TPTP Problem File: SWW472^2.p
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%------------------------------------------------------------------------------
% File : SWW472^2 : TPTP v8.2.0. Released v5.3.0.
% Domain : Software Verification
% Problem : Hoare's Logic with Procedures line 327, 500 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : hoare_500_thf_l327 [Bla11]
% Status : Theorem
% Rating : 1.00 v5.3.0
% Syntax : Number of formulae : 370 ( 41 unt; 52 typ; 0 def)
% Number of atoms : 1185 ( 184 equ; 5 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 2830 ( 103 ~; 12 |; 37 &;2115 @)
% ( 84 <=>; 479 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 8 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 847 ( 847 >; 0 *; 0 +; 0 <<)
% Number of symbols : 54 ( 48 usr; 8 con; 0-4 aty)
% Number of variables : 924 ( 4 ^; 904 !; 16 ?; 924 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 19:32:32
%------------------------------------------------------------------------------
%----Should-be-implicit typings (4)
thf(ty_ty_tc__Com__Ocom,type,
com: $tType ).
thf(ty_ty_tc__Com__Ostate,type,
state: $tType ).
thf(ty_ty_tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostate_J,type,
hoare_1167836817_state: $tType ).
thf(ty_ty_tc__Nat__Onat,type,
nat: $tType ).
%----Explicit typings (48)
thf(sy_c_All2,type,
all2: ( ( hoare_1167836817_state > $o ) > $o ) > $o ).
thf(sy_c_All1,type,
all1: ( $o > $o ) > $o ).
thf(sy_c_Big__Operators_Ocomm__monoid__add__class_Osetsum_000tc__Hoare____Mirabelle_,type,
big_co337839062te_nat: ( hoare_1167836817_state > nat ) > ( hoare_1167836817_state > $o ) > nat ).
thf(sy_c_Big__Operators_Osemilattice__big_000tc__Hoare____Mirabelle____srushsumbx__O,type,
big_se1603066171_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > ( ( hoare_1167836817_state > $o ) > hoare_1167836817_state ) > $o ).
thf(sy_c_Com_Ocom_OSKIP,type,
skip: com ).
thf(sy_c_Com_Ocom_OSemi,type,
semi: com > com > com ).
thf(sy_c_Ex,type,
ex: ( hoare_1167836817_state > $o ) > $o ).
thf(sy_c_Finite__Set_Ocomp__fun__commute_000tc__Hoare____Mirabelle____srushsumbx__Ot,type,
finite1091222817_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > $o ).
thf(sy_c_Finite__Set_Ocomp__fun__idem_000tc__Hoare____Mirabelle____srushsumbx__Otrip,type,
finite856902323tate_o: ( hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ) > $o ).
thf(sy_c_Finite__Set_Ocomp__fun__idem_000tc__Hoare____Mirabelle____srushsumbx__Otrip_001,type,
finite1900754844_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__C,type,
finite1084549118_state: ( hoare_1167836817_state > $o ) > $o ).
thf(sy_c_Finite__Set_Ofold1Set_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc_,type,
finite309220289_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(sy_c_Finite__Set_Ofold1_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Co,type,
finite1646097201_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state ).
thf(sy_c_Finite__Set_Ofold_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com,type,
finite291020855tate_o: ( hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(sy_c_Finite__Set_Ofold_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com_002,type,
finite1731015960_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state ).
thf(sy_c_Finite__Set_Ofold__graph_000tc__Hoare____Mirabelle____srushsumbx__Otriple_I,type,
finite1316643734_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(sy_c_Finite__Set_Ofolding__one_000tc__Hoare____Mirabelle____srushsumbx__Otriple_,type,
finite1074406356_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > ( ( hoare_1167836817_state > $o ) > hoare_1167836817_state ) > $o ).
thf(sy_c_Finite__Set_Ofolding__one__idem_000tc__Hoare____Mirabelle____srushsumbx__Ot,type,
finite806517911_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > ( ( hoare_1167836817_state > $o ) > hoare_1167836817_state ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_000_062_Itc__Hoare____Mirabelle____srushsumbx__,type,
minus_2107060239tate_o: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_000_Eo,type,
minus_minus_o: $o > $o > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_000tc__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_000tc__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Hoare__Mirabelle__srushsumbx_OMGT,type,
hoare_Mirabelle_MGT: com > hoare_1167836817_state ).
thf(sy_c_Hoare__Mirabelle__srushsumbx_Ohoare__derivs_000tc__Com__Ostate,type,
hoare_123228589_state: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > $o ).
thf(sy_c_Hoare__Mirabelle__srushsumbx_Ohoare__valids_000tc__Com__Ostate,type,
hoare_529639851_state: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > $o ).
thf(sy_c_Hoare__Mirabelle__srushsumbx_Otriple_Otriple_000tc__Com__Ostate,type,
hoare_908217195_state: ( state > state > $o ) > com > ( state > state > $o ) > hoare_1167836817_state ).
thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Hoare____Mirabelle____srushsumbx__O,type,
bot_bo70021908tate_o: hoare_1167836817_state > $o ).
thf(sy_c_Orderings_Obot__class_Obot_000_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Oord_Omax_000_062_Itc__Hoare____Mirabelle____srushsumbx__Otriple_,type,
max_Ho421493569tate_o: ( ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > $o ) > ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(sy_c_Orderings_Oord_Omax_000_Eo,type,
max_o: ( $o > $o > $o ) > $o > $o > $o ).
thf(sy_c_Orderings_Oord_Omin_000_062_Itc__Hoare____Mirabelle____srushsumbx__Otriple_,type,
min_Ho1955171539tate_o: ( ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > $o ) > ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(sy_c_Orderings_Oord_Omin_000_Eo,type,
min_o: ( $o > $o > $o ) > $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless_000_062_Itc__Hoare____Mirabelle____srushsumbx__,type,
ord_le65125204tate_o: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_000_Eo,type,
ord_less_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless_000tc__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__Hoare____Mirabelle____srushsum,type,
ord_le827224136tate_o: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000_Eo,type,
ord_less_eq_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Omax_000_062_Itc__Hoare____Mirabelle____srushsumbx__O,type,
ord_ma164008317tate_o: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(sy_c_Orderings_Oord__class_Omax_000_Eo,type,
ord_max_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Omin_000_062_Itc__Hoare____Mirabelle____srushsumbx__O,type,
ord_mi1697686287tate_o: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(sy_c_Orderings_Oord__class_Omin_000_Eo,type,
ord_min_o: $o > $o > $o ).
thf(sy_c_Set_OCollect_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ost,type,
collec1027672124_state: ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(sy_c_Set_Oinsert_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Osta,type,
insert2134838167_state: hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(sy_c_Set_Othe__elem_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__O,type,
the_el323660082_state: ( hoare_1167836817_state > $o ) > hoare_1167836817_state ).
thf(sy_c_member_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostate_J,type,
member2058392318_state: hoare_1167836817_state > ( hoare_1167836817_state > $o ) > $o ).
thf(sy_v_P,type,
p: state > state > $o ).
thf(sy_v_Q,type,
q: state > state > $o ).
thf(sy_v_c,type,
c: com ).
%----Relevant facts (315)
thf(fact_0_empty,axiom,
! [G_29: hoare_1167836817_state > $o] : ( hoare_123228589_state @ G_29 @ bot_bo70021908tate_o ) ).
thf(fact_1_triple_Oinject,axiom,
! [Fun1_2: state > state > $o,Com_2: com,Fun2_2: state > state > $o,Fun1_1: state > state > $o,Com_1: com,Fun2_1: state > state > $o] :
( ( ( hoare_908217195_state @ Fun1_2 @ Com_2 @ Fun2_2 )
= ( hoare_908217195_state @ Fun1_1 @ Com_1 @ Fun2_1 ) )
<=> ( ( Fun1_2 = Fun1_1 )
& ( Com_2 = Com_1 )
& ( Fun2_2 = Fun2_1 ) ) ) ).
thf(fact_2_hoare__sound,axiom,
! [G_28: hoare_1167836817_state > $o,Ts_7: hoare_1167836817_state > $o] :
( ( hoare_123228589_state @ G_28 @ Ts_7 )
=> ( hoare_529639851_state @ G_28 @ Ts_7 ) ) ).
thf(fact_3_cut,axiom,
! [G_27: hoare_1167836817_state > $o,G_26: hoare_1167836817_state > $o,Ts_6: hoare_1167836817_state > $o] :
( ( hoare_123228589_state @ G_26 @ Ts_6 )
=> ( ( hoare_123228589_state @ G_27 @ G_26 )
=> ( hoare_123228589_state @ G_27 @ Ts_6 ) ) ) ).
thf(fact_4_hoare__derivs_Oinsert,axiom,
! [Ts_5: hoare_1167836817_state > $o,G_25: hoare_1167836817_state > $o,T_1: hoare_1167836817_state] :
( ( hoare_123228589_state @ G_25 @ ( insert2134838167_state @ T_1 @ bot_bo70021908tate_o ) )
=> ( ( hoare_123228589_state @ G_25 @ Ts_5 )
=> ( hoare_123228589_state @ G_25 @ ( insert2134838167_state @ T_1 @ Ts_5 ) ) ) ) ).
thf(fact_5_derivs__insertD,axiom,
! [G_24: hoare_1167836817_state > $o,T: hoare_1167836817_state,Ts_4: hoare_1167836817_state > $o] :
( ( hoare_123228589_state @ G_24 @ ( insert2134838167_state @ T @ Ts_4 ) )
=> ( ( hoare_123228589_state @ G_24 @ ( insert2134838167_state @ T @ bot_bo70021908tate_o ) )
& ( hoare_123228589_state @ G_24 @ Ts_4 ) ) ) ).
thf(fact_6_conseq2,axiom,
! [Q_11: state > state > $o,G_23: hoare_1167836817_state > $o,P_21: state > state > $o,C_39: com,Q_10: state > state > $o] :
( ( hoare_123228589_state @ G_23 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_21 @ C_39 @ Q_10 ) @ bot_bo70021908tate_o ) )
=> ( ! [Z_28: state,S: state] :
( ( Q_10 @ Z_28 @ S )
=> ( Q_11 @ Z_28 @ S ) )
=> ( hoare_123228589_state @ G_23 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_21 @ C_39 @ Q_11 ) @ bot_bo70021908tate_o ) ) ) ) ).
thf(fact_7_conseq1,axiom,
! [P_20: state > state > $o,G_22: hoare_1167836817_state > $o,P_19: state > state > $o,C_38: com,Q_9: state > state > $o] :
( ( hoare_123228589_state @ G_22 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_19 @ C_38 @ Q_9 ) @ bot_bo70021908tate_o ) )
=> ( ! [Z_28: state,S: state] :
( ( P_20 @ Z_28 @ S )
=> ( P_19 @ Z_28 @ S ) )
=> ( hoare_123228589_state @ G_22 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_20 @ C_38 @ Q_9 ) @ bot_bo70021908tate_o ) ) ) ) ).
thf(fact_8_insertE,axiom,
! [A_185: hoare_1167836817_state,B_92: hoare_1167836817_state,A_184: hoare_1167836817_state > $o] :
( ( member2058392318_state @ A_185 @ ( insert2134838167_state @ B_92 @ A_184 ) )
=> ( ( A_185 != B_92 )
=> ( member2058392318_state @ A_185 @ A_184 ) ) ) ).
thf(fact_9_insertCI,axiom,
! [B_91: hoare_1167836817_state,A_183: hoare_1167836817_state,B_90: hoare_1167836817_state > $o] :
( ( ~ ( member2058392318_state @ A_183 @ B_90 )
=> ( A_183 = B_91 ) )
=> ( member2058392318_state @ A_183 @ ( insert2134838167_state @ B_91 @ B_90 ) ) ) ).
thf(fact_10_conseq12,axiom,
! [Q_8: state > state > $o,P_18: state > state > $o,G_21: hoare_1167836817_state > $o,P_17: state > state > $o,C_37: com,Q_7: state > state > $o] :
( ( hoare_123228589_state @ G_21 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_17 @ C_37 @ Q_7 ) @ bot_bo70021908tate_o ) )
=> ( ! [Z_28: state,S: state] :
( ( P_18 @ Z_28 @ S )
=> ! [S_1: state] :
( ! [Z_29: state] :
( ( P_17 @ Z_29 @ S )
=> ( Q_7 @ Z_29 @ S_1 ) )
=> ( Q_8 @ Z_28 @ S_1 ) ) )
=> ( hoare_123228589_state @ G_21 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_18 @ C_37 @ Q_8 ) @ bot_bo70021908tate_o ) ) ) ) ).
thf(fact_11_emptyE,axiom,
! [A_182: hoare_1167836817_state] :
~ ( member2058392318_state @ A_182 @ bot_bo70021908tate_o ) ).
thf(fact_12_empty__not__insert,axiom,
! [A_181: hoare_1167836817_state,A_180: hoare_1167836817_state > $o] :
( bot_bo70021908tate_o
!= ( insert2134838167_state @ A_181 @ A_180 ) ) ).
thf(fact_13_insert__not__empty,axiom,
! [A_179: hoare_1167836817_state,A_178: hoare_1167836817_state > $o] :
( ( insert2134838167_state @ A_179 @ A_178 )
!= bot_bo70021908tate_o ) ).
thf(fact_14_singleton__iff,axiom,
! [B_89: hoare_1167836817_state,A_177: hoare_1167836817_state] :
( ( member2058392318_state @ B_89 @ ( insert2134838167_state @ A_177 @ bot_bo70021908tate_o ) )
<=> ( B_89 = A_177 ) ) ).
thf(fact_15_doubleton__eq__iff,axiom,
! [A_176: hoare_1167836817_state,B_88: hoare_1167836817_state,C_36: hoare_1167836817_state,D_3: hoare_1167836817_state] :
( ( ( insert2134838167_state @ A_176 @ ( insert2134838167_state @ B_88 @ bot_bo70021908tate_o ) )
= ( insert2134838167_state @ C_36 @ ( insert2134838167_state @ D_3 @ bot_bo70021908tate_o ) ) )
<=> ( ( ( A_176 = C_36 )
& ( B_88 = D_3 ) )
| ( ( A_176 = D_3 )
& ( B_88 = C_36 ) ) ) ) ).
thf(fact_16_equals0D,axiom,
! [A_175: hoare_1167836817_state,A_174: hoare_1167836817_state > $o] :
( ( A_174 = bot_bo70021908tate_o )
=> ~ ( member2058392318_state @ A_175 @ A_174 ) ) ).
thf(fact_17_Collect__empty__eq,axiom,
! [P_16: hoare_1167836817_state > $o] :
( ( ( collec1027672124_state @ P_16 )
= bot_bo70021908tate_o )
<=> ! [X: hoare_1167836817_state] :
~ ( P_16 @ X ) ) ).
thf(fact_18_empty__iff,axiom,
! [C_35: hoare_1167836817_state] :
~ ( member2058392318_state @ C_35 @ bot_bo70021908tate_o ) ).
thf(fact_19_empty__Collect__eq,axiom,
! [P_15: hoare_1167836817_state > $o] :
( ( bot_bo70021908tate_o
= ( collec1027672124_state @ P_15 ) )
<=> ! [X: hoare_1167836817_state] :
~ ( P_15 @ X ) ) ).
thf(fact_20_ex__in__conv,axiom,
! [A_173: hoare_1167836817_state > $o] :
( ? [X: hoare_1167836817_state] : ( member2058392318_state @ X @ A_173 )
<=> ( A_173 != bot_bo70021908tate_o ) ) ).
thf(fact_21_all__not__in__conv,axiom,
! [A_172: hoare_1167836817_state > $o] :
( ! [X: hoare_1167836817_state] :
~ ( member2058392318_state @ X @ A_172 )
<=> ( A_172 = bot_bo70021908tate_o ) ) ).
thf(fact_22_empty__def,axiom,
( bot_bo70021908tate_o
= ( collec1027672124_state
@ ^ [X: hoare_1167836817_state] : $false ) ) ).
thf(fact_23_insert__absorb,axiom,
! [A_171: hoare_1167836817_state,A_170: hoare_1167836817_state > $o] :
( ( member2058392318_state @ A_171 @ A_170 )
=> ( ( insert2134838167_state @ A_171 @ A_170 )
= A_170 ) ) ).
thf(fact_24_insertI2,axiom,
! [B_87: hoare_1167836817_state,A_169: hoare_1167836817_state,B_86: hoare_1167836817_state > $o] :
( ( member2058392318_state @ A_169 @ B_86 )
=> ( member2058392318_state @ A_169 @ ( insert2134838167_state @ B_87 @ B_86 ) ) ) ).
thf(fact_25_insert__ident,axiom,
! [B_85: hoare_1167836817_state > $o,X_85: hoare_1167836817_state,A_168: hoare_1167836817_state > $o] :
( ~ ( member2058392318_state @ X_85 @ A_168 )
=> ( ~ ( member2058392318_state @ X_85 @ B_85 )
=> ( ( ( insert2134838167_state @ X_85 @ A_168 )
= ( insert2134838167_state @ X_85 @ B_85 ) )
<=> ( A_168 = B_85 ) ) ) ) ).
thf(fact_26_insert__code,axiom,
! [Y_34: hoare_1167836817_state,A_167: hoare_1167836817_state > $o,X_84: hoare_1167836817_state] :
( ( insert2134838167_state @ Y_34 @ A_167 @ X_84 )
<=> ( ( Y_34 = X_84 )
| ( A_167 @ X_84 ) ) ) ).
thf(fact_27_insert__iff,axiom,
! [A_166: hoare_1167836817_state,B_84: hoare_1167836817_state,A_165: hoare_1167836817_state > $o] :
( ( member2058392318_state @ A_166 @ ( insert2134838167_state @ B_84 @ A_165 ) )
<=> ( ( A_166 = B_84 )
| ( member2058392318_state @ A_166 @ A_165 ) ) ) ).
thf(fact_28_insert__commute,axiom,
! [X_83: hoare_1167836817_state,Y_33: hoare_1167836817_state,A_164: hoare_1167836817_state > $o] :
( ( insert2134838167_state @ X_83 @ ( insert2134838167_state @ Y_33 @ A_164 ) )
= ( insert2134838167_state @ Y_33 @ ( insert2134838167_state @ X_83 @ A_164 ) ) ) ).
thf(fact_29_insert__absorb2,axiom,
! [X_82: hoare_1167836817_state,A_163: hoare_1167836817_state > $o] :
( ( insert2134838167_state @ X_82 @ ( insert2134838167_state @ X_82 @ A_163 ) )
= ( insert2134838167_state @ X_82 @ A_163 ) ) ).
thf(fact_30_insert__Collect,axiom,
! [A_162: hoare_1167836817_state,P_14: hoare_1167836817_state > $o] :
( ( insert2134838167_state @ A_162 @ ( collec1027672124_state @ P_14 ) )
= ( collec1027672124_state
@ ^ [U: hoare_1167836817_state] : ( (=>) @ ( (~) @ ( U = A_162 ) ) @ ( P_14 @ U ) ) ) ) ).
thf(fact_31_insert__compr,axiom,
! [A_161: hoare_1167836817_state,B_83: hoare_1167836817_state > $o] :
( ( insert2134838167_state @ A_161 @ B_83 )
= ( collec1027672124_state
@ ^ [X: hoare_1167836817_state] : ( (|) @ ( X = A_161 ) @ ( member2058392318_state @ X @ B_83 ) ) ) ) ).
thf(fact_32_insertI1,axiom,
! [A_160: hoare_1167836817_state,B_82: hoare_1167836817_state > $o] : ( member2058392318_state @ A_160 @ ( insert2134838167_state @ A_160 @ B_82 ) ) ).
thf(fact_33_singleton__inject,axiom,
! [A_159: hoare_1167836817_state,B_81: hoare_1167836817_state] :
( ( ( insert2134838167_state @ A_159 @ bot_bo70021908tate_o )
= ( insert2134838167_state @ B_81 @ bot_bo70021908tate_o ) )
=> ( A_159 = B_81 ) ) ).
thf(fact_34_singletonE,axiom,
! [B_80: hoare_1167836817_state,A_158: hoare_1167836817_state] :
( ( member2058392318_state @ B_80 @ ( insert2134838167_state @ A_158 @ bot_bo70021908tate_o ) )
=> ( B_80 = A_158 ) ) ).
thf(fact_35_the__elem__eq,axiom,
! [X_81: hoare_1167836817_state] :
( ( the_el323660082_state @ ( insert2134838167_state @ X_81 @ bot_bo70021908tate_o ) )
= X_81 ) ).
thf(fact_36_bot__apply,axiom,
! [X_80: hoare_1167836817_state] :
( ( bot_bo70021908tate_o @ X_80 )
<=> bot_bot_o ) ).
thf(fact_37_bot__fun__def,axiom,
! [X: hoare_1167836817_state] :
( ( bot_bo70021908tate_o @ X )
<=> bot_bot_o ) ).
thf(fact_38_hoare__derivs_OSkip,axiom,
! [G_20: hoare_1167836817_state > $o,P_13: state > state > $o] : ( hoare_123228589_state @ G_20 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_13 @ skip @ P_13 ) @ bot_bo70021908tate_o ) ) ).
thf(fact_39_Comp,axiom,
! [D_2: com,R: state > state > $o,G_19: hoare_1167836817_state > $o,P_12: state > state > $o,C_34: com,Q_6: state > state > $o] :
( ( hoare_123228589_state @ G_19 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_12 @ C_34 @ Q_6 ) @ bot_bo70021908tate_o ) )
=> ( ( hoare_123228589_state @ G_19 @ ( insert2134838167_state @ ( hoare_908217195_state @ Q_6 @ D_2 @ R ) @ bot_bo70021908tate_o ) )
=> ( hoare_123228589_state @ G_19 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_12 @ ( semi @ C_34 @ D_2 ) @ R ) @ bot_bo70021908tate_o ) ) ) ) ).
thf(fact_40_triple_Oexhaust,axiom,
! [Y_32: hoare_1167836817_state] :
~ ! [Fun1: state > state > $o,Com: com,Fun2: state > state > $o] :
( Y_32
!= ( hoare_908217195_state @ Fun1 @ Com @ Fun2 ) ) ).
thf(fact_41_Set_Oset__insert,axiom,
! [X_79: hoare_1167836817_state,A_157: hoare_1167836817_state > $o] :
( ( member2058392318_state @ X_79 @ A_157 )
=> ~ ! [B_79: hoare_1167836817_state > $o] :
( ( A_157
= ( insert2134838167_state @ X_79 @ B_79 ) )
=> ( member2058392318_state @ X_79 @ B_79 ) ) ) ).
thf(fact_42_mk__disjoint__insert,axiom,
! [A_156: hoare_1167836817_state,A_155: hoare_1167836817_state > $o] :
( ( member2058392318_state @ A_156 @ A_155 )
=> ? [B_79: hoare_1167836817_state > $o] :
( ( A_155
= ( insert2134838167_state @ A_156 @ B_79 ) )
& ~ ( member2058392318_state @ A_156 @ B_79 ) ) ) ).
thf(fact_43_equals0I,axiom,
! [A_154: hoare_1167836817_state > $o] :
( ! [Y: hoare_1167836817_state] :
~ ( member2058392318_state @ Y @ A_154 )
=> ( A_154 = bot_bo70021908tate_o ) ) ).
thf(fact_44_conseq,axiom,
! [Q_4: state > state > $o,G_18: hoare_1167836817_state > $o,C_33: com,P_10: state > state > $o] :
( ! [Z_28: state,S: state] :
( ( P_10 @ Z_28 @ S )
=> ? [P_11: state > state > $o,Q_5: state > state > $o] :
( ( hoare_123228589_state @ G_18 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_11 @ C_33 @ Q_5 ) @ bot_bo70021908tate_o ) )
& ! [S_1: state] :
( ! [Z_29: state] :
( ( P_11 @ Z_29 @ S )
=> ( Q_5 @ Z_29 @ S_1 ) )
=> ( Q_4 @ Z_28 @ S_1 ) ) ) )
=> ( hoare_123228589_state @ G_18 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_10 @ C_33 @ Q_4 ) @ bot_bo70021908tate_o ) ) ) ).
thf(fact_45_nonempty__iff,axiom,
! [A_153: hoare_1167836817_state > $o] :
( ( A_153 != bot_bo70021908tate_o )
<=> ? [X: hoare_1167836817_state,B_79: hoare_1167836817_state > $o] :
( ( A_153
= ( insert2134838167_state @ X @ B_79 ) )
& ~ ( member2058392318_state @ X @ B_79 ) ) ) ).
thf(fact_46_com_Osimps_I13_J,axiom,
! [Com1: com,Com2: com] :
( ( semi @ Com1 @ Com2 )
!= skip ) ).
thf(fact_47_com_Osimps_I12_J,axiom,
! [Com1: com,Com2: com] :
( skip
!= ( semi @ Com1 @ Com2 ) ) ).
thf(fact_48_com_Osimps_I3_J,axiom,
! [Com1_1: com,Com2_1: com,Com1: com,Com2: com] :
( ( ( semi @ Com1_1 @ Com2_1 )
= ( semi @ Com1 @ Com2 ) )
<=> ( ( Com1_1 = Com1 )
& ( Com2_1 = Com2 ) ) ) ).
thf(fact_49_fold1Set__sing,axiom,
! [F_82: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,A_152: hoare_1167836817_state,B_78: hoare_1167836817_state] :
( ( finite309220289_state @ F_82 @ ( insert2134838167_state @ A_152 @ bot_bo70021908tate_o ) @ B_78 )
<=> ( A_152 = B_78 ) ) ).
thf(fact_50_folding__one_Osingleton,axiom,
! [X_78: hoare_1167836817_state,F_81: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_80: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
( ( finite1074406356_state @ F_81 @ F_80 )
=> ( ( F_80 @ ( insert2134838167_state @ X_78 @ bot_bo70021908tate_o ) )
= X_78 ) ) ).
thf(fact_51_bot__empty__eq,axiom,
! [X: hoare_1167836817_state] :
( ( bot_bo70021908tate_o @ X )
<=> ( member2058392318_state @ X @ bot_bo70021908tate_o ) ) ).
thf(fact_52_fold1__singleton,axiom,
! [F_79: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,A_151: hoare_1167836817_state] :
( ( finite1646097201_state @ F_79 @ ( insert2134838167_state @ A_151 @ bot_bo70021908tate_o ) )
= A_151 ) ).
thf(fact_53_fold1__singleton__def,axiom,
! [A_150: hoare_1167836817_state,G_17: ( hoare_1167836817_state > $o ) > hoare_1167836817_state,F_78: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state] :
( ( G_17
= ( finite1646097201_state @ F_78 ) )
=> ( ( G_17 @ ( insert2134838167_state @ A_150 @ bot_bo70021908tate_o ) )
= A_150 ) ) ).
thf(fact_54_empty__fold1SetE,axiom,
! [F_77: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,X_77: hoare_1167836817_state] :
~ ( finite309220289_state @ F_77 @ bot_bo70021908tate_o @ X_77 ) ).
thf(fact_55_fold1Set__nonempty,axiom,
! [F_76: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,A_149: hoare_1167836817_state > $o,X_76: hoare_1167836817_state] :
( ( finite309220289_state @ F_76 @ A_149 @ X_76 )
=> ( A_149 != bot_bo70021908tate_o ) ) ).
thf(fact_56_fold1Set_Ointros,axiom,
! [F_75: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,A_148: hoare_1167836817_state,A_147: hoare_1167836817_state > $o,X_75: hoare_1167836817_state] :
( ( finite1316643734_state @ F_75 @ A_148 @ A_147 @ X_75 )
=> ( ~ ( member2058392318_state @ A_148 @ A_147 )
=> ( finite309220289_state @ F_75 @ ( insert2134838167_state @ A_148 @ A_147 ) @ X_75 ) ) ) ).
thf(fact_57_folding__one_Oinsert,axiom,
! [X_74: hoare_1167836817_state,A_146: hoare_1167836817_state > $o,F_74: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_73: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
( ( finite1074406356_state @ F_74 @ F_73 )
=> ( ( finite1084549118_state @ A_146 )
=> ( ~ ( member2058392318_state @ X_74 @ A_146 )
=> ( ( A_146 != bot_bo70021908tate_o )
=> ( ( F_73 @ ( insert2134838167_state @ X_74 @ A_146 ) )
= ( F_74 @ X_74 @ ( F_73 @ A_146 ) ) ) ) ) ) ) ).
thf(fact_58_folding__one_Oeq__fold,axiom,
! [A_145: hoare_1167836817_state > $o,F_72: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_71: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
( ( finite1074406356_state @ F_72 @ F_71 )
=> ( ( finite1084549118_state @ A_145 )
=> ( ( F_71 @ A_145 )
= ( finite1646097201_state @ F_72 @ A_145 ) ) ) ) ).
thf(fact_59_folding__one_Oclosed,axiom,
! [A_144: hoare_1167836817_state > $o,F_70: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_69: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
( ( finite1074406356_state @ F_70 @ F_69 )
=> ( ( finite1084549118_state @ A_144 )
=> ( ( A_144 != bot_bo70021908tate_o )
=> ( ! [X: hoare_1167836817_state,Y: hoare_1167836817_state] : ( member2058392318_state @ ( F_70 @ X @ Y ) @ ( insert2134838167_state @ X @ ( insert2134838167_state @ Y @ bot_bo70021908tate_o ) ) )
=> ( member2058392318_state @ ( F_69 @ A_144 ) @ A_144 ) ) ) ) ) ).
thf(fact_60_insert__fold1SetE,axiom,
! [F_68: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,A_143: hoare_1167836817_state,X_73: hoare_1167836817_state > $o,X_72: hoare_1167836817_state] :
( ( finite309220289_state @ F_68 @ ( insert2134838167_state @ A_143 @ X_73 ) @ X_72 )
=> ~ ! [A_59: hoare_1167836817_state,A_58: hoare_1167836817_state > $o] :
( ( ( insert2134838167_state @ A_143 @ X_73 )
= ( insert2134838167_state @ A_59 @ A_58 ) )
=> ( ( finite1316643734_state @ F_68 @ A_59 @ A_58 @ X_72 )
=> ( member2058392318_state @ A_59 @ A_58 ) ) ) ) ).
thf(fact_61_subset__singletonD,axiom,
! [A_142: hoare_1167836817_state > $o,X_71: hoare_1167836817_state] :
( ( ord_le827224136tate_o @ A_142 @ ( insert2134838167_state @ X_71 @ bot_bo70021908tate_o ) )
=> ( ( A_142 = bot_bo70021908tate_o )
| ( A_142
= ( insert2134838167_state @ X_71 @ bot_bo70021908tate_o ) ) ) ) ).
thf(fact_62_order__refl,axiom,
! [X_70: $o] : ( ord_less_eq_o @ X_70 @ X_70 ) ).
thf(fact_63_order__refl,axiom,
! [X_70: hoare_1167836817_state > $o] : ( ord_le827224136tate_o @ X_70 @ X_70 ) ).
thf(fact_64_subsetD,axiom,
! [C_32: hoare_1167836817_state,A_141: hoare_1167836817_state > $o,B_77: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ A_141 @ B_77 )
=> ( ( member2058392318_state @ C_32 @ A_141 )
=> ( member2058392318_state @ C_32 @ B_77 ) ) ) ).
thf(fact_65_equalityI,axiom,
! [A_140: hoare_1167836817_state > $o,B_76: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ A_140 @ B_76 )
=> ( ( ord_le827224136tate_o @ B_76 @ A_140 )
=> ( A_140 = B_76 ) ) ) ).
thf(fact_66_finite_OemptyI,axiom,
finite1084549118_state @ bot_bo70021908tate_o ).
thf(fact_67_finite_OinsertI,axiom,
! [A_139: hoare_1167836817_state,A_138: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ A_138 )
=> ( finite1084549118_state @ ( insert2134838167_state @ A_139 @ A_138 ) ) ) ).
thf(fact_68_empty__subsetI,axiom,
! [A_137: hoare_1167836817_state > $o] : ( ord_le827224136tate_o @ bot_bo70021908tate_o @ A_137 ) ).
thf(fact_69_rev__predicate1D,axiom,
! [Q_3: hoare_1167836817_state > $o,P_9: hoare_1167836817_state > $o,X_69: hoare_1167836817_state] :
( ( P_9 @ X_69 )
=> ( ( ord_le827224136tate_o @ P_9 @ Q_3 )
=> ( Q_3 @ X_69 ) ) ) ).
thf(fact_70_predicate1D,axiom,
! [X_68: hoare_1167836817_state,P_8: hoare_1167836817_state > $o,Q_2: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ P_8 @ Q_2 )
=> ( ( P_8 @ X_68 )
=> ( Q_2 @ X_68 ) ) ) ).
thf(fact_71_mem__def,axiom,
! [X_67: hoare_1167836817_state,A_136: hoare_1167836817_state > $o] :
( ( member2058392318_state @ X_67 @ A_136 )
<=> ( A_136 @ X_67 ) ) ).
thf(fact_72_Collect__def,axiom,
! [P_7: hoare_1167836817_state > $o] :
( ( collec1027672124_state @ P_7 )
= P_7 ) ).
thf(fact_73_subset__refl,axiom,
! [A_135: hoare_1167836817_state > $o] : ( ord_le827224136tate_o @ A_135 @ A_135 ) ).
thf(fact_74_set__eq__subset,axiom,
! [A_134: hoare_1167836817_state > $o,B_75: hoare_1167836817_state > $o] :
( ( A_134 = B_75 )
<=> ( ( ord_le827224136tate_o @ A_134 @ B_75 )
& ( ord_le827224136tate_o @ B_75 @ A_134 ) ) ) ).
thf(fact_75_equalityD1,axiom,
! [A_133: hoare_1167836817_state > $o,B_74: hoare_1167836817_state > $o] :
( ( A_133 = B_74 )
=> ( ord_le827224136tate_o @ A_133 @ B_74 ) ) ).
thf(fact_76_equalityD2,axiom,
! [A_132: hoare_1167836817_state > $o,B_73: hoare_1167836817_state > $o] :
( ( A_132 = B_73 )
=> ( ord_le827224136tate_o @ B_73 @ A_132 ) ) ).
thf(fact_77_in__mono,axiom,
! [X_66: hoare_1167836817_state,A_131: hoare_1167836817_state > $o,B_72: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ A_131 @ B_72 )
=> ( ( member2058392318_state @ X_66 @ A_131 )
=> ( member2058392318_state @ X_66 @ B_72 ) ) ) ).
thf(fact_78_set__rev__mp,axiom,
! [B_71: hoare_1167836817_state > $o,X_65: hoare_1167836817_state,A_130: hoare_1167836817_state > $o] :
( ( member2058392318_state @ X_65 @ A_130 )
=> ( ( ord_le827224136tate_o @ A_130 @ B_71 )
=> ( member2058392318_state @ X_65 @ B_71 ) ) ) ).
thf(fact_79_set__mp,axiom,
! [X_64: hoare_1167836817_state,A_129: hoare_1167836817_state > $o,B_70: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ A_129 @ B_70 )
=> ( ( member2058392318_state @ X_64 @ A_129 )
=> ( member2058392318_state @ X_64 @ B_70 ) ) ) ).
thf(fact_80_subset__trans,axiom,
! [C_31: hoare_1167836817_state > $o,A_128: hoare_1167836817_state > $o,B_69: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ A_128 @ B_69 )
=> ( ( ord_le827224136tate_o @ B_69 @ C_31 )
=> ( ord_le827224136tate_o @ A_128 @ C_31 ) ) ) ).
thf(fact_81_equalityE,axiom,
! [A_127: hoare_1167836817_state > $o,B_68: hoare_1167836817_state > $o] :
( ( A_127 = B_68 )
=> ~ ( ( ord_le827224136tate_o @ A_127 @ B_68 )
=> ~ ( ord_le827224136tate_o @ B_68 @ A_127 ) ) ) ).
thf(fact_82_le__fun__def,axiom,
! [F_67: hoare_1167836817_state > $o,G_16: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ F_67 @ G_16 )
<=> ! [X: hoare_1167836817_state] : ( ord_less_eq_o @ ( F_67 @ X ) @ ( G_16 @ X ) ) ) ).
thf(fact_83_order__eq__iff,axiom,
! [Y_31: $o,X_63: $o] :
( ( X_63
<=> Y_31 )
<=> ( ( ord_less_eq_o @ X_63 @ Y_31 )
& ( ord_less_eq_o @ Y_31 @ X_63 ) ) ) ).
thf(fact_84_order__eq__iff,axiom,
! [X_63: hoare_1167836817_state > $o,Y_31: hoare_1167836817_state > $o] :
( ( X_63 = Y_31 )
<=> ( ( ord_le827224136tate_o @ X_63 @ Y_31 )
& ( ord_le827224136tate_o @ Y_31 @ X_63 ) ) ) ).
thf(fact_85_rev__finite__subset,axiom,
! [A_126: hoare_1167836817_state > $o,B_67: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ B_67 )
=> ( ( ord_le827224136tate_o @ A_126 @ B_67 )
=> ( finite1084549118_state @ A_126 ) ) ) ).
thf(fact_86_order__eq__refl,axiom,
! [Y_30: $o,X_62: $o] :
( ( X_62
<=> Y_30 )
=> ( ord_less_eq_o @ X_62 @ Y_30 ) ) ).
thf(fact_87_order__eq__refl,axiom,
! [X_62: hoare_1167836817_state > $o,Y_30: hoare_1167836817_state > $o] :
( ( X_62 = Y_30 )
=> ( ord_le827224136tate_o @ X_62 @ Y_30 ) ) ).
thf(fact_88_le__funD,axiom,
! [X_61: hoare_1167836817_state,F_66: hoare_1167836817_state > $o,G_15: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ F_66 @ G_15 )
=> ( ord_less_eq_o @ ( F_66 @ X_61 ) @ ( G_15 @ X_61 ) ) ) ).
thf(fact_89_order__antisym__conv,axiom,
! [Y_29: $o,X_60: $o] :
( ( ord_less_eq_o @ Y_29 @ X_60 )
=> ( ( ord_less_eq_o @ X_60 @ Y_29 )
<=> ( X_60
<=> Y_29 ) ) ) ).
thf(fact_90_order__antisym__conv,axiom,
! [Y_29: hoare_1167836817_state > $o,X_60: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ Y_29 @ X_60 )
=> ( ( ord_le827224136tate_o @ X_60 @ Y_29 )
<=> ( X_60 = Y_29 ) ) ) ).
thf(fact_91_finite__subset,axiom,
! [A_125: hoare_1167836817_state > $o,B_66: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ A_125 @ B_66 )
=> ( ( finite1084549118_state @ B_66 )
=> ( finite1084549118_state @ A_125 ) ) ) ).
thf(fact_92_ord__eq__le__trans,axiom,
! [C_30: $o,B_65: $o,A_124: $o] :
( ( A_124
<=> B_65 )
=> ( ( ord_less_eq_o @ B_65 @ C_30 )
=> ( ord_less_eq_o @ A_124 @ C_30 ) ) ) ).
thf(fact_93_ord__eq__le__trans,axiom,
! [C_30: hoare_1167836817_state > $o,A_124: hoare_1167836817_state > $o,B_65: hoare_1167836817_state > $o] :
( ( A_124 = B_65 )
=> ( ( ord_le827224136tate_o @ B_65 @ C_30 )
=> ( ord_le827224136tate_o @ A_124 @ C_30 ) ) ) ).
thf(fact_94_xt1_I3_J,axiom,
! [C_29: $o,B_64: $o,A_123: $o] :
( ( A_123
<=> B_64 )
=> ( ( ord_less_eq_o @ C_29 @ B_64 )
=> ( ord_less_eq_o @ C_29 @ A_123 ) ) ) ).
thf(fact_95_xt1_I3_J,axiom,
! [C_29: hoare_1167836817_state > $o,A_123: hoare_1167836817_state > $o,B_64: hoare_1167836817_state > $o] :
( ( A_123 = B_64 )
=> ( ( ord_le827224136tate_o @ C_29 @ B_64 )
=> ( ord_le827224136tate_o @ C_29 @ A_123 ) ) ) ).
thf(fact_96_ord__le__eq__trans,axiom,
! [C_28: $o,A_122: $o,B_63: $o] :
( ( ord_less_eq_o @ A_122 @ B_63 )
=> ( ( B_63
<=> C_28 )
=> ( ord_less_eq_o @ A_122 @ C_28 ) ) ) ).
thf(fact_97_ord__le__eq__trans,axiom,
! [C_28: hoare_1167836817_state > $o,A_122: hoare_1167836817_state > $o,B_63: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ A_122 @ B_63 )
=> ( ( B_63 = C_28 )
=> ( ord_le827224136tate_o @ A_122 @ C_28 ) ) ) ).
thf(fact_98_xt1_I4_J,axiom,
! [C_27: $o,B_62: $o,A_121: $o] :
( ( ord_less_eq_o @ B_62 @ A_121 )
=> ( ( B_62
<=> C_27 )
=> ( ord_less_eq_o @ C_27 @ A_121 ) ) ) ).
thf(fact_99_xt1_I4_J,axiom,
! [C_27: hoare_1167836817_state > $o,B_62: hoare_1167836817_state > $o,A_121: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ B_62 @ A_121 )
=> ( ( B_62 = C_27 )
=> ( ord_le827224136tate_o @ C_27 @ A_121 ) ) ) ).
thf(fact_100_order__antisym,axiom,
! [X_59: $o,Y_28: $o] :
( ( ord_less_eq_o @ X_59 @ Y_28 )
=> ( ( ord_less_eq_o @ Y_28 @ X_59 )
=> ( X_59
<=> Y_28 ) ) ) ).
thf(fact_101_order__antisym,axiom,
! [X_59: hoare_1167836817_state > $o,Y_28: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ X_59 @ Y_28 )
=> ( ( ord_le827224136tate_o @ Y_28 @ X_59 )
=> ( X_59 = Y_28 ) ) ) ).
thf(fact_102_order__trans,axiom,
! [Z_27: $o,X_58: $o,Y_27: $o] :
( ( ord_less_eq_o @ X_58 @ Y_27 )
=> ( ( ord_less_eq_o @ Y_27 @ Z_27 )
=> ( ord_less_eq_o @ X_58 @ Z_27 ) ) ) ).
thf(fact_103_order__trans,axiom,
! [Z_27: hoare_1167836817_state > $o,X_58: hoare_1167836817_state > $o,Y_27: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ X_58 @ Y_27 )
=> ( ( ord_le827224136tate_o @ Y_27 @ Z_27 )
=> ( ord_le827224136tate_o @ X_58 @ Z_27 ) ) ) ).
thf(fact_104_xt1_I5_J,axiom,
! [Y_26: $o,X_57: $o] :
( ( ord_less_eq_o @ Y_26 @ X_57 )
=> ( ( ord_less_eq_o @ X_57 @ Y_26 )
=> ( X_57
<=> Y_26 ) ) ) ).
thf(fact_105_xt1_I5_J,axiom,
! [Y_26: hoare_1167836817_state > $o,X_57: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ Y_26 @ X_57 )
=> ( ( ord_le827224136tate_o @ X_57 @ Y_26 )
=> ( X_57 = Y_26 ) ) ) ).
thf(fact_106_xt1_I6_J,axiom,
! [Z_26: $o,Y_25: $o,X_56: $o] :
( ( ord_less_eq_o @ Y_25 @ X_56 )
=> ( ( ord_less_eq_o @ Z_26 @ Y_25 )
=> ( ord_less_eq_o @ Z_26 @ X_56 ) ) ) ).
thf(fact_107_xt1_I6_J,axiom,
! [Z_26: hoare_1167836817_state > $o,Y_25: hoare_1167836817_state > $o,X_56: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ Y_25 @ X_56 )
=> ( ( ord_le827224136tate_o @ Z_26 @ Y_25 )
=> ( ord_le827224136tate_o @ Z_26 @ X_56 ) ) ) ).
thf(fact_108_le__funE,axiom,
! [X_55: hoare_1167836817_state,F_65: hoare_1167836817_state > $o,G_14: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ F_65 @ G_14 )
=> ( ord_less_eq_o @ ( F_65 @ X_55 ) @ ( G_14 @ X_55 ) ) ) ).
thf(fact_109_bot__least,axiom,
! [A_120: hoare_1167836817_state > $o] : ( ord_le827224136tate_o @ bot_bo70021908tate_o @ A_120 ) ).
thf(fact_110_bot__least,axiom,
! [A_120: $o] : ( ord_less_eq_o @ bot_bot_o @ A_120 ) ).
thf(fact_111_bot__unique,axiom,
! [A_119: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ A_119 @ bot_bo70021908tate_o )
<=> ( A_119 = bot_bo70021908tate_o ) ) ).
thf(fact_112_bot__unique,axiom,
! [A_119: $o] :
( ( ord_less_eq_o @ A_119 @ bot_bot_o )
<=> ( A_119
<=> bot_bot_o ) ) ).
thf(fact_113_le__bot,axiom,
! [A_118: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ A_118 @ bot_bo70021908tate_o )
=> ( A_118 = bot_bo70021908tate_o ) ) ).
thf(fact_114_le__bot,axiom,
! [A_118: $o] :
( ( ord_less_eq_o @ A_118 @ bot_bot_o )
=> ( A_118
<=> bot_bot_o ) ) ).
thf(fact_115_subset__empty,axiom,
! [A_117: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ A_117 @ bot_bo70021908tate_o )
<=> ( A_117 = bot_bo70021908tate_o ) ) ).
thf(fact_116_subset__insertI,axiom,
! [B_61: hoare_1167836817_state > $o,A_116: hoare_1167836817_state] : ( ord_le827224136tate_o @ B_61 @ ( insert2134838167_state @ A_116 @ B_61 ) ) ).
thf(fact_117_insert__subset,axiom,
! [X_54: hoare_1167836817_state,A_115: hoare_1167836817_state > $o,B_60: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ ( insert2134838167_state @ X_54 @ A_115 ) @ B_60 )
<=> ( ( member2058392318_state @ X_54 @ B_60 )
& ( ord_le827224136tate_o @ A_115 @ B_60 ) ) ) ).
thf(fact_118_subset__insert,axiom,
! [B_59: hoare_1167836817_state > $o,X_53: hoare_1167836817_state,A_114: hoare_1167836817_state > $o] :
( ~ ( member2058392318_state @ X_53 @ A_114 )
=> ( ( ord_le827224136tate_o @ A_114 @ ( insert2134838167_state @ X_53 @ B_59 ) )
<=> ( ord_le827224136tate_o @ A_114 @ B_59 ) ) ) ).
thf(fact_119_subset__insertI2,axiom,
! [B_58: hoare_1167836817_state,A_113: hoare_1167836817_state > $o,B_57: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ A_113 @ B_57 )
=> ( ord_le827224136tate_o @ A_113 @ ( insert2134838167_state @ B_58 @ B_57 ) ) ) ).
thf(fact_120_insert__mono,axiom,
! [A_112: hoare_1167836817_state,C_26: hoare_1167836817_state > $o,D_1: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ C_26 @ D_1 )
=> ( ord_le827224136tate_o @ ( insert2134838167_state @ A_112 @ C_26 ) @ ( insert2134838167_state @ A_112 @ D_1 ) ) ) ).
thf(fact_121_finite__insert,axiom,
! [A_111: hoare_1167836817_state,A_110: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ ( insert2134838167_state @ A_111 @ A_110 ) )
<=> ( finite1084549118_state @ A_110 ) ) ).
thf(fact_122_asm,axiom,
! [Ts_3: hoare_1167836817_state > $o,G_13: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ Ts_3 @ G_13 )
=> ( hoare_123228589_state @ G_13 @ Ts_3 ) ) ).
thf(fact_123_weaken,axiom,
! [Ts_2: hoare_1167836817_state > $o,G_12: hoare_1167836817_state > $o,Ts_1: hoare_1167836817_state > $o] :
( ( hoare_123228589_state @ G_12 @ Ts_1 )
=> ( ( ord_le827224136tate_o @ Ts_2 @ Ts_1 )
=> ( hoare_123228589_state @ G_12 @ Ts_2 ) ) ) ).
thf(fact_124_thin,axiom,
! [G_11: hoare_1167836817_state > $o,G_10: hoare_1167836817_state > $o,Ts: hoare_1167836817_state > $o] :
( ( hoare_123228589_state @ G_10 @ Ts )
=> ( ( ord_le827224136tate_o @ G_10 @ G_11 )
=> ( hoare_123228589_state @ G_11 @ Ts ) ) ) ).
thf(fact_125_fold__graph_OemptyI,axiom,
! [F_64: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,Z_25: hoare_1167836817_state] : ( finite1316643734_state @ F_64 @ Z_25 @ bot_bo70021908tate_o @ Z_25 ) ).
thf(fact_126_empty__fold__graphE,axiom,
! [F_63: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,Z_24: hoare_1167836817_state,X_52: hoare_1167836817_state] :
( ( finite1316643734_state @ F_63 @ Z_24 @ bot_bo70021908tate_o @ X_52 )
=> ( X_52 = Z_24 ) ) ).
thf(fact_127_fold__graph_OinsertI,axiom,
! [F_62: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,Z_23: hoare_1167836817_state,Y_24: hoare_1167836817_state,X_51: hoare_1167836817_state,A_109: hoare_1167836817_state > $o] :
( ~ ( member2058392318_state @ X_51 @ A_109 )
=> ( ( finite1316643734_state @ F_62 @ Z_23 @ A_109 @ Y_24 )
=> ( finite1316643734_state @ F_62 @ Z_23 @ ( insert2134838167_state @ X_51 @ A_109 ) @ ( F_62 @ X_51 @ Y_24 ) ) ) ) ).
thf(fact_128_finite__subset__induct,axiom,
! [P_6: ( hoare_1167836817_state > $o ) > $o,A_108: hoare_1167836817_state > $o,F_61: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ F_61 )
=> ( ( ord_le827224136tate_o @ F_61 @ A_108 )
=> ( ( P_6 @ bot_bo70021908tate_o )
=> ( ! [A_59: hoare_1167836817_state,F_50: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ F_50 )
=> ( ( member2058392318_state @ A_59 @ A_108 )
=> ( ~ ( member2058392318_state @ A_59 @ F_50 )
=> ( ( P_6 @ F_50 )
=> ( P_6 @ ( insert2134838167_state @ A_59 @ F_50 ) ) ) ) ) )
=> ( P_6 @ F_61 ) ) ) ) ) ).
thf(fact_129_finite__nonempty__imp__fold1Set,axiom,
! [F_60: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,A_107: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ A_107 )
=> ( ( A_107 != bot_bo70021908tate_o )
=> ( ex @ ( finite309220289_state @ F_60 @ A_107 ) ) ) ) ).
thf(fact_130_subsetI,axiom,
! [B_56: hoare_1167836817_state > $o,A_106: hoare_1167836817_state > $o] :
( ! [X: hoare_1167836817_state] :
( ( member2058392318_state @ X @ A_106 )
=> ( member2058392318_state @ X @ B_56 ) )
=> ( ord_le827224136tate_o @ A_106 @ B_56 ) ) ).
thf(fact_131_finite_Osimps,axiom,
! [A_105: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ A_105 )
<=> ( ( A_105 = bot_bo70021908tate_o )
| ? [A_58: hoare_1167836817_state > $o,A_59: hoare_1167836817_state] :
( ( A_105
= ( insert2134838167_state @ A_59 @ A_58 ) )
& ( finite1084549118_state @ A_58 ) ) ) ) ).
thf(fact_132_finite__induct,axiom,
! [P_5: ( hoare_1167836817_state > $o ) > $o,F_59: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ F_59 )
=> ( ( P_5 @ bot_bo70021908tate_o )
=> ( ! [X: hoare_1167836817_state,F_50: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ F_50 )
=> ( ~ ( member2058392318_state @ X @ F_50 )
=> ( ( P_5 @ F_50 )
=> ( P_5 @ ( insert2134838167_state @ X @ F_50 ) ) ) ) )
=> ( P_5 @ F_59 ) ) ) ) ).
thf(fact_133_finite__imp__fold__graph,axiom,
! [F_58: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,Z_22: hoare_1167836817_state,A_104: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ A_104 )
=> ( ex @ ( finite1316643734_state @ F_58 @ Z_22 @ A_104 ) ) ) ).
thf(fact_134_le__funI,axiom,
! [F_57: hoare_1167836817_state > $o,G_9: hoare_1167836817_state > $o] :
( ! [X: hoare_1167836817_state] : ( ord_less_eq_o @ ( F_57 @ X ) @ ( G_9 @ X ) )
=> ( ord_le827224136tate_o @ F_57 @ G_9 ) ) ).
thf(fact_135_fold1Set_Osimps,axiom,
! [F_56: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,A1_1: hoare_1167836817_state > $o,A2_1: hoare_1167836817_state] :
( ( finite309220289_state @ F_56 @ A1_1 @ A2_1 )
<=> ? [A_59: hoare_1167836817_state,A_58: hoare_1167836817_state > $o,X: hoare_1167836817_state] :
( ( A1_1
= ( insert2134838167_state @ A_59 @ A_58 ) )
& ( A2_1 = X )
& ( finite1316643734_state @ F_56 @ A_59 @ A_58 @ X )
& ~ ( member2058392318_state @ A_59 @ A_58 ) ) ) ).
thf(fact_136_fold__graph_Osimps,axiom,
! [F_55: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,Z_21: hoare_1167836817_state,A1: hoare_1167836817_state > $o,A2: hoare_1167836817_state] :
( ( finite1316643734_state @ F_55 @ Z_21 @ A1 @ A2 )
<=> ( ( ( A1 = bot_bo70021908tate_o )
& ( A2 = Z_21 ) )
| ? [X: hoare_1167836817_state,A_58: hoare_1167836817_state > $o,Y: hoare_1167836817_state] :
( ( A1
= ( insert2134838167_state @ X @ A_58 ) )
& ( A2
= ( F_55 @ X @ Y ) )
& ~ ( member2058392318_state @ X @ A_58 )
& ( finite1316643734_state @ F_55 @ Z_21 @ A_58 @ Y ) ) ) ) ).
thf(fact_137_folding__one__idem_Osubset__idem,axiom,
! [B_55: hoare_1167836817_state > $o,A_103: hoare_1167836817_state > $o,F_54: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_53: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
( ( finite806517911_state @ F_54 @ F_53 )
=> ( ( finite1084549118_state @ A_103 )
=> ( ( B_55 != bot_bo70021908tate_o )
=> ( ( ord_le827224136tate_o @ B_55 @ A_103 )
=> ( ( F_54 @ ( F_53 @ B_55 ) @ ( F_53 @ A_103 ) )
= ( F_53 @ A_103 ) ) ) ) ) ) ).
thf(fact_138_folding__one__idem_Oinsert__idem,axiom,
! [X_50: hoare_1167836817_state,A_102: hoare_1167836817_state > $o,F_52: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_51: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
( ( finite806517911_state @ F_52 @ F_51 )
=> ( ( finite1084549118_state @ A_102 )
=> ( ( A_102 != bot_bo70021908tate_o )
=> ( ( F_51 @ ( insert2134838167_state @ X_50 @ A_102 ) )
= ( F_52 @ X_50 @ ( F_51 @ A_102 ) ) ) ) ) ) ).
thf(fact_139_finite__ne__induct,axiom,
! [P_4: ( hoare_1167836817_state > $o ) > $o,F_49: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ F_49 )
=> ( ( F_49 != bot_bo70021908tate_o )
=> ( ! [X: hoare_1167836817_state] : ( P_4 @ ( insert2134838167_state @ X @ bot_bo70021908tate_o ) )
=> ( ! [X: hoare_1167836817_state,F_50: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ F_50 )
=> ( ( F_50 != bot_bo70021908tate_o )
=> ( ~ ( member2058392318_state @ X @ F_50 )
=> ( ( P_4 @ F_50 )
=> ( P_4 @ ( insert2134838167_state @ X @ F_50 ) ) ) ) ) )
=> ( P_4 @ F_49 ) ) ) ) ) ).
thf(fact_140_Collect__mono,axiom,
! [Q_1: hoare_1167836817_state > $o,P_3: hoare_1167836817_state > $o] :
( ! [X: hoare_1167836817_state] :
( ( P_3 @ X )
=> ( Q_1 @ X ) )
=> ( ord_le827224136tate_o @ ( collec1027672124_state @ P_3 ) @ ( collec1027672124_state @ Q_1 ) ) ) ).
thf(fact_141_folding__one__idem_Oidem,axiom,
! [X_49: hoare_1167836817_state,F_48: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_47: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
( ( finite806517911_state @ F_48 @ F_47 )
=> ( ( F_48 @ X_49 @ X_49 )
= X_49 ) ) ).
thf(fact_142_folding__one__idem_Oin__idem,axiom,
! [X_48: hoare_1167836817_state,A_101: hoare_1167836817_state > $o,F_46: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_45: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
( ( finite806517911_state @ F_46 @ F_45 )
=> ( ( finite1084549118_state @ A_101 )
=> ( ( member2058392318_state @ X_48 @ A_101 )
=> ( ( F_46 @ X_48 @ ( F_45 @ A_101 ) )
= ( F_45 @ A_101 ) ) ) ) ) ).
thf(fact_143_predicate1I,axiom,
! [Q: hoare_1167836817_state > $o,P_2: hoare_1167836817_state > $o] :
( ! [X: hoare_1167836817_state] :
( ( P_2 @ X )
=> ( Q @ X ) )
=> ( ord_le827224136tate_o @ P_2 @ Q ) ) ).
thf(fact_144_xt3,axiom,
! [C_25: $o,F_44: ( hoare_1167836817_state > $o ) > $o,B_54: hoare_1167836817_state > $o,A_100: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ B_54 @ A_100 )
=> ( ( ord_less_eq_o @ C_25 @ ( F_44 @ B_54 ) )
=> ( ! [X: hoare_1167836817_state > $o,Y: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ Y @ X )
=> ( ord_less_eq_o @ ( F_44 @ Y ) @ ( F_44 @ X ) ) )
=> ( ord_less_eq_o @ C_25 @ ( F_44 @ A_100 ) ) ) ) ) ).
thf(fact_145_xt3,axiom,
! [C_25: hoare_1167836817_state > $o,F_44: $o > hoare_1167836817_state > $o,B_54: $o,A_100: $o] :
( ( ord_less_eq_o @ B_54 @ A_100 )
=> ( ( ord_le827224136tate_o @ C_25 @ ( F_44 @ B_54 ) )
=> ( ! [X: $o,Y: $o] :
( ( ord_less_eq_o @ Y @ X )
=> ( ord_le827224136tate_o @ ( F_44 @ Y ) @ ( F_44 @ X ) ) )
=> ( ord_le827224136tate_o @ C_25 @ ( F_44 @ A_100 ) ) ) ) ) ).
thf(fact_146_xt1_I16_J,axiom,
! [C_24: $o,F_43: $o > $o,B_53: $o,A_99: $o] :
( ( ord_less_eq_o @ B_53 @ A_99 )
=> ( ( ( F_43 @ B_53 )
<=> C_24 )
=> ( ! [X: $o,Y: $o] :
( ( ord_less_eq_o @ Y @ X )
=> ( ord_less_eq_o @ ( F_43 @ Y ) @ ( F_43 @ X ) ) )
=> ( ord_less_eq_o @ C_24 @ ( F_43 @ A_99 ) ) ) ) ) ).
thf(fact_147_xt1_I16_J,axiom,
! [F_43: ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o,C_24: hoare_1167836817_state > $o,B_53: hoare_1167836817_state > $o,A_99: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ B_53 @ A_99 )
=> ( ( ( F_43 @ B_53 )
= C_24 )
=> ( ! [X: hoare_1167836817_state > $o,Y: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ Y @ X )
=> ( ord_le827224136tate_o @ ( F_43 @ Y ) @ ( F_43 @ X ) ) )
=> ( ord_le827224136tate_o @ C_24 @ ( F_43 @ A_99 ) ) ) ) ) ).
thf(fact_148_ord__le__eq__subst,axiom,
! [C_23: $o,F_42: ( hoare_1167836817_state > $o ) > $o,A_98: hoare_1167836817_state > $o,B_52: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ A_98 @ B_52 )
=> ( ( ( F_42 @ B_52 )
<=> C_23 )
=> ( ! [X: hoare_1167836817_state > $o,Y: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ X @ Y )
=> ( ord_less_eq_o @ ( F_42 @ X ) @ ( F_42 @ Y ) ) )
=> ( ord_less_eq_o @ ( F_42 @ A_98 ) @ C_23 ) ) ) ) ).
thf(fact_149_ord__le__eq__subst,axiom,
! [F_42: $o > hoare_1167836817_state > $o,C_23: hoare_1167836817_state > $o,A_98: $o,B_52: $o] :
( ( ord_less_eq_o @ A_98 @ B_52 )
=> ( ( ( F_42 @ B_52 )
= C_23 )
=> ( ! [X: $o,Y: $o] :
( ( ord_less_eq_o @ X @ Y )
=> ( ord_le827224136tate_o @ ( F_42 @ X ) @ ( F_42 @ Y ) ) )
=> ( ord_le827224136tate_o @ ( F_42 @ A_98 ) @ C_23 ) ) ) ) ).
thf(fact_150_order__subst2,axiom,
! [F_41: ( hoare_1167836817_state > $o ) > $o,C_22: $o,A_97: hoare_1167836817_state > $o,B_51: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ A_97 @ B_51 )
=> ( ( ord_less_eq_o @ ( F_41 @ B_51 ) @ C_22 )
=> ( ! [X: hoare_1167836817_state > $o,Y: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ X @ Y )
=> ( ord_less_eq_o @ ( F_41 @ X ) @ ( F_41 @ Y ) ) )
=> ( ord_less_eq_o @ ( F_41 @ A_97 ) @ C_22 ) ) ) ) ).
thf(fact_151_order__subst2,axiom,
! [F_41: $o > hoare_1167836817_state > $o,C_22: hoare_1167836817_state > $o,A_97: $o,B_51: $o] :
( ( ord_less_eq_o @ A_97 @ B_51 )
=> ( ( ord_le827224136tate_o @ ( F_41 @ B_51 ) @ C_22 )
=> ( ! [X: $o,Y: $o] :
( ( ord_less_eq_o @ X @ Y )
=> ( ord_le827224136tate_o @ ( F_41 @ X ) @ ( F_41 @ Y ) ) )
=> ( ord_le827224136tate_o @ ( F_41 @ A_97 ) @ C_22 ) ) ) ) ).
thf(fact_152_ord__eq__le__subst,axiom,
! [C_21: hoare_1167836817_state > $o,F_40: ( hoare_1167836817_state > $o ) > $o,B_50: hoare_1167836817_state > $o,A_96: $o] :
( ( A_96
<=> ( F_40 @ B_50 ) )
=> ( ( ord_le827224136tate_o @ B_50 @ C_21 )
=> ( ! [X: hoare_1167836817_state > $o,Y: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ X @ Y )
=> ( ord_less_eq_o @ ( F_40 @ X ) @ ( F_40 @ Y ) ) )
=> ( ord_less_eq_o @ A_96 @ ( F_40 @ C_21 ) ) ) ) ) ).
thf(fact_153_ord__eq__le__subst,axiom,
! [C_21: $o,A_96: hoare_1167836817_state > $o,F_40: $o > hoare_1167836817_state > $o,B_50: $o] :
( ( A_96
= ( F_40 @ B_50 ) )
=> ( ( ord_less_eq_o @ B_50 @ C_21 )
=> ( ! [X: $o,Y: $o] :
( ( ord_less_eq_o @ X @ Y )
=> ( ord_le827224136tate_o @ ( F_40 @ X ) @ ( F_40 @ Y ) ) )
=> ( ord_le827224136tate_o @ A_96 @ ( F_40 @ C_21 ) ) ) ) ) ).
thf(fact_154_xt2,axiom,
! [C_20: $o,F_39: $o > hoare_1167836817_state > $o,B_49: $o,A_95: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ ( F_39 @ B_49 ) @ A_95 )
=> ( ( ord_less_eq_o @ C_20 @ B_49 )
=> ( ! [X: $o,Y: $o] :
( ( ord_less_eq_o @ Y @ X )
=> ( ord_le827224136tate_o @ ( F_39 @ Y ) @ ( F_39 @ X ) ) )
=> ( ord_le827224136tate_o @ ( F_39 @ C_20 ) @ A_95 ) ) ) ) ).
thf(fact_155_xt2,axiom,
! [C_20: hoare_1167836817_state > $o,F_39: ( hoare_1167836817_state > $o ) > $o,B_49: hoare_1167836817_state > $o,A_95: $o] :
( ( ord_less_eq_o @ ( F_39 @ B_49 ) @ A_95 )
=> ( ( ord_le827224136tate_o @ C_20 @ B_49 )
=> ( ! [X: hoare_1167836817_state > $o,Y: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ Y @ X )
=> ( ord_less_eq_o @ ( F_39 @ Y ) @ ( F_39 @ X ) ) )
=> ( ord_less_eq_o @ ( F_39 @ C_20 ) @ A_95 ) ) ) ) ).
thf(fact_156_xt1_I15_J,axiom,
! [C_19: $o,F_38: $o > $o,B_48: $o,A_94: $o] :
( ( A_94
<=> ( F_38 @ B_48 ) )
=> ( ( ord_less_eq_o @ C_19 @ B_48 )
=> ( ! [X: $o,Y: $o] :
( ( ord_less_eq_o @ Y @ X )
=> ( ord_less_eq_o @ ( F_38 @ Y ) @ ( F_38 @ X ) ) )
=> ( ord_less_eq_o @ ( F_38 @ C_19 ) @ A_94 ) ) ) ) ).
thf(fact_157_xt1_I15_J,axiom,
! [C_19: hoare_1167836817_state > $o,A_94: hoare_1167836817_state > $o,F_38: ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o,B_48: hoare_1167836817_state > $o] :
( ( A_94
= ( F_38 @ B_48 ) )
=> ( ( ord_le827224136tate_o @ C_19 @ B_48 )
=> ( ! [X: hoare_1167836817_state > $o,Y: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ Y @ X )
=> ( ord_le827224136tate_o @ ( F_38 @ Y ) @ ( F_38 @ X ) ) )
=> ( ord_le827224136tate_o @ ( F_38 @ C_19 ) @ A_94 ) ) ) ) ).
thf(fact_158_order__subst1,axiom,
! [C_18: $o,A_93: hoare_1167836817_state > $o,F_37: $o > hoare_1167836817_state > $o,B_47: $o] :
( ( ord_le827224136tate_o @ A_93 @ ( F_37 @ B_47 ) )
=> ( ( ord_less_eq_o @ B_47 @ C_18 )
=> ( ! [X: $o,Y: $o] :
( ( ord_less_eq_o @ X @ Y )
=> ( ord_le827224136tate_o @ ( F_37 @ X ) @ ( F_37 @ Y ) ) )
=> ( ord_le827224136tate_o @ A_93 @ ( F_37 @ C_18 ) ) ) ) ) ).
thf(fact_159_order__subst1,axiom,
! [C_18: hoare_1167836817_state > $o,A_93: $o,F_37: ( hoare_1167836817_state > $o ) > $o,B_47: hoare_1167836817_state > $o] :
( ( ord_less_eq_o @ A_93 @ ( F_37 @ B_47 ) )
=> ( ( ord_le827224136tate_o @ B_47 @ C_18 )
=> ( ! [X: hoare_1167836817_state > $o,Y: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ X @ Y )
=> ( ord_less_eq_o @ ( F_37 @ X ) @ ( F_37 @ Y ) ) )
=> ( ord_less_eq_o @ A_93 @ ( F_37 @ C_18 ) ) ) ) ) ).
thf(fact_160_semilattice__big_OF__eq,axiom,
! [A_92: hoare_1167836817_state > $o,F_36: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_35: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
( ( big_se1603066171_state @ F_36 @ F_35 )
=> ( ( finite1084549118_state @ A_92 )
=> ( ( F_35 @ A_92 )
= ( finite1646097201_state @ F_36 @ A_92 ) ) ) ) ).
thf(fact_161_folding__one_Oremove,axiom,
! [X_47: hoare_1167836817_state,A_91: hoare_1167836817_state > $o,F_34: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_33: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
( ( finite1074406356_state @ F_34 @ F_33 )
=> ( ( finite1084549118_state @ A_91 )
=> ( ( member2058392318_state @ X_47 @ A_91 )
=> ( ( ( ( minus_2107060239tate_o @ A_91 @ ( insert2134838167_state @ X_47 @ bot_bo70021908tate_o ) )
= bot_bo70021908tate_o )
=> ( ( F_33 @ A_91 )
= X_47 ) )
& ( ( ( minus_2107060239tate_o @ A_91 @ ( insert2134838167_state @ X_47 @ bot_bo70021908tate_o ) )
!= bot_bo70021908tate_o )
=> ( ( F_33 @ A_91 )
= ( F_34 @ X_47 @ ( F_33 @ ( minus_2107060239tate_o @ A_91 @ ( insert2134838167_state @ X_47 @ bot_bo70021908tate_o ) ) ) ) ) ) ) ) ) ) ).
thf(fact_162_folding__one_Oinsert__remove,axiom,
! [X_46: hoare_1167836817_state,A_90: hoare_1167836817_state > $o,F_32: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_31: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
( ( finite1074406356_state @ F_32 @ F_31 )
=> ( ( finite1084549118_state @ A_90 )
=> ( ( ( ( minus_2107060239tate_o @ A_90 @ ( insert2134838167_state @ X_46 @ bot_bo70021908tate_o ) )
= bot_bo70021908tate_o )
=> ( ( F_31 @ ( insert2134838167_state @ X_46 @ A_90 ) )
= X_46 ) )
& ( ( ( minus_2107060239tate_o @ A_90 @ ( insert2134838167_state @ X_46 @ bot_bo70021908tate_o ) )
!= bot_bo70021908tate_o )
=> ( ( F_31 @ ( insert2134838167_state @ X_46 @ A_90 ) )
= ( F_32 @ X_46 @ ( F_31 @ ( minus_2107060239tate_o @ A_90 @ ( insert2134838167_state @ X_46 @ bot_bo70021908tate_o ) ) ) ) ) ) ) ) ) ).
thf(fact_163_DiffI,axiom,
! [B_46: hoare_1167836817_state > $o,C_17: hoare_1167836817_state,A_89: hoare_1167836817_state > $o] :
( ( member2058392318_state @ C_17 @ A_89 )
=> ( ~ ( member2058392318_state @ C_17 @ B_46 )
=> ( member2058392318_state @ C_17 @ ( minus_2107060239tate_o @ A_89 @ B_46 ) ) ) ) ).
thf(fact_164_DiffE,axiom,
! [C_16: hoare_1167836817_state,A_88: hoare_1167836817_state > $o,B_45: hoare_1167836817_state > $o] :
( ( member2058392318_state @ C_16 @ ( minus_2107060239tate_o @ A_88 @ B_45 ) )
=> ~ ( ( member2058392318_state @ C_16 @ A_88 )
=> ( member2058392318_state @ C_16 @ B_45 ) ) ) ).
thf(fact_165_finite__Diff,axiom,
! [B_44: hoare_1167836817_state > $o,A_87: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ A_87 )
=> ( finite1084549118_state @ ( minus_2107060239tate_o @ A_87 @ B_44 ) ) ) ).
thf(fact_166_set__diff__eq,axiom,
! [A_86: hoare_1167836817_state > $o,B_43: hoare_1167836817_state > $o] :
( ( minus_2107060239tate_o @ A_86 @ B_43 )
= ( collec1027672124_state
@ ^ [X: hoare_1167836817_state] : ( (&) @ ( member2058392318_state @ X @ A_86 ) @ ( (~) @ ( member2058392318_state @ X @ B_43 ) ) ) ) ) ).
thf(fact_167_Diff__iff,axiom,
! [C_15: hoare_1167836817_state,A_85: hoare_1167836817_state > $o,B_42: hoare_1167836817_state > $o] :
( ( member2058392318_state @ C_15 @ ( minus_2107060239tate_o @ A_85 @ B_42 ) )
<=> ( ( member2058392318_state @ C_15 @ A_85 )
& ~ ( member2058392318_state @ C_15 @ B_42 ) ) ) ).
thf(fact_168_Diff__idemp,axiom,
! [A_84: hoare_1167836817_state > $o,B_41: hoare_1167836817_state > $o] :
( ( minus_2107060239tate_o @ ( minus_2107060239tate_o @ A_84 @ B_41 ) @ B_41 )
= ( minus_2107060239tate_o @ A_84 @ B_41 ) ) ).
thf(fact_169_DiffD1,axiom,
! [C_14: hoare_1167836817_state,A_83: hoare_1167836817_state > $o,B_40: hoare_1167836817_state > $o] :
( ( member2058392318_state @ C_14 @ ( minus_2107060239tate_o @ A_83 @ B_40 ) )
=> ( member2058392318_state @ C_14 @ A_83 ) ) ).
thf(fact_170_DiffD2,axiom,
! [C_13: hoare_1167836817_state,A_82: hoare_1167836817_state > $o,B_39: hoare_1167836817_state > $o] :
( ( member2058392318_state @ C_13 @ ( minus_2107060239tate_o @ A_82 @ B_39 ) )
=> ~ ( member2058392318_state @ C_13 @ B_39 ) ) ).
thf(fact_171_empty__Diff,axiom,
! [A_81: hoare_1167836817_state > $o] :
( ( minus_2107060239tate_o @ bot_bo70021908tate_o @ A_81 )
= bot_bo70021908tate_o ) ).
thf(fact_172_Diff__empty,axiom,
! [A_80: hoare_1167836817_state > $o] :
( ( minus_2107060239tate_o @ A_80 @ bot_bo70021908tate_o )
= A_80 ) ).
thf(fact_173_Diff__cancel,axiom,
! [A_79: hoare_1167836817_state > $o] :
( ( minus_2107060239tate_o @ A_79 @ A_79 )
= bot_bo70021908tate_o ) ).
thf(fact_174_finite__Diff2,axiom,
! [A_78: hoare_1167836817_state > $o,B_38: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ B_38 )
=> ( ( finite1084549118_state @ ( minus_2107060239tate_o @ A_78 @ B_38 ) )
<=> ( finite1084549118_state @ A_78 ) ) ) ).
thf(fact_175_insert__Diff__if,axiom,
! [A_77: hoare_1167836817_state > $o,X_45: hoare_1167836817_state,B_37: hoare_1167836817_state > $o] :
( ( ( member2058392318_state @ X_45 @ B_37 )
=> ( ( minus_2107060239tate_o @ ( insert2134838167_state @ X_45 @ A_77 ) @ B_37 )
= ( minus_2107060239tate_o @ A_77 @ B_37 ) ) )
& ( ~ ( member2058392318_state @ X_45 @ B_37 )
=> ( ( minus_2107060239tate_o @ ( insert2134838167_state @ X_45 @ A_77 ) @ B_37 )
= ( insert2134838167_state @ X_45 @ ( minus_2107060239tate_o @ A_77 @ B_37 ) ) ) ) ) ).
thf(fact_176_insert__Diff1,axiom,
! [A_76: hoare_1167836817_state > $o,X_44: hoare_1167836817_state,B_36: hoare_1167836817_state > $o] :
( ( member2058392318_state @ X_44 @ B_36 )
=> ( ( minus_2107060239tate_o @ ( insert2134838167_state @ X_44 @ A_76 ) @ B_36 )
= ( minus_2107060239tate_o @ A_76 @ B_36 ) ) ) ).
thf(fact_177_Diff__subset,axiom,
! [A_75: hoare_1167836817_state > $o,B_35: hoare_1167836817_state > $o] : ( ord_le827224136tate_o @ ( minus_2107060239tate_o @ A_75 @ B_35 ) @ A_75 ) ).
thf(fact_178_Diff__mono,axiom,
! [D: hoare_1167836817_state > $o,B_34: hoare_1167836817_state > $o,A_74: hoare_1167836817_state > $o,C_12: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ A_74 @ C_12 )
=> ( ( ord_le827224136tate_o @ D @ B_34 )
=> ( ord_le827224136tate_o @ ( minus_2107060239tate_o @ A_74 @ B_34 ) @ ( minus_2107060239tate_o @ C_12 @ D ) ) ) ) ).
thf(fact_179_double__diff,axiom,
! [C_11: hoare_1167836817_state > $o,A_73: hoare_1167836817_state > $o,B_33: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ A_73 @ B_33 )
=> ( ( ord_le827224136tate_o @ B_33 @ C_11 )
=> ( ( minus_2107060239tate_o @ B_33 @ ( minus_2107060239tate_o @ C_11 @ A_73 ) )
= A_73 ) ) ) ).
thf(fact_180_Diff__insert,axiom,
! [A_72: hoare_1167836817_state > $o,A_71: hoare_1167836817_state,B_32: hoare_1167836817_state > $o] :
( ( minus_2107060239tate_o @ A_72 @ ( insert2134838167_state @ A_71 @ B_32 ) )
= ( minus_2107060239tate_o @ ( minus_2107060239tate_o @ A_72 @ B_32 ) @ ( insert2134838167_state @ A_71 @ bot_bo70021908tate_o ) ) ) ).
thf(fact_181_Diff__insert2,axiom,
! [A_70: hoare_1167836817_state > $o,A_69: hoare_1167836817_state,B_31: hoare_1167836817_state > $o] :
( ( minus_2107060239tate_o @ A_70 @ ( insert2134838167_state @ A_69 @ B_31 ) )
= ( minus_2107060239tate_o @ ( minus_2107060239tate_o @ A_70 @ ( insert2134838167_state @ A_69 @ bot_bo70021908tate_o ) ) @ B_31 ) ) ).
thf(fact_182_insert__Diff__single,axiom,
! [A_68: hoare_1167836817_state,A_67: hoare_1167836817_state > $o] :
( ( insert2134838167_state @ A_68 @ ( minus_2107060239tate_o @ A_67 @ ( insert2134838167_state @ A_68 @ bot_bo70021908tate_o ) ) )
= ( insert2134838167_state @ A_68 @ A_67 ) ) ).
thf(fact_183_Diff__insert__absorb,axiom,
! [X_43: hoare_1167836817_state,A_66: hoare_1167836817_state > $o] :
( ~ ( member2058392318_state @ X_43 @ A_66 )
=> ( ( minus_2107060239tate_o @ ( insert2134838167_state @ X_43 @ A_66 ) @ ( insert2134838167_state @ X_43 @ bot_bo70021908tate_o ) )
= A_66 ) ) ).
thf(fact_184_insert__Diff,axiom,
! [A_65: hoare_1167836817_state,A_64: hoare_1167836817_state > $o] :
( ( member2058392318_state @ A_65 @ A_64 )
=> ( ( insert2134838167_state @ A_65 @ ( minus_2107060239tate_o @ A_64 @ ( insert2134838167_state @ A_65 @ bot_bo70021908tate_o ) ) )
= A_64 ) ) ).
thf(fact_185_finite__Diff__insert,axiom,
! [A_63: hoare_1167836817_state > $o,A_62: hoare_1167836817_state,B_30: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ ( minus_2107060239tate_o @ A_63 @ ( insert2134838167_state @ A_62 @ B_30 ) ) )
<=> ( finite1084549118_state @ ( minus_2107060239tate_o @ A_63 @ B_30 ) ) ) ).
thf(fact_186_subset__insert__iff,axiom,
! [A_61: hoare_1167836817_state > $o,X_42: hoare_1167836817_state,B_29: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ A_61 @ ( insert2134838167_state @ X_42 @ B_29 ) )
<=> ( ( ( member2058392318_state @ X_42 @ A_61 )
=> ( ord_le827224136tate_o @ ( minus_2107060239tate_o @ A_61 @ ( insert2134838167_state @ X_42 @ bot_bo70021908tate_o ) ) @ B_29 ) )
& ( ~ ( member2058392318_state @ X_42 @ A_61 )
=> ( ord_le827224136tate_o @ A_61 @ B_29 ) ) ) ) ).
thf(fact_187_diff__single__insert,axiom,
! [A_60: hoare_1167836817_state > $o,X_41: hoare_1167836817_state,B_28: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ ( minus_2107060239tate_o @ A_60 @ ( insert2134838167_state @ X_41 @ bot_bo70021908tate_o ) ) @ B_28 )
=> ( ( member2058392318_state @ X_41 @ A_60 )
=> ( ord_le827224136tate_o @ A_60 @ ( insert2134838167_state @ X_41 @ B_28 ) ) ) ) ).
thf(fact_188_finite__empty__induct,axiom,
! [P_1: ( hoare_1167836817_state > $o ) > $o,A_57: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ A_57 )
=> ( ( P_1 @ A_57 )
=> ( ! [A_59: hoare_1167836817_state,A_58: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ A_58 )
=> ( ( member2058392318_state @ A_59 @ A_58 )
=> ( ( P_1 @ A_58 )
=> ( P_1 @ ( minus_2107060239tate_o @ A_58 @ ( insert2134838167_state @ A_59 @ bot_bo70021908tate_o ) ) ) ) ) )
=> ( P_1 @ bot_bo70021908tate_o ) ) ) ) ).
thf(fact_189_minus__apply,axiom,
! [A_56: hoare_1167836817_state > $o,B_27: hoare_1167836817_state > $o,X_40: hoare_1167836817_state] :
( ( minus_2107060239tate_o @ A_56 @ B_27 @ X_40 )
<=> ( minus_minus_o @ ( A_56 @ X_40 ) @ ( B_27 @ X_40 ) ) ) ).
thf(fact_190_fun__diff__def,axiom,
! [A_55: hoare_1167836817_state > $o,B_26: hoare_1167836817_state > $o,X: hoare_1167836817_state] :
( ( minus_2107060239tate_o @ A_55 @ B_26 @ X )
<=> ( minus_minus_o @ ( A_55 @ X ) @ ( B_26 @ X ) ) ) ).
thf(fact_191_comp__fun__commute_Ofold__graph__insertE__aux,axiom,
! [A_54: hoare_1167836817_state,Z_20: hoare_1167836817_state,A_53: hoare_1167836817_state > $o,Y_22: hoare_1167836817_state,F_30: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state] :
( ( finite1091222817_state @ F_30 )
=> ( ( finite1316643734_state @ F_30 @ Z_20 @ A_53 @ Y_22 )
=> ( ( member2058392318_state @ A_54 @ A_53 )
=> ? [Y_23: hoare_1167836817_state] :
( ( Y_22
= ( F_30 @ A_54 @ Y_23 ) )
& ( finite1316643734_state @ F_30 @ Z_20 @ ( minus_2107060239tate_o @ A_53 @ ( insert2134838167_state @ A_54 @ bot_bo70021908tate_o ) ) @ Y_23 ) ) ) ) ) ).
thf(fact_192_comp__fun__commute_Ofun__left__comm,axiom,
! [X_39: hoare_1167836817_state,Y_21: hoare_1167836817_state,Z_19: hoare_1167836817_state,F_29: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state] :
( ( finite1091222817_state @ F_29 )
=> ( ( F_29 @ X_39 @ ( F_29 @ Y_21 @ Z_19 ) )
= ( F_29 @ Y_21 @ ( F_29 @ X_39 @ Z_19 ) ) ) ) ).
thf(fact_193_comp__fun__commute_Ofold__graph__determ,axiom,
! [Y_20: hoare_1167836817_state,Z_18: hoare_1167836817_state,A_52: hoare_1167836817_state > $o,X_38: hoare_1167836817_state,F_28: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state] :
( ( finite1091222817_state @ F_28 )
=> ( ( finite1316643734_state @ F_28 @ Z_18 @ A_52 @ X_38 )
=> ( ( finite1316643734_state @ F_28 @ Z_18 @ A_52 @ Y_20 )
=> ( Y_20 = X_38 ) ) ) ) ).
thf(fact_194_comp__fun__commute_Ofold__graph__insertE,axiom,
! [Z_17: hoare_1167836817_state,X_37: hoare_1167836817_state,A_51: hoare_1167836817_state > $o,V: hoare_1167836817_state,F_27: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state] :
( ( finite1091222817_state @ F_27 )
=> ( ( finite1316643734_state @ F_27 @ Z_17 @ ( insert2134838167_state @ X_37 @ A_51 ) @ V )
=> ( ~ ( member2058392318_state @ X_37 @ A_51 )
=> ~ ! [Y: hoare_1167836817_state] :
( ( V
= ( F_27 @ X_37 @ Y ) )
=> ~ ( finite1316643734_state @ F_27 @ Z_17 @ A_51 @ Y ) ) ) ) ) ).
thf(fact_195_min__leastR,axiom,
! [X_36: $o,Least_3: $o] :
( ( all1 @ ( ord_less_eq_o @ Least_3 ) )
=> ( ( ord_min_o @ X_36 @ Least_3 )
<=> Least_3 ) ) ).
thf(fact_196_min__leastR,axiom,
! [X_36: hoare_1167836817_state > $o,Least_3: hoare_1167836817_state > $o] :
( ( all2 @ ( ord_le827224136tate_o @ Least_3 ) )
=> ( ( ord_mi1697686287tate_o @ X_36 @ Least_3 )
= Least_3 ) ) ).
thf(fact_197_min__leastL,axiom,
! [X_35: $o,Least_2: $o] :
( ( all1 @ ( ord_less_eq_o @ Least_2 ) )
=> ( ( ord_min_o @ Least_2 @ X_35 )
<=> Least_2 ) ) ).
thf(fact_198_min__leastL,axiom,
! [X_35: hoare_1167836817_state > $o,Least_2: hoare_1167836817_state > $o] :
( ( all2 @ ( ord_le827224136tate_o @ Least_2 ) )
=> ( ( ord_mi1697686287tate_o @ Least_2 @ X_35 )
= Least_2 ) ) ).
thf(fact_199_min__ord__min,axiom,
( ord_min_o
= ( min_o @ ord_less_eq_o ) ) ).
thf(fact_200_min__ord__min,axiom,
( ord_mi1697686287tate_o
= ( min_Ho1955171539tate_o @ ord_le827224136tate_o ) ) ).
thf(fact_201_comp__fun__commute_Ofold__insert__remove,axiom,
! [Z_16: hoare_1167836817_state,X_34: hoare_1167836817_state,A_50: hoare_1167836817_state > $o,F_26: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state] :
( ( finite1091222817_state @ F_26 )
=> ( ( finite1084549118_state @ A_50 )
=> ( ( finite1731015960_state @ F_26 @ Z_16 @ ( insert2134838167_state @ X_34 @ A_50 ) )
= ( F_26 @ X_34 @ ( finite1731015960_state @ F_26 @ Z_16 @ ( minus_2107060239tate_o @ A_50 @ ( insert2134838167_state @ X_34 @ bot_bo70021908tate_o ) ) ) ) ) ) ) ).
thf(fact_202_comp__fun__commute_Ofold__rec,axiom,
! [Z_15: hoare_1167836817_state,X_33: hoare_1167836817_state,A_49: hoare_1167836817_state > $o,F_25: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state] :
( ( finite1091222817_state @ F_25 )
=> ( ( finite1084549118_state @ A_49 )
=> ( ( member2058392318_state @ X_33 @ A_49 )
=> ( ( finite1731015960_state @ F_25 @ Z_15 @ A_49 )
= ( F_25 @ X_33 @ ( finite1731015960_state @ F_25 @ Z_15 @ ( minus_2107060239tate_o @ A_49 @ ( insert2134838167_state @ X_33 @ bot_bo70021908tate_o ) ) ) ) ) ) ) ) ).
thf(fact_203_fold__empty,axiom,
! [F_24: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,Z_14: hoare_1167836817_state] :
( ( finite1731015960_state @ F_24 @ Z_14 @ bot_bo70021908tate_o )
= Z_14 ) ).
thf(fact_204_comp__fun__commute_Ofold__fun__comm,axiom,
! [X_32: hoare_1167836817_state,Z_13: hoare_1167836817_state,A_48: hoare_1167836817_state > $o,F_23: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state] :
( ( finite1091222817_state @ F_23 )
=> ( ( finite1084549118_state @ A_48 )
=> ( ( F_23 @ X_32 @ ( finite1731015960_state @ F_23 @ Z_13 @ A_48 ) )
= ( finite1731015960_state @ F_23 @ ( F_23 @ X_32 @ Z_13 ) @ A_48 ) ) ) ) ).
thf(fact_205_comp__fun__commute_Ofold__equality,axiom,
! [Z_12: hoare_1167836817_state,A_47: hoare_1167836817_state > $o,Y_19: hoare_1167836817_state,F_22: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state] :
( ( finite1091222817_state @ F_22 )
=> ( ( finite1316643734_state @ F_22 @ Z_12 @ A_47 @ Y_19 )
=> ( ( finite1731015960_state @ F_22 @ Z_12 @ A_47 )
= Y_19 ) ) ) ).
thf(fact_206_comp__fun__commute_Ofold__insert2,axiom,
! [Z_11: hoare_1167836817_state,X_31: hoare_1167836817_state,A_46: hoare_1167836817_state > $o,F_21: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state] :
( ( finite1091222817_state @ F_21 )
=> ( ( finite1084549118_state @ A_46 )
=> ( ~ ( member2058392318_state @ X_31 @ A_46 )
=> ( ( finite1731015960_state @ F_21 @ Z_11 @ ( insert2134838167_state @ X_31 @ A_46 ) )
= ( finite1731015960_state @ F_21 @ ( F_21 @ X_31 @ Z_11 ) @ A_46 ) ) ) ) ) ).
thf(fact_207_comp__fun__commute_Ofold__insert,axiom,
! [Z_10: hoare_1167836817_state,X_30: hoare_1167836817_state,A_45: hoare_1167836817_state > $o,F_20: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state] :
( ( finite1091222817_state @ F_20 )
=> ( ( finite1084549118_state @ A_45 )
=> ( ~ ( member2058392318_state @ X_30 @ A_45 )
=> ( ( finite1731015960_state @ F_20 @ Z_10 @ ( insert2134838167_state @ X_30 @ A_45 ) )
= ( F_20 @ X_30 @ ( finite1731015960_state @ F_20 @ Z_10 @ A_45 ) ) ) ) ) ) ).
thf(fact_208_folding__one_Oeq__fold_H,axiom,
! [X_29: hoare_1167836817_state,A_44: hoare_1167836817_state > $o,F_19: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_18: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
( ( finite1074406356_state @ F_19 @ F_18 )
=> ( ( finite1084549118_state @ A_44 )
=> ( ~ ( member2058392318_state @ X_29 @ A_44 )
=> ( ( F_18 @ ( insert2134838167_state @ X_29 @ A_44 ) )
= ( finite1731015960_state @ F_19 @ X_29 @ A_44 ) ) ) ) ) ).
thf(fact_209_folding__one__idem_Oeq__fold__idem_H,axiom,
! [A_43: hoare_1167836817_state,A_42: hoare_1167836817_state > $o,F_17: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_16: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
( ( finite806517911_state @ F_17 @ F_16 )
=> ( ( finite1084549118_state @ A_42 )
=> ( ( F_16 @ ( insert2134838167_state @ A_43 @ A_42 ) )
= ( finite1731015960_state @ F_17 @ A_43 @ A_42 ) ) ) ) ).
thf(fact_210_comp__fun__commute_Ofold__graph__fold,axiom,
! [Z_9: hoare_1167836817_state,A_41: hoare_1167836817_state > $o,F_15: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state] :
( ( finite1091222817_state @ F_15 )
=> ( ( finite1084549118_state @ A_41 )
=> ( finite1316643734_state @ F_15 @ Z_9 @ A_41 @ ( finite1731015960_state @ F_15 @ Z_9 @ A_41 ) ) ) ) ).
thf(fact_211_comp__fun__idem_Ofold__insert__idem,axiom,
! [Z_8: hoare_1167836817_state,X_28: hoare_1167836817_state,A_40: hoare_1167836817_state > $o,F_14: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state] :
( ( finite1900754844_state @ F_14 )
=> ( ( finite1084549118_state @ A_40 )
=> ( ( finite1731015960_state @ F_14 @ Z_8 @ ( insert2134838167_state @ X_28 @ A_40 ) )
= ( F_14 @ X_28 @ ( finite1731015960_state @ F_14 @ Z_8 @ A_40 ) ) ) ) ) ).
thf(fact_212_comp__fun__idem_Ofold__insert__idem,axiom,
! [Z_8: hoare_1167836817_state > $o,X_28: hoare_1167836817_state,A_40: hoare_1167836817_state > $o,F_14: hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o] :
( ( finite856902323tate_o @ F_14 )
=> ( ( finite1084549118_state @ A_40 )
=> ( ( finite291020855tate_o @ F_14 @ Z_8 @ ( insert2134838167_state @ X_28 @ A_40 ) )
= ( F_14 @ X_28 @ ( finite291020855tate_o @ F_14 @ Z_8 @ A_40 ) ) ) ) ) ).
thf(fact_213_comp__fun__idem_Ofold__insert__idem2,axiom,
! [Z_7: hoare_1167836817_state,X_27: hoare_1167836817_state,A_39: hoare_1167836817_state > $o,F_13: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state] :
( ( finite1900754844_state @ F_13 )
=> ( ( finite1084549118_state @ A_39 )
=> ( ( finite1731015960_state @ F_13 @ Z_7 @ ( insert2134838167_state @ X_27 @ A_39 ) )
= ( finite1731015960_state @ F_13 @ ( F_13 @ X_27 @ Z_7 ) @ A_39 ) ) ) ) ).
thf(fact_214_comp__fun__idem_Ofold__insert__idem2,axiom,
! [Z_7: hoare_1167836817_state > $o,X_27: hoare_1167836817_state,A_39: hoare_1167836817_state > $o,F_13: hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o] :
( ( finite856902323tate_o @ F_13 )
=> ( ( finite1084549118_state @ A_39 )
=> ( ( finite291020855tate_o @ F_13 @ Z_7 @ ( insert2134838167_state @ X_27 @ A_39 ) )
= ( finite291020855tate_o @ F_13 @ ( F_13 @ X_27 @ Z_7 ) @ A_39 ) ) ) ) ).
thf(fact_215_comp__fun__idem_Ofun__left__idem,axiom,
! [X_26: hoare_1167836817_state,Z_6: hoare_1167836817_state > $o,F_12: hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o] :
( ( finite856902323tate_o @ F_12 )
=> ( ( F_12 @ X_26 @ ( F_12 @ X_26 @ Z_6 ) )
= ( F_12 @ X_26 @ Z_6 ) ) ) ).
thf(fact_216_comp__fun__idem__insert,axiom,
finite856902323tate_o @ insert2134838167_state ).
thf(fact_217_setsum__diff1__nat,axiom,
! [F_11: hoare_1167836817_state > nat,A_38: hoare_1167836817_state,A_37: hoare_1167836817_state > $o] :
( ( ( member2058392318_state @ A_38 @ A_37 )
=> ( ( big_co337839062te_nat @ F_11 @ ( minus_2107060239tate_o @ A_37 @ ( insert2134838167_state @ A_38 @ bot_bo70021908tate_o ) ) )
= ( minus_minus_nat @ ( big_co337839062te_nat @ F_11 @ A_37 ) @ ( F_11 @ A_38 ) ) ) )
& ( ~ ( member2058392318_state @ A_38 @ A_37 )
=> ( ( big_co337839062te_nat @ F_11 @ ( minus_2107060239tate_o @ A_37 @ ( insert2134838167_state @ A_38 @ bot_bo70021908tate_o ) ) )
= ( big_co337839062te_nat @ F_11 @ A_37 ) ) ) ) ).
thf(fact_218_setsum__diff__nat,axiom,
! [F_10: hoare_1167836817_state > nat,A_36: hoare_1167836817_state > $o,B_25: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ B_25 )
=> ( ( ord_le827224136tate_o @ B_25 @ A_36 )
=> ( ( big_co337839062te_nat @ F_10 @ ( minus_2107060239tate_o @ A_36 @ B_25 ) )
= ( minus_minus_nat @ ( big_co337839062te_nat @ F_10 @ A_36 ) @ ( big_co337839062te_nat @ F_10 @ B_25 ) ) ) ) ) ).
thf(fact_219_setsum_Ocong,axiom,
! [G_8: hoare_1167836817_state > nat,H_1: hoare_1167836817_state > nat,A_35: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ A_35 )
=> ( ! [X: hoare_1167836817_state] :
( ( member2058392318_state @ X @ A_35 )
=> ( ( G_8 @ X )
= ( H_1 @ X ) ) )
=> ( ( big_co337839062te_nat @ G_8 @ A_35 )
= ( big_co337839062te_nat @ H_1 @ A_35 ) ) ) ) ).
thf(fact_220_max__leastR,axiom,
! [X_25: $o,Least_1: $o] :
( ( all1 @ ( ord_less_eq_o @ Least_1 ) )
=> ( ( ord_max_o @ X_25 @ Least_1 )
<=> X_25 ) ) ).
thf(fact_221_max__leastR,axiom,
! [X_25: hoare_1167836817_state > $o,Least_1: hoare_1167836817_state > $o] :
( ( all2 @ ( ord_le827224136tate_o @ Least_1 ) )
=> ( ( ord_ma164008317tate_o @ X_25 @ Least_1 )
= X_25 ) ) ).
thf(fact_222_max__leastL,axiom,
! [X_24: $o,Least: $o] :
( ( all1 @ ( ord_less_eq_o @ Least ) )
=> ( ( ord_max_o @ Least @ X_24 )
<=> X_24 ) ) ).
thf(fact_223_max__leastL,axiom,
! [X_24: hoare_1167836817_state > $o,Least: hoare_1167836817_state > $o] :
( ( all2 @ ( ord_le827224136tate_o @ Least ) )
=> ( ( ord_ma164008317tate_o @ Least @ X_24 )
= X_24 ) ) ).
thf(fact_224_max__ord__max,axiom,
( ord_max_o
= ( max_o @ ord_less_eq_o ) ) ).
thf(fact_225_max__ord__max,axiom,
( ord_ma164008317tate_o
= ( max_Ho421493569tate_o @ ord_le827224136tate_o ) ) ).
thf(fact_226_setsum_Oremove,axiom,
! [G_7: hoare_1167836817_state > nat,X_23: hoare_1167836817_state,A_34: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ A_34 )
=> ( ( member2058392318_state @ X_23 @ A_34 )
=> ( ( big_co337839062te_nat @ G_7 @ A_34 )
= ( plus_plus_nat @ ( G_7 @ X_23 ) @ ( big_co337839062te_nat @ G_7 @ ( minus_2107060239tate_o @ A_34 @ ( insert2134838167_state @ X_23 @ bot_bo70021908tate_o ) ) ) ) ) ) ) ).
thf(fact_227_setsum__diff1_H,axiom,
! [F_9: hoare_1167836817_state > nat,A_33: hoare_1167836817_state,A_32: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ A_32 )
=> ( ( member2058392318_state @ A_33 @ A_32 )
=> ( ( big_co337839062te_nat @ F_9 @ A_32 )
= ( plus_plus_nat @ ( F_9 @ A_33 ) @ ( big_co337839062te_nat @ F_9 @ ( minus_2107060239tate_o @ A_32 @ ( insert2134838167_state @ A_33 @ bot_bo70021908tate_o ) ) ) ) ) ) ) ).
thf(fact_228_setsum_Oinsert,axiom,
! [G_6: hoare_1167836817_state > nat,X_22: hoare_1167836817_state,A_31: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ A_31 )
=> ( ~ ( member2058392318_state @ X_22 @ A_31 )
=> ( ( big_co337839062te_nat @ G_6 @ ( insert2134838167_state @ X_22 @ A_31 ) )
= ( plus_plus_nat @ ( G_6 @ X_22 ) @ ( big_co337839062te_nat @ G_6 @ A_31 ) ) ) ) ) ).
thf(fact_229_setsum__insert,axiom,
! [F_8: hoare_1167836817_state > nat,A_30: hoare_1167836817_state,F_7: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ F_7 )
=> ( ~ ( member2058392318_state @ A_30 @ F_7 )
=> ( ( big_co337839062te_nat @ F_8 @ ( insert2134838167_state @ A_30 @ F_7 ) )
= ( plus_plus_nat @ ( F_8 @ A_30 ) @ ( big_co337839062te_nat @ F_8 @ F_7 ) ) ) ) ) ).
thf(fact_230_setsum_Oinsert__remove,axiom,
! [G_5: hoare_1167836817_state > nat,X_21: hoare_1167836817_state,A_29: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ A_29 )
=> ( ( big_co337839062te_nat @ G_5 @ ( insert2134838167_state @ X_21 @ A_29 ) )
= ( plus_plus_nat @ ( G_5 @ X_21 ) @ ( big_co337839062te_nat @ G_5 @ ( minus_2107060239tate_o @ A_29 @ ( insert2134838167_state @ X_21 @ bot_bo70021908tate_o ) ) ) ) ) ) ).
thf(fact_231_setsum__cong2,axiom,
! [F_6: hoare_1167836817_state > nat,G_4: hoare_1167836817_state > nat,A_28: hoare_1167836817_state > $o] :
( ! [X: hoare_1167836817_state] :
( ( member2058392318_state @ X @ A_28 )
=> ( ( F_6 @ X )
= ( G_4 @ X ) ) )
=> ( ( big_co337839062te_nat @ F_6 @ A_28 )
= ( big_co337839062te_nat @ G_4 @ A_28 ) ) ) ).
thf(fact_232_setsum__cong,axiom,
! [F_5: hoare_1167836817_state > nat,G_3: hoare_1167836817_state > nat,A_27: hoare_1167836817_state > $o,B_24: hoare_1167836817_state > $o] :
( ( A_27 = B_24 )
=> ( ! [X: hoare_1167836817_state] :
( ( member2058392318_state @ X @ B_24 )
=> ( ( F_5 @ X )
= ( G_3 @ X ) ) )
=> ( ( big_co337839062te_nat @ F_5 @ A_27 )
= ( big_co337839062te_nat @ G_3 @ B_24 ) ) ) ) ).
thf(fact_233_setsum_OF__cong,axiom,
! [H: hoare_1167836817_state > nat,G_2: hoare_1167836817_state > nat,A_26: hoare_1167836817_state > $o,B_23: hoare_1167836817_state > $o] :
( ( A_26 = B_23 )
=> ( ! [X: hoare_1167836817_state] :
( ( member2058392318_state @ X @ B_23 )
=> ( ( H @ X )
= ( G_2 @ X ) ) )
=> ( ( big_co337839062te_nat @ H @ A_26 )
= ( big_co337839062te_nat @ G_2 @ B_23 ) ) ) ) ).
thf(fact_234_not__less__bot,axiom,
! [A_25: hoare_1167836817_state > $o] :
~ ( ord_le65125204tate_o @ A_25 @ bot_bo70021908tate_o ) ).
thf(fact_235_not__less__bot,axiom,
! [A_25: $o] :
~ ( ord_less_o @ A_25 @ bot_bot_o ) ).
thf(fact_236_bot__less,axiom,
! [A_24: hoare_1167836817_state > $o] :
( ( A_24 != bot_bo70021908tate_o )
<=> ( ord_le65125204tate_o @ bot_bo70021908tate_o @ A_24 ) ) ).
thf(fact_237_bot__less,axiom,
! [A_24: $o] :
( ~ ( A_24
<=> bot_bot_o )
<=> ( ord_less_o @ bot_bot_o @ A_24 ) ) ).
thf(fact_238_not__psubset__empty,axiom,
! [A_23: hoare_1167836817_state > $o] :
~ ( ord_le65125204tate_o @ A_23 @ bot_bo70021908tate_o ) ).
thf(fact_239_psubset__eq,axiom,
! [A_22: hoare_1167836817_state > $o,B_22: hoare_1167836817_state > $o] :
( ( ord_le65125204tate_o @ A_22 @ B_22 )
<=> ( ( ord_le827224136tate_o @ A_22 @ B_22 )
& ( A_22 != B_22 ) ) ) ).
thf(fact_240_subset__iff__psubset__eq,axiom,
! [A_21: hoare_1167836817_state > $o,B_21: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ A_21 @ B_21 )
<=> ( ( ord_le65125204tate_o @ A_21 @ B_21 )
| ( A_21 = B_21 ) ) ) ).
thf(fact_241_psubset__imp__subset,axiom,
! [A_20: hoare_1167836817_state > $o,B_20: hoare_1167836817_state > $o] :
( ( ord_le65125204tate_o @ A_20 @ B_20 )
=> ( ord_le827224136tate_o @ A_20 @ B_20 ) ) ).
thf(fact_242_psubset__subset__trans,axiom,
! [C_10: hoare_1167836817_state > $o,A_19: hoare_1167836817_state > $o,B_19: hoare_1167836817_state > $o] :
( ( ord_le65125204tate_o @ A_19 @ B_19 )
=> ( ( ord_le827224136tate_o @ B_19 @ C_10 )
=> ( ord_le65125204tate_o @ A_19 @ C_10 ) ) ) ).
thf(fact_243_subset__psubset__trans,axiom,
! [C_9: hoare_1167836817_state > $o,A_18: hoare_1167836817_state > $o,B_18: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ A_18 @ B_18 )
=> ( ( ord_le65125204tate_o @ B_18 @ C_9 )
=> ( ord_le65125204tate_o @ A_18 @ C_9 ) ) ) ).
thf(fact_244_order__less__asym,axiom,
! [X_20: hoare_1167836817_state > $o,Y_18: hoare_1167836817_state > $o] :
( ( ord_le65125204tate_o @ X_20 @ Y_18 )
=> ~ ( ord_le65125204tate_o @ Y_18 @ X_20 ) ) ).
thf(fact_245_order__less__asym,axiom,
! [X_20: $o,Y_18: $o] :
( ( ord_less_o @ X_20 @ Y_18 )
=> ~ ( ord_less_o @ Y_18 @ X_20 ) ) ).
thf(fact_246_xt1_I10_J,axiom,
! [Z_5: hoare_1167836817_state > $o,Y_17: hoare_1167836817_state > $o,X_19: hoare_1167836817_state > $o] :
( ( ord_le65125204tate_o @ Y_17 @ X_19 )
=> ( ( ord_le65125204tate_o @ Z_5 @ Y_17 )
=> ( ord_le65125204tate_o @ Z_5 @ X_19 ) ) ) ).
thf(fact_247_xt1_I10_J,axiom,
! [Z_5: $o,Y_17: $o,X_19: $o] :
( ( ord_less_o @ Y_17 @ X_19 )
=> ( ( ord_less_o @ Z_5 @ Y_17 )
=> ( ord_less_o @ Z_5 @ X_19 ) ) ) ).
thf(fact_248_order__less__trans,axiom,
! [Z_4: hoare_1167836817_state > $o,X_18: hoare_1167836817_state > $o,Y_16: hoare_1167836817_state > $o] :
( ( ord_le65125204tate_o @ X_18 @ Y_16 )
=> ( ( ord_le65125204tate_o @ Y_16 @ Z_4 )
=> ( ord_le65125204tate_o @ X_18 @ Z_4 ) ) ) ).
thf(fact_249_order__less__trans,axiom,
! [Z_4: $o,X_18: $o,Y_16: $o] :
( ( ord_less_o @ X_18 @ Y_16 )
=> ( ( ord_less_o @ Y_16 @ Z_4 )
=> ( ord_less_o @ X_18 @ Z_4 ) ) ) ).
thf(fact_250_xt1_I2_J,axiom,
! [C_8: hoare_1167836817_state > $o,B_17: hoare_1167836817_state > $o,A_17: hoare_1167836817_state > $o] :
( ( ord_le65125204tate_o @ B_17 @ A_17 )
=> ( ( B_17 = C_8 )
=> ( ord_le65125204tate_o @ C_8 @ A_17 ) ) ) ).
thf(fact_251_xt1_I2_J,axiom,
! [C_8: $o,B_17: $o,A_17: $o] :
( ( ord_less_o @ B_17 @ A_17 )
=> ( ( B_17
<=> C_8 )
=> ( ord_less_o @ C_8 @ A_17 ) ) ) ).
thf(fact_252_ord__less__eq__trans,axiom,
! [C_7: hoare_1167836817_state > $o,A_16: hoare_1167836817_state > $o,B_16: hoare_1167836817_state > $o] :
( ( ord_le65125204tate_o @ A_16 @ B_16 )
=> ( ( B_16 = C_7 )
=> ( ord_le65125204tate_o @ A_16 @ C_7 ) ) ) ).
thf(fact_253_ord__less__eq__trans,axiom,
! [C_7: $o,A_16: $o,B_16: $o] :
( ( ord_less_o @ A_16 @ B_16 )
=> ( ( B_16
<=> C_7 )
=> ( ord_less_o @ A_16 @ C_7 ) ) ) ).
thf(fact_254_xt1_I1_J,axiom,
! [C_6: hoare_1167836817_state > $o,A_15: hoare_1167836817_state > $o,B_15: hoare_1167836817_state > $o] :
( ( A_15 = B_15 )
=> ( ( ord_le65125204tate_o @ C_6 @ B_15 )
=> ( ord_le65125204tate_o @ C_6 @ A_15 ) ) ) ).
thf(fact_255_xt1_I1_J,axiom,
! [C_6: $o,B_15: $o,A_15: $o] :
( ( A_15
<=> B_15 )
=> ( ( ord_less_o @ C_6 @ B_15 )
=> ( ord_less_o @ C_6 @ A_15 ) ) ) ).
thf(fact_256_ord__eq__less__trans,axiom,
! [C_5: hoare_1167836817_state > $o,A_14: hoare_1167836817_state > $o,B_14: hoare_1167836817_state > $o] :
( ( A_14 = B_14 )
=> ( ( ord_le65125204tate_o @ B_14 @ C_5 )
=> ( ord_le65125204tate_o @ A_14 @ C_5 ) ) ) ).
thf(fact_257_ord__eq__less__trans,axiom,
! [C_5: $o,B_14: $o,A_14: $o] :
( ( A_14
<=> B_14 )
=> ( ( ord_less_o @ B_14 @ C_5 )
=> ( ord_less_o @ A_14 @ C_5 ) ) ) ).
thf(fact_258_xt1_I9_J,axiom,
! [B_13: hoare_1167836817_state > $o,A_13: hoare_1167836817_state > $o] :
( ( ord_le65125204tate_o @ B_13 @ A_13 )
=> ~ ( ord_le65125204tate_o @ A_13 @ B_13 ) ) ).
thf(fact_259_xt1_I9_J,axiom,
! [B_13: $o,A_13: $o] :
( ( ord_less_o @ B_13 @ A_13 )
=> ~ ( ord_less_o @ A_13 @ B_13 ) ) ).
thf(fact_260_order__less__asym_H,axiom,
! [A_12: hoare_1167836817_state > $o,B_12: hoare_1167836817_state > $o] :
( ( ord_le65125204tate_o @ A_12 @ B_12 )
=> ~ ( ord_le65125204tate_o @ B_12 @ A_12 ) ) ).
thf(fact_261_order__less__asym_H,axiom,
! [A_12: $o,B_12: $o] :
( ( ord_less_o @ A_12 @ B_12 )
=> ~ ( ord_less_o @ B_12 @ A_12 ) ) ).
thf(fact_262_order__less__imp__triv,axiom,
! [P: $o,X_17: hoare_1167836817_state > $o,Y_15: hoare_1167836817_state > $o] :
( ( ord_le65125204tate_o @ X_17 @ Y_15 )
=> ( ( ord_le65125204tate_o @ Y_15 @ X_17 )
=> P ) ) ).
thf(fact_263_order__less__imp__triv,axiom,
! [P: $o,X_17: $o,Y_15: $o] :
( ( ord_less_o @ X_17 @ Y_15 )
=> ( ( ord_less_o @ Y_15 @ X_17 )
=> P ) ) ).
thf(fact_264_order__less__imp__not__eq2,axiom,
! [X_16: hoare_1167836817_state > $o,Y_14: hoare_1167836817_state > $o] :
( ( ord_le65125204tate_o @ X_16 @ Y_14 )
=> ( Y_14 != X_16 ) ) ).
thf(fact_265_order__less__imp__not__eq2,axiom,
! [X_16: $o,Y_14: $o] :
( ( ord_less_o @ X_16 @ Y_14 )
=> ( Y_14
<=> ~ X_16 ) ) ).
thf(fact_266_order__less__imp__not__eq,axiom,
! [X_15: hoare_1167836817_state > $o,Y_13: hoare_1167836817_state > $o] :
( ( ord_le65125204tate_o @ X_15 @ Y_13 )
=> ( X_15 != Y_13 ) ) ).
thf(fact_267_order__less__imp__not__eq,axiom,
! [X_15: $o,Y_13: $o] :
( ( ord_less_o @ X_15 @ Y_13 )
=> ( X_15
<=> ~ Y_13 ) ) ).
thf(fact_268_order__less__imp__not__less,axiom,
! [X_14: hoare_1167836817_state > $o,Y_12: hoare_1167836817_state > $o] :
( ( ord_le65125204tate_o @ X_14 @ Y_12 )
=> ~ ( ord_le65125204tate_o @ Y_12 @ X_14 ) ) ).
thf(fact_269_order__less__imp__not__less,axiom,
! [X_14: $o,Y_12: $o] :
( ( ord_less_o @ X_14 @ Y_12 )
=> ~ ( ord_less_o @ Y_12 @ X_14 ) ) ).
thf(fact_270_order__less__not__sym,axiom,
! [X_13: hoare_1167836817_state > $o,Y_11: hoare_1167836817_state > $o] :
( ( ord_le65125204tate_o @ X_13 @ Y_11 )
=> ~ ( ord_le65125204tate_o @ Y_11 @ X_13 ) ) ).
thf(fact_271_order__less__not__sym,axiom,
! [X_13: $o,Y_11: $o] :
( ( ord_less_o @ X_13 @ Y_11 )
=> ~ ( ord_less_o @ Y_11 @ X_13 ) ) ).
thf(fact_272_less__imp__neq,axiom,
! [X_12: hoare_1167836817_state > $o,Y_10: hoare_1167836817_state > $o] :
( ( ord_le65125204tate_o @ X_12 @ Y_10 )
=> ( X_12 != Y_10 ) ) ).
thf(fact_273_less__imp__neq,axiom,
! [X_12: $o,Y_10: $o] :
( ( ord_less_o @ X_12 @ Y_10 )
=> ~ ( X_12
<=> Y_10 ) ) ).
thf(fact_274_order__less__irrefl,axiom,
! [X_11: hoare_1167836817_state > $o] :
~ ( ord_le65125204tate_o @ X_11 @ X_11 ) ).
thf(fact_275_order__less__irrefl,axiom,
! [X_11: $o] :
~ ( ord_less_o @ X_11 @ X_11 ) ).
thf(fact_276_order__less__le,axiom,
! [X_10: $o,Y_9: $o] :
( ( ord_less_o @ X_10 @ Y_9 )
<=> ( ( ord_less_eq_o @ X_10 @ Y_9 )
& ~ ( X_10
<=> Y_9 ) ) ) ).
thf(fact_277_order__less__le,axiom,
! [X_10: hoare_1167836817_state > $o,Y_9: hoare_1167836817_state > $o] :
( ( ord_le65125204tate_o @ X_10 @ Y_9 )
<=> ( ( ord_le827224136tate_o @ X_10 @ Y_9 )
& ( X_10 != Y_9 ) ) ) ).
thf(fact_278_less__le__not__le,axiom,
! [X_9: $o,Y_8: $o] :
( ( ord_less_o @ X_9 @ Y_8 )
<=> ( ( ord_less_eq_o @ X_9 @ Y_8 )
& ~ ( ord_less_eq_o @ Y_8 @ X_9 ) ) ) ).
thf(fact_279_less__le__not__le,axiom,
! [X_9: hoare_1167836817_state > $o,Y_8: hoare_1167836817_state > $o] :
( ( ord_le65125204tate_o @ X_9 @ Y_8 )
<=> ( ( ord_le827224136tate_o @ X_9 @ Y_8 )
& ~ ( ord_le827224136tate_o @ Y_8 @ X_9 ) ) ) ).
thf(fact_280_order__le__less,axiom,
! [X_8: $o,Y_7: $o] :
( ( ord_less_eq_o @ X_8 @ Y_7 )
<=> ( ( ord_less_o @ X_8 @ Y_7 )
| ( X_8
<=> Y_7 ) ) ) ).
thf(fact_281_order__le__less,axiom,
! [X_8: hoare_1167836817_state > $o,Y_7: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ X_8 @ Y_7 )
<=> ( ( ord_le65125204tate_o @ X_8 @ Y_7 )
| ( X_8 = Y_7 ) ) ) ).
thf(fact_282_order__neq__le__trans,axiom,
! [B_11: $o,A_11: $o] :
( ~ ( A_11
<=> B_11 )
=> ( ( ord_less_eq_o @ A_11 @ B_11 )
=> ( ord_less_o @ A_11 @ B_11 ) ) ) ).
thf(fact_283_order__neq__le__trans,axiom,
! [A_11: hoare_1167836817_state > $o,B_11: hoare_1167836817_state > $o] :
( ( A_11 != B_11 )
=> ( ( ord_le827224136tate_o @ A_11 @ B_11 )
=> ( ord_le65125204tate_o @ A_11 @ B_11 ) ) ) ).
thf(fact_284_xt1_I12_J,axiom,
! [B_10: $o,A_10: $o] :
( ~ ( A_10
<=> B_10 )
=> ( ( ord_less_eq_o @ B_10 @ A_10 )
=> ( ord_less_o @ B_10 @ A_10 ) ) ) ).
thf(fact_285_xt1_I12_J,axiom,
! [A_10: hoare_1167836817_state > $o,B_10: hoare_1167836817_state > $o] :
( ( A_10 != B_10 )
=> ( ( ord_le827224136tate_o @ B_10 @ A_10 )
=> ( ord_le65125204tate_o @ B_10 @ A_10 ) ) ) ).
thf(fact_286_order__less__imp__le,axiom,
! [X_7: $o,Y_6: $o] :
( ( ord_less_o @ X_7 @ Y_6 )
=> ( ord_less_eq_o @ X_7 @ Y_6 ) ) ).
thf(fact_287_order__less__imp__le,axiom,
! [X_7: hoare_1167836817_state > $o,Y_6: hoare_1167836817_state > $o] :
( ( ord_le65125204tate_o @ X_7 @ Y_6 )
=> ( ord_le827224136tate_o @ X_7 @ Y_6 ) ) ).
thf(fact_288_order__le__imp__less__or__eq,axiom,
! [X_6: $o,Y_5: $o] :
( ( ord_less_eq_o @ X_6 @ Y_5 )
=> ( ( ord_less_o @ X_6 @ Y_5 )
| ( X_6
<=> Y_5 ) ) ) ).
thf(fact_289_order__le__imp__less__or__eq,axiom,
! [X_6: hoare_1167836817_state > $o,Y_5: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ X_6 @ Y_5 )
=> ( ( ord_le65125204tate_o @ X_6 @ Y_5 )
| ( X_6 = Y_5 ) ) ) ).
thf(fact_290_order__le__neq__trans,axiom,
! [A_9: $o,B_9: $o] :
( ( ord_less_eq_o @ A_9 @ B_9 )
=> ( ~ ( A_9
<=> B_9 )
=> ( ord_less_o @ A_9 @ B_9 ) ) ) ).
thf(fact_291_order__le__neq__trans,axiom,
! [A_9: hoare_1167836817_state > $o,B_9: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ A_9 @ B_9 )
=> ( ( A_9 != B_9 )
=> ( ord_le65125204tate_o @ A_9 @ B_9 ) ) ) ).
thf(fact_292_xt1_I11_J,axiom,
! [B_8: $o,A_8: $o] :
( ( ord_less_eq_o @ B_8 @ A_8 )
=> ( ~ ( A_8
<=> B_8 )
=> ( ord_less_o @ B_8 @ A_8 ) ) ) ).
thf(fact_293_xt1_I11_J,axiom,
! [B_8: hoare_1167836817_state > $o,A_8: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ B_8 @ A_8 )
=> ( ( A_8 != B_8 )
=> ( ord_le65125204tate_o @ B_8 @ A_8 ) ) ) ).
thf(fact_294_order__less__le__trans,axiom,
! [Z_3: $o,X_5: $o,Y_4: $o] :
( ( ord_less_o @ X_5 @ Y_4 )
=> ( ( ord_less_eq_o @ Y_4 @ Z_3 )
=> ( ord_less_o @ X_5 @ Z_3 ) ) ) ).
thf(fact_295_order__less__le__trans,axiom,
! [Z_3: hoare_1167836817_state > $o,X_5: hoare_1167836817_state > $o,Y_4: hoare_1167836817_state > $o] :
( ( ord_le65125204tate_o @ X_5 @ Y_4 )
=> ( ( ord_le827224136tate_o @ Y_4 @ Z_3 )
=> ( ord_le65125204tate_o @ X_5 @ Z_3 ) ) ) ).
thf(fact_296_xt1_I7_J,axiom,
! [Z_2: $o,Y_3: $o,X_4: $o] :
( ( ord_less_o @ Y_3 @ X_4 )
=> ( ( ord_less_eq_o @ Z_2 @ Y_3 )
=> ( ord_less_o @ Z_2 @ X_4 ) ) ) ).
thf(fact_297_xt1_I7_J,axiom,
! [Z_2: hoare_1167836817_state > $o,Y_3: hoare_1167836817_state > $o,X_4: hoare_1167836817_state > $o] :
( ( ord_le65125204tate_o @ Y_3 @ X_4 )
=> ( ( ord_le827224136tate_o @ Z_2 @ Y_3 )
=> ( ord_le65125204tate_o @ Z_2 @ X_4 ) ) ) ).
thf(fact_298_order__le__less__trans,axiom,
! [Z_1: $o,X_3: $o,Y_2: $o] :
( ( ord_less_eq_o @ X_3 @ Y_2 )
=> ( ( ord_less_o @ Y_2 @ Z_1 )
=> ( ord_less_o @ X_3 @ Z_1 ) ) ) ).
thf(fact_299_order__le__less__trans,axiom,
! [Z_1: hoare_1167836817_state > $o,X_3: hoare_1167836817_state > $o,Y_2: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ X_3 @ Y_2 )
=> ( ( ord_le65125204tate_o @ Y_2 @ Z_1 )
=> ( ord_le65125204tate_o @ X_3 @ Z_1 ) ) ) ).
thf(fact_300_xt1_I8_J,axiom,
! [Z: $o,Y_1: $o,X_2: $o] :
( ( ord_less_eq_o @ Y_1 @ X_2 )
=> ( ( ord_less_o @ Z @ Y_1 )
=> ( ord_less_o @ Z @ X_2 ) ) ) ).
thf(fact_301_xt1_I8_J,axiom,
! [Z: hoare_1167836817_state > $o,Y_1: hoare_1167836817_state > $o,X_2: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ Y_1 @ X_2 )
=> ( ( ord_le65125204tate_o @ Z @ Y_1 )
=> ( ord_le65125204tate_o @ Z @ X_2 ) ) ) ).
thf(fact_302_less__fun__def,axiom,
! [F_4: hoare_1167836817_state > $o,G_1: hoare_1167836817_state > $o] :
( ( ord_le65125204tate_o @ F_4 @ G_1 )
<=> ( ( ord_le827224136tate_o @ F_4 @ G_1 )
& ~ ( ord_le827224136tate_o @ G_1 @ F_4 ) ) ) ).
thf(fact_303_psubset__insert__iff,axiom,
! [A_7: hoare_1167836817_state > $o,X_1: hoare_1167836817_state,B_7: hoare_1167836817_state > $o] :
( ( ord_le65125204tate_o @ A_7 @ ( insert2134838167_state @ X_1 @ B_7 ) )
<=> ( ( ( member2058392318_state @ X_1 @ B_7 )
=> ( ord_le65125204tate_o @ A_7 @ B_7 ) )
& ( ~ ( member2058392318_state @ X_1 @ B_7 )
=> ( ( ( member2058392318_state @ X_1 @ A_7 )
=> ( ord_le65125204tate_o @ ( minus_2107060239tate_o @ A_7 @ ( insert2134838167_state @ X_1 @ bot_bo70021908tate_o ) ) @ B_7 ) )
& ( ~ ( member2058392318_state @ X_1 @ A_7 )
=> ( ord_le827224136tate_o @ A_7 @ B_7 ) ) ) ) ) ) ).
thf(fact_304_setsum__strict__mono,axiom,
! [F_3: hoare_1167836817_state > nat,G: hoare_1167836817_state > nat,A_6: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ A_6 )
=> ( ( A_6 != bot_bo70021908tate_o )
=> ( ! [X: hoare_1167836817_state] :
( ( member2058392318_state @ X @ A_6 )
=> ( ord_less_nat @ ( F_3 @ X ) @ ( G @ X ) ) )
=> ( ord_less_nat @ ( big_co337839062te_nat @ F_3 @ A_6 ) @ ( big_co337839062te_nat @ G @ A_6 ) ) ) ) ) ).
thf(fact_305_psubset__trans,axiom,
! [C_4: hoare_1167836817_state > $o,A_5: hoare_1167836817_state > $o,B_6: hoare_1167836817_state > $o] :
( ( ord_le65125204tate_o @ A_5 @ B_6 )
=> ( ( ord_le65125204tate_o @ B_6 @ C_4 )
=> ( ord_le65125204tate_o @ A_5 @ C_4 ) ) ) ).
thf(fact_306_psubsetD,axiom,
! [C_3: hoare_1167836817_state,A_4: hoare_1167836817_state > $o,B_5: hoare_1167836817_state > $o] :
( ( ord_le65125204tate_o @ A_4 @ B_5 )
=> ( ( member2058392318_state @ C_3 @ A_4 )
=> ( member2058392318_state @ C_3 @ B_5 ) ) ) ).
thf(fact_307_psubset__imp__ex__mem,axiom,
! [A_3: hoare_1167836817_state > $o,B_3: hoare_1167836817_state > $o] :
( ( ord_le65125204tate_o @ A_3 @ B_3 )
=> ? [B_4: hoare_1167836817_state] : ( member2058392318_state @ B_4 @ ( minus_2107060239tate_o @ B_3 @ A_3 ) ) ) ).
thf(fact_308_xt6,axiom,
! [C_2: hoare_1167836817_state > $o,F_2: ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o,B_2: hoare_1167836817_state > $o,A_2: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ ( F_2 @ B_2 ) @ A_2 )
=> ( ( ord_le65125204tate_o @ C_2 @ B_2 )
=> ( ! [X: hoare_1167836817_state > $o,Y: hoare_1167836817_state > $o] :
( ( ord_le65125204tate_o @ Y @ X )
=> ( ord_le65125204tate_o @ ( F_2 @ Y ) @ ( F_2 @ X ) ) )
=> ( ord_le65125204tate_o @ ( F_2 @ C_2 ) @ A_2 ) ) ) ) ).
thf(fact_309_xt6,axiom,
! [C_2: $o,F_2: $o > hoare_1167836817_state > $o,B_2: $o,A_2: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ ( F_2 @ B_2 ) @ A_2 )
=> ( ( ord_less_o @ C_2 @ B_2 )
=> ( ! [X: $o,Y: $o] :
( ( ord_less_o @ Y @ X )
=> ( ord_le65125204tate_o @ ( F_2 @ Y ) @ ( F_2 @ X ) ) )
=> ( ord_le65125204tate_o @ ( F_2 @ C_2 ) @ A_2 ) ) ) ) ).
thf(fact_310_xt5,axiom,
! [C_1: hoare_1167836817_state > $o,F_1: ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o,B_1: hoare_1167836817_state > $o,A_1: hoare_1167836817_state > $o] :
( ( ord_le65125204tate_o @ B_1 @ A_1 )
=> ( ( ord_le827224136tate_o @ C_1 @ ( F_1 @ B_1 ) )
=> ( ! [X: hoare_1167836817_state > $o,Y: hoare_1167836817_state > $o] :
( ( ord_le65125204tate_o @ Y @ X )
=> ( ord_le65125204tate_o @ ( F_1 @ Y ) @ ( F_1 @ X ) ) )
=> ( ord_le65125204tate_o @ C_1 @ ( F_1 @ A_1 ) ) ) ) ) ).
thf(fact_311_xt5,axiom,
! [C_1: hoare_1167836817_state > $o,F_1: $o > hoare_1167836817_state > $o,B_1: $o,A_1: $o] :
( ( ord_less_o @ B_1 @ A_1 )
=> ( ( ord_le827224136tate_o @ C_1 @ ( F_1 @ B_1 ) )
=> ( ! [X: $o,Y: $o] :
( ( ord_less_o @ Y @ X )
=> ( ord_le65125204tate_o @ ( F_1 @ Y ) @ ( F_1 @ X ) ) )
=> ( ord_le65125204tate_o @ C_1 @ ( F_1 @ A_1 ) ) ) ) ) ).
thf(fact_312_xt4,axiom,
! [C: hoare_1167836817_state > $o,F: ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o,B: hoare_1167836817_state > $o,A: hoare_1167836817_state > $o] :
( ( ord_le65125204tate_o @ ( F @ B ) @ A )
=> ( ( ord_le827224136tate_o @ C @ B )
=> ( ! [X: hoare_1167836817_state > $o,Y: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ Y @ X )
=> ( ord_le827224136tate_o @ ( F @ Y ) @ ( F @ X ) ) )
=> ( ord_le65125204tate_o @ ( F @ C ) @ A ) ) ) ) ).
thf(fact_313_xt4,axiom,
! [C: hoare_1167836817_state > $o,F: ( hoare_1167836817_state > $o ) > $o,B: hoare_1167836817_state > $o,A: $o] :
( ( ord_less_o @ ( F @ B ) @ A )
=> ( ( ord_le827224136tate_o @ C @ B )
=> ( ! [X: hoare_1167836817_state > $o,Y: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ Y @ X )
=> ( ord_less_eq_o @ ( F @ Y ) @ ( F @ X ) ) )
=> ( ord_less_o @ ( F @ C ) @ A ) ) ) ) ).
thf(fact_314_xt4,axiom,
! [C: $o,F: $o > hoare_1167836817_state > $o,B: $o,A: hoare_1167836817_state > $o] :
( ( ord_le65125204tate_o @ ( F @ B ) @ A )
=> ( ( ord_less_eq_o @ C @ B )
=> ( ! [X: $o,Y: $o] :
( ( ord_less_eq_o @ Y @ X )
=> ( ord_le827224136tate_o @ ( F @ Y ) @ ( F @ X ) ) )
=> ( ord_le65125204tate_o @ ( F @ C ) @ A ) ) ) ) ).
%----Conjectures (3)
thf(conj_0,hypothesis,
hoare_123228589_state @ bot_bo70021908tate_o @ ( insert2134838167_state @ ( hoare_Mirabelle_MGT @ c ) @ bot_bo70021908tate_o ) ).
thf(conj_1,hypothesis,
hoare_529639851_state @ bot_bo70021908tate_o @ ( insert2134838167_state @ ( hoare_908217195_state @ p @ c @ q ) @ bot_bo70021908tate_o ) ).
thf(conj_2,conjecture,
hoare_123228589_state @ bot_bo70021908tate_o @ ( insert2134838167_state @ ( hoare_908217195_state @ p @ c @ q ) @ bot_bo70021908tate_o ) ).
%------------------------------------------------------------------------------