TPTP Problem File: SWW862+1.p
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%------------------------------------------------------------------------------
% File : SWW862+1 : TPTP v8.2.0. Released v7.3.0.
% Domain : Software Verification
% Problem : namespaceProps__78__nsAll2_nsAppend__dep
% Version : Especial.
% English :
% Refs : [Kum18] Kumar (2018), Email to Geoff Sutcliffe
% Source : [Kum18]
% Names : namespaceProps__78__nsAll2_nsAppend__dep [Kum18]
% Status : Theorem
% Rating : 1.00 v7.3.0
% Syntax : Number of formulae : 43 ( 6 unt; 0 def)
% Number of atoms : 215 ( 42 equ)
% Maximal formula atoms : 15 ( 5 avg)
% Number of connectives : 220 ( 48 ~; 37 |; 47 &)
% ( 42 <=>; 46 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 7 avg)
% Maximal term depth : 14 ( 2 avg)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 22 ( 22 usr; 5 con; 0-3 aty)
% Number of variables : 137 ( 132 !; 5 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments :
%------------------------------------------------------------------------------
fof('HL_TRUTH',axiom,
p__01(s__02(cbool__00,cT__00)) ).
fof('HL_FALSITY',axiom,
~ p__01(s__02(cbool__00,cF__00)) ).
fof('HL_BOOL_CASES',axiom,
! [Vt] :
( s__02(cbool__00,Vt) = s__02(cbool__00,cT__00)
| s__02(cbool__00,Vt) = s__02(cbool__00,cF__00) ) ).
fof('HL_EXT',axiom,
! [V_3f2384,V_3f2380,Vf,Vg] :
( ! [Vx] : s__02(V_3f2380,chapp__02(s__02(cfun__02(V_3f2384,V_3f2380),Vf),s__02(V_3f2384,Vx))) = s__02(V_3f2380,chapp__02(s__02(cfun__02(V_3f2384,V_3f2380),Vg),s__02(V_3f2384,Vx)))
=> s__02(cfun__02(V_3f2384,V_3f2380),Vf) = s__02(cfun__02(V_3f2384,V_3f2380),Vg) ) ).
fof('thm.bool.EXISTS_DEF',axiom,
! [V_27A_27,Vx] : s__02(cbool__00,c_24exists__01(s__02(cfun__02(V_27A_27,cbool__00),Vx))) = s__02(cbool__00,chapp__02(s__02(cfun__02(V_27A_27,cbool__00),Vx),s__02(V_27A_27,c_27const_2emin_2e_40_27__01(s__02(cfun__02(V_27A_27,cbool__00),Vx))))) ).
fof('thm.bool.SELECT_AX',axiom,
! [V_27A_27,V_27P_27,V_27x_27] :
( p__01(s__02(cbool__00,chapp__02(s__02(cfun__02(V_27A_27,cbool__00),V_27P_27),s__02(V_27A_27,V_27x_27))))
=> p__01(s__02(cbool__00,chapp__02(s__02(cfun__02(V_27A_27,cbool__00),V_27P_27),s__02(V_27A_27,c_27const_2emin_2e_40_27__01(s__02(cfun__02(V_27A_27,cbool__00),V_27P_27)))))) ) ).
fof('thm.bool.TRUTH',axiom,
p__01(s__02(cbool__00,cT__00)) ).
fof('thm.bool.IMP_ANTISYM_AX',axiom,
! [V_27t1_27,V_27t2_27] :
( ( p__01(s__02(cbool__00,V_27t1_27))
=> p__01(s__02(cbool__00,V_27t2_27)) )
=> ( ( p__01(s__02(cbool__00,V_27t2_27))
=> p__01(s__02(cbool__00,V_27t1_27)) )
=> s__02(cbool__00,V_27t1_27) = s__02(cbool__00,V_27t2_27) ) ) ).
fof('thm.bool.AND_CLAUSES',axiom,
! [V_27t_27] :
( ( ( p__01(s__02(cbool__00,cT__00))
& p__01(s__02(cbool__00,V_27t_27)) )
<=> p__01(s__02(cbool__00,V_27t_27)) )
& ( ( p__01(s__02(cbool__00,V_27t_27))
& p__01(s__02(cbool__00,cT__00)) )
<=> p__01(s__02(cbool__00,V_27t_27)) )
& ( ( p__01(s__02(cbool__00,cF__00))
& p__01(s__02(cbool__00,V_27t_27)) )
<=> p__01(s__02(cbool__00,cF__00)) )
& ( ( p__01(s__02(cbool__00,V_27t_27))
& p__01(s__02(cbool__00,cF__00)) )
<=> p__01(s__02(cbool__00,cF__00)) )
& ( ( p__01(s__02(cbool__00,V_27t_27))
& p__01(s__02(cbool__00,V_27t_27)) )
<=> p__01(s__02(cbool__00,V_27t_27)) ) ) ).
fof('thm.bool.IMP_CLAUSES',axiom,
! [V_27t_27] :
( ( ( p__01(s__02(cbool__00,cT__00))
=> p__01(s__02(cbool__00,V_27t_27)) )
<=> p__01(s__02(cbool__00,V_27t_27)) )
& ( ( p__01(s__02(cbool__00,V_27t_27))
=> p__01(s__02(cbool__00,cT__00)) )
<=> p__01(s__02(cbool__00,cT__00)) )
& ( ( p__01(s__02(cbool__00,cF__00))
=> p__01(s__02(cbool__00,V_27t_27)) )
<=> p__01(s__02(cbool__00,cT__00)) )
& ( ( p__01(s__02(cbool__00,V_27t_27))
=> p__01(s__02(cbool__00,V_27t_27)) )
<=> p__01(s__02(cbool__00,cT__00)) )
& ( ( p__01(s__02(cbool__00,V_27t_27))
=> p__01(s__02(cbool__00,cF__00)) )
<=> ~ p__01(s__02(cbool__00,V_27t_27)) ) ) ).
fof('thm.bool.NOT_CLAUSES',axiom,
( ! [V_27t_27] :
( ~ ~ p__01(s__02(cbool__00,V_27t_27))
<=> p__01(s__02(cbool__00,V_27t_27)) )
& ( ~ p__01(s__02(cbool__00,cT__00))
<=> p__01(s__02(cbool__00,cF__00)) )
& ( ~ p__01(s__02(cbool__00,cF__00))
<=> p__01(s__02(cbool__00,cT__00)) ) ) ).
fof('thm.bool.EQ_REFL',axiom,
! [V_27A_27,V_27x_27] : s__02(V_27A_27,V_27x_27) = s__02(V_27A_27,V_27x_27) ).
fof('thm.bool.REFL_CLAUSE',axiom,
! [V_27A_27,V_27x_27] :
( s__02(V_27A_27,V_27x_27) = s__02(V_27A_27,V_27x_27)
<=> p__01(s__02(cbool__00,cT__00)) ) ).
fof('thm.bool.EQ_SYM_EQ',axiom,
! [V_27A_27,V_27x_27,V_27y_27] :
( s__02(V_27A_27,V_27x_27) = s__02(V_27A_27,V_27y_27)
<=> s__02(V_27A_27,V_27y_27) = s__02(V_27A_27,V_27x_27) ) ).
fof('thm.bool.EQ_CLAUSES',axiom,
! [V_27t_27] :
( ( s__02(cbool__00,cT__00) = s__02(cbool__00,V_27t_27)
<=> p__01(s__02(cbool__00,V_27t_27)) )
& ( s__02(cbool__00,V_27t_27) = s__02(cbool__00,cT__00)
<=> p__01(s__02(cbool__00,V_27t_27)) )
& ( s__02(cbool__00,cF__00) = s__02(cbool__00,V_27t_27)
<=> ~ p__01(s__02(cbool__00,V_27t_27)) )
& ( s__02(cbool__00,V_27t_27) = s__02(cbool__00,cF__00)
<=> ~ p__01(s__02(cbool__00,V_27t_27)) ) ) ).
fof('thm.bool.RIGHT_AND_FORALL_THM',axiom,
! [V_27A_27,V_27P_27,V_27Q_27] :
( ( p__01(s__02(cbool__00,V_27P_27))
& ! [V_27x_27] : p__01(s__02(cbool__00,chapp__02(s__02(cfun__02(V_27A_27,cbool__00),V_27Q_27),s__02(V_27A_27,V_27x_27)))) )
<=> ! [V_27x_27] :
( p__01(s__02(cbool__00,V_27P_27))
& p__01(s__02(cbool__00,chapp__02(s__02(cfun__02(V_27A_27,cbool__00),V_27Q_27),s__02(V_27A_27,V_27x_27)))) ) ) ).
fof('thm.bool.LEFT_OR_OVER_AND',axiom,
! [V_27A_270,V_27B_270,V_27C_270] :
( ( p__01(s__02(cbool__00,V_27A_270))
| ( p__01(s__02(cbool__00,V_27B_270))
& p__01(s__02(cbool__00,V_27C_270)) ) )
<=> ( ( p__01(s__02(cbool__00,V_27A_270))
| p__01(s__02(cbool__00,V_27B_270)) )
& ( p__01(s__02(cbool__00,V_27A_270))
| p__01(s__02(cbool__00,V_27C_270)) ) ) ) ).
fof('thm.bool.RIGHT_OR_OVER_AND',axiom,
! [V_27A_270,V_27B_270,V_27C_270] :
( ( ( p__01(s__02(cbool__00,V_27B_270))
& p__01(s__02(cbool__00,V_27C_270)) )
| p__01(s__02(cbool__00,V_27A_270)) )
<=> ( ( p__01(s__02(cbool__00,V_27B_270))
| p__01(s__02(cbool__00,V_27A_270)) )
& ( p__01(s__02(cbool__00,V_27C_270))
| p__01(s__02(cbool__00,V_27A_270)) ) ) ) ).
fof('thm.bool.AND_IMP_INTRO',axiom,
! [V_27t1_27,V_27t2_27,V_27t3_27] :
( ( p__01(s__02(cbool__00,V_27t1_27))
=> ( p__01(s__02(cbool__00,V_27t2_27))
=> p__01(s__02(cbool__00,V_27t3_27)) ) )
<=> ( ( p__01(s__02(cbool__00,V_27t1_27))
& p__01(s__02(cbool__00,V_27t2_27)) )
=> p__01(s__02(cbool__00,V_27t3_27)) ) ) ).
fof('thm.bool.IMP_CONG',axiom,
! [V_27x_27,V_27x_7c39_7c_27,V_27y_27,V_27y_7c39_7c_27] :
( ( s__02(cbool__00,V_27x_27) = s__02(cbool__00,V_27x_7c39_7c_27)
& ( p__01(s__02(cbool__00,V_27x_7c39_7c_27))
=> s__02(cbool__00,V_27y_27) = s__02(cbool__00,V_27y_7c39_7c_27) ) )
=> ( ( p__01(s__02(cbool__00,V_27x_27))
=> p__01(s__02(cbool__00,V_27y_27)) )
<=> ( p__01(s__02(cbool__00,V_27x_7c39_7c_27))
=> p__01(s__02(cbool__00,V_27y_7c39_7c_27)) ) ) ) ).
fof('thm.sat.NOT_NOT',axiom,
! [V_27t_27] :
( ~ ~ p__01(s__02(cbool__00,V_27t_27))
<=> p__01(s__02(cbool__00,V_27t_27)) ) ).
fof('thm.sat.AND_INV_IMP',axiom,
! [V_27A_270] :
( p__01(s__02(cbool__00,V_27A_270))
=> ( ~ p__01(s__02(cbool__00,V_27A_270))
=> p__01(s__02(cbool__00,cF__00)) ) ) ).
fof('thm.sat.OR_DUAL2',axiom,
! [V_27B_270,V_27A_270] :
( ( ~ ( p__01(s__02(cbool__00,V_27A_270))
| p__01(s__02(cbool__00,V_27B_270)) )
=> p__01(s__02(cbool__00,cF__00)) )
<=> ( ( p__01(s__02(cbool__00,V_27A_270))
=> p__01(s__02(cbool__00,cF__00)) )
=> ( ~ p__01(s__02(cbool__00,V_27B_270))
=> p__01(s__02(cbool__00,cF__00)) ) ) ) ).
fof('thm.sat.OR_DUAL3',axiom,
! [V_27B_270,V_27A_270] :
( ( ~ ( ~ p__01(s__02(cbool__00,V_27A_270))
| p__01(s__02(cbool__00,V_27B_270)) )
=> p__01(s__02(cbool__00,cF__00)) )
<=> ( p__01(s__02(cbool__00,V_27A_270))
=> ( ~ p__01(s__02(cbool__00,V_27B_270))
=> p__01(s__02(cbool__00,cF__00)) ) ) ) ).
fof('thm.sat.AND_INV2',axiom,
! [V_27A_270] :
( ( ~ p__01(s__02(cbool__00,V_27A_270))
=> p__01(s__02(cbool__00,cF__00)) )
=> ( ( p__01(s__02(cbool__00,V_27A_270))
=> p__01(s__02(cbool__00,cF__00)) )
=> p__01(s__02(cbool__00,cF__00)) ) ) ).
fof('thm.sat.dc_eq',axiom,
! [V_27r_27,V_27q_27,V_27p_27] :
( ( p__01(s__02(cbool__00,V_27p_27))
<=> s__02(cbool__00,V_27q_27) = s__02(cbool__00,V_27r_27) )
<=> ( ( p__01(s__02(cbool__00,V_27p_27))
| p__01(s__02(cbool__00,V_27q_27))
| p__01(s__02(cbool__00,V_27r_27)) )
& ( p__01(s__02(cbool__00,V_27p_27))
| ~ p__01(s__02(cbool__00,V_27r_27))
| ~ p__01(s__02(cbool__00,V_27q_27)) )
& ( p__01(s__02(cbool__00,V_27q_27))
| ~ p__01(s__02(cbool__00,V_27r_27))
| ~ p__01(s__02(cbool__00,V_27p_27)) )
& ( p__01(s__02(cbool__00,V_27r_27))
| ~ p__01(s__02(cbool__00,V_27q_27))
| ~ p__01(s__02(cbool__00,V_27p_27)) ) ) ) ).
fof('thm.sat.dc_conj',axiom,
! [V_27r_27,V_27q_27,V_27p_27] :
( ( p__01(s__02(cbool__00,V_27p_27))
<=> ( p__01(s__02(cbool__00,V_27q_27))
& p__01(s__02(cbool__00,V_27r_27)) ) )
<=> ( ( p__01(s__02(cbool__00,V_27p_27))
| ~ p__01(s__02(cbool__00,V_27q_27))
| ~ p__01(s__02(cbool__00,V_27r_27)) )
& ( p__01(s__02(cbool__00,V_27q_27))
| ~ p__01(s__02(cbool__00,V_27p_27)) )
& ( p__01(s__02(cbool__00,V_27r_27))
| ~ p__01(s__02(cbool__00,V_27p_27)) ) ) ) ).
fof('thm.sat.dc_disj',axiom,
! [V_27r_27,V_27q_27,V_27p_27] :
( ( p__01(s__02(cbool__00,V_27p_27))
<=> ( p__01(s__02(cbool__00,V_27q_27))
| p__01(s__02(cbool__00,V_27r_27)) ) )
<=> ( ( p__01(s__02(cbool__00,V_27p_27))
| ~ p__01(s__02(cbool__00,V_27q_27)) )
& ( p__01(s__02(cbool__00,V_27p_27))
| ~ p__01(s__02(cbool__00,V_27r_27)) )
& ( p__01(s__02(cbool__00,V_27q_27))
| p__01(s__02(cbool__00,V_27r_27))
| ~ p__01(s__02(cbool__00,V_27p_27)) ) ) ) ).
fof('thm.sat.dc_imp',axiom,
! [V_27r_27,V_27q_27,V_27p_27] :
( ( p__01(s__02(cbool__00,V_27p_27))
<=> ( p__01(s__02(cbool__00,V_27q_27))
=> p__01(s__02(cbool__00,V_27r_27)) ) )
<=> ( ( p__01(s__02(cbool__00,V_27p_27))
| p__01(s__02(cbool__00,V_27q_27)) )
& ( p__01(s__02(cbool__00,V_27p_27))
| ~ p__01(s__02(cbool__00,V_27r_27)) )
& ( ~ p__01(s__02(cbool__00,V_27q_27))
| p__01(s__02(cbool__00,V_27r_27))
| ~ p__01(s__02(cbool__00,V_27p_27)) ) ) ) ).
fof('thm.sat.dc_neg',axiom,
! [V_27q_27,V_27p_27] :
( ( p__01(s__02(cbool__00,V_27p_27))
<=> ~ p__01(s__02(cbool__00,V_27q_27)) )
<=> ( ( p__01(s__02(cbool__00,V_27p_27))
| p__01(s__02(cbool__00,V_27q_27)) )
& ( ~ p__01(s__02(cbool__00,V_27q_27))
| ~ p__01(s__02(cbool__00,V_27p_27)) ) ) ) ).
fof('thm.sat.pth_ni1',axiom,
! [V_27q_27,V_27p_27] :
( ~ ( p__01(s__02(cbool__00,V_27p_27))
=> p__01(s__02(cbool__00,V_27q_27)) )
=> p__01(s__02(cbool__00,V_27p_27)) ) ).
fof('thm.sat.pth_ni2',axiom,
! [V_27q_27,V_27p_27] :
( ~ ( p__01(s__02(cbool__00,V_27p_27))
=> p__01(s__02(cbool__00,V_27q_27)) )
=> ~ p__01(s__02(cbool__00,V_27q_27)) ) ).
fof('thm.sat.pth_no1',axiom,
! [V_27q_27,V_27p_27] :
( ~ ( p__01(s__02(cbool__00,V_27p_27))
| p__01(s__02(cbool__00,V_27q_27)) )
=> ~ p__01(s__02(cbool__00,V_27p_27)) ) ).
fof('thm.sat.pth_no2',axiom,
! [V_27q_27,V_27p_27] :
( ~ ( p__01(s__02(cbool__00,V_27p_27))
| p__01(s__02(cbool__00,V_27q_27)) )
=> ~ p__01(s__02(cbool__00,V_27q_27)) ) ).
fof('thm.sat.pth_nn',axiom,
! [V_27p_27] :
( ~ ~ p__01(s__02(cbool__00,V_27p_27))
=> p__01(s__02(cbool__00,V_27p_27)) ) ).
fof('thm.option.option_nchotomy',axiom,
! [V_27A_27,V_27opt_27] :
( s__02(c_27type_2eoption_2eoption_27__01(V_27A_27),V_27opt_27) = s__02(c_27type_2eoption_2eoption_27__01(V_27A_27),c_27const_2eoption_2eNONE_27__00)
| ? [V_27x_27] : s__02(c_27type_2eoption_2eoption_27__01(V_27A_27),V_27opt_27) = s__02(c_27type_2eoption_2eoption_27__01(V_27A_27),c_27const_2eoption_2eSOME_27__01(s__02(V_27A_27,V_27x_27))) ) ).
fof('thm.option.SOME_11',axiom,
! [V_27A_27,V_27x_27,V_27y_27] :
( s__02(c_27type_2eoption_2eoption_27__01(V_27A_27),c_27const_2eoption_2eSOME_27__01(s__02(V_27A_27,V_27x_27))) = s__02(c_27type_2eoption_2eoption_27__01(V_27A_27),c_27const_2eoption_2eSOME_27__01(s__02(V_27A_27,V_27y_27)))
<=> s__02(V_27A_27,V_27x_27) = s__02(V_27A_27,V_27y_27) ) ).
fof('thm.option.NOT_SOME_NONE',axiom,
! [V_27A_27,V_27x_27] : s__02(c_27type_2eoption_2eoption_27__01(V_27A_27),c_27const_2eoption_2eSOME_27__01(s__02(V_27A_27,V_27x_27))) != s__02(c_27type_2eoption_2eoption_27__01(V_27A_27),c_27const_2eoption_2eNONE_27__00) ).
fof('thm.namespace.nsSub_def',axiom,
! [V_27D_27,V_27A_27,V_27B_27,V_27C_27,V_27r_27,V_27env1_27,V_27env2_27] :
( p__01(s__02(cbool__00,c_27const_2enamespace_2ensSub_27__03(s__02(cfun__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),cfun__02(V_27C_27,cfun__02(V_27D_27,cbool__00))),V_27r_27),s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27),V_27env1_27),s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27D_27),V_27env2_27))))
<=> ( ! [V_27id_27,V_27v1_27] :
( s__02(c_27type_2eoption_2eoption_27__01(V_27C_27),c_27const_2enamespace_2ensLookup_27__02(s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27),V_27env1_27),s__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),V_27id_27))) = s__02(c_27type_2eoption_2eoption_27__01(V_27C_27),c_27const_2eoption_2eSOME_27__01(s__02(V_27C_27,V_27v1_27)))
=> ? [V_27v2_27] :
( s__02(c_27type_2eoption_2eoption_27__01(V_27D_27),c_27const_2enamespace_2ensLookup_27__02(s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27D_27),V_27env2_27),s__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),V_27id_27))) = s__02(c_27type_2eoption_2eoption_27__01(V_27D_27),c_27const_2eoption_2eSOME_27__01(s__02(V_27D_27,V_27v2_27)))
& p__01(s__02(cbool__00,chapp__02(s__02(cfun__02(V_27D_27,cbool__00),chapp__02(s__02(cfun__02(V_27C_27,cfun__02(V_27D_27,cbool__00)),chapp__02(s__02(cfun__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),cfun__02(V_27C_27,cfun__02(V_27D_27,cbool__00))),V_27r_27),s__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),V_27id_27))),s__02(V_27C_27,V_27v1_27))),s__02(V_27D_27,V_27v2_27)))) ) )
& ! [V_27path_27] :
( s__02(c_27type_2eoption_2eoption_27__01(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27D_27)),c_27const_2enamespace_2ensLookupMod_27__02(s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27D_27),V_27env2_27),s__02(c_27type_2elist_2elist_27__01(V_27A_27),V_27path_27))) = s__02(c_27type_2eoption_2eoption_27__01(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27D_27)),c_27const_2eoption_2eNONE_27__00)
=> s__02(c_27type_2eoption_2eoption_27__01(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27)),c_27const_2enamespace_2ensLookupMod_27__02(s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27),V_27env1_27),s__02(c_27type_2elist_2elist_27__01(V_27A_27),V_27path_27))) = s__02(c_27type_2eoption_2eoption_27__01(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27)),c_27const_2eoption_2eNONE_27__00) ) ) ) ).
fof('thm.namespace.nsAll2_def',axiom,
! [V_27D_27,V_27A_27,V_27B_27,V_27C_27,V__2] :
( ! [V_27r_27,V_27x_27,V_27y_27,V_27z_27] : s__02(cbool__00,chapp__02(s__02(cfun__02(V_27C_27,cbool__00),chapp__02(s__02(cfun__02(V_27D_27,cfun__02(V_27C_27,cbool__00)),chapp__02(s__02(cfun__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),cfun__02(V_27D_27,cfun__02(V_27C_27,cbool__00))),chapp__02(s__02(cfun__02(cfun__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),cfun__02(V_27C_27,cfun__02(V_27D_27,cbool__00))),cfun__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),cfun__02(V_27D_27,cfun__02(V_27C_27,cbool__00)))),V__2),s__02(cfun__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),cfun__02(V_27C_27,cfun__02(V_27D_27,cbool__00))),V_27r_27))),s__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),V_27x_27))),s__02(V_27D_27,V_27y_27))),s__02(V_27C_27,V_27z_27))) = s__02(cbool__00,chapp__02(s__02(cfun__02(V_27D_27,cbool__00),chapp__02(s__02(cfun__02(V_27C_27,cfun__02(V_27D_27,cbool__00)),chapp__02(s__02(cfun__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),cfun__02(V_27C_27,cfun__02(V_27D_27,cbool__00))),V_27r_27),s__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),V_27x_27))),s__02(V_27C_27,V_27z_27))),s__02(V_27D_27,V_27y_27)))
=> ! [V__1] :
( ! [V_27r_27,V_27x_27,V_27y_27] : s__02(cfun__02(V_27C_27,cbool__00),chapp__02(s__02(cfun__02(V_27D_27,cfun__02(V_27C_27,cbool__00)),chapp__02(s__02(cfun__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),cfun__02(V_27D_27,cfun__02(V_27C_27,cbool__00))),chapp__02(s__02(cfun__02(cfun__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),cfun__02(V_27C_27,cfun__02(V_27D_27,cbool__00))),cfun__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),cfun__02(V_27D_27,cfun__02(V_27C_27,cbool__00)))),V__1),s__02(cfun__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),cfun__02(V_27C_27,cfun__02(V_27D_27,cbool__00))),V_27r_27))),s__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),V_27x_27))),s__02(V_27D_27,V_27y_27))) = s__02(cfun__02(V_27C_27,cbool__00),chapp__02(s__02(cfun__02(V_27D_27,cfun__02(V_27C_27,cbool__00)),chapp__02(s__02(cfun__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),cfun__02(V_27D_27,cfun__02(V_27C_27,cbool__00))),chapp__02(s__02(cfun__02(cfun__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),cfun__02(V_27C_27,cfun__02(V_27D_27,cbool__00))),cfun__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),cfun__02(V_27D_27,cfun__02(V_27C_27,cbool__00)))),V__2),s__02(cfun__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),cfun__02(V_27C_27,cfun__02(V_27D_27,cbool__00))),V_27r_27))),s__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),V_27x_27))),s__02(V_27D_27,V_27y_27)))
=> ! [V__0] :
( ! [V_27r_27,V_27x_27] : s__02(cfun__02(V_27D_27,cfun__02(V_27C_27,cbool__00)),chapp__02(s__02(cfun__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),cfun__02(V_27D_27,cfun__02(V_27C_27,cbool__00))),chapp__02(s__02(cfun__02(cfun__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),cfun__02(V_27C_27,cfun__02(V_27D_27,cbool__00))),cfun__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),cfun__02(V_27D_27,cfun__02(V_27C_27,cbool__00)))),V__0),s__02(cfun__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),cfun__02(V_27C_27,cfun__02(V_27D_27,cbool__00))),V_27r_27))),s__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),V_27x_27))) = s__02(cfun__02(V_27D_27,cfun__02(V_27C_27,cbool__00)),chapp__02(s__02(cfun__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),cfun__02(V_27D_27,cfun__02(V_27C_27,cbool__00))),chapp__02(s__02(cfun__02(cfun__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),cfun__02(V_27C_27,cfun__02(V_27D_27,cbool__00))),cfun__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),cfun__02(V_27D_27,cfun__02(V_27C_27,cbool__00)))),V__1),s__02(cfun__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),cfun__02(V_27C_27,cfun__02(V_27D_27,cbool__00))),V_27r_27))),s__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),V_27x_27)))
=> ! [V_27r_27,V_27env1_27,V_27env2_27] :
( p__01(s__02(cbool__00,c_27const_2enamespace_2ensAll2_27__03(s__02(cfun__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),cfun__02(V_27C_27,cfun__02(V_27D_27,cbool__00))),V_27r_27),s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27),V_27env1_27),s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27D_27),V_27env2_27))))
<=> ( p__01(s__02(cbool__00,c_27const_2enamespace_2ensSub_27__03(s__02(cfun__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),cfun__02(V_27C_27,cfun__02(V_27D_27,cbool__00))),V_27r_27),s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27),V_27env1_27),s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27D_27),V_27env2_27))))
& p__01(s__02(cbool__00,c_27const_2enamespace_2ensSub_27__03(s__02(cfun__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),cfun__02(V_27D_27,cfun__02(V_27C_27,cbool__00))),chapp__02(s__02(cfun__02(cfun__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),cfun__02(V_27C_27,cfun__02(V_27D_27,cbool__00))),cfun__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),cfun__02(V_27D_27,cfun__02(V_27C_27,cbool__00)))),V__0),s__02(cfun__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),cfun__02(V_27C_27,cfun__02(V_27D_27,cbool__00))),V_27r_27))),s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27D_27),V_27env2_27),s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27),V_27env1_27)))) ) ) ) ) ) ).
fof('thm.namespaceProps.nsLookup_nsAppend_some',axiom,
! [V_27A_27,V_27B_27,V_27C_27,V_27e1_27,V_27id_27,V_27e2_27,V_27v_27] :
( s__02(c_27type_2eoption_2eoption_27__01(V_27C_27),c_27const_2enamespace_2ensLookup_27__02(s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27),c_27const_2enamespace_2ensAppend_27__02(s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27),V_27e1_27),s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27),V_27e2_27))),s__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),V_27id_27))) = s__02(c_27type_2eoption_2eoption_27__01(V_27C_27),c_27const_2eoption_2eSOME_27__01(s__02(V_27C_27,V_27v_27)))
<=> ( s__02(c_27type_2eoption_2eoption_27__01(V_27C_27),c_27const_2enamespace_2ensLookup_27__02(s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27),V_27e1_27),s__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),V_27id_27))) = s__02(c_27type_2eoption_2eoption_27__01(V_27C_27),c_27const_2eoption_2eSOME_27__01(s__02(V_27C_27,V_27v_27)))
| ( s__02(c_27type_2eoption_2eoption_27__01(V_27C_27),c_27const_2enamespace_2ensLookup_27__02(s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27),V_27e1_27),s__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),V_27id_27))) = s__02(c_27type_2eoption_2eoption_27__01(V_27C_27),c_27const_2eoption_2eNONE_27__00)
& s__02(c_27type_2eoption_2eoption_27__01(V_27C_27),c_27const_2enamespace_2ensLookup_27__02(s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27),V_27e2_27),s__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),V_27id_27))) = s__02(c_27type_2eoption_2eoption_27__01(V_27C_27),c_27const_2eoption_2eSOME_27__01(s__02(V_27C_27,V_27v_27)))
& ! [V_27p1_27,V_27p2_27] :
( ( s__02(c_27type_2elist_2elist_27__01(V_27A_27),V_27p1_27) != s__02(c_27type_2elist_2elist_27__01(V_27A_27),c_27const_2elist_2eNIL_27__00)
& s__02(c_27type_2elist_2elist_27__01(V_27A_27),c_27const_2enamespace_2eid__to__mods_27__01(s__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),V_27id_27))) = s__02(c_27type_2elist_2elist_27__01(V_27A_27),c_27const_2elist_2eAPPEND_27__02(s__02(c_27type_2elist_2elist_27__01(V_27A_27),V_27p1_27),s__02(c_27type_2elist_2elist_27__01(V_27A_27),V_27p2_27))) )
=> s__02(c_27type_2eoption_2eoption_27__01(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27)),c_27const_2enamespace_2ensLookupMod_27__02(s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27),V_27e1_27),s__02(c_27type_2elist_2elist_27__01(V_27A_27),V_27p1_27))) = s__02(c_27type_2eoption_2eoption_27__01(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27)),c_27const_2eoption_2eNONE_27__00) ) ) ) ) ).
fof('thm.namespaceProps.nsLookupMod_nsAppend_none',axiom,
! [V_27A_27,V_27B_27,V_27C_27,V_27e1_27,V_27e2_27,V_27path_27] :
( s__02(c_27type_2eoption_2eoption_27__01(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27)),c_27const_2enamespace_2ensLookupMod_27__02(s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27),c_27const_2enamespace_2ensAppend_27__02(s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27),V_27e1_27),s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27),V_27e2_27))),s__02(c_27type_2elist_2elist_27__01(V_27A_27),V_27path_27))) = s__02(c_27type_2eoption_2eoption_27__01(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27)),c_27const_2eoption_2eNONE_27__00)
<=> ( s__02(c_27type_2eoption_2eoption_27__01(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27)),c_27const_2enamespace_2ensLookupMod_27__02(s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27),V_27e1_27),s__02(c_27type_2elist_2elist_27__01(V_27A_27),V_27path_27))) = s__02(c_27type_2eoption_2eoption_27__01(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27)),c_27const_2eoption_2eNONE_27__00)
& ( s__02(c_27type_2eoption_2eoption_27__01(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27)),c_27const_2enamespace_2ensLookupMod_27__02(s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27),V_27e2_27),s__02(c_27type_2elist_2elist_27__01(V_27A_27),V_27path_27))) = s__02(c_27type_2eoption_2eoption_27__01(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27)),c_27const_2eoption_2eNONE_27__00)
| ? [V_27p1_27,V_27p2_27,V_27e3_27] :
( s__02(c_27type_2elist_2elist_27__01(V_27A_27),V_27p1_27) != s__02(c_27type_2elist_2elist_27__01(V_27A_27),c_27const_2elist_2eNIL_27__00)
& s__02(c_27type_2elist_2elist_27__01(V_27A_27),V_27path_27) = s__02(c_27type_2elist_2elist_27__01(V_27A_27),c_27const_2elist_2eAPPEND_27__02(s__02(c_27type_2elist_2elist_27__01(V_27A_27),V_27p1_27),s__02(c_27type_2elist_2elist_27__01(V_27A_27),V_27p2_27)))
& s__02(c_27type_2eoption_2eoption_27__01(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27)),c_27const_2enamespace_2ensLookupMod_27__02(s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27),V_27e1_27),s__02(c_27type_2elist_2elist_27__01(V_27A_27),V_27p1_27))) = s__02(c_27type_2eoption_2eoption_27__01(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27)),c_27const_2eoption_2eSOME_27__01(s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27),V_27e3_27))) ) ) ) ) ).
fof(conjecture,conjecture,
! [V_27C_27,V_27A_27,V_27B_27,V_27D_27,V_27R_27,V_27e1_27,V_27e1_7c39_7c_27,V_27e2_27,V_27e2_7c39_7c_27] :
( ( p__01(s__02(cbool__00,c_27const_2enamespace_2ensAll2_27__03(s__02(cfun__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),cfun__02(V_27C_27,cfun__02(V_27D_27,cbool__00))),V_27R_27),s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27),V_27e1_27),s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27D_27),V_27e2_27))))
& p__01(s__02(cbool__00,c_27const_2enamespace_2ensAll2_27__03(s__02(cfun__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),cfun__02(V_27C_27,cfun__02(V_27D_27,cbool__00))),V_27R_27),s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27),V_27e1_7c39_7c_27),s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27D_27),V_27e2_7c39_7c_27)))) )
=> p__01(s__02(cbool__00,c_27const_2enamespace_2ensAll2_27__03(s__02(cfun__02(c_27type_2enamespace_2eid_27__02(V_27A_27,V_27B_27),cfun__02(V_27C_27,cfun__02(V_27D_27,cbool__00))),V_27R_27),s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27),c_27const_2enamespace_2ensAppend_27__02(s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27),V_27e1_27),s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27C_27),V_27e1_7c39_7c_27))),s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27D_27),c_27const_2enamespace_2ensAppend_27__02(s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27D_27),V_27e2_27),s__02(c_27type_2enamespace_2enamespace_27__03(V_27A_27,V_27B_27,V_27D_27),V_27e2_7c39_7c_27)))))) ) ).
%------------------------------------------------------------------------------