) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 401 (all X_b (finite_finite(X_b) -> (all A_1 hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1))))) # label(fact_143_finite__code) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 402 (all X_b all N_3 all Pa all Ca all Q_1 (hBOOL(hAPP(hoare_1656922687triple(X_b),bool,hAPP(nat,fun(hoare_1656922687triple(X_b),bool),hoare_920331057_valid(X_b),N_3),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Pa),Ca),Q_1))) <-> (all Z_2 all S_2 (hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),Pa,Z_2),S_2)) -> (all S_3 (hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,Ca),S_2),N_3),S_3)) -> hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),Q_1,Z_2),S_3)))))))) # label(fact_135_triple__valid__def2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 403 (all X_a ti(fun(X_a,fun(X_a,bool)),fequal(X_a)) = fequal(X_a)) # label(tsy_c_fequal_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 404 (all X_b all A_3 all A_1 bot_bot(fun(X_b,bool)) != hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),A_1)) # label(fact_39_insert__not__empty) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 405 (all C1 all S2 all C0 all S0 all S1 (hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C0),S0),S1)) -> (hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C1),S1),S2)) -> hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,hAPP(com,com,hAPP(com,fun(com,com),semi,C0),C1)),S0),S2))))) # label(fact_110_evalc_OSemi) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 406 (all X_b all A_1 all B all C hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),C)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),B),C))) # label(fact_496_Un__Int__distrib) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 407 (all X_b (finite_finite(X_b) -> (all A_1 hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1))))) # label(fact_144_finite) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 408 (all X_b all A_1 all F all F_1 (hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),big_semilattice_big(X_b),F),F_1)) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> hAPP(fun(X_b,bool),X_b,F_1,A_1) = hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),F),A_1)))) # label(fact_233_semilattice__big_OF__eq) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 409 (all X_a all X all Y (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),fequal(X_a),X),Y)) | ti(X_a,X) != ti(X_a,Y))) # label(help_fequal_2_1_T) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 410 (all X_b all N all H all F all F_1 (hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite2073411215e_idem(X_b),F),F_1)) -> ((all X_2 all Y_2 hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(X_b,X_b,H,X_2)),hAPP(X_b,X_b,H,Y_2)) = hAPP(X_b,X_b,H,hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,X_2),Y_2))) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),N)) -> (ti(fun(X_b,bool),N) != bot_bot(fun(X_b,bool)) -> hAPP(X_b,X_b,H,hAPP(fun(X_b,bool),X_b,F_1,N)) = hAPP(fun(X_b,bool),X_b,F_1,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,X_b),fun(fun(X_b,bool),fun(X_b,bool)),image(X_b,X_b),H),N))))))) # label(fact_184_folding__one__idem_Ohom__commute) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 411 (all X_a all X_c all B_1_1 all B_2_1 hAPP(X_a,X_c,B_1_1,ti(X_a,B_2_1)) = hAPP(X_a,X_c,B_1_1,B_2_1)) # label(tsy_c_hAPP_arg2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 412 (all T_2 all T_1 (ord(T_1) -> ord(fun(T_2,T_1)))) # label(arity_fun___Orderings_Oord) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 413 (all X_a (semilattice_sup(X_a) -> (all B_1 all A_2 all C_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),A_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),B_1),C_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),B_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),A_2),C_1))))) # label(fact_279_sup_Oleft__commute) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 414 (all X_b all X_c all Z_1 all A_1 all F (hBOOL(hAPP(fun(X_b,fun(X_c,X_c)),bool,finite100568337ommute(X_b,X_c),F)) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),finite_fold_graph(X_b,X_c),F),Z_1),A_1),hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c)),finite_fold(X_b,X_c),F),Z_1),A_1)))))) # label(fact_228_comp__fun__commute_Ofold__graph__fold) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 415 (all X_c (minus(X_c) -> ti(fun(X_c,fun(X_c,X_c)),minus_minus(X_c)) = minus_minus(X_c))) # label(tsy_c_Groups_Ominus__class_Ominus_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 416 (all X_b all A_3 all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),A_1))))) # label(fact_139_finite_OinsertI) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 417 (all X_b all A_1 ti(fun(X_b,bool),A_1) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),A_1)) # label(fact_456_Int__absorb) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 418 (all X_b all A_1 all B all C (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),B)),C)) <-> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),B),C))))) # label(fact_263_Diff__subset__conv) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 419 (all X_a (order(X_a) -> (all C_1 all B_1 all A_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_1),A_2)) -> (ti(X_a,C_1) = ti(X_a,B_1) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),A_2))))))) # label(fact_308_xt1_I4_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 420 (all X_b all A_1 all B (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),B)) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B),A_1)) -> ti(fun(X_b,bool),A_1) = ti(fun(X_b,bool),B)))) # label(fact_237_equalityI) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 421 (all X_c all X_b all F1 all Fun1_1 all Com_1 all Fun2_1 hAPP(hoare_1656922687triple(X_c),X_b,hAPP(fun(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),X_b))),fun(hoare_1656922687triple(X_c),X_b),hoare_1632998903le_rec(X_c,X_b),F1),hAPP(fun(X_c,fun(state,bool)),hoare_1656922687triple(X_c),hAPP(com,fun(fun(X_c,fun(state,bool)),hoare_1656922687triple(X_c)),hAPP(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),hoare_1656922687triple(X_c))),hoare_246368825triple(X_c),Fun1_1),Com_1),Fun2_1)) = hAPP(fun(X_c,fun(state,bool)),X_b,hAPP(com,fun(fun(X_c,fun(state,bool)),X_b),hAPP(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),X_b)),F1,Fun1_1),Com_1),Fun2_1)) # label(fact_16_triple_Orecs) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 422 (all X_b all X_c all X_1 all Z_1 all A_1 all F (hBOOL(hAPP(fun(X_b,fun(X_c,X_c)),bool,finite100568337ommute(X_b,X_c),F)) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_1),hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c)),finite_fold(X_b,X_c),F),Z_1),A_1)) = hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c)),finite_fold(X_b,X_c),F),hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_1),Z_1)),A_1)))) # label(fact_216_comp__fun__commute_Ofold__fun__comm) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 423 (all X_b all Pa all A_3 (hBOOL(hAPP(X_b,bool,Pa,A_3)) -> ((all X_2 (hBOOL(hAPP(X_b,bool,Pa,X_2)) -> ti(X_b,X_2) = ti(X_b,A_3))) -> hBOOL(hAPP(X_b,bool,Pa,hAPP(fun(X_b,bool),X_b,the(X_b),Pa)))))) # label(fact_99_theI) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 424 (all X_b all X_c all Z_1 all X_1 all A_1 all F (hBOOL(hAPP(fun(X_b,fun(X_c,X_c)),bool,finite100568337ommute(X_b,X_c),F)) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_1)) -> hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_1),hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c)),finite_fold(X_b,X_c),F),Z_1),A_1)) = hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c)),finite_fold(X_b,X_c),F),Z_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),A_1)))))) # label(fact_223_comp__fun__commute_Ofold__insert) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 425 (all Ca all S_4 all T_5 ((exists N_1 hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,Ca),S_4),N_1),T_5))) <-> hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,Ca),S_4),T_5)))) # label(fact_117_eval__eq) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 426 (all X_b all Ca all A_1 all B (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B))) <-> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),B)) | hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),A_1)))) # label(fact_336_Un__iff) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 427 (all Y ((all Glb ti(vname,Y) != hAPP(glb_1,vname,glb,Glb)) -> -(all Loc ti(vname,Y) != hAPP(loc_1,vname,loc,Loc)))) # label(fact_156_vname_Oexhaust) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 428 (all X_b ti(fun(fun(X_b,bool),bool),finite_finite_1(X_b)) = finite_finite_1(X_b)) # label(tsy_c_Finite__Set_Ofinite_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 429 (all X_b (bot(X_b) -> ti(X_b,bot_bot(X_b)) = bot_bot(X_b))) # label(tsy_c_Orderings_Obot__class_Obot_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 430 (all X_b all Pa all Ga all P_2 all Ca all Q_1 (hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),P_2),Ca),Q_1)),bot_bot(fun(hoare_1656922687triple(X_b),bool))))) -> ((all Z_2 all S_2 (hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),Pa,Z_2),S_2)) -> hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),P_2,Z_2),S_2)))) -> hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Pa),Ca),Q_1)),bot_bot(fun(hoare_1656922687triple(X_b),bool)))))))) # label(fact_7_conseq1) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 431 (all T_2 all T_1 (minus(T_1) -> minus(fun(T_2,T_1)))) # label(arity_fun___Groups_Ominus) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 432 (all X_b (semilattice_sup(X_b) -> (all Ba all A_3 all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_3),A_1)) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),A_3),Ba)),hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),X_b)),finite_fold(X_b,X_b),semilattice_sup_sup(X_b)),Ba),A_1)))))))) # label(fact_245_sup__le__fold__sup) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 433 (all X_b all F all A_3 ti(X_b,A_3) = hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),F),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),bot_bot(fun(X_b,bool))))) # label(fact_217_fold1__singleton) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 434 (all X_b all Ca -hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),bot_bot(fun(X_b,bool))))) # label(fact_19_empty__iff) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 435 (all X_b all A_1 ((all X_2 -hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_1))) <-> bot_bot(fun(X_b,bool)) = ti(fun(X_b,bool),A_1))) # label(fact_22_all__not__in__conv) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 436 (all X_3 all A_3 all S_4 all N_3 all T_5 (hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,hAPP(fun(state,nat),com,hAPP(vname,fun(fun(state,nat),com),ass,X_3),A_3)),S_4),N_3),T_5)) -> hAPP(nat,state,hAPP(vname,fun(nat,state),hAPP(state,fun(vname,fun(nat,state)),update,S_4),X_3),hAPP(state,nat,A_3,S_4)) = T_5)) # label(fact_114_evaln__elim__cases_I2_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 437 (all X_b all X_1 all F all F_1 (hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite2073411215e_idem(X_b),F),F_1)) -> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,X_1),X_1) = ti(X_b,X_1))) # label(fact_169_folding__one__idem_Oidem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 438 (all X_b all A_1 hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),bot_bot(fun(X_b,bool))) = ti(fun(X_b,bool),A_1)) # label(fact_171_Diff__empty) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 439 (all Com1_2 all Com2_2 hAPP(com,com,hAPP(com,fun(com,com),semi,Com1_2),Com2_2) != skip) # label(fact_52_com_Osimps_I13_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 440 (all X_b all F all A_1 all X_1 (hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),fun(X_b,bool)),finite_fold1Set(X_b),F),A_1),X_1)) -> ti(fun(X_b,bool),A_1) != bot_bot(fun(X_b,bool)))) # label(fact_98_fold1Set__nonempty) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 441 (all X_b all A_1 all B all C hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),B),C))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),C),A_1)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),B),C))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),C),A_1))) # label(fact_499_Un__Int__crazy) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 442 (all X_b (semilattice_sup(X_b) -> (all Ba all A_3 all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),X_b)),finite_fold(X_b,X_b),semilattice_sup_sup(X_b)),Ba),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),A_1)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),A_3),hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),X_b)),finite_fold(X_b,X_b),semilattice_sup_sup(X_b)),Ba),A_1)))))) # label(fact_364_fold__sup__insert) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 443 (all X_a (lattice(X_a) -> (all X ti(X_a,X) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X),X)))) # label(fact_319_Sup__fin_Oidem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 444 (all X_a (semilattice_sup(X_a) -> (all A_2 all B_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),A_2),B_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),A_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),A_2),B_1))))) # label(fact_282_sup_Oleft__idem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 445 (all T_2 all T_1 (lattice(T_1) -> semilattice_sup(fun(T_2,T_1)))) # label(arity_fun___Lattices_Osemilattice__sup) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 446 (all X_b (ab_sem1668676832m_mult(X_b) -> (all B all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (bot_bot(fun(X_b,bool)) != ti(fun(X_b,bool),A_1) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B)) -> (ti(fun(X_b,bool),B) != bot_bot(fun(X_b,bool)) -> hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),A_1)),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),B))))))))) # label(fact_234_fold1__Un2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 447 (all X_a ti(fun(X_a,X_a),combi(X_a)) = combi(X_a)) # label(tsy_c_COMBI_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 448 (all X_c all X_b all F all A_1 (bot_bot(fun(X_c,bool)) = ti(fun(X_c,bool),A_1) <-> bot_bot(fun(X_b,bool)) = hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),A_1))) # label(fact_72_image__is__empty) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 449 (all X_b (lattice(X_b) -> (all X_1 all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_1)) -> hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),X_b)),finite_fold(X_b,X_b),semilattice_sup_sup(X_b)),X_1),A_1) = hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),A_1))))))) # label(fact_407_Sup__fin_Oeq__fold_H) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 450 (all X_b finite_fold1Set(X_b) = ti(fun(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),fun(X_b,bool))),finite_fold1Set(X_b))) # label(tsy_c_Finite__Set_Ofold1Set_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 451 (all X_b all Fun1_1 all Com_1 all Fun2_1 all Fun1_2 all Com all Fun2_2 (hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Fun1_2),Com),Fun2_2) = hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Fun1_1),Com_1),Fun2_1) <-> Fun2_1 = Fun2_2 & Com = Com_1 & Fun1_1 = Fun1_2)) # label(fact_1_triple_Oinject) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 452 (all X_b all X_1 all A_1 all F all F_1 (hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite_folding_one(X_b),F),F_1)) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_1)) -> (bot_bot(fun(X_b,bool)) != ti(fun(X_b,bool),A_1) -> hAPP(fun(X_b,bool),X_b,F_1,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),A_1)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,X_1),hAPP(fun(X_b,bool),X_b,F_1,A_1))))))) # label(fact_136_folding__one_Oinsert) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 453 (all X_b (ab_group_add(X_b) -> (all A_3 all Ba all Ca all D_2 (hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),Ca),D_2) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),A_3),Ba) -> (ti(X_b,D_2) = ti(X_b,Ca) <-> ti(X_b,A_3) = ti(X_b,Ba)))))) # label(fact_385_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 454 (all X_b all A_1 all B all C (hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),B),C)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B)),C) <-> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),C),A_1)))) # label(fact_441_Un__Int__assoc__eq) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 455 (all X_b all Ts all Ga all Ts_1 (hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),Ts_1)) -> (hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),ord_less_eq(fun(hoare_1656922687triple(X_b),bool)),Ts),Ts_1)) -> hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),Ts))))) # label(fact_359_weaken) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 456 (all X_b all A_3 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3)) <-> (exists A_5 exists A_4 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_5)) & hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_4),A_5) = ti(fun(X_b,bool),A_3))) | ti(fun(X_b,bool),A_3) = bot_bot(fun(X_b,bool)))) # label(fact_151_finite_Osimps) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 457 (all X_b all X_c all Z_1 all X_1 all A_1 all F (hBOOL(hAPP(fun(X_b,fun(X_c,X_c)),bool,finite100568337ommute(X_b,X_c),F)) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_1)) -> hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c)),finite_fold(X_b,X_c),F),Z_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),A_1)) = hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c)),finite_fold(X_b,X_c),F),hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_1),Z_1)),A_1))))) # label(fact_222_comp__fun__commute_Ofold__insert2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 458 (all X_b all A_1 hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),A_1))) # label(fact_346_subset__refl) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 459 (all X_b all A_1 all X_1 all B ((hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),B)) -> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),A_1)),B) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),B)) & (-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),B)) -> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),B)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),A_1)),B)))) # label(fact_175_insert__Diff__if) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 460 (all X_b all A_1 all B bot_bot(fun(X_b,bool)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),B),A_1))) # label(fact_494_Diff__disjoint) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 461 (all X_a (semilattice_inf(X_a) -> (all B_1 all D_1 all A_2 all C_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),C_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_1),D_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),A_2),B_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),C_1),D_1)))))))) # label(fact_419_inf__mono) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 462 (all Loc_3 all Fun all Com hAPP(com,com,hAPP(fun(state,nat),fun(com,com),hAPP(loc_1,fun(fun(state,nat),fun(com,com)),local,Loc_3),Fun),Com) != skip) # label(fact_95_com_Osimps_I11_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 463 (all T_2 all T_1 (bounded_lattice(T_1) -> bounded_lattice(fun(T_2,T_1)))) # label(arity_fun___Lattices_Obounded__lattice) # label(axiom) # label(non_clause). [assumption]. 2.22/2.63 464 (all X_b all F all A1 all A2 (hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),fun(X_b,bool)),finite_fold1Set(X_b),F),A1),A2)) <-> (exists A_4 exists A_5 exists X_2 (ti(fun(X_b,bool),A1) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_4),A_5) & ti(X_b,X_2) = ti(X_b,A2) & -hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_4),A_5)) & hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),finite_fold_graph(X_b,X_b),F),A_4),A_5),X_2)))))) # label(fact_130_fold1Set_Osimps) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 465 (all X_a (semilattice_inf(X_a) -> (all X all Y hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X),Y) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),Y),X)))) # label(fact_475_inf__commute) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 466 (all X_b all Ga all F_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),F_1)) | hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),Ga)) -> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),F_1),Ga))))) # label(fact_417_finite__Int) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 467 (all X_b all X_c ti(fun(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c))),finite_fold(X_b,X_c)) = finite_fold(X_b,X_c)) # label(tsy_c_Finite__Set_Ofold_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 468 (all X_a all P hAPP(X_a,X_a,combi(X_a),P) = ti(X_a,P)) # label(help_COMBI_1_1_U) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 469 (all X_b all A_1 all X_1 all B (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),B))) <-> (-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_1)) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),B))) & (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_1)) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),bot_bot(fun(X_b,bool))))),B))))) # label(fact_372_subset__insert__iff) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 470 (all X_b all X_c (minus(X_c) -> (all A_1 all B all X_2 hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),minus_minus(X_c),hAPP(X_b,X_c,A_1,X_2)),hAPP(X_b,X_c,B,X_2)) = hAPP(X_b,X_c,hAPP(fun(X_b,X_c),fun(X_b,X_c),hAPP(fun(X_b,X_c),fun(fun(X_b,X_c),fun(X_b,X_c)),minus_minus(fun(X_b,X_c)),A_1),B),X_2)))) # label(fact_188_fun__diff__def) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 471 (all Loc_2 all Fun_1 all Com_1 all Loc_3 all Fun all Com (hAPP(com,com,hAPP(fun(state,nat),fun(com,com),hAPP(loc_1,fun(fun(state,nat),fun(com,com)),local,Loc_2),Fun_1),Com_1) = hAPP(com,com,hAPP(fun(state,nat),fun(com,com),hAPP(loc_1,fun(fun(state,nat),fun(com,com)),local,Loc_3),Fun),Com) <-> ti(loc_1,Loc_3) = ti(loc_1,Loc_2) & Fun = Fun_1 & Com_1 = Com)) # label(fact_90_com_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 472 (all N ((exists M all X_2 (hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),X_2),N)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X_2),M)))) <-> hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),N)))) # label(fact_386_finite__nat__set__iff__bounded__le) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 473 (all X_b all Q_1 all Pa all Ga all P_2 all Ca all Q_3 (hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),P_2),Ca),Q_3)),bot_bot(fun(hoare_1656922687triple(X_b),bool))))) -> ((all Z_2 all S_2 (hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),Pa,Z_2),S_2)) -> (all S_3 ((all Z_3 (hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),P_2,Z_3),S_2)) -> hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),Q_3,Z_3),S_3)))) -> hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),Q_1,Z_2),S_3)))))) -> hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Pa),Ca),Q_1)),bot_bot(fun(hoare_1656922687triple(X_b),bool)))))))) # label(fact_8_conseq12) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 474 (all X_b all D all B all A_1 all C (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),C)) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),D),B)) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),B)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),C),D)))))) # label(fact_260_Diff__mono) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 475 (all X_b all Pa all A_3 (hBOOL(hAPP(X_b,bool,Pa,A_3)) -> ((all X_2 (hBOOL(hAPP(X_b,bool,Pa,X_2)) -> ti(X_b,A_3) = ti(X_b,X_2))) -> hAPP(fun(X_b,bool),X_b,the(X_b),Pa) = ti(X_b,A_3)))) # label(fact_86_the__equality) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 476 (all X_b all A_1 all B all C hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),B),C)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),C))) # label(fact_495_Int__Un__distrib) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 477 (all X_b ti(fun(fun(glb_1,X_b),fun(fun(loc_1,X_b),fun(vname,X_b))),vname_case(X_b)) = vname_case(X_b)) # label(tsy_c_Com_Ovname_Ovname__case_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 478 (all P all Q (hBOOL(P) | -hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,P),Q)))) # label(help_fconj_2_1_U) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 479 (all X_a (semilattice_inf(X_a) -> (all B_1 all X all A_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),A_2)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),B_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),A_2),B_1)))))))) # label(fact_421_le__infI) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 480 (all X_c all X_b all F all Z_1 all G all F_1 (hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite1357897459simple(X_b,X_c),F),Z_1),G),F_1)) -> hAPP(fun(X_c,bool),X_b,F_1,bot_bot(fun(X_c,bool))) = ti(X_b,Z_1))) # label(fact_190_folding__image__simple_Oempty) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 481 (all X_b all Pa all Q_1 hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fdisj),Pa)),Q_1)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Pa)),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Q_1))) # label(fact_318_Collect__disj__eq) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 482 (all U all F ((all N_1 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),hAPP(nat,nat,F,N_1)))) -> hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),hAPP(fun(nat,nat),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combb(nat,fun(nat,bool),nat),ord_less_eq(nat)),F)),U)))))) # label(fact_393_finite__less__ub) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 483 (all X_c all X_b all X_1 all A_1 all F all Z_1 all G all F_1 (hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite908156982e_idem(X_b,X_c),F),Z_1),G),F_1)) -> (hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_1)) -> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(X_c,X_b,G,X_1)),hAPP(fun(X_c,bool),X_b,F_1,A_1)) = hAPP(fun(X_c,bool),X_b,F_1,hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_c,bool),fun(X_c,bool)),insert(X_c),X_1),A_1))))) # label(fact_157_folding__image__simple__idem_Oinsert__idem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 484 (all X_b all X_1 ti(X_b,X_1) = hAPP(fun(X_b,bool),X_b,the_elem(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),bot_bot(fun(X_b,bool))))) # label(fact_41_the__elem__eq) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 485 (all X_b all A_3 all Ba all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Ba),A_1))) <-> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_3),A_1)) | ti(X_b,Ba) = ti(X_b,A_3))) # label(fact_28_insert__iff) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 486 (all T_2 all T_1 (preorder(T_1) -> preorder(fun(T_2,T_1)))) # label(arity_fun___Orderings_Opreorder) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 487 (all X_a (semilattice_inf(X_a) -> (all X all Y hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X),Y)),X))))) # label(fact_430_inf__le1) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 488 (all Ca all S0_1 all Y_4 all A_3 all S1_2 (hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,Ca),hAPP(nat,state,hAPP(vname,fun(nat,state),hAPP(state,fun(vname,fun(nat,state)),update,S0_1),hAPP(loc_1,vname,loc,Y_4)),hAPP(state,nat,A_3,S0_1))),S1_2)) -> hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,hAPP(com,com,hAPP(fun(state,nat),fun(com,com),hAPP(loc_1,fun(fun(state,nat),fun(com,com)),local,Y_4),A_3),Ca)),S0_1),hAPP(nat,state,hAPP(vname,fun(nat,state),hAPP(state,fun(vname,fun(nat,state)),update,S1_2),hAPP(loc_1,vname,loc,Y_4)),hAPP(loc_1,nat,hAPP(state,fun(loc_1,nat),getlocs,S0_1),Y_4)))))) # label(fact_104_evalc_OLocal) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 489 (all Com1 all Com2 all Loc_2 all Fun_1 all Com_1 hAPP(com,com,hAPP(fun(state,nat),fun(com,com),hAPP(loc_1,fun(fun(state,nat),fun(com,com)),local,Loc_2),Fun_1),Com_1) != hAPP(com,com,hAPP(com,fun(com,com),semi,Com1),Com2)) # label(fact_92_com_Osimps_I35_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 490 (all X_a (preorder(X_a) -> (all X all Y (X = Y -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)))))) # label(fact_314_order__eq__refl) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 491 (all X_b (semilattice_sup(X_b) -> hBOOL(hAPP(fun(X_b,fun(X_b,X_b)),bool,finite_comp_fun_idem(X_b,X_b),semilattice_sup_sup(X_b))))) # label(fact_361_comp__fun__idem__sup) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 492 (all X_b all B all Ca all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),A_1)) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),B)) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B)))))) # label(fact_415_IntI) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 493 (all X_b all F1 all F2 all Glb_3 hAPP(glb_1,X_b,F1,Glb_3) = hAPP(vname,X_b,hAPP(fun(loc_1,X_b),fun(vname,X_b),hAPP(fun(glb_1,X_b),fun(fun(loc_1,X_b),fun(vname,X_b)),vname_case(X_b),F1),F2),hAPP(glb_1,vname,glb,Glb_3))) # label(fact_134_vname_Osimps_I5_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 494 (all X_c all X_b all Ca all A_1 ((ti(fun(X_b,bool),A_1) = bot_bot(fun(X_b,bool)) -> bot_bot(fun(X_c,bool)) = hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),hAPP(X_c,fun(X_b,X_c),combk(X_c,X_b),Ca)),A_1)) & (bot_bot(fun(X_b,bool)) != ti(fun(X_b,bool),A_1) -> hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),hAPP(X_c,fun(X_b,X_c),combk(X_c,X_b),Ca)),A_1) = hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_c,bool),fun(X_c,bool)),insert(X_c),Ca),bot_bot(fun(X_c,bool)))))) # label(fact_59_image__constant__conv) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 495 (all X_b all A_3 all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_3),A_1)) -> (exists B_2 (-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_3),B_2)) & hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),B_2) = ti(fun(X_b,bool),A_1))))) # label(fact_49_mk__disjoint__insert) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 496 (all X_a (semilattice_sup(X_a) -> (all X all Y all Z hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),Y),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),Y),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X),Z))))) # label(fact_277_sup__left__commute) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 497 (all X_a (bounded_lattice_bot(X_a) -> (all X ti(X_a,X) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X),bot_bot(X_a))))) # label(fact_295_sup__bot__right) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 498 (all X_b all A_3 all A_1 hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),A_1) != bot_bot(fun(X_b,bool))) # label(fact_40_empty__not__insert) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 499 (all X_c all X_b all F all X_1 all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_1)) -> hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_c,bool),fun(X_c,bool)),insert(X_c),hAPP(X_b,X_c,F,X_1)),hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),A_1)) = hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),A_1))) # label(fact_78_insert__image) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 500 (all X_b all B all C all A_1 hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),B),A_1)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),C),A_1)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),B),C)),A_1)) # label(fact_497_Int__Un__distrib2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 501 (all X_b all A_1 all B (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),B)) & hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B),A_1)) <-> ti(fun(X_b,bool),A_1) = ti(fun(X_b,bool),B))) # label(fact_339_set__eq__subset) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 502 (all X_a (semilattice_sup(X_a) -> (all B_1 all X all A_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),A_2)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),A_2),B_1))))))) # label(fact_271_le__supI1) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 503 (all X_b (ab_semigroup_mult(X_b) -> (all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (ti(fun(X_b,bool),A_1) != bot_bot(fun(X_b,bool)) -> ((all X_2 all Y_2 hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),X_2),Y_2)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Y_2),bot_bot(fun(X_b,bool))))))) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),A_1)),A_1)))))))) # label(fact_232_fold1__in) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 504 (all X_b ti(fun(fun(glb_1,X_b),fun(fun(loc_1,X_b),fun(vname,X_b))),vname_rec(X_b)) = vname_rec(X_b)) # label(tsy_c_Com_Ovname_Ovname__rec_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 505 (all X_c all X_b ti(fun(fun(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),X_b))),fun(hoare_1656922687triple(X_c),X_b)),hoare_1632998903le_rec(X_c,X_b)) = hoare_1632998903le_rec(X_c,X_b)) # label(tsy_c_Hoare__Mirabelle__nrugjuseim_Otriple_Otriple__rec_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 506 (all X_b all X_2 all Xa hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fdisj),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),X_2))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),Xa))) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),Xa)) # label(fact_34_insert__compr__raw) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 507 (all Loc_2 all Fun_1 all Com_1 all Com1 all Com2 hAPP(com,com,hAPP(com,fun(com,com),semi,Com1),Com2) != hAPP(com,com,hAPP(fun(state,nat),fun(com,com),hAPP(loc_1,fun(fun(state,nat),fun(com,com)),local,Loc_2),Fun_1),Com_1)) # label(fact_91_com_Osimps_I34_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 508 (all X_b all B all A_1 all F all F_1 (hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite2073411215e_idem(X_b),F),F_1)) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (ti(fun(X_b,bool),B) != bot_bot(fun(X_b,bool)) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B),A_1)) -> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(fun(X_b,bool),X_b,F_1,B)),hAPP(fun(X_b,bool),X_b,F_1,A_1)) = hAPP(fun(X_b,bool),X_b,F_1,A_1)))))) # label(fact_235_folding__one__idem_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 509 (all X_a all X_b all X_c combs(X_a,X_b,X_c) = ti(fun(fun(X_a,fun(X_b,X_c)),fun(fun(X_a,X_b),fun(X_a,X_c))),combs(X_a,X_b,X_c))) # label(tsy_c_COMBS_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 510 (all P all Q (-hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fimplies,P),Q)) | -hBOOL(P) | hBOOL(Q))) # label(help_fimplies_3_1_U) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 511 (all X_a all X all Y (ti(X_a,X) = ti(X_a,Y) | -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),fequal(X_a),X),Y)))) # label(help_fequal_1_1_T) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 512 (all X_c all X_b all F all Z_1 ti(X_b,Z_1) = hAPP(fun(X_c,bool),X_b,hAPP(X_b,fun(fun(X_c,bool),X_b),hAPP(fun(X_c,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,bool),X_b)),finite_fold(X_c,X_b),F),Z_1),bot_bot(fun(X_c,bool)))) # label(fact_213_fold__empty) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 513 (all X_b (ab_semigroup_mult(X_b) -> (all A_3 all Ba all A_1 all X_1 (hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),finite_fold_graph(X_b,X_b),times_times(X_b)),Ba),A_1),X_1)) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_3),A_1)) -> (-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ba),A_1)) -> hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),finite_fold_graph(X_b,X_b),times_times(X_b)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Ba),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),bot_bot(fun(X_b,bool)))))),X_1)))))))) # label(fact_196_fold__graph__permute__diff) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 514 (all Com1_2 all Com2_2 hAPP(com,com,hAPP(com,fun(com,com),semi,Com1_2),Com2_2) != skip) # label(fact_53_com_Osimps_I12_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 515 (all X_b all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> hBOOL(hAPP(fun(fun(X_b,bool),bool),bool,finite_finite_1(fun(X_b,bool)),hAPP(fun(fun(X_b,bool),bool),fun(fun(X_b,bool),bool),collect(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),hAPP(fun(fun(X_b,bool),fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(fun(X_b,bool),bool)),combc(fun(X_b,bool),fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool))),A_1)))))) # label(fact_244_finite__Collect__subsets) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 516 (all X_b all A_1 all B all X_1 (hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B),X_1)) -> hBOOL(hAPP(X_b,bool,A_1,X_1)))) # label(fact_452_inf1D1) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 517 (all X_b all Ba all A_3 all B ((-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_3),B)) -> ti(X_b,Ba) = ti(X_b,A_3)) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Ba),B))))) # label(fact_10_insertCI) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 518 (all X_b all Ts all Ga (hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),ord_less_eq(fun(hoare_1656922687triple(X_b),bool)),Ts),Ga)) -> hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),Ts)))) # label(fact_360_asm) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 519 (all X_b hBOOL(hAPP(fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),bool,finite_comp_fun_idem(X_b,fun(X_b,bool)),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),hAPP(fun(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool)))),combb(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),X_b),hAPP(fun(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool))),combc(fun(X_b,bool),fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)))),hAPP(fun(X_b,bool),fun(X_b,fun(X_b,bool)),hAPP(fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,bool),fun(X_b,fun(X_b,bool))),combc(X_b,fun(X_b,bool),fun(X_b,bool)),insert(X_b)),bot_bot(fun(X_b,bool))))))) # label(fact_193_comp__fun__idem__remove) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 520 (all X_b all B all C all A_1 hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),B),A_1)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),C),A_1)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),B),C)),A_1)) # label(fact_498_Un__Int__distrib2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 521 (all X_b all A_1 all B hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B)),A_1))) # label(fact_437_Int__lower1) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 522 (all X_a (semilattice_inf(X_a) -> (all Z all X all Y (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Z)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),Y),Z)))))))) # label(fact_420_inf__greatest) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 523 (all X_a (semilattice_sup(X_a) -> (all A_2 all B_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),B_1),A_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),A_2),B_1)))) # label(fact_286_sup_Ocommute) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 524 (all X_c all X_b all A_1 all F all Z_1 all G all F_1 (hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite1357897459simple(X_b,X_c),F),Z_1),G),F_1)) -> (hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_1)) -> ((all X_2 (hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),X_2),A_1)) -> hAPP(X_c,X_b,G,X_2) = ti(X_b,Z_1))) -> hAPP(fun(X_c,bool),X_b,F_1,A_1) = ti(X_b,Z_1))))) # label(fact_192_folding__image__simple_Oneutral) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 525 (all X_b all X_1 all Y_1 all Pa ((-hBOOL(Pa) -> hAPP(fun(X_b,bool),X_b,the(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),combb(bool,bool,X_b),hAPP(bool,fun(bool,bool),fimplies,Pa)),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),X_1)))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),combb(bool,bool,X_b),hAPP(bool,fun(bool,bool),fimplies,hAPP(bool,bool,fNot,Pa))),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),Y_1)))) = ti(X_b,Y_1)) & (hBOOL(Pa) -> hAPP(fun(X_b,bool),X_b,the(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),combb(bool,bool,X_b),hAPP(bool,fun(bool,bool),fimplies,Pa)),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),X_1)))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),combb(bool,bool,X_b),hAPP(bool,fun(bool,bool),fimplies,hAPP(bool,bool,fNot,Pa))),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),Y_1)))) = ti(X_b,X_1)))) # label(fact_83_If__def) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 526 (all X_c all X_b all F all A_1 (-hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),A_1))) -> (exists X_2 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_1)) & -hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),A_1))),hAPP(X_c,fun(X_b,bool),hAPP(fun(X_b,fun(X_c,bool)),fun(X_c,fun(X_b,bool)),combc(X_b,X_c,bool),hAPP(fun(X_b,X_c),fun(X_b,fun(X_c,bool)),hAPP(fun(X_c,fun(X_c,bool)),fun(fun(X_b,X_c),fun(X_b,fun(X_c,bool))),combb(X_c,fun(X_c,bool),X_b),fequal(X_c)),F)),hAPP(X_b,X_c,F,X_2))))))))))) # label(fact_152_pigeonhole__infinite) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 527 (all X_a (ab_sem1668676832m_mult(X_a) -> (all A_2 all B_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_2),B_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_2),B_1))))) # label(fact_197_mult__left__idem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 528 (all C_1 all S_1 all N_2 all T_4 (hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C_1),S_1),N_2),T_4)) -> hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C_1),S_1),T_4)))) # label(fact_119_evaln__evalc) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 529 (all X_b all A_1 all B (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B)) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) <-> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),B)))))) # label(fact_173_finite__Diff2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 530 (all X_b all Ca all A_1 all B (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),B))) -> -hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),B)))) # label(fact_164_DiffD2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 531 (all X_a (semilattice_sup(X_a) -> (all Y all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X)) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X),Y) = ti(X_a,X))))) # label(fact_268_sup__absorb1) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 532 (all X_b all X_c (ord(X_c) -> (all X_1 all F all G (hBOOL(hAPP(fun(X_b,X_c),bool,hAPP(fun(X_b,X_c),fun(fun(X_b,X_c),bool),ord_less_eq(fun(X_b,X_c)),F),G)) -> hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less_eq(X_c),hAPP(X_b,X_c,F,X_1)),hAPP(X_b,X_c,G,X_1))))))) # label(fact_303_le__funE) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 533 (all X_c all X_b all A_1 all F all Z_1 all G all F_1 (hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite1357897459simple(X_b,X_c),F),Z_1),G),F_1)) -> (hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_1)) -> hAPP(fun(X_c,bool),X_b,F_1,A_1) = hAPP(fun(X_c,bool),X_b,hAPP(X_b,fun(fun(X_c,bool),X_b),hAPP(fun(X_c,X_b),fun(X_b,fun(fun(X_c,bool),X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_c,X_b),fun(X_b,fun(fun(X_c,bool),X_b))),finite_fold_image(X_b,X_c),F),G),Z_1),A_1)))) # label(fact_390_folding__image__simple_Oeq__fold__g) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 534 (all X_b all X_c (ab_semigroup_mult(X_c) -> (all Z_1 all G all H all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> ((all X_2 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_1)) -> hAPP(X_b,X_c,H,X_2) = hAPP(X_b,X_c,G,X_2))) -> hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c)),hAPP(fun(X_c,fun(X_c,X_c)),fun(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c))),finite_fold_image(X_c,X_b),times_times(X_c)),G),Z_1),A_1) = hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c)),hAPP(fun(X_c,fun(X_c,X_c)),fun(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c))),finite_fold_image(X_c,X_b),times_times(X_c)),H),Z_1),A_1)))))) # label(fact_392_fold__image__cong) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 535 (all X_b all A_1 all X_1 all B (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),B)) -> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),B) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),A_1)),B))) # label(fact_174_insert__Diff1) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 536 (all X_b all X_1 all A_1 (-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_1)) -> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),A_1)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),bot_bot(fun(X_b,bool)))) = ti(fun(X_b,bool),A_1))) # label(fact_177_Diff__insert__absorb) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 537 (all X_b all B all Ca all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),A_1)) -> (-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),B)) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),B)))))) # label(fact_161_DiffI) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 538 (all X_a (lattice(X_a) -> (all X all Y hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X),Y)))))) # label(fact_293_inf__sup__ord_I3_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 539 (all X_b all Ba all A_3 all B (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_3),B)) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Ba),B))))) # label(fact_25_insertI2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 540 (all X_b all X_c all X_a ti(fun(fun(X_b,X_c),fun(fun(X_a,X_b),fun(X_a,X_c))),combb(X_b,X_c,X_a)) = combb(X_b,X_c,X_a)) # label(tsy_c_COMBB_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 541 (all X_c all X_a all B_1_1 all B_2_1 ti(X_c,hAPP(X_a,X_c,B_1_1,B_2_1)) = hAPP(X_a,X_c,B_1_1,B_2_1)) # label(tsy_c_hAPP_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 542 (all X_b all Pa ((exists X_2 (hBOOL(hAPP(X_b,bool,Pa,X_2)) & (all Y_2 (hBOOL(hAPP(X_b,bool,Pa,Y_2)) -> ti(X_b,Y_2) = ti(X_b,X_2))))) -> hBOOL(hAPP(X_b,bool,Pa,hAPP(fun(X_b,bool),X_b,the(X_b),Pa))))) # label(fact_101_theI_H) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 543 (all X_a (semilattice_sup(X_a) -> (all X all Y (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)) -> ti(X_a,Y) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X),Y))))) # label(fact_269_sup__absorb2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 544 (all X_b all A_1 all Ca all B (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),B)) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B))))) # label(fact_329_UnI2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 545 (all X_b all A_3 all Ba all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Ba),A_1))) -> (ti(X_b,A_3) != ti(X_b,Ba) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_3),A_1))))) # label(fact_9_insertE) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 546 (all Q all P (-hBOOL(P) | hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fdisj,P),Q)))) # label(help_fdisj_1_1_U) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 547 (all X_b (lattice(X_b) -> (all B all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (bot_bot(fun(X_b,bool)) != ti(fun(X_b,bool),A_1) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B)) -> (ti(fun(X_b,bool),B) != bot_bot(fun(X_b,bool)) -> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),A_1)),hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),B)) = hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B))))))))) # label(fact_405_Sup__fin_Ounion__idem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.64 548 (all X_b all B all D all A_1 all C (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),C)) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B),D)) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),C),D)))))) # label(fact_321_Un__mono) # label(axiom) # label(non_clause). [assumption]. 2.22/2.65 549 (all X_b all A_1 all F all F_1 (hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite_folding_one(X_b),F),F_1)) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (bot_bot(fun(X_b,bool)) != ti(fun(X_b,bool),A_1) -> ((all X_2 all Y_2 hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,X_2),Y_2)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Y_2),bot_bot(fun(X_b,bool))))))) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(fun(X_b,bool),X_b,F_1,A_1)),A_1))))))) # label(fact_148_folding__one_Oclosed) # label(axiom) # label(non_clause). [assumption]. 2.22/2.65 550 (all X_b all A_3 all Pa ((exists X_2 ((all Y_2 (hBOOL(hAPP(X_b,bool,Pa,Y_2)) -> ti(X_b,X_2) = ti(X_b,Y_2))) & hBOOL(hAPP(X_b,bool,Pa,X_2)))) -> (hBOOL(hAPP(X_b,bool,Pa,A_3)) -> ti(X_b,A_3) = hAPP(fun(X_b,bool),X_b,the(X_b),Pa)))) # label(fact_100_the1__equality) # label(axiom) # label(non_clause). [assumption]. 2.22/2.65 551 (all X_3 all A_3 all S_4 all T_5 (hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,hAPP(fun(state,nat),com,hAPP(vname,fun(fun(state,nat),com),ass,X_3),A_3)),S_4),T_5)) -> hAPP(nat,state,hAPP(vname,fun(nat,state),hAPP(state,fun(vname,fun(nat,state)),update,S_4),X_3),hAPP(state,nat,A_3,S_4)) = T_5)) # label(fact_116_evalc__elim__cases_I2_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.65 552 (all X_b ti(fun(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool)),big_semilattice_big(X_b)) = big_semilattice_big(X_b)) # label(tsy_c_Big__Operators_Osemilattice__big_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.65 553 (all X_b all X_c finite_fold_graph(X_b,X_c) = ti(fun(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool)))),finite_fold_graph(X_b,X_c))) # label(tsy_c_Finite__Set_Ofold__graph_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.65 554 (all X_b all B all X_1 all A_1 (-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_1)) -> (-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),B)) -> (hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),B) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),A_1) <-> ti(fun(X_b,bool),A_1) = ti(fun(X_b,bool),B))))) # label(fact_26_insert__ident) # label(axiom) # label(non_clause). [assumption]. 2.22/2.65 555 (all X_b all B all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool))),finite_fold(X_b,fun(X_b,bool)),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),hAPP(fun(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool)))),combb(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),X_b),hAPP(fun(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool))),combc(fun(X_b,bool),fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)))),hAPP(fun(X_b,bool),fun(X_b,fun(X_b,bool)),hAPP(fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,bool),fun(X_b,fun(X_b,bool))),combc(X_b,fun(X_b,bool),fun(X_b,bool)),insert(X_b)),bot_bot(fun(X_b,bool))))),B),A_1) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),B),A_1))) # label(fact_230_minus__fold__remove) # label(axiom) # label(non_clause). [assumption]. 2.22/2.65 556 (all X_b all A_3 -hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_3),bot_bot(fun(X_b,bool))))) # label(fact_11_emptyE) # label(axiom) # label(non_clause). [assumption]. 2.22/2.65 557 (all X_c all X_b all F all Z_1 all A_1 hAPP(fun(X_c,bool),X_b,hAPP(X_b,fun(fun(X_c,bool),X_b),hAPP(fun(X_c,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,bool),X_b)),finite_fold(X_c,X_b),F),Z_1),A_1) = hAPP(fun(X_b,bool),X_b,the(X_b),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_c,bool),fun(X_b,bool)),hAPP(fun(X_c,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,bool),fun(X_b,bool))),finite_fold_graph(X_c,X_b),F),Z_1),A_1))) # label(fact_220_fold__def) # label(axiom) # label(non_clause). [assumption]. 2.22/2.65 558 (all X_b all R_1 all S all X_2 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),R_1),S))) <-> hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),R_1)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),S)),X_2)))) # label(fact_451_inf__Int__eq) # label(axiom) # label(non_clause). [assumption]. 2.22/2.65 559 (all X_b (lattice(X_b) -> (all X_1 all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (bot_bot(fun(X_b,bool)) != ti(fun(X_b,bool),A_1) -> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),X_1),hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),A_1)) = hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),A_1))))))) # label(fact_402_Sup__fin_Oinsert__idem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.65 560 (all X_c (ord(X_c) -> ord_less_eq(X_c) = ti(fun(X_c,fun(X_c,bool)),ord_less_eq(X_c)))) # label(tsy_c_Orderings_Oord__class_Oless__eq_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.65 561 (all Glb_1 all Loc_1 hAPP(loc_1,vname,loc,Loc_1) != hAPP(glb_1,vname,glb,Glb_1)) # label(fact_147_vname_Osimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.65 562 (all X_b all Pa all A_1 all B ((all X_2 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B))) -> hBOOL(hAPP(X_b,bool,Pa,X_2)))) <-> (all X_2 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),B)) -> hBOOL(hAPP(X_b,bool,Pa,X_2)))) & (all X_2 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_1)) -> hBOOL(hAPP(X_b,bool,Pa,X_2)))))) # label(fact_333_ball__Un) # label(axiom) # label(non_clause). [assumption]. 2.22/2.65 563 (all X_a (semilattice_sup(X_a) -> (all A_2 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),A_2),A_2) = ti(X_a,A_2)))) # label(fact_289_sup_Oidem) # label(axiom) # label(non_clause). [assumption]. 2.22/2.65 564 (all X_b all A_1 all B hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B)) # label(fact_459_Int__left__absorb) # label(axiom) # label(non_clause). [assumption]. 2.22/2.65 565 (all X_b all X_c all F all A_1 (ti(fun(X_c,bool),A_1) = bot_bot(fun(X_c,bool)) <-> hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),A_1) = bot_bot(fun(X_b,bool)))) # label(fact_74_empty__is__image) # label(axiom) # label(non_clause). [assumption]. 2.22/2.65 566 (all X_a (lattice(X_a) -> (all X all Y hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),Y),X) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X),Y)))) # label(fact_474_inf__sup__aci_I1_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.65 567 (all X_b all A_1 all Ba ((-hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Ba),bot_bot(fun(X_b,bool))))) -> hAPP(fun(X_b,bool),X_b,the(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Ba),bot_bot(fun(X_b,bool)))))) = hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),partial_flat_lub(X_b),Ba),A_1)) & (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Ba),bot_bot(fun(X_b,bool))))) -> hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),partial_flat_lub(X_b),Ba),A_1) = ti(X_b,Ba)))) # label(fact_383_flat__lub__def) # label(axiom) # label(non_clause). [assumption]. 2.22/2.65 568 (all X_b all Ga hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),bot_bot(fun(hoare_1656922687triple(X_b),bool))))) # label(fact_0_empty) # label(axiom) # label(non_clause). [assumption]. 2.22/2.65 569 (all X_b all A_1 all B (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_1),B)) -> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B) = ti(fun(X_b,bool),A_1))) # label(fact_435_Int__absorb2) # label(axiom) # label(non_clause). [assumption]. 2.22/2.65 570 (all X_b ti(fun(nat,fun(hoare_1656922687triple(X_b),bool)),hoare_920331057_valid(X_b)) = hoare_920331057_valid(X_b)) # label(tsy_c_Hoare__Mirabelle__nrugjuseim_Otriple__valid_res) # label(axiom) # label(non_clause). [assumption]. 2.22/2.65 571 (all X_a (semilattice_sup(X_a) -> (all B_1 all A_2 all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),X)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_1),X)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),A_2),B_1)),X))))))) # label(fact_267_le__supI) # label(axiom) # label(non_clause). [assumption]. 2.22/2.65 572 (all Com1_1 all Com2_1 all Com1 all Com2 (Com2_1 = Com2 & Com1_1 = Com1 <-> hAPP(com,com,hAPP(com,fun(com,com),semi,Com1),Com2) = hAPP(com,com,hAPP(com,fun(com,com),semi,Com1_1),Com2_1))) # label(fact_55_com_Osimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.65 573 (all X_b (lattice(X_b) -> (all B all A_1 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_1)) -> (ti(fun(X_b,bool),A_1) != bot_bot(fun(X_b,bool)) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B)) -> (ti(fun(X_b,bool),B) != bot_bot(fun(X_b,bool)) -> (bot_bot(fun(X_b,bool)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_1),B) -> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),A_1)),hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),B)) = hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_1),B)))))))))) # label(fact_412_Sup__fin_Ounion__disjoint) # label(axiom) # label(non_clause). [assumption]. 2.22/2.65 574 (all X_c all X_b all F all A_1 all B (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B)) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),A_1))) -> (exists C_2 (hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),C_2) = ti(fun(X_b,bool),B) & hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),C_2)) & hBOOL(hAPP(fun(X_c,bool),bool,hAPP(fun(X_c,bool),fun(fun(X_c,bool),bool),ord_less_eq(fun(X_c,bool)),C_2),A_1))))))) # label(fact_377_finite__subset__image) # label(axiom) # label(non_clause). [assumption]. 2.22/2.66 575 (all P all Q (hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fimplies,P),Q)) | -hBOOL(Q))) # label(help_fimplies_2_1_U) # label(axiom) # label(non_clause). [assumption]. 2.22/2.66 576 (all X_b all F1 all F2 all Loc_2 hAPP(loc_1,X_b,F2,Loc_2) = hAPP(vname,X_b,hAPP(fun(loc_1,X_b),fun(vname,X_b),hAPP(fun(glb_1,X_b),fun(fun(loc_1,X_b),fun(vname,X_b)),vname_case(X_b),F1),F2),hAPP(loc_1,vname,loc,Loc_2))) # label(fact_103_vname_Osimps_I6_J) # label(axiom) # label(non_clause). [assumption]. 2.22/2.66 577 (all X_b all A_3 all A_1 (ti(fun(X_b,bool),A_1) = bot_bot(fun(X_b,bool)) -> -hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_3),A_1)))) # label(fact_17_equals0D) # label(axiom) # label(non_clause). [assumption]. 2.22/2.66 578 (all X_b all Ba all A_3 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ba),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),bot_bot(fun(X_b,bool))))) <-> ti(X_b,A_3) = ti(X_b,Ba))) # label(fact_38_singleton__iff) # label(axiom) # label(non_clause). [assumption]. 2.22/2.66 2.22/2.66 ============================== end of process non-clausal formulas === 2.22/2.66 2.22/2.66 ============================== PROCESS INITIAL CLAUSES =============== 2.22/2.66 2.22/2.66 ============================== PREDICATE ELIMINATION ================= 2.22/2.66 579 -semilattice_inf(A) | hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),B),C) = hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),C),B) # label(fact_473_inf_Ocommute) # label(axiom). [clausify(27)]. 2.22/2.66 580 -lattice(A) | semilattice_inf(fun(B,A)) # label(arity_fun___Lattices_Osemilattice__inf) # label(axiom). [clausify(12)]. 2.22/2.66 Derived: hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_inf_inf(fun(A,B)),C),D) = hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_inf_inf(fun(A,B)),D),C) | -lattice(B). [resolve(579,a,580,b)]. 2.22/2.66 581 -semilattice_inf(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),B),C) # label(fact_427_le__iff__inf) # label(axiom). [clausify(32)]. 2.22/2.66 Derived: -hBOOL(hAPP(fun(A,B),bool,hAPP(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),C),D)) | ti(fun(A,B),C) = hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_inf_inf(fun(A,B)),C),D) | -lattice(B). [resolve(581,a,580,b)]. 2.22/2.66 582 -semilattice_inf(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | ti(A,B) != hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),B),C) # label(fact_427_le__iff__inf) # label(axiom). [clausify(32)]. 2.22/2.66 Derived: hBOOL(hAPP(fun(A,B),bool,hAPP(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),C),D)) | ti(fun(A,B),C) != hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_inf_inf(fun(A,B)),C),D) | -lattice(B). [resolve(582,a,580,b)]. 2.22/2.66 583 -semilattice_inf(A) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),B),B) # label(fact_470_inf_Oidem) # label(axiom). [clausify(34)]. 2.22/2.66 Derived: ti(fun(A,B),C) = hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_inf_inf(fun(A,B)),C),C) | -lattice(B). [resolve(583,a,580,b)]. 2.22/2.66 584 semilattice_inf(nat) # label(arity_Nat_Onat___Lattices_Osemilattice__inf) # label(axiom). [assumption]. 2.22/2.66 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_inf_inf(nat),A),B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_inf_inf(nat),B),A). [resolve(584,a,579,a)]. 2.22/2.66 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | ti(nat,A) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_inf_inf(nat),A),B). [resolve(584,a,581,a)]. 2.22/2.66 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | ti(nat,A) != hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_inf_inf(nat),A),B). [resolve(584,a,582,a)]. 2.22/2.66 Derived: ti(nat,A) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_inf_inf(nat),A),A). [resolve(584,a,583,a)]. 2.22/2.66 585 -semilattice_inf(A) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),B),B) # label(fact_471_inf__idem) # label(axiom). [clausify(97)]. 2.22/2.66 586 -semilattice_inf(A) | hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),B),hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),B),C)) = hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),B),C) # label(fact_478_inf__left__idem) # label(axiom). [clausify(103)]. 2.22/2.66 Derived: hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_inf_inf(fun(A,B)),C),hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_inf_inf(fun(A,B)),C),D)) = hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_inf_inf(fun(A,B)),C),D) | -lattice(B). [resolve(586,a,580,b)]. 2.22/2.66 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_inf_inf(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_inf_inf(nat),A),B)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_inf_inf(nat),A),B). [resolve(586,a,584,a)]. 2.22/2.66 587 -semilattice_inf(A) | hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),B),hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),C),D)) = hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),C),hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),B),D)) # label(fact_479_inf_Oleft__commute) # label(axiom). [clausify(137)]. 2.22/2.66 Derived: hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_inf_inf(fun(A,B)),C),hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_inf_inf(fun(A,B)),D),E)) = hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_inf_inf(fun(A,B)),D),hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_inf_inf(fun(A,B)),C),E)) | -lattice(B). [resolve(587,a,580,b)]. 2.22/2.66 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_inf_inf(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_inf_inf(nat),B),C)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_inf_inf(nat),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_inf_inf(nat),A),C)). [resolve(587,a,584,a)]. 2.22/2.66 588 -semilattice_inf(A) | hBOOL(hAPP(fun(A,fun(A,A)),bool,finite_comp_fun_idem(A,A),semilattice_inf_inf(A))) # label(fact_443_comp__fun__idem__inf) # label(axiom). [clausify(138)]. 2.22/2.66 Derived: hBOOL(hAPP(fun(fun(A,B),fun(fun(A,B),fun(A,B))),bool,finite_comp_fun_idem(fun(A,B),fun(A,B)),semilattice_inf_inf(fun(A,B)))) | -lattice(B). [resolve(588,a,580,b)]. 2.22/2.66 Derived: hBOOL(hAPP(fun(nat,fun(nat,nat)),bool,finite_comp_fun_idem(nat,nat),semilattice_inf_inf(nat))). [resolve(588,a,584,a)]. 2.22/2.66 589 -semilattice_inf(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),D),B)),C)) # label(fact_424_le__infI2) # label(axiom). [clausify(178)]. 2.22/2.66 Derived: -hBOOL(hAPP(fun(A,B),bool,hAPP(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),C),D)) | hBOOL(hAPP(fun(A,B),bool,hAPP(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_inf_inf(fun(A,B)),E),C)),D)) | -lattice(B). [resolve(589,a,580,b)]. 2.22/2.66 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_inf_inf(nat),C),A)),B)). [resolve(589,a,584,a)]. 2.22/2.66 590 -semilattice_inf(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),B),C) # label(fact_423_inf__absorb1) # label(axiom). [clausify(191)]. 2.22/2.66 591 -semilattice_inf(A) | hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),B),hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),B),C)) = hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),B),C) # label(fact_476_inf_Oleft__idem) # label(axiom). [clausify(202)]. 2.22/2.66 592 -semilattice_inf(A) | hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),B),hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),C),D)) # label(fact_482_inf_Oassoc) # label(axiom). [clausify(257)]. 2.22/2.67 Derived: hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_inf_inf(fun(A,B)),hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_inf_inf(fun(A,B)),C),D)),E) = hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_inf_inf(fun(A,B)),C),hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_inf_inf(fun(A,B)),D),E)) | -lattice(B). [resolve(592,a,580,b)]. 2.22/2.67 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_inf_inf(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_inf_inf(nat),A),B)),C) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_inf_inf(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_inf_inf(nat),B),C)). [resolve(592,a,584,a)]. 2.22/2.67 593 -semilattice_inf(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),D),C))) # label(fact_426_le__inf__iff) # label(axiom). [clausify(269)]. 2.22/2.67 Derived: -hBOOL(hAPP(fun(A,B),bool,hAPP(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),C),D)) | -hBOOL(hAPP(fun(A,B),bool,hAPP(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),C),E)) | hBOOL(hAPP(fun(A,B),bool,hAPP(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),C),hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_inf_inf(fun(A,B)),E),D))) | -lattice(B). [resolve(593,a,580,b)]. 2.22/2.67 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_inf_inf(nat),C),B))). [resolve(593,a,584,a)]. 2.22/2.67 594 -semilattice_inf(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),D),C))) # label(fact_426_le__inf__iff) # label(axiom). [clausify(269)]. 2.22/2.67 Derived: hBOOL(hAPP(fun(A,B),bool,hAPP(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),C),D)) | -hBOOL(hAPP(fun(A,B),bool,hAPP(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),C),hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_inf_inf(fun(A,B)),E),D))) | -lattice(B). [resolve(594,a,580,b)]. 2.22/2.67 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_inf_inf(nat),C),B))). [resolve(594,a,584,a)]. 2.22/2.67 595 -semilattice_inf(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),C),D))) # label(fact_426_le__inf__iff) # label(axiom). [clausify(269)]. 2.22/2.67 Derived: hBOOL(hAPP(fun(A,B),bool,hAPP(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),C),D)) | -hBOOL(hAPP(fun(A,B),bool,hAPP(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),C),hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_inf_inf(fun(A,B)),D),E))) | -lattice(B). [resolve(595,a,580,b)]. 2.22/2.67 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_inf_inf(nat),B),C))). [resolve(595,a,584,a)]. 2.22/2.67 596 -semilattice_inf(A) | hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),B),hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),C),D)) # label(fact_484_inf__assoc) # label(axiom). [clausify(315)]. 2.22/2.67 597 -semilattice_inf(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),C),B) # label(fact_422_inf__absorb2) # label(axiom). [clausify(333)]. 2.22/2.67 Derived: -hBOOL(hAPP(fun(A,B),bool,hAPP(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),C),D)) | ti(fun(A,B),C) = hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_inf_inf(fun(A,B)),D),C) | -lattice(B). [resolve(597,a,580,b)]. 2.22/2.68 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | ti(nat,A) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_inf_inf(nat),B),A). [resolve(597,a,584,a)]. 2.22/2.68 598 -semilattice_inf(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),C),D))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) # label(fact_418_le__infE) # label(axiom). [clausify(334)]. 2.22/2.68 599 -semilattice_inf(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),C),D))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),D)) # label(fact_418_le__infE) # label(axiom). [clausify(334)]. 2.22/2.68 600 -semilattice_inf(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),B),D)),C)) # label(fact_425_le__infI1) # label(axiom). [clausify(351)]. 2.22/2.68 Derived: -hBOOL(hAPP(fun(A,B),bool,hAPP(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),C),D)) | hBOOL(hAPP(fun(A,B),bool,hAPP(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_inf_inf(fun(A,B)),C),E)),D)) | -lattice(B). [resolve(600,a,580,b)]. 2.22/2.68 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_inf_inf(nat),A),C)),B)). [resolve(600,a,584,a)]. 2.22/2.68 601 -semilattice_inf(A) | semilattice_inf_inf(A) = ti(fun(A,fun(A,A)),semilattice_inf_inf(A)) # label(tsy_c_Lattices_Osemilattice__inf__class_Oinf_res) # label(axiom). [clausify(358)]. 2.22/2.68 Derived: semilattice_inf_inf(fun(A,B)) = ti(fun(fun(A,B),fun(fun(A,B),fun(A,B))),semilattice_inf_inf(fun(A,B))) | -lattice(B). [resolve(601,a,580,b)]. 2.22/2.68 Derived: semilattice_inf_inf(nat) = ti(fun(nat,fun(nat,nat)),semilattice_inf_inf(nat)). [resolve(601,a,584,a)]. 2.22/2.68 602 -semilattice_inf(A) | hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),B),hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),C),D)) = hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),C),hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),B),D)) # label(fact_481_inf__left__commute) # label(axiom). [clausify(389)]. 2.22/2.68 603 -semilattice_inf(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),B),C)),C)) # label(fact_428_inf__le2) # label(axiom). [clausify(395)]. 2.22/2.68 Derived: hBOOL(hAPP(fun(A,B),bool,hAPP(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_inf_inf(fun(A,B)),C),D)),D)) | -lattice(B). [resolve(603,a,580,b)]. 2.22/2.68 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_inf_inf(nat),A),B)),B)). [resolve(603,a,584,a)]. 2.22/2.68 604 -semilattice_inf(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),E)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),B),D)),hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),C),E))) # label(fact_419_inf__mono) # label(axiom). [clausify(461)]. 2.22/2.68 Derived: -hBOOL(hAPP(fun(A,B),bool,hAPP(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),C),D)) | -hBOOL(hAPP(fun(A,B),bool,hAPP(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),E),F)) | hBOOL(hAPP(fun(A,B),bool,hAPP(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_inf_inf(fun(A,B)),C),E)),hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_inf_inf(fun(A,B)),D),F))) | -lattice(B). [resolve(604,a,580,b)]. 2.22/2.68 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),D)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_inf_inf(nat),A),C)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_inf_inf(nat),B),D))). [resolve(604,a,584,a)]. 2.22/2.70 605 -semilattice_inf(A) | hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),B),C) = hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),C),B) # label(fact_475_inf__commute) # label(axiom). [clausify(465)]. 2.22/2.70 606 -semilattice_inf(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),C),D))) # label(fact_421_le__infI) # label(axiom). [clausify(479)]. 2.22/2.70 607 -semilattice_inf(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),B),C)),B)) # label(fact_430_inf__le1) # label(axiom). [clausify(487)]. 2.22/2.70 Derived: hBOOL(hAPP(fun(A,B),bool,hAPP(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_inf_inf(fun(A,B)),C),D)),C)) | -lattice(B). [resolve(607,a,580,b)]. 2.22/2.70 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_inf_inf(nat),A),B)),A)). [resolve(607,a,584,a)]. 2.22/2.70 608 -semilattice_inf(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),C),D))) # label(fact_420_inf__greatest) # label(axiom). [clausify(522)]. 2.22/2.70 609 semilattice_inf(bool) # label(arity_HOL_Obool___Lattices_Osemilattice__inf) # label(axiom). [assumption]. 2.22/2.70 Derived: hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_inf_inf(bool),A),B) = hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_inf_inf(bool),B),A). [resolve(609,a,579,a)]. 2.22/2.70 Derived: -hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),A),B)) | ti(bool,A) = hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_inf_inf(bool),A),B). [resolve(609,a,581,a)]. 2.22/2.70 Derived: hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),A),B)) | ti(bool,A) != hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_inf_inf(bool),A),B). [resolve(609,a,582,a)]. 2.22/2.70 Derived: ti(bool,A) = hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_inf_inf(bool),A),A). [resolve(609,a,583,a)]. 2.22/2.70 Derived: hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_inf_inf(bool),A),hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_inf_inf(bool),A),B)) = hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_inf_inf(bool),A),B). [resolve(609,a,586,a)]. 2.22/2.70 Derived: hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_inf_inf(bool),A),hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_inf_inf(bool),B),C)) = hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_inf_inf(bool),B),hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_inf_inf(bool),A),C)). [resolve(609,a,587,a)]. 2.22/2.70 Derived: hBOOL(hAPP(fun(bool,fun(bool,bool)),bool,finite_comp_fun_idem(bool,bool),semilattice_inf_inf(bool))). [resolve(609,a,588,a)]. 2.22/2.70 Derived: -hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),A),B)) | hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_inf_inf(bool),C),A)),B)). [resolve(609,a,589,a)]. 2.22/2.70 Derived: hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_inf_inf(bool),hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_inf_inf(bool),A),B)),C) = hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_inf_inf(bool),A),hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_inf_inf(bool),B),C)). [resolve(609,a,592,a)]. 2.22/2.70 Derived: -hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),A),B)) | -hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),A),C)) | hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),A),hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_inf_inf(bool),C),B))). [resolve(609,a,593,a)]. 2.22/2.70 Derived: hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),A),B)) | -hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),A),hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_inf_inf(bool),C),B))). [resolve(609,a,594,a)]. 2.22/2.75 Derived: hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),A),B)) | -hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),A),hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_inf_inf(bool),B),C))). [resolve(609,a,595,a)]. 2.22/2.75 Derived: -hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),A),B)) | ti(bool,A) = hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_inf_inf(bool),B),A). [resolve(609,a,597,a)]. 2.22/2.75 Derived: -hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),A),B)) | hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_inf_inf(bool),A),C)),B)). [resolve(609,a,600,a)]. 2.22/2.75 Derived: semilattice_inf_inf(bool) = ti(fun(bool,fun(bool,bool)),semilattice_inf_inf(bool)). [resolve(609,a,601,a)]. 2.22/2.75 Derived: hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_inf_inf(bool),A),B)),B)). [resolve(609,a,603,a)]. 2.22/2.75 Derived: -hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),A),B)) | -hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),C),D)) | hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_inf_inf(bool),A),C)),hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_inf_inf(bool),B),D))). [resolve(609,a,604,a)]. 2.22/2.75 Derived: hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_inf_inf(bool),A),B)),A)). [resolve(609,a,607,a)]. 2.22/2.75 610 semilattice_sup(bool) # label(arity_HOL_Obool___Lattices_Osemilattice__sup) # label(axiom). [assumption]. 2.22/2.75 611 -semilattice_sup(A) | hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),B),C) = hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),C),B) # label(fact_284_sup__commute) # label(axiom). [clausify(39)]. 2.22/2.75 612 -semilattice_sup(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),B),C))) # label(fact_292_sup__ge1) # label(axiom). [clausify(44)]. 2.22/2.75 613 -semilattice_sup(A) | hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),B),hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),B),C)) = hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),B),C) # label(fact_280_sup__left__idem) # label(axiom). [clausify(49)]. 2.22/2.75 614 -semilattice_sup(A) | hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),B),hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),C),D)) # label(fact_276_sup_Oassoc) # label(axiom). [clausify(71)]. 2.22/2.75 615 -semilattice_sup(A) | ti(fun(A,fun(A,A)),semilattice_sup_sup(A)) = semilattice_sup_sup(A) # label(tsy_c_Lattices_Osemilattice__sup__class_Osup_res) # label(axiom). [clausify(73)]. 2.22/2.75 Derived: hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),A),B) = hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),B),A). [resolve(610,a,611,a)]. 2.22/2.75 Derived: hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),A),hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),A),B))). [resolve(610,a,612,a)]. 2.22/2.75 Derived: hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),A),hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),A),B)) = hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),A),B). [resolve(610,a,613,a)]. 2.22/2.75 Derived: hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),A),B)),C) = hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),A),hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),B),C)). [resolve(610,a,614,a)]. 2.22/2.75 Derived: ti(fun(bool,fun(bool,bool)),semilattice_sup_sup(bool)) = semilattice_sup_sup(bool). [resolve(610,a,615,a)]. 2.22/2.75 616 -semilattice_sup(A) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),B),B) # label(fact_288_sup__idem) # label(axiom). [clausify(77)]. 2.22/2.75 Derived: ti(bool,A) = hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),A),A). [resolve(616,a,610,a)]. 2.22/2.75 617 -semilattice_sup(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),B),C)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),D)) # label(fact_264_le__supE) # label(axiom). [clausify(124)]. 2.22/2.75 Derived: -hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),A),B)),C)) | hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),A),C)). [resolve(617,a,610,a)]. 2.22/2.75 618 -semilattice_sup(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),B),C)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),D)) # label(fact_264_le__supE) # label(axiom). [clausify(124)]. 2.22/2.75 Derived: -hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),A),B)),C)) | hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),B),C)). [resolve(618,a,610,a)]. 2.22/2.75 619 -semilattice_sup(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),D),C))) # label(fact_270_le__supI2) # label(axiom). [clausify(140)]. 2.22/2.75 Derived: -hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),A),B)) | hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),A),hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),C),B))). [resolve(619,a,610,a)]. 2.22/2.75 620 -semilattice_sup(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),E)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),B),D)),hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),C),E))) # label(fact_265_sup__mono) # label(axiom). [clausify(152)]. 2.22/2.75 Derived: -hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),A),B)) | -hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),C),D)) | hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),A),C)),hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),B),D))). [resolve(620,a,610,a)]. 2.22/2.75 621 -semilattice_sup(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),D),B)),C)) # label(fact_273_le__sup__iff) # label(axiom). [clausify(161)]. 2.22/2.75 Derived: -hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),A),B)) | -hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),C),B)) | hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),C),A)),B)). [resolve(621,a,610,a)]. 2.22/2.75 622 -semilattice_sup(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),D),B)),C)) # label(fact_273_le__sup__iff) # label(axiom). [clausify(161)]. 2.22/2.75 623 -semilattice_sup(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),B),D)),C)) # label(fact_273_le__sup__iff) # label(axiom). [clausify(161)]. 2.22/2.75 624 semilattice_sup(nat) # label(arity_Nat_Onat___Lattices_Osemilattice__sup) # label(axiom). [assumption]. 2.22/2.75 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_sup_sup(nat),A),B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_sup_sup(nat),B),A). [resolve(624,a,611,a)]. 2.22/2.75 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_sup_sup(nat),A),B))). [resolve(624,a,612,a)]. 2.22/2.76 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_sup_sup(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_sup_sup(nat),A),B)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_sup_sup(nat),A),B). [resolve(624,a,613,a)]. 2.22/2.76 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_sup_sup(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_sup_sup(nat),A),B)),C) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_sup_sup(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_sup_sup(nat),B),C)). [resolve(624,a,614,a)]. 2.22/2.76 Derived: ti(fun(nat,fun(nat,nat)),semilattice_sup_sup(nat)) = semilattice_sup_sup(nat). [resolve(624,a,615,a)]. 2.22/2.76 Derived: ti(nat,A) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_sup_sup(nat),A),A). [resolve(624,a,616,a)]. 2.22/2.76 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_sup_sup(nat),A),B)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),C)). [resolve(624,a,617,a)]. 2.22/2.76 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_sup_sup(nat),A),B)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)). [resolve(624,a,618,a)]. 2.22/2.76 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_sup_sup(nat),C),B))). [resolve(624,a,619,a)]. 2.22/2.76 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),D)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_sup_sup(nat),A),C)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_sup_sup(nat),B),D))). [resolve(624,a,620,a)]. 2.22/2.76 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_sup_sup(nat),C),A)),B)). [resolve(624,a,621,a)]. 2.22/2.76 625 -semilattice_sup(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),C),B))) # label(fact_290_sup__ge2) # label(axiom). [clausify(200)]. 2.22/2.76 Derived: hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),A),hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),B),A))). [resolve(625,a,610,a)]. 2.22/2.76 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_sup_sup(nat),B),A))). [resolve(625,a,624,a)]. 2.22/2.76 626 -semilattice_sup(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),B),D)),C)) # label(fact_266_sup__least) # label(axiom). [clausify(303)]. 2.22/2.76 627 -semilattice_sup(A) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f60(A,C,D,B)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(fun(A,bool),A,hAPP(A,fun(fun(A,bool),A),hAPP(fun(A,fun(A,A)),fun(A,fun(fun(A,bool),A)),finite_fold(A,A),semilattice_sup_sup(A)),C),B)),hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),D),C))) # label(fact_374_fold__sup__le__sup) # label(axiom). [clausify(321)]. 2.22/2.76 Derived: -hBOOL(hAPP(fun(bool,bool),bool,finite_finite_1(bool),A)) | hBOOL(hAPP(fun(bool,bool),bool,hAPP(bool,fun(fun(bool,bool),bool),member(bool),f60(bool,B,C,A)),A)) | hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),hAPP(fun(bool,bool),bool,hAPP(bool,fun(fun(bool,bool),bool),hAPP(fun(bool,fun(bool,bool)),fun(bool,fun(fun(bool,bool),bool)),finite_fold(bool,bool),semilattice_sup_sup(bool)),B),A)),hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),C),B))). [resolve(627,a,610,a)]. 2.22/2.76 Derived: -hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),A)) | hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),f60(nat,B,C,A)),A)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(nat,bool),nat,hAPP(nat,fun(fun(nat,bool),nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(fun(nat,bool),nat)),finite_fold(nat,nat),semilattice_sup_sup(nat)),B),A)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_sup_sup(nat),C),B))). [resolve(627,a,624,a)]. 2.22/2.77 628 -semilattice_sup(A) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),f60(A,C,D,B)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(fun(A,bool),A,hAPP(A,fun(fun(A,bool),A),hAPP(fun(A,fun(A,A)),fun(A,fun(fun(A,bool),A)),finite_fold(A,A),semilattice_sup_sup(A)),C),B)),hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),D),C))) # label(fact_374_fold__sup__le__sup) # label(axiom). [clausify(321)]. 2.22/2.77 Derived: -hBOOL(hAPP(fun(bool,bool),bool,finite_finite_1(bool),A)) | -hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),f60(bool,B,C,A)),C)) | hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),hAPP(fun(bool,bool),bool,hAPP(bool,fun(fun(bool,bool),bool),hAPP(fun(bool,fun(bool,bool)),fun(bool,fun(fun(bool,bool),bool)),finite_fold(bool,bool),semilattice_sup_sup(bool)),B),A)),hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),C),B))). [resolve(628,a,610,a)]. 2.22/2.77 Derived: -hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),f60(nat,B,C,A)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(nat,bool),nat,hAPP(nat,fun(fun(nat,bool),nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(fun(nat,bool),nat)),finite_fold(nat,nat),semilattice_sup_sup(nat)),B),A)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_sup_sup(nat),C),B))). [resolve(628,a,624,a)]. 2.22/2.77 629 -semilattice_sup(A) | hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),B),hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),C),D)) # label(fact_274_sup__assoc) # label(axiom). [clausify(345)]. 2.22/2.77 630 -semilattice_sup(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | ti(A,C) = hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),B),C) # label(fact_283_le__iff__sup) # label(axiom). [clausify(385)]. 2.22/2.77 Derived: -hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),A),B)) | ti(bool,B) = hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),A),B). [resolve(630,a,610,a)]. 2.22/2.77 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | ti(nat,B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_sup_sup(nat),A),B). [resolve(630,a,624,a)]. 2.22/2.77 631 -semilattice_sup(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | ti(A,C) != hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),B),C) # label(fact_283_le__iff__sup) # label(axiom). [clausify(385)]. 2.22/2.77 Derived: hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),A),B)) | ti(bool,B) != hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),A),B). [resolve(631,a,610,a)]. 2.22/2.77 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | ti(nat,B) != hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_sup_sup(nat),A),B). [resolve(631,a,624,a)]. 2.22/2.77 632 -semilattice_sup(A) | hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),B),hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),C),D)) = hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),C),hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),B),D)) # label(fact_279_sup_Oleft__commute) # label(axiom). [clausify(413)]. 2.22/2.77 Derived: hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),A),hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),B),C)) = hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),B),hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),A),C)). [resolve(632,a,610,a)]. 2.22/2.77 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_sup_sup(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_sup_sup(nat),B),C)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_sup_sup(nat),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_sup_sup(nat),A),C)). [resolve(632,a,624,a)]. 2.22/2.77 633 -semilattice_sup(A) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),C),D)),hAPP(fun(A,bool),A,hAPP(A,fun(fun(A,bool),A),hAPP(fun(A,fun(A,A)),fun(A,fun(fun(A,bool),A)),finite_fold(A,A),semilattice_sup_sup(A)),D),B))) # label(fact_245_sup__le__fold__sup) # label(axiom). [clausify(432)]. 2.22/2.77 Derived: -hBOOL(hAPP(fun(bool,bool),bool,finite_finite_1(bool),A)) | -hBOOL(hAPP(fun(bool,bool),bool,hAPP(bool,fun(fun(bool,bool),bool),member(bool),B),A)) | hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),B),C)),hAPP(fun(bool,bool),bool,hAPP(bool,fun(fun(bool,bool),bool),hAPP(fun(bool,fun(bool,bool)),fun(bool,fun(fun(bool,bool),bool)),finite_fold(bool,bool),semilattice_sup_sup(bool)),C),A))). [resolve(633,a,610,a)]. 2.22/2.77 Derived: -hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),A)) | -hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),B),A)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_sup_sup(nat),B),C)),hAPP(fun(nat,bool),nat,hAPP(nat,fun(fun(nat,bool),nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(fun(nat,bool),nat)),finite_fold(nat,nat),semilattice_sup_sup(nat)),C),A))). [resolve(633,a,624,a)]. 2.22/2.77 634 -semilattice_sup(A) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hAPP(fun(A,bool),A,hAPP(A,fun(fun(A,bool),A),hAPP(fun(A,fun(A,A)),fun(A,fun(fun(A,bool),A)),finite_fold(A,A),semilattice_sup_sup(A)),C),hAPP(fun(A,bool),fun(A,bool),hAPP(A,fun(fun(A,bool),fun(A,bool)),insert(A),D),B)) = hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),D),hAPP(fun(A,bool),A,hAPP(A,fun(fun(A,bool),A),hAPP(fun(A,fun(A,A)),fun(A,fun(fun(A,bool),A)),finite_fold(A,A),semilattice_sup_sup(A)),C),B)) # label(fact_364_fold__sup__insert) # label(axiom). [clausify(442)]. 2.22/2.77 Derived: -hBOOL(hAPP(fun(bool,bool),bool,finite_finite_1(bool),A)) | hAPP(fun(bool,bool),bool,hAPP(bool,fun(fun(bool,bool),bool),hAPP(fun(bool,fun(bool,bool)),fun(bool,fun(fun(bool,bool),bool)),finite_fold(bool,bool),semilattice_sup_sup(bool)),B),hAPP(fun(bool,bool),fun(bool,bool),hAPP(bool,fun(fun(bool,bool),fun(bool,bool)),insert(bool),C),A)) = hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),C),hAPP(fun(bool,bool),bool,hAPP(bool,fun(fun(bool,bool),bool),hAPP(fun(bool,fun(bool,bool)),fun(bool,fun(fun(bool,bool),bool)),finite_fold(bool,bool),semilattice_sup_sup(bool)),B),A)). [resolve(634,a,610,a)]. 2.22/2.77 Derived: -hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),A)) | hAPP(fun(nat,bool),nat,hAPP(nat,fun(fun(nat,bool),nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(fun(nat,bool),nat)),finite_fold(nat,nat),semilattice_sup_sup(nat)),B),hAPP(fun(nat,bool),fun(nat,bool),hAPP(nat,fun(fun(nat,bool),fun(nat,bool)),insert(nat),C),A)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_sup_sup(nat),C),hAPP(fun(nat,bool),nat,hAPP(nat,fun(fun(nat,bool),nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(fun(nat,bool),nat)),finite_fold(nat,nat),semilattice_sup_sup(nat)),B),A)). [resolve(634,a,624,a)]. 2.22/2.77 635 -semilattice_sup(A) | hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),B),hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),B),C)) = hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),B),C) # label(fact_282_sup_Oleft__idem) # label(axiom). [clausify(444)]. 2.22/2.77 636 -lattice(A) | semilattice_sup(fun(B,A)) # label(arity_fun___Lattices_Osemilattice__sup) # label(axiom). [clausify(445)]. 2.22/2.77 Derived: -lattice(A) | hAPP(fun(B,A),fun(B,A),hAPP(fun(B,A),fun(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A)),C),D) = hAPP(fun(B,A),fun(B,A),hAPP(fun(B,A),fun(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A)),D),C). [resolve(636,b,611,a)]. 2.22/2.77 Derived: -lattice(A) | hBOOL(hAPP(fun(B,A),bool,hAPP(fun(B,A),fun(fun(B,A),bool),ord_less_eq(fun(B,A)),C),hAPP(fun(B,A),fun(B,A),hAPP(fun(B,A),fun(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A)),C),D))). [resolve(636,b,612,a)]. 2.22/2.78 Derived: -lattice(A) | hAPP(fun(B,A),fun(B,A),hAPP(fun(B,A),fun(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A)),C),hAPP(fun(B,A),fun(B,A),hAPP(fun(B,A),fun(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A)),C),D)) = hAPP(fun(B,A),fun(B,A),hAPP(fun(B,A),fun(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A)),C),D). [resolve(636,b,613,a)]. 2.22/2.78 Derived: -lattice(A) | hAPP(fun(B,A),fun(B,A),hAPP(fun(B,A),fun(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A)),hAPP(fun(B,A),fun(B,A),hAPP(fun(B,A),fun(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A)),C),D)),E) = hAPP(fun(B,A),fun(B,A),hAPP(fun(B,A),fun(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A)),C),hAPP(fun(B,A),fun(B,A),hAPP(fun(B,A),fun(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A)),D),E)). [resolve(636,b,614,a)]. 2.22/2.78 Derived: -lattice(A) | ti(fun(fun(B,A),fun(fun(B,A),fun(B,A))),semilattice_sup_sup(fun(B,A))) = semilattice_sup_sup(fun(B,A)). [resolve(636,b,615,a)]. 2.22/2.78 Derived: -lattice(A) | ti(fun(B,A),C) = hAPP(fun(B,A),fun(B,A),hAPP(fun(B,A),fun(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A)),C),C). [resolve(636,b,616,a)]. 2.22/2.78 Derived: -lattice(A) | -hBOOL(hAPP(fun(B,A),bool,hAPP(fun(B,A),fun(fun(B,A),bool),ord_less_eq(fun(B,A)),hAPP(fun(B,A),fun(B,A),hAPP(fun(B,A),fun(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A)),C),D)),E)) | hBOOL(hAPP(fun(B,A),bool,hAPP(fun(B,A),fun(fun(B,A),bool),ord_less_eq(fun(B,A)),C),E)). [resolve(636,b,617,a)]. 2.22/2.78 Derived: -lattice(A) | -hBOOL(hAPP(fun(B,A),bool,hAPP(fun(B,A),fun(fun(B,A),bool),ord_less_eq(fun(B,A)),hAPP(fun(B,A),fun(B,A),hAPP(fun(B,A),fun(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A)),C),D)),E)) | hBOOL(hAPP(fun(B,A),bool,hAPP(fun(B,A),fun(fun(B,A),bool),ord_less_eq(fun(B,A)),D),E)). [resolve(636,b,618,a)]. 2.22/2.78 Derived: -lattice(A) | -hBOOL(hAPP(fun(B,A),bool,hAPP(fun(B,A),fun(fun(B,A),bool),ord_less_eq(fun(B,A)),C),D)) | hBOOL(hAPP(fun(B,A),bool,hAPP(fun(B,A),fun(fun(B,A),bool),ord_less_eq(fun(B,A)),C),hAPP(fun(B,A),fun(B,A),hAPP(fun(B,A),fun(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A)),E),D))). [resolve(636,b,619,a)]. 2.22/2.78 Derived: -lattice(A) | -hBOOL(hAPP(fun(B,A),bool,hAPP(fun(B,A),fun(fun(B,A),bool),ord_less_eq(fun(B,A)),C),D)) | -hBOOL(hAPP(fun(B,A),bool,hAPP(fun(B,A),fun(fun(B,A),bool),ord_less_eq(fun(B,A)),E),F)) | hBOOL(hAPP(fun(B,A),bool,hAPP(fun(B,A),fun(fun(B,A),bool),ord_less_eq(fun(B,A)),hAPP(fun(B,A),fun(B,A),hAPP(fun(B,A),fun(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A)),C),E)),hAPP(fun(B,A),fun(B,A),hAPP(fun(B,A),fun(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A)),D),F))). [resolve(636,b,620,a)]. 2.22/2.78 Derived: -lattice(A) | -hBOOL(hAPP(fun(B,A),bool,hAPP(fun(B,A),fun(fun(B,A),bool),ord_less_eq(fun(B,A)),C),D)) | -hBOOL(hAPP(fun(B,A),bool,hAPP(fun(B,A),fun(fun(B,A),bool),ord_less_eq(fun(B,A)),E),D)) | hBOOL(hAPP(fun(B,A),bool,hAPP(fun(B,A),fun(fun(B,A),bool),ord_less_eq(fun(B,A)),hAPP(fun(B,A),fun(B,A),hAPP(fun(B,A),fun(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A)),E),C)),D)). [resolve(636,b,621,a)]. 2.22/2.78 Derived: -lattice(A) | hBOOL(hAPP(fun(B,A),bool,hAPP(fun(B,A),fun(fun(B,A),bool),ord_less_eq(fun(B,A)),C),hAPP(fun(B,A),fun(B,A),hAPP(fun(B,A),fun(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A)),D),C))). [resolve(636,b,625,a)]. 2.22/2.78 Derived: -lattice(A) | -hBOOL(hAPP(fun(fun(B,A),bool),bool,finite_finite_1(fun(B,A)),C)) | hBOOL(hAPP(fun(fun(B,A),bool),bool,hAPP(fun(B,A),fun(fun(fun(B,A),bool),bool),member(fun(B,A)),f60(fun(B,A),D,E,C)),C)) | hBOOL(hAPP(fun(B,A),bool,hAPP(fun(B,A),fun(fun(B,A),bool),ord_less_eq(fun(B,A)),hAPP(fun(fun(B,A),bool),fun(B,A),hAPP(fun(B,A),fun(fun(fun(B,A),bool),fun(B,A)),hAPP(fun(fun(B,A),fun(fun(B,A),fun(B,A))),fun(fun(B,A),fun(fun(fun(B,A),bool),fun(B,A))),finite_fold(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A))),D),C)),hAPP(fun(B,A),fun(B,A),hAPP(fun(B,A),fun(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A)),E),D))). [resolve(636,b,627,a)]. 2.22/2.78 Derived: -lattice(A) | -hBOOL(hAPP(fun(fun(B,A),bool),bool,finite_finite_1(fun(B,A)),C)) | -hBOOL(hAPP(fun(B,A),bool,hAPP(fun(B,A),fun(fun(B,A),bool),ord_less_eq(fun(B,A)),f60(fun(B,A),D,E,C)),E)) | hBOOL(hAPP(fun(B,A),bool,hAPP(fun(B,A),fun(fun(B,A),bool),ord_less_eq(fun(B,A)),hAPP(fun(fun(B,A),bool),fun(B,A),hAPP(fun(B,A),fun(fun(fun(B,A),bool),fun(B,A)),hAPP(fun(fun(B,A),fun(fun(B,A),fun(B,A))),fun(fun(B,A),fun(fun(fun(B,A),bool),fun(B,A))),finite_fold(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A))),D),C)),hAPP(fun(B,A),fun(B,A),hAPP(fun(B,A),fun(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A)),E),D))). [resolve(636,b,628,a)]. 2.22/2.78 Derived: -lattice(A) | -hBOOL(hAPP(fun(B,A),bool,hAPP(fun(B,A),fun(fun(B,A),bool),ord_less_eq(fun(B,A)),C),D)) | ti(fun(B,A),D) = hAPP(fun(B,A),fun(B,A),hAPP(fun(B,A),fun(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A)),C),D). [resolve(636,b,630,a)]. 2.22/2.78 Derived: -lattice(A) | hBOOL(hAPP(fun(B,A),bool,hAPP(fun(B,A),fun(fun(B,A),bool),ord_less_eq(fun(B,A)),C),D)) | ti(fun(B,A),D) != hAPP(fun(B,A),fun(B,A),hAPP(fun(B,A),fun(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A)),C),D). [resolve(636,b,631,a)]. 2.22/2.78 Derived: -lattice(A) | hAPP(fun(B,A),fun(B,A),hAPP(fun(B,A),fun(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A)),C),hAPP(fun(B,A),fun(B,A),hAPP(fun(B,A),fun(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A)),D),E)) = hAPP(fun(B,A),fun(B,A),hAPP(fun(B,A),fun(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A)),D),hAPP(fun(B,A),fun(B,A),hAPP(fun(B,A),fun(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A)),C),E)). [resolve(636,b,632,a)]. 2.22/2.78 Derived: -lattice(A) | -hBOOL(hAPP(fun(fun(B,A),bool),bool,finite_finite_1(fun(B,A)),C)) | -hBOOL(hAPP(fun(fun(B,A),bool),bool,hAPP(fun(B,A),fun(fun(fun(B,A),bool),bool),member(fun(B,A)),D),C)) | hBOOL(hAPP(fun(B,A),bool,hAPP(fun(B,A),fun(fun(B,A),bool),ord_less_eq(fun(B,A)),hAPP(fun(B,A),fun(B,A),hAPP(fun(B,A),fun(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A)),D),E)),hAPP(fun(fun(B,A),bool),fun(B,A),hAPP(fun(B,A),fun(fun(fun(B,A),bool),fun(B,A)),hAPP(fun(fun(B,A),fun(fun(B,A),fun(B,A))),fun(fun(B,A),fun(fun(fun(B,A),bool),fun(B,A))),finite_fold(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A))),E),C))). [resolve(636,b,633,a)]. 2.22/2.78 Derived: -lattice(A) | -hBOOL(hAPP(fun(fun(B,A),bool),bool,finite_finite_1(fun(B,A)),C)) | hAPP(fun(fun(B,A),bool),fun(B,A),hAPP(fun(B,A),fun(fun(fun(B,A),bool),fun(B,A)),hAPP(fun(fun(B,A),fun(fun(B,A),fun(B,A))),fun(fun(B,A),fun(fun(fun(B,A),bool),fun(B,A))),finite_fold(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A))),D),hAPP(fun(fun(B,A),bool),fun(fun(B,A),bool),hAPP(fun(B,A),fun(fun(fun(B,A),bool),fun(fun(B,A),bool)),insert(fun(B,A)),E),C)) = hAPP(fun(B,A),fun(B,A),hAPP(fun(B,A),fun(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A)),E),hAPP(fun(fun(B,A),bool),fun(B,A),hAPP(fun(B,A),fun(fun(fun(B,A),bool),fun(B,A)),hAPP(fun(fun(B,A),fun(fun(B,A),fun(B,A))),fun(fun(B,A),fun(fun(fun(B,A),bool),fun(B,A))),finite_fold(fun(B,A),fun(B,A)),semilattice_sup_sup(fun(B,A))),D),C)). [resolve(636,b,634,a)]. 2.22/2.78 637 -semilattice_sup(A) | hBOOL(hAPP(fun(A,fun(A,A)),bool,finite_comp_fun_idem(A,A),semilattice_sup_sup(A))) # label(fact_361_comp__fun__idem__sup) # label(axiom). [clausify(491)]. 2.22/2.78 Derived: hBOOL(hAPP(fun(bool,fun(bool,bool)),bool,finite_comp_fun_idem(bool,bool),semilattice_sup_sup(bool))). [resolve(637,a,610,a)]. 2.22/2.78 Derived: hBOOL(hAPP(fun(nat,fun(nat,nat)),bool,finite_comp_fun_idem(nat,nat),semilattice_sup_sup(nat))). [resolve(637,a,624,a)]. 2.22/2.78 Derived: hBOOL(hAPP(fun(fun(A,B),fun(fun(A,B),fun(A,B))),bool,finite_comp_fun_idem(fun(A,B),fun(A,B)),semilattice_sup_sup(fun(A,B)))) | -lattice(B). [resolve(637,a,636,b)]. 2.22/2.78 638 -semilattice_sup(A) | hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),B),hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),C),D)) = hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),C),hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),B),D)) # label(fact_277_sup__left__commute) # label(axiom). [clausify(496)]. 2.22/2.78 639 -semilattice_sup(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),C),D))) # label(fact_271_le__supI1) # label(axiom). [clausify(502)]. 2.58/2.82 Derived: -hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),A),B)) | hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),A),hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),B),C))). [resolve(639,a,610,a)]. 2.58/2.82 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_sup_sup(nat),B),C))). [resolve(639,a,624,a)]. 2.58/2.82 Derived: -hBOOL(hAPP(fun(A,B),bool,hAPP(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),C),D)) | hBOOL(hAPP(fun(A,B),bool,hAPP(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),C),hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_sup_sup(fun(A,B)),D),E))) | -lattice(B). [resolve(639,a,636,b)]. 2.58/2.82 640 -semilattice_sup(A) | hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),B),C) = hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),C),B) # label(fact_286_sup_Ocommute) # label(axiom). [clausify(523)]. 2.58/2.82 641 -semilattice_sup(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | ti(A,C) = hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),C),B) # label(fact_268_sup__absorb1) # label(axiom). [clausify(531)]. 2.58/2.82 Derived: -hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),A),B)) | ti(bool,B) = hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),B),A). [resolve(641,a,610,a)]. 2.58/2.82 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | ti(nat,B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),semilattice_sup_sup(nat),B),A). [resolve(641,a,624,a)]. 2.58/2.82 Derived: -hBOOL(hAPP(fun(A,B),bool,hAPP(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),C),D)) | ti(fun(A,B),D) = hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_sup_sup(fun(A,B)),D),C) | -lattice(B). [resolve(641,a,636,b)]. 2.58/2.82 642 -semilattice_sup(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | ti(A,C) = hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),B),C) # label(fact_269_sup__absorb2) # label(axiom). [clausify(543)]. 2.58/2.82 643 -semilattice_sup(A) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),B),B) # label(fact_289_sup_Oidem) # label(axiom). [clausify(563)]. 2.58/2.82 644 -semilattice_sup(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),B),D)),C)) # label(fact_267_le__supI) # label(axiom). [clausify(571)]. 2.58/2.82 645 ab_semigroup_mult(nat) # label(arity_Nat_Onat___Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 2.58/2.82 646 -ab_semigroup_mult(A) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hAPP(fun(A,bool),A,hAPP(fun(A,fun(A,A)),fun(fun(A,bool),A),finite_fold1(A),times_times(A)),hAPP(fun(A,bool),fun(A,bool),hAPP(A,fun(fun(A,bool),fun(A,bool)),insert(A),C),B)) = hAPP(fun(A,bool),A,hAPP(A,fun(fun(A,bool),A),hAPP(fun(A,fun(A,A)),fun(A,fun(fun(A,bool),A)),finite_fold(A,A),times_times(A)),C),B) # label(fact_215_fold1__eq__fold) # label(axiom). [clausify(48)]. 2.58/2.82 647 -ab_semigroup_mult(A) | times_times(A) = ti(fun(A,fun(A,A)),times_times(A)) # label(tsy_c_Groups_Otimes__class_Otimes_res) # label(axiom). [clausify(62)]. 2.58/2.82 648 -ab_semigroup_mult(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),D),C)) | hAPP(fun(B,bool),A,hAPP(A,fun(fun(B,bool),A),hAPP(fun(B,A),fun(A,fun(fun(B,bool),A)),hAPP(fun(A,fun(A,A)),fun(fun(B,A),fun(A,fun(fun(B,bool),A))),finite_fold_image(A,B),times_times(A)),E),F),hAPP(fun(B,bool),fun(B,bool),hAPP(B,fun(fun(B,bool),fun(B,bool)),insert(B),D),C)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(B,A,E,D)),hAPP(fun(B,bool),A,hAPP(A,fun(fun(B,bool),A),hAPP(fun(B,A),fun(A,fun(fun(B,bool),A)),hAPP(fun(A,fun(A,A)),fun(fun(B,A),fun(A,fun(fun(B,bool),A))),finite_fold_image(A,B),times_times(A)),E),F),C)) # label(fact_391_fold__image__insert) # label(axiom). [clausify(111)]. 2.58/2.82 649 -ab_semigroup_mult(A) | -hBOOL(hAPP(A,bool,hAPP(fun(A,bool),fun(A,bool),hAPP(A,fun(fun(A,bool),fun(A,bool)),hAPP(fun(A,fun(A,A)),fun(A,fun(fun(A,bool),fun(A,bool))),finite_fold_graph(A,A),times_times(A)),B),C),D)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(fun(A,bool),fun(A,bool),hAPP(A,fun(fun(A,bool),fun(A,bool)),hAPP(fun(A,fun(A,A)),fun(A,fun(fun(A,bool),fun(A,bool))),finite_fold_graph(A,A),times_times(A)),E),hAPP(fun(A,bool),fun(A,bool),hAPP(A,fun(fun(A,bool),fun(A,bool)),insert(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),E),D))) # label(fact_207_fold__graph__insert__swap) # label(axiom). [clausify(214)]. 2.58/2.82 650 -ab_semigroup_mult(A) | bot_bot(fun(A,bool)) = ti(fun(A,bool),B) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hAPP(fun(A,bool),A,hAPP(fun(A,fun(A,A)),fun(fun(A,bool),A),finite_fold1(A),times_times(A)),hAPP(fun(A,bool),fun(A,bool),hAPP(A,fun(fun(A,bool),fun(A,bool)),insert(A),C),B)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),hAPP(fun(A,bool),A,hAPP(fun(A,fun(A,A)),fun(fun(A,bool),A),finite_fold1(A),times_times(A)),B)) # label(fact_211_fold1__insert) # label(axiom). [clausify(295)]. 2.58/2.82 651 -ab_semigroup_mult(A) | hBOOL(hAPP(fun(A,fun(A,A)),bool,finite100568337ommute(A,A),times_times(A))) # label(fact_202_comp__fun__commute) # label(axiom). [clausify(311)]. 2.58/2.82 652 -ab_semigroup_mult(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),D)) # label(fact_384_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom). [clausify(329)]. 2.58/2.82 Derived: -hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),A)) | hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),B),A)) | hAPP(fun(nat,bool),nat,hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,bool),nat),finite_fold1(nat),times_times(nat)),hAPP(fun(nat,bool),fun(nat,bool),hAPP(nat,fun(fun(nat,bool),fun(nat,bool)),insert(nat),B),A)) = hAPP(fun(nat,bool),nat,hAPP(nat,fun(fun(nat,bool),nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(fun(nat,bool),nat)),finite_fold(nat,nat),times_times(nat)),B),A). [resolve(645,a,646,a)]. 2.58/2.82 Derived: times_times(nat) = ti(fun(nat,fun(nat,nat)),times_times(nat)). [resolve(645,a,647,a)]. 2.58/2.82 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hAPP(fun(A,bool),nat,hAPP(nat,fun(fun(A,bool),nat),hAPP(fun(A,nat),fun(nat,fun(fun(A,bool),nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(A,nat),fun(nat,fun(fun(A,bool),nat))),finite_fold_image(nat,A),times_times(nat)),D),E),hAPP(fun(A,bool),fun(A,bool),hAPP(A,fun(fun(A,bool),fun(A,bool)),insert(A),C),B)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(A,nat,D,C)),hAPP(fun(A,bool),nat,hAPP(nat,fun(fun(A,bool),nat),hAPP(fun(A,nat),fun(nat,fun(fun(A,bool),nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(A,nat),fun(nat,fun(fun(A,bool),nat))),finite_fold_image(nat,A),times_times(nat)),D),E),B)). [resolve(645,a,648,a)]. 2.58/2.82 Derived: -hBOOL(hAPP(nat,bool,hAPP(fun(nat,bool),fun(nat,bool),hAPP(nat,fun(fun(nat,bool),fun(nat,bool)),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(fun(nat,bool),fun(nat,bool))),finite_fold_graph(nat,nat),times_times(nat)),A),B),C)) | hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),A),B)) | hBOOL(hAPP(nat,bool,hAPP(fun(nat,bool),fun(nat,bool),hAPP(nat,fun(fun(nat,bool),fun(nat,bool)),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(fun(nat,bool),fun(nat,bool))),finite_fold_graph(nat,nat),times_times(nat)),D),hAPP(fun(nat,bool),fun(nat,bool),hAPP(nat,fun(fun(nat,bool),fun(nat,bool)),insert(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),D),C))). [resolve(645,a,649,a)]. 2.58/2.82 Derived: bot_bot(fun(nat,bool)) = ti(fun(nat,bool),A) | -hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),A)) | hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),B),A)) | hAPP(fun(nat,bool),nat,hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,bool),nat),finite_fold1(nat),times_times(nat)),hAPP(fun(nat,bool),fun(nat,bool),hAPP(nat,fun(fun(nat,bool),fun(nat,bool)),insert(nat),B),A)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),hAPP(fun(nat,bool),nat,hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,bool),nat),finite_fold1(nat),times_times(nat)),A)). [resolve(645,a,650,a)]. 2.59/2.83 Derived: hBOOL(hAPP(fun(nat,fun(nat,nat)),bool,finite100568337ommute(nat,nat),times_times(nat))). [resolve(645,a,651,a)]. 2.59/2.83 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),B)),C) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),C)). [resolve(645,a,652,a)]. 2.59/2.83 653 -ab_semigroup_mult(A) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | bot_bot(fun(A,bool)) = ti(fun(A,bool),B) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),f96(A,B)),f97(A,B))),hAPP(fun(A,bool),fun(A,bool),hAPP(A,fun(fun(A,bool),fun(A,bool)),insert(A),f96(A,B)),hAPP(fun(A,bool),fun(A,bool),hAPP(A,fun(fun(A,bool),fun(A,bool)),insert(A),f97(A,B)),bot_bot(fun(A,bool)))))) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),hAPP(fun(A,bool),A,hAPP(fun(A,fun(A,A)),fun(fun(A,bool),A),finite_fold1(A),times_times(A)),B)),B)) # label(fact_232_fold1__in) # label(axiom). [clausify(503)]. 2.59/2.83 Derived: -hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),A)) | bot_bot(fun(nat,bool)) = ti(fun(nat,bool),A) | -hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),f96(nat,A)),f97(nat,A))),hAPP(fun(nat,bool),fun(nat,bool),hAPP(nat,fun(fun(nat,bool),fun(nat,bool)),insert(nat),f96(nat,A)),hAPP(fun(nat,bool),fun(nat,bool),hAPP(nat,fun(fun(nat,bool),fun(nat,bool)),insert(nat),f97(nat,A)),bot_bot(fun(nat,bool)))))) | hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),hAPP(fun(nat,bool),nat,hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,bool),nat),finite_fold1(nat),times_times(nat)),A)),A)). [resolve(653,a,645,a)]. 2.59/2.83 654 -ab_semigroup_mult(A) | -hBOOL(hAPP(A,bool,hAPP(fun(A,bool),fun(A,bool),hAPP(A,fun(fun(A,bool),fun(A,bool)),hAPP(fun(A,fun(A,A)),fun(A,fun(fun(A,bool),fun(A,bool))),finite_fold_graph(A,A),times_times(A)),B),C),D)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),E),C)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(fun(A,bool),fun(A,bool),hAPP(A,fun(fun(A,bool),fun(A,bool)),hAPP(fun(A,fun(A,A)),fun(A,fun(fun(A,bool),fun(A,bool))),finite_fold_graph(A,A),times_times(A)),E),hAPP(fun(A,bool),fun(A,bool),hAPP(A,fun(fun(A,bool),fun(A,bool)),insert(A),B),hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),C),hAPP(fun(A,bool),fun(A,bool),hAPP(A,fun(fun(A,bool),fun(A,bool)),insert(A),E),bot_bot(fun(A,bool)))))),D)) # label(fact_196_fold__graph__permute__diff) # label(axiom). [clausify(513)]. 2.59/2.83 Derived: -hBOOL(hAPP(nat,bool,hAPP(fun(nat,bool),fun(nat,bool),hAPP(nat,fun(fun(nat,bool),fun(nat,bool)),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(fun(nat,bool),fun(nat,bool))),finite_fold_graph(nat,nat),times_times(nat)),A),B),C)) | -hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),D),B)) | hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),A),B)) | hBOOL(hAPP(nat,bool,hAPP(fun(nat,bool),fun(nat,bool),hAPP(nat,fun(fun(nat,bool),fun(nat,bool)),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(fun(nat,bool),fun(nat,bool))),finite_fold_graph(nat,nat),times_times(nat)),D),hAPP(fun(nat,bool),fun(nat,bool),hAPP(nat,fun(fun(nat,bool),fun(nat,bool)),insert(nat),A),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,bool),fun(fun(nat,bool),fun(nat,bool)),minus_minus(fun(nat,bool)),B),hAPP(fun(nat,bool),fun(nat,bool),hAPP(nat,fun(fun(nat,bool),fun(nat,bool)),insert(nat),D),bot_bot(fun(nat,bool)))))),C)). [resolve(654,a,645,a)]. 2.59/2.83 655 -ab_semigroup_mult(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f100(B,A,D,E,F,C)),C)) | hAPP(fun(B,bool),A,hAPP(A,fun(fun(B,bool),A),hAPP(fun(B,A),fun(A,fun(fun(B,bool),A)),hAPP(fun(A,fun(A,A)),fun(fun(B,A),fun(A,fun(fun(B,bool),A))),finite_fold_image(A,B),times_times(A)),F),D),C) = hAPP(fun(B,bool),A,hAPP(A,fun(fun(B,bool),A),hAPP(fun(B,A),fun(A,fun(fun(B,bool),A)),hAPP(fun(A,fun(A,A)),fun(fun(B,A),fun(A,fun(fun(B,bool),A))),finite_fold_image(A,B),times_times(A)),E),D),C) # label(fact_392_fold__image__cong) # label(axiom). [clausify(534)]. 2.59/2.90 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f100(A,nat,C,D,E,B)),B)) | hAPP(fun(A,bool),nat,hAPP(nat,fun(fun(A,bool),nat),hAPP(fun(A,nat),fun(nat,fun(fun(A,bool),nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(A,nat),fun(nat,fun(fun(A,bool),nat))),finite_fold_image(nat,A),times_times(nat)),E),C),B) = hAPP(fun(A,bool),nat,hAPP(nat,fun(fun(A,bool),nat),hAPP(fun(A,nat),fun(nat,fun(fun(A,bool),nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(A,nat),fun(nat,fun(fun(A,bool),nat))),finite_fold_image(nat,A),times_times(nat)),D),C),B). [resolve(655,a,645,a)]. 2.59/2.90 656 -ab_semigroup_mult(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | hAPP(B,A,D,f100(B,A,E,F,D,C)) != hAPP(B,A,F,f100(B,A,E,F,D,C)) | hAPP(fun(B,bool),A,hAPP(A,fun(fun(B,bool),A),hAPP(fun(B,A),fun(A,fun(fun(B,bool),A)),hAPP(fun(A,fun(A,A)),fun(fun(B,A),fun(A,fun(fun(B,bool),A))),finite_fold_image(A,B),times_times(A)),D),E),C) = hAPP(fun(B,bool),A,hAPP(A,fun(fun(B,bool),A),hAPP(fun(B,A),fun(A,fun(fun(B,bool),A)),hAPP(fun(A,fun(A,A)),fun(fun(B,A),fun(A,fun(fun(B,bool),A))),finite_fold_image(A,B),times_times(A)),F),E),C) # label(fact_392_fold__image__cong) # label(axiom). [clausify(534)]. 2.59/2.90 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hAPP(A,nat,C,f100(A,nat,D,E,C,B)) != hAPP(A,nat,E,f100(A,nat,D,E,C,B)) | hAPP(fun(A,bool),nat,hAPP(nat,fun(fun(A,bool),nat),hAPP(fun(A,nat),fun(nat,fun(fun(A,bool),nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(A,nat),fun(nat,fun(fun(A,bool),nat))),finite_fold_image(nat,A),times_times(nat)),C),D),B) = hAPP(fun(A,bool),nat,hAPP(nat,fun(fun(A,bool),nat),hAPP(fun(A,nat),fun(nat,fun(fun(A,bool),nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(A,nat),fun(nat,fun(fun(A,bool),nat))),finite_fold_image(nat,A),times_times(nat)),E),D),B). [resolve(656,a,645,a)]. 2.59/2.90 657 linorder(nat) # label(arity_Nat_Onat___Orderings_Olinorder) # label(axiom). [assumption]. 2.59/2.90 658 -linorder(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),B)) # label(fact_302_linorder__le__cases) # label(axiom). [clausify(57)]. 2.59/2.90 659 -linorder(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),B)) # label(fact_316_linorder__linear) # label(axiom). [clausify(387)]. 2.59/2.90 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),A)). [resolve(657,a,658,a)]. 2.59/2.90 660 comm_monoid_mult(nat) # label(arity_Nat_Onat___Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 2.59/2.90 661 -comm_monoid_mult(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | hBOOL(hAPP(fun(D,bool),bool,hAPP(D,fun(fun(D,bool),bool),member(D),f27(D,B,A,E,F,V6,V7,V8,V9,C)),V9)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f28(D,B,A,E,F,V6,V7,V8,V9,C)),C)) | hAPP(fun(D,bool),A,hAPP(A,fun(fun(D,bool),A),hAPP(fun(D,A),fun(A,fun(fun(D,bool),A)),hAPP(fun(A,fun(A,A)),fun(fun(D,A),fun(A,fun(fun(D,bool),A))),finite_fold_image(A,D),times_times(A)),F),E),V9) = hAPP(fun(B,bool),A,hAPP(A,fun(fun(B,bool),A),hAPP(fun(B,A),fun(A,fun(fun(B,bool),A)),hAPP(fun(A,fun(A,A)),fun(fun(B,A),fun(A,fun(fun(B,bool),A))),finite_fold_image(A,B),times_times(A)),V6),E),C) # label(fact_394_fold__image__eq__general__inverses) # label(axiom). [clausify(144)]. 2.59/2.90 662 -comm_monoid_mult(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | hBOOL(hAPP(fun(D,bool),bool,hAPP(D,fun(fun(D,bool),bool),member(D),f27(D,B,A,E,F,V6,V7,V8,V9,C)),V9)) | hAPP(D,A,F,hAPP(B,D,V7,f28(D,B,A,E,F,V6,V7,V8,V9,C))) != hAPP(B,A,V6,f28(D,B,A,E,F,V6,V7,V8,V9,C)) | ti(B,f28(D,B,A,E,F,V6,V7,V8,V9,C)) != hAPP(D,B,V8,hAPP(B,D,V7,f28(D,B,A,E,F,V6,V7,V8,V9,C))) | -hBOOL(hAPP(fun(D,bool),bool,hAPP(D,fun(fun(D,bool),bool),member(D),hAPP(B,D,V7,f28(D,B,A,E,F,V6,V7,V8,V9,C))),V9)) | hAPP(fun(D,bool),A,hAPP(A,fun(fun(D,bool),A),hAPP(fun(D,A),fun(A,fun(fun(D,bool),A)),hAPP(fun(A,fun(A,A)),fun(fun(D,A),fun(A,fun(fun(D,bool),A))),finite_fold_image(A,D),times_times(A)),F),E),V9) = hAPP(fun(B,bool),A,hAPP(A,fun(fun(B,bool),A),hAPP(fun(B,A),fun(A,fun(fun(B,bool),A)),hAPP(fun(A,fun(A,A)),fun(fun(B,A),fun(A,fun(fun(B,bool),A))),finite_fold_image(A,B),times_times(A)),V6),E),C) # label(fact_394_fold__image__eq__general__inverses) # label(axiom). [clausify(144)]. 2.59/2.90 663 -comm_monoid_mult(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | -hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),hAPP(D,B,E,f27(D,B,A,F,V6,V7,V8,E,V9,C))),C)) | ti(D,f27(D,B,A,F,V6,V7,V8,E,V9,C)) != hAPP(B,D,V8,hAPP(D,B,E,f27(D,B,A,F,V6,V7,V8,E,V9,C))) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f28(D,B,A,F,V6,V7,V8,E,V9,C)),C)) | hAPP(fun(D,bool),A,hAPP(A,fun(fun(D,bool),A),hAPP(fun(D,A),fun(A,fun(fun(D,bool),A)),hAPP(fun(A,fun(A,A)),fun(fun(D,A),fun(A,fun(fun(D,bool),A))),finite_fold_image(A,D),times_times(A)),V6),F),V9) = hAPP(fun(B,bool),A,hAPP(A,fun(fun(B,bool),A),hAPP(fun(B,A),fun(A,fun(fun(B,bool),A)),hAPP(fun(A,fun(A,A)),fun(fun(B,A),fun(A,fun(fun(B,bool),A))),finite_fold_image(A,B),times_times(A)),V7),F),C) # label(fact_394_fold__image__eq__general__inverses) # label(axiom). [clausify(144)]. 2.59/2.90 664 -comm_monoid_mult(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | -hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),hAPP(D,B,E,f27(D,B,A,F,V6,V7,V8,E,V9,C))),C)) | ti(D,f27(D,B,A,F,V6,V7,V8,E,V9,C)) != hAPP(B,D,V8,hAPP(D,B,E,f27(D,B,A,F,V6,V7,V8,E,V9,C))) | hAPP(D,A,V6,hAPP(B,D,V8,f28(D,B,A,F,V6,V7,V8,E,V9,C))) != hAPP(B,A,V7,f28(D,B,A,F,V6,V7,V8,E,V9,C)) | ti(B,f28(D,B,A,F,V6,V7,V8,E,V9,C)) != hAPP(D,B,E,hAPP(B,D,V8,f28(D,B,A,F,V6,V7,V8,E,V9,C))) | -hBOOL(hAPP(fun(D,bool),bool,hAPP(D,fun(fun(D,bool),bool),member(D),hAPP(B,D,V8,f28(D,B,A,F,V6,V7,V8,E,V9,C))),V9)) | hAPP(fun(D,bool),A,hAPP(A,fun(fun(D,bool),A),hAPP(fun(D,A),fun(A,fun(fun(D,bool),A)),hAPP(fun(A,fun(A,A)),fun(fun(D,A),fun(A,fun(fun(D,bool),A))),finite_fold_image(A,D),times_times(A)),V6),F),V9) = hAPP(fun(B,bool),A,hAPP(A,fun(fun(B,bool),A),hAPP(fun(B,A),fun(A,fun(fun(B,bool),A)),hAPP(fun(A,fun(A,A)),fun(fun(B,A),fun(A,fun(fun(B,bool),A))),finite_fold_image(A,B),times_times(A)),V7),F),C) # label(fact_394_fold__image__eq__general__inverses) # label(axiom). [clausify(144)]. 2.59/2.90 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(C,bool),bool,hAPP(C,fun(fun(C,bool),bool),member(C),f27(C,A,nat,D,E,F,V6,V7,V8,B)),V8)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f28(C,A,nat,D,E,F,V6,V7,V8,B)),B)) | hAPP(fun(C,bool),nat,hAPP(nat,fun(fun(C,bool),nat),hAPP(fun(C,nat),fun(nat,fun(fun(C,bool),nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(C,nat),fun(nat,fun(fun(C,bool),nat))),finite_fold_image(nat,C),times_times(nat)),E),D),V8) = hAPP(fun(A,bool),nat,hAPP(nat,fun(fun(A,bool),nat),hAPP(fun(A,nat),fun(nat,fun(fun(A,bool),nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(A,nat),fun(nat,fun(fun(A,bool),nat))),finite_fold_image(nat,A),times_times(nat)),F),D),B). [resolve(660,a,661,a)]. 2.59/2.90 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(C,bool),bool,hAPP(C,fun(fun(C,bool),bool),member(C),f27(C,A,nat,D,E,F,V6,V7,V8,B)),V8)) | hAPP(C,nat,E,hAPP(A,C,V6,f28(C,A,nat,D,E,F,V6,V7,V8,B))) != hAPP(A,nat,F,f28(C,A,nat,D,E,F,V6,V7,V8,B)) | ti(A,f28(C,A,nat,D,E,F,V6,V7,V8,B)) != hAPP(C,A,V7,hAPP(A,C,V6,f28(C,A,nat,D,E,F,V6,V7,V8,B))) | -hBOOL(hAPP(fun(C,bool),bool,hAPP(C,fun(fun(C,bool),bool),member(C),hAPP(A,C,V6,f28(C,A,nat,D,E,F,V6,V7,V8,B))),V8)) | hAPP(fun(C,bool),nat,hAPP(nat,fun(fun(C,bool),nat),hAPP(fun(C,nat),fun(nat,fun(fun(C,bool),nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(C,nat),fun(nat,fun(fun(C,bool),nat))),finite_fold_image(nat,C),times_times(nat)),E),D),V8) = hAPP(fun(A,bool),nat,hAPP(nat,fun(fun(A,bool),nat),hAPP(fun(A,nat),fun(nat,fun(fun(A,bool),nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(A,nat),fun(nat,fun(fun(A,bool),nat))),finite_fold_image(nat,A),times_times(nat)),F),D),B). [resolve(660,a,662,a)]. 2.59/2.90 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),hAPP(C,A,D,f27(C,A,nat,E,F,V6,V7,D,V8,B))),B)) | ti(C,f27(C,A,nat,E,F,V6,V7,D,V8,B)) != hAPP(A,C,V7,hAPP(C,A,D,f27(C,A,nat,E,F,V6,V7,D,V8,B))) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f28(C,A,nat,E,F,V6,V7,D,V8,B)),B)) | hAPP(fun(C,bool),nat,hAPP(nat,fun(fun(C,bool),nat),hAPP(fun(C,nat),fun(nat,fun(fun(C,bool),nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(C,nat),fun(nat,fun(fun(C,bool),nat))),finite_fold_image(nat,C),times_times(nat)),F),E),V8) = hAPP(fun(A,bool),nat,hAPP(nat,fun(fun(A,bool),nat),hAPP(fun(A,nat),fun(nat,fun(fun(A,bool),nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(A,nat),fun(nat,fun(fun(A,bool),nat))),finite_fold_image(nat,A),times_times(nat)),V6),E),B). [resolve(660,a,663,a)]. 2.59/2.90 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),hAPP(C,A,D,f27(C,A,nat,E,F,V6,V7,D,V8,B))),B)) | ti(C,f27(C,A,nat,E,F,V6,V7,D,V8,B)) != hAPP(A,C,V7,hAPP(C,A,D,f27(C,A,nat,E,F,V6,V7,D,V8,B))) | hAPP(C,nat,F,hAPP(A,C,V7,f28(C,A,nat,E,F,V6,V7,D,V8,B))) != hAPP(A,nat,V6,f28(C,A,nat,E,F,V6,V7,D,V8,B)) | ti(A,f28(C,A,nat,E,F,V6,V7,D,V8,B)) != hAPP(C,A,D,hAPP(A,C,V7,f28(C,A,nat,E,F,V6,V7,D,V8,B))) | -hBOOL(hAPP(fun(C,bool),bool,hAPP(C,fun(fun(C,bool),bool),member(C),hAPP(A,C,V7,f28(C,A,nat,E,F,V6,V7,D,V8,B))),V8)) | hAPP(fun(C,bool),nat,hAPP(nat,fun(fun(C,bool),nat),hAPP(fun(C,nat),fun(nat,fun(fun(C,bool),nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(C,nat),fun(nat,fun(fun(C,bool),nat))),finite_fold_image(nat,C),times_times(nat)),F),E),V8) = hAPP(fun(A,bool),nat,hAPP(nat,fun(fun(A,bool),nat),hAPP(fun(A,nat),fun(nat,fun(fun(A,bool),nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(A,nat),fun(nat,fun(fun(A,bool),nat))),finite_fold_image(nat,A),times_times(nat)),V6),E),B). [resolve(660,a,664,a)]. 2.59/2.90 665 -comm_monoid_mult(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,C),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,f54(D,A,E,F,V6,B,C)),f56(D,A,E,F,V6,B,C))) | -hBOOL(hAPP(fun(D,bool),bool,finite_finite_1(D),V6)) | hBOOL(hAPP(fun(D,bool),bool,hAPP(D,fun(fun(D,bool),bool),member(D),f57(D,A,E,F,V6,B,C)),V6)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,hAPP(fun(D,bool),A,hAPP(A,fun(fun(D,bool),A),hAPP(fun(D,A),fun(A,fun(fun(D,bool),A)),hAPP(fun(A,fun(A,A)),fun(fun(D,A),fun(A,fun(fun(D,bool),A))),finite_fold_image(A,D),times_times(A)),E),C),V6)),hAPP(fun(D,bool),A,hAPP(A,fun(fun(D,bool),A),hAPP(fun(D,A),fun(A,fun(fun(D,bool),A)),hAPP(fun(A,fun(A,A)),fun(fun(D,A),fun(A,fun(fun(D,bool),A))),finite_fold_image(A,D),times_times(A)),F),C),V6))) # label(fact_395_fold__image__related) # label(axiom). [clausify(283)]. 2.59/2.90 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,B),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,f54(C,nat,D,E,F,A,B)),f56(C,nat,D,E,F,A,B))) | -hBOOL(hAPP(fun(C,bool),bool,finite_finite_1(C),F)) | hBOOL(hAPP(fun(C,bool),bool,hAPP(C,fun(fun(C,bool),bool),member(C),f57(C,nat,D,E,F,A,B)),F)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,hAPP(fun(C,bool),nat,hAPP(nat,fun(fun(C,bool),nat),hAPP(fun(C,nat),fun(nat,fun(fun(C,bool),nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(C,nat),fun(nat,fun(fun(C,bool),nat))),finite_fold_image(nat,C),times_times(nat)),D),B),F)),hAPP(fun(C,bool),nat,hAPP(nat,fun(fun(C,bool),nat),hAPP(fun(C,nat),fun(nat,fun(fun(C,bool),nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(C,nat),fun(nat,fun(fun(C,bool),nat))),finite_fold_image(nat,C),times_times(nat)),E),B),F))). [resolve(665,a,660,a)]. 2.59/2.90 666 -comm_monoid_mult(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,C),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,f54(D,A,E,F,V6,B,C)),f56(D,A,E,F,V6,B,C))) | -hBOOL(hAPP(fun(D,bool),bool,finite_finite_1(D),V6)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,hAPP(D,A,E,f57(D,A,E,F,V6,B,C))),hAPP(D,A,F,f57(D,A,E,F,V6,B,C)))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,hAPP(fun(D,bool),A,hAPP(A,fun(fun(D,bool),A),hAPP(fun(D,A),fun(A,fun(fun(D,bool),A)),hAPP(fun(A,fun(A,A)),fun(fun(D,A),fun(A,fun(fun(D,bool),A))),finite_fold_image(A,D),times_times(A)),E),C),V6)),hAPP(fun(D,bool),A,hAPP(A,fun(fun(D,bool),A),hAPP(fun(D,A),fun(A,fun(fun(D,bool),A)),hAPP(fun(A,fun(A,A)),fun(fun(D,A),fun(A,fun(fun(D,bool),A))),finite_fold_image(A,D),times_times(A)),F),C),V6))) # label(fact_395_fold__image__related) # label(axiom). [clausify(283)]. 2.59/2.91 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,B),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,f54(C,nat,D,E,F,A,B)),f56(C,nat,D,E,F,A,B))) | -hBOOL(hAPP(fun(C,bool),bool,finite_finite_1(C),F)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,hAPP(C,nat,D,f57(C,nat,D,E,F,A,B))),hAPP(C,nat,E,f57(C,nat,D,E,F,A,B)))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,hAPP(fun(C,bool),nat,hAPP(nat,fun(fun(C,bool),nat),hAPP(fun(C,nat),fun(nat,fun(fun(C,bool),nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(C,nat),fun(nat,fun(fun(C,bool),nat))),finite_fold_image(nat,C),times_times(nat)),D),B),F)),hAPP(fun(C,bool),nat,hAPP(nat,fun(fun(C,bool),nat),hAPP(fun(C,nat),fun(nat,fun(fun(C,bool),nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(C,nat),fun(nat,fun(fun(C,bool),nat))),finite_fold_image(nat,C),times_times(nat)),E),B),F))). [resolve(666,a,660,a)]. 2.59/2.91 667 -comm_monoid_mult(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,C),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,f53(D,A,E,F,V6,B,C)),f55(D,A,E,F,V6,B,C))) | -hBOOL(hAPP(fun(D,bool),bool,finite_finite_1(D),V6)) | hBOOL(hAPP(fun(D,bool),bool,hAPP(D,fun(fun(D,bool),bool),member(D),f57(D,A,E,F,V6,B,C)),V6)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,hAPP(fun(D,bool),A,hAPP(A,fun(fun(D,bool),A),hAPP(fun(D,A),fun(A,fun(fun(D,bool),A)),hAPP(fun(A,fun(A,A)),fun(fun(D,A),fun(A,fun(fun(D,bool),A))),finite_fold_image(A,D),times_times(A)),E),C),V6)),hAPP(fun(D,bool),A,hAPP(A,fun(fun(D,bool),A),hAPP(fun(D,A),fun(A,fun(fun(D,bool),A)),hAPP(fun(A,fun(A,A)),fun(fun(D,A),fun(A,fun(fun(D,bool),A))),finite_fold_image(A,D),times_times(A)),F),C),V6))) # label(fact_395_fold__image__related) # label(axiom). [clausify(283)]. 2.59/2.91 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,B),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,f53(C,nat,D,E,F,A,B)),f55(C,nat,D,E,F,A,B))) | -hBOOL(hAPP(fun(C,bool),bool,finite_finite_1(C),F)) | hBOOL(hAPP(fun(C,bool),bool,hAPP(C,fun(fun(C,bool),bool),member(C),f57(C,nat,D,E,F,A,B)),F)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,hAPP(fun(C,bool),nat,hAPP(nat,fun(fun(C,bool),nat),hAPP(fun(C,nat),fun(nat,fun(fun(C,bool),nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(C,nat),fun(nat,fun(fun(C,bool),nat))),finite_fold_image(nat,C),times_times(nat)),D),B),F)),hAPP(fun(C,bool),nat,hAPP(nat,fun(fun(C,bool),nat),hAPP(fun(C,nat),fun(nat,fun(fun(C,bool),nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(C,nat),fun(nat,fun(fun(C,bool),nat))),finite_fold_image(nat,C),times_times(nat)),E),B),F))). [resolve(667,a,660,a)]. 2.59/2.91 668 -comm_monoid_mult(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,C),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,f53(D,A,E,F,V6,B,C)),f55(D,A,E,F,V6,B,C))) | -hBOOL(hAPP(fun(D,bool),bool,finite_finite_1(D),V6)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,hAPP(D,A,E,f57(D,A,E,F,V6,B,C))),hAPP(D,A,F,f57(D,A,E,F,V6,B,C)))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,hAPP(fun(D,bool),A,hAPP(A,fun(fun(D,bool),A),hAPP(fun(D,A),fun(A,fun(fun(D,bool),A)),hAPP(fun(A,fun(A,A)),fun(fun(D,A),fun(A,fun(fun(D,bool),A))),finite_fold_image(A,D),times_times(A)),E),C),V6)),hAPP(fun(D,bool),A,hAPP(A,fun(fun(D,bool),A),hAPP(fun(D,A),fun(A,fun(fun(D,bool),A)),hAPP(fun(A,fun(A,A)),fun(fun(D,A),fun(A,fun(fun(D,bool),A))),finite_fold_image(A,D),times_times(A)),F),C),V6))) # label(fact_395_fold__image__related) # label(axiom). [clausify(283)]. 2.59/2.91 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,B),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,f53(C,nat,D,E,F,A,B)),f55(C,nat,D,E,F,A,B))) | -hBOOL(hAPP(fun(C,bool),bool,finite_finite_1(C),F)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,hAPP(C,nat,D,f57(C,nat,D,E,F,A,B))),hAPP(C,nat,E,f57(C,nat,D,E,F,A,B)))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,hAPP(fun(C,bool),nat,hAPP(nat,fun(fun(C,bool),nat),hAPP(fun(C,nat),fun(nat,fun(fun(C,bool),nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(C,nat),fun(nat,fun(fun(C,bool),nat))),finite_fold_image(nat,C),times_times(nat)),D),B),F)),hAPP(fun(C,bool),nat,hAPP(nat,fun(fun(C,bool),nat),hAPP(fun(C,nat),fun(nat,fun(fun(C,bool),nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(C,nat),fun(nat,fun(fun(C,bool),nat))),finite_fold_image(nat,C),times_times(nat)),E),B),F))). [resolve(668,a,660,a)]. 2.59/2.91 669 -comm_monoid_mult(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,C),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,hAPP(A,A,hAPP(A,fun(A,A),times_times(A),f53(D,A,E,F,V6,B,C)),f54(D,A,E,F,V6,B,C))),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),f55(D,A,E,F,V6,B,C)),f56(D,A,E,F,V6,B,C)))) | -hBOOL(hAPP(fun(D,bool),bool,finite_finite_1(D),V6)) | hBOOL(hAPP(fun(D,bool),bool,hAPP(D,fun(fun(D,bool),bool),member(D),f57(D,A,E,F,V6,B,C)),V6)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,hAPP(fun(D,bool),A,hAPP(A,fun(fun(D,bool),A),hAPP(fun(D,A),fun(A,fun(fun(D,bool),A)),hAPP(fun(A,fun(A,A)),fun(fun(D,A),fun(A,fun(fun(D,bool),A))),finite_fold_image(A,D),times_times(A)),E),C),V6)),hAPP(fun(D,bool),A,hAPP(A,fun(fun(D,bool),A),hAPP(fun(D,A),fun(A,fun(fun(D,bool),A)),hAPP(fun(A,fun(A,A)),fun(fun(D,A),fun(A,fun(fun(D,bool),A))),finite_fold_image(A,D),times_times(A)),F),C),V6))) # label(fact_395_fold__image__related) # label(axiom). [clausify(283)]. 2.59/2.91 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,B),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),f53(C,nat,D,E,F,A,B)),f54(C,nat,D,E,F,A,B))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),f55(C,nat,D,E,F,A,B)),f56(C,nat,D,E,F,A,B)))) | -hBOOL(hAPP(fun(C,bool),bool,finite_finite_1(C),F)) | hBOOL(hAPP(fun(C,bool),bool,hAPP(C,fun(fun(C,bool),bool),member(C),f57(C,nat,D,E,F,A,B)),F)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,hAPP(fun(C,bool),nat,hAPP(nat,fun(fun(C,bool),nat),hAPP(fun(C,nat),fun(nat,fun(fun(C,bool),nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(C,nat),fun(nat,fun(fun(C,bool),nat))),finite_fold_image(nat,C),times_times(nat)),D),B),F)),hAPP(fun(C,bool),nat,hAPP(nat,fun(fun(C,bool),nat),hAPP(fun(C,nat),fun(nat,fun(fun(C,bool),nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(C,nat),fun(nat,fun(fun(C,bool),nat))),finite_fold_image(nat,C),times_times(nat)),E),B),F))). [resolve(669,a,660,a)]. 2.59/2.91 670 -comm_monoid_mult(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,C),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,hAPP(A,A,hAPP(A,fun(A,A),times_times(A),f53(D,A,E,F,V6,B,C)),f54(D,A,E,F,V6,B,C))),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),f55(D,A,E,F,V6,B,C)),f56(D,A,E,F,V6,B,C)))) | -hBOOL(hAPP(fun(D,bool),bool,finite_finite_1(D),V6)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,hAPP(D,A,E,f57(D,A,E,F,V6,B,C))),hAPP(D,A,F,f57(D,A,E,F,V6,B,C)))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,hAPP(fun(D,bool),A,hAPP(A,fun(fun(D,bool),A),hAPP(fun(D,A),fun(A,fun(fun(D,bool),A)),hAPP(fun(A,fun(A,A)),fun(fun(D,A),fun(A,fun(fun(D,bool),A))),finite_fold_image(A,D),times_times(A)),E),C),V6)),hAPP(fun(D,bool),A,hAPP(A,fun(fun(D,bool),A),hAPP(fun(D,A),fun(A,fun(fun(D,bool),A)),hAPP(fun(A,fun(A,A)),fun(fun(D,A),fun(A,fun(fun(D,bool),A))),finite_fold_image(A,D),times_times(A)),F),C),V6))) # label(fact_395_fold__image__related) # label(axiom). [clausify(283)]. 2.59/2.91 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,B),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),f53(C,nat,D,E,F,A,B)),f54(C,nat,D,E,F,A,B))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),f55(C,nat,D,E,F,A,B)),f56(C,nat,D,E,F,A,B)))) | -hBOOL(hAPP(fun(C,bool),bool,finite_finite_1(C),F)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,hAPP(C,nat,D,f57(C,nat,D,E,F,A,B))),hAPP(C,nat,E,f57(C,nat,D,E,F,A,B)))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,hAPP(fun(C,bool),nat,hAPP(nat,fun(fun(C,bool),nat),hAPP(fun(C,nat),fun(nat,fun(fun(C,bool),nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(C,nat),fun(nat,fun(fun(C,bool),nat))),finite_fold_image(nat,C),times_times(nat)),D),B),F)),hAPP(fun(C,bool),nat,hAPP(nat,fun(fun(C,bool),nat),hAPP(fun(C,nat),fun(nat,fun(fun(C,bool),nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(C,nat),fun(nat,fun(fun(C,bool),nat))),finite_fold_image(nat,C),times_times(nat)),E),B),F))). [resolve(670,a,660,a)]. 2.69/2.96 671 -bounded_lattice_bot(A) | bot_bot(A) = hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),bot_bot(A)),B) # label(fact_468_inf__bot__left) # label(axiom). [clausify(213)]. 2.69/2.96 672 -bounded_lattice(A) | bounded_lattice_bot(fun(B,A)) # label(arity_fun___Lattices_Obounded__lattice__bot) # label(axiom). [clausify(145)]. 2.69/2.96 Derived: bot_bot(fun(A,B)) = hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_inf_inf(fun(A,B)),bot_bot(fun(A,B))),C) | -bounded_lattice(B). [resolve(671,a,672,b)]. 2.69/2.96 673 -bounded_lattice_bot(A) | bot_bot(A) = hAPP(A,A,hAPP(A,fun(A,A),semilattice_inf_inf(A),B),bot_bot(A)) # label(fact_469_inf__bot__right) # label(axiom). [clausify(260)]. 2.69/2.96 Derived: bot_bot(fun(A,B)) = hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_inf_inf(fun(A,B)),C),bot_bot(fun(A,B))) | -bounded_lattice(B). [resolve(673,a,672,b)]. 2.69/2.96 674 -bounded_lattice_bot(A) | bot_bot(A) != hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),B),C) | bot_bot(A) = ti(A,C) # label(fact_294_sup__eq__bot__iff) # label(axiom). [clausify(394)]. 2.69/2.96 Derived: bot_bot(fun(A,B)) != hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_sup_sup(fun(A,B)),C),D) | bot_bot(fun(A,B)) = ti(fun(A,B),D) | -bounded_lattice(B). [resolve(674,a,672,b)]. 2.69/2.96 675 -bounded_lattice_bot(A) | bot_bot(A) != hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),B),C) | bot_bot(A) = ti(A,B) # label(fact_294_sup__eq__bot__iff) # label(axiom). [clausify(394)]. 2.69/2.96 Derived: bot_bot(fun(A,B)) != hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_sup_sup(fun(A,B)),C),D) | bot_bot(fun(A,B)) = ti(fun(A,B),C) | -bounded_lattice(B). [resolve(675,a,672,b)]. 2.69/2.96 676 -bounded_lattice_bot(A) | bot_bot(A) = hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),B),C) | bot_bot(A) != ti(A,C) | bot_bot(A) != ti(A,B) # label(fact_294_sup__eq__bot__iff) # label(axiom). [clausify(394)]. 2.69/2.96 Derived: bot_bot(fun(A,B)) = hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_sup_sup(fun(A,B)),C),D) | bot_bot(fun(A,B)) != ti(fun(A,B),D) | bot_bot(fun(A,B)) != ti(fun(A,B),C) | -bounded_lattice(B). [resolve(676,a,672,b)]. 2.69/2.96 677 -bounded_lattice_bot(A) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),bot_bot(A)),B) # label(fact_296_sup__bot__left) # label(axiom). [clausify(400)]. 2.69/2.96 Derived: ti(fun(A,B),C) = hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_sup_sup(fun(A,B)),bot_bot(fun(A,B))),C) | -bounded_lattice(B). [resolve(677,a,672,b)]. 2.69/2.96 678 bounded_lattice_bot(bool) # label(arity_HOL_Obool___Lattices_Obounded__lattice__bot) # label(axiom). [assumption]. 2.69/2.96 Derived: bot_bot(bool) = hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_inf_inf(bool),bot_bot(bool)),A). [resolve(678,a,671,a)]. 2.69/2.96 Derived: bot_bot(bool) = hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_inf_inf(bool),A),bot_bot(bool)). [resolve(678,a,673,a)]. 2.69/2.96 Derived: bot_bot(bool) != hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),A),B) | bot_bot(bool) = ti(bool,B). [resolve(678,a,674,a)]. 2.69/2.96 Derived: bot_bot(bool) != hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),A),B) | bot_bot(bool) = ti(bool,A). [resolve(678,a,675,a)]. 2.69/2.96 Derived: bot_bot(bool) = hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),A),B) | bot_bot(bool) != ti(bool,B) | bot_bot(bool) != ti(bool,A). [resolve(678,a,676,a)]. 2.69/2.96 Derived: ti(bool,A) = hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),bot_bot(bool)),A). [resolve(678,a,677,a)]. 6.88/7.18 679 -bounded_lattice_bot(A) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),semilattice_sup_sup(A),B),bot_bot(A)) # label(fact_295_sup__bot__right) # label(axiom). [clausify(497)]. 6.88/7.18 Derived: ti(fun(A,B),C) = hAPP(fun(A,B),fun(A,B),hAPP(fun(A,B),fun(fun(A,B),fun(A,B)),semilattice_sup_sup(fun(A,B)),C),bot_bot(fun(A,B))) | -bounded_lattice(B). [resolve(679,a,672,b)]. 6.88/7.18 Derived: ti(bool,A) = hAPP(bool,bool,hAPP(bool,fun(bool,bool),semilattice_sup_sup(bool),A),bot_bot(bool)). [resolve(679,a,678,a)]. 6.88/7.18 6.88/7.18 ============================== end predicate elimination ============= 6.88/7.18 6.88/7.18 Auto_denials: (non-Horn, no changes). 6.88/7.18 6.88/7.18 Term ordering decisions: 6.88/7.18 Function symbol KB weights: bool=1. nat=1. state=1. com=1. vname=1. loc_1=1. evaln=1. evalc=1. fconj=1. glb_1=1. loc=1. local=1. update=1. semi=1. ass=1. fdisj=1. fimplies=1. glb=1. fNot=1. getlocs=1. skip=1. fFalse=1. fTrue=1. hoare_Mirabelle_MGT=1. b=1. c=1. g=1. p=1. x_a=1. fun=1. ti=1. image=1. finite_fold=1. finite_fold_graph=1. finite_fold_image=1. finite100568337ommute=1. finite_comp_fun_idem=1. combk=1. finite1357897459simple=1. big_comm_monoid_big=1. finite908156982e_idem=1. fold_graph=1. hoare_1312322281e_case=1. hoare_1632998903le_rec=1. f22=1. f24=1. f25=1. f39=1. f40=1. f41=1. f42=1. f51=1. f52=1. f62=1. f68=1. f82=1. f83=1. f84=1. f88=1. f94=1. f96=1. f97=1. hoare_1656922687triple=1. ord_less_eq=1. semilattice_sup_sup=1. bot_bot=1. member=1. insert=1. semilattice_inf_inf=1. finite_finite_1=1. minus_minus=1. hoare_246368825triple=1. times_times=1. collect=1. hoare_279057269derivs=1. big_lattice_Sup_fin=1. fequal=1. the=1. finite_fold1Set=1. finite_fold1=1. finite_folding_one=1. finite2073411215e_idem=1. hoare_920331057_valid=1. combi=1. partial_flat_lub=1. the_elem=1. vname_case=1. vname_rec=1. big_semilattice_big=1. undefined=1. f78=1. f79=1. f89=1. combc=1. combb=1. combs=1. f3=1. f8=1. f9=1. f11=1. f12=1. f13=1. f14=1. f23=1. f36=1. f37=1. f50=1. f65=1. f66=1. f67=1. f69=1. f76=1. f77=1. f93=1. f95=1. f101=1. hAPP=1. f6=1. f7=1. f21=1. f30=1. f31=1. f32=1. f35=1. f38=1. f47=1. f60=1. f85=1. f86=1. f87=1. f99=1. f102=1. f103=1. f104=1. f105=1. f106=1. f107=1. f1=1. f10=1. f16=1. f20=1. f29=1. f43=1. f44=1. f48=1. f49=1. f58=1. f61=1. f63=1. f64=1. f71=1. f72=1. f73=1. f74=1. f75=1. f108=1. f4=1. f5=1. f17=1. f18=1. f19=1. f26=1. f70=1. f80=1. f81=1. f100=1. f2=1. f45=1. f46=1. f53=1. f54=1. f55=1. f56=1. f57=1. f90=1. f91=1. f92=1. f98=1. f15=1. f59=1. f27=1. f28=1. 6.88/7.18 6.88/7.18 ============================== end of process initial clauses ======== 6.88/7.18 6.88/7.18 ============================== CLAUSES FOR SEARCH ==================== 6.88/7.18 6.88/7.18 ============================== end of clauses for search ============= 6.88/7.18 6.88/7.18 ============================== SEARCH ================================ 6.88/7.18 6.88/7.18 % Starting search at 1.03 seconds. 6.88/7.18 6.88/7.18 Low Water (keep): wt=189.000, iters=3597 6.88/7.18 6.88/7.18 Low Water (keep): wt=172.000, iters=3408 6.88/7.18 6.88/7.18 Low Water (keep): wt=168.000, iters=3357 6.88/7.18 6.88/7.18 Low Water (keep): wt=167.000, iters=3405 6.88/7.18 6.88/7.18 Low Water (keep): wt=163.000, iters=3369 6.88/7.18 6.88/7.18 Low Water (keep): wt=159.000, iters=3337 6.88/7.18 6.88/7.18 Low Water (keep): wt=157.000, iters=3373 6.88/7.18 6.88/7.18 Low Water (keep): wt=156.000, iters=3352 6.88/7.18 6.88/7.18 Low Water (keep): wt=154.000, iters=3359 6.88/7.18 6.88/7.18 Low Water (keep): wt=153.000, iters=3352 6.88/7.18 6.88/7.18 Low Water (keep): wt=152.000, iters=3388 6.88/7.18 6.88/7.18 Low Water (keep): wt=149.000, iters=3342 6.88/7.18 6.88/7.18 Low Water (keep): wt=144.000, iters=3342 6.88/7.18 6.88/7.18 Low Water (keep): wt=139.000, iters=3361 6.88/7.18 6.88/7.18 Low Water (keep): wt=138.000, iters=3342 6.88/7.18 6.88/7.18 Low Water (keep): wt=137.000, iters=3470 6.88/7.18 6.88/7.18 Low Water (keep): wt=133.000, iters=3383 6.88/7.18 6.88/7.18 Low Water (keep): wt=131.000, iters=3336 6.88/7.18 6.88/7.18 Low Water (keep): wt=129.000, iters=3371 6.88/7.18 6.88/7.18 Low Water (keep): wt=127.000, iters=3334 6.88/7.18 6.88/7.18 Low Water (keep): wt=126.000, iters=3349 6.88/7.18 6.88/7.18 Low Water (keep): wt=125.000, iters=3407 6.88/7.18 6.88/7.18 Low Water (keep): wt=124.000, iters=3469 6.88/7.18 6.88/7.18 Low Water (keep): wt=122.000, iters=3398 6.88/7.18 6.88/7.18 Low Water (keep): wt=120.000, iters=3334 6.88/7.18 6.88/7.18 Low Water (keep): wt=119.000, iters=3351 6.88/7.18 6.88/7.18 Low Water (keep): wt=118.000, iters=3406 6.88/7.18 6.88/7.18 Low Water (keep): wt=115.000, iters=3400 6.88/7.18 6.88/7.18 Low Water (keep): wt=113.000, iters=3351 6.88/7.18 6.88/7.18 ============================== PROOF ================================= 6.88/7.18 % SZS status Theorem 6.88/7.18 % SZS output start Refutation 6.88/7.18 6.88/7.18 % Proof 1 at 4.67 (+ 0.04) seconds. 6.88/7.18 % Length of proof is 25. 6.88/7.18 % Level of proof is 4. 6.88/7.18 % Maximum clause weight is 298.000. 6.88/7.18 % Given clauses 445. 6.88/7.18 6.88/7.18 29 (all X_b all X_a all P all Q ti(X_a,P) = hAPP(X_b,X_a,hAPP(X_a,fun(X_b,X_a),combk(X_a,X_b),P),Q)) # label(help_COMBK_1_1_U) # label(axiom) # label(non_clause). [assumption]. 6.88/7.18 94 (all X_b all X_1 hAPP(fun(X_b,bool),X_b,the(X_b),hAPP(X_b,fun(X_b,bool),fequal(X_b),X_1)) = ti(X_b,X_1)) # label(fact_81_the__sym__eq__trivial) # label(axiom) # label(non_clause). [assumption]. 6.88/7.18 175 (all X_b all A_3 hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_3),bot_bot(fun(X_b,bool))) = hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(X_b,fun(X_b,bool),fequal(X_b),A_3))) # label(fact_12_singleton__conv2) # label(axiom) # label(non_clause). [assumption]. 6.88/7.18 360 (all X_b all Ga all Ca all Q_1 all Pa ((all Z_2 all S_2 (hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),Pa,Z_2),S_2)) -> hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),hAPP(fun(state,bool),fun(X_b,fun(state,bool)),combk(fun(state,bool),X_b),hAPP(state,fun(state,bool),hAPP(fun(state,fun(state,bool)),fun(state,fun(state,bool)),combc(state,state,bool),fequal(state)),S_2))),Ca),hAPP(fun(state,bool),fun(X_b,fun(state,bool)),combk(fun(state,bool),X_b),hAPP(X_b,fun(state,bool),Q_1,Z_2)))),bot_bot(fun(hoare_1656922687triple(X_b),bool))))))) -> hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Pa),Ca),Q_1)),bot_bot(fun(hoare_1656922687triple(X_b),bool))))))) # label(fact_5_escape) # label(axiom) # label(non_clause). [assumption]. 6.88/7.18 468 (all X_a all P hAPP(X_a,X_a,combi(X_a),P) = ti(X_a,P)) # label(help_COMBI_1_1_U) # label(axiom) # label(non_clause). [assumption]. 6.88/7.18 541 (all X_c all X_a all B_1_1 all B_2_1 ti(X_c,hAPP(X_a,X_c,B_1_1,B_2_1)) = hAPP(X_a,X_c,B_1_1,B_2_1)) # label(tsy_c_hAPP_res) # label(axiom) # label(non_clause). [assumption]. 6.88/7.18 730 ti(A,B) = hAPP(C,A,hAPP(A,fun(C,A),combk(A,C),B),D) # label(help_COMBK_1_1_U) # label(axiom). [clausify(29)]. 6.88/7.18 731 hAPP(A,B,hAPP(B,fun(A,B),combk(B,A),C),D) = ti(B,C). [copy(730),flip(a)]. 6.88/7.18 823 ti(A,B) = hAPP(fun(A,bool),A,the(A),hAPP(A,fun(A,bool),fequal(A),B)) # label(fact_81_the__sym__eq__trivial) # label(axiom). [clausify(94)]. 6.88/7.18 887 fFalse = ti(bool,fFalse) # label(tsy_c_fFalse_res) # label(hypothesis). [assumption]. 6.88/7.18 888 hAPP(fun(bool,bool),bool,the(bool),hAPP(bool,fun(bool,bool),fequal(bool),fFalse)) = fFalse. [copy(887),rewrite([823(4)]),flip(a)]. 6.88/7.18 973 hAPP(fun(A,bool),fun(A,bool),collect(A),hAPP(A,fun(A,bool),fequal(A),B)) = hAPP(fun(A,bool),fun(A,bool),hAPP(A,fun(fun(A,bool),fun(A,bool)),insert(A),B),bot_bot(fun(A,bool))) # label(fact_12_singleton__conv2) # label(axiom). [clausify(175)]. 6.88/7.18 974 hAPP(fun(A,bool),fun(A,bool),hAPP(A,fun(fun(A,bool),fun(A,bool)),insert(A),B),bot_bot(fun(A,bool))) = hAPP(fun(A,bool),fun(A,bool),collect(A),hAPP(A,fun(A,bool),fequal(A),B)). [copy(973),flip(a)]. 6.88/7.18 1089 -hBOOL(fFalse) # label(help_fFalse_1_1_U) # label(axiom). [assumption]. 6.88/7.18 1268 hBOOL(hAPP(state,bool,hAPP(A,fun(state,bool),B,f63(A,C,D,E,B)),f64(A,C,D,E,B))) | hBOOL(hAPP(fun(hoare_1656922687triple(A),bool),bool,hAPP(fun(hoare_1656922687triple(A),bool),fun(fun(hoare_1656922687triple(A),bool),bool),hoare_279057269derivs(A),C),hAPP(fun(hoare_1656922687triple(A),bool),fun(hoare_1656922687triple(A),bool),hAPP(hoare_1656922687triple(A),fun(fun(hoare_1656922687triple(A),bool),fun(hoare_1656922687triple(A),bool)),insert(hoare_1656922687triple(A)),hAPP(fun(A,fun(state,bool)),hoare_1656922687triple(A),hAPP(com,fun(fun(A,fun(state,bool)),hoare_1656922687triple(A)),hAPP(fun(A,fun(state,bool)),fun(com,fun(fun(A,fun(state,bool)),hoare_1656922687triple(A))),hoare_246368825triple(A),B),D),E)),bot_bot(fun(hoare_1656922687triple(A),bool))))) # label(fact_5_escape) # label(axiom). [clausify(360)]. 6.88/7.18 1269 hBOOL(hAPP(state,bool,hAPP(A,fun(state,bool),B,f63(A,C,D,E,B)),f64(A,C,D,E,B))) | hBOOL(hAPP(fun(hoare_1656922687triple(A),bool),bool,hAPP(fun(hoare_1656922687triple(A),bool),fun(fun(hoare_1656922687triple(A),bool),bool),hoare_279057269derivs(A),C),hAPP(fun(hoare_1656922687triple(A),bool),fun(hoare_1656922687triple(A),bool),collect(hoare_1656922687triple(A)),hAPP(hoare_1656922687triple(A),fun(hoare_1656922687triple(A),bool),fequal(hoare_1656922687triple(A)),hAPP(fun(A,fun(state,bool)),hoare_1656922687triple(A),hAPP(com,fun(fun(A,fun(state,bool)),hoare_1656922687triple(A)),hAPP(fun(A,fun(state,bool)),fun(com,fun(fun(A,fun(state,bool)),hoare_1656922687triple(A))),hoare_246368825triple(A),B),D),E))))). [copy(1268),rewrite([974(74)])]. 6.88/7.18 1456 ti(A,B) = hAPP(A,A,combi(A),B) # label(help_COMBI_1_1_U) # label(axiom). [clausify(468)]. 6.88/7.18 1457 hAPP(fun(A,bool),A,the(A),hAPP(A,fun(A,bool),fequal(A),B)) = hAPP(A,A,combi(A),B). [copy(1456),rewrite([823(1)])]. 6.88/7.18 1571 ti(A,hAPP(B,A,C,D)) = hAPP(B,A,C,D) # label(tsy_c_hAPP_res) # label(axiom). [clausify(541)]. 6.88/7.18 1572 hAPP(A,A,combi(A),hAPP(B,A,C,D)) = hAPP(B,A,C,D). [copy(1571),rewrite([823(2),1457(9)])]. 6.88/7.18 1651 -hBOOL(hAPP(fun(hoare_1656922687triple(x_a),bool),bool,hAPP(fun(hoare_1656922687triple(x_a),bool),fun(fun(hoare_1656922687triple(x_a),bool),bool),hoare_279057269derivs(x_a),g),hAPP(fun(hoare_1656922687triple(x_a),bool),fun(hoare_1656922687triple(x_a),bool),hAPP(hoare_1656922687triple(x_a),fun(fun(hoare_1656922687triple(x_a),bool),fun(hoare_1656922687triple(x_a),bool)),insert(hoare_1656922687triple(x_a)),hAPP(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a),hAPP(com,fun(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a)),hAPP(fun(x_a,fun(state,bool)),fun(com,fun(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a))),hoare_246368825triple(x_a),hAPP(fun(state,bool),fun(x_a,fun(state,bool)),combk(fun(state,bool),x_a),hAPP(bool,fun(state,bool),combk(bool,state),fFalse))),c),hAPP(fun(state,bool),fun(x_a,fun(state,bool)),hAPP(fun(x_a,fun(fun(state,bool),fun(state,bool))),fun(fun(state,bool),fun(x_a,fun(state,bool))),combc(x_a,fun(state,bool),fun(state,bool)),hAPP(fun(x_a,fun(state,fun(bool,bool))),fun(x_a,fun(fun(state,bool),fun(state,bool))),hAPP(fun(fun(state,fun(bool,bool)),fun(fun(state,bool),fun(state,bool))),fun(fun(x_a,fun(state,fun(bool,bool))),fun(x_a,fun(fun(state,bool),fun(state,bool)))),combb(fun(state,fun(bool,bool)),fun(fun(state,bool),fun(state,bool)),x_a),combs(state,bool,bool)),hAPP(fun(x_a,fun(state,bool)),fun(x_a,fun(state,fun(bool,bool))),hAPP(fun(fun(state,bool),fun(state,fun(bool,bool))),fun(fun(x_a,fun(state,bool)),fun(x_a,fun(state,fun(bool,bool)))),combb(fun(state,bool),fun(state,fun(bool,bool)),x_a),hAPP(fun(bool,fun(bool,bool)),fun(fun(state,bool),fun(state,fun(bool,bool))),combb(bool,fun(bool,bool),state),fconj)),p))),hAPP(fun(state,bool),fun(state,bool),hAPP(fun(bool,bool),fun(fun(state,bool),fun(state,bool)),combb(bool,bool,state),fNot),b)))),bot_bot(fun(hoare_1656922687triple(x_a),bool))))) # label(conj_0) # label(negated_conjecture). [assumption]. 6.88/7.18 1652 -hBOOL(hAPP(fun(hoare_1656922687triple(x_a),bool),bool,hAPP(fun(hoare_1656922687triple(x_a),bool),fun(fun(hoare_1656922687triple(x_a),bool),bool),hoare_279057269derivs(x_a),g),hAPP(fun(hoare_1656922687triple(x_a),bool),fun(hoare_1656922687triple(x_a),bool),collect(hoare_1656922687triple(x_a)),hAPP(hoare_1656922687triple(x_a),fun(hoare_1656922687triple(x_a),bool),fequal(hoare_1656922687triple(x_a)),hAPP(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a),hAPP(com,fun(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a)),hAPP(fun(x_a,fun(state,bool)),fun(com,fun(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a))),hoare_246368825triple(x_a),hAPP(fun(state,bool),fun(x_a,fun(state,bool)),combk(fun(state,bool),x_a),hAPP(bool,fun(state,bool),combk(bool,state),fFalse))),c),hAPP(fun(state,bool),fun(x_a,fun(state,bool)),hAPP(fun(x_a,fun(fun(state,bool),fun(state,bool))),fun(fun(state,bool),fun(x_a,fun(state,bool))),combc(x_a,fun(state,bool),fun(state,bool)),hAPP(fun(x_a,fun(state,fun(bool,bool))),fun(x_a,fun(fun(state,bool),fun(state,bool))),hAPP(fun(fun(state,fun(bool,bool)),fun(fun(state,bool),fun(state,bool))),fun(fun(x_a,fun(state,fun(bool,bool))),fun(x_a,fun(fun(state,bool),fun(state,bool)))),combb(fun(state,fun(bool,bool)),fun(fun(state,bool),fun(state,bool)),x_a),combs(state,bool,bool)),hAPP(fun(x_a,fun(state,bool)),fun(x_a,fun(state,fun(bool,bool))),hAPP(fun(fun(state,bool),fun(state,fun(bool,bool))),fun(fun(x_a,fun(state,bool)),fun(x_a,fun(state,fun(bool,bool)))),combb(fun(state,bool),fun(state,fun(bool,bool)),x_a),hAPP(fun(bool,fun(bool,bool)),fun(fun(state,bool),fun(state,fun(bool,bool))),combb(bool,fun(bool,bool),state),fconj)),p))),hAPP(fun(state,bool),fun(state,bool),hAPP(fun(bool,bool),fun(fun(state,bool),fun(state,bool)),combb(bool,bool,state),fNot),b))))))). [copy(1651),rewrite([974(303)])]. 6.88/7.18 1888 hAPP(A,B,hAPP(B,fun(A,B),combk(B,A),C),D) = hAPP(B,B,combi(B),C). [back_rewrite(731),rewrite([823(5),1457(12)])]. 6.88/7.18 2023 hAPP(bool,bool,combi(bool),fFalse) = fFalse. [back_rewrite(888),rewrite([1457(15)])]. 6.88/7.18 7151 $F. [resolve(1652,a,1269,b),rewrite([1888(252),1572(22),1888(234),2023(6)]),unit_del(a,1089)]. 6.88/7.18 6.88/7.18 % SZS output end Refutation 6.88/7.18 ============================== end of proof ========================== 6.88/7.18 6.88/7.18 ============================== STATISTICS ============================ 6.88/7.18 6.88/7.18 Given=445. Generated=11254. Kept=6092. proofs=1. 6.88/7.18 Usable=434. Sos=4738. Demods=406. Limbo=0, Disabled=1905. Hints=0. 6.88/7.18 Megabytes=55.42. 6.88/7.18 User_CPU=4.68, System_CPU=0.04, Wall_clock=5. 6.88/7.18 6.88/7.18 ============================== end of statistics ===================== 6.88/7.18 6.88/7.18 ============================== end of search ========================= 6.88/7.18 6.88/7.18 THEOREM PROVED 6.88/7.18 % SZS status Theorem 6.88/7.18 6.88/7.18 Exiting with 1 proof. 6.88/7.18 6.88/7.18 Process 20774 exit (max_proofs) Mon Jul 3 08:51:15 2023 6.88/7.18 Prover9 interrupted 6.88/7.18 EOF