%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP074^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n002.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:20 AM UTC 2026
% Result : Theorem 2.69s 1.63s
% Output : Refutation 2.69s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP074^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n002.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 08:25:18 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.20/0.46 ---- Original DTF file ---
% 0.20/0.46 thf(spec,logic,$$dhol).
% 0.20/0.46 %------------------------------------------------------------------------------
% 0.20/0.46 % File : ITP074^2 : TPTP v9.2.1. Released v7.5.0.
% 0.20/0.46 % Domain : Interactive Theorem Proving
% 0.20/0.46 % Problem : Sledgehammer HF problem prob_612__5334652_1
% 0.20/0.46 % Version : Especial.
% 0.20/0.46 % English :
% 0.20/0.46
% 0.20/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.20/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.20/0.46 % Source : [Des21]
% 0.20/0.46 % Names : HF/prob_612__5334652_1 [Des21]
% 0.20/0.46
% 0.20/0.46 % Status : Theorem
% 0.20/0.46 % Rating : 0.33 v8.1.0, 0.25 v7.5.0
% 0.20/0.46 % Syntax : Number of formulae : 348 ( 113 unt; 48 typ; 0 def)
% 0.20/0.46 % Number of atoms : 732 ( 269 equ; 0 cnn)
% 0.20/0.46 % Maximal formula atoms : 7 ( 2 avg)
% 0.20/0.46 % Number of connectives : 2623 ( 34 ~; 7 |; 29 &;2255 @)
% 0.20/0.46 % ( 0 <=>; 298 =>; 0 <=; 0 <~>)
% 0.20/0.46 % Maximal formula depth : 18 ( 6 avg)
% 0.20/0.46 % Number of types : 3 ( 2 usr)
% 0.20/0.46 % Number of type conns : 113 ( 113 >; 0 *; 0 +; 0 <<)
% 0.20/0.46 % Number of symbols : 47 ( 46 usr; 2 con; 0-4 aty)
% 0.20/0.46 % Number of variables : 843 ( 71 ^; 728 !; 15 ?; 843 :)
% 0.20/0.46 % ( 29 !>; 0 ?*; 0 @-; 0 @+)
% 0.20/0.46 % SPC : TH1_THM_EQU_NAR
% 0.20/0.46
% 0.20/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 16:22:40.568
% 0.20/0.46 %------------------------------------------------------------------------------
% 0.20/0.46 % Could-be-implicit typings (3)
% 0.20/0.46 thf(ty_t_HF__Mirabelle__fsbjehakzm_Ohf,type,
% 0.20/0.46 hF_Mirabelle_hf: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_Set_Oset,type,
% 0.20/0.46 set: $tType > $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_Nat_Onat,type,
% 0.20/0.46 nat: $tType ).
% 0.20/0.46
% 0.20/0.46 % Explicit typings (45)
% 0.20/0.46 thf(sy_cl_HOL_Otype,type,
% 0.20/0.46 type:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Ozero,type,
% 0.20/0.46 zero:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Ominus,type,
% 0.20/0.46 minus:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Oorder,type,
% 0.20/0.46 order:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Ogroup__add,type,
% 0.20/0.46 group_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Lattices_Olattice,type,
% 0.20/0.46 lattice:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Osemiring__1,type,
% 0.20/0.46 semiring_1:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Opreorder,type,
% 0.20/0.46 preorder:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Nat_Osemiring__char__0,type,
% 0.20/0.46 semiring_char_0:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Ocomm__monoid__diff,type,
% 0.20/0.46 comm_monoid_diff:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Osemiring__1__cancel,type,
% 0.20/0.46 semiring_1_cancel:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Lattices_Odistrib__lattice,type,
% 0.20/0.46 distrib_lattice:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Lattices_Osemilattice__inf,type,
% 0.20/0.46 semilattice_inf:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 0.20/0.46 semilattice_sup:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.20/0.46 ordered_ab_group_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 0.20/0.46 cancel1352612707id_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 0.20/0.46 cancel146912293up_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.20/0.46 linord1659791738miring:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.20/0.46 canoni770627133id_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.20/0.46 minus_minus:
% 0.20/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.20/0.46 zero_zero:
% 0.20/0.46 !>[A: $tType] : A ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHCollect,type,
% 0.20/0.46 hF_Mir1687042746ollect: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHF,type,
% 0.20/0.46 hF_Mirabelle_HF: ( set @ hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHInter,type,
% 0.20/0.46 hF_Mirabelle_HInter: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHUnion,type,
% 0.20/0.46 hF_Mirabelle_HUnion: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_HF__Mirabelle__fsbjehakzm_OPrimReplace,type,
% 0.20/0.46 hF_Mir569462966eplace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_HF__Mirabelle__fsbjehakzm_ORepFun,type,
% 0.20/0.46 hF_Mirabelle_RepFun: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_HF__Mirabelle__fsbjehakzm_OReplace,type,
% 0.20/0.46 hF_Mirabelle_Replace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohf_OAbs__hf,type,
% 0.20/0.46 hF_Mirabelle_Abs_hf: nat > hF_Mirabelle_hf ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohfset,type,
% 0.20/0.46 hF_Mirabelle_hfset: hF_Mirabelle_hf > ( set @ hF_Mirabelle_hf ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohfst,type,
% 0.20/0.46 hF_Mirabelle_hfst: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohinsert,type,
% 0.20/0.46 hF_Mirabelle_hinsert: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohmem,type,
% 0.20/0.46 hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohpair,type,
% 0.20/0.46 hF_Mirabelle_hpair: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohsnd,type,
% 0.20/0.46 hF_Mirabelle_hsnd: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohsplit,type,
% 0.20/0.46 hF_Mirabelle_hsplit:
% 0.20/0.46 !>[A: $tType] : ( ( hF_Mirabelle_hf > hF_Mirabelle_hf > A ) > hF_Mirabelle_hf > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 0.20/0.46 inf_inf:
% 0.20/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Lattices_Osup__class_Osup,type,
% 0.20/0.46 sup_sup:
% 0.20/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
% 0.20/0.46 semiring_1_of_nat:
% 0.20/0.46 !>[A: $tType] : ( nat > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
% 0.20/0.46 semiri532925092at_aux:
% 0.20/0.46 !>[A: $tType] : ( ( A > A ) > nat > A > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.20/0.46 ord_less_eq:
% 0.20/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Set_OCollect,type,
% 0.20/0.46 collect:
% 0.20/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_member,type,
% 0.20/0.46 member:
% 0.20/0.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_v_x,type,
% 0.20/0.46 x: hF_Mirabelle_hf ).
% 0.20/0.46
% 0.20/0.46 thf(sy_v_y,type,
% 0.20/0.46 y: hF_Mirabelle_hf ).
% 0.20/0.46
% 0.20/0.46 % Relevant facts (254)
% 0.20/0.46 thf(fact_0_hpair__iff,axiom,
% 0.20/0.46 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( ( hF_Mirabelle_hpair @ A2 @ B )
% 0.20/0.46 = ( hF_Mirabelle_hpair @ A3 @ B2 ) )
% 0.20/0.46 = ( ( A2 = A3 )
% 0.20/0.46 & ( B = B2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hpair_iff
% 0.20/0.46 thf(fact_1_hpair__inject,axiom,
% 0.20/0.46 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( ( hF_Mirabelle_hpair @ A2 @ B )
% 0.20/0.46 = ( hF_Mirabelle_hpair @ A3 @ B2 ) )
% 0.20/0.46 => ~ ( ( A2 = A3 )
% 0.20/0.46 => ( B != B2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hpair_inject
% 0.20/0.46 thf(fact_2_hpair__neq__fst,axiom,
% 0.20/0.46 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hpair @ A2 @ B )
% 0.20/0.46 != A2 ) ).
% 0.20/0.46
% 0.20/0.46 % hpair_neq_fst
% 0.20/0.46 thf(fact_3_hpair__neq__snd,axiom,
% 0.20/0.46 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hpair @ A2 @ B )
% 0.20/0.46 != B ) ).
% 0.20/0.46
% 0.20/0.46 % hpair_neq_snd
% 0.20/0.46 thf(fact_4_hsplit,axiom,
% 0.20/0.46 ! [A: $tType,C: hF_Mirabelle_hf > hF_Mirabelle_hf > A,A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hsplit @ A @ C @ ( hF_Mirabelle_hpair @ A2 @ B ) )
% 0.20/0.46 = ( C @ A2 @ B ) ) ).
% 0.20/0.46
% 0.20/0.46 % hsplit
% 0.20/0.46 thf(fact_5_hfst__conv,axiom,
% 0.20/0.46 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hfst @ ( hF_Mirabelle_hpair @ A2 @ B ) )
% 0.20/0.46 = A2 ) ).
% 0.20/0.46
% 0.20/0.46 % hfst_conv
% 0.20/0.46 thf(fact_6_hsnd__conv,axiom,
% 0.20/0.46 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hsnd @ ( hF_Mirabelle_hpair @ A2 @ B ) )
% 0.20/0.46 = B ) ).
% 0.20/0.46
% 0.20/0.46 % hsnd_conv
% 0.20/0.46 thf(fact_7_HInter__hempty,axiom,
% 0.20/0.46 ( ( hF_Mirabelle_HInter @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.20/0.46 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.20/0.46
% 0.20/0.46 % HInter_hempty
% 0.20/0.46 thf(fact_8_HCollect__hempty,axiom,
% 0.20/0.46 ! [P: hF_Mirabelle_hf > $o] :
% 0.20/0.46 ( ( hF_Mir1687042746ollect @ P @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.20/0.46 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.20/0.46
% 0.20/0.46 % HCollect_hempty
% 0.20/0.46 thf(fact_9_Replace__0,axiom,
% 0.20/0.46 ! [R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.20/0.46 ( ( hF_Mirabelle_Replace @ ( zero_zero @ hF_Mirabelle_hf ) @ R )
% 0.20/0.46 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.20/0.46
% 0.20/0.46 % Replace_0
% 0.20/0.46 thf(fact_10_HUnion__hempty,axiom,
% 0.20/0.46 ( ( hF_Mirabelle_HUnion @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.20/0.46 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.20/0.46
% 0.20/0.46 % HUnion_hempty
% 0.20/0.46 thf(fact_11_zero__reorient,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( zero @ A )
% 0.20/0.46 => ! [X: A] :
% 0.20/0.46 ( ( ( zero_zero @ A )
% 0.20/0.46 = X )
% 0.20/0.46 = ( X
% 0.20/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % zero_reorient
% 0.20/0.46 thf(fact_12_RepFun__0,axiom,
% 0.20/0.46 ! [F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_RepFun @ ( zero_zero @ hF_Mirabelle_hf ) @ F )
% 0.20/0.46 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.20/0.46
% 0.20/0.46 % RepFun_0
% 0.20/0.46 thf(fact_13_hpair__def,axiom,
% 0.20/0.46 ( hF_Mirabelle_hpair
% 0.20/0.46 = ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A4 @ ( zero_zero @ hF_Mirabelle_hf ) ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A4 @ ( hF_Mirabelle_hinsert @ B3 @ ( zero_zero @ hF_Mirabelle_hf ) ) ) @ ( zero_zero @ hF_Mirabelle_hf ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hpair_def
% 0.20/0.46 thf(fact_14_hpair__def_H,axiom,
% 0.20/0.46 ( hF_Mirabelle_hpair
% 0.20/0.46 = ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A4 @ ( hF_Mirabelle_hinsert @ A4 @ ( zero_zero @ hF_Mirabelle_hf ) ) ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A4 @ ( hF_Mirabelle_hinsert @ B3 @ ( zero_zero @ hF_Mirabelle_hf ) ) ) @ ( zero_zero @ hF_Mirabelle_hf ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hpair_def'
% 0.20/0.46 thf(fact_15_singleton__eq__iff,axiom,
% 0.20/0.46 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( ( hF_Mirabelle_hinsert @ A2 @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.20/0.46 = ( hF_Mirabelle_hinsert @ B @ ( zero_zero @ hF_Mirabelle_hf ) ) )
% 0.20/0.46 = ( A2 = B ) ) ).
% 0.20/0.46
% 0.20/0.46 % singleton_eq_iff
% 0.20/0.46 thf(fact_16_RepFun__hinsert,axiom,
% 0.20/0.46 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_RepFun @ ( hF_Mirabelle_hinsert @ A2 @ B ) @ F )
% 0.20/0.46 = ( hF_Mirabelle_hinsert @ ( F @ A2 ) @ ( hF_Mirabelle_RepFun @ B @ F ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % RepFun_hinsert
% 0.20/0.46 thf(fact_17_hinsert__commute,axiom,
% 0.20/0.46 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hinsert @ X @ ( hF_Mirabelle_hinsert @ Y @ Z ) )
% 0.20/0.46 = ( hF_Mirabelle_hinsert @ Y @ ( hF_Mirabelle_hinsert @ X @ Z ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hinsert_commute
% 0.20/0.46 thf(fact_18_hinsert__nonempty,axiom,
% 0.20/0.46 ! [A2: hF_Mirabelle_hf,A5: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hinsert @ A2 @ A5 )
% 0.20/0.46 != ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.20/0.46
% 0.20/0.46 % hinsert_nonempty
% 0.20/0.46 thf(fact_19_HF__Mirabelle__fsbjehakzm_Odoubleton__eq__iff,axiom,
% 0.20/0.46 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf,D: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( ( hF_Mirabelle_hinsert @ A2 @ ( hF_Mirabelle_hinsert @ B @ ( zero_zero @ hF_Mirabelle_hf ) ) )
% 0.20/0.46 = ( hF_Mirabelle_hinsert @ C @ ( hF_Mirabelle_hinsert @ D @ ( zero_zero @ hF_Mirabelle_hf ) ) ) )
% 0.20/0.46 = ( ( ( A2 = C )
% 0.20/0.46 & ( B = D ) )
% 0.20/0.46 | ( ( A2 = D )
% 0.20/0.46 & ( B = C ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % HF_Mirabelle_fsbjehakzm.doubleton_eq_iff
% 0.20/0.46 thf(fact_20_hf__induct__ax,axiom,
% 0.20/0.46 ! [P: hF_Mirabelle_hf > $o,X: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( P @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.20/0.46 => ( ! [X2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( P @ X2 )
% 0.20/0.46 => ! [Y2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( P @ Y2 )
% 0.20/0.46 => ( P @ ( hF_Mirabelle_hinsert @ Y2 @ X2 ) ) ) )
% 0.20/0.46 => ( P @ X ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hf_induct_ax
% 0.20/0.46 thf(fact_21_HInter__hinsert,axiom,
% 0.20/0.46 ! [A5: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( A5
% 0.20/0.46 != ( zero_zero @ hF_Mirabelle_hf ) )
% 0.20/0.46 => ( ( hF_Mirabelle_HInter @ ( hF_Mirabelle_hinsert @ A2 @ A5 ) )
% 0.20/0.46 = ( inf_inf @ hF_Mirabelle_hf @ A2 @ ( hF_Mirabelle_HInter @ A5 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % HInter_hinsert
% 0.20/0.46 thf(fact_22_Abs__hf__0,axiom,
% 0.20/0.46 ( ( hF_Mirabelle_Abs_hf @ ( zero_zero @ nat ) )
% 0.20/0.46 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.20/0.46
% 0.20/0.46 % Abs_hf_0
% 0.20/0.46 thf(fact_23_HInter__iff,axiom,
% 0.20/0.46 ! [A5: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( A5
% 0.20/0.46 != ( zero_zero @ hF_Mirabelle_hf ) )
% 0.20/0.46 => ( ( hF_Mirabelle_hmem @ X @ ( hF_Mirabelle_HInter @ A5 ) )
% 0.20/0.46 = ( ! [Y3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ Y3 @ A5 )
% 0.20/0.46 => ( hF_Mirabelle_hmem @ X @ Y3 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % HInter_iff
% 0.20/0.46 thf(fact_24_HUnion__hinsert,axiom,
% 0.20/0.46 ! [A2: hF_Mirabelle_hf,A5: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_HUnion @ ( hF_Mirabelle_hinsert @ A2 @ A5 ) )
% 0.20/0.46 = ( sup_sup @ hF_Mirabelle_hf @ A2 @ ( hF_Mirabelle_HUnion @ A5 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % HUnion_hinsert
% 0.20/0.46 thf(fact_25_hdiff__insert,axiom,
% 0.20/0.46 ! [A5: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( minus_minus @ hF_Mirabelle_hf @ A5 @ ( hF_Mirabelle_hinsert @ A2 @ B4 ) )
% 0.20/0.46 = ( minus_minus @ hF_Mirabelle_hf @ ( minus_minus @ hF_Mirabelle_hf @ A5 @ B4 ) @ ( hF_Mirabelle_hinsert @ A2 @ ( zero_zero @ hF_Mirabelle_hf ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hdiff_insert
% 0.20/0.46 thf(fact_26_hinsert__eq__sup,axiom,
% 0.20/0.46 ( hF_Mirabelle_hinsert
% 0.20/0.46 = ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( sup_sup @ hF_Mirabelle_hf @ B3 @ ( hF_Mirabelle_hinsert @ A4 @ ( zero_zero @ hF_Mirabelle_hf ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hinsert_eq_sup
% 0.20/0.46 thf(fact_27_hf__induct,axiom,
% 0.20/0.46 ! [P: hF_Mirabelle_hf > $o,Z: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( P @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.20/0.46 => ( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( P @ X2 )
% 0.20/0.46 => ( ( P @ Y2 )
% 0.20/0.46 => ( ~ ( hF_Mirabelle_hmem @ X2 @ Y2 )
% 0.20/0.46 => ( P @ ( hF_Mirabelle_hinsert @ X2 @ Y2 ) ) ) ) )
% 0.20/0.46 => ( P @ Z ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hf_induct
% 0.20/0.46 thf(fact_28_hf__cases,axiom,
% 0.20/0.46 ! [Y: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( Y
% 0.20/0.46 != ( zero_zero @ hF_Mirabelle_hf ) )
% 0.20/0.46 => ~ ! [A6: hF_Mirabelle_hf,B5: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( Y
% 0.20/0.46 = ( hF_Mirabelle_hinsert @ A6 @ B5 ) )
% 0.20/0.46 => ( hF_Mirabelle_hmem @ A6 @ B5 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hf_cases
% 0.20/0.46 thf(fact_29_Replace__hunion,axiom,
% 0.20/0.46 ! [A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.20/0.46 ( ( hF_Mirabelle_Replace @ ( sup_sup @ hF_Mirabelle_hf @ A5 @ B4 ) @ R )
% 0.20/0.46 = ( sup_sup @ hF_Mirabelle_hf @ ( hF_Mirabelle_Replace @ A5 @ R ) @ ( hF_Mirabelle_Replace @ B4 @ R ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Replace_hunion
% 0.20/0.46 thf(fact_30_HCollect__iff,axiom,
% 0.20/0.46 ! [X: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,A5: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ X @ ( hF_Mir1687042746ollect @ P @ A5 ) )
% 0.20/0.46 = ( ( P @ X )
% 0.20/0.46 & ( hF_Mirabelle_hmem @ X @ A5 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % HCollect_iff
% 0.20/0.46 thf(fact_31_hf__equalityI,axiom,
% 0.20/0.46 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.20/0.46 ( ! [X2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ X2 @ A2 )
% 0.20/0.46 = ( hF_Mirabelle_hmem @ X2 @ B ) )
% 0.20/0.46 => ( A2 = B ) ) ).
% 0.20/0.46
% 0.20/0.46 % hf_equalityI
% 0.20/0.46 thf(fact_32_diff__self,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( group_add @ A )
% 0.20/0.46 => ! [A2: A] :
% 0.20/0.46 ( ( minus_minus @ A @ A2 @ A2 )
% 0.20/0.46 = ( zero_zero @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % diff_self
% 0.20/0.46 thf(fact_33_diff__0__right,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( group_add @ A )
% 0.20/0.46 => ! [A2: A] :
% 0.20/0.46 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 0.20/0.46 = A2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % diff_0_right
% 0.20/0.46 thf(fact_34_zero__diff,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( comm_monoid_diff @ A )
% 0.20/0.46 => ! [A2: A] :
% 0.20/0.46 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
% 0.20/0.46 = ( zero_zero @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % zero_diff
% 0.20/0.46 thf(fact_35_diff__zero,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( cancel1352612707id_add @ A )
% 0.20/0.46 => ! [A2: A] :
% 0.20/0.46 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 0.20/0.46 = A2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % diff_zero
% 0.20/0.46 thf(fact_36_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( cancel1352612707id_add @ A )
% 0.20/0.46 => ! [A2: A] :
% 0.20/0.46 ( ( minus_minus @ A @ A2 @ A2 )
% 0.20/0.46 = ( zero_zero @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % cancel_comm_monoid_add_class.diff_cancel
% 0.20/0.46 thf(fact_37_hmem__hinsert,axiom,
% 0.20/0.46 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ A2 @ ( hF_Mirabelle_hinsert @ B @ C ) )
% 0.20/0.46 = ( ( A2 = B )
% 0.20/0.46 | ( hF_Mirabelle_hmem @ A2 @ C ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hmem_hinsert
% 0.20/0.46 thf(fact_38_hunion__iff,axiom,
% 0.20/0.46 ! [X: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ X @ ( sup_sup @ hF_Mirabelle_hf @ A2 @ B ) )
% 0.20/0.46 = ( ( hF_Mirabelle_hmem @ X @ A2 )
% 0.20/0.46 | ( hF_Mirabelle_hmem @ X @ B ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hunion_iff
% 0.20/0.46 thf(fact_39_hinter__iff,axiom,
% 0.20/0.46 ! [U: hF_Mirabelle_hf,X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ U @ ( inf_inf @ hF_Mirabelle_hf @ X @ Y ) )
% 0.20/0.46 = ( ( hF_Mirabelle_hmem @ U @ X )
% 0.20/0.46 & ( hF_Mirabelle_hmem @ U @ Y ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hinter_iff
% 0.20/0.46 thf(fact_40_RepFun__iff,axiom,
% 0.20/0.46 ! [V: hF_Mirabelle_hf,A5: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ V @ ( hF_Mirabelle_RepFun @ A5 @ F ) )
% 0.20/0.46 = ( ? [U2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ U2 @ A5 )
% 0.20/0.46 & ( V
% 0.20/0.46 = ( F @ U2 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % RepFun_iff
% 0.20/0.46 thf(fact_41_hunion__hempty__left,axiom,
% 0.20/0.46 ! [A5: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( sup_sup @ hF_Mirabelle_hf @ ( zero_zero @ hF_Mirabelle_hf ) @ A5 )
% 0.20/0.46 = A5 ) ).
% 0.20/0.46
% 0.20/0.46 % hunion_hempty_left
% 0.20/0.46 thf(fact_42_hunion__hempty__right,axiom,
% 0.20/0.46 ! [A5: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( sup_sup @ hF_Mirabelle_hf @ A5 @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.20/0.46 = A5 ) ).
% 0.20/0.46
% 0.20/0.46 % hunion_hempty_right
% 0.20/0.46 thf(fact_43_HUnion__iff,axiom,
% 0.20/0.46 ! [X: hF_Mirabelle_hf,A5: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ X @ ( hF_Mirabelle_HUnion @ A5 ) )
% 0.20/0.46 = ( ? [Y3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ Y3 @ A5 )
% 0.20/0.46 & ( hF_Mirabelle_hmem @ X @ Y3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % HUnion_iff
% 0.20/0.46 thf(fact_44_mem__Collect__eq,axiom,
% 0.20/0.46 ! [A: $tType,A2: A,P: A > $o] :
% 0.20/0.46 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.20/0.46 = ( P @ A2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % mem_Collect_eq
% 0.20/0.46 thf(fact_45_Collect__mem__eq,axiom,
% 0.20/0.46 ! [A: $tType,A5: set @ A] :
% 0.20/0.46 ( ( collect @ A
% 0.20/0.46 @ ^ [X3: A] : ( member @ A @ X3 @ A5 ) )
% 0.20/0.46 = A5 ) ).
% 0.20/0.46
% 0.20/0.46 % Collect_mem_eq
% 0.20/0.46 thf(fact_46_Collect__cong,axiom,
% 0.20/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.20/0.46 ( ! [X2: A] :
% 0.20/0.46 ( ( P @ X2 )
% 0.20/0.46 = ( Q @ X2 ) )
% 0.20/0.46 => ( ( collect @ A @ P )
% 0.20/0.46 = ( collect @ A @ Q ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Collect_cong
% 0.20/0.46 thf(fact_47_ext,axiom,
% 0.20/0.46 ! [B6: $tType,A: $tType,F: A > B6,G: A > B6] :
% 0.20/0.46 ( ! [X2: A] :
% 0.20/0.46 ( ( F @ X2 )
% 0.20/0.46 = ( G @ X2 ) )
% 0.20/0.46 => ( F = G ) ) ).
% 0.20/0.46
% 0.20/0.46 % ext
% 0.20/0.46 thf(fact_48_hdiff__iff,axiom,
% 0.20/0.46 ! [U: hF_Mirabelle_hf,X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ U @ ( minus_minus @ hF_Mirabelle_hf @ X @ Y ) )
% 0.20/0.46 = ( ( hF_Mirabelle_hmem @ U @ X )
% 0.20/0.46 & ~ ( hF_Mirabelle_hmem @ U @ Y ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hdiff_iff
% 0.20/0.46 thf(fact_49_hinter__hempty__left,axiom,
% 0.20/0.46 ! [A5: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( inf_inf @ hF_Mirabelle_hf @ ( zero_zero @ hF_Mirabelle_hf ) @ A5 )
% 0.20/0.46 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.20/0.46
% 0.20/0.46 % hinter_hempty_left
% 0.20/0.46 thf(fact_50_hinter__hempty__right,axiom,
% 0.20/0.46 ! [A5: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( inf_inf @ hF_Mirabelle_hf @ A5 @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.20/0.46 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.20/0.46
% 0.20/0.46 % hinter_hempty_right
% 0.20/0.46 thf(fact_51_Replace__iff,axiom,
% 0.20/0.46 ! [V: hF_Mirabelle_hf,A5: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ V @ ( hF_Mirabelle_Replace @ A5 @ R ) )
% 0.20/0.46 = ( ? [U2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ U2 @ A5 )
% 0.20/0.46 & ( R @ U2 @ V )
% 0.20/0.46 & ! [Y3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( R @ U2 @ Y3 )
% 0.20/0.46 => ( Y3 = V ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Replace_iff
% 0.20/0.46 thf(fact_52_hdiff__zero,axiom,
% 0.20/0.46 ! [X: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( minus_minus @ hF_Mirabelle_hf @ X @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.20/0.46 = X ) ).
% 0.20/0.46
% 0.20/0.46 % hdiff_zero
% 0.20/0.46 thf(fact_53_zero__hdiff,axiom,
% 0.20/0.46 ! [X: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( minus_minus @ hF_Mirabelle_hf @ ( zero_zero @ hF_Mirabelle_hf ) @ X )
% 0.20/0.46 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.20/0.46
% 0.20/0.46 % zero_hdiff
% 0.20/0.46 thf(fact_54_RepFun__hunion,axiom,
% 0.20/0.46 ! [A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_RepFun @ ( sup_sup @ hF_Mirabelle_hf @ A5 @ B4 ) @ F )
% 0.20/0.46 = ( sup_sup @ hF_Mirabelle_hf @ ( hF_Mirabelle_RepFun @ A5 @ F ) @ ( hF_Mirabelle_RepFun @ B4 @ F ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % RepFun_hunion
% 0.20/0.46 thf(fact_55_HUnion__hunion,axiom,
% 0.20/0.46 ! [A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_HUnion @ ( sup_sup @ hF_Mirabelle_hf @ A5 @ B4 ) )
% 0.20/0.46 = ( sup_sup @ hF_Mirabelle_hf @ ( hF_Mirabelle_HUnion @ A5 ) @ ( hF_Mirabelle_HUnion @ B4 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % HUnion_hunion
% 0.20/0.46 thf(fact_56_diff__eq__diff__eq,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( group_add @ A )
% 0.20/0.46 => ! [A2: A,B: A,C: A,D: A] :
% 0.20/0.46 ( ( ( minus_minus @ A @ A2 @ B )
% 0.20/0.46 = ( minus_minus @ A @ C @ D ) )
% 0.20/0.46 => ( ( A2 = B )
% 0.20/0.46 = ( C = D ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % diff_eq_diff_eq
% 0.20/0.46 thf(fact_57_diff__right__commute,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( cancel146912293up_add @ A )
% 0.20/0.46 => ! [A2: A,C: A,B: A] :
% 0.20/0.46 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B )
% 0.20/0.46 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B ) @ C ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % diff_right_commute
% 0.20/0.46 thf(fact_58_replacement__fun,axiom,
% 0.20/0.46 ! [X: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.20/0.46 ? [Z2: hF_Mirabelle_hf] :
% 0.20/0.46 ! [V2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ V2 @ Z2 )
% 0.20/0.46 = ( ? [U2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ U2 @ X )
% 0.20/0.46 & ( V2
% 0.20/0.46 = ( F @ U2 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % replacement_fun
% 0.20/0.46 thf(fact_59_hmem__not__refl,axiom,
% 0.20/0.46 ! [X: hF_Mirabelle_hf] :
% 0.20/0.46 ~ ( hF_Mirabelle_hmem @ X @ X ) ).
% 0.20/0.46
% 0.20/0.46 % hmem_not_refl
% 0.20/0.46 thf(fact_60_comprehension,axiom,
% 0.20/0.46 ! [X: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 0.20/0.46 ? [Z2: hF_Mirabelle_hf] :
% 0.20/0.46 ! [U3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ U3 @ Z2 )
% 0.20/0.46 = ( ( hF_Mirabelle_hmem @ U3 @ X )
% 0.20/0.46 & ( P @ U3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % comprehension
% 0.20/0.46 thf(fact_61_union__of__set,axiom,
% 0.20/0.46 ! [X: hF_Mirabelle_hf] :
% 0.20/0.46 ? [Z2: hF_Mirabelle_hf] :
% 0.20/0.46 ! [U3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ U3 @ Z2 )
% 0.20/0.46 = ( ? [Y3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ Y3 @ X )
% 0.20/0.46 & ( hF_Mirabelle_hmem @ U3 @ Y3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % union_of_set
% 0.20/0.46 thf(fact_62_hmem__not__sym,axiom,
% 0.20/0.46 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.20/0.46 ~ ( ( hF_Mirabelle_hmem @ X @ Y )
% 0.20/0.46 & ( hF_Mirabelle_hmem @ Y @ X ) ) ).
% 0.20/0.46
% 0.20/0.46 % hmem_not_sym
% 0.20/0.46 thf(fact_63_binary__union,axiom,
% 0.20/0.46 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.20/0.46 ? [Z2: hF_Mirabelle_hf] :
% 0.20/0.46 ! [U3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ U3 @ Z2 )
% 0.20/0.46 = ( ( hF_Mirabelle_hmem @ U3 @ X )
% 0.20/0.46 | ( hF_Mirabelle_hmem @ U3 @ Y ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % binary_union
% 0.20/0.46 thf(fact_64_replacement,axiom,
% 0.20/0.46 ! [X: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.20/0.46 ( ! [U4: hF_Mirabelle_hf,V3: hF_Mirabelle_hf,V4: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ U4 @ X )
% 0.20/0.46 => ( ( R @ U4 @ V3 )
% 0.20/0.46 => ( ( R @ U4 @ V4 )
% 0.20/0.46 => ( V4 = V3 ) ) ) )
% 0.20/0.46 => ? [Z2: hF_Mirabelle_hf] :
% 0.20/0.46 ! [V2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ V2 @ Z2 )
% 0.20/0.46 = ( ? [U2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ U2 @ X )
% 0.20/0.46 & ( R @ U2 @ V2 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % replacement
% 0.20/0.46 thf(fact_65_hmem__ne,axiom,
% 0.20/0.46 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ X @ Y )
% 0.20/0.46 => ( X != Y ) ) ).
% 0.20/0.46
% 0.20/0.46 % hmem_ne
% 0.20/0.46 thf(fact_66_hf__ext,axiom,
% 0.20/0.46 ( ( ^ [Y4: hF_Mirabelle_hf,Z3: hF_Mirabelle_hf] : ( Y4 = Z3 ) )
% 0.20/0.46 = ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 0.20/0.46 ! [X3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ X3 @ A4 )
% 0.20/0.46 = ( hF_Mirabelle_hmem @ X3 @ B3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hf_ext
% 0.20/0.46 thf(fact_67_hinter__hinsert__right,axiom,
% 0.20/0.46 ! [X: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,A5: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( ( hF_Mirabelle_hmem @ X @ B4 )
% 0.20/0.46 => ( ( inf_inf @ hF_Mirabelle_hf @ B4 @ ( hF_Mirabelle_hinsert @ X @ A5 ) )
% 0.20/0.46 = ( hF_Mirabelle_hinsert @ X @ ( inf_inf @ hF_Mirabelle_hf @ B4 @ A5 ) ) ) )
% 0.20/0.46 & ( ~ ( hF_Mirabelle_hmem @ X @ B4 )
% 0.20/0.46 => ( ( inf_inf @ hF_Mirabelle_hf @ B4 @ ( hF_Mirabelle_hinsert @ X @ A5 ) )
% 0.20/0.46 = ( inf_inf @ hF_Mirabelle_hf @ B4 @ A5 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hinter_hinsert_right
% 0.20/0.46 thf(fact_68_hinter__hinsert__left,axiom,
% 0.20/0.46 ! [X: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,A5: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( ( hF_Mirabelle_hmem @ X @ B4 )
% 0.20/0.46 => ( ( inf_inf @ hF_Mirabelle_hf @ ( hF_Mirabelle_hinsert @ X @ A5 ) @ B4 )
% 0.20/0.46 = ( hF_Mirabelle_hinsert @ X @ ( inf_inf @ hF_Mirabelle_hf @ A5 @ B4 ) ) ) )
% 0.20/0.46 & ( ~ ( hF_Mirabelle_hmem @ X @ B4 )
% 0.20/0.46 => ( ( inf_inf @ hF_Mirabelle_hf @ ( hF_Mirabelle_hinsert @ X @ A5 ) @ B4 )
% 0.20/0.46 = ( inf_inf @ hF_Mirabelle_hf @ A5 @ B4 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hinter_hinsert_left
% 0.20/0.46 thf(fact_69_foundation,axiom,
% 0.20/0.46 ! [Z: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( Z
% 0.20/0.46 != ( zero_zero @ hF_Mirabelle_hf ) )
% 0.20/0.46 => ? [W: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ W @ Z )
% 0.20/0.46 & ( ( inf_inf @ hF_Mirabelle_hf @ W @ Z )
% 0.20/0.46 = ( zero_zero @ hF_Mirabelle_hf ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % foundation
% 0.20/0.46 thf(fact_70_hinsert__hdiff__if,axiom,
% 0.20/0.46 ! [X: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,A5: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( ( hF_Mirabelle_hmem @ X @ B4 )
% 0.20/0.46 => ( ( minus_minus @ hF_Mirabelle_hf @ ( hF_Mirabelle_hinsert @ X @ A5 ) @ B4 )
% 0.20/0.46 = ( minus_minus @ hF_Mirabelle_hf @ A5 @ B4 ) ) )
% 0.20/0.46 & ( ~ ( hF_Mirabelle_hmem @ X @ B4 )
% 0.20/0.46 => ( ( minus_minus @ hF_Mirabelle_hf @ ( hF_Mirabelle_hinsert @ X @ A5 ) @ B4 )
% 0.20/0.46 = ( hF_Mirabelle_hinsert @ X @ ( minus_minus @ hF_Mirabelle_hf @ A5 @ B4 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hinsert_hdiff_if
% 0.20/0.46 thf(fact_71_eq__iff__diff__eq__0,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( group_add @ A )
% 0.20/0.46 => ( ( ^ [Y4: A,Z3: A] : ( Y4 = Z3 ) )
% 0.20/0.46 = ( ^ [A4: A,B3: A] :
% 0.20/0.46 ( ( minus_minus @ A @ A4 @ B3 )
% 0.20/0.46 = ( zero_zero @ A ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % eq_iff_diff_eq_0
% 0.20/0.46 thf(fact_72_hinsert__iff,axiom,
% 0.20/0.46 ! [Z: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( Z
% 0.20/0.46 = ( hF_Mirabelle_hinsert @ Y @ X ) )
% 0.20/0.46 = ( ! [U2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ U2 @ Z )
% 0.20/0.46 = ( ( hF_Mirabelle_hmem @ U2 @ X )
% 0.20/0.46 | ( U2 = Y ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hinsert_iff
% 0.20/0.46 thf(fact_73_hemptyE,axiom,
% 0.20/0.46 ! [A2: hF_Mirabelle_hf] :
% 0.20/0.46 ~ ( hF_Mirabelle_hmem @ A2 @ ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.20/0.46
% 0.20/0.46 % hemptyE
% 0.20/0.46 thf(fact_74_hempty__iff,axiom,
% 0.20/0.46 ! [Z: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( Z
% 0.20/0.46 = ( zero_zero @ hF_Mirabelle_hf ) )
% 0.20/0.46 = ( ! [X3: hF_Mirabelle_hf] :
% 0.20/0.46 ~ ( hF_Mirabelle_hmem @ X3 @ Z ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hempty_iff
% 0.20/0.46 thf(fact_75_hmem__hempty,axiom,
% 0.20/0.46 ! [A2: hF_Mirabelle_hf] :
% 0.20/0.46 ~ ( hF_Mirabelle_hmem @ A2 @ ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.20/0.46
% 0.20/0.46 % hmem_hempty
% 0.20/0.46 thf(fact_76_hunion__hinsert__right,axiom,
% 0.20/0.46 ! [B4: hF_Mirabelle_hf,X: hF_Mirabelle_hf,A5: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( sup_sup @ hF_Mirabelle_hf @ B4 @ ( hF_Mirabelle_hinsert @ X @ A5 ) )
% 0.20/0.46 = ( hF_Mirabelle_hinsert @ X @ ( sup_sup @ hF_Mirabelle_hf @ B4 @ A5 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hunion_hinsert_right
% 0.20/0.46 thf(fact_77_hunion__hinsert__left,axiom,
% 0.20/0.46 ! [X: hF_Mirabelle_hf,A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( sup_sup @ hF_Mirabelle_hf @ ( hF_Mirabelle_hinsert @ X @ A5 ) @ B4 )
% 0.20/0.46 = ( hF_Mirabelle_hinsert @ X @ ( sup_sup @ hF_Mirabelle_hf @ A5 @ B4 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hunion_hinsert_left
% 0.20/0.46 thf(fact_78_RepFun__cong,axiom,
% 0.20/0.46 ! [A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,G: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.20/0.46 ( ( A5 = B4 )
% 0.20/0.46 => ( ! [X2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ X2 @ B4 )
% 0.20/0.46 => ( ( F @ X2 )
% 0.20/0.46 = ( G @ X2 ) ) )
% 0.20/0.46 => ( ( hF_Mirabelle_RepFun @ A5 @ F )
% 0.20/0.46 = ( hF_Mirabelle_RepFun @ B4 @ G ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % RepFun_cong
% 0.20/0.46 thf(fact_79_hmem__Sup__ne,axiom,
% 0.20/0.46 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ X @ Y )
% 0.20/0.46 => ( ( hF_Mirabelle_HUnion @ X )
% 0.20/0.46 != Y ) ) ).
% 0.20/0.46
% 0.20/0.46 % hmem_Sup_ne
% 0.20/0.46 thf(fact_80_Replace__cong,axiom,
% 0.20/0.46 ! [A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,P: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.20/0.46 ( ( A5 = B4 )
% 0.20/0.46 => ( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ X2 @ B4 )
% 0.20/0.46 => ( ( P @ X2 @ Y2 )
% 0.20/0.46 = ( Q @ X2 @ Y2 ) ) )
% 0.20/0.46 => ( ( hF_Mirabelle_Replace @ A5 @ P )
% 0.20/0.46 = ( hF_Mirabelle_Replace @ B4 @ Q ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Replace_cong
% 0.20/0.46 thf(fact_81_sup__inf__absorb,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( lattice @ A )
% 0.20/0.46 => ! [X: A,Y: A] :
% 0.20/0.46 ( ( sup_sup @ A @ X @ ( inf_inf @ A @ X @ Y ) )
% 0.20/0.46 = X ) ) ).
% 0.20/0.46
% 0.20/0.46 % sup_inf_absorb
% 0.20/0.46 thf(fact_82_inf__sup__absorb,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( lattice @ A )
% 0.20/0.46 => ! [X: A,Y: A] :
% 0.20/0.46 ( ( inf_inf @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 0.20/0.46 = X ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_sup_absorb
% 0.20/0.46 thf(fact_83_sup_Oright__idem,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_sup @ A )
% 0.20/0.46 => ! [A2: A,B: A] :
% 0.20/0.46 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B ) @ B )
% 0.20/0.46 = ( sup_sup @ A @ A2 @ B ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sup.right_idem
% 0.20/0.46 thf(fact_84_sup__left__idem,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_sup @ A )
% 0.20/0.46 => ! [X: A,Y: A] :
% 0.20/0.46 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 0.20/0.46 = ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sup_left_idem
% 0.20/0.46 thf(fact_85_sup_Oleft__idem,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_sup @ A )
% 0.20/0.46 => ! [A2: A,B: A] :
% 0.20/0.46 ( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B ) )
% 0.20/0.46 = ( sup_sup @ A @ A2 @ B ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sup.left_idem
% 0.20/0.46 thf(fact_86_sup__idem,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_sup @ A )
% 0.20/0.46 => ! [X: A] :
% 0.20/0.46 ( ( sup_sup @ A @ X @ X )
% 0.20/0.46 = X ) ) ).
% 0.20/0.46
% 0.20/0.46 % sup_idem
% 0.20/0.46 thf(fact_87_sup_Oidem,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_sup @ A )
% 0.20/0.46 => ! [A2: A] :
% 0.20/0.46 ( ( sup_sup @ A @ A2 @ A2 )
% 0.20/0.46 = A2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % sup.idem
% 0.20/0.46 thf(fact_88_sup__apply,axiom,
% 0.20/0.46 ! [B6: $tType,A: $tType] :
% 0.20/0.46 ( ( semilattice_sup @ B6 )
% 0.20/0.46 => ( ( sup_sup @ ( A > B6 ) )
% 0.20/0.46 = ( ^ [F2: A > B6,G2: A > B6,X3: A] : ( sup_sup @ B6 @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sup_apply
% 0.20/0.46 thf(fact_89_inf__right__idem,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ! [X: A,Y: A] :
% 0.20/0.46 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Y )
% 0.20/0.46 = ( inf_inf @ A @ X @ Y ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_right_idem
% 0.20/0.46 thf(fact_90_minus__apply,axiom,
% 0.20/0.46 ! [B6: $tType,A: $tType] :
% 0.20/0.46 ( ( minus @ B6 )
% 0.20/0.46 => ( ( minus_minus @ ( A > B6 ) )
% 0.20/0.46 = ( ^ [A7: A > B6,B7: A > B6,X3: A] : ( minus_minus @ B6 @ ( A7 @ X3 ) @ ( B7 @ X3 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % minus_apply
% 0.20/0.46 thf(fact_91_inf__apply,axiom,
% 0.20/0.46 ! [B6: $tType,A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ B6 )
% 0.20/0.46 => ( ( inf_inf @ ( A > B6 ) )
% 0.20/0.46 = ( ^ [F2: A > B6,G2: A > B6,X3: A] : ( inf_inf @ B6 @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_apply
% 0.20/0.46 thf(fact_92_inf_Oidem,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ! [A2: A] :
% 0.20/0.46 ( ( inf_inf @ A @ A2 @ A2 )
% 0.20/0.46 = A2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf.idem
% 0.20/0.46 thf(fact_93_inf__idem,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ! [X: A] :
% 0.20/0.46 ( ( inf_inf @ A @ X @ X )
% 0.20/0.46 = X ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_idem
% 0.20/0.46 thf(fact_94_inf_Oleft__idem,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ! [A2: A,B: A] :
% 0.20/0.46 ( ( inf_inf @ A @ A2 @ ( inf_inf @ A @ A2 @ B ) )
% 0.20/0.46 = ( inf_inf @ A @ A2 @ B ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf.left_idem
% 0.20/0.46 thf(fact_95_inf__left__idem,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ! [X: A,Y: A] :
% 0.20/0.46 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y ) )
% 0.20/0.46 = ( inf_inf @ A @ X @ Y ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_left_idem
% 0.20/0.46 thf(fact_96_inf_Oright__idem,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ! [A2: A,B: A] :
% 0.20/0.46 ( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B ) @ B )
% 0.20/0.46 = ( inf_inf @ A @ A2 @ B ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf.right_idem
% 0.20/0.46 thf(fact_97_fun__diff__def,axiom,
% 0.20/0.46 ! [B6: $tType,A: $tType] :
% 0.20/0.46 ( ( minus @ B6 )
% 0.20/0.46 => ( ( minus_minus @ ( A > B6 ) )
% 0.20/0.46 = ( ^ [A7: A > B6,B7: A > B6,X3: A] : ( minus_minus @ B6 @ ( A7 @ X3 ) @ ( B7 @ X3 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % fun_diff_def
% 0.20/0.46 thf(fact_98_inf__sup__aci_I4_J,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( lattice @ A )
% 0.20/0.46 => ! [X: A,Y: A] :
% 0.20/0.46 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y ) )
% 0.20/0.46 = ( inf_inf @ A @ X @ Y ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_sup_aci(4)
% 0.20/0.46 thf(fact_99_inf__sup__aci_I3_J,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( lattice @ A )
% 0.20/0.46 => ! [X: A,Y: A,Z: A] :
% 0.20/0.46 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
% 0.20/0.46 = ( inf_inf @ A @ Y @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_sup_aci(3)
% 0.20/0.46 thf(fact_100_inf__sup__aci_I2_J,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( lattice @ A )
% 0.20/0.46 => ! [X: A,Y: A,Z: A] :
% 0.20/0.46 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Z )
% 0.20/0.46 = ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_sup_aci(2)
% 0.20/0.46 thf(fact_101_inf__sup__aci_I1_J,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( lattice @ A )
% 0.20/0.46 => ( ( inf_inf @ A )
% 0.20/0.46 = ( ^ [X3: A,Y3: A] : ( inf_inf @ A @ Y3 @ X3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_sup_aci(1)
% 0.20/0.46 thf(fact_102_inf__fun__def,axiom,
% 0.20/0.46 ! [B6: $tType,A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ B6 )
% 0.20/0.46 => ( ( inf_inf @ ( A > B6 ) )
% 0.20/0.46 = ( ^ [F2: A > B6,G2: A > B6,X3: A] : ( inf_inf @ B6 @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_fun_def
% 0.20/0.46 thf(fact_103_boolean__algebra__cancel_Oinf1,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ! [A5: A,K: A,A2: A,B: A] :
% 0.20/0.46 ( ( A5
% 0.20/0.46 = ( inf_inf @ A @ K @ A2 ) )
% 0.20/0.46 => ( ( inf_inf @ A @ A5 @ B )
% 0.20/0.46 = ( inf_inf @ A @ K @ ( inf_inf @ A @ A2 @ B ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % boolean_algebra_cancel.inf1
% 0.20/0.46 thf(fact_104_boolean__algebra__cancel_Oinf2,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ! [B4: A,K: A,B: A,A2: A] :
% 0.20/0.46 ( ( B4
% 0.20/0.46 = ( inf_inf @ A @ K @ B ) )
% 0.20/0.46 => ( ( inf_inf @ A @ A2 @ B4 )
% 0.20/0.46 = ( inf_inf @ A @ K @ ( inf_inf @ A @ A2 @ B ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % boolean_algebra_cancel.inf2
% 0.20/0.46 thf(fact_105_inf_Oassoc,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ! [A2: A,B: A,C: A] :
% 0.20/0.46 ( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B ) @ C )
% 0.20/0.46 = ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B @ C ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf.assoc
% 0.20/0.46 thf(fact_106_inf__assoc,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ! [X: A,Y: A,Z: A] :
% 0.20/0.46 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Z )
% 0.20/0.46 = ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_assoc
% 0.20/0.46 thf(fact_107_inf_Ocommute,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ( ( inf_inf @ A )
% 0.20/0.46 = ( ^ [A4: A,B3: A] : ( inf_inf @ A @ B3 @ A4 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf.commute
% 0.20/0.46 thf(fact_108_inf__commute,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ( ( inf_inf @ A )
% 0.20/0.46 = ( ^ [X3: A,Y3: A] : ( inf_inf @ A @ Y3 @ X3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_commute
% 0.20/0.46 thf(fact_109_inf_Oleft__commute,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ! [B: A,A2: A,C: A] :
% 0.20/0.46 ( ( inf_inf @ A @ B @ ( inf_inf @ A @ A2 @ C ) )
% 0.20/0.46 = ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B @ C ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf.left_commute
% 0.20/0.46 thf(fact_110_inf__left__commute,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ! [X: A,Y: A,Z: A] :
% 0.20/0.46 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
% 0.20/0.46 = ( inf_inf @ A @ Y @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_left_commute
% 0.20/0.46 thf(fact_111_inf__sup__aci_I8_J,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( lattice @ A )
% 0.20/0.46 => ! [X: A,Y: A] :
% 0.20/0.46 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 0.20/0.46 = ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_sup_aci(8)
% 0.20/0.46 thf(fact_112_inf__sup__aci_I7_J,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( lattice @ A )
% 0.20/0.46 => ! [X: A,Y: A,Z: A] :
% 0.20/0.46 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
% 0.20/0.46 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_sup_aci(7)
% 0.20/0.46 thf(fact_113_inf__sup__aci_I6_J,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( lattice @ A )
% 0.20/0.46 => ! [X: A,Y: A,Z: A] :
% 0.20/0.46 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
% 0.20/0.46 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_sup_aci(6)
% 0.20/0.46 thf(fact_114_inf__sup__aci_I5_J,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( lattice @ A )
% 0.20/0.46 => ( ( sup_sup @ A )
% 0.20/0.46 = ( ^ [X3: A,Y3: A] : ( sup_sup @ A @ Y3 @ X3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_sup_aci(5)
% 0.20/0.46 thf(fact_115_sup__fun__def,axiom,
% 0.20/0.46 ! [B6: $tType,A: $tType] :
% 0.20/0.46 ( ( semilattice_sup @ B6 )
% 0.20/0.46 => ( ( sup_sup @ ( A > B6 ) )
% 0.20/0.46 = ( ^ [F2: A > B6,G2: A > B6,X3: A] : ( sup_sup @ B6 @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sup_fun_def
% 0.20/0.46 thf(fact_116_boolean__algebra__cancel_Osup1,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_sup @ A )
% 0.20/0.46 => ! [A5: A,K: A,A2: A,B: A] :
% 0.20/0.46 ( ( A5
% 0.20/0.46 = ( sup_sup @ A @ K @ A2 ) )
% 0.20/0.46 => ( ( sup_sup @ A @ A5 @ B )
% 0.20/0.46 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A2 @ B ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % boolean_algebra_cancel.sup1
% 0.20/0.46 thf(fact_117_boolean__algebra__cancel_Osup2,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_sup @ A )
% 0.20/0.46 => ! [B4: A,K: A,B: A,A2: A] :
% 0.20/0.46 ( ( B4
% 0.20/0.46 = ( sup_sup @ A @ K @ B ) )
% 0.20/0.46 => ( ( sup_sup @ A @ A2 @ B4 )
% 0.20/0.46 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A2 @ B ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % boolean_algebra_cancel.sup2
% 0.20/0.46 thf(fact_118_sup_Oassoc,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_sup @ A )
% 0.20/0.46 => ! [A2: A,B: A,C: A] :
% 0.20/0.46 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B ) @ C )
% 0.20/0.46 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B @ C ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sup.assoc
% 0.20/0.46 thf(fact_119_sup__assoc,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_sup @ A )
% 0.20/0.46 => ! [X: A,Y: A,Z: A] :
% 0.20/0.46 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
% 0.20/0.46 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sup_assoc
% 0.20/0.46 thf(fact_120_sup_Ocommute,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_sup @ A )
% 0.20/0.46 => ( ( sup_sup @ A )
% 0.20/0.46 = ( ^ [A4: A,B3: A] : ( sup_sup @ A @ B3 @ A4 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sup.commute
% 0.20/0.46 thf(fact_121_sup__commute,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_sup @ A )
% 0.20/0.46 => ( ( sup_sup @ A )
% 0.20/0.46 = ( ^ [X3: A,Y3: A] : ( sup_sup @ A @ Y3 @ X3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sup_commute
% 0.20/0.46 thf(fact_122_sup_Oleft__commute,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_sup @ A )
% 0.20/0.46 => ! [B: A,A2: A,C: A] :
% 0.20/0.46 ( ( sup_sup @ A @ B @ ( sup_sup @ A @ A2 @ C ) )
% 0.20/0.46 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B @ C ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sup.left_commute
% 0.20/0.46 thf(fact_123_sup__left__commute,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_sup @ A )
% 0.20/0.46 => ! [X: A,Y: A,Z: A] :
% 0.20/0.46 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
% 0.20/0.46 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sup_left_commute
% 0.20/0.46 thf(fact_124_distrib__imp1,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( lattice @ A )
% 0.20/0.46 => ! [X: A,Y: A,Z: A] :
% 0.20/0.46 ( ! [X2: A,Y2: A,Z2: A] :
% 0.20/0.46 ( ( inf_inf @ A @ X2 @ ( sup_sup @ A @ Y2 @ Z2 ) )
% 0.20/0.46 = ( sup_sup @ A @ ( inf_inf @ A @ X2 @ Y2 ) @ ( inf_inf @ A @ X2 @ Z2 ) ) )
% 0.20/0.46 => ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
% 0.20/0.46 = ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % distrib_imp1
% 0.20/0.46 thf(fact_125_distrib__imp2,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( lattice @ A )
% 0.20/0.46 => ! [X: A,Y: A,Z: A] :
% 0.20/0.46 ( ! [X2: A,Y2: A,Z2: A] :
% 0.20/0.46 ( ( sup_sup @ A @ X2 @ ( inf_inf @ A @ Y2 @ Z2 ) )
% 0.20/0.46 = ( inf_inf @ A @ ( sup_sup @ A @ X2 @ Y2 ) @ ( sup_sup @ A @ X2 @ Z2 ) ) )
% 0.20/0.46 => ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
% 0.20/0.46 = ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % distrib_imp2
% 0.20/0.46 thf(fact_126_inf__sup__distrib1,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( distrib_lattice @ A )
% 0.20/0.46 => ! [X: A,Y: A,Z: A] :
% 0.20/0.46 ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
% 0.20/0.46 = ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_sup_distrib1
% 0.20/0.46 thf(fact_127_inf__sup__distrib2,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( distrib_lattice @ A )
% 0.20/0.46 => ! [Y: A,Z: A,X: A] :
% 0.20/0.46 ( ( inf_inf @ A @ ( sup_sup @ A @ Y @ Z ) @ X )
% 0.20/0.46 = ( sup_sup @ A @ ( inf_inf @ A @ Y @ X ) @ ( inf_inf @ A @ Z @ X ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_sup_distrib2
% 0.20/0.46 thf(fact_128_sup__inf__distrib1,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( distrib_lattice @ A )
% 0.20/0.46 => ! [X: A,Y: A,Z: A] :
% 0.20/0.46 ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
% 0.20/0.46 = ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sup_inf_distrib1
% 0.20/0.46 thf(fact_129_sup__inf__distrib2,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( distrib_lattice @ A )
% 0.20/0.46 => ! [Y: A,Z: A,X: A] :
% 0.20/0.46 ( ( sup_sup @ A @ ( inf_inf @ A @ Y @ Z ) @ X )
% 0.20/0.46 = ( inf_inf @ A @ ( sup_sup @ A @ Y @ X ) @ ( sup_sup @ A @ Z @ X ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % sup_inf_distrib2
% 0.20/0.46 thf(fact_130_diff__self__eq__0,axiom,
% 0.20/0.46 ! [M: nat] :
% 0.20/0.46 ( ( minus_minus @ nat @ M @ M )
% 0.20/0.46 = ( zero_zero @ nat ) ) ).
% 0.20/0.46
% 0.20/0.46 % diff_self_eq_0
% 0.20/0.46 thf(fact_131_diff__0__eq__0,axiom,
% 0.20/0.46 ! [N: nat] :
% 0.20/0.46 ( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
% 0.20/0.46 = ( zero_zero @ nat ) ) ).
% 0.20/0.46
% 0.20/0.46 % diff_0_eq_0
% 0.20/0.46 thf(fact_132_PrimReplace__iff,axiom,
% 0.20/0.46 ! [A5: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,V: hF_Mirabelle_hf] :
% 0.20/0.46 ( ! [U4: hF_Mirabelle_hf,V3: hF_Mirabelle_hf,V4: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ U4 @ A5 )
% 0.20/0.46 => ( ( R @ U4 @ V3 )
% 0.20/0.46 => ( ( R @ U4 @ V4 )
% 0.20/0.46 => ( V4 = V3 ) ) ) )
% 0.20/0.46 => ( ( hF_Mirabelle_hmem @ V @ ( hF_Mir569462966eplace @ A5 @ R ) )
% 0.20/0.46 = ( ? [U2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ U2 @ A5 )
% 0.20/0.46 & ( R @ U2 @ V ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % PrimReplace_iff
% 0.20/0.46 thf(fact_133_diffs0__imp__equal,axiom,
% 0.20/0.46 ! [M: nat,N: nat] :
% 0.20/0.46 ( ( ( minus_minus @ nat @ M @ N )
% 0.20/0.46 = ( zero_zero @ nat ) )
% 0.20/0.46 => ( ( ( minus_minus @ nat @ N @ M )
% 0.20/0.46 = ( zero_zero @ nat ) )
% 0.20/0.46 => ( M = N ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % diffs0_imp_equal
% 0.20/0.46 thf(fact_134_minus__nat_Odiff__0,axiom,
% 0.20/0.46 ! [M: nat] :
% 0.20/0.46 ( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
% 0.32/0.46 = M ) ).
% 0.32/0.46
% 0.32/0.46 % minus_nat.diff_0
% 0.32/0.46 thf(fact_135_zero__natural_Orsp,axiom,
% 0.32/0.46 ( ( zero_zero @ nat )
% 0.32/0.46 = ( zero_zero @ nat ) ) ).
% 0.32/0.46
% 0.32/0.46 % zero_natural.rsp
% 0.32/0.46 thf(fact_136_diff__commute,axiom,
% 0.32/0.46 ! [I: nat,J: nat,K: nat] :
% 0.32/0.46 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
% 0.32/0.46 = ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K ) @ J ) ) ).
% 0.32/0.46
% 0.32/0.46 % diff_commute
% 0.32/0.46 thf(fact_137_of__nat__aux_Osimps_I1_J,axiom,
% 0.32/0.46 ! [A: $tType] :
% 0.32/0.46 ( ( semiring_1 @ A )
% 0.32/0.46 => ! [Inc: A > A,I: A] :
% 0.32/0.46 ( ( semiri532925092at_aux @ A @ Inc @ ( zero_zero @ nat ) @ I )
% 0.32/0.46 = I ) ) ).
% 0.32/0.46
% 0.32/0.46 % of_nat_aux.simps(1)
% 0.32/0.46 thf(fact_138_less__eq__insert1__iff,axiom,
% 0.32/0.46 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.32/0.46 ( ( ord_less_eq @ hF_Mirabelle_hf @ ( hF_Mirabelle_hinsert @ X @ Y ) @ Z )
% 0.32/0.46 = ( ( hF_Mirabelle_hmem @ X @ Z )
% 0.32/0.46 & ( ord_less_eq @ hF_Mirabelle_hf @ Y @ Z ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % less_eq_insert1_iff
% 0.32/0.46 thf(fact_139_hmem__def,axiom,
% 0.32/0.46 ( hF_Mirabelle_hmem
% 0.32/0.46 = ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( member @ hF_Mirabelle_hf @ A4 @ ( hF_Mirabelle_hfset @ B3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % hmem_def
% 0.32/0.47 thf(fact_140_diff__is__0__eq,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ( minus_minus @ nat @ M @ N )
% 0.32/0.47 = ( zero_zero @ nat ) )
% 0.32/0.47 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_is_0_eq
% 0.32/0.47 thf(fact_141_le__zero__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( canoni770627133id_add @ A )
% 0.32/0.47 => ! [N: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.32/0.47 = ( N
% 0.32/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_zero_eq
% 0.32/0.47 thf(fact_142_inf_Obounded__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_inf @ A )
% 0.32/0.47 => ! [A2: A,B: A,C: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B @ C ) )
% 0.32/0.47 = ( ( ord_less_eq @ A @ A2 @ B )
% 0.32/0.47 & ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inf.bounded_iff
% 0.32/0.47 thf(fact_143_le__inf__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_inf @ A )
% 0.32/0.47 => ! [X: A,Y: A,Z: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
% 0.32/0.47 = ( ( ord_less_eq @ A @ X @ Y )
% 0.32/0.47 & ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_inf_iff
% 0.32/0.47 thf(fact_144_sup_Obounded__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [B: A,C: A,A2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B @ C ) @ A2 )
% 0.32/0.47 = ( ( ord_less_eq @ A @ B @ A2 )
% 0.32/0.47 & ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.bounded_iff
% 0.32/0.47 thf(fact_145_le__sup__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [X: A,Y: A,Z: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
% 0.32/0.47 = ( ( ord_less_eq @ A @ X @ Z )
% 0.32/0.47 & ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_sup_iff
% 0.32/0.47 thf(fact_146_bot__nat__0_Oextremum,axiom,
% 0.32/0.47 ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% 0.32/0.47
% 0.32/0.47 % bot_nat_0.extremum
% 0.32/0.47 thf(fact_147_le0,axiom,
% 0.32/0.47 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.32/0.47
% 0.32/0.47 % le0
% 0.32/0.47 thf(fact_148_diff__diff__cancel,axiom,
% 0.32/0.47 ! [I: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ I @ N )
% 0.32/0.47 => ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I ) )
% 0.32/0.47 = I ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_diff_cancel
% 0.32/0.47 thf(fact_149_hsubsetI,axiom,
% 0.32/0.47 ! [A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 0.32/0.47 ( ! [X2: hF_Mirabelle_hf] :
% 0.32/0.47 ( ( hF_Mirabelle_hmem @ X2 @ A5 )
% 0.32/0.47 => ( hF_Mirabelle_hmem @ X2 @ B4 ) )
% 0.32/0.47 => ( ord_less_eq @ hF_Mirabelle_hf @ A5 @ B4 ) ) ).
% 0.32/0.47
% 0.32/0.47 % hsubsetI
% 0.32/0.47 thf(fact_150_less__eq__hempty,axiom,
% 0.32/0.47 ! [U: hF_Mirabelle_hf] :
% 0.32/0.47 ( ( ord_less_eq @ hF_Mirabelle_hf @ U @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.32/0.47 = ( U
% 0.32/0.47 = ( zero_zero @ hF_Mirabelle_hf ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_eq_hempty
% 0.32/0.47 thf(fact_151_diff__ge__0__iff__ge,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordered_ab_group_add @ A )
% 0.32/0.47 => ! [A2: A,B: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B ) )
% 0.32/0.47 = ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_ge_0_iff_ge
% 0.32/0.47 thf(fact_152_diff__is__0__eq_H,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.32/0.47 => ( ( minus_minus @ nat @ M @ N )
% 0.32/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_is_0_eq'
% 0.32/0.47 thf(fact_153_diff__le__mono2,axiom,
% 0.32/0.47 ! [M: nat,N: nat,L: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.32/0.47 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_le_mono2
% 0.32/0.47 thf(fact_154_le__diff__iff_H,axiom,
% 0.32/0.47 ! [A2: nat,C: nat,B: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ A2 @ C )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ B @ C )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ A2 ) @ ( minus_minus @ nat @ C @ B ) )
% 0.32/0.47 = ( ord_less_eq @ nat @ B @ A2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_diff_iff'
% 0.32/0.47 thf(fact_155_diff__le__self,axiom,
% 0.32/0.47 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% 0.32/0.47
% 0.32/0.47 % diff_le_self
% 0.32/0.47 thf(fact_156_diff__le__mono,axiom,
% 0.32/0.47 ! [M: nat,N: nat,L: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.32/0.47 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_le_mono
% 0.32/0.47 thf(fact_157_Nat_Odiff__diff__eq,axiom,
% 0.32/0.47 ! [K: nat,M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ K @ M )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ K @ N )
% 0.32/0.47 => ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 0.32/0.47 = ( minus_minus @ nat @ M @ N ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Nat.diff_diff_eq
% 0.32/0.47 thf(fact_158_le__diff__iff,axiom,
% 0.32/0.47 ! [K: nat,M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ K @ M )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ K @ N )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 0.32/0.47 = ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_diff_iff
% 0.32/0.47 thf(fact_159_eq__diff__iff,axiom,
% 0.32/0.47 ! [K: nat,M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ K @ M )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ K @ N )
% 0.32/0.47 => ( ( ( minus_minus @ nat @ M @ K )
% 0.32/0.47 = ( minus_minus @ nat @ N @ K ) )
% 0.32/0.47 = ( M = N ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_diff_iff
% 0.32/0.47 thf(fact_160_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.32/0.47 ! [A2: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 0.32/0.47 => ( A2
% 0.32/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % bot_nat_0.extremum_uniqueI
% 0.32/0.47 thf(fact_161_bot__nat__0_Oextremum__unique,axiom,
% 0.32/0.47 ! [A2: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 0.32/0.47 = ( A2
% 0.32/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % bot_nat_0.extremum_unique
% 0.32/0.47 thf(fact_162_le__0__eq,axiom,
% 0.32/0.47 ! [N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 0.32/0.47 = ( N
% 0.32/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_0_eq
% 0.32/0.47 thf(fact_163_less__eq__nat_Osimps_I1_J,axiom,
% 0.32/0.47 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.32/0.47
% 0.32/0.47 % less_eq_nat.simps(1)
% 0.32/0.47 thf(fact_164_Nat_Oex__has__greatest__nat,axiom,
% 0.32/0.47 ! [P: nat > $o,K: nat,B: nat] :
% 0.32/0.47 ( ( P @ K )
% 0.32/0.47 => ( ! [Y2: nat] :
% 0.32/0.47 ( ( P @ Y2 )
% 0.32/0.47 => ( ord_less_eq @ nat @ Y2 @ B ) )
% 0.32/0.47 => ? [X2: nat] :
% 0.32/0.47 ( ( P @ X2 )
% 0.32/0.47 & ! [Y5: nat] :
% 0.32/0.47 ( ( P @ Y5 )
% 0.32/0.47 => ( ord_less_eq @ nat @ Y5 @ X2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Nat.ex_has_greatest_nat
% 0.32/0.47 thf(fact_165_nat__le__linear,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.32/0.47 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.32/0.47
% 0.32/0.47 % nat_le_linear
% 0.32/0.47 thf(fact_166_le__antisym,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ N @ M )
% 0.32/0.47 => ( M = N ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_antisym
% 0.32/0.47 thf(fact_167_eq__imp__le,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( M = N )
% 0.32/0.47 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_imp_le
% 0.32/0.47 thf(fact_168_le__trans,axiom,
% 0.32/0.47 ! [I: nat,J: nat,K: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ I @ J )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ J @ K )
% 0.32/0.47 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_trans
% 0.32/0.47 thf(fact_169_le__refl,axiom,
% 0.32/0.47 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 0.32/0.47
% 0.32/0.47 % le_refl
% 0.32/0.47 thf(fact_170_canonically__ordered__monoid__add__class_Ozero__le,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( canoni770627133id_add @ A )
% 0.32/0.47 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 0.32/0.47
% 0.32/0.47 % canonically_ordered_monoid_add_class.zero_le
% 0.32/0.47 thf(fact_171_diff__mono,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordered_ab_group_add @ A )
% 0.32/0.47 => ! [A2: A,B: A,D: A,C: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.32/0.47 => ( ( ord_less_eq @ A @ D @ C )
% 0.32/0.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ D ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_mono
% 0.32/0.47 thf(fact_172_diff__left__mono,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordered_ab_group_add @ A )
% 0.32/0.47 => ! [B: A,A2: A,C: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ B @ A2 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_left_mono
% 0.32/0.47 thf(fact_173_diff__right__mono,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordered_ab_group_add @ A )
% 0.32/0.47 => ! [A2: A,B: A,C: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.32/0.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_right_mono
% 0.32/0.47 thf(fact_174_diff__eq__diff__less__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordered_ab_group_add @ A )
% 0.32/0.47 => ! [A2: A,B: A,C: A,D: A] :
% 0.32/0.47 ( ( ( minus_minus @ A @ A2 @ B )
% 0.32/0.47 = ( minus_minus @ A @ C @ D ) )
% 0.32/0.47 => ( ( ord_less_eq @ A @ A2 @ B )
% 0.32/0.47 = ( ord_less_eq @ A @ C @ D ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_eq_diff_less_eq
% 0.32/0.47 thf(fact_175_inf_OcoboundedI2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_inf @ A )
% 0.32/0.47 => ! [B: A,C: A,A2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ B @ C )
% 0.32/0.47 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B ) @ C ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inf.coboundedI2
% 0.32/0.47 thf(fact_176_inf_OcoboundedI1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_inf @ A )
% 0.32/0.47 => ! [A2: A,C: A,B: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A2 @ C )
% 0.32/0.47 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B ) @ C ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inf.coboundedI1
% 0.32/0.47 thf(fact_177_inf_Oabsorb__iff2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_inf @ A )
% 0.32/0.47 => ( ( ord_less_eq @ A )
% 0.32/0.47 = ( ^ [B3: A,A4: A] :
% 0.32/0.47 ( ( inf_inf @ A @ A4 @ B3 )
% 0.32/0.47 = B3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inf.absorb_iff2
% 0.32/0.47 thf(fact_178_inf_Oabsorb__iff1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_inf @ A )
% 0.32/0.47 => ( ( ord_less_eq @ A )
% 0.32/0.47 = ( ^ [A4: A,B3: A] :
% 0.32/0.47 ( ( inf_inf @ A @ A4 @ B3 )
% 0.32/0.47 = A4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inf.absorb_iff1
% 0.32/0.47 thf(fact_179_inf_Ocobounded2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_inf @ A )
% 0.32/0.47 => ! [A2: A,B: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B ) @ B ) ) ).
% 0.32/0.47
% 0.32/0.47 % inf.cobounded2
% 0.32/0.47 thf(fact_180_inf_Ocobounded1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_inf @ A )
% 0.32/0.47 => ! [A2: A,B: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B ) @ A2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % inf.cobounded1
% 0.32/0.47 thf(fact_181_inf_Oorder__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_inf @ A )
% 0.32/0.47 => ( ( ord_less_eq @ A )
% 0.32/0.47 = ( ^ [A4: A,B3: A] :
% 0.32/0.47 ( A4
% 0.32/0.47 = ( inf_inf @ A @ A4 @ B3 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inf.order_iff
% 0.32/0.47 thf(fact_182_inf__greatest,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_inf @ A )
% 0.32/0.47 => ! [X: A,Y: A,Z: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.32/0.47 => ( ( ord_less_eq @ A @ X @ Z )
% 0.32/0.47 => ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inf_greatest
% 0.32/0.47 thf(fact_183_inf_OboundedI,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_inf @ A )
% 0.32/0.47 => ! [A2: A,B: A,C: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.32/0.47 => ( ( ord_less_eq @ A @ A2 @ C )
% 0.32/0.47 => ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B @ C ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inf.boundedI
% 0.32/0.47 thf(fact_184_inf_OboundedE,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_inf @ A )
% 0.32/0.47 => ! [A2: A,B: A,C: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B @ C ) )
% 0.32/0.47 => ~ ( ( ord_less_eq @ A @ A2 @ B )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inf.boundedE
% 0.32/0.47 thf(fact_185_inf__absorb2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_inf @ A )
% 0.32/0.47 => ! [Y: A,X: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ Y @ X )
% 0.32/0.47 => ( ( inf_inf @ A @ X @ Y )
% 0.32/0.47 = Y ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inf_absorb2
% 0.32/0.47 thf(fact_186_inf__absorb1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_inf @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.32/0.47 => ( ( inf_inf @ A @ X @ Y )
% 0.32/0.47 = X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inf_absorb1
% 0.32/0.47 thf(fact_187_inf_Oabsorb2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_inf @ A )
% 0.32/0.47 => ! [B: A,A2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ B @ A2 )
% 0.32/0.47 => ( ( inf_inf @ A @ A2 @ B )
% 0.32/0.47 = B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inf.absorb2
% 0.32/0.47 thf(fact_188_inf_Oabsorb1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_inf @ A )
% 0.32/0.47 => ! [A2: A,B: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.32/0.47 => ( ( inf_inf @ A @ A2 @ B )
% 0.32/0.47 = A2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inf.absorb1
% 0.32/0.47 thf(fact_189_le__iff__inf,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_inf @ A )
% 0.32/0.47 => ( ( ord_less_eq @ A )
% 0.32/0.47 = ( ^ [X3: A,Y3: A] :
% 0.32/0.47 ( ( inf_inf @ A @ X3 @ Y3 )
% 0.32/0.47 = X3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_iff_inf
% 0.32/0.47 thf(fact_190_inf__unique,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_inf @ A )
% 0.32/0.47 => ! [F: A > A > A,X: A,Y: A] :
% 0.32/0.47 ( ! [X2: A,Y2: A] : ( ord_less_eq @ A @ ( F @ X2 @ Y2 ) @ X2 )
% 0.32/0.47 => ( ! [X2: A,Y2: A] : ( ord_less_eq @ A @ ( F @ X2 @ Y2 ) @ Y2 )
% 0.32/0.47 => ( ! [X2: A,Y2: A,Z2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ X2 @ Z2 )
% 0.32/0.47 => ( ord_less_eq @ A @ X2 @ ( F @ Y2 @ Z2 ) ) ) )
% 0.32/0.47 => ( ( inf_inf @ A @ X @ Y )
% 0.32/0.47 = ( F @ X @ Y ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inf_unique
% 0.32/0.47 thf(fact_191_inf_OorderI,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_inf @ A )
% 0.32/0.47 => ! [A2: A,B: A] :
% 0.32/0.47 ( ( A2
% 0.32/0.47 = ( inf_inf @ A @ A2 @ B ) )
% 0.32/0.47 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inf.orderI
% 0.32/0.47 thf(fact_192_inf_OorderE,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_inf @ A )
% 0.32/0.47 => ! [A2: A,B: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.32/0.47 => ( A2
% 0.32/0.47 = ( inf_inf @ A @ A2 @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inf.orderE
% 0.32/0.47 thf(fact_193_le__infI2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_inf @ A )
% 0.32/0.47 => ! [B: A,X: A,A2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ B @ X )
% 0.32/0.47 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B ) @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_infI2
% 0.32/0.47 thf(fact_194_le__infI1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_inf @ A )
% 0.32/0.47 => ! [A2: A,X: A,B: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A2 @ X )
% 0.32/0.47 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B ) @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_infI1
% 0.32/0.47 thf(fact_195_inf__mono,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_inf @ A )
% 0.32/0.47 => ! [A2: A,C: A,B: A,D: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A2 @ C )
% 0.32/0.47 => ( ( ord_less_eq @ A @ B @ D )
% 0.32/0.47 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B ) @ ( inf_inf @ A @ C @ D ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inf_mono
% 0.32/0.47 thf(fact_196_le__infI,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_inf @ A )
% 0.32/0.47 => ! [X: A,A2: A,B: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ A2 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ X @ B )
% 0.32/0.47 => ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A2 @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_infI
% 0.32/0.47 thf(fact_197_le__infE,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_inf @ A )
% 0.32/0.47 => ! [X: A,A2: A,B: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A2 @ B ) )
% 0.32/0.47 => ~ ( ( ord_less_eq @ A @ X @ A2 )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ X @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_infE
% 0.32/0.47 thf(fact_198_inf__le2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_inf @ A )
% 0.32/0.47 => ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Y ) ) ).
% 0.32/0.47
% 0.32/0.47 % inf_le2
% 0.32/0.47 thf(fact_199_inf__le1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_inf @ A )
% 0.32/0.47 => ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ X ) ) ).
% 0.32/0.47
% 0.32/0.47 % inf_le1
% 0.32/0.47 thf(fact_200_inf__sup__ord_I1_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( lattice @ A )
% 0.32/0.47 => ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ X ) ) ).
% 0.32/0.47
% 0.32/0.47 % inf_sup_ord(1)
% 0.32/0.47 thf(fact_201_inf__sup__ord_I2_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( lattice @ A )
% 0.32/0.47 => ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Y ) ) ).
% 0.32/0.47
% 0.32/0.47 % inf_sup_ord(2)
% 0.32/0.47 thf(fact_202_sup_OcoboundedI2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [C: A,B: A,A2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ C @ B )
% 0.32/0.47 => ( ord_less_eq @ A @ C @ ( sup_sup @ A @ A2 @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.coboundedI2
% 0.32/0.47 thf(fact_203_sup_OcoboundedI1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [C: A,A2: A,B: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ C @ A2 )
% 0.32/0.47 => ( ord_less_eq @ A @ C @ ( sup_sup @ A @ A2 @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.coboundedI1
% 0.32/0.47 thf(fact_204_sup_Oabsorb__iff2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ( ( ord_less_eq @ A )
% 0.32/0.47 = ( ^ [A4: A,B3: A] :
% 0.32/0.47 ( ( sup_sup @ A @ A4 @ B3 )
% 0.32/0.47 = B3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.absorb_iff2
% 0.32/0.47 thf(fact_205_sup_Oabsorb__iff1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ( ( ord_less_eq @ A )
% 0.32/0.47 = ( ^ [B3: A,A4: A] :
% 0.32/0.47 ( ( sup_sup @ A @ A4 @ B3 )
% 0.32/0.47 = A4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.absorb_iff1
% 0.32/0.47 thf(fact_206_sup_Ocobounded2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [B: A,A2: A] : ( ord_less_eq @ A @ B @ ( sup_sup @ A @ A2 @ B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.cobounded2
% 0.32/0.47 thf(fact_207_sup_Ocobounded1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [A2: A,B: A] : ( ord_less_eq @ A @ A2 @ ( sup_sup @ A @ A2 @ B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.cobounded1
% 0.32/0.47 thf(fact_208_sup_Oorder__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ( ( ord_less_eq @ A )
% 0.32/0.47 = ( ^ [B3: A,A4: A] :
% 0.32/0.47 ( A4
% 0.32/0.47 = ( sup_sup @ A @ A4 @ B3 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.order_iff
% 0.32/0.47 thf(fact_209_sup_OboundedI,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [B: A,A2: A,C: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ B @ A2 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ C @ A2 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( sup_sup @ A @ B @ C ) @ A2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.boundedI
% 0.32/0.47 thf(fact_210_sup_OboundedE,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [B: A,C: A,A2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B @ C ) @ A2 )
% 0.32/0.47 => ~ ( ( ord_less_eq @ A @ B @ A2 )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.boundedE
% 0.32/0.47 thf(fact_211_sup__absorb2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [X: A,Y: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.32/0.47 => ( ( sup_sup @ A @ X @ Y )
% 0.32/0.47 = Y ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup_absorb2
% 0.32/0.47 thf(fact_212_sup__absorb1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [Y: A,X: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ Y @ X )
% 0.32/0.47 => ( ( sup_sup @ A @ X @ Y )
% 0.32/0.47 = X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup_absorb1
% 0.32/0.47 thf(fact_213_sup_Oabsorb2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [A2: A,B: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.32/0.47 => ( ( sup_sup @ A @ A2 @ B )
% 0.32/0.47 = B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.absorb2
% 0.32/0.47 thf(fact_214_sup_Oabsorb1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [B: A,A2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ B @ A2 )
% 0.32/0.47 => ( ( sup_sup @ A @ A2 @ B )
% 0.32/0.47 = A2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.absorb1
% 0.32/0.47 thf(fact_215_sup__unique,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [F: A > A > A,X: A,Y: A] :
% 0.32/0.47 ( ! [X2: A,Y2: A] : ( ord_less_eq @ A @ X2 @ ( F @ X2 @ Y2 ) )
% 0.32/0.47 => ( ! [X2: A,Y2: A] : ( ord_less_eq @ A @ Y2 @ ( F @ X2 @ Y2 ) )
% 0.32/0.47 => ( ! [X2: A,Y2: A,Z2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ Y2 @ X2 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ Z2 @ X2 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( F @ Y2 @ Z2 ) @ X2 ) ) )
% 0.32/0.47 => ( ( sup_sup @ A @ X @ Y )
% 0.32/0.47 = ( F @ X @ Y ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup_unique
% 0.32/0.47 thf(fact_216_sup_OorderI,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [A2: A,B: A] :
% 0.32/0.47 ( ( A2
% 0.32/0.47 = ( sup_sup @ A @ A2 @ B ) )
% 0.32/0.47 => ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.orderI
% 0.32/0.47 thf(fact_217_sup_OorderE,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [B: A,A2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ B @ A2 )
% 0.32/0.47 => ( A2
% 0.32/0.47 = ( sup_sup @ A @ A2 @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.orderE
% 0.32/0.47 thf(fact_218_le__iff__sup,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ( ( ord_less_eq @ A )
% 0.32/0.47 = ( ^ [X3: A,Y3: A] :
% 0.32/0.47 ( ( sup_sup @ A @ X3 @ Y3 )
% 0.32/0.47 = Y3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_iff_sup
% 0.32/0.47 thf(fact_219_sup__least,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [Y: A,X: A,Z: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ Y @ X )
% 0.32/0.47 => ( ( ord_less_eq @ A @ Z @ X )
% 0.32/0.47 => ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z ) @ X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup_least
% 0.32/0.47 thf(fact_220_sup__mono,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [A2: A,C: A,B: A,D: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A2 @ C )
% 0.32/0.47 => ( ( ord_less_eq @ A @ B @ D )
% 0.32/0.47 => ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B ) @ ( sup_sup @ A @ C @ D ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup_mono
% 0.32/0.47 thf(fact_221_sup_Omono,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [C: A,A2: A,D: A,B: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ C @ A2 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ D @ B )
% 0.32/0.47 => ( ord_less_eq @ A @ ( sup_sup @ A @ C @ D ) @ ( sup_sup @ A @ A2 @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup.mono
% 0.32/0.47 thf(fact_222_le__supI2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [X: A,B: A,A2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ B )
% 0.32/0.47 => ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A2 @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_supI2
% 0.32/0.47 thf(fact_223_le__supI1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [X: A,A2: A,B: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ A2 )
% 0.32/0.47 => ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A2 @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_supI1
% 0.32/0.47 thf(fact_224_sup__ge2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup_ge2
% 0.32/0.47 thf(fact_225_sup__ge1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sup_ge1
% 0.32/0.47 thf(fact_226_le__supI,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [A2: A,X: A,B: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A2 @ X )
% 0.32/0.47 => ( ( ord_less_eq @ A @ B @ X )
% 0.32/0.47 => ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B ) @ X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_supI
% 0.32/0.47 thf(fact_227_le__supE,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A )
% 0.32/0.47 => ! [A2: A,B: A,X: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B ) @ X )
% 0.32/0.47 => ~ ( ( ord_less_eq @ A @ A2 @ X )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ B @ X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_supE
% 0.32/0.47 thf(fact_228_inf__sup__ord_I3_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( lattice @ A )
% 0.32/0.47 => ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inf_sup_ord(3)
% 0.32/0.47 thf(fact_229_inf__sup__ord_I4_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( lattice @ A )
% 0.32/0.47 => ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inf_sup_ord(4)
% 0.32/0.47 thf(fact_230_less__eq__hf__def,axiom,
% 0.32/0.47 ( ( ord_less_eq @ hF_Mirabelle_hf )
% 0.32/0.47 = ( ^ [A7: hF_Mirabelle_hf,B7: hF_Mirabelle_hf] :
% 0.32/0.47 ! [X3: hF_Mirabelle_hf] :
% 0.32/0.47 ( ( hF_Mirabelle_hmem @ X3 @ A7 )
% 0.32/0.47 => ( hF_Mirabelle_hmem @ X3 @ B7 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_eq_hf_def
% 0.32/0.47 thf(fact_231_rev__hsubsetD,axiom,
% 0.32/0.47 ! [C: hF_Mirabelle_hf,A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 0.32/0.47 ( ( hF_Mirabelle_hmem @ C @ A5 )
% 0.32/0.47 => ( ( ord_less_eq @ hF_Mirabelle_hf @ A5 @ B4 )
% 0.32/0.47 => ( hF_Mirabelle_hmem @ C @ B4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % rev_hsubsetD
% 0.32/0.47 thf(fact_232_hsubsetCE,axiom,
% 0.32/0.47 ! [A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.32/0.47 ( ( ord_less_eq @ hF_Mirabelle_hf @ A5 @ B4 )
% 0.32/0.47 => ( ( hF_Mirabelle_hmem @ C @ A5 )
% 0.32/0.47 => ( hF_Mirabelle_hmem @ C @ B4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % hsubsetCE
% 0.32/0.47 thf(fact_233_hsubsetD,axiom,
% 0.32/0.47 ! [A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.32/0.47 ( ( ord_less_eq @ hF_Mirabelle_hf @ A5 @ B4 )
% 0.32/0.47 => ( ( hF_Mirabelle_hmem @ C @ A5 )
% 0.32/0.47 => ( hF_Mirabelle_hmem @ C @ B4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % hsubsetD
% 0.32/0.47 thf(fact_234_HF__Mirabelle__fsbjehakzm_Ozero__le,axiom,
% 0.32/0.47 ! [X: hF_Mirabelle_hf] : ( ord_less_eq @ hF_Mirabelle_hf @ ( zero_zero @ hF_Mirabelle_hf ) @ X ) ).
% 0.32/0.47
% 0.32/0.47 % HF_Mirabelle_fsbjehakzm.zero_le
% 0.32/0.47 thf(fact_235_hf__equalityE,axiom,
% 0.32/0.47 ! [A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 0.32/0.47 ( ( A5 = B4 )
% 0.32/0.47 => ~ ( ( ord_less_eq @ hF_Mirabelle_hf @ A5 @ B4 )
% 0.32/0.47 => ~ ( ord_less_eq @ hF_Mirabelle_hf @ B4 @ A5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % hf_equalityE
% 0.32/0.47 thf(fact_236_le__iff__diff__le__0,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordered_ab_group_add @ A )
% 0.32/0.47 => ( ( ord_less_eq @ A )
% 0.32/0.47 = ( ^ [A4: A,B3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_iff_diff_le_0
% 0.32/0.47 thf(fact_237_distrib__inf__le,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( lattice @ A )
% 0.32/0.47 => ! [X: A,Y: A,Z: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z ) ) @ ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % distrib_inf_le
% 0.32/0.47 thf(fact_238_distrib__sup__le,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( lattice @ A )
% 0.32/0.47 => ! [X: A,Y: A,Z: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) @ ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % distrib_sup_le
% 0.32/0.47 thf(fact_239_less__eq__insert2__iff,axiom,
% 0.32/0.47 ! [Z: hF_Mirabelle_hf,X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.32/0.47 ( ( ord_less_eq @ hF_Mirabelle_hf @ Z @ ( hF_Mirabelle_hinsert @ X @ Y ) )
% 0.32/0.47 = ( ( ord_less_eq @ hF_Mirabelle_hf @ Z @ Y )
% 0.32/0.47 | ? [U2: hF_Mirabelle_hf] :
% 0.32/0.47 ( ( ( hF_Mirabelle_hinsert @ X @ U2 )
% 0.32/0.47 = Z )
% 0.32/0.47 & ~ ( hF_Mirabelle_hmem @ X @ U2 )
% 0.32/0.47 & ( ord_less_eq @ hF_Mirabelle_hf @ U2 @ Y ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_eq_insert2_iff
% 0.32/0.47 thf(fact_240_order__refl,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_refl
% 0.32/0.47 thf(fact_241_le__numeral__extra_I3_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linord1659791738miring @ A )
% 0.32/0.47 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_numeral_extra(3)
% 0.32/0.47 thf(fact_242_HF__hfset,axiom,
% 0.32/0.47 ! [A2: hF_Mirabelle_hf] :
% 0.32/0.47 ( ( hF_Mirabelle_HF @ ( hF_Mirabelle_hfset @ A2 ) )
% 0.32/0.47 = A2 ) ).
% 0.32/0.47
% 0.32/0.47 % HF_hfset
% 0.32/0.47 thf(fact_243_of__nat__le__0__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linord1659791738miring @ A )
% 0.32/0.47 => ! [M: nat] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) )
% 0.32/0.47 = ( M
% 0.32/0.47 = ( zero_zero @ nat ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % of_nat_le_0_iff
% 0.32/0.47 thf(fact_244_of__nat__eq__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semiring_char_0 @ A )
% 0.32/0.47 => ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ( semiring_1_of_nat @ A @ M )
% 0.32/0.47 = ( semiring_1_of_nat @ A @ N ) )
% 0.32/0.47 = ( M = N ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % of_nat_eq_iff
% 0.32/0.47 thf(fact_245_of__nat__0,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semiring_1 @ A )
% 0.32/0.47 => ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
% 0.32/0.47 = ( zero_zero @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % of_nat_0
% 0.32/0.47 thf(fact_246_of__nat__0__eq__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semiring_char_0 @ A )
% 0.32/0.47 => ! [N: nat] :
% 0.32/0.47 ( ( ( zero_zero @ A )
% 0.32/0.47 = ( semiring_1_of_nat @ A @ N ) )
% 0.32/0.47 = ( ( zero_zero @ nat )
% 0.32/0.47 = N ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % of_nat_0_eq_iff
% 0.32/0.47 thf(fact_247_of__nat__eq__0__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semiring_char_0 @ A )
% 0.32/0.47 => ! [M: nat] :
% 0.32/0.47 ( ( ( semiring_1_of_nat @ A @ M )
% 0.32/0.47 = ( zero_zero @ A ) )
% 0.32/0.47 = ( M
% 0.32/0.47 = ( zero_zero @ nat ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % of_nat_eq_0_iff
% 0.32/0.47 thf(fact_248_of__nat__le__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linord1659791738miring @ A )
% 0.32/0.47 => ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.32/0.47 = ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % of_nat_le_iff
% 0.32/0.47 thf(fact_249_of__nat__0__le__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linord1659791738miring @ A )
% 0.32/0.47 => ! [N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % of_nat_0_le_iff
% 0.32/0.47 thf(fact_250_of__nat__mono,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linord1659791738miring @ A )
% 0.32/0.47 => ! [I: nat,J: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ I @ J )
% 0.32/0.47 => ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ I ) @ ( semiring_1_of_nat @ A @ J ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % of_nat_mono
% 0.32/0.47 thf(fact_251_of__nat__diff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semiring_1_cancel @ A )
% 0.32/0.47 => ! [N: nat,M: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ N @ M )
% 0.32/0.47 => ( ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ M @ N ) )
% 0.32/0.47 = ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % of_nat_diff
% 0.32/0.47 thf(fact_252_dual__order_Oantisym,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [B: A,A2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ B @ A2 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ A2 @ B )
% 0.32/0.47 => ( A2 = B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.antisym
% 0.32/0.47 thf(fact_253_dual__order_Oeq__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ( ( ^ [Y4: A,Z3: A] : ( Y4 = Z3 ) )
% 0.32/0.47 = ( ^ [A4: A,B3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ B3 @ A4 )
% 0.32/0.47 & ( ord_less_eq @ A @ A4 @ B3 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.eq_iff
% 0.32/0.47
% 0.32/0.47 % Type constructors (45)
% 0.32/0.47 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 0.32/0.47 ! [A8: $tType,A9: $tType] :
% 0.32/0.47 ( ( semilattice_sup @ A9 )
% 0.32/0.47 => ( semilattice_sup @ ( A8 > A9 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
% 0.32/0.47 ! [A8: $tType,A9: $tType] :
% 0.32/0.47 ( ( semilattice_inf @ A9 )
% 0.32/0.47 => ( semilattice_inf @ ( A8 > A9 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_fun___Lattices_Odistrib__lattice,axiom,
% 0.32/0.47 ! [A8: $tType,A9: $tType] :
% 0.32/0.47 ( ( distrib_lattice @ A9 )
% 0.32/0.47 => ( distrib_lattice @ ( A8 > A9 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.32/0.47 ! [A8: $tType,A9: $tType] :
% 0.32/0.47 ( ( preorder @ A9 )
% 0.32/0.47 => ( preorder @ ( A8 > A9 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_fun___Lattices_Olattice,axiom,
% 0.32/0.47 ! [A8: $tType,A9: $tType] :
% 0.32/0.47 ( ( lattice @ A9 )
% 0.32/0.47 => ( lattice @ ( A8 > A9 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.32/0.47 ! [A8: $tType,A9: $tType] :
% 0.32/0.47 ( ( order @ A9 )
% 0.32/0.47 => ( order @ ( A8 > A9 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_fun___Groups_Ominus,axiom,
% 0.32/0.47 ! [A8: $tType,A9: $tType] :
% 0.32/0.47 ( ( minus @ A9 )
% 0.32/0.47 => ( minus @ ( A8 > A9 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.32/0.47 canoni770627133id_add @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
% 0.32/0.47 linord1659791738miring @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
% 0.32/0.47 cancel146912293up_add @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
% 0.32/0.47 cancel1352612707id_add @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_1,axiom,
% 0.32/0.47 semilattice_sup @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_2,axiom,
% 0.32/0.47 semilattice_inf @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Lattices_Odistrib__lattice_3,axiom,
% 0.32/0.47 distrib_lattice @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Rings_Osemiring__1__cancel,axiom,
% 0.32/0.47 semiring_1_cancel @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
% 0.32/0.47 comm_monoid_diff @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Nat_Osemiring__char__0,axiom,
% 0.32/0.47 semiring_char_0 @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Orderings_Opreorder_4,axiom,
% 0.32/0.47 preorder @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Rings_Osemiring__1,axiom,
% 0.32/0.47 semiring_1 @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Lattices_Olattice_5,axiom,
% 0.32/0.47 lattice @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Orderings_Oorder_6,axiom,
% 0.32/0.47 order @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Groups_Ominus_7,axiom,
% 0.32/0.47 minus @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 0.32/0.47 zero @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_8,axiom,
% 0.32/0.47 ! [A8: $tType] : ( semilattice_sup @ ( set @ A8 ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_9,axiom,
% 0.32/0.47 ! [A8: $tType] : ( semilattice_inf @ ( set @ A8 ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Set_Oset___Lattices_Odistrib__lattice_10,axiom,
% 0.32/0.47 ! [A8: $tType] : ( distrib_lattice @ ( set @ A8 ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Set_Oset___Orderings_Opreorder_11,axiom,
% 0.32/0.47 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Set_Oset___Lattices_Olattice_12,axiom,
% 0.32/0.47 ! [A8: $tType] : ( lattice @ ( set @ A8 ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Set_Oset___Orderings_Oorder_13,axiom,
% 0.32/0.47 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Set_Oset___Groups_Ominus_14,axiom,
% 0.32/0.47 ! [A8: $tType] : ( minus @ ( set @ A8 ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_15,axiom,
% 0.32/0.47 semilattice_sup @ $o ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_16,axiom,
% 0.32/0.47 semilattice_inf @ $o ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_17,axiom,
% 0.32/0.47 distrib_lattice @ $o ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Orderings_Opreorder_18,axiom,
% 0.32/0.47 preorder @ $o ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Lattices_Olattice_19,axiom,
% 0.32/0.47 lattice @ $o ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Orderings_Oorder_20,axiom,
% 0.32/0.47 order @ $o ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Groups_Ominus_21,axiom,
% 0.32/0.47 minus @ $o ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Osemilattice__sup_22,axiom,
% 0.32/0.47 semilattice_sup @ hF_Mirabelle_hf ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Osemilattice__inf_23,axiom,
% 0.32/0.47 semilattice_inf @ hF_Mirabelle_hf ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Odistrib__lattice_24,axiom,
% 0.32/0.47 distrib_lattice @ hF_Mirabelle_hf ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Orderings_Opreorder_25,axiom,
% 0.32/0.47 preorder @ hF_Mirabelle_hf ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Olattice_26,axiom,
% 0.32/0.47 lattice @ hF_Mirabelle_hf ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Orderings_Oorder_27,axiom,
% 0.32/0.47 order @ hF_Mirabelle_hf ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Groups_Ominus_28,axiom,
% 0.32/0.47 minus @ hF_Mirabelle_hf ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Groups_Ozero_29,axiom,
% 0.32/0.47 zero @ hF_Mirabelle_hf ).
% 0.32/0.47
% 0.32/0.47 % Conjectures (1)
% 0.32/0.47 thf(conj_0,conjecture,
% 0.32/0.47 ( ( hF_Mirabelle_hpair @ x @ y )
% 0.32/0.47 != ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.32/0.47
% 0.32/0.47 %------------------------------------------------------------------------------
% 0.32/0.47 ------------------------
% 2.47/1.56 ---- Embedded in THF ---
% 2.47/1.56 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.47/1.56 %%% Generated on Tue Feb 24 08:25:19 EST 2026
% 2.47/1.56 %%% using '$$dhol' embedding, version 1.3.0.
% 2.47/1.56 %%% Logic specification used:
% 2.47/1.56 %%% thf(spec, logic, $$dhol).
% 2.47/1.56
% 2.47/1.56 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.dWcERH6OXC/DTF2DTF7787.p
% 2.47/1.56 thf(ty_t_HF__Mirabelle__fsbjehakzm_Ohf, type, hF_Mirabelle_hf: $tType).
% 2.47/1.56 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.47/1.56 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.47/1.56 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.47/1.56 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: ($o))).
% 2.47/1.56 thf(sy_cl_Groups_Ominus, type, minus: (!> [A:$tType]: ($o))).
% 2.47/1.56 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.47/1.56 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 2.47/1.56 thf(sy_cl_Lattices_Olattice, type, lattice: (!> [A:$tType]: ($o))).
% 2.47/1.56 thf(sy_cl_Rings_Osemiring__1, type, semiring_1: (!> [A:$tType]: ($o))).
% 2.47/1.56 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.47/1.56 thf(sy_cl_Nat_Osemiring__char__0, type, semiring_char_0: (!> [A:$tType]: ($o))).
% 2.47/1.56 thf(sy_cl_Groups_Ocomm__monoid__diff, type, comm_monoid_diff: (!> [A:$tType]: ($o))).
% 2.47/1.56 thf(sy_cl_Rings_Osemiring__1__cancel, type, semiring_1_cancel: (!> [A:$tType]: ($o))).
% 2.47/1.56 thf(sy_cl_Lattices_Odistrib__lattice, type, distrib_lattice: (!> [A:$tType]: ($o))).
% 2.47/1.56 thf(sy_cl_Lattices_Osemilattice__inf, type, semilattice_inf: (!> [A:$tType]: ($o))).
% 2.47/1.56 thf(sy_cl_Lattices_Osemilattice__sup, type, semilattice_sup: (!> [A:$tType]: ($o))).
% 2.47/1.56 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: ($o))).
% 2.47/1.56 thf(sy_cl_Groups_Ocancel__comm__monoid__add, type, cancel1352612707id_add: (!> [A:$tType]: ($o))).
% 2.47/1.56 thf(sy_cl_Groups_Ocancel__ab__semigroup__add, type, cancel146912293up_add: (!> [A:$tType]: ($o))).
% 2.47/1.56 thf(sy_cl_Rings_Olinordered__nonzero__semiring, type, linord1659791738miring: (!> [A:$tType]: ($o))).
% 2.47/1.56 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 2.47/1.56 thf(sy_c_Groups_Ominus__class_Ominus, type, minus_minus: (!> [A:$tType]: ((A > (A > A))))).
% 2.47/1.56 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.47/1.56 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHCollect, type, hF_Mir1687042746ollect: ((hF_Mirabelle_hf > $o) > (hF_Mirabelle_hf > hF_Mirabelle_hf))).
% 2.47/1.56 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHF, type, hF_Mirabelle_HF: ((set @ hF_Mirabelle_hf) > hF_Mirabelle_hf)).
% 2.47/1.56 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHInter, type, hF_Mirabelle_HInter: (hF_Mirabelle_hf > hF_Mirabelle_hf)).
% 2.47/1.56 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHUnion, type, hF_Mirabelle_HUnion: (hF_Mirabelle_hf > hF_Mirabelle_hf)).
% 2.47/1.56 thf(sy_c_HF__Mirabelle__fsbjehakzm_OPrimReplace, type, hF_Mir569462966eplace: (hF_Mirabelle_hf > ((hF_Mirabelle_hf > (hF_Mirabelle_hf > $o)) > hF_Mirabelle_hf))).
% 2.47/1.56 thf(sy_c_HF__Mirabelle__fsbjehakzm_ORepFun, type, hF_Mirabelle_RepFun: (hF_Mirabelle_hf > ((hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf))).
% 2.47/1.56 thf(sy_c_HF__Mirabelle__fsbjehakzm_OReplace, type, hF_Mirabelle_Replace: (hF_Mirabelle_hf > ((hF_Mirabelle_hf > (hF_Mirabelle_hf > $o)) > hF_Mirabelle_hf))).
% 2.47/1.56 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohf_OAbs__hf, type, hF_Mirabelle_Abs_hf: (nat > hF_Mirabelle_hf)).
% 2.47/1.56 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohfset, type, hF_Mirabelle_hfset: (hF_Mirabelle_hf > (set @ hF_Mirabelle_hf))).
% 2.47/1.56 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohfst, type, hF_Mirabelle_hfst: (hF_Mirabelle_hf > hF_Mirabelle_hf)).
% 2.47/1.56 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohinsert, type, hF_Mirabelle_hinsert: (hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf))).
% 2.47/1.56 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohmem, type, hF_Mirabelle_hmem: (hF_Mirabelle_hf > (hF_Mirabelle_hf > $o))).
% 2.47/1.56 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohpair, type, hF_Mirabelle_hpair: (hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf))).
% 2.47/1.56 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohsnd, type, hF_Mirabelle_hsnd: (hF_Mirabelle_hf > hF_Mirabelle_hf)).
% 2.47/1.56 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohsplit, type, hF_Mirabelle_hsplit: (!> [A:$tType]: (((hF_Mirabelle_hf > (hF_Mirabelle_hf > A)) > (hF_Mirabelle_hf > A))))).
% 2.47/1.56 thf(sy_c_Lattices_Oinf__class_Oinf, type, inf_inf: (!> [A:$tType]: ((A > (A > A))))).
% 2.47/1.56 thf(sy_c_Lattices_Osup__class_Osup, type, sup_sup: (!> [A:$tType]: ((A > (A > A))))).
% 2.47/1.56 thf(sy_c_Nat_Osemiring__1__class_Oof__nat, type, semiring_1_of_nat: (!> [A:$tType]: ((nat > A)))).
% 2.47/1.56 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux, type, semiri532925092at_aux: (!> [A:$tType]: (((A > A) > (nat > (A > A)))))).
% 2.47/1.56 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.47/1.56 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.47/1.56 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.47/1.56 thf(sy_v_x, type, x: hF_Mirabelle_hf).
% 2.47/1.56 thf(sy_v_y, type, y: hF_Mirabelle_hf).
% 2.47/1.56 thf(fact_0_hpair__iff, axiom, (! [A2:hF_Mirabelle_hf,B:hF_Mirabelle_hf,A3:hF_Mirabelle_hf,B2:hF_Mirabelle_hf]: (((((hF_Mirabelle_hpair @ A2) @ B) = ((hF_Mirabelle_hpair @ A3) @ B2)) = ((A2 = A3) & (B = B2)))))).
% 2.47/1.56 thf(fact_1_hpair__inject, axiom, (! [A2:hF_Mirabelle_hf,B:hF_Mirabelle_hf,A3:hF_Mirabelle_hf,B2:hF_Mirabelle_hf]: (((((hF_Mirabelle_hpair @ A2) @ B) = ((hF_Mirabelle_hpair @ A3) @ B2)) => (~ (((A2 = A3) => (B != B2)))))))).
% 2.47/1.56 thf(fact_2_hpair__neq__fst, axiom, (! [A2:hF_Mirabelle_hf,B:hF_Mirabelle_hf]: ((((hF_Mirabelle_hpair @ A2) @ B) != A2)))).
% 2.47/1.56 thf(fact_3_hpair__neq__snd, axiom, (! [A2:hF_Mirabelle_hf,B:hF_Mirabelle_hf]: ((((hF_Mirabelle_hpair @ A2) @ B) != B)))).
% 2.47/1.56 thf(fact_4_hsplit, axiom, (! [A:$tType,C:(hF_Mirabelle_hf > (hF_Mirabelle_hf > A)),A2:hF_Mirabelle_hf,B:hF_Mirabelle_hf]: (((((hF_Mirabelle_hsplit @ A) @ C) @ ((hF_Mirabelle_hpair @ A2) @ B)) = ((C @ A2) @ B))))).
% 2.47/1.56 thf(fact_5_hfst__conv, axiom, (! [A2:hF_Mirabelle_hf,B:hF_Mirabelle_hf]: (((hF_Mirabelle_hfst @ ((hF_Mirabelle_hpair @ A2) @ B)) = A2)))).
% 2.47/1.56 thf(fact_6_hsnd__conv, axiom, (! [A2:hF_Mirabelle_hf,B:hF_Mirabelle_hf]: (((hF_Mirabelle_hsnd @ ((hF_Mirabelle_hpair @ A2) @ B)) = B)))).
% 2.47/1.56 thf(fact_7_HInter__hempty, axiom, ((hF_Mirabelle_HInter @ (zero_zero @ hF_Mirabelle_hf)) = (zero_zero @ hF_Mirabelle_hf))).
% 2.47/1.56 thf(fact_8_HCollect__hempty, axiom, (! [P:(hF_Mirabelle_hf > $o)]: ((((hF_Mir1687042746ollect @ P) @ (zero_zero @ hF_Mirabelle_hf)) = (zero_zero @ hF_Mirabelle_hf))))).
% 2.47/1.56 thf(fact_9_Replace__0, axiom, (! [R:(hF_Mirabelle_hf > (hF_Mirabelle_hf > $o))]: ((((hF_Mirabelle_Replace @ (zero_zero @ hF_Mirabelle_hf)) @ R) = (zero_zero @ hF_Mirabelle_hf))))).
% 2.47/1.56 thf(fact_10_HUnion__hempty, axiom, ((hF_Mirabelle_HUnion @ (zero_zero @ hF_Mirabelle_hf)) = (zero_zero @ hF_Mirabelle_hf))).
% 2.47/1.56 thf(fact_11_zero__reorient, axiom, (! [A:$tType]: (((zero @ A) => (! [X:A]: ((((zero_zero @ A) = X) = (X = (zero_zero @ A))))))))).
% 2.47/1.56 thf(fact_12_RepFun__0, axiom, (! [F:(hF_Mirabelle_hf > hF_Mirabelle_hf)]: ((((hF_Mirabelle_RepFun @ (zero_zero @ hF_Mirabelle_hf)) @ F) = (zero_zero @ hF_Mirabelle_hf))))).
% 2.47/1.56 thf(fact_13_hpair__def, axiom, (hF_Mirabelle_hpair = (^ [A4:hF_Mirabelle_hf,B3:hF_Mirabelle_hf]: (((hF_Mirabelle_hinsert @ ((hF_Mirabelle_hinsert @ A4) @ (zero_zero @ hF_Mirabelle_hf))) @ ((hF_Mirabelle_hinsert @ ((hF_Mirabelle_hinsert @ A4) @ ((hF_Mirabelle_hinsert @ B3) @ (zero_zero @ hF_Mirabelle_hf)))) @ (zero_zero @ hF_Mirabelle_hf))))))).
% 2.47/1.56 thf(fact_14_hpair__def_H, axiom, (hF_Mirabelle_hpair = (^ [A4:hF_Mirabelle_hf,B3:hF_Mirabelle_hf]: (((hF_Mirabelle_hinsert @ ((hF_Mirabelle_hinsert @ A4) @ ((hF_Mirabelle_hinsert @ A4) @ (zero_zero @ hF_Mirabelle_hf)))) @ ((hF_Mirabelle_hinsert @ ((hF_Mirabelle_hinsert @ A4) @ ((hF_Mirabelle_hinsert @ B3) @ (zero_zero @ hF_Mirabelle_hf)))) @ (zero_zero @ hF_Mirabelle_hf))))))).
% 2.47/1.56 thf(fact_15_singleton__eq__iff, axiom, (! [A2:hF_Mirabelle_hf,B:hF_Mirabelle_hf]: (((((hF_Mirabelle_hinsert @ A2) @ (zero_zero @ hF_Mirabelle_hf)) = ((hF_Mirabelle_hinsert @ B) @ (zero_zero @ hF_Mirabelle_hf))) = (A2 = B))))).
% 2.47/1.56 thf(fact_16_RepFun__hinsert, axiom, (! [A2:hF_Mirabelle_hf,B:hF_Mirabelle_hf,F:(hF_Mirabelle_hf > hF_Mirabelle_hf)]: ((((hF_Mirabelle_RepFun @ ((hF_Mirabelle_hinsert @ A2) @ B)) @ F) = ((hF_Mirabelle_hinsert @ (F @ A2)) @ ((hF_Mirabelle_RepFun @ B) @ F)))))).
% 2.47/1.56 thf(fact_17_hinsert__commute, axiom, (! [X:hF_Mirabelle_hf,Y:hF_Mirabelle_hf,Z:hF_Mirabelle_hf]: ((((hF_Mirabelle_hinsert @ X) @ ((hF_Mirabelle_hinsert @ Y) @ Z)) = ((hF_Mirabelle_hinsert @ Y) @ ((hF_Mirabelle_hinsert @ X) @ Z)))))).
% 2.47/1.56 thf(fact_18_hinsert__nonempty, axiom, (! [A2:hF_Mirabelle_hf,A5:hF_Mirabelle_hf]: ((((hF_Mirabelle_hinsert @ A2) @ A5) != (zero_zero @ hF_Mirabelle_hf))))).
% 2.47/1.56 thf(fact_19_HF__Mirabelle__fsbjehakzm_Odoubleton__eq__iff, axiom, (! [A2:hF_Mirabelle_hf,B:hF_Mirabelle_hf,C:hF_Mirabelle_hf,D:hF_Mirabelle_hf]: (((((hF_Mirabelle_hinsert @ A2) @ ((hF_Mirabelle_hinsert @ B) @ (zero_zero @ hF_Mirabelle_hf))) = ((hF_Mirabelle_hinsert @ C) @ ((hF_Mirabelle_hinsert @ D) @ (zero_zero @ hF_Mirabelle_hf)))) = (((A2 = C) & (B = D)) | ((A2 = D) & (B = C))))))).
% 2.47/1.56 thf(fact_20_hf__induct__ax, axiom, (! [P:(hF_Mirabelle_hf > $o),X:hF_Mirabelle_hf]: (((P @ (zero_zero @ hF_Mirabelle_hf)) => ((! [X2:hF_Mirabelle_hf]: (((P @ X2) => (! [Y2:hF_Mirabelle_hf]: (((P @ Y2) => (P @ ((hF_Mirabelle_hinsert @ Y2) @ X2)))))))) => (P @ X)))))).
% 2.47/1.56 thf(fact_21_HInter__hinsert, axiom, (! [A5:hF_Mirabelle_hf,A2:hF_Mirabelle_hf]: (((A5 != (zero_zero @ hF_Mirabelle_hf)) => ((hF_Mirabelle_HInter @ ((hF_Mirabelle_hinsert @ A2) @ A5)) = (((inf_inf @ hF_Mirabelle_hf) @ A2) @ (hF_Mirabelle_HInter @ A5))))))).
% 2.47/1.56 thf(fact_22_Abs__hf__0, axiom, ((hF_Mirabelle_Abs_hf @ (zero_zero @ nat)) = (zero_zero @ hF_Mirabelle_hf))).
% 2.47/1.56 thf(fact_23_HInter__iff, axiom, (! [A5:hF_Mirabelle_hf,X:hF_Mirabelle_hf]: (((A5 != (zero_zero @ hF_Mirabelle_hf)) => (((hF_Mirabelle_hmem @ X) @ (hF_Mirabelle_HInter @ A5)) = (! [Y3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ Y3) @ A5) => ((hF_Mirabelle_hmem @ X) @ Y3))))))))).
% 2.47/1.56 thf(fact_24_HUnion__hinsert, axiom, (! [A2:hF_Mirabelle_hf,A5:hF_Mirabelle_hf]: (((hF_Mirabelle_HUnion @ ((hF_Mirabelle_hinsert @ A2) @ A5)) = (((sup_sup @ hF_Mirabelle_hf) @ A2) @ (hF_Mirabelle_HUnion @ A5)))))).
% 2.47/1.56 thf(fact_25_hdiff__insert, axiom, (! [A5:hF_Mirabelle_hf,A2:hF_Mirabelle_hf,B4:hF_Mirabelle_hf]: (((((minus_minus @ hF_Mirabelle_hf) @ A5) @ ((hF_Mirabelle_hinsert @ A2) @ B4)) = (((minus_minus @ hF_Mirabelle_hf) @ (((minus_minus @ hF_Mirabelle_hf) @ A5) @ B4)) @ ((hF_Mirabelle_hinsert @ A2) @ (zero_zero @ hF_Mirabelle_hf))))))).
% 2.47/1.56 thf(fact_26_hinsert__eq__sup, axiom, (hF_Mirabelle_hinsert = (^ [A4:hF_Mirabelle_hf,B3:hF_Mirabelle_hf]: ((((sup_sup @ hF_Mirabelle_hf) @ B3) @ ((hF_Mirabelle_hinsert @ A4) @ (zero_zero @ hF_Mirabelle_hf))))))).
% 2.47/1.56 thf(fact_27_hf__induct, axiom, (! [P:(hF_Mirabelle_hf > $o),Z:hF_Mirabelle_hf]: (((P @ (zero_zero @ hF_Mirabelle_hf)) => ((! [X2:hF_Mirabelle_hf,Y2:hF_Mirabelle_hf]: (((P @ X2) => ((P @ Y2) => ((~ (((hF_Mirabelle_hmem @ X2) @ Y2))) => (P @ ((hF_Mirabelle_hinsert @ X2) @ Y2))))))) => (P @ Z)))))).
% 2.47/1.56 thf(fact_28_hf__cases, axiom, (! [Y:hF_Mirabelle_hf]: (((Y != (zero_zero @ hF_Mirabelle_hf)) => (~ ((! [A6:hF_Mirabelle_hf,B5:hF_Mirabelle_hf]: (((Y = ((hF_Mirabelle_hinsert @ A6) @ B5)) => ((hF_Mirabelle_hmem @ A6) @ B5)))))))))).
% 2.47/1.56 thf(fact_29_Replace__hunion, axiom, (! [A5:hF_Mirabelle_hf,B4:hF_Mirabelle_hf,R:(hF_Mirabelle_hf > (hF_Mirabelle_hf > $o))]: ((((hF_Mirabelle_Replace @ (((sup_sup @ hF_Mirabelle_hf) @ A5) @ B4)) @ R) = (((sup_sup @ hF_Mirabelle_hf) @ ((hF_Mirabelle_Replace @ A5) @ R)) @ ((hF_Mirabelle_Replace @ B4) @ R)))))).
% 2.47/1.56 thf(fact_30_HCollect__iff, axiom, (! [X:hF_Mirabelle_hf,P:(hF_Mirabelle_hf > $o),A5:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X) @ ((hF_Mir1687042746ollect @ P) @ A5)) = ((P @ X) & ((hF_Mirabelle_hmem @ X) @ A5)))))).
% 2.47/1.56 thf(fact_31_hf__equalityI, axiom, (! [A2:hF_Mirabelle_hf,B:hF_Mirabelle_hf]: (((! [X2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X2) @ A2) = ((hF_Mirabelle_hmem @ X2) @ B)))) => (A2 = B))))).
% 2.47/1.56 thf(fact_32_diff__self, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ A2) = (zero_zero @ A)))))))).
% 2.47/1.56 thf(fact_33_diff__0__right, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ (zero_zero @ A)) = A2))))))).
% 2.47/1.56 thf(fact_34_zero__diff, axiom, (! [A:$tType]: (((comm_monoid_diff @ A) => (! [A2:A]: (((((minus_minus @ A) @ (zero_zero @ A)) @ A2) = (zero_zero @ A)))))))).
% 2.47/1.56 thf(fact_35_diff__zero, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ (zero_zero @ A)) = A2))))))).
% 2.47/1.56 thf(fact_36_cancel__comm__monoid__add__class_Odiff__cancel, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ A2) = (zero_zero @ A)))))))).
% 2.47/1.56 thf(fact_37_hmem__hinsert, axiom, (! [A2:hF_Mirabelle_hf,B:hF_Mirabelle_hf,C:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ A2) @ ((hF_Mirabelle_hinsert @ B) @ C)) = ((A2 = B) | ((hF_Mirabelle_hmem @ A2) @ C)))))).
% 2.47/1.56 thf(fact_38_hunion__iff, axiom, (! [X:hF_Mirabelle_hf,A2:hF_Mirabelle_hf,B:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X) @ (((sup_sup @ hF_Mirabelle_hf) @ A2) @ B)) = (((hF_Mirabelle_hmem @ X) @ A2) | ((hF_Mirabelle_hmem @ X) @ B)))))).
% 2.47/1.56 thf(fact_39_hinter__iff, axiom, (! [U:hF_Mirabelle_hf,X:hF_Mirabelle_hf,Y:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U) @ (((inf_inf @ hF_Mirabelle_hf) @ X) @ Y)) = (((hF_Mirabelle_hmem @ U) @ X) & ((hF_Mirabelle_hmem @ U) @ Y)))))).
% 2.47/1.56 thf(fact_40_RepFun__iff, axiom, (! [V:hF_Mirabelle_hf,A5:hF_Mirabelle_hf,F:(hF_Mirabelle_hf > hF_Mirabelle_hf)]: ((((hF_Mirabelle_hmem @ V) @ ((hF_Mirabelle_RepFun @ A5) @ F)) = (? [U2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U2) @ A5) & (V = (F @ U2))))))))).
% 2.47/1.56 thf(fact_41_hunion__hempty__left, axiom, (! [A5:hF_Mirabelle_hf]: (((((sup_sup @ hF_Mirabelle_hf) @ (zero_zero @ hF_Mirabelle_hf)) @ A5) = A5)))).
% 2.47/1.56 thf(fact_42_hunion__hempty__right, axiom, (! [A5:hF_Mirabelle_hf]: (((((sup_sup @ hF_Mirabelle_hf) @ A5) @ (zero_zero @ hF_Mirabelle_hf)) = A5)))).
% 2.47/1.56 thf(fact_43_HUnion__iff, axiom, (! [X:hF_Mirabelle_hf,A5:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X) @ (hF_Mirabelle_HUnion @ A5)) = (? [Y3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ Y3) @ A5) & ((hF_Mirabelle_hmem @ X) @ Y3)))))))).
% 2.47/1.56 thf(fact_44_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.47/1.56 thf(fact_45_Collect__mem__eq, axiom, (! [A:$tType,A5:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ A5)))) = A5)))).
% 2.47/1.56 thf(fact_46_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P @ X2) = (Q @ X2)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.47/1.56 thf(fact_47_ext, axiom, (! [B6:$tType,A:$tType,F:(A > B6),G:(A > B6)]: (((! [X2:A]: (((F @ X2) = (G @ X2)))) => (F = G))))).
% 2.47/1.56 thf(fact_48_hdiff__iff, axiom, (! [U:hF_Mirabelle_hf,X:hF_Mirabelle_hf,Y:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U) @ (((minus_minus @ hF_Mirabelle_hf) @ X) @ Y)) = (((hF_Mirabelle_hmem @ U) @ X) & (~ (((hF_Mirabelle_hmem @ U) @ Y)))))))).
% 2.47/1.56 thf(fact_49_hinter__hempty__left, axiom, (! [A5:hF_Mirabelle_hf]: (((((inf_inf @ hF_Mirabelle_hf) @ (zero_zero @ hF_Mirabelle_hf)) @ A5) = (zero_zero @ hF_Mirabelle_hf))))).
% 2.47/1.56 thf(fact_50_hinter__hempty__right, axiom, (! [A5:hF_Mirabelle_hf]: (((((inf_inf @ hF_Mirabelle_hf) @ A5) @ (zero_zero @ hF_Mirabelle_hf)) = (zero_zero @ hF_Mirabelle_hf))))).
% 2.47/1.56 thf(fact_51_Replace__iff, axiom, (! [V:hF_Mirabelle_hf,A5:hF_Mirabelle_hf,R:(hF_Mirabelle_hf > (hF_Mirabelle_hf > $o))]: ((((hF_Mirabelle_hmem @ V) @ ((hF_Mirabelle_Replace @ A5) @ R)) = (? [U2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U2) @ A5) & (((R @ U2) @ V) & (! [Y3:hF_Mirabelle_hf]: ((((R @ U2) @ Y3) => (Y3 = V)))))))))))).
% 2.47/1.56 thf(fact_52_hdiff__zero, axiom, (! [X:hF_Mirabelle_hf]: (((((minus_minus @ hF_Mirabelle_hf) @ X) @ (zero_zero @ hF_Mirabelle_hf)) = X)))).
% 2.47/1.56 thf(fact_53_zero__hdiff, axiom, (! [X:hF_Mirabelle_hf]: (((((minus_minus @ hF_Mirabelle_hf) @ (zero_zero @ hF_Mirabelle_hf)) @ X) = (zero_zero @ hF_Mirabelle_hf))))).
% 2.47/1.56 thf(fact_54_RepFun__hunion, axiom, (! [A5:hF_Mirabelle_hf,B4:hF_Mirabelle_hf,F:(hF_Mirabelle_hf > hF_Mirabelle_hf)]: ((((hF_Mirabelle_RepFun @ (((sup_sup @ hF_Mirabelle_hf) @ A5) @ B4)) @ F) = (((sup_sup @ hF_Mirabelle_hf) @ ((hF_Mirabelle_RepFun @ A5) @ F)) @ ((hF_Mirabelle_RepFun @ B4) @ F)))))).
% 2.47/1.56 thf(fact_55_HUnion__hunion, axiom, (! [A5:hF_Mirabelle_hf,B4:hF_Mirabelle_hf]: (((hF_Mirabelle_HUnion @ (((sup_sup @ hF_Mirabelle_hf) @ A5) @ B4)) = (((sup_sup @ hF_Mirabelle_hf) @ (hF_Mirabelle_HUnion @ A5)) @ (hF_Mirabelle_HUnion @ B4)))))).
% 2.47/1.56 thf(fact_56_diff__eq__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B:A,C:A,D:A]: ((((((minus_minus @ A) @ A2) @ B) = (((minus_minus @ A) @ C) @ D)) => ((A2 = B) = (C = D))))))))).
% 2.47/1.56 thf(fact_57_diff__right__commute, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A2:A,C:A,B:A]: (((((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ C)) @ B) = (((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ B)) @ C)))))))).
% 2.47/1.56 thf(fact_58_replacement__fun, axiom, (! [X:hF_Mirabelle_hf,F:(hF_Mirabelle_hf > hF_Mirabelle_hf)]: ((? [Z2:hF_Mirabelle_hf]: ((! [V2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ V2) @ Z2) = (? [U2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U2) @ X) & (V2 = (F @ U2))))))))))))).
% 2.47/1.56 thf(fact_59_hmem__not__refl, axiom, (! [X:hF_Mirabelle_hf]: ((~ (((hF_Mirabelle_hmem @ X) @ X)))))).
% 2.47/1.56 thf(fact_60_comprehension, axiom, (! [X:hF_Mirabelle_hf,P:(hF_Mirabelle_hf > $o)]: ((? [Z2:hF_Mirabelle_hf]: ((! [U3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U3) @ Z2) = (((hF_Mirabelle_hmem @ U3) @ X) & (P @ U3)))))))))).
% 2.47/1.56 thf(fact_61_union__of__set, axiom, (! [X:hF_Mirabelle_hf]: ((? [Z2:hF_Mirabelle_hf]: ((! [U3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U3) @ Z2) = (? [Y3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ Y3) @ X) & ((hF_Mirabelle_hmem @ U3) @ Y3)))))))))))).
% 2.47/1.56 thf(fact_62_hmem__not__sym, axiom, (! [X:hF_Mirabelle_hf,Y:hF_Mirabelle_hf]: ((~ ((((hF_Mirabelle_hmem @ X) @ Y) & ((hF_Mirabelle_hmem @ Y) @ X))))))).
% 2.47/1.56 thf(fact_63_binary__union, axiom, (! [X:hF_Mirabelle_hf,Y:hF_Mirabelle_hf]: ((? [Z2:hF_Mirabelle_hf]: ((! [U3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U3) @ Z2) = (((hF_Mirabelle_hmem @ U3) @ X) | ((hF_Mirabelle_hmem @ U3) @ Y)))))))))).
% 2.47/1.56 thf(fact_64_replacement, axiom, (! [X:hF_Mirabelle_hf,R:(hF_Mirabelle_hf > (hF_Mirabelle_hf > $o))]: (((! [U4:hF_Mirabelle_hf,V3:hF_Mirabelle_hf,V4:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U4) @ X) => (((R @ U4) @ V3) => (((R @ U4) @ V4) => (V4 = V3)))))) => (? [Z2:hF_Mirabelle_hf]: ((! [V2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ V2) @ Z2) = (? [U2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U2) @ X) & ((R @ U2) @ V2))))))))))))).
% 2.47/1.56 thf(fact_65_hmem__ne, axiom, (! [X:hF_Mirabelle_hf,Y:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X) @ Y) => (X != Y))))).
% 2.47/1.56 thf(fact_66_hf__ext, axiom, ((^ [Y4:hF_Mirabelle_hf,Z3:hF_Mirabelle_hf]: ((Y4 = Z3))) = (^ [A4:hF_Mirabelle_hf,B3:hF_Mirabelle_hf]: ((! [X3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X3) @ A4) = ((hF_Mirabelle_hmem @ X3) @ B3)))))))).
% 2.47/1.56 thf(fact_67_hinter__hinsert__right, axiom, (! [X:hF_Mirabelle_hf,B4:hF_Mirabelle_hf,A5:hF_Mirabelle_hf]: (((((hF_Mirabelle_hmem @ X) @ B4) => ((((inf_inf @ hF_Mirabelle_hf) @ B4) @ ((hF_Mirabelle_hinsert @ X) @ A5)) = ((hF_Mirabelle_hinsert @ X) @ (((inf_inf @ hF_Mirabelle_hf) @ B4) @ A5)))) & ((~ (((hF_Mirabelle_hmem @ X) @ B4))) => ((((inf_inf @ hF_Mirabelle_hf) @ B4) @ ((hF_Mirabelle_hinsert @ X) @ A5)) = (((inf_inf @ hF_Mirabelle_hf) @ B4) @ A5))))))).
% 2.47/1.56 thf(fact_68_hinter__hinsert__left, axiom, (! [X:hF_Mirabelle_hf,B4:hF_Mirabelle_hf,A5:hF_Mirabelle_hf]: (((((hF_Mirabelle_hmem @ X) @ B4) => ((((inf_inf @ hF_Mirabelle_hf) @ ((hF_Mirabelle_hinsert @ X) @ A5)) @ B4) = ((hF_Mirabelle_hinsert @ X) @ (((inf_inf @ hF_Mirabelle_hf) @ A5) @ B4)))) & ((~ (((hF_Mirabelle_hmem @ X) @ B4))) => ((((inf_inf @ hF_Mirabelle_hf) @ ((hF_Mirabelle_hinsert @ X) @ A5)) @ B4) = (((inf_inf @ hF_Mirabelle_hf) @ A5) @ B4))))))).
% 2.47/1.56 thf(fact_69_foundation, axiom, (! [Z:hF_Mirabelle_hf]: (((Z != (zero_zero @ hF_Mirabelle_hf)) => (? [W:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ W) @ Z) & ((((inf_inf @ hF_Mirabelle_hf) @ W) @ Z) = (zero_zero @ hF_Mirabelle_hf))))))))).
% 2.47/1.56 thf(fact_70_hinsert__hdiff__if, axiom, (! [X:hF_Mirabelle_hf,B4:hF_Mirabelle_hf,A5:hF_Mirabelle_hf]: (((((hF_Mirabelle_hmem @ X) @ B4) => ((((minus_minus @ hF_Mirabelle_hf) @ ((hF_Mirabelle_hinsert @ X) @ A5)) @ B4) = (((minus_minus @ hF_Mirabelle_hf) @ A5) @ B4))) & ((~ (((hF_Mirabelle_hmem @ X) @ B4))) => ((((minus_minus @ hF_Mirabelle_hf) @ ((hF_Mirabelle_hinsert @ X) @ A5)) @ B4) = ((hF_Mirabelle_hinsert @ X) @ (((minus_minus @ hF_Mirabelle_hf) @ A5) @ B4)))))))).
% 2.47/1.56 thf(fact_71_eq__iff__diff__eq__0, axiom, (! [A:$tType]: (((group_add @ A) => ((^ [Y4:A,Z3:A]: ((Y4 = Z3))) = (^ [A4:A,B3:A]: (((((minus_minus @ A) @ A4) @ B3) = (zero_zero @ A))))))))).
% 2.47/1.56 thf(fact_72_hinsert__iff, axiom, (! [Z:hF_Mirabelle_hf,Y:hF_Mirabelle_hf,X:hF_Mirabelle_hf]: (((Z = ((hF_Mirabelle_hinsert @ Y) @ X)) = (! [U2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U2) @ Z) = (((hF_Mirabelle_hmem @ U2) @ X) | (U2 = Y))))))))).
% 2.47/1.56 thf(fact_73_hemptyE, axiom, (! [A2:hF_Mirabelle_hf]: ((~ (((hF_Mirabelle_hmem @ A2) @ (zero_zero @ hF_Mirabelle_hf))))))).
% 2.47/1.56 thf(fact_74_hempty__iff, axiom, (! [Z:hF_Mirabelle_hf]: (((Z = (zero_zero @ hF_Mirabelle_hf)) = (! [X3:hF_Mirabelle_hf]: ((~ (((hF_Mirabelle_hmem @ X3) @ Z))))))))).
% 2.47/1.56 thf(fact_75_hmem__hempty, axiom, (! [A2:hF_Mirabelle_hf]: ((~ (((hF_Mirabelle_hmem @ A2) @ (zero_zero @ hF_Mirabelle_hf))))))).
% 2.47/1.56 thf(fact_76_hunion__hinsert__right, axiom, (! [B4:hF_Mirabelle_hf,X:hF_Mirabelle_hf,A5:hF_Mirabelle_hf]: (((((sup_sup @ hF_Mirabelle_hf) @ B4) @ ((hF_Mirabelle_hinsert @ X) @ A5)) = ((hF_Mirabelle_hinsert @ X) @ (((sup_sup @ hF_Mirabelle_hf) @ B4) @ A5)))))).
% 2.47/1.56 thf(fact_77_hunion__hinsert__left, axiom, (! [X:hF_Mirabelle_hf,A5:hF_Mirabelle_hf,B4:hF_Mirabelle_hf]: (((((sup_sup @ hF_Mirabelle_hf) @ ((hF_Mirabelle_hinsert @ X) @ A5)) @ B4) = ((hF_Mirabelle_hinsert @ X) @ (((sup_sup @ hF_Mirabelle_hf) @ A5) @ B4)))))).
% 2.47/1.56 thf(fact_78_RepFun__cong, axiom, (! [A5:hF_Mirabelle_hf,B4:hF_Mirabelle_hf,F:(hF_Mirabelle_hf > hF_Mirabelle_hf),G:(hF_Mirabelle_hf > hF_Mirabelle_hf)]: (((A5 = B4) => ((! [X2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X2) @ B4) => ((F @ X2) = (G @ X2))))) => (((hF_Mirabelle_RepFun @ A5) @ F) = ((hF_Mirabelle_RepFun @ B4) @ G))))))).
% 2.47/1.56 thf(fact_79_hmem__Sup__ne, axiom, (! [X:hF_Mirabelle_hf,Y:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X) @ Y) => ((hF_Mirabelle_HUnion @ X) != Y))))).
% 2.47/1.56 thf(fact_80_Replace__cong, axiom, (! [A5:hF_Mirabelle_hf,B4:hF_Mirabelle_hf,P:(hF_Mirabelle_hf > (hF_Mirabelle_hf > $o)),Q:(hF_Mirabelle_hf > (hF_Mirabelle_hf > $o))]: (((A5 = B4) => ((! [X2:hF_Mirabelle_hf,Y2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X2) @ B4) => (((P @ X2) @ Y2) = ((Q @ X2) @ Y2))))) => (((hF_Mirabelle_Replace @ A5) @ P) = ((hF_Mirabelle_Replace @ B4) @ Q))))))).
% 2.47/1.56 thf(fact_81_sup__inf__absorb, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y:A]: (((((sup_sup @ A) @ X) @ (((inf_inf @ A) @ X) @ Y)) = X))))))).
% 2.47/1.56 thf(fact_82_inf__sup__absorb, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y:A]: (((((inf_inf @ A) @ X) @ (((sup_sup @ A) @ X) @ Y)) = X))))))).
% 2.47/1.56 thf(fact_83_sup_Oright__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,B:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A2) @ B)) @ B) = (((sup_sup @ A) @ A2) @ B)))))))).
% 2.47/1.56 thf(fact_84_sup__left__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A,Y:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ X) @ Y)) = (((sup_sup @ A) @ X) @ Y)))))))).
% 2.47/1.56 thf(fact_85_sup_Oleft__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,B:A]: (((((sup_sup @ A) @ A2) @ (((sup_sup @ A) @ A2) @ B)) = (((sup_sup @ A) @ A2) @ B)))))))).
% 2.47/1.56 thf(fact_86_sup__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A]: (((((sup_sup @ A) @ X) @ X) = X))))))).
% 2.47/1.56 thf(fact_87_sup_Oidem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A]: (((((sup_sup @ A) @ A2) @ A2) = A2))))))).
% 2.47/1.56 thf(fact_88_sup__apply, axiom, (! [B6:$tType,A:$tType]: (((semilattice_sup @ B6) => ((sup_sup @ (A > B6)) = (^ [F2:(A > B6),G2:(A > B6),X3:A]: ((((sup_sup @ B6) @ (F2 @ X3)) @ (G2 @ X3))))))))).
% 2.47/1.56 thf(fact_89_inf__right__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,Y:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ X) @ Y)) @ Y) = (((inf_inf @ A) @ X) @ Y)))))))).
% 2.47/1.56 thf(fact_90_minus__apply, axiom, (! [B6:$tType,A:$tType]: (((minus @ B6) => ((minus_minus @ (A > B6)) = (^ [A7:(A > B6),B7:(A > B6),X3:A]: ((((minus_minus @ B6) @ (A7 @ X3)) @ (B7 @ X3))))))))).
% 2.47/1.56 thf(fact_91_inf__apply, axiom, (! [B6:$tType,A:$tType]: (((semilattice_inf @ B6) => ((inf_inf @ (A > B6)) = (^ [F2:(A > B6),G2:(A > B6),X3:A]: ((((inf_inf @ B6) @ (F2 @ X3)) @ (G2 @ X3))))))))).
% 2.47/1.56 thf(fact_92_inf_Oidem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A]: (((((inf_inf @ A) @ A2) @ A2) = A2))))))).
% 2.47/1.56 thf(fact_93_inf__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A]: (((((inf_inf @ A) @ X) @ X) = X))))))).
% 2.47/1.56 thf(fact_94_inf_Oleft__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B:A]: (((((inf_inf @ A) @ A2) @ (((inf_inf @ A) @ A2) @ B)) = (((inf_inf @ A) @ A2) @ B)))))))).
% 2.47/1.56 thf(fact_95_inf__left__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,Y:A]: (((((inf_inf @ A) @ X) @ (((inf_inf @ A) @ X) @ Y)) = (((inf_inf @ A) @ X) @ Y)))))))).
% 2.47/1.56 thf(fact_96_inf_Oright__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ A2) @ B)) @ B) = (((inf_inf @ A) @ A2) @ B)))))))).
% 2.47/1.56 thf(fact_97_fun__diff__def, axiom, (! [B6:$tType,A:$tType]: (((minus @ B6) => ((minus_minus @ (A > B6)) = (^ [A7:(A > B6),B7:(A > B6),X3:A]: ((((minus_minus @ B6) @ (A7 @ X3)) @ (B7 @ X3))))))))).
% 2.47/1.56 thf(fact_98_inf__sup__aci_I4_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y:A]: (((((inf_inf @ A) @ X) @ (((inf_inf @ A) @ X) @ Y)) = (((inf_inf @ A) @ X) @ Y)))))))).
% 2.47/1.56 thf(fact_99_inf__sup__aci_I3_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y:A,Z:A]: (((((inf_inf @ A) @ X) @ (((inf_inf @ A) @ Y) @ Z)) = (((inf_inf @ A) @ Y) @ (((inf_inf @ A) @ X) @ Z))))))))).
% 2.47/1.56 thf(fact_100_inf__sup__aci_I2_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y:A,Z:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ X) @ Y)) @ Z) = (((inf_inf @ A) @ X) @ (((inf_inf @ A) @ Y) @ Z))))))))).
% 2.47/1.56 thf(fact_101_inf__sup__aci_I1_J, axiom, (! [A:$tType]: (((lattice @ A) => ((inf_inf @ A) = (^ [X3:A,Y3:A]: ((((inf_inf @ A) @ Y3) @ X3)))))))).
% 2.47/1.56 thf(fact_102_inf__fun__def, axiom, (! [B6:$tType,A:$tType]: (((semilattice_inf @ B6) => ((inf_inf @ (A > B6)) = (^ [F2:(A > B6),G2:(A > B6),X3:A]: ((((inf_inf @ B6) @ (F2 @ X3)) @ (G2 @ X3))))))))).
% 2.47/1.56 thf(fact_103_boolean__algebra__cancel_Oinf1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A5:A,K:A,A2:A,B:A]: (((A5 = (((inf_inf @ A) @ K) @ A2)) => ((((inf_inf @ A) @ A5) @ B) = (((inf_inf @ A) @ K) @ (((inf_inf @ A) @ A2) @ B)))))))))).
% 2.47/1.56 thf(fact_104_boolean__algebra__cancel_Oinf2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B4:A,K:A,B:A,A2:A]: (((B4 = (((inf_inf @ A) @ K) @ B)) => ((((inf_inf @ A) @ A2) @ B4) = (((inf_inf @ A) @ K) @ (((inf_inf @ A) @ A2) @ B)))))))))).
% 2.47/1.56 thf(fact_105_inf_Oassoc, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B:A,C:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ A2) @ B)) @ C) = (((inf_inf @ A) @ A2) @ (((inf_inf @ A) @ B) @ C))))))))).
% 2.47/1.56 thf(fact_106_inf__assoc, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,Y:A,Z:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ X) @ Y)) @ Z) = (((inf_inf @ A) @ X) @ (((inf_inf @ A) @ Y) @ Z))))))))).
% 2.47/1.56 thf(fact_107_inf_Ocommute, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((inf_inf @ A) = (^ [A4:A,B3:A]: ((((inf_inf @ A) @ B3) @ A4)))))))).
% 2.47/1.56 thf(fact_108_inf__commute, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((inf_inf @ A) = (^ [X3:A,Y3:A]: ((((inf_inf @ A) @ Y3) @ X3)))))))).
% 2.47/1.56 thf(fact_109_inf_Oleft__commute, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B:A,A2:A,C:A]: (((((inf_inf @ A) @ B) @ (((inf_inf @ A) @ A2) @ C)) = (((inf_inf @ A) @ A2) @ (((inf_inf @ A) @ B) @ C))))))))).
% 2.47/1.56 thf(fact_110_inf__left__commute, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,Y:A,Z:A]: (((((inf_inf @ A) @ X) @ (((inf_inf @ A) @ Y) @ Z)) = (((inf_inf @ A) @ Y) @ (((inf_inf @ A) @ X) @ Z))))))))).
% 2.47/1.56 thf(fact_111_inf__sup__aci_I8_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ X) @ Y)) = (((sup_sup @ A) @ X) @ Y)))))))).
% 2.47/1.56 thf(fact_112_inf__sup__aci_I7_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y:A,Z:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z)) = (((sup_sup @ A) @ Y) @ (((sup_sup @ A) @ X) @ Z))))))))).
% 2.47/1.56 thf(fact_113_inf__sup__aci_I6_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y:A,Z:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X) @ Y)) @ Z) = (((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z))))))))).
% 2.47/1.56 thf(fact_114_inf__sup__aci_I5_J, axiom, (! [A:$tType]: (((lattice @ A) => ((sup_sup @ A) = (^ [X3:A,Y3:A]: ((((sup_sup @ A) @ Y3) @ X3)))))))).
% 2.47/1.56 thf(fact_115_sup__fun__def, axiom, (! [B6:$tType,A:$tType]: (((semilattice_sup @ B6) => ((sup_sup @ (A > B6)) = (^ [F2:(A > B6),G2:(A > B6),X3:A]: ((((sup_sup @ B6) @ (F2 @ X3)) @ (G2 @ X3))))))))).
% 2.47/1.56 thf(fact_116_boolean__algebra__cancel_Osup1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A5:A,K:A,A2:A,B:A]: (((A5 = (((sup_sup @ A) @ K) @ A2)) => ((((sup_sup @ A) @ A5) @ B) = (((sup_sup @ A) @ K) @ (((sup_sup @ A) @ A2) @ B)))))))))).
% 2.47/1.56 thf(fact_117_boolean__algebra__cancel_Osup2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B4:A,K:A,B:A,A2:A]: (((B4 = (((sup_sup @ A) @ K) @ B)) => ((((sup_sup @ A) @ A2) @ B4) = (((sup_sup @ A) @ K) @ (((sup_sup @ A) @ A2) @ B)))))))))).
% 2.47/1.56 thf(fact_118_sup_Oassoc, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,B:A,C:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A2) @ B)) @ C) = (((sup_sup @ A) @ A2) @ (((sup_sup @ A) @ B) @ C))))))))).
% 2.47/1.56 thf(fact_119_sup__assoc, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A,Y:A,Z:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X) @ Y)) @ Z) = (((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z))))))))).
% 2.47/1.56 thf(fact_120_sup_Ocommute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((sup_sup @ A) = (^ [A4:A,B3:A]: ((((sup_sup @ A) @ B3) @ A4)))))))).
% 2.47/1.56 thf(fact_121_sup__commute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((sup_sup @ A) = (^ [X3:A,Y3:A]: ((((sup_sup @ A) @ Y3) @ X3)))))))).
% 2.47/1.56 thf(fact_122_sup_Oleft__commute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B:A,A2:A,C:A]: (((((sup_sup @ A) @ B) @ (((sup_sup @ A) @ A2) @ C)) = (((sup_sup @ A) @ A2) @ (((sup_sup @ A) @ B) @ C))))))))).
% 2.47/1.56 thf(fact_123_sup__left__commute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A,Y:A,Z:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z)) = (((sup_sup @ A) @ Y) @ (((sup_sup @ A) @ X) @ Z))))))))).
% 2.47/1.56 thf(fact_124_distrib__imp1, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y:A,Z:A]: (((! [X2:A,Y2:A,Z2:A]: (((((inf_inf @ A) @ X2) @ (((sup_sup @ A) @ Y2) @ Z2)) = (((sup_sup @ A) @ (((inf_inf @ A) @ X2) @ Y2)) @ (((inf_inf @ A) @ X2) @ Z2))))) => ((((sup_sup @ A) @ X) @ (((inf_inf @ A) @ Y) @ Z)) = (((inf_inf @ A) @ (((sup_sup @ A) @ X) @ Y)) @ (((sup_sup @ A) @ X) @ Z)))))))))).
% 2.47/1.56 thf(fact_125_distrib__imp2, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y:A,Z:A]: (((! [X2:A,Y2:A,Z2:A]: (((((sup_sup @ A) @ X2) @ (((inf_inf @ A) @ Y2) @ Z2)) = (((inf_inf @ A) @ (((sup_sup @ A) @ X2) @ Y2)) @ (((sup_sup @ A) @ X2) @ Z2))))) => ((((inf_inf @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z)) = (((sup_sup @ A) @ (((inf_inf @ A) @ X) @ Y)) @ (((inf_inf @ A) @ X) @ Z)))))))))).
% 2.47/1.56 thf(fact_126_inf__sup__distrib1, axiom, (! [A:$tType]: (((distrib_lattice @ A) => (! [X:A,Y:A,Z:A]: (((((inf_inf @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z)) = (((sup_sup @ A) @ (((inf_inf @ A) @ X) @ Y)) @ (((inf_inf @ A) @ X) @ Z))))))))).
% 2.47/1.56 thf(fact_127_inf__sup__distrib2, axiom, (! [A:$tType]: (((distrib_lattice @ A) => (! [Y:A,Z:A,X:A]: (((((inf_inf @ A) @ (((sup_sup @ A) @ Y) @ Z)) @ X) = (((sup_sup @ A) @ (((inf_inf @ A) @ Y) @ X)) @ (((inf_inf @ A) @ Z) @ X))))))))).
% 2.47/1.56 thf(fact_128_sup__inf__distrib1, axiom, (! [A:$tType]: (((distrib_lattice @ A) => (! [X:A,Y:A,Z:A]: (((((sup_sup @ A) @ X) @ (((inf_inf @ A) @ Y) @ Z)) = (((inf_inf @ A) @ (((sup_sup @ A) @ X) @ Y)) @ (((sup_sup @ A) @ X) @ Z))))))))).
% 2.47/1.56 thf(fact_129_sup__inf__distrib2, axiom, (! [A:$tType]: (((distrib_lattice @ A) => (! [Y:A,Z:A,X:A]: (((((sup_sup @ A) @ (((inf_inf @ A) @ Y) @ Z)) @ X) = (((inf_inf @ A) @ (((sup_sup @ A) @ Y) @ X)) @ (((sup_sup @ A) @ Z) @ X))))))))).
% 2.47/1.56 thf(fact_130_diff__self__eq__0, axiom, (! [M:nat]: (((((minus_minus @ nat) @ M) @ M) = (zero_zero @ nat))))).
% 2.47/1.56 thf(fact_131_diff__0__eq__0, axiom, (! [N:nat]: (((((minus_minus @ nat) @ (zero_zero @ nat)) @ N) = (zero_zero @ nat))))).
% 2.47/1.56 thf(fact_132_PrimReplace__iff, axiom, (! [A5:hF_Mirabelle_hf,R:(hF_Mirabelle_hf > (hF_Mirabelle_hf > $o)),V:hF_Mirabelle_hf]: (((! [U4:hF_Mirabelle_hf,V3:hF_Mirabelle_hf,V4:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U4) @ A5) => (((R @ U4) @ V3) => (((R @ U4) @ V4) => (V4 = V3)))))) => (((hF_Mirabelle_hmem @ V) @ ((hF_Mir569462966eplace @ A5) @ R)) = (? [U2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U2) @ A5) & ((R @ U2) @ V))))))))).
% 2.47/1.56 thf(fact_133_diffs0__imp__equal, axiom, (! [M:nat,N:nat]: ((((((minus_minus @ nat) @ M) @ N) = (zero_zero @ nat)) => (((((minus_minus @ nat) @ N) @ M) = (zero_zero @ nat)) => (M = N)))))).
% 2.47/1.56 thf(fact_134_minus__nat_Odiff__0, axiom, (! [M:nat]: (((((minus_minus @ nat) @ M) @ (zero_zero @ nat)) = M)))).
% 2.47/1.56 thf(fact_135_zero__natural_Orsp, axiom, ((zero_zero @ nat) = (zero_zero @ nat))).
% 2.47/1.56 thf(fact_136_diff__commute, axiom, (! [I:nat,J:nat,K:nat]: (((((minus_minus @ nat) @ (((minus_minus @ nat) @ I) @ J)) @ K) = (((minus_minus @ nat) @ (((minus_minus @ nat) @ I) @ K)) @ J))))).
% 2.47/1.56 thf(fact_137_of__nat__aux_Osimps_I1_J, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [Inc:(A > A),I:A]: ((((((semiri532925092at_aux @ A) @ Inc) @ (zero_zero @ nat)) @ I) = I))))))).
% 2.47/1.56 thf(fact_138_less__eq__insert1__iff, axiom, (! [X:hF_Mirabelle_hf,Y:hF_Mirabelle_hf,Z:hF_Mirabelle_hf]: (((((ord_less_eq @ hF_Mirabelle_hf) @ ((hF_Mirabelle_hinsert @ X) @ Y)) @ Z) = (((hF_Mirabelle_hmem @ X) @ Z) & (((ord_less_eq @ hF_Mirabelle_hf) @ Y) @ Z)))))).
% 2.47/1.56 thf(fact_139_hmem__def, axiom, (hF_Mirabelle_hmem = (^ [A4:hF_Mirabelle_hf,B3:hF_Mirabelle_hf]: ((((member @ hF_Mirabelle_hf) @ A4) @ (hF_Mirabelle_hfset @ B3)))))).
% 2.47/1.56 thf(fact_140_diff__is__0__eq, axiom, (! [M:nat,N:nat]: ((((((minus_minus @ nat) @ M) @ N) = (zero_zero @ nat)) = (((ord_less_eq @ nat) @ M) @ N))))).
% 2.47/1.56 thf(fact_141_le__zero__eq, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 2.47/1.56 thf(fact_142_inf_Obounded__iff, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ (((inf_inf @ A) @ B) @ C)) = ((((ord_less_eq @ A) @ A2) @ B) & (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.47/1.56 thf(fact_143_le__inf__iff, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ X) @ (((inf_inf @ A) @ Y) @ Z)) = ((((ord_less_eq @ A) @ X) @ Y) & (((ord_less_eq @ A) @ X) @ Z))))))))).
% 2.47/1.56 thf(fact_144_sup_Obounded__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B:A,C:A,A2:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ B) @ C)) @ A2) = ((((ord_less_eq @ A) @ B) @ A2) & (((ord_less_eq @ A) @ C) @ A2))))))))).
% 2.47/1.56 thf(fact_145_le__sup__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ X) @ Y)) @ Z) = ((((ord_less_eq @ A) @ X) @ Z) & (((ord_less_eq @ A) @ Y) @ Z))))))))).
% 2.47/1.56 thf(fact_146_bot__nat__0_Oextremum, axiom, (! [A2:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ A2)))).
% 2.47/1.56 thf(fact_147_le0, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.47/1.56 thf(fact_148_diff__diff__cancel, axiom, (! [I:nat,N:nat]: (((((ord_less_eq @ nat) @ I) @ N) => ((((minus_minus @ nat) @ N) @ (((minus_minus @ nat) @ N) @ I)) = I))))).
% 2.47/1.56 thf(fact_149_hsubsetI, axiom, (! [A5:hF_Mirabelle_hf,B4:hF_Mirabelle_hf]: (((! [X2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X2) @ A5) => ((hF_Mirabelle_hmem @ X2) @ B4)))) => (((ord_less_eq @ hF_Mirabelle_hf) @ A5) @ B4))))).
% 2.47/1.56 thf(fact_150_less__eq__hempty, axiom, (! [U:hF_Mirabelle_hf]: (((((ord_less_eq @ hF_Mirabelle_hf) @ U) @ (zero_zero @ hF_Mirabelle_hf)) = (U = (zero_zero @ hF_Mirabelle_hf)))))).
% 2.47/1.56 thf(fact_151_diff__ge__0__iff__ge, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((minus_minus @ A) @ A2) @ B)) = (((ord_less_eq @ A) @ B) @ A2)))))))).
% 2.47/1.56 thf(fact_152_diff__is__0__eq_H, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((minus_minus @ nat) @ M) @ N) = (zero_zero @ nat)))))).
% 2.47/1.56 thf(fact_153_diff__le__mono2, axiom, (! [M:nat,N:nat,L:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ (((minus_minus @ nat) @ L) @ N)) @ (((minus_minus @ nat) @ L) @ M)))))).
% 2.47/1.56 thf(fact_154_le__diff__iff_H, axiom, (! [A2:nat,C:nat,B:nat]: (((((ord_less_eq @ nat) @ A2) @ C) => ((((ord_less_eq @ nat) @ B) @ C) => ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ C) @ A2)) @ (((minus_minus @ nat) @ C) @ B)) = (((ord_less_eq @ nat) @ B) @ A2))))))).
% 2.47/1.56 thf(fact_155_diff__le__self, axiom, (! [M:nat,N:nat]: ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M) @ N)) @ M)))).
% 2.47/1.56 thf(fact_156_diff__le__mono, axiom, (! [M:nat,N:nat,L:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M) @ L)) @ (((minus_minus @ nat) @ N) @ L)))))).
% 2.47/1.56 thf(fact_157_Nat_Odiff__diff__eq, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ K) @ N) => ((((minus_minus @ nat) @ (((minus_minus @ nat) @ M) @ K)) @ (((minus_minus @ nat) @ N) @ K)) = (((minus_minus @ nat) @ M) @ N))))))).
% 2.47/1.56 thf(fact_158_le__diff__iff, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ K) @ N) => ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M) @ K)) @ (((minus_minus @ nat) @ N) @ K)) = (((ord_less_eq @ nat) @ M) @ N))))))).
% 2.47/1.56 thf(fact_159_eq__diff__iff, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ K) @ N) => (((((minus_minus @ nat) @ M) @ K) = (((minus_minus @ nat) @ N) @ K)) = (M = N))))))).
% 2.47/1.56 thf(fact_160_bot__nat__0_Oextremum__uniqueI, axiom, (! [A2:nat]: (((((ord_less_eq @ nat) @ A2) @ (zero_zero @ nat)) => (A2 = (zero_zero @ nat)))))).
% 2.47/1.56 thf(fact_161_bot__nat__0_Oextremum__unique, axiom, (! [A2:nat]: (((((ord_less_eq @ nat) @ A2) @ (zero_zero @ nat)) = (A2 = (zero_zero @ nat)))))).
% 2.47/1.56 thf(fact_162_le__0__eq, axiom, (! [N:nat]: (((((ord_less_eq @ nat) @ N) @ (zero_zero @ nat)) = (N = (zero_zero @ nat)))))).
% 2.47/1.56 thf(fact_163_less__eq__nat_Osimps_I1_J, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.47/1.56 thf(fact_164_Nat_Oex__has__greatest__nat, axiom, (! [P:(nat > $o),K:nat,B:nat]: (((P @ K) => ((! [Y2:nat]: (((P @ Y2) => (((ord_less_eq @ nat) @ Y2) @ B)))) => (? [X2:nat]: (((P @ X2) & (! [Y5:nat]: (((P @ Y5) => (((ord_less_eq @ nat) @ Y5) @ X2)))))))))))).
% 2.47/1.56 thf(fact_165_nat__le__linear, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) | (((ord_less_eq @ nat) @ N) @ M))))).
% 2.47/1.56 thf(fact_166_le__antisym, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((ord_less_eq @ nat) @ N) @ M) => (M = N)))))).
% 2.47/1.56 thf(fact_167_eq__imp__le, axiom, (! [M:nat,N:nat]: (((M = N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.47/1.56 thf(fact_168_le__trans, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ J) @ K) => (((ord_less_eq @ nat) @ I) @ K)))))).
% 2.47/1.56 thf(fact_169_le__refl, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ N) @ N)))).
% 2.47/1.56 thf(fact_170_canonically__ordered__monoid__add__class_Ozero__le, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [X:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X))))))).
% 2.47/1.56 thf(fact_171_diff__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A,D:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ D) @ C) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A2) @ C)) @ (((minus_minus @ A) @ B) @ D)))))))))).
% 2.47/1.56 thf(fact_172_diff__left__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B) @ A2) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ C) @ A2)) @ (((minus_minus @ A) @ C) @ B))))))))).
% 2.47/1.56 thf(fact_173_diff__right__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A2) @ C)) @ (((minus_minus @ A) @ B) @ C))))))))).
% 2.47/1.56 thf(fact_174_diff__eq__diff__less__eq, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A,C:A,D:A]: ((((((minus_minus @ A) @ A2) @ B) = (((minus_minus @ A) @ C) @ D)) => ((((ord_less_eq @ A) @ A2) @ B) = (((ord_less_eq @ A) @ C) @ D))))))))).
% 2.47/1.56 thf(fact_175_inf_OcoboundedI2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B:A,C:A,A2:A]: (((((ord_less_eq @ A) @ B) @ C) => (((ord_less_eq @ A) @ (((inf_inf @ A) @ A2) @ B)) @ C)))))))).
% 2.47/1.56 thf(fact_176_inf_OcoboundedI1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,C:A,B:A]: (((((ord_less_eq @ A) @ A2) @ C) => (((ord_less_eq @ A) @ (((inf_inf @ A) @ A2) @ B)) @ C)))))))).
% 2.47/1.56 thf(fact_177_inf_Oabsorb__iff2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((ord_less_eq @ A) = (^ [B3:A,A4:A]: (((((inf_inf @ A) @ A4) @ B3) = B3)))))))).
% 2.47/1.56 thf(fact_178_inf_Oabsorb__iff1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((ord_less_eq @ A) = (^ [A4:A,B3:A]: (((((inf_inf @ A) @ A4) @ B3) = A4)))))))).
% 2.47/1.56 thf(fact_179_inf_Ocobounded2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ A2) @ B)) @ B))))))).
% 2.47/1.56 thf(fact_180_inf_Ocobounded1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ A2) @ B)) @ A2))))))).
% 2.47/1.56 thf(fact_181_inf_Oorder__iff, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((ord_less_eq @ A) = (^ [A4:A,B3:A]: ((A4 = (((inf_inf @ A) @ A4) @ B3))))))))).
% 2.47/1.56 thf(fact_182_inf__greatest, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ X) @ Z) => (((ord_less_eq @ A) @ X) @ (((inf_inf @ A) @ Y) @ Z)))))))))).
% 2.47/1.56 thf(fact_183_inf_OboundedI, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ A2) @ C) => (((ord_less_eq @ A) @ A2) @ (((inf_inf @ A) @ B) @ C)))))))))).
% 2.47/1.56 thf(fact_184_inf_OboundedE, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ (((inf_inf @ A) @ B) @ C)) => (~ (((((ord_less_eq @ A) @ A2) @ B) => (~ ((((ord_less_eq @ A) @ A2) @ C))))))))))))).
% 2.47/1.56 thf(fact_185_inf__absorb2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((inf_inf @ A) @ X) @ Y) = Y)))))))).
% 2.47/1.56 thf(fact_186_inf__absorb1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((inf_inf @ A) @ X) @ Y) = X)))))))).
% 2.47/1.56 thf(fact_187_inf_Oabsorb2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B:A,A2:A]: (((((ord_less_eq @ A) @ B) @ A2) => ((((inf_inf @ A) @ A2) @ B) = B)))))))).
% 2.47/1.56 thf(fact_188_inf_Oabsorb1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((inf_inf @ A) @ A2) @ B) = A2)))))))).
% 2.47/1.56 thf(fact_189_le__iff__inf, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((ord_less_eq @ A) = (^ [X3:A,Y3:A]: (((((inf_inf @ A) @ X3) @ Y3) = X3)))))))).
% 2.47/1.56 thf(fact_190_inf__unique, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [F:(A > (A > A)),X:A,Y:A]: (((! [X2:A,Y2:A]: ((((ord_less_eq @ A) @ ((F @ X2) @ Y2)) @ X2))) => ((! [X2:A,Y2:A]: ((((ord_less_eq @ A) @ ((F @ X2) @ Y2)) @ Y2))) => ((! [X2:A,Y2:A,Z2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => ((((ord_less_eq @ A) @ X2) @ Z2) => (((ord_less_eq @ A) @ X2) @ ((F @ Y2) @ Z2)))))) => ((((inf_inf @ A) @ X) @ Y) = ((F @ X) @ Y))))))))))).
% 2.47/1.56 thf(fact_191_inf_OorderI, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B:A]: (((A2 = (((inf_inf @ A) @ A2) @ B)) => (((ord_less_eq @ A) @ A2) @ B)))))))).
% 2.47/1.56 thf(fact_192_inf_OorderE, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ A2) @ B) => (A2 = (((inf_inf @ A) @ A2) @ B))))))))).
% 2.47/1.56 thf(fact_193_le__infI2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B:A,X:A,A2:A]: (((((ord_less_eq @ A) @ B) @ X) => (((ord_less_eq @ A) @ (((inf_inf @ A) @ A2) @ B)) @ X)))))))).
% 2.47/1.56 thf(fact_194_le__infI1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,X:A,B:A]: (((((ord_less_eq @ A) @ A2) @ X) => (((ord_less_eq @ A) @ (((inf_inf @ A) @ A2) @ B)) @ X)))))))).
% 2.47/1.56 thf(fact_195_inf__mono, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,C:A,B:A,D:A]: (((((ord_less_eq @ A) @ A2) @ C) => ((((ord_less_eq @ A) @ B) @ D) => (((ord_less_eq @ A) @ (((inf_inf @ A) @ A2) @ B)) @ (((inf_inf @ A) @ C) @ D)))))))))).
% 2.47/1.56 thf(fact_196_le__infI, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,A2:A,B:A]: (((((ord_less_eq @ A) @ X) @ A2) => ((((ord_less_eq @ A) @ X) @ B) => (((ord_less_eq @ A) @ X) @ (((inf_inf @ A) @ A2) @ B)))))))))).
% 2.47/1.56 thf(fact_197_le__infE, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,A2:A,B:A]: (((((ord_less_eq @ A) @ X) @ (((inf_inf @ A) @ A2) @ B)) => (~ (((((ord_less_eq @ A) @ X) @ A2) => (~ ((((ord_less_eq @ A) @ X) @ B))))))))))))).
% 2.47/1.56 thf(fact_198_inf__le2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,Y:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ X) @ Y)) @ Y))))))).
% 2.47/1.56 thf(fact_199_inf__le1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,Y:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ X) @ Y)) @ X))))))).
% 2.47/1.56 thf(fact_200_inf__sup__ord_I1_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ X) @ Y)) @ X))))))).
% 2.47/1.56 thf(fact_201_inf__sup__ord_I2_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ X) @ Y)) @ Y))))))).
% 2.47/1.56 thf(fact_202_sup_OcoboundedI2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [C:A,B:A,A2:A]: (((((ord_less_eq @ A) @ C) @ B) => (((ord_less_eq @ A) @ C) @ (((sup_sup @ A) @ A2) @ B))))))))).
% 2.47/1.56 thf(fact_203_sup_OcoboundedI1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [C:A,A2:A,B:A]: (((((ord_less_eq @ A) @ C) @ A2) => (((ord_less_eq @ A) @ C) @ (((sup_sup @ A) @ A2) @ B))))))))).
% 2.47/1.56 thf(fact_204_sup_Oabsorb__iff2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less_eq @ A) = (^ [A4:A,B3:A]: (((((sup_sup @ A) @ A4) @ B3) = B3)))))))).
% 2.47/1.56 thf(fact_205_sup_Oabsorb__iff1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less_eq @ A) = (^ [B3:A,A4:A]: (((((sup_sup @ A) @ A4) @ B3) = A4)))))))).
% 2.47/1.56 thf(fact_206_sup_Ocobounded2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B:A,A2:A]: ((((ord_less_eq @ A) @ B) @ (((sup_sup @ A) @ A2) @ B)))))))).
% 2.47/1.56 thf(fact_207_sup_Ocobounded1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,B:A]: ((((ord_less_eq @ A) @ A2) @ (((sup_sup @ A) @ A2) @ B)))))))).
% 2.47/1.56 thf(fact_208_sup_Oorder__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less_eq @ A) = (^ [B3:A,A4:A]: ((A4 = (((sup_sup @ A) @ A4) @ B3))))))))).
% 2.47/1.56 thf(fact_209_sup_OboundedI, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B) @ A2) => ((((ord_less_eq @ A) @ C) @ A2) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ B) @ C)) @ A2))))))))).
% 2.47/1.56 thf(fact_210_sup_OboundedE, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B:A,C:A,A2:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ B) @ C)) @ A2) => (~ (((((ord_less_eq @ A) @ B) @ A2) => (~ ((((ord_less_eq @ A) @ C) @ A2))))))))))))).
% 2.47/1.56 thf(fact_211_sup__absorb2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((sup_sup @ A) @ X) @ Y) = Y)))))))).
% 2.47/1.56 thf(fact_212_sup__absorb1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((sup_sup @ A) @ X) @ Y) = X)))))))).
% 2.47/1.56 thf(fact_213_sup_Oabsorb2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((sup_sup @ A) @ A2) @ B) = B)))))))).
% 2.47/1.56 thf(fact_214_sup_Oabsorb1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B:A,A2:A]: (((((ord_less_eq @ A) @ B) @ A2) => ((((sup_sup @ A) @ A2) @ B) = A2)))))))).
% 2.47/1.56 thf(fact_215_sup__unique, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [F:(A > (A > A)),X:A,Y:A]: (((! [X2:A,Y2:A]: ((((ord_less_eq @ A) @ X2) @ ((F @ X2) @ Y2)))) => ((! [X2:A,Y2:A]: ((((ord_less_eq @ A) @ Y2) @ ((F @ X2) @ Y2)))) => ((! [X2:A,Y2:A,Z2:A]: (((((ord_less_eq @ A) @ Y2) @ X2) => ((((ord_less_eq @ A) @ Z2) @ X2) => (((ord_less_eq @ A) @ ((F @ Y2) @ Z2)) @ X2))))) => ((((sup_sup @ A) @ X) @ Y) = ((F @ X) @ Y))))))))))).
% 2.47/1.56 thf(fact_216_sup_OorderI, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,B:A]: (((A2 = (((sup_sup @ A) @ A2) @ B)) => (((ord_less_eq @ A) @ B) @ A2)))))))).
% 2.47/1.56 thf(fact_217_sup_OorderE, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B:A,A2:A]: (((((ord_less_eq @ A) @ B) @ A2) => (A2 = (((sup_sup @ A) @ A2) @ B))))))))).
% 2.47/1.56 thf(fact_218_le__iff__sup, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less_eq @ A) = (^ [X3:A,Y3:A]: (((((sup_sup @ A) @ X3) @ Y3) = Y3)))))))).
% 2.47/1.56 thf(fact_219_sup__least, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [Y:A,X:A,Z:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ Z) @ X) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ Y) @ Z)) @ X))))))))).
% 2.47/1.56 thf(fact_220_sup__mono, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,C:A,B:A,D:A]: (((((ord_less_eq @ A) @ A2) @ C) => ((((ord_less_eq @ A) @ B) @ D) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ A2) @ B)) @ (((sup_sup @ A) @ C) @ D)))))))))).
% 2.47/1.56 thf(fact_221_sup_Omono, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [C:A,A2:A,D:A,B:A]: (((((ord_less_eq @ A) @ C) @ A2) => ((((ord_less_eq @ A) @ D) @ B) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ C) @ D)) @ (((sup_sup @ A) @ A2) @ B)))))))))).
% 2.47/1.56 thf(fact_222_le__supI2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A,B:A,A2:A]: (((((ord_less_eq @ A) @ X) @ B) => (((ord_less_eq @ A) @ X) @ (((sup_sup @ A) @ A2) @ B))))))))).
% 2.47/1.56 thf(fact_223_le__supI1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A,A2:A,B:A]: (((((ord_less_eq @ A) @ X) @ A2) => (((ord_less_eq @ A) @ X) @ (((sup_sup @ A) @ A2) @ B))))))))).
% 2.47/1.56 thf(fact_224_sup__ge2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [Y:A,X:A]: ((((ord_less_eq @ A) @ Y) @ (((sup_sup @ A) @ X) @ Y)))))))).
% 2.47/1.56 thf(fact_225_sup__ge1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A,Y:A]: ((((ord_less_eq @ A) @ X) @ (((sup_sup @ A) @ X) @ Y)))))))).
% 2.47/1.56 thf(fact_226_le__supI, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,X:A,B:A]: (((((ord_less_eq @ A) @ A2) @ X) => ((((ord_less_eq @ A) @ B) @ X) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ A2) @ B)) @ X))))))))).
% 2.47/1.56 thf(fact_227_le__supE, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,B:A,X:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ A2) @ B)) @ X) => (~ (((((ord_less_eq @ A) @ A2) @ X) => (~ ((((ord_less_eq @ A) @ B) @ X))))))))))))).
% 2.47/1.56 thf(fact_228_inf__sup__ord_I3_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y:A]: ((((ord_less_eq @ A) @ X) @ (((sup_sup @ A) @ X) @ Y)))))))).
% 2.47/1.56 thf(fact_229_inf__sup__ord_I4_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [Y:A,X:A]: ((((ord_less_eq @ A) @ Y) @ (((sup_sup @ A) @ X) @ Y)))))))).
% 2.47/1.56 thf(fact_230_less__eq__hf__def, axiom, ((ord_less_eq @ hF_Mirabelle_hf) = (^ [A7:hF_Mirabelle_hf,B7:hF_Mirabelle_hf]: ((! [X3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X3) @ A7) => ((hF_Mirabelle_hmem @ X3) @ B7)))))))).
% 2.47/1.56 thf(fact_231_rev__hsubsetD, axiom, (! [C:hF_Mirabelle_hf,A5:hF_Mirabelle_hf,B4:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ C) @ A5) => ((((ord_less_eq @ hF_Mirabelle_hf) @ A5) @ B4) => ((hF_Mirabelle_hmem @ C) @ B4)))))).
% 2.47/1.56 thf(fact_232_hsubsetCE, axiom, (! [A5:hF_Mirabelle_hf,B4:hF_Mirabelle_hf,C:hF_Mirabelle_hf]: (((((ord_less_eq @ hF_Mirabelle_hf) @ A5) @ B4) => (((hF_Mirabelle_hmem @ C) @ A5) => ((hF_Mirabelle_hmem @ C) @ B4)))))).
% 2.47/1.56 thf(fact_233_hsubsetD, axiom, (! [A5:hF_Mirabelle_hf,B4:hF_Mirabelle_hf,C:hF_Mirabelle_hf]: (((((ord_less_eq @ hF_Mirabelle_hf) @ A5) @ B4) => (((hF_Mirabelle_hmem @ C) @ A5) => ((hF_Mirabelle_hmem @ C) @ B4)))))).
% 2.47/1.56 thf(fact_234_HF__Mirabelle__fsbjehakzm_Ozero__le, axiom, (! [X:hF_Mirabelle_hf]: ((((ord_less_eq @ hF_Mirabelle_hf) @ (zero_zero @ hF_Mirabelle_hf)) @ X)))).
% 2.47/1.56 thf(fact_235_hf__equalityE, axiom, (! [A5:hF_Mirabelle_hf,B4:hF_Mirabelle_hf]: (((A5 = B4) => (~ (((((ord_less_eq @ hF_Mirabelle_hf) @ A5) @ B4) => (~ ((((ord_less_eq @ hF_Mirabelle_hf) @ B4) @ A5)))))))))).
% 2.47/1.56 thf(fact_236_le__iff__diff__le__0, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => ((ord_less_eq @ A) = (^ [A4:A,B3:A]: ((((ord_less_eq @ A) @ (((minus_minus @ A) @ A4) @ B3)) @ (zero_zero @ A))))))))).
% 2.47/1.56 thf(fact_237_distrib__inf__le, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y:A,Z:A]: ((((ord_less_eq @ A) @ (((sup_sup @ A) @ (((inf_inf @ A) @ X) @ Y)) @ (((inf_inf @ A) @ X) @ Z))) @ (((inf_inf @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z))))))))).
% 2.47/1.56 thf(fact_238_distrib__sup__le, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y:A,Z:A]: ((((ord_less_eq @ A) @ (((sup_sup @ A) @ X) @ (((inf_inf @ A) @ Y) @ Z))) @ (((inf_inf @ A) @ (((sup_sup @ A) @ X) @ Y)) @ (((sup_sup @ A) @ X) @ Z))))))))).
% 2.47/1.56 thf(fact_239_less__eq__insert2__iff, axiom, (! [Z:hF_Mirabelle_hf,X:hF_Mirabelle_hf,Y:hF_Mirabelle_hf]: (((((ord_less_eq @ hF_Mirabelle_hf) @ Z) @ ((hF_Mirabelle_hinsert @ X) @ Y)) = ((((ord_less_eq @ hF_Mirabelle_hf) @ Z) @ Y) | (? [U2:hF_Mirabelle_hf]: (((((hF_Mirabelle_hinsert @ X) @ U2) = Z) & ((~ (((hF_Mirabelle_hmem @ X) @ U2))) & (((ord_less_eq @ hF_Mirabelle_hf) @ U2) @ Y)))))))))).
% 2.47/1.56 thf(fact_240_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.47/1.56 thf(fact_241_le__numeral__extra_I3_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))).
% 2.47/1.56 thf(fact_242_HF__hfset, axiom, (! [A2:hF_Mirabelle_hf]: (((hF_Mirabelle_HF @ (hF_Mirabelle_hfset @ A2)) = A2)))).
% 2.47/1.56 thf(fact_243_of__nat__le__0__iff, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [M:nat]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ M)) @ (zero_zero @ A)) = (M = (zero_zero @ nat))))))))).
% 2.47/1.56 thf(fact_244_of__nat__eq__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [M:nat,N:nat]: (((((semiring_1_of_nat @ A) @ M) = ((semiring_1_of_nat @ A) @ N)) = (M = N)))))))).
% 2.47/1.56 thf(fact_245_of__nat__0, axiom, (! [A:$tType]: (((semiring_1 @ A) => (((semiring_1_of_nat @ A) @ (zero_zero @ nat)) = (zero_zero @ A)))))).
% 2.47/1.56 thf(fact_246_of__nat__0__eq__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:nat]: ((((zero_zero @ A) = ((semiring_1_of_nat @ A) @ N)) = ((zero_zero @ nat) = N)))))))).
% 2.47/1.56 thf(fact_247_of__nat__eq__0__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [M:nat]: (((((semiring_1_of_nat @ A) @ M) = (zero_zero @ A)) = (M = (zero_zero @ nat))))))))).
% 2.47/1.56 thf(fact_248_of__nat__le__iff, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [M:nat,N:nat]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N)) = (((ord_less_eq @ nat) @ M) @ N)))))))).
% 2.47/1.56 thf(fact_249_of__nat__0__le__iff, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [N:nat]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((semiring_1_of_nat @ A) @ N)))))))).
% 2.47/1.56 thf(fact_250_of__nat__mono, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [I:nat,J:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ I)) @ ((semiring_1_of_nat @ A) @ J))))))))).
% 2.47/1.56 thf(fact_251_of__nat__diff, axiom, (! [A:$tType]: (((semiring_1_cancel @ A) => (! [N:nat,M:nat]: (((((ord_less_eq @ nat) @ N) @ M) => (((semiring_1_of_nat @ A) @ (((minus_minus @ nat) @ M) @ N)) = (((minus_minus @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N)))))))))).
% 2.47/1.56 thf(fact_252_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B:A,A2:A]: (((((ord_less_eq @ A) @ B) @ A2) => ((((ord_less_eq @ A) @ A2) @ B) => (A2 = B))))))))).
% 2.47/1.56 thf(fact_253_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z3:A]: ((Y4 = Z3))) = (^ [A4:A,B3:A]: (((((ord_less_eq @ A) @ B3) @ A4) & (((ord_less_eq @ A) @ A4) @ B3))))))))).
% 2.47/1.56 thf(tcon_fun___Lattices_Osemilattice__sup, axiom, (! [A8:$tType,A9:$tType]: (((semilattice_sup @ A9) => (semilattice_sup @ (A8 > A9)))))).
% 2.47/1.56 thf(tcon_fun___Lattices_Osemilattice__inf, axiom, (! [A8:$tType,A9:$tType]: (((semilattice_inf @ A9) => (semilattice_inf @ (A8 > A9)))))).
% 2.47/1.56 thf(tcon_fun___Lattices_Odistrib__lattice, axiom, (! [A8:$tType,A9:$tType]: (((distrib_lattice @ A9) => (distrib_lattice @ (A8 > A9)))))).
% 2.47/1.56 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A8:$tType,A9:$tType]: (((preorder @ A9) => (preorder @ (A8 > A9)))))).
% 2.47/1.56 thf(tcon_fun___Lattices_Olattice, axiom, (! [A8:$tType,A9:$tType]: (((lattice @ A9) => (lattice @ (A8 > A9)))))).
% 2.47/1.56 thf(tcon_fun___Orderings_Oorder, axiom, (! [A8:$tType,A9:$tType]: (((order @ A9) => (order @ (A8 > A9)))))).
% 2.47/1.56 thf(tcon_fun___Groups_Ominus, axiom, (! [A8:$tType,A9:$tType]: (((minus @ A9) => (minus @ (A8 > A9)))))).
% 2.47/1.56 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, (canoni770627133id_add @ nat)).
% 2.47/1.56 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring, axiom, (linord1659791738miring @ nat)).
% 2.47/1.56 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add, axiom, (cancel146912293up_add @ nat)).
% 2.47/1.56 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add, axiom, (cancel1352612707id_add @ nat)).
% 2.47/1.56 thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_1, axiom, (semilattice_sup @ nat)).
% 2.47/1.56 thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_2, axiom, (semilattice_inf @ nat)).
% 2.47/1.56 thf(tcon_Nat_Onat___Lattices_Odistrib__lattice_3, axiom, (distrib_lattice @ nat)).
% 2.47/1.56 thf(tcon_Nat_Onat___Rings_Osemiring__1__cancel, axiom, (semiring_1_cancel @ nat)).
% 2.47/1.56 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff, axiom, (comm_monoid_diff @ nat)).
% 2.47/1.56 thf(tcon_Nat_Onat___Nat_Osemiring__char__0, axiom, (semiring_char_0 @ nat)).
% 2.47/1.56 thf(tcon_Nat_Onat___Orderings_Opreorder_4, axiom, (preorder @ nat)).
% 2.47/1.56 thf(tcon_Nat_Onat___Rings_Osemiring__1, axiom, (semiring_1 @ nat)).
% 2.47/1.56 thf(tcon_Nat_Onat___Lattices_Olattice_5, axiom, (lattice @ nat)).
% 2.47/1.56 thf(tcon_Nat_Onat___Orderings_Oorder_6, axiom, (order @ nat)).
% 2.47/1.56 thf(tcon_Nat_Onat___Groups_Ominus_7, axiom, (minus @ nat)).
% 2.47/1.56 thf(tcon_Nat_Onat___Groups_Ozero, axiom, (zero @ nat)).
% 2.47/1.56 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_8, axiom, (! [A8:$tType]: ((semilattice_sup @ (set @ A8))))).
% 2.47/1.56 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_9, axiom, (! [A8:$tType]: ((semilattice_inf @ (set @ A8))))).
% 2.47/1.56 thf(tcon_Set_Oset___Lattices_Odistrib__lattice_10, axiom, (! [A8:$tType]: ((distrib_lattice @ (set @ A8))))).
% 2.47/1.56 thf(tcon_Set_Oset___Orderings_Opreorder_11, axiom, (! [A8:$tType]: ((preorder @ (set @ A8))))).
% 2.47/1.56 thf(tcon_Set_Oset___Lattices_Olattice_12, axiom, (! [A8:$tType]: ((lattice @ (set @ A8))))).
% 2.47/1.56 thf(tcon_Set_Oset___Orderings_Oorder_13, axiom, (! [A8:$tType]: ((order @ (set @ A8))))).
% 2.47/1.56 thf(tcon_Set_Oset___Groups_Ominus_14, axiom, (! [A8:$tType]: ((minus @ (set @ A8))))).
% 2.47/1.56 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_15, axiom, (semilattice_sup @ $o)).
% 2.47/1.56 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_16, axiom, (semilattice_inf @ $o)).
% 2.47/1.56 thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_17, axiom, (distrib_lattice @ $o)).
% 2.47/1.56 thf(tcon_HOL_Obool___Orderings_Opreorder_18, axiom, (preorder @ $o)).
% 2.47/1.56 thf(tcon_HOL_Obool___Lattices_Olattice_19, axiom, (lattice @ $o)).
% 2.47/1.56 thf(tcon_HOL_Obool___Orderings_Oorder_20, axiom, (order @ $o)).
% 2.47/1.56 thf(tcon_HOL_Obool___Groups_Ominus_21, axiom, (minus @ $o)).
% 2.47/1.56 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Osemilattice__sup_22, axiom, (semilattice_sup @ hF_Mirabelle_hf)).
% 2.47/1.56 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Osemilattice__inf_23, axiom, (semilattice_inf @ hF_Mirabelle_hf)).
% 2.47/1.56 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Odistrib__lattice_24, axiom, (distrib_lattice @ hF_Mirabelle_hf)).
% 2.47/1.56 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Orderings_Opreorder_25, axiom, (preorder @ hF_Mirabelle_hf)).
% 2.47/1.56 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Olattice_26, axiom, (lattice @ hF_Mirabelle_hf)).
% 2.47/1.56 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Orderings_Oorder_27, axiom, (order @ hF_Mirabelle_hf)).
% 2.47/1.56 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Groups_Ominus_28, axiom, (minus @ hF_Mirabelle_hf)).
% 2.47/1.56 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Groups_Ozero_29, axiom, (zero @ hF_Mirabelle_hf)).
% 2.47/1.56 thf(conj_0, conjecture, (((hF_Mirabelle_hpair @ x) @ y) != (zero_zero @ hF_Mirabelle_hf))).
% 2.47/1.56 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.dWcERH6OXC/DTF2DTF7787.p
% 2.47/1.56 ------------------------
% 2.47/1.58 ---- Cleaned THF ---
% 2.47/1.58 thf(ty_t_HF__Mirabelle__fsbjehakzm_Ohf,type,
% 2.47/1.58 hF_Mirabelle_hf: $tType ).
% 2.47/1.58
% 2.47/1.58 thf(ty_t_Set_Oset,type,
% 2.47/1.58 set: $tType > $tType ).
% 2.47/1.58
% 2.47/1.58 thf(ty_t_Nat_Onat,type,
% 2.47/1.58 nat: $tType ).
% 2.47/1.58
% 2.47/1.58 thf(sy_cl_HOL_Otype,type,
% 2.47/1.58 type:
% 2.47/1.58 !>[A: $tType] : $o ).
% 2.47/1.58
% 2.47/1.58 thf(sy_cl_Groups_Ozero,type,
% 2.47/1.58 zero:
% 2.47/1.58 !>[A: $tType] : $o ).
% 2.47/1.58
% 2.47/1.58 thf(sy_cl_Groups_Ominus,type,
% 2.47/1.58 minus:
% 2.47/1.58 !>[A: $tType] : $o ).
% 2.47/1.58
% 2.47/1.58 thf(sy_cl_Orderings_Oorder,type,
% 2.47/1.58 order:
% 2.47/1.58 !>[A: $tType] : $o ).
% 2.47/1.58
% 2.47/1.58 thf(sy_cl_Groups_Ogroup__add,type,
% 2.47/1.58 group_add:
% 2.47/1.58 !>[A: $tType] : $o ).
% 2.47/1.58
% 2.47/1.58 thf(sy_cl_Lattices_Olattice,type,
% 2.47/1.58 lattice:
% 2.47/1.58 !>[A: $tType] : $o ).
% 2.47/1.58
% 2.47/1.58 thf(sy_cl_Rings_Osemiring__1,type,
% 2.47/1.58 semiring_1:
% 2.47/1.58 !>[A: $tType] : $o ).
% 2.47/1.58
% 2.47/1.58 thf(sy_cl_Orderings_Opreorder,type,
% 2.47/1.58 preorder:
% 2.47/1.58 !>[A: $tType] : $o ).
% 2.47/1.58
% 2.47/1.58 thf(sy_cl_Nat_Osemiring__char__0,type,
% 2.47/1.58 semiring_char_0:
% 2.47/1.58 !>[A: $tType] : $o ).
% 2.47/1.58
% 2.47/1.58 thf(sy_cl_Groups_Ocomm__monoid__diff,type,
% 2.47/1.58 comm_monoid_diff:
% 2.47/1.58 !>[A: $tType] : $o ).
% 2.47/1.58
% 2.47/1.58 thf(sy_cl_Rings_Osemiring__1__cancel,type,
% 2.47/1.58 semiring_1_cancel:
% 2.47/1.58 !>[A: $tType] : $o ).
% 2.47/1.58
% 2.47/1.58 thf(sy_cl_Lattices_Odistrib__lattice,type,
% 2.47/1.58 distrib_lattice:
% 2.47/1.58 !>[A: $tType] : $o ).
% 2.47/1.58
% 2.47/1.58 thf(sy_cl_Lattices_Osemilattice__inf,type,
% 2.47/1.58 semilattice_inf:
% 2.47/1.58 !>[A: $tType] : $o ).
% 2.47/1.58
% 2.47/1.58 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 2.47/1.58 semilattice_sup:
% 2.47/1.58 !>[A: $tType] : $o ).
% 2.47/1.58
% 2.47/1.58 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 2.47/1.58 ordered_ab_group_add:
% 2.47/1.58 !>[A: $tType] : $o ).
% 2.47/1.58
% 2.47/1.58 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 2.47/1.58 cancel1352612707id_add:
% 2.47/1.58 !>[A: $tType] : $o ).
% 2.47/1.58
% 2.47/1.58 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 2.47/1.58 cancel146912293up_add:
% 2.47/1.58 !>[A: $tType] : $o ).
% 2.47/1.58
% 2.47/1.58 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 2.47/1.58 linord1659791738miring:
% 2.47/1.58 !>[A: $tType] : $o ).
% 2.47/1.58
% 2.47/1.58 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.47/1.58 canoni770627133id_add:
% 2.47/1.58 !>[A: $tType] : $o ).
% 2.47/1.58
% 2.47/1.58 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 2.47/1.58 minus_minus:
% 2.47/1.58 !>[A: $tType] : ( A > A > A ) ).
% 2.47/1.58
% 2.47/1.58 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 2.47/1.58 zero_zero:
% 2.47/1.58 !>[A: $tType] : A ).
% 2.47/1.58
% 2.47/1.58 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHCollect,type,
% 2.47/1.58 hF_Mir1687042746ollect: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 2.47/1.58
% 2.47/1.58 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHF,type,
% 2.47/1.58 hF_Mirabelle_HF: ( set @ hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
% 2.47/1.58
% 2.47/1.58 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHInter,type,
% 2.47/1.58 hF_Mirabelle_HInter: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 2.47/1.58
% 2.47/1.58 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHUnion,type,
% 2.47/1.58 hF_Mirabelle_HUnion: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 2.47/1.58
% 2.47/1.58 thf(sy_c_HF__Mirabelle__fsbjehakzm_OPrimReplace,type,
% 2.47/1.58 hF_Mir569462966eplace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 2.47/1.58
% 2.47/1.58 thf(sy_c_HF__Mirabelle__fsbjehakzm_ORepFun,type,
% 2.47/1.58 hF_Mirabelle_RepFun: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
% 2.47/1.58
% 2.47/1.58 thf(sy_c_HF__Mirabelle__fsbjehakzm_OReplace,type,
% 2.47/1.58 hF_Mirabelle_Replace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 2.47/1.58
% 2.47/1.58 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohf_OAbs__hf,type,
% 2.47/1.58 hF_Mirabelle_Abs_hf: nat > hF_Mirabelle_hf ).
% 2.47/1.58
% 2.47/1.58 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohfset,type,
% 2.47/1.58 hF_Mirabelle_hfset: hF_Mirabelle_hf > ( set @ hF_Mirabelle_hf ) ).
% 2.47/1.58
% 2.47/1.58 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohfst,type,
% 2.47/1.58 hF_Mirabelle_hfst: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 2.47/1.58
% 2.47/1.58 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohinsert,type,
% 2.47/1.58 hF_Mirabelle_hinsert: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 2.47/1.58
% 2.47/1.58 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohmem,type,
% 2.47/1.58 hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
% 2.47/1.58
% 2.47/1.58 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohpair,type,
% 2.47/1.58 hF_Mirabelle_hpair: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 2.47/1.58
% 2.47/1.58 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohsnd,type,
% 2.47/1.58 hF_Mirabelle_hsnd: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 2.47/1.58
% 2.47/1.58 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohsplit,type,
% 2.47/1.58 hF_Mirabelle_hsplit:
% 2.47/1.58 !>[A: $tType] : ( ( hF_Mirabelle_hf > hF_Mirabelle_hf > A ) > hF_Mirabelle_hf > A ) ).
% 2.47/1.58
% 2.47/1.58 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 2.47/1.58 inf_inf:
% 2.47/1.58 !>[A: $tType] : ( A > A > A ) ).
% 2.47/1.58
% 2.47/1.58 thf(sy_c_Lattices_Osup__class_Osup,type,
% 2.47/1.58 sup_sup:
% 2.47/1.58 !>[A: $tType] : ( A > A > A ) ).
% 2.47/1.58
% 2.47/1.58 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
% 2.47/1.58 semiring_1_of_nat:
% 2.47/1.58 !>[A: $tType] : ( nat > A ) ).
% 2.47/1.58
% 2.47/1.58 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
% 2.47/1.58 semiri532925092at_aux:
% 2.47/1.58 !>[A: $tType] : ( ( A > A ) > nat > A > A ) ).
% 2.47/1.58
% 2.47/1.58 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.47/1.58 ord_less_eq:
% 2.47/1.58 !>[A: $tType] : ( A > A > $o ) ).
% 2.47/1.58
% 2.47/1.58 thf(sy_c_Set_OCollect,type,
% 2.47/1.58 collect:
% 2.47/1.58 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(sy_c_member,type,
% 2.47/1.58 member:
% 2.47/1.58 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.47/1.58
% 2.47/1.58 thf(sy_v_x,type,
% 2.47/1.58 x: hF_Mirabelle_hf ).
% 2.47/1.58
% 2.47/1.58 thf(sy_v_y,type,
% 2.47/1.58 y: hF_Mirabelle_hf ).
% 2.47/1.58
% 2.47/1.58 thf(fact_0_hpair__iff,axiom,
% 2.47/1.58 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( ( hF_Mirabelle_hpair @ A2 @ B )
% 2.47/1.58 = ( hF_Mirabelle_hpair @ A3 @ B2 ) )
% 2.47/1.58 = ( ( A2 = A3 )
% 2.47/1.58 & ( B = B2 ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_1_hpair__inject,axiom,
% 2.47/1.58 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( ( hF_Mirabelle_hpair @ A2 @ B )
% 2.47/1.58 = ( hF_Mirabelle_hpair @ A3 @ B2 ) )
% 2.47/1.58 => ~ ( ( A2 = A3 )
% 2.47/1.58 => ( B != B2 ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_2_hpair__neq__fst,axiom,
% 2.47/1.58 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hpair @ A2 @ B )
% 2.47/1.58 != A2 ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_3_hpair__neq__snd,axiom,
% 2.47/1.58 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hpair @ A2 @ B )
% 2.47/1.58 != B ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_4_hsplit,axiom,
% 2.47/1.58 ! [A: $tType,C: hF_Mirabelle_hf > hF_Mirabelle_hf > A,A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hsplit @ A @ C @ ( hF_Mirabelle_hpair @ A2 @ B ) )
% 2.47/1.58 = ( C @ A2 @ B ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_5_hfst__conv,axiom,
% 2.47/1.58 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hfst @ ( hF_Mirabelle_hpair @ A2 @ B ) )
% 2.47/1.58 = A2 ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_6_hsnd__conv,axiom,
% 2.47/1.58 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hsnd @ ( hF_Mirabelle_hpair @ A2 @ B ) )
% 2.47/1.58 = B ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_7_HInter__hempty,axiom,
% 2.47/1.58 ( ( hF_Mirabelle_HInter @ ( zero_zero @ hF_Mirabelle_hf ) )
% 2.47/1.58 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_8_HCollect__hempty,axiom,
% 2.47/1.58 ! [P: hF_Mirabelle_hf > $o] :
% 2.47/1.58 ( ( hF_Mir1687042746ollect @ P @ ( zero_zero @ hF_Mirabelle_hf ) )
% 2.47/1.58 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_9_Replace__0,axiom,
% 2.47/1.58 ! [R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 2.47/1.58 ( ( hF_Mirabelle_Replace @ ( zero_zero @ hF_Mirabelle_hf ) @ R )
% 2.47/1.58 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_10_HUnion__hempty,axiom,
% 2.47/1.58 ( ( hF_Mirabelle_HUnion @ ( zero_zero @ hF_Mirabelle_hf ) )
% 2.47/1.58 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_11_zero__reorient,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( zero @ A )
% 2.47/1.58 => ! [X: A] :
% 2.47/1.58 ( ( ( zero_zero @ A )
% 2.47/1.58 = X )
% 2.47/1.58 = ( X
% 2.47/1.58 = ( zero_zero @ A ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_12_RepFun__0,axiom,
% 2.47/1.58 ! [F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_RepFun @ ( zero_zero @ hF_Mirabelle_hf ) @ F )
% 2.47/1.58 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_13_hpair__def,axiom,
% 2.47/1.58 ( hF_Mirabelle_hpair
% 2.47/1.58 = ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A4 @ ( zero_zero @ hF_Mirabelle_hf ) ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A4 @ ( hF_Mirabelle_hinsert @ B3 @ ( zero_zero @ hF_Mirabelle_hf ) ) ) @ ( zero_zero @ hF_Mirabelle_hf ) ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_14_hpair__def_H,axiom,
% 2.47/1.58 ( hF_Mirabelle_hpair
% 2.47/1.58 = ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A4 @ ( hF_Mirabelle_hinsert @ A4 @ ( zero_zero @ hF_Mirabelle_hf ) ) ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A4 @ ( hF_Mirabelle_hinsert @ B3 @ ( zero_zero @ hF_Mirabelle_hf ) ) ) @ ( zero_zero @ hF_Mirabelle_hf ) ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_15_singleton__eq__iff,axiom,
% 2.47/1.58 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( ( hF_Mirabelle_hinsert @ A2 @ ( zero_zero @ hF_Mirabelle_hf ) )
% 2.47/1.58 = ( hF_Mirabelle_hinsert @ B @ ( zero_zero @ hF_Mirabelle_hf ) ) )
% 2.47/1.58 = ( A2 = B ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_16_RepFun__hinsert,axiom,
% 2.47/1.58 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_RepFun @ ( hF_Mirabelle_hinsert @ A2 @ B ) @ F )
% 2.47/1.58 = ( hF_Mirabelle_hinsert @ ( F @ A2 ) @ ( hF_Mirabelle_RepFun @ B @ F ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_17_hinsert__commute,axiom,
% 2.47/1.58 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hinsert @ X @ ( hF_Mirabelle_hinsert @ Y @ Z ) )
% 2.47/1.58 = ( hF_Mirabelle_hinsert @ Y @ ( hF_Mirabelle_hinsert @ X @ Z ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_18_hinsert__nonempty,axiom,
% 2.47/1.58 ! [A2: hF_Mirabelle_hf,A5: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hinsert @ A2 @ A5 )
% 2.47/1.58 != ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_19_HF__Mirabelle__fsbjehakzm_Odoubleton__eq__iff,axiom,
% 2.47/1.58 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf,D: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( ( hF_Mirabelle_hinsert @ A2 @ ( hF_Mirabelle_hinsert @ B @ ( zero_zero @ hF_Mirabelle_hf ) ) )
% 2.47/1.58 = ( hF_Mirabelle_hinsert @ C @ ( hF_Mirabelle_hinsert @ D @ ( zero_zero @ hF_Mirabelle_hf ) ) ) )
% 2.47/1.58 = ( ( ( A2 = C )
% 2.47/1.58 & ( B = D ) )
% 2.47/1.58 | ( ( A2 = D )
% 2.47/1.58 & ( B = C ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_20_hf__induct__ax,axiom,
% 2.47/1.58 ! [P: hF_Mirabelle_hf > $o,X: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( P @ ( zero_zero @ hF_Mirabelle_hf ) )
% 2.47/1.58 => ( ! [X2: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( P @ X2 )
% 2.47/1.58 => ! [Y2: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( P @ Y2 )
% 2.47/1.58 => ( P @ ( hF_Mirabelle_hinsert @ Y2 @ X2 ) ) ) )
% 2.47/1.58 => ( P @ X ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_21_HInter__hinsert,axiom,
% 2.47/1.58 ! [A5: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( A5
% 2.47/1.58 != ( zero_zero @ hF_Mirabelle_hf ) )
% 2.47/1.58 => ( ( hF_Mirabelle_HInter @ ( hF_Mirabelle_hinsert @ A2 @ A5 ) )
% 2.47/1.58 = ( inf_inf @ hF_Mirabelle_hf @ A2 @ ( hF_Mirabelle_HInter @ A5 ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_22_Abs__hf__0,axiom,
% 2.47/1.58 ( ( hF_Mirabelle_Abs_hf @ ( zero_zero @ nat ) )
% 2.47/1.58 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_23_HInter__iff,axiom,
% 2.47/1.58 ! [A5: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( A5
% 2.47/1.58 != ( zero_zero @ hF_Mirabelle_hf ) )
% 2.47/1.58 => ( ( hF_Mirabelle_hmem @ X @ ( hF_Mirabelle_HInter @ A5 ) )
% 2.47/1.58 = ( ! [Y3: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hmem @ Y3 @ A5 )
% 2.47/1.58 => ( hF_Mirabelle_hmem @ X @ Y3 ) ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_24_HUnion__hinsert,axiom,
% 2.47/1.58 ! [A2: hF_Mirabelle_hf,A5: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_HUnion @ ( hF_Mirabelle_hinsert @ A2 @ A5 ) )
% 2.47/1.58 = ( sup_sup @ hF_Mirabelle_hf @ A2 @ ( hF_Mirabelle_HUnion @ A5 ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_25_hdiff__insert,axiom,
% 2.47/1.58 ! [A5: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( minus_minus @ hF_Mirabelle_hf @ A5 @ ( hF_Mirabelle_hinsert @ A2 @ B4 ) )
% 2.47/1.58 = ( minus_minus @ hF_Mirabelle_hf @ ( minus_minus @ hF_Mirabelle_hf @ A5 @ B4 ) @ ( hF_Mirabelle_hinsert @ A2 @ ( zero_zero @ hF_Mirabelle_hf ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_26_hinsert__eq__sup,axiom,
% 2.47/1.58 ( hF_Mirabelle_hinsert
% 2.47/1.58 = ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( sup_sup @ hF_Mirabelle_hf @ B3 @ ( hF_Mirabelle_hinsert @ A4 @ ( zero_zero @ hF_Mirabelle_hf ) ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_27_hf__induct,axiom,
% 2.47/1.58 ! [P: hF_Mirabelle_hf > $o,Z: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( P @ ( zero_zero @ hF_Mirabelle_hf ) )
% 2.47/1.58 => ( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( P @ X2 )
% 2.47/1.58 => ( ( P @ Y2 )
% 2.47/1.58 => ( ~ ( hF_Mirabelle_hmem @ X2 @ Y2 )
% 2.47/1.58 => ( P @ ( hF_Mirabelle_hinsert @ X2 @ Y2 ) ) ) ) )
% 2.47/1.58 => ( P @ Z ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_28_hf__cases,axiom,
% 2.47/1.58 ! [Y: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( Y
% 2.47/1.58 != ( zero_zero @ hF_Mirabelle_hf ) )
% 2.47/1.58 => ~ ! [A6: hF_Mirabelle_hf,B5: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( Y
% 2.47/1.58 = ( hF_Mirabelle_hinsert @ A6 @ B5 ) )
% 2.47/1.58 => ( hF_Mirabelle_hmem @ A6 @ B5 ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_29_Replace__hunion,axiom,
% 2.47/1.58 ! [A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 2.47/1.58 ( ( hF_Mirabelle_Replace @ ( sup_sup @ hF_Mirabelle_hf @ A5 @ B4 ) @ R )
% 2.47/1.58 = ( sup_sup @ hF_Mirabelle_hf @ ( hF_Mirabelle_Replace @ A5 @ R ) @ ( hF_Mirabelle_Replace @ B4 @ R ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_30_HCollect__iff,axiom,
% 2.47/1.58 ! [X: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,A5: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hmem @ X @ ( hF_Mir1687042746ollect @ P @ A5 ) )
% 2.47/1.58 = ( ( P @ X )
% 2.47/1.58 & ( hF_Mirabelle_hmem @ X @ A5 ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_31_hf__equalityI,axiom,
% 2.47/1.58 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.47/1.58 ( ! [X2: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hmem @ X2 @ A2 )
% 2.47/1.58 = ( hF_Mirabelle_hmem @ X2 @ B ) )
% 2.47/1.58 => ( A2 = B ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_32_diff__self,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( group_add @ A )
% 2.47/1.58 => ! [A2: A] :
% 2.47/1.58 ( ( minus_minus @ A @ A2 @ A2 )
% 2.47/1.58 = ( zero_zero @ A ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_33_diff__0__right,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( group_add @ A )
% 2.47/1.58 => ! [A2: A] :
% 2.47/1.58 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 2.47/1.58 = A2 ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_34_zero__diff,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( comm_monoid_diff @ A )
% 2.47/1.58 => ! [A2: A] :
% 2.47/1.58 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
% 2.47/1.58 = ( zero_zero @ A ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_35_diff__zero,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( cancel1352612707id_add @ A )
% 2.47/1.58 => ! [A2: A] :
% 2.47/1.58 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 2.47/1.58 = A2 ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_36_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( cancel1352612707id_add @ A )
% 2.47/1.58 => ! [A2: A] :
% 2.47/1.58 ( ( minus_minus @ A @ A2 @ A2 )
% 2.47/1.58 = ( zero_zero @ A ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_37_hmem__hinsert,axiom,
% 2.47/1.58 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hmem @ A2 @ ( hF_Mirabelle_hinsert @ B @ C ) )
% 2.47/1.58 = ( ( A2 = B )
% 2.47/1.58 | ( hF_Mirabelle_hmem @ A2 @ C ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_38_hunion__iff,axiom,
% 2.47/1.58 ! [X: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hmem @ X @ ( sup_sup @ hF_Mirabelle_hf @ A2 @ B ) )
% 2.47/1.58 = ( ( hF_Mirabelle_hmem @ X @ A2 )
% 2.47/1.58 | ( hF_Mirabelle_hmem @ X @ B ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_39_hinter__iff,axiom,
% 2.47/1.58 ! [U: hF_Mirabelle_hf,X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hmem @ U @ ( inf_inf @ hF_Mirabelle_hf @ X @ Y ) )
% 2.47/1.58 = ( ( hF_Mirabelle_hmem @ U @ X )
% 2.47/1.58 & ( hF_Mirabelle_hmem @ U @ Y ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_40_RepFun__iff,axiom,
% 2.47/1.58 ! [V: hF_Mirabelle_hf,A5: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hmem @ V @ ( hF_Mirabelle_RepFun @ A5 @ F ) )
% 2.47/1.58 = ( ? [U2: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hmem @ U2 @ A5 )
% 2.47/1.58 & ( V
% 2.47/1.58 = ( F @ U2 ) ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_41_hunion__hempty__left,axiom,
% 2.47/1.58 ! [A5: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( sup_sup @ hF_Mirabelle_hf @ ( zero_zero @ hF_Mirabelle_hf ) @ A5 )
% 2.47/1.58 = A5 ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_42_hunion__hempty__right,axiom,
% 2.47/1.58 ! [A5: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( sup_sup @ hF_Mirabelle_hf @ A5 @ ( zero_zero @ hF_Mirabelle_hf ) )
% 2.47/1.58 = A5 ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_43_HUnion__iff,axiom,
% 2.47/1.58 ! [X: hF_Mirabelle_hf,A5: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hmem @ X @ ( hF_Mirabelle_HUnion @ A5 ) )
% 2.47/1.58 = ( ? [Y3: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hmem @ Y3 @ A5 )
% 2.47/1.58 & ( hF_Mirabelle_hmem @ X @ Y3 ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_44_mem__Collect__eq,axiom,
% 2.47/1.58 ! [A: $tType,A2: A,P: A > $o] :
% 2.47/1.58 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.47/1.58 = ( P @ A2 ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_45_Collect__mem__eq,axiom,
% 2.47/1.58 ! [A: $tType,A5: set @ A] :
% 2.47/1.58 ( ( collect @ A
% 2.47/1.58 @ ^ [X3: A] : ( member @ A @ X3 @ A5 ) )
% 2.47/1.58 = A5 ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_46_Collect__cong,axiom,
% 2.47/1.58 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.47/1.58 ( ! [X2: A] :
% 2.47/1.58 ( ( P @ X2 )
% 2.47/1.58 = ( Q @ X2 ) )
% 2.47/1.58 => ( ( collect @ A @ P )
% 2.47/1.58 = ( collect @ A @ Q ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_47_ext,axiom,
% 2.47/1.58 ! [B6: $tType,A: $tType,F: A > B6,G: A > B6] :
% 2.47/1.58 ( ! [X2: A] :
% 2.47/1.58 ( ( F @ X2 )
% 2.47/1.58 = ( G @ X2 ) )
% 2.47/1.58 => ( F = G ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_48_hdiff__iff,axiom,
% 2.47/1.58 ! [U: hF_Mirabelle_hf,X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hmem @ U @ ( minus_minus @ hF_Mirabelle_hf @ X @ Y ) )
% 2.47/1.58 = ( ( hF_Mirabelle_hmem @ U @ X )
% 2.47/1.58 & ~ ( hF_Mirabelle_hmem @ U @ Y ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_49_hinter__hempty__left,axiom,
% 2.47/1.58 ! [A5: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( inf_inf @ hF_Mirabelle_hf @ ( zero_zero @ hF_Mirabelle_hf ) @ A5 )
% 2.47/1.58 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_50_hinter__hempty__right,axiom,
% 2.47/1.58 ! [A5: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( inf_inf @ hF_Mirabelle_hf @ A5 @ ( zero_zero @ hF_Mirabelle_hf ) )
% 2.47/1.58 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_51_Replace__iff,axiom,
% 2.47/1.58 ! [V: hF_Mirabelle_hf,A5: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 2.47/1.58 ( ( hF_Mirabelle_hmem @ V @ ( hF_Mirabelle_Replace @ A5 @ R ) )
% 2.47/1.58 = ( ? [U2: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hmem @ U2 @ A5 )
% 2.47/1.58 & ( R @ U2 @ V )
% 2.47/1.58 & ! [Y3: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( R @ U2 @ Y3 )
% 2.47/1.58 => ( Y3 = V ) ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_52_hdiff__zero,axiom,
% 2.47/1.58 ! [X: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( minus_minus @ hF_Mirabelle_hf @ X @ ( zero_zero @ hF_Mirabelle_hf ) )
% 2.47/1.58 = X ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_53_zero__hdiff,axiom,
% 2.47/1.58 ! [X: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( minus_minus @ hF_Mirabelle_hf @ ( zero_zero @ hF_Mirabelle_hf ) @ X )
% 2.47/1.58 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_54_RepFun__hunion,axiom,
% 2.47/1.58 ! [A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_RepFun @ ( sup_sup @ hF_Mirabelle_hf @ A5 @ B4 ) @ F )
% 2.47/1.58 = ( sup_sup @ hF_Mirabelle_hf @ ( hF_Mirabelle_RepFun @ A5 @ F ) @ ( hF_Mirabelle_RepFun @ B4 @ F ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_55_HUnion__hunion,axiom,
% 2.47/1.58 ! [A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_HUnion @ ( sup_sup @ hF_Mirabelle_hf @ A5 @ B4 ) )
% 2.47/1.58 = ( sup_sup @ hF_Mirabelle_hf @ ( hF_Mirabelle_HUnion @ A5 ) @ ( hF_Mirabelle_HUnion @ B4 ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_56_diff__eq__diff__eq,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( group_add @ A )
% 2.47/1.58 => ! [A2: A,B: A,C: A,D: A] :
% 2.47/1.58 ( ( ( minus_minus @ A @ A2 @ B )
% 2.47/1.58 = ( minus_minus @ A @ C @ D ) )
% 2.47/1.58 => ( ( A2 = B )
% 2.47/1.58 = ( C = D ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_57_diff__right__commute,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( cancel146912293up_add @ A )
% 2.47/1.58 => ! [A2: A,C: A,B: A] :
% 2.47/1.58 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B )
% 2.47/1.58 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B ) @ C ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_58_replacement__fun,axiom,
% 2.47/1.58 ! [X: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 2.47/1.58 ? [Z2: hF_Mirabelle_hf] :
% 2.47/1.58 ! [V2: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hmem @ V2 @ Z2 )
% 2.47/1.58 = ( ? [U2: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hmem @ U2 @ X )
% 2.47/1.58 & ( V2
% 2.47/1.58 = ( F @ U2 ) ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_59_hmem__not__refl,axiom,
% 2.47/1.58 ! [X: hF_Mirabelle_hf] :
% 2.47/1.58 ~ ( hF_Mirabelle_hmem @ X @ X ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_60_comprehension,axiom,
% 2.47/1.58 ! [X: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 2.47/1.58 ? [Z2: hF_Mirabelle_hf] :
% 2.47/1.58 ! [U3: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hmem @ U3 @ Z2 )
% 2.47/1.58 = ( ( hF_Mirabelle_hmem @ U3 @ X )
% 2.47/1.58 & ( P @ U3 ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_61_union__of__set,axiom,
% 2.47/1.58 ! [X: hF_Mirabelle_hf] :
% 2.47/1.58 ? [Z2: hF_Mirabelle_hf] :
% 2.47/1.58 ! [U3: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hmem @ U3 @ Z2 )
% 2.47/1.58 = ( ? [Y3: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hmem @ Y3 @ X )
% 2.47/1.58 & ( hF_Mirabelle_hmem @ U3 @ Y3 ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_62_hmem__not__sym,axiom,
% 2.47/1.58 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 2.47/1.58 ~ ( ( hF_Mirabelle_hmem @ X @ Y )
% 2.47/1.58 & ( hF_Mirabelle_hmem @ Y @ X ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_63_binary__union,axiom,
% 2.47/1.58 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 2.47/1.58 ? [Z2: hF_Mirabelle_hf] :
% 2.47/1.58 ! [U3: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hmem @ U3 @ Z2 )
% 2.47/1.58 = ( ( hF_Mirabelle_hmem @ U3 @ X )
% 2.47/1.58 | ( hF_Mirabelle_hmem @ U3 @ Y ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_64_replacement,axiom,
% 2.47/1.58 ! [X: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 2.47/1.58 ( ! [U4: hF_Mirabelle_hf,V3: hF_Mirabelle_hf,V4: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hmem @ U4 @ X )
% 2.47/1.58 => ( ( R @ U4 @ V3 )
% 2.47/1.58 => ( ( R @ U4 @ V4 )
% 2.47/1.58 => ( V4 = V3 ) ) ) )
% 2.47/1.58 => ? [Z2: hF_Mirabelle_hf] :
% 2.47/1.58 ! [V2: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hmem @ V2 @ Z2 )
% 2.47/1.58 = ( ? [U2: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hmem @ U2 @ X )
% 2.47/1.58 & ( R @ U2 @ V2 ) ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_65_hmem__ne,axiom,
% 2.47/1.58 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hmem @ X @ Y )
% 2.47/1.58 => ( X != Y ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_66_hf__ext,axiom,
% 2.47/1.58 ( ( ^ [Y4: hF_Mirabelle_hf,Z3: hF_Mirabelle_hf] : ( Y4 = Z3 ) )
% 2.47/1.58 = ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 2.47/1.58 ! [X3: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hmem @ X3 @ A4 )
% 2.47/1.58 = ( hF_Mirabelle_hmem @ X3 @ B3 ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_67_hinter__hinsert__right,axiom,
% 2.47/1.58 ! [X: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,A5: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( ( hF_Mirabelle_hmem @ X @ B4 )
% 2.47/1.58 => ( ( inf_inf @ hF_Mirabelle_hf @ B4 @ ( hF_Mirabelle_hinsert @ X @ A5 ) )
% 2.47/1.58 = ( hF_Mirabelle_hinsert @ X @ ( inf_inf @ hF_Mirabelle_hf @ B4 @ A5 ) ) ) )
% 2.47/1.58 & ( ~ ( hF_Mirabelle_hmem @ X @ B4 )
% 2.47/1.58 => ( ( inf_inf @ hF_Mirabelle_hf @ B4 @ ( hF_Mirabelle_hinsert @ X @ A5 ) )
% 2.47/1.58 = ( inf_inf @ hF_Mirabelle_hf @ B4 @ A5 ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_68_hinter__hinsert__left,axiom,
% 2.47/1.58 ! [X: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,A5: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( ( hF_Mirabelle_hmem @ X @ B4 )
% 2.47/1.58 => ( ( inf_inf @ hF_Mirabelle_hf @ ( hF_Mirabelle_hinsert @ X @ A5 ) @ B4 )
% 2.47/1.58 = ( hF_Mirabelle_hinsert @ X @ ( inf_inf @ hF_Mirabelle_hf @ A5 @ B4 ) ) ) )
% 2.47/1.58 & ( ~ ( hF_Mirabelle_hmem @ X @ B4 )
% 2.47/1.58 => ( ( inf_inf @ hF_Mirabelle_hf @ ( hF_Mirabelle_hinsert @ X @ A5 ) @ B4 )
% 2.47/1.58 = ( inf_inf @ hF_Mirabelle_hf @ A5 @ B4 ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_69_foundation,axiom,
% 2.47/1.58 ! [Z: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( Z
% 2.47/1.58 != ( zero_zero @ hF_Mirabelle_hf ) )
% 2.47/1.58 => ? [W: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hmem @ W @ Z )
% 2.47/1.58 & ( ( inf_inf @ hF_Mirabelle_hf @ W @ Z )
% 2.47/1.58 = ( zero_zero @ hF_Mirabelle_hf ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_70_hinsert__hdiff__if,axiom,
% 2.47/1.58 ! [X: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,A5: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( ( hF_Mirabelle_hmem @ X @ B4 )
% 2.47/1.58 => ( ( minus_minus @ hF_Mirabelle_hf @ ( hF_Mirabelle_hinsert @ X @ A5 ) @ B4 )
% 2.47/1.58 = ( minus_minus @ hF_Mirabelle_hf @ A5 @ B4 ) ) )
% 2.47/1.58 & ( ~ ( hF_Mirabelle_hmem @ X @ B4 )
% 2.47/1.58 => ( ( minus_minus @ hF_Mirabelle_hf @ ( hF_Mirabelle_hinsert @ X @ A5 ) @ B4 )
% 2.47/1.58 = ( hF_Mirabelle_hinsert @ X @ ( minus_minus @ hF_Mirabelle_hf @ A5 @ B4 ) ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_71_eq__iff__diff__eq__0,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( group_add @ A )
% 2.47/1.58 => ( ( ^ [Y4: A,Z3: A] : ( Y4 = Z3 ) )
% 2.47/1.58 = ( ^ [A4: A,B3: A] :
% 2.47/1.58 ( ( minus_minus @ A @ A4 @ B3 )
% 2.47/1.58 = ( zero_zero @ A ) ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_72_hinsert__iff,axiom,
% 2.47/1.58 ! [Z: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( Z
% 2.47/1.58 = ( hF_Mirabelle_hinsert @ Y @ X ) )
% 2.47/1.58 = ( ! [U2: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hmem @ U2 @ Z )
% 2.47/1.58 = ( ( hF_Mirabelle_hmem @ U2 @ X )
% 2.47/1.58 | ( U2 = Y ) ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_73_hemptyE,axiom,
% 2.47/1.58 ! [A2: hF_Mirabelle_hf] :
% 2.47/1.58 ~ ( hF_Mirabelle_hmem @ A2 @ ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_74_hempty__iff,axiom,
% 2.47/1.58 ! [Z: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( Z
% 2.47/1.58 = ( zero_zero @ hF_Mirabelle_hf ) )
% 2.47/1.58 = ( ! [X3: hF_Mirabelle_hf] :
% 2.47/1.58 ~ ( hF_Mirabelle_hmem @ X3 @ Z ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_75_hmem__hempty,axiom,
% 2.47/1.58 ! [A2: hF_Mirabelle_hf] :
% 2.47/1.58 ~ ( hF_Mirabelle_hmem @ A2 @ ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_76_hunion__hinsert__right,axiom,
% 2.47/1.58 ! [B4: hF_Mirabelle_hf,X: hF_Mirabelle_hf,A5: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( sup_sup @ hF_Mirabelle_hf @ B4 @ ( hF_Mirabelle_hinsert @ X @ A5 ) )
% 2.47/1.58 = ( hF_Mirabelle_hinsert @ X @ ( sup_sup @ hF_Mirabelle_hf @ B4 @ A5 ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_77_hunion__hinsert__left,axiom,
% 2.47/1.58 ! [X: hF_Mirabelle_hf,A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( sup_sup @ hF_Mirabelle_hf @ ( hF_Mirabelle_hinsert @ X @ A5 ) @ B4 )
% 2.47/1.58 = ( hF_Mirabelle_hinsert @ X @ ( sup_sup @ hF_Mirabelle_hf @ A5 @ B4 ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_78_RepFun__cong,axiom,
% 2.47/1.58 ! [A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,G: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 2.47/1.58 ( ( A5 = B4 )
% 2.47/1.58 => ( ! [X2: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hmem @ X2 @ B4 )
% 2.47/1.58 => ( ( F @ X2 )
% 2.47/1.58 = ( G @ X2 ) ) )
% 2.47/1.58 => ( ( hF_Mirabelle_RepFun @ A5 @ F )
% 2.47/1.58 = ( hF_Mirabelle_RepFun @ B4 @ G ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_79_hmem__Sup__ne,axiom,
% 2.47/1.58 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hmem @ X @ Y )
% 2.47/1.58 => ( ( hF_Mirabelle_HUnion @ X )
% 2.47/1.58 != Y ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_80_Replace__cong,axiom,
% 2.47/1.58 ! [A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,P: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 2.47/1.58 ( ( A5 = B4 )
% 2.47/1.58 => ( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 2.47/1.58 ( ( hF_Mirabelle_hmem @ X2 @ B4 )
% 2.47/1.58 => ( ( P @ X2 @ Y2 )
% 2.47/1.58 = ( Q @ X2 @ Y2 ) ) )
% 2.47/1.58 => ( ( hF_Mirabelle_Replace @ A5 @ P )
% 2.47/1.58 = ( hF_Mirabelle_Replace @ B4 @ Q ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_81_sup__inf__absorb,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( lattice @ A )
% 2.47/1.58 => ! [X: A,Y: A] :
% 2.47/1.58 ( ( sup_sup @ A @ X @ ( inf_inf @ A @ X @ Y ) )
% 2.47/1.58 = X ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_82_inf__sup__absorb,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( lattice @ A )
% 2.47/1.58 => ! [X: A,Y: A] :
% 2.47/1.58 ( ( inf_inf @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 2.47/1.58 = X ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_83_sup_Oright__idem,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( semilattice_sup @ A )
% 2.47/1.58 => ! [A2: A,B: A] :
% 2.47/1.58 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B ) @ B )
% 2.47/1.58 = ( sup_sup @ A @ A2 @ B ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_84_sup__left__idem,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( semilattice_sup @ A )
% 2.47/1.58 => ! [X: A,Y: A] :
% 2.47/1.58 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 2.47/1.58 = ( sup_sup @ A @ X @ Y ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_85_sup_Oleft__idem,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( semilattice_sup @ A )
% 2.47/1.58 => ! [A2: A,B: A] :
% 2.47/1.58 ( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B ) )
% 2.47/1.58 = ( sup_sup @ A @ A2 @ B ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_86_sup__idem,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( semilattice_sup @ A )
% 2.47/1.58 => ! [X: A] :
% 2.47/1.58 ( ( sup_sup @ A @ X @ X )
% 2.47/1.58 = X ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_87_sup_Oidem,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( semilattice_sup @ A )
% 2.47/1.58 => ! [A2: A] :
% 2.47/1.58 ( ( sup_sup @ A @ A2 @ A2 )
% 2.47/1.58 = A2 ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_88_sup__apply,axiom,
% 2.47/1.58 ! [B6: $tType,A: $tType] :
% 2.47/1.58 ( ( semilattice_sup @ B6 )
% 2.47/1.58 => ( ( sup_sup @ ( A > B6 ) )
% 2.47/1.58 = ( ^ [F2: A > B6,G2: A > B6,X3: A] : ( sup_sup @ B6 @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_89_inf__right__idem,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( semilattice_inf @ A )
% 2.47/1.58 => ! [X: A,Y: A] :
% 2.47/1.58 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Y )
% 2.47/1.58 = ( inf_inf @ A @ X @ Y ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_90_minus__apply,axiom,
% 2.47/1.58 ! [B6: $tType,A: $tType] :
% 2.47/1.58 ( ( minus @ B6 )
% 2.47/1.58 => ( ( minus_minus @ ( A > B6 ) )
% 2.47/1.58 = ( ^ [A7: A > B6,B7: A > B6,X3: A] : ( minus_minus @ B6 @ ( A7 @ X3 ) @ ( B7 @ X3 ) ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_91_inf__apply,axiom,
% 2.47/1.58 ! [B6: $tType,A: $tType] :
% 2.47/1.58 ( ( semilattice_inf @ B6 )
% 2.47/1.58 => ( ( inf_inf @ ( A > B6 ) )
% 2.47/1.58 = ( ^ [F2: A > B6,G2: A > B6,X3: A] : ( inf_inf @ B6 @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_92_inf_Oidem,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( semilattice_inf @ A )
% 2.47/1.58 => ! [A2: A] :
% 2.47/1.58 ( ( inf_inf @ A @ A2 @ A2 )
% 2.47/1.58 = A2 ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_93_inf__idem,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( semilattice_inf @ A )
% 2.47/1.58 => ! [X: A] :
% 2.47/1.58 ( ( inf_inf @ A @ X @ X )
% 2.47/1.58 = X ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_94_inf_Oleft__idem,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( semilattice_inf @ A )
% 2.47/1.58 => ! [A2: A,B: A] :
% 2.47/1.58 ( ( inf_inf @ A @ A2 @ ( inf_inf @ A @ A2 @ B ) )
% 2.47/1.58 = ( inf_inf @ A @ A2 @ B ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_95_inf__left__idem,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( semilattice_inf @ A )
% 2.47/1.58 => ! [X: A,Y: A] :
% 2.47/1.58 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y ) )
% 2.47/1.58 = ( inf_inf @ A @ X @ Y ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_96_inf_Oright__idem,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( semilattice_inf @ A )
% 2.47/1.58 => ! [A2: A,B: A] :
% 2.47/1.58 ( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B ) @ B )
% 2.47/1.58 = ( inf_inf @ A @ A2 @ B ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_97_fun__diff__def,axiom,
% 2.47/1.58 ! [B6: $tType,A: $tType] :
% 2.47/1.58 ( ( minus @ B6 )
% 2.47/1.58 => ( ( minus_minus @ ( A > B6 ) )
% 2.47/1.58 = ( ^ [A7: A > B6,B7: A > B6,X3: A] : ( minus_minus @ B6 @ ( A7 @ X3 ) @ ( B7 @ X3 ) ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_98_inf__sup__aci_I4_J,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( lattice @ A )
% 2.47/1.58 => ! [X: A,Y: A] :
% 2.47/1.58 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y ) )
% 2.47/1.58 = ( inf_inf @ A @ X @ Y ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_99_inf__sup__aci_I3_J,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( lattice @ A )
% 2.47/1.58 => ! [X: A,Y: A,Z: A] :
% 2.47/1.58 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
% 2.47/1.58 = ( inf_inf @ A @ Y @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_100_inf__sup__aci_I2_J,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( lattice @ A )
% 2.47/1.58 => ! [X: A,Y: A,Z: A] :
% 2.47/1.58 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Z )
% 2.47/1.58 = ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_101_inf__sup__aci_I1_J,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( lattice @ A )
% 2.47/1.58 => ( ( inf_inf @ A )
% 2.47/1.58 = ( ^ [X3: A,Y3: A] : ( inf_inf @ A @ Y3 @ X3 ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_102_inf__fun__def,axiom,
% 2.47/1.58 ! [B6: $tType,A: $tType] :
% 2.47/1.58 ( ( semilattice_inf @ B6 )
% 2.47/1.58 => ( ( inf_inf @ ( A > B6 ) )
% 2.47/1.58 = ( ^ [F2: A > B6,G2: A > B6,X3: A] : ( inf_inf @ B6 @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_103_boolean__algebra__cancel_Oinf1,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( semilattice_inf @ A )
% 2.47/1.58 => ! [A5: A,K: A,A2: A,B: A] :
% 2.47/1.58 ( ( A5
% 2.47/1.58 = ( inf_inf @ A @ K @ A2 ) )
% 2.47/1.58 => ( ( inf_inf @ A @ A5 @ B )
% 2.47/1.58 = ( inf_inf @ A @ K @ ( inf_inf @ A @ A2 @ B ) ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_104_boolean__algebra__cancel_Oinf2,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( semilattice_inf @ A )
% 2.47/1.58 => ! [B4: A,K: A,B: A,A2: A] :
% 2.47/1.58 ( ( B4
% 2.47/1.58 = ( inf_inf @ A @ K @ B ) )
% 2.47/1.58 => ( ( inf_inf @ A @ A2 @ B4 )
% 2.47/1.58 = ( inf_inf @ A @ K @ ( inf_inf @ A @ A2 @ B ) ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_105_inf_Oassoc,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( semilattice_inf @ A )
% 2.47/1.58 => ! [A2: A,B: A,C: A] :
% 2.47/1.58 ( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B ) @ C )
% 2.47/1.58 = ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B @ C ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_106_inf__assoc,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( semilattice_inf @ A )
% 2.47/1.58 => ! [X: A,Y: A,Z: A] :
% 2.47/1.58 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Z )
% 2.47/1.58 = ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_107_inf_Ocommute,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( semilattice_inf @ A )
% 2.47/1.58 => ( ( inf_inf @ A )
% 2.47/1.58 = ( ^ [A4: A,B3: A] : ( inf_inf @ A @ B3 @ A4 ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_108_inf__commute,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( semilattice_inf @ A )
% 2.47/1.58 => ( ( inf_inf @ A )
% 2.47/1.58 = ( ^ [X3: A,Y3: A] : ( inf_inf @ A @ Y3 @ X3 ) ) ) ) ).
% 2.47/1.58
% 2.47/1.58 thf(fact_109_inf_Oleft__commute,axiom,
% 2.47/1.58 ! [A: $tType] :
% 2.47/1.58 ( ( semilattice_inf @ A )
% 2.47/1.58 => ! [B: A,A2: A,C: A] :
% 2.47/1.58 ( ( inf_inf @ A @ B @ ( inf_inf @ A @ A2 @ C ) )
% 2.47/1.59 = ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B @ C ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_110_inf__left__commute,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_inf @ A )
% 2.47/1.59 => ! [X: A,Y: A,Z: A] :
% 2.47/1.59 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
% 2.47/1.59 = ( inf_inf @ A @ Y @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_111_inf__sup__aci_I8_J,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( lattice @ A )
% 2.47/1.59 => ! [X: A,Y: A] :
% 2.47/1.59 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 2.47/1.59 = ( sup_sup @ A @ X @ Y ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_112_inf__sup__aci_I7_J,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( lattice @ A )
% 2.47/1.59 => ! [X: A,Y: A,Z: A] :
% 2.47/1.59 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
% 2.47/1.59 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_113_inf__sup__aci_I6_J,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( lattice @ A )
% 2.47/1.59 => ! [X: A,Y: A,Z: A] :
% 2.47/1.59 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
% 2.47/1.59 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_114_inf__sup__aci_I5_J,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( lattice @ A )
% 2.47/1.59 => ( ( sup_sup @ A )
% 2.47/1.59 = ( ^ [X3: A,Y3: A] : ( sup_sup @ A @ Y3 @ X3 ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_115_sup__fun__def,axiom,
% 2.47/1.59 ! [B6: $tType,A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ B6 )
% 2.47/1.59 => ( ( sup_sup @ ( A > B6 ) )
% 2.47/1.59 = ( ^ [F2: A > B6,G2: A > B6,X3: A] : ( sup_sup @ B6 @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_116_boolean__algebra__cancel_Osup1,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ! [A5: A,K: A,A2: A,B: A] :
% 2.47/1.59 ( ( A5
% 2.47/1.59 = ( sup_sup @ A @ K @ A2 ) )
% 2.47/1.59 => ( ( sup_sup @ A @ A5 @ B )
% 2.47/1.59 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A2 @ B ) ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_117_boolean__algebra__cancel_Osup2,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ! [B4: A,K: A,B: A,A2: A] :
% 2.47/1.59 ( ( B4
% 2.47/1.59 = ( sup_sup @ A @ K @ B ) )
% 2.47/1.59 => ( ( sup_sup @ A @ A2 @ B4 )
% 2.47/1.59 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A2 @ B ) ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_118_sup_Oassoc,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ! [A2: A,B: A,C: A] :
% 2.47/1.59 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B ) @ C )
% 2.47/1.59 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B @ C ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_119_sup__assoc,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ! [X: A,Y: A,Z: A] :
% 2.47/1.59 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
% 2.47/1.59 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_120_sup_Ocommute,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ( ( sup_sup @ A )
% 2.47/1.59 = ( ^ [A4: A,B3: A] : ( sup_sup @ A @ B3 @ A4 ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_121_sup__commute,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ( ( sup_sup @ A )
% 2.47/1.59 = ( ^ [X3: A,Y3: A] : ( sup_sup @ A @ Y3 @ X3 ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_122_sup_Oleft__commute,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ! [B: A,A2: A,C: A] :
% 2.47/1.59 ( ( sup_sup @ A @ B @ ( sup_sup @ A @ A2 @ C ) )
% 2.47/1.59 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B @ C ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_123_sup__left__commute,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ! [X: A,Y: A,Z: A] :
% 2.47/1.59 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
% 2.47/1.59 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_124_distrib__imp1,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( lattice @ A )
% 2.47/1.59 => ! [X: A,Y: A,Z: A] :
% 2.47/1.59 ( ! [X2: A,Y2: A,Z2: A] :
% 2.47/1.59 ( ( inf_inf @ A @ X2 @ ( sup_sup @ A @ Y2 @ Z2 ) )
% 2.47/1.59 = ( sup_sup @ A @ ( inf_inf @ A @ X2 @ Y2 ) @ ( inf_inf @ A @ X2 @ Z2 ) ) )
% 2.47/1.59 => ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
% 2.47/1.59 = ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z ) ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_125_distrib__imp2,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( lattice @ A )
% 2.47/1.59 => ! [X: A,Y: A,Z: A] :
% 2.47/1.59 ( ! [X2: A,Y2: A,Z2: A] :
% 2.47/1.59 ( ( sup_sup @ A @ X2 @ ( inf_inf @ A @ Y2 @ Z2 ) )
% 2.47/1.59 = ( inf_inf @ A @ ( sup_sup @ A @ X2 @ Y2 ) @ ( sup_sup @ A @ X2 @ Z2 ) ) )
% 2.47/1.59 => ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
% 2.47/1.59 = ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z ) ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_126_inf__sup__distrib1,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( distrib_lattice @ A )
% 2.47/1.59 => ! [X: A,Y: A,Z: A] :
% 2.47/1.59 ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
% 2.47/1.59 = ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_127_inf__sup__distrib2,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( distrib_lattice @ A )
% 2.47/1.59 => ! [Y: A,Z: A,X: A] :
% 2.47/1.59 ( ( inf_inf @ A @ ( sup_sup @ A @ Y @ Z ) @ X )
% 2.47/1.59 = ( sup_sup @ A @ ( inf_inf @ A @ Y @ X ) @ ( inf_inf @ A @ Z @ X ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_128_sup__inf__distrib1,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( distrib_lattice @ A )
% 2.47/1.59 => ! [X: A,Y: A,Z: A] :
% 2.47/1.59 ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
% 2.47/1.59 = ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_129_sup__inf__distrib2,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( distrib_lattice @ A )
% 2.47/1.59 => ! [Y: A,Z: A,X: A] :
% 2.47/1.59 ( ( sup_sup @ A @ ( inf_inf @ A @ Y @ Z ) @ X )
% 2.47/1.59 = ( inf_inf @ A @ ( sup_sup @ A @ Y @ X ) @ ( sup_sup @ A @ Z @ X ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_130_diff__self__eq__0,axiom,
% 2.47/1.59 ! [M: nat] :
% 2.47/1.59 ( ( minus_minus @ nat @ M @ M )
% 2.47/1.59 = ( zero_zero @ nat ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_131_diff__0__eq__0,axiom,
% 2.47/1.59 ! [N: nat] :
% 2.47/1.59 ( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
% 2.47/1.59 = ( zero_zero @ nat ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_132_PrimReplace__iff,axiom,
% 2.47/1.59 ! [A5: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,V: hF_Mirabelle_hf] :
% 2.47/1.59 ( ! [U4: hF_Mirabelle_hf,V3: hF_Mirabelle_hf,V4: hF_Mirabelle_hf] :
% 2.47/1.59 ( ( hF_Mirabelle_hmem @ U4 @ A5 )
% 2.47/1.59 => ( ( R @ U4 @ V3 )
% 2.47/1.59 => ( ( R @ U4 @ V4 )
% 2.47/1.59 => ( V4 = V3 ) ) ) )
% 2.47/1.59 => ( ( hF_Mirabelle_hmem @ V @ ( hF_Mir569462966eplace @ A5 @ R ) )
% 2.47/1.59 = ( ? [U2: hF_Mirabelle_hf] :
% 2.47/1.59 ( ( hF_Mirabelle_hmem @ U2 @ A5 )
% 2.47/1.59 & ( R @ U2 @ V ) ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_133_diffs0__imp__equal,axiom,
% 2.47/1.59 ! [M: nat,N: nat] :
% 2.47/1.59 ( ( ( minus_minus @ nat @ M @ N )
% 2.47/1.59 = ( zero_zero @ nat ) )
% 2.47/1.59 => ( ( ( minus_minus @ nat @ N @ M )
% 2.47/1.59 = ( zero_zero @ nat ) )
% 2.47/1.59 => ( M = N ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_134_minus__nat_Odiff__0,axiom,
% 2.47/1.59 ! [M: nat] :
% 2.47/1.59 ( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
% 2.47/1.59 = M ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_135_zero__natural_Orsp,axiom,
% 2.47/1.59 ( ( zero_zero @ nat )
% 2.47/1.59 = ( zero_zero @ nat ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_136_diff__commute,axiom,
% 2.47/1.59 ! [I: nat,J: nat,K: nat] :
% 2.47/1.59 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
% 2.47/1.59 = ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K ) @ J ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_137_of__nat__aux_Osimps_I1_J,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semiring_1 @ A )
% 2.47/1.59 => ! [Inc: A > A,I: A] :
% 2.47/1.59 ( ( semiri532925092at_aux @ A @ Inc @ ( zero_zero @ nat ) @ I )
% 2.47/1.59 = I ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_138_less__eq__insert1__iff,axiom,
% 2.47/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 2.47/1.59 ( ( ord_less_eq @ hF_Mirabelle_hf @ ( hF_Mirabelle_hinsert @ X @ Y ) @ Z )
% 2.47/1.59 = ( ( hF_Mirabelle_hmem @ X @ Z )
% 2.47/1.59 & ( ord_less_eq @ hF_Mirabelle_hf @ Y @ Z ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_139_hmem__def,axiom,
% 2.47/1.59 ( hF_Mirabelle_hmem
% 2.47/1.59 = ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( member @ hF_Mirabelle_hf @ A4 @ ( hF_Mirabelle_hfset @ B3 ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_140_diff__is__0__eq,axiom,
% 2.47/1.59 ! [M: nat,N: nat] :
% 2.47/1.59 ( ( ( minus_minus @ nat @ M @ N )
% 2.47/1.59 = ( zero_zero @ nat ) )
% 2.47/1.59 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_141_le__zero__eq,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( canoni770627133id_add @ A )
% 2.47/1.59 => ! [N: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 2.47/1.59 = ( N
% 2.47/1.59 = ( zero_zero @ A ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_142_inf_Obounded__iff,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_inf @ A )
% 2.47/1.59 => ! [A2: A,B: A,C: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B @ C ) )
% 2.47/1.59 = ( ( ord_less_eq @ A @ A2 @ B )
% 2.47/1.59 & ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_143_le__inf__iff,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_inf @ A )
% 2.47/1.59 => ! [X: A,Y: A,Z: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
% 2.47/1.59 = ( ( ord_less_eq @ A @ X @ Y )
% 2.47/1.59 & ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_144_sup_Obounded__iff,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ! [B: A,C: A,A2: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B @ C ) @ A2 )
% 2.47/1.59 = ( ( ord_less_eq @ A @ B @ A2 )
% 2.47/1.59 & ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_145_le__sup__iff,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ! [X: A,Y: A,Z: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
% 2.47/1.59 = ( ( ord_less_eq @ A @ X @ Z )
% 2.47/1.59 & ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_146_bot__nat__0_Oextremum,axiom,
% 2.47/1.59 ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_147_le0,axiom,
% 2.47/1.59 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_148_diff__diff__cancel,axiom,
% 2.47/1.59 ! [I: nat,N: nat] :
% 2.47/1.59 ( ( ord_less_eq @ nat @ I @ N )
% 2.47/1.59 => ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I ) )
% 2.47/1.59 = I ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_149_hsubsetI,axiom,
% 2.47/1.59 ! [A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 2.47/1.59 ( ! [X2: hF_Mirabelle_hf] :
% 2.47/1.59 ( ( hF_Mirabelle_hmem @ X2 @ A5 )
% 2.47/1.59 => ( hF_Mirabelle_hmem @ X2 @ B4 ) )
% 2.47/1.59 => ( ord_less_eq @ hF_Mirabelle_hf @ A5 @ B4 ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_150_less__eq__hempty,axiom,
% 2.47/1.59 ! [U: hF_Mirabelle_hf] :
% 2.47/1.59 ( ( ord_less_eq @ hF_Mirabelle_hf @ U @ ( zero_zero @ hF_Mirabelle_hf ) )
% 2.47/1.59 = ( U
% 2.47/1.59 = ( zero_zero @ hF_Mirabelle_hf ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_151_diff__ge__0__iff__ge,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( ordered_ab_group_add @ A )
% 2.47/1.59 => ! [A2: A,B: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B ) )
% 2.47/1.59 = ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_152_diff__is__0__eq_H,axiom,
% 2.47/1.59 ! [M: nat,N: nat] :
% 2.47/1.59 ( ( ord_less_eq @ nat @ M @ N )
% 2.47/1.59 => ( ( minus_minus @ nat @ M @ N )
% 2.47/1.59 = ( zero_zero @ nat ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_153_diff__le__mono2,axiom,
% 2.47/1.59 ! [M: nat,N: nat,L: nat] :
% 2.47/1.59 ( ( ord_less_eq @ nat @ M @ N )
% 2.47/1.59 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_154_le__diff__iff_H,axiom,
% 2.47/1.59 ! [A2: nat,C: nat,B: nat] :
% 2.47/1.59 ( ( ord_less_eq @ nat @ A2 @ C )
% 2.47/1.59 => ( ( ord_less_eq @ nat @ B @ C )
% 2.47/1.59 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ A2 ) @ ( minus_minus @ nat @ C @ B ) )
% 2.47/1.59 = ( ord_less_eq @ nat @ B @ A2 ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_155_diff__le__self,axiom,
% 2.47/1.59 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_156_diff__le__mono,axiom,
% 2.47/1.59 ! [M: nat,N: nat,L: nat] :
% 2.47/1.59 ( ( ord_less_eq @ nat @ M @ N )
% 2.47/1.59 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_157_Nat_Odiff__diff__eq,axiom,
% 2.47/1.59 ! [K: nat,M: nat,N: nat] :
% 2.47/1.59 ( ( ord_less_eq @ nat @ K @ M )
% 2.47/1.59 => ( ( ord_less_eq @ nat @ K @ N )
% 2.47/1.59 => ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 2.47/1.59 = ( minus_minus @ nat @ M @ N ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_158_le__diff__iff,axiom,
% 2.47/1.59 ! [K: nat,M: nat,N: nat] :
% 2.47/1.59 ( ( ord_less_eq @ nat @ K @ M )
% 2.47/1.59 => ( ( ord_less_eq @ nat @ K @ N )
% 2.47/1.59 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 2.47/1.59 = ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_159_eq__diff__iff,axiom,
% 2.47/1.59 ! [K: nat,M: nat,N: nat] :
% 2.47/1.59 ( ( ord_less_eq @ nat @ K @ M )
% 2.47/1.59 => ( ( ord_less_eq @ nat @ K @ N )
% 2.47/1.59 => ( ( ( minus_minus @ nat @ M @ K )
% 2.47/1.59 = ( minus_minus @ nat @ N @ K ) )
% 2.47/1.59 = ( M = N ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_160_bot__nat__0_Oextremum__uniqueI,axiom,
% 2.47/1.59 ! [A2: nat] :
% 2.47/1.59 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 2.47/1.59 => ( A2
% 2.47/1.59 = ( zero_zero @ nat ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_161_bot__nat__0_Oextremum__unique,axiom,
% 2.47/1.59 ! [A2: nat] :
% 2.47/1.59 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 2.47/1.59 = ( A2
% 2.47/1.59 = ( zero_zero @ nat ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_162_le__0__eq,axiom,
% 2.47/1.59 ! [N: nat] :
% 2.47/1.59 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 2.47/1.59 = ( N
% 2.47/1.59 = ( zero_zero @ nat ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_163_less__eq__nat_Osimps_I1_J,axiom,
% 2.47/1.59 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_164_Nat_Oex__has__greatest__nat,axiom,
% 2.47/1.59 ! [P: nat > $o,K: nat,B: nat] :
% 2.47/1.59 ( ( P @ K )
% 2.47/1.59 => ( ! [Y2: nat] :
% 2.47/1.59 ( ( P @ Y2 )
% 2.47/1.59 => ( ord_less_eq @ nat @ Y2 @ B ) )
% 2.47/1.59 => ? [X2: nat] :
% 2.47/1.59 ( ( P @ X2 )
% 2.47/1.59 & ! [Y5: nat] :
% 2.47/1.59 ( ( P @ Y5 )
% 2.47/1.59 => ( ord_less_eq @ nat @ Y5 @ X2 ) ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_165_nat__le__linear,axiom,
% 2.47/1.59 ! [M: nat,N: nat] :
% 2.47/1.59 ( ( ord_less_eq @ nat @ M @ N )
% 2.47/1.59 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_166_le__antisym,axiom,
% 2.47/1.59 ! [M: nat,N: nat] :
% 2.47/1.59 ( ( ord_less_eq @ nat @ M @ N )
% 2.47/1.59 => ( ( ord_less_eq @ nat @ N @ M )
% 2.47/1.59 => ( M = N ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_167_eq__imp__le,axiom,
% 2.47/1.59 ! [M: nat,N: nat] :
% 2.47/1.59 ( ( M = N )
% 2.47/1.59 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_168_le__trans,axiom,
% 2.47/1.59 ! [I: nat,J: nat,K: nat] :
% 2.47/1.59 ( ( ord_less_eq @ nat @ I @ J )
% 2.47/1.59 => ( ( ord_less_eq @ nat @ J @ K )
% 2.47/1.59 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_169_le__refl,axiom,
% 2.47/1.59 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_170_canonically__ordered__monoid__add__class_Ozero__le,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( canoni770627133id_add @ A )
% 2.47/1.59 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_171_diff__mono,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( ordered_ab_group_add @ A )
% 2.47/1.59 => ! [A2: A,B: A,D: A,C: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ A2 @ B )
% 2.47/1.59 => ( ( ord_less_eq @ A @ D @ C )
% 2.47/1.59 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ D ) ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_172_diff__left__mono,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( ordered_ab_group_add @ A )
% 2.47/1.59 => ! [B: A,A2: A,C: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ B @ A2 )
% 2.47/1.59 => ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_173_diff__right__mono,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( ordered_ab_group_add @ A )
% 2.47/1.59 => ! [A2: A,B: A,C: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ A2 @ B )
% 2.47/1.59 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ C ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_174_diff__eq__diff__less__eq,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( ordered_ab_group_add @ A )
% 2.47/1.59 => ! [A2: A,B: A,C: A,D: A] :
% 2.47/1.59 ( ( ( minus_minus @ A @ A2 @ B )
% 2.47/1.59 = ( minus_minus @ A @ C @ D ) )
% 2.47/1.59 => ( ( ord_less_eq @ A @ A2 @ B )
% 2.47/1.59 = ( ord_less_eq @ A @ C @ D ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_175_inf_OcoboundedI2,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_inf @ A )
% 2.47/1.59 => ! [B: A,C: A,A2: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ B @ C )
% 2.47/1.59 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B ) @ C ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_176_inf_OcoboundedI1,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_inf @ A )
% 2.47/1.59 => ! [A2: A,C: A,B: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ A2 @ C )
% 2.47/1.59 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B ) @ C ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_177_inf_Oabsorb__iff2,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_inf @ A )
% 2.47/1.59 => ( ( ord_less_eq @ A )
% 2.47/1.59 = ( ^ [B3: A,A4: A] :
% 2.47/1.59 ( ( inf_inf @ A @ A4 @ B3 )
% 2.47/1.59 = B3 ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_178_inf_Oabsorb__iff1,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_inf @ A )
% 2.47/1.59 => ( ( ord_less_eq @ A )
% 2.47/1.59 = ( ^ [A4: A,B3: A] :
% 2.47/1.59 ( ( inf_inf @ A @ A4 @ B3 )
% 2.47/1.59 = A4 ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_179_inf_Ocobounded2,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_inf @ A )
% 2.47/1.59 => ! [A2: A,B: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B ) @ B ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_180_inf_Ocobounded1,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_inf @ A )
% 2.47/1.59 => ! [A2: A,B: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B ) @ A2 ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_181_inf_Oorder__iff,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_inf @ A )
% 2.47/1.59 => ( ( ord_less_eq @ A )
% 2.47/1.59 = ( ^ [A4: A,B3: A] :
% 2.47/1.59 ( A4
% 2.47/1.59 = ( inf_inf @ A @ A4 @ B3 ) ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_182_inf__greatest,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_inf @ A )
% 2.47/1.59 => ! [X: A,Y: A,Z: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ X @ Y )
% 2.47/1.59 => ( ( ord_less_eq @ A @ X @ Z )
% 2.47/1.59 => ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_183_inf_OboundedI,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_inf @ A )
% 2.47/1.59 => ! [A2: A,B: A,C: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ A2 @ B )
% 2.47/1.59 => ( ( ord_less_eq @ A @ A2 @ C )
% 2.47/1.59 => ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B @ C ) ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_184_inf_OboundedE,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_inf @ A )
% 2.47/1.59 => ! [A2: A,B: A,C: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B @ C ) )
% 2.47/1.59 => ~ ( ( ord_less_eq @ A @ A2 @ B )
% 2.47/1.59 => ~ ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_185_inf__absorb2,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_inf @ A )
% 2.47/1.59 => ! [Y: A,X: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ Y @ X )
% 2.47/1.59 => ( ( inf_inf @ A @ X @ Y )
% 2.47/1.59 = Y ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_186_inf__absorb1,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_inf @ A )
% 2.47/1.59 => ! [X: A,Y: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ X @ Y )
% 2.47/1.59 => ( ( inf_inf @ A @ X @ Y )
% 2.47/1.59 = X ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_187_inf_Oabsorb2,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_inf @ A )
% 2.47/1.59 => ! [B: A,A2: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ B @ A2 )
% 2.47/1.59 => ( ( inf_inf @ A @ A2 @ B )
% 2.47/1.59 = B ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_188_inf_Oabsorb1,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_inf @ A )
% 2.47/1.59 => ! [A2: A,B: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ A2 @ B )
% 2.47/1.59 => ( ( inf_inf @ A @ A2 @ B )
% 2.47/1.59 = A2 ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_189_le__iff__inf,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_inf @ A )
% 2.47/1.59 => ( ( ord_less_eq @ A )
% 2.47/1.59 = ( ^ [X3: A,Y3: A] :
% 2.47/1.59 ( ( inf_inf @ A @ X3 @ Y3 )
% 2.47/1.59 = X3 ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_190_inf__unique,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_inf @ A )
% 2.47/1.59 => ! [F: A > A > A,X: A,Y: A] :
% 2.47/1.59 ( ! [X2: A,Y2: A] : ( ord_less_eq @ A @ ( F @ X2 @ Y2 ) @ X2 )
% 2.47/1.59 => ( ! [X2: A,Y2: A] : ( ord_less_eq @ A @ ( F @ X2 @ Y2 ) @ Y2 )
% 2.47/1.59 => ( ! [X2: A,Y2: A,Z2: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.47/1.59 => ( ( ord_less_eq @ A @ X2 @ Z2 )
% 2.47/1.59 => ( ord_less_eq @ A @ X2 @ ( F @ Y2 @ Z2 ) ) ) )
% 2.47/1.59 => ( ( inf_inf @ A @ X @ Y )
% 2.47/1.59 = ( F @ X @ Y ) ) ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_191_inf_OorderI,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_inf @ A )
% 2.47/1.59 => ! [A2: A,B: A] :
% 2.47/1.59 ( ( A2
% 2.47/1.59 = ( inf_inf @ A @ A2 @ B ) )
% 2.47/1.59 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_192_inf_OorderE,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_inf @ A )
% 2.47/1.59 => ! [A2: A,B: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ A2 @ B )
% 2.47/1.59 => ( A2
% 2.47/1.59 = ( inf_inf @ A @ A2 @ B ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_193_le__infI2,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_inf @ A )
% 2.47/1.59 => ! [B: A,X: A,A2: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ B @ X )
% 2.47/1.59 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B ) @ X ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_194_le__infI1,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_inf @ A )
% 2.47/1.59 => ! [A2: A,X: A,B: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ A2 @ X )
% 2.47/1.59 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B ) @ X ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_195_inf__mono,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_inf @ A )
% 2.47/1.59 => ! [A2: A,C: A,B: A,D: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ A2 @ C )
% 2.47/1.59 => ( ( ord_less_eq @ A @ B @ D )
% 2.47/1.59 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B ) @ ( inf_inf @ A @ C @ D ) ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_196_le__infI,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_inf @ A )
% 2.47/1.59 => ! [X: A,A2: A,B: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ X @ A2 )
% 2.47/1.59 => ( ( ord_less_eq @ A @ X @ B )
% 2.47/1.59 => ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A2 @ B ) ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_197_le__infE,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_inf @ A )
% 2.47/1.59 => ! [X: A,A2: A,B: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A2 @ B ) )
% 2.47/1.59 => ~ ( ( ord_less_eq @ A @ X @ A2 )
% 2.47/1.59 => ~ ( ord_less_eq @ A @ X @ B ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_198_inf__le2,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_inf @ A )
% 2.47/1.59 => ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Y ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_199_inf__le1,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_inf @ A )
% 2.47/1.59 => ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ X ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_200_inf__sup__ord_I1_J,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( lattice @ A )
% 2.47/1.59 => ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ X ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_201_inf__sup__ord_I2_J,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( lattice @ A )
% 2.47/1.59 => ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Y ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_202_sup_OcoboundedI2,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ! [C: A,B: A,A2: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ C @ B )
% 2.47/1.59 => ( ord_less_eq @ A @ C @ ( sup_sup @ A @ A2 @ B ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_203_sup_OcoboundedI1,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ! [C: A,A2: A,B: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ C @ A2 )
% 2.47/1.59 => ( ord_less_eq @ A @ C @ ( sup_sup @ A @ A2 @ B ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_204_sup_Oabsorb__iff2,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ( ( ord_less_eq @ A )
% 2.47/1.59 = ( ^ [A4: A,B3: A] :
% 2.47/1.59 ( ( sup_sup @ A @ A4 @ B3 )
% 2.47/1.59 = B3 ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_205_sup_Oabsorb__iff1,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ( ( ord_less_eq @ A )
% 2.47/1.59 = ( ^ [B3: A,A4: A] :
% 2.47/1.59 ( ( sup_sup @ A @ A4 @ B3 )
% 2.47/1.59 = A4 ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_206_sup_Ocobounded2,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ! [B: A,A2: A] : ( ord_less_eq @ A @ B @ ( sup_sup @ A @ A2 @ B ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_207_sup_Ocobounded1,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ! [A2: A,B: A] : ( ord_less_eq @ A @ A2 @ ( sup_sup @ A @ A2 @ B ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_208_sup_Oorder__iff,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ( ( ord_less_eq @ A )
% 2.47/1.59 = ( ^ [B3: A,A4: A] :
% 2.47/1.59 ( A4
% 2.47/1.59 = ( sup_sup @ A @ A4 @ B3 ) ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_209_sup_OboundedI,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ! [B: A,A2: A,C: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ B @ A2 )
% 2.47/1.59 => ( ( ord_less_eq @ A @ C @ A2 )
% 2.47/1.59 => ( ord_less_eq @ A @ ( sup_sup @ A @ B @ C ) @ A2 ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_210_sup_OboundedE,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ! [B: A,C: A,A2: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B @ C ) @ A2 )
% 2.47/1.59 => ~ ( ( ord_less_eq @ A @ B @ A2 )
% 2.47/1.59 => ~ ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_211_sup__absorb2,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ! [X: A,Y: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ X @ Y )
% 2.47/1.59 => ( ( sup_sup @ A @ X @ Y )
% 2.47/1.59 = Y ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_212_sup__absorb1,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ! [Y: A,X: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ Y @ X )
% 2.47/1.59 => ( ( sup_sup @ A @ X @ Y )
% 2.47/1.59 = X ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_213_sup_Oabsorb2,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ! [A2: A,B: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ A2 @ B )
% 2.47/1.59 => ( ( sup_sup @ A @ A2 @ B )
% 2.47/1.59 = B ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_214_sup_Oabsorb1,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ! [B: A,A2: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ B @ A2 )
% 2.47/1.59 => ( ( sup_sup @ A @ A2 @ B )
% 2.47/1.59 = A2 ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_215_sup__unique,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ! [F: A > A > A,X: A,Y: A] :
% 2.47/1.59 ( ! [X2: A,Y2: A] : ( ord_less_eq @ A @ X2 @ ( F @ X2 @ Y2 ) )
% 2.47/1.59 => ( ! [X2: A,Y2: A] : ( ord_less_eq @ A @ Y2 @ ( F @ X2 @ Y2 ) )
% 2.47/1.59 => ( ! [X2: A,Y2: A,Z2: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ Y2 @ X2 )
% 2.47/1.59 => ( ( ord_less_eq @ A @ Z2 @ X2 )
% 2.47/1.59 => ( ord_less_eq @ A @ ( F @ Y2 @ Z2 ) @ X2 ) ) )
% 2.47/1.59 => ( ( sup_sup @ A @ X @ Y )
% 2.47/1.59 = ( F @ X @ Y ) ) ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_216_sup_OorderI,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ! [A2: A,B: A] :
% 2.47/1.59 ( ( A2
% 2.47/1.59 = ( sup_sup @ A @ A2 @ B ) )
% 2.47/1.59 => ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_217_sup_OorderE,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ! [B: A,A2: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ B @ A2 )
% 2.47/1.59 => ( A2
% 2.47/1.59 = ( sup_sup @ A @ A2 @ B ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_218_le__iff__sup,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ( ( ord_less_eq @ A )
% 2.47/1.59 = ( ^ [X3: A,Y3: A] :
% 2.47/1.59 ( ( sup_sup @ A @ X3 @ Y3 )
% 2.47/1.59 = Y3 ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_219_sup__least,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ! [Y: A,X: A,Z: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ Y @ X )
% 2.47/1.59 => ( ( ord_less_eq @ A @ Z @ X )
% 2.47/1.59 => ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z ) @ X ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_220_sup__mono,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ! [A2: A,C: A,B: A,D: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ A2 @ C )
% 2.47/1.59 => ( ( ord_less_eq @ A @ B @ D )
% 2.47/1.59 => ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B ) @ ( sup_sup @ A @ C @ D ) ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_221_sup_Omono,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ! [C: A,A2: A,D: A,B: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ C @ A2 )
% 2.47/1.59 => ( ( ord_less_eq @ A @ D @ B )
% 2.47/1.59 => ( ord_less_eq @ A @ ( sup_sup @ A @ C @ D ) @ ( sup_sup @ A @ A2 @ B ) ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_222_le__supI2,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ! [X: A,B: A,A2: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ X @ B )
% 2.47/1.59 => ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A2 @ B ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_223_le__supI1,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ! [X: A,A2: A,B: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ X @ A2 )
% 2.47/1.59 => ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A2 @ B ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_224_sup__ge2,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_225_sup__ge1,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_226_le__supI,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ! [A2: A,X: A,B: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ A2 @ X )
% 2.47/1.59 => ( ( ord_less_eq @ A @ B @ X )
% 2.47/1.59 => ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B ) @ X ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_227_le__supE,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A )
% 2.47/1.59 => ! [A2: A,B: A,X: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B ) @ X )
% 2.47/1.59 => ~ ( ( ord_less_eq @ A @ A2 @ X )
% 2.47/1.59 => ~ ( ord_less_eq @ A @ B @ X ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_228_inf__sup__ord_I3_J,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( lattice @ A )
% 2.47/1.59 => ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_229_inf__sup__ord_I4_J,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( lattice @ A )
% 2.47/1.59 => ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_230_less__eq__hf__def,axiom,
% 2.47/1.59 ( ( ord_less_eq @ hF_Mirabelle_hf )
% 2.47/1.59 = ( ^ [A7: hF_Mirabelle_hf,B7: hF_Mirabelle_hf] :
% 2.47/1.59 ! [X3: hF_Mirabelle_hf] :
% 2.47/1.59 ( ( hF_Mirabelle_hmem @ X3 @ A7 )
% 2.47/1.59 => ( hF_Mirabelle_hmem @ X3 @ B7 ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_231_rev__hsubsetD,axiom,
% 2.47/1.59 ! [C: hF_Mirabelle_hf,A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 2.47/1.59 ( ( hF_Mirabelle_hmem @ C @ A5 )
% 2.47/1.59 => ( ( ord_less_eq @ hF_Mirabelle_hf @ A5 @ B4 )
% 2.47/1.59 => ( hF_Mirabelle_hmem @ C @ B4 ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_232_hsubsetCE,axiom,
% 2.47/1.59 ! [A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.47/1.59 ( ( ord_less_eq @ hF_Mirabelle_hf @ A5 @ B4 )
% 2.47/1.59 => ( ( hF_Mirabelle_hmem @ C @ A5 )
% 2.47/1.59 => ( hF_Mirabelle_hmem @ C @ B4 ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_233_hsubsetD,axiom,
% 2.47/1.59 ! [A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.47/1.59 ( ( ord_less_eq @ hF_Mirabelle_hf @ A5 @ B4 )
% 2.47/1.59 => ( ( hF_Mirabelle_hmem @ C @ A5 )
% 2.47/1.59 => ( hF_Mirabelle_hmem @ C @ B4 ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_234_HF__Mirabelle__fsbjehakzm_Ozero__le,axiom,
% 2.47/1.59 ! [X: hF_Mirabelle_hf] : ( ord_less_eq @ hF_Mirabelle_hf @ ( zero_zero @ hF_Mirabelle_hf ) @ X ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_235_hf__equalityE,axiom,
% 2.47/1.59 ! [A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 2.47/1.59 ( ( A5 = B4 )
% 2.47/1.59 => ~ ( ( ord_less_eq @ hF_Mirabelle_hf @ A5 @ B4 )
% 2.47/1.59 => ~ ( ord_less_eq @ hF_Mirabelle_hf @ B4 @ A5 ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_236_le__iff__diff__le__0,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( ordered_ab_group_add @ A )
% 2.47/1.59 => ( ( ord_less_eq @ A )
% 2.47/1.59 = ( ^ [A4: A,B3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_237_distrib__inf__le,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( lattice @ A )
% 2.47/1.59 => ! [X: A,Y: A,Z: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z ) ) @ ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_238_distrib__sup__le,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( lattice @ A )
% 2.47/1.59 => ! [X: A,Y: A,Z: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) @ ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_239_less__eq__insert2__iff,axiom,
% 2.47/1.59 ! [Z: hF_Mirabelle_hf,X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 2.47/1.59 ( ( ord_less_eq @ hF_Mirabelle_hf @ Z @ ( hF_Mirabelle_hinsert @ X @ Y ) )
% 2.47/1.59 = ( ( ord_less_eq @ hF_Mirabelle_hf @ Z @ Y )
% 2.47/1.59 | ? [U2: hF_Mirabelle_hf] :
% 2.47/1.59 ( ( ( hF_Mirabelle_hinsert @ X @ U2 )
% 2.47/1.59 = Z )
% 2.47/1.59 & ~ ( hF_Mirabelle_hmem @ X @ U2 )
% 2.47/1.59 & ( ord_less_eq @ hF_Mirabelle_hf @ U2 @ Y ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_240_order__refl,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( preorder @ A )
% 2.47/1.59 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_241_le__numeral__extra_I3_J,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( linord1659791738miring @ A )
% 2.47/1.59 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_242_HF__hfset,axiom,
% 2.47/1.59 ! [A2: hF_Mirabelle_hf] :
% 2.47/1.59 ( ( hF_Mirabelle_HF @ ( hF_Mirabelle_hfset @ A2 ) )
% 2.47/1.59 = A2 ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_243_of__nat__le__0__iff,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( linord1659791738miring @ A )
% 2.47/1.59 => ! [M: nat] :
% 2.47/1.59 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) )
% 2.47/1.59 = ( M
% 2.47/1.59 = ( zero_zero @ nat ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_244_of__nat__eq__iff,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semiring_char_0 @ A )
% 2.47/1.59 => ! [M: nat,N: nat] :
% 2.47/1.59 ( ( ( semiring_1_of_nat @ A @ M )
% 2.47/1.59 = ( semiring_1_of_nat @ A @ N ) )
% 2.47/1.59 = ( M = N ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_245_of__nat__0,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semiring_1 @ A )
% 2.47/1.59 => ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
% 2.47/1.59 = ( zero_zero @ A ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_246_of__nat__0__eq__iff,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semiring_char_0 @ A )
% 2.47/1.59 => ! [N: nat] :
% 2.47/1.59 ( ( ( zero_zero @ A )
% 2.47/1.59 = ( semiring_1_of_nat @ A @ N ) )
% 2.47/1.59 = ( ( zero_zero @ nat )
% 2.47/1.59 = N ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_247_of__nat__eq__0__iff,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semiring_char_0 @ A )
% 2.47/1.59 => ! [M: nat] :
% 2.47/1.59 ( ( ( semiring_1_of_nat @ A @ M )
% 2.47/1.59 = ( zero_zero @ A ) )
% 2.47/1.59 = ( M
% 2.47/1.59 = ( zero_zero @ nat ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_248_of__nat__le__iff,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( linord1659791738miring @ A )
% 2.47/1.59 => ! [M: nat,N: nat] :
% 2.47/1.59 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 2.47/1.59 = ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_249_of__nat__0__le__iff,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( linord1659791738miring @ A )
% 2.47/1.59 => ! [N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_250_of__nat__mono,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( linord1659791738miring @ A )
% 2.47/1.59 => ! [I: nat,J: nat] :
% 2.47/1.59 ( ( ord_less_eq @ nat @ I @ J )
% 2.47/1.59 => ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ I ) @ ( semiring_1_of_nat @ A @ J ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_251_of__nat__diff,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( semiring_1_cancel @ A )
% 2.47/1.59 => ! [N: nat,M: nat] :
% 2.47/1.59 ( ( ord_less_eq @ nat @ N @ M )
% 2.47/1.59 => ( ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ M @ N ) )
% 2.47/1.59 = ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_252_dual__order_Oantisym,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( order @ A )
% 2.47/1.59 => ! [B: A,A2: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ B @ A2 )
% 2.47/1.59 => ( ( ord_less_eq @ A @ A2 @ B )
% 2.47/1.59 => ( A2 = B ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(fact_253_dual__order_Oeq__iff,axiom,
% 2.47/1.59 ! [A: $tType] :
% 2.47/1.59 ( ( order @ A )
% 2.47/1.59 => ( ( ^ [Y4: A,Z3: A] : ( Y4 = Z3 ) )
% 2.47/1.59 = ( ^ [A4: A,B3: A] :
% 2.47/1.59 ( ( ord_less_eq @ A @ B3 @ A4 )
% 2.47/1.59 & ( ord_less_eq @ A @ A4 @ B3 ) ) ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 2.47/1.59 ! [A8: $tType,A9: $tType] :
% 2.47/1.59 ( ( semilattice_sup @ A9 )
% 2.47/1.59 => ( semilattice_sup @ ( A8 > A9 ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
% 2.47/1.59 ! [A8: $tType,A9: $tType] :
% 2.47/1.59 ( ( semilattice_inf @ A9 )
% 2.47/1.59 => ( semilattice_inf @ ( A8 > A9 ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_fun___Lattices_Odistrib__lattice,axiom,
% 2.47/1.59 ! [A8: $tType,A9: $tType] :
% 2.47/1.59 ( ( distrib_lattice @ A9 )
% 2.47/1.59 => ( distrib_lattice @ ( A8 > A9 ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.47/1.59 ! [A8: $tType,A9: $tType] :
% 2.47/1.59 ( ( preorder @ A9 )
% 2.47/1.59 => ( preorder @ ( A8 > A9 ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_fun___Lattices_Olattice,axiom,
% 2.47/1.59 ! [A8: $tType,A9: $tType] :
% 2.47/1.59 ( ( lattice @ A9 )
% 2.47/1.59 => ( lattice @ ( A8 > A9 ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.47/1.59 ! [A8: $tType,A9: $tType] :
% 2.47/1.59 ( ( order @ A9 )
% 2.47/1.59 => ( order @ ( A8 > A9 ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_fun___Groups_Ominus,axiom,
% 2.47/1.59 ! [A8: $tType,A9: $tType] :
% 2.47/1.59 ( ( minus @ A9 )
% 2.47/1.59 => ( minus @ ( A8 > A9 ) ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 2.47/1.59 canoni770627133id_add @ nat ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
% 2.47/1.59 linord1659791738miring @ nat ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
% 2.47/1.59 cancel146912293up_add @ nat ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
% 2.47/1.59 cancel1352612707id_add @ nat ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_1,axiom,
% 2.47/1.59 semilattice_sup @ nat ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_2,axiom,
% 2.47/1.59 semilattice_inf @ nat ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_Nat_Onat___Lattices_Odistrib__lattice_3,axiom,
% 2.47/1.59 distrib_lattice @ nat ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_Nat_Onat___Rings_Osemiring__1__cancel,axiom,
% 2.47/1.59 semiring_1_cancel @ nat ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
% 2.47/1.59 comm_monoid_diff @ nat ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_Nat_Onat___Nat_Osemiring__char__0,axiom,
% 2.47/1.59 semiring_char_0 @ nat ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_Nat_Onat___Orderings_Opreorder_4,axiom,
% 2.47/1.59 preorder @ nat ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_Nat_Onat___Rings_Osemiring__1,axiom,
% 2.47/1.59 semiring_1 @ nat ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_Nat_Onat___Lattices_Olattice_5,axiom,
% 2.47/1.59 lattice @ nat ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_Nat_Onat___Orderings_Oorder_6,axiom,
% 2.47/1.59 order @ nat ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_Nat_Onat___Groups_Ominus_7,axiom,
% 2.47/1.59 minus @ nat ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 2.47/1.59 zero @ nat ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_8,axiom,
% 2.47/1.59 ! [A8: $tType] : ( semilattice_sup @ ( set @ A8 ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_9,axiom,
% 2.47/1.59 ! [A8: $tType] : ( semilattice_inf @ ( set @ A8 ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_Set_Oset___Lattices_Odistrib__lattice_10,axiom,
% 2.47/1.59 ! [A8: $tType] : ( distrib_lattice @ ( set @ A8 ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_Set_Oset___Orderings_Opreorder_11,axiom,
% 2.47/1.59 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_Set_Oset___Lattices_Olattice_12,axiom,
% 2.47/1.59 ! [A8: $tType] : ( lattice @ ( set @ A8 ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_Set_Oset___Orderings_Oorder_13,axiom,
% 2.47/1.59 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_Set_Oset___Groups_Ominus_14,axiom,
% 2.47/1.59 ! [A8: $tType] : ( minus @ ( set @ A8 ) ) ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_15,axiom,
% 2.47/1.59 semilattice_sup @ $o ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_16,axiom,
% 2.47/1.59 semilattice_inf @ $o ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_17,axiom,
% 2.47/1.59 distrib_lattice @ $o ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_HOL_Obool___Orderings_Opreorder_18,axiom,
% 2.47/1.59 preorder @ $o ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_HOL_Obool___Lattices_Olattice_19,axiom,
% 2.47/1.59 lattice @ $o ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_HOL_Obool___Orderings_Oorder_20,axiom,
% 2.47/1.59 order @ $o ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_HOL_Obool___Groups_Ominus_21,axiom,
% 2.47/1.59 minus @ $o ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Osemilattice__sup_22,axiom,
% 2.47/1.59 semilattice_sup @ hF_Mirabelle_hf ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Osemilattice__inf_23,axiom,
% 2.47/1.59 semilattice_inf @ hF_Mirabelle_hf ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Odistrib__lattice_24,axiom,
% 2.47/1.59 distrib_lattice @ hF_Mirabelle_hf ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Orderings_Opreorder_25,axiom,
% 2.47/1.59 preorder @ hF_Mirabelle_hf ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Olattice_26,axiom,
% 2.47/1.59 lattice @ hF_Mirabelle_hf ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Orderings_Oorder_27,axiom,
% 2.47/1.59 order @ hF_Mirabelle_hf ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Groups_Ominus_28,axiom,
% 2.47/1.59 minus @ hF_Mirabelle_hf ).
% 2.47/1.59
% 2.47/1.59 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Groups_Ozero_29,axiom,
% 2.47/1.59 zero @ hF_Mirabelle_hf ).
% 2.47/1.59
% 2.47/1.59 thf(conj_0,conjecture,
% 2.47/1.59 ( ( hF_Mirabelle_hpair @ x @ y )
% 2.47/1.59 != ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 2.47/1.59 ------------------------
% 2.69/1.60 % (7917)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_7787 for (2999ds/4Mi)
% 2.69/1.60 % (7918)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_7787 for (2999ds/27Mi)
% 2.69/1.60 % (7919)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_7787 for (2999ds/2Mi)
% 2.69/1.60 % (7922)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_7787 for (2999ds/18Mi)
% 2.69/1.60 % (7920)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_7787 for (2999ds/2Mi)
% 2.69/1.60 % (7919)Instruction limit reached!
% 2.69/1.60 % (7919)------------------------------
% 2.69/1.60 % (7919)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.60 % (7919)Termination reason: Unknown
% 2.69/1.60 % (7919)Termination phase: shuffling
% 2.69/1.60
% 2.69/1.60 % (7919)Memory used [KB]: 1279
% 2.69/1.60 % (7919)Time elapsed: 0.002 s
% 2.69/1.60 % (7919)Instructions burned: 3 (million)
% 2.69/1.60 % (7919)------------------------------
% 2.69/1.60 % (7919)------------------------------
% 2.69/1.60 % (7920)Instruction limit reached!
% 2.69/1.60 % (7920)------------------------------
% 2.69/1.60 % (7920)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.60 % (7920)Termination reason: Unknown
% 2.69/1.60 % (7920)Termination phase: shuffling
% 2.69/1.60
% 2.69/1.60 % (7920)Memory used [KB]: 1279
% 2.69/1.60 % (7920)Time elapsed: 0.002 s
% 2.69/1.60 % (7920)Instructions burned: 2 (million)
% 2.69/1.60 % (7920)------------------------------
% 2.69/1.60 % (7920)------------------------------
% 2.69/1.60 % (7917)Instruction limit reached!
% 2.69/1.60 % (7917)------------------------------
% 2.69/1.60 % (7917)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.60 % (7917)Termination reason: Unknown
% 2.69/1.60 % (7917)Termination phase: shuffling
% 2.69/1.60
% 2.69/1.60 % (7917)Memory used [KB]: 1407
% 2.69/1.60 % (7917)Time elapsed: 0.003 s
% 2.69/1.60 % (7917)Instructions burned: 6 (million)
% 2.69/1.60 % (7917)------------------------------
% 2.69/1.60 % (7917)------------------------------
% 2.69/1.61 % (7921)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_7787 for (2999ds/275Mi)
% 2.69/1.61 % (7922)Instruction limit reached!
% 2.69/1.61 % (7922)------------------------------
% 2.69/1.61 % (7922)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.61 % (7922)Termination reason: Unknown
% 2.69/1.61 % (7922)Termination phase: Property scanning
% 2.69/1.61
% 2.69/1.61 % (7922)Memory used [KB]: 1535
% 2.69/1.61 % (7922)Time elapsed: 0.007 s
% 2.69/1.61 % (7922)Instructions burned: 18 (million)
% 2.69/1.61 % (7922)------------------------------
% 2.69/1.61 % (7922)------------------------------
% 2.69/1.61 % (7916)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_7787 for (2999ds/183Mi)
% 2.69/1.61 % (7918)Instruction limit reached!
% 2.69/1.61 % (7918)------------------------------
% 2.69/1.61 % (7918)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.61 % (7918)Termination reason: Unknown
% 2.69/1.61 % (7918)Termination phase: Preprocessing 2
% 2.69/1.61
% 2.69/1.61 % (7918)Memory used [KB]: 1663
% 2.69/1.61 % (7918)Time elapsed: 0.010 s
% 2.69/1.61 % (7918)Instructions burned: 29 (million)
% 2.69/1.61 % (7918)------------------------------
% 2.69/1.61 % (7918)------------------------------
% 2.69/1.61 % (7927)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_7787 for (2999ds/3Mi)
% 2.69/1.61 % (7928)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_7787 for (2999ds/37Mi)
% 2.69/1.61 % (7929)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_7787 for (2999ds/15Mi)
% 2.69/1.61 % (7927)Instruction limit reached!
% 2.69/1.61 % (7927)------------------------------
% 2.69/1.61 % (7927)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.61 % (7927)Termination reason: Unknown
% 2.69/1.61 % (7927)Termination phase: shuffling
% 2.69/1.61
% 2.69/1.61 % (7927)Memory used [KB]: 1279
% 2.69/1.61 % (7927)Time elapsed: 0.002 s
% 2.69/1.61 % (7927)Instructions burned: 3 (million)
% 2.69/1.61 % (7927)------------------------------
% 2.69/1.61 % (7927)------------------------------
% 2.69/1.62 % (7931)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_7787 for (2999ds/3Mi)
% 2.69/1.62 % (7929)Instruction limit reached!
% 2.69/1.62 % (7929)------------------------------
% 2.69/1.62 % (7929)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.62 % (7929)Termination reason: Unknown
% 2.69/1.62 % (7929)Termination phase: shuffling
% 2.69/1.62
% 2.69/1.62 % (7929)Memory used [KB]: 1535
% 2.69/1.62 % (7929)Time elapsed: 0.006 s
% 2.69/1.62 % (7929)Instructions burned: 15 (million)
% 2.69/1.62 % (7929)------------------------------
% 2.69/1.62 % (7929)------------------------------
% 2.69/1.62 % (7931)Instruction limit reached!
% 2.69/1.62 % (7931)------------------------------
% 2.69/1.62 % (7931)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.62 % (7931)Termination reason: Unknown
% 2.69/1.62 % (7931)Termination phase: shuffling
% 2.69/1.62
% 2.69/1.62 % (7931)Memory used [KB]: 1279
% 2.69/1.62 % (7931)Time elapsed: 0.002 s
% 2.69/1.62 % (7931)Instructions burned: 3 (million)
% 2.69/1.62 % (7931)------------------------------
% 2.69/1.62 % (7931)------------------------------
% 2.69/1.62 % (7935)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_7787 for (2999ds/1041Mi)
% 2.69/1.62 % (7938)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_7787 for (2999ds/7Mi)
% 2.69/1.62 % (7928)Instruction limit reached!
% 2.69/1.62 % (7928)------------------------------
% 2.69/1.62 % (7928)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.62 % (7928)Termination reason: Unknown
% 2.69/1.62 % (7928)Termination phase: Property scanning
% 2.69/1.62
% 2.69/1.62 % (7928)Memory used [KB]: 1918
% 2.69/1.62 % (7928)Time elapsed: 0.012 s
% 2.69/1.62 % (7928)Instructions burned: 40 (million)
% 2.69/1.62 % (7928)------------------------------
% 2.69/1.62 % (7928)------------------------------
% 2.69/1.63 % (7938)Instruction limit reached!
% 2.69/1.63 % (7938)------------------------------
% 2.69/1.63 % (7938)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.63 % (7938)Termination reason: Unknown
% 2.69/1.63 % (7938)Termination phase: shuffling
% 2.69/1.63
% 2.69/1.63 % (7938)Memory used [KB]: 1407
% 2.69/1.63 % (7938)Time elapsed: 0.004 s
% 2.69/1.63 % (7938)Instructions burned: 8 (million)
% 2.69/1.63 % (7938)------------------------------
% 2.69/1.63 % (7938)------------------------------
% 2.69/1.63 % (7941)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_7787 for (2999ds/16Mi)
% 2.69/1.63 % (7942)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_7787 for (2999ds/3Mi)
% 2.69/1.63 % (7921)First to succeed.
% 2.69/1.63 % (7942)Instruction limit reached!
% 2.69/1.63 % (7942)------------------------------
% 2.69/1.63 % (7942)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.63 % (7942)Termination reason: Unknown
% 2.69/1.63 % (7942)Termination phase: shuffling
% 2.69/1.63
% 2.69/1.63 % (7942)Memory used [KB]: 1407
% 2.69/1.63 % (7942)Time elapsed: 0.003 s
% 2.69/1.63 % (7942)Instructions burned: 6 (million)
% 2.69/1.63 % (7942)------------------------------
% 2.69/1.63 % (7942)------------------------------
% 2.69/1.63 % (7921)Refutation found. Thanks to Tanya!
% 2.69/1.63 % SZS status Theorem for DTF2THF_7787
% 2.69/1.63 % SZS output start Proof for DTF2THF_7787
% 2.69/1.63 thf(type_def_6, type, hF_Mirabelle_hf: $tType).
% 2.69/1.63 thf(type_def_7, type, set: ($tType) > $tType).
% 2.69/1.63 thf(type_def_8, type, nat: $tType).
% 2.69/1.63 thf(func_def_0, type, hF_Mirabelle_hf: $tType).
% 2.69/1.63 thf(func_def_1, type, set: ($tType) > $tType).
% 2.69/1.63 thf(func_def_2, type, nat: $tType).
% 2.69/1.63 thf(func_def_3, type, type: !>[X0: $tType]:($o)).
% 2.69/1.63 thf(func_def_4, type, zero: !>[X0: $tType]:($o)).
% 2.69/1.63 thf(func_def_5, type, minus: !>[X0: $tType]:($o)).
% 2.69/1.63 thf(func_def_6, type, order: !>[X0: $tType]:($o)).
% 2.69/1.63 thf(func_def_7, type, group_add: !>[X0: $tType]:($o)).
% 2.69/1.63 thf(func_def_8, type, lattice: !>[X0: $tType]:($o)).
% 2.69/1.63 thf(func_def_9, type, semiring_1: !>[X0: $tType]:($o)).
% 2.69/1.63 thf(func_def_10, type, preorder: !>[X0: $tType]:($o)).
% 2.69/1.63 thf(func_def_11, type, semiring_char_0: !>[X0: $tType]:($o)).
% 2.69/1.63 thf(func_def_12, type, comm_monoid_diff: !>[X0: $tType]:($o)).
% 2.69/1.63 thf(func_def_13, type, semiring_1_cancel: !>[X0: $tType]:($o)).
% 2.69/1.63 thf(func_def_14, type, distrib_lattice: !>[X0: $tType]:($o)).
% 2.69/1.63 thf(func_def_15, type, semilattice_inf: !>[X0: $tType]:($o)).
% 2.69/1.63 thf(func_def_16, type, semilattice_sup: !>[X0: $tType]:($o)).
% 2.69/1.63 thf(func_def_17, type, ordered_ab_group_add: !>[X0: $tType]:($o)).
% 2.69/1.63 thf(func_def_18, type, cancel1352612707id_add: !>[X0: $tType]:($o)).
% 2.69/1.63 thf(func_def_19, type, cancel146912293up_add: !>[X0: $tType]:($o)).
% 2.69/1.63 thf(func_def_20, type, linord1659791738miring: !>[X0: $tType]:($o)).
% 2.69/1.63 thf(func_def_21, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 2.69/1.63 thf(func_def_22, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.69/1.63 thf(func_def_23, type, zero_zero: !>[X0: $tType]:(X0)).
% 2.69/1.63 thf(func_def_24, type, hF_Mir1687042746ollect: (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.69/1.63 thf(func_def_25, type, hF_Mirabelle_HF: set @ hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.69/1.63 thf(func_def_26, type, hF_Mirabelle_HInter: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.69/1.63 thf(func_def_27, type, hF_Mirabelle_HUnion: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.69/1.63 thf(func_def_28, type, hF_Mir569462966eplace: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.69/1.63 thf(func_def_29, type, hF_Mirabelle_RepFun: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf).
% 2.69/1.63 thf(func_def_30, type, hF_Mirabelle_Replace: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.69/1.63 thf(func_def_31, type, hF_Mirabelle_Abs_hf: nat > hF_Mirabelle_hf).
% 2.69/1.63 thf(func_def_32, type, hF_Mirabelle_hfset: hF_Mirabelle_hf > set @ hF_Mirabelle_hf).
% 2.69/1.63 thf(func_def_33, type, hF_Mirabelle_hfst: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.69/1.63 thf(func_def_34, type, hF_Mirabelle_hinsert: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.69/1.63 thf(func_def_35, type, hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o).
% 2.69/1.63 thf(func_def_36, type, hF_Mirabelle_hpair: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.69/1.63 thf(func_def_37, type, hF_Mirabelle_hsnd: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.69/1.63 thf(func_def_38, type, hF_Mirabelle_hsplit: !>[X0: $tType]:((hF_Mirabelle_hf > hF_Mirabelle_hf > X0) > hF_Mirabelle_hf > X0)).
% 2.69/1.63 thf(func_def_39, type, inf_inf: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.69/1.63 thf(func_def_40, type, sup_sup: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.69/1.63 thf(func_def_41, type, semiring_1_of_nat: !>[X0: $tType]:(nat > X0)).
% 2.69/1.63 thf(func_def_42, type, semiri532925092at_aux: !>[X0: $tType]:((X0 > X0) > nat > X0 > X0)).
% 2.69/1.63 thf(func_def_43, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.69/1.63 thf(func_def_44, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.69/1.63 thf(func_def_45, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.69/1.63 thf(func_def_46, type, x: hF_Mirabelle_hf).
% 2.69/1.63 thf(func_def_47, type, y: hF_Mirabelle_hf).
% 2.69/1.63 thf(func_def_59, type, sK0: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.69/1.63 thf(func_def_60, type, sK1: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.69/1.63 thf(func_def_61, type, sK2: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.69/1.63 thf(func_def_62, type, sK3: (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.69/1.63 thf(func_def_63, type, sK4: (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.69/1.63 thf(func_def_64, type, sK5: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.69/1.63 thf(func_def_65, type, sK6: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.69/1.63 thf(func_def_66, type, sK7: (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.69/1.63 thf(func_def_67, type, sK8: (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.69/1.63 thf(func_def_69, type, ph10: !>[X0: $tType]:(X0)).
% 2.69/1.63 thf(f987,plain,(
% 2.69/1.63 $false),
% 2.69/1.63 inference(subsumption_resolution,[status(thm)],[f986,f944])).
% 2.69/1.63 thf(f944,plain,(
% 2.69/1.63 ( ! [X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : ((zero_zero @ hF_Mirabelle_hf != (hF_Mirabelle_hinsert @ X0 @ X1))) )),
% 2.69/1.63 inference(cnf_transformation,[status(thm)],[f836])).
% 2.69/1.63 thf(f836,plain,(
% 2.69/1.63 ! [X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : (zero_zero @ hF_Mirabelle_hf != (hF_Mirabelle_hinsert @ X0 @ X1))),
% 2.69/1.63 inference(rectify,[status(thm)],[f19])).
% 2.69/1.63 thf(f19,axiom,(
% 2.69/1.63 ! [X1 : hF_Mirabelle_hf,X14 : hF_Mirabelle_hf] : (zero_zero @ hF_Mirabelle_hf != (hF_Mirabelle_hinsert @ X1 @ X14))),
% 2.69/1.63 file('/export/starexec/sandbox/tmp/tmp.dWcERH6OXC/DTF2THF_7787.p',fact_18_hinsert__nonempty)).
% 2.69/1.63 thf(f986,plain,(
% 2.69/1.63 (zero_zero @ hF_Mirabelle_hf = (hF_Mirabelle_hinsert @ (hF_Mirabelle_hinsert @ x @ (hF_Mirabelle_hinsert @ x @ zero_zero @ hF_Mirabelle_hf)) @ (hF_Mirabelle_hinsert @ (hF_Mirabelle_hinsert @ x @ (hF_Mirabelle_hinsert @ y @ zero_zero @ hF_Mirabelle_hf)) @ zero_zero @ hF_Mirabelle_hf)))),
% 2.69/1.63 inference(beta_eta_normalization,[status(thm)],[f963])).
% 2.69/1.63 thf(f963,plain,(
% 2.69/1.63 (zero_zero @ hF_Mirabelle_hf = ((^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf]: (hF_Mirabelle_hinsert @ (hF_Mirabelle_hinsert @ Y0 @ (hF_Mirabelle_hinsert @ Y0 @ zero_zero @ hF_Mirabelle_hf)) @ (hF_Mirabelle_hinsert @ (hF_Mirabelle_hinsert @ Y0 @ (hF_Mirabelle_hinsert @ Y1 @ zero_zero @ hF_Mirabelle_hf)) @ zero_zero @ hF_Mirabelle_hf))))) @ x @ y))),
% 2.69/1.63 inference(definition_unfolding,[status(thm)],[f929,f885])).
% 2.69/1.63 thf(f885,plain,(
% 2.69/1.63 (hF_Mirabelle_hpair = (^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf]: (hF_Mirabelle_hinsert @ (hF_Mirabelle_hinsert @ Y0 @ (hF_Mirabelle_hinsert @ Y0 @ zero_zero @ hF_Mirabelle_hf)) @ (hF_Mirabelle_hinsert @ (hF_Mirabelle_hinsert @ Y0 @ (hF_Mirabelle_hinsert @ Y1 @ zero_zero @ hF_Mirabelle_hf)) @ zero_zero @ hF_Mirabelle_hf))))))),
% 2.69/1.63 inference(cnf_transformation,[status(thm)],[f718])).
% 2.69/1.63 thf(f718,plain,(
% 2.69/1.63 (hF_Mirabelle_hpair = (^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf]: (hF_Mirabelle_hinsert @ (hF_Mirabelle_hinsert @ Y0 @ (hF_Mirabelle_hinsert @ Y0 @ zero_zero @ hF_Mirabelle_hf)) @ (hF_Mirabelle_hinsert @ (hF_Mirabelle_hinsert @ Y0 @ (hF_Mirabelle_hinsert @ Y1 @ zero_zero @ hF_Mirabelle_hf)) @ zero_zero @ hF_Mirabelle_hf))))))),
% 2.69/1.63 inference(fool_elimination,[status(thm)],[f717])).
% 2.69/1.63 thf(f717,plain,(
% 2.69/1.63 ((^[X0 : hF_Mirabelle_hf, X1 : hF_Mirabelle_hf] : (hF_Mirabelle_hinsert @ (hF_Mirabelle_hinsert @ X0 @ (hF_Mirabelle_hinsert @ X0 @ zero_zero @ hF_Mirabelle_hf)) @ (hF_Mirabelle_hinsert @ (hF_Mirabelle_hinsert @ X0 @ (hF_Mirabelle_hinsert @ X1 @ zero_zero @ hF_Mirabelle_hf)) @ zero_zero @ hF_Mirabelle_hf))) = hF_Mirabelle_hpair)),
% 2.69/1.63 inference(rectify,[status(thm)],[f15])).
% 2.69/1.63 thf(f15,axiom,(
% 2.69/1.63 ((^[X10 : hF_Mirabelle_hf, X11 : hF_Mirabelle_hf] : (hF_Mirabelle_hinsert @ (hF_Mirabelle_hinsert @ X10 @ (hF_Mirabelle_hinsert @ X10 @ zero_zero @ hF_Mirabelle_hf)) @ (hF_Mirabelle_hinsert @ (hF_Mirabelle_hinsert @ X10 @ (hF_Mirabelle_hinsert @ X11 @ zero_zero @ hF_Mirabelle_hf)) @ zero_zero @ hF_Mirabelle_hf))) = hF_Mirabelle_hpair)),
% 2.69/1.63 file('/export/starexec/sandbox/tmp/tmp.dWcERH6OXC/DTF2THF_7787.p',fact_14_hpair__def_H)).
% 2.69/1.63 thf(f929,plain,(
% 2.69/1.63 (zero_zero @ hF_Mirabelle_hf = (hF_Mirabelle_hpair @ x @ y))),
% 2.69/1.63 inference(cnf_transformation,[status(thm)],[f821])).
% 2.69/1.63 thf(f821,plain,(
% 2.69/1.63 (zero_zero @ hF_Mirabelle_hf = (hF_Mirabelle_hpair @ x @ y))),
% 2.69/1.63 inference(flattening,[status(thm)],[f301])).
% 2.69/1.63 thf(f301,negated_conjecture,(
% 2.69/1.63 ~(zero_zero @ hF_Mirabelle_hf != (hF_Mirabelle_hpair @ x @ y))),
% 2.69/1.63 inference(negated_conjecture,[status(cth)],[f300])).
% 2.69/1.63 thf(f300,conjecture,(
% 2.69/1.63 (zero_zero @ hF_Mirabelle_hf != (hF_Mirabelle_hpair @ x @ y))),
% 2.69/1.63 file('/export/starexec/sandbox/tmp/tmp.dWcERH6OXC/DTF2THF_7787.p',conj_0)).
% 2.69/1.63 % SZS output end Proof for DTF2THF_7787
% 2.69/1.63 % (7921)------------------------------
% 2.69/1.63 % (7921)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.63 % (7921)Termination reason: Refutation
% 2.69/1.63
% 2.69/1.63 % (7921)Memory used [KB]: 6268
% 2.69/1.63 % (7921)Time elapsed: 0.023 s
% 2.69/1.63 % (7921)Instructions burned: 40 (million)
% 2.69/1.63 % (7921)------------------------------
% 2.69/1.63 % (7921)------------------------------
% 2.69/1.63 % (7912)Success in time 0.04 s
%------------------------------------------------------------------------------