TPTP Problem File: SLH0973^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Safe_Range_RC/0023_Restrict_Bounds_Impl/prob_00049_002115__17716086_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1683 ( 756 unt; 416 typ; 0 def)
% Number of atoms : 3858 (2153 equ; 0 cnn)
% Maximal formula atoms : 60 ( 3 avg)
% Number of connectives : 12828 ( 933 ~; 107 |; 339 &;9922 @)
% ( 0 <=>;1527 =>; 0 <=; 0 <~>)
% Maximal formula depth : 37 ( 6 avg)
% Number of types : 83 ( 82 usr)
% Number of type conns : 1131 (1131 >; 0 *; 0 +; 0 <<)
% Number of symbols : 337 ( 334 usr; 28 con; 0-4 aty)
% Number of variables : 4217 ( 168 ^;3864 !; 185 ?;4217 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:27:02.202
%------------------------------------------------------------------------------
% Could-be-implicit typings (82)
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collec2099942116761351594la_a_b: ( set_Re381260168593705685la_a_b > $o ) > set_se6865892389300016395la_a_b ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J,type,
collec7641561117972753376la_a_b: ( set_se6865892389300016395la_a_b > $o ) > set_se1626149436229564481la_a_b ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OCollect_001tf__b,type,
collect_b: ( b > $o ) > set_b ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
image_1636186708463624715la_a_b: ( list_R8263082107343818799la_a_b > set_Re381260168593705685la_a_b ) > set_li8334890247616788453la_a_b > set_se6865892389300016395la_a_b ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J_001t__Set__Oset_It__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J,type,
image_6556082427161849975la_a_b: ( list_s3671366213839843941la_a_b > set_se6865892389300016395la_a_b ) > set_li7574107037410898715la_a_b > set_se1626149436229564481la_a_b ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
image_nat_list_nat: ( nat > list_nat ) > set_nat > set_list_nat ).
thf(sy_c_Set_Oimage_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
image_6790371041703824709la_a_b: ( relational_fmla_a_b > relational_fmla_a_b ) > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_Oimage_001t__Relational____Calculus__Oterm_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_1080223610614215258_set_a: ( relational_term_a > set_a ) > set_Re5178783185447174953term_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
image_7051608999182166449la_a_b: ( set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ) > set_se6865892389300016395la_a_b > set_se6865892389300016395la_a_b ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J_001t__Set__Oset_It__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J,type,
image_5985962561035126813la_a_b: ( set_se6865892389300016395la_a_b > set_se6865892389300016395la_a_b ) > set_se1626149436229564481la_a_b > set_se1626149436229564481la_a_b ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
insert_list_nat2: list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
insert3617074100887873119la_a_b: list_R8263082107343818799la_a_b > set_li8334890247616788453la_a_b > set_li8334890247616788453la_a_b ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J,type,
insert6225987573742651541la_a_b: list_s3671366213839843941la_a_b > set_li7574107037410898715la_a_b > set_li7574107037410898715la_a_b ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat2: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
insert7010464514620295119la_a_b: relational_fmla_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
insert2023870700798818565la_a_b: set_Re381260168593705685la_a_b > set_se6865892389300016395la_a_b > set_se6865892389300016395la_a_b ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J,type,
insert1549238021488837435la_a_b: set_se6865892389300016395la_a_b > set_se1626149436229564481la_a_b > set_se1626149436229564481la_a_b ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Oinsert_001tf__b,type,
insert_b2: b > set_b > set_b ).
thf(sy_c_Set_Ois__singleton_001t__List__Olist_It__Nat__Onat_J,type,
is_sin2641923865335537900st_nat: set_list_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__List__Olist_It__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J,type,
is_sin5582224557664405617la_a_b: set_li7574107037410898715la_a_b > $o ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
is_sin6594375743535830443la_a_b: set_Re381260168593705685la_a_b > $o ).
thf(sy_c_Set_Ois__singleton_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
is_sin1114528679824004833la_a_b: set_se6865892389300016395la_a_b > $o ).
thf(sy_c_Set_Ois__singleton_001t__Set__Oset_It__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J,type,
is_sin4154956242601603351la_a_b: set_se1626149436229564481la_a_b > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Ois__singleton_001tf__b,type,
is_singleton_b: set_b > $o ).
thf(sy_c_Set_Oremove_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
remove6344774115224635824la_a_b: set_Re381260168593705685la_a_b > set_se6865892389300016395la_a_b > set_se6865892389300016395la_a_b ).
thf(sy_c_Set_Othe__elem_001t__List__Olist_It__Nat__Onat_J,type,
the_elem_list_nat: set_list_nat > list_nat ).
thf(sy_c_Set_Othe__elem_001t__List__Olist_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
the_el2253512510949844602la_a_b: set_li8334890247616788453la_a_b > list_R8263082107343818799la_a_b ).
thf(sy_c_Set_Othe__elem_001t__List__Olist_It__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J,type,
the_el1595598750643379376la_a_b: set_li7574107037410898715la_a_b > list_s3671366213839843941la_a_b ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set_Othe__elem_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
the_el6350558617753882986la_a_b: set_Re381260168593705685la_a_b > relational_fmla_a_b ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_It__Nat__Onat_J,type,
the_elem_set_nat: set_set_nat > set_nat ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
the_el7486773796875720352la_a_b: set_se6865892389300016395la_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
accp_P8429596082290424793la_a_b: ( produc2220957400383678800la_a_b > produc2220957400383678800la_a_b > $o ) > produc2220957400383678800la_a_b > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Nat__Onat_J,type,
accp_P4351966040938400857_b_nat: ( produc7366699395886430672_b_nat > produc7366699395886430672_b_nat > $o ) > produc7366699395886430672_b_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
accp_P4977102363926644935st_nat: ( produc2715289222843590288st_nat > produc2715289222843590288st_nat > $o ) > produc2715289222843590288st_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
accp_P6432500237705381767st_nat: ( produc1024650801244601168st_nat > produc1024650801244601168st_nat > $o ) > produc1024650801244601168st_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
accp_R989495437599811158la_a_b: ( relational_fmla_a_b > relational_fmla_a_b > $o ) > relational_fmla_a_b > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
member8712135081189583430la_a_b: list_R8263082107343818799la_a_b > set_li8334890247616788453la_a_b > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J,type,
member987187531357909116la_a_b: list_s3671366213839843941la_a_b > set_li7574107037410898715la_a_b > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Diff____Array__Oarray_It__Option__Ooption_It__Product____Type__Ounit_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member3320643659109938320st_nat: produc2026718404927347961st_nat > set_Pr5509071407266205487st_nat > $o ).
thf(sy_c_member_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
member4680049679412964150la_a_b: relational_fmla_a_b > set_Re381260168593705685la_a_b > $o ).
thf(sy_c_member_001t__Relational____Calculus__Oterm_Itf__a_J,type,
member5466445112152945266term_a: relational_term_a > set_Re5178783185447174953term_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
member3481406638322139244la_a_b: set_Re381260168593705685la_a_b > set_se6865892389300016395la_a_b > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J,type,
member1693404252451860130la_a_b: set_se6865892389300016395la_a_b > set_se1626149436229564481la_a_b > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_Ga____,type,
ga: set_Re381260168593705685la_a_b ).
thf(sy_v_Q1____,type,
q1: relational_fmla_a_b ).
thf(sy_v_Q2____,type,
q2: relational_fmla_a_b ).
thf(sy_v_xa____,type,
xa: nat ).
% Relevant facts (1249)
thf(fact_0__C7_C,axiom,
( ( ( relational_gen_a_b @ xa @ q1 @ ga )
& ( member3481406638322139244la_a_b @ ga @ ( set_se5111286776497155840la_a_b @ ( restri4663030114664890836pl_a_b @ xa @ q1 ) ) ) )
| ( ( relational_gen_a_b @ xa @ q2 @ ga )
& ( member3481406638322139244la_a_b @ ga @ ( set_se5111286776497155840la_a_b @ ( restri4663030114664890836pl_a_b @ xa @ q2 ) ) ) ) ) ).
% "7"
thf(fact_1_gen__impl__gen,axiom,
! [G: set_Re381260168593705685la_a_b,X: nat,Q: relational_fmla_a_b] :
( ( member3481406638322139244la_a_b @ G @ ( set_se5111286776497155840la_a_b @ ( restri4663030114664890836pl_a_b @ X @ Q ) ) )
=> ( relational_gen_a_b @ X @ Q @ G ) ) ).
% gen_impl_gen
thf(fact_2_fmla_Oinject_I5_J,axiom,
! [X51: relational_fmla_a_b,X52: relational_fmla_a_b,Y51: relational_fmla_a_b,Y52: relational_fmla_a_b] :
( ( ( relational_Conj_a_b @ X51 @ X52 )
= ( relational_Conj_a_b @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% fmla.inject(5)
thf(fact_3_gen__impl_Osimps_I19_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,V: relational_fmla_a_b,Va: relational_fmla_a_b] :
( ( restri4663030114664890836pl_a_b @ X @ ( relational_Conj_a_b @ Q1 @ ( relational_Conj_a_b @ V @ Va ) ) )
= ( union_7072582408929332827la_a_b @ ( restri4663030114664890836pl_a_b @ X @ Q1 ) @ ( restri4663030114664890836pl_a_b @ X @ ( relational_Conj_a_b @ V @ Va ) ) ) ) ).
% gen_impl.simps(19)
thf(fact_4_Gen__Conj_I2_J,axiom,
! [X: nat,Q2: relational_fmla_a_b,Q1: relational_fmla_a_b] :
( ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ Q2 @ X_1 )
=> ? [X_12: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ X_12 ) ) ).
% Gen_Conj(2)
thf(fact_5_Gen__Conj_I1_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ Q1 @ X_1 )
=> ? [X_12: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ X_12 ) ) ).
% Gen_Conj(1)
thf(fact_6_gen_Ointros_I7_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q2: relational_fmla_a_b] :
( ( ( relational_gen_a_b @ X @ Q1 @ G )
| ( relational_gen_a_b @ X @ Q2 @ G ) )
=> ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ G ) ) ).
% gen.intros(7)
thf(fact_7_list__ex__rec__eq,axiom,
! [L: list_R8263082107343818799la_a_b,P: relational_fmla_a_b > $o] :
( ( ? [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ ( set_Re9104216502384355786la_a_b @ L ) )
& ( P @ X2 ) ) )
= ( param_230422583733286043la_a_b @ P @ L ) ) ).
% list_ex_rec_eq
thf(fact_8_list__ex__rec__eq,axiom,
! [L: list_l3216422893793986623la_a_b,P: list_R8263082107343818799la_a_b > $o] :
( ( ? [X2: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ X2 @ ( set_li8988895567068466906la_a_b @ L ) )
& ( P @ X2 ) ) )
= ( param_5570067113634626859la_a_b @ P @ L ) ) ).
% list_ex_rec_eq
thf(fact_9_list__ex__rec__eq,axiom,
! [L: list_l1512381142989314933la_a_b,P: list_s3671366213839843941la_a_b > $o] :
( ( ? [X2: list_s3671366213839843941la_a_b] :
( ( member987187531357909116la_a_b @ X2 @ ( set_li3253934503784678672la_a_b @ L ) )
& ( P @ X2 ) ) )
= ( param_7264837599873937249la_a_b @ P @ L ) ) ).
% list_ex_rec_eq
thf(fact_10_list__ex__rec__eq,axiom,
! [L: list_list_nat,P: list_nat > $o] :
( ( ? [X2: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ L ) )
& ( P @ X2 ) ) )
= ( param_519148586836064220st_nat @ P @ L ) ) ).
% list_ex_rec_eq
thf(fact_11_list__ex__rec__eq,axiom,
! [L: list_nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ L ) )
& ( P @ X2 ) ) )
= ( param_7592452054083848908ec_nat @ P @ L ) ) ).
% list_ex_rec_eq
thf(fact_12_list__ex__rec__eq,axiom,
! [L: list_s3671366213839843941la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( ? [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ ( set_se5111286776497155840la_a_b @ L ) )
& ( P @ X2 ) ) )
= ( param_2860366322937624017la_a_b @ P @ L ) ) ).
% list_ex_rec_eq
thf(fact_13_list__all__rec__eq,axiom,
! [L: list_R8263082107343818799la_a_b,P: relational_fmla_a_b > $o] :
( ( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ ( set_Re9104216502384355786la_a_b @ L ) )
=> ( P @ X2 ) ) )
= ( param_5591567056726616081la_a_b @ P @ L ) ) ).
% list_all_rec_eq
thf(fact_14_list__all__rec__eq,axiom,
! [L: list_l3216422893793986623la_a_b,P: list_R8263082107343818799la_a_b > $o] :
( ( ! [X2: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ X2 @ ( set_li8988895567068466906la_a_b @ L ) )
=> ( P @ X2 ) ) )
= ( param_6917107903037750689la_a_b @ P @ L ) ) ).
% list_all_rec_eq
thf(fact_15_list__all__rec__eq,axiom,
! [L: list_l1512381142989314933la_a_b,P: list_s3671366213839843941la_a_b > $o] :
( ( ! [X2: list_s3671366213839843941la_a_b] :
( ( member987187531357909116la_a_b @ X2 @ ( set_li3253934503784678672la_a_b @ L ) )
=> ( P @ X2 ) ) )
= ( param_3827852918142015447la_a_b @ P @ L ) ) ).
% list_all_rec_eq
thf(fact_16_list__all__rec__eq,axiom,
! [L: list_list_nat,P: list_nat > $o] :
( ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ L ) )
=> ( P @ X2 ) ) )
= ( param_5680023212210265682st_nat @ P @ L ) ) ).
% list_all_rec_eq
thf(fact_17_list__all__rec__eq,axiom,
! [L: list_nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ L ) )
=> ( P @ X2 ) ) )
= ( param_3668158194536031298ec_nat @ P @ L ) ) ).
% list_all_rec_eq
thf(fact_18_list__all__rec__eq,axiom,
! [L: list_s3671366213839843941la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( ! [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ ( set_se5111286776497155840la_a_b @ L ) )
=> ( P @ X2 ) ) )
= ( param_8621423519915839815la_a_b @ P @ L ) ) ).
% list_all_rec_eq
thf(fact_19_gen__impl_Osimps_I6_J,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ( restri4663030114664890836pl_a_b @ X @ ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q ) ) )
= ( restri4663030114664890836pl_a_b @ X @ Q ) ) ).
% gen_impl.simps(6)
thf(fact_20_list__ex1__iff,axiom,
( list_e5987854106142083202la_a_b
= ( ^ [P2: set_se6865892389300016395la_a_b > $o,Xs: list_s5517785546295487899la_a_b] :
? [X2: set_se6865892389300016395la_a_b] :
( ( member1693404252451860130la_a_b @ X2 @ ( set_se4873105269812759862la_a_b @ Xs ) )
& ( P2 @ X2 )
& ! [Y: set_se6865892389300016395la_a_b] :
( ( ( member1693404252451860130la_a_b @ Y @ ( set_se4873105269812759862la_a_b @ Xs ) )
& ( P2 @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_21_list__ex1__iff,axiom,
( list_ex1_b
= ( ^ [P2: b > $o,Xs: list_b] :
? [X2: b] :
( ( member_b @ X2 @ ( set_b2 @ Xs ) )
& ( P2 @ X2 )
& ! [Y: b] :
( ( ( member_b @ Y @ ( set_b2 @ Xs ) )
& ( P2 @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_22_list__ex1__iff,axiom,
( list_e1892966175763416342la_a_b
= ( ^ [P2: relational_fmla_a_b > $o,Xs: list_R8263082107343818799la_a_b] :
? [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ ( set_Re9104216502384355786la_a_b @ Xs ) )
& ( P2 @ X2 )
& ! [Y: relational_fmla_a_b] :
( ( ( member4680049679412964150la_a_b @ Y @ ( set_Re9104216502384355786la_a_b @ Xs ) )
& ( P2 @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_23_list__ex1__iff,axiom,
( list_e3575657358457426982la_a_b
= ( ^ [P2: list_R8263082107343818799la_a_b > $o,Xs: list_l3216422893793986623la_a_b] :
? [X2: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ X2 @ ( set_li8988895567068466906la_a_b @ Xs ) )
& ( P2 @ X2 )
& ! [Y: list_R8263082107343818799la_a_b] :
( ( ( member8712135081189583430la_a_b @ Y @ ( set_li8988895567068466906la_a_b @ Xs ) )
& ( P2 @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_24_list__ex1__iff,axiom,
( list_e7636442695929993820la_a_b
= ( ^ [P2: list_s3671366213839843941la_a_b > $o,Xs: list_l1512381142989314933la_a_b] :
? [X2: list_s3671366213839843941la_a_b] :
( ( member987187531357909116la_a_b @ X2 @ ( set_li3253934503784678672la_a_b @ Xs ) )
& ( P2 @ X2 )
& ! [Y: list_s3671366213839843941la_a_b] :
( ( ( member987187531357909116la_a_b @ Y @ ( set_li3253934503784678672la_a_b @ Xs ) )
& ( P2 @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_25_list__ex1__iff,axiom,
( list_ex1_list_nat
= ( ^ [P2: list_nat > $o,Xs: list_list_nat] :
? [X2: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
& ( P2 @ X2 )
& ! [Y: list_nat] :
( ( ( member_list_nat @ Y @ ( set_list_nat2 @ Xs ) )
& ( P2 @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_26_list__ex1__iff,axiom,
( list_ex1_nat
= ( ^ [P2: nat > $o,Xs: list_nat] :
? [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
& ( P2 @ X2 )
& ! [Y: nat] :
( ( ( member_nat @ Y @ ( set_nat2 @ Xs ) )
& ( P2 @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_27_list__ex1__iff,axiom,
( list_e6968285794140641868la_a_b
= ( ^ [P2: set_Re381260168593705685la_a_b > $o,Xs: list_s3671366213839843941la_a_b] :
? [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ ( set_se5111286776497155840la_a_b @ Xs ) )
& ( P2 @ X2 )
& ! [Y: set_Re381260168593705685la_a_b] :
( ( ( member3481406638322139244la_a_b @ Y @ ( set_se5111286776497155840la_a_b @ Xs ) )
& ( P2 @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_28_gen__impl_Osimps_I18_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,V: relational_fmla_a_b] :
( ( restri4663030114664890836pl_a_b @ X @ ( relational_Conj_a_b @ Q1 @ ( relational_Neg_a_b @ V ) ) )
= ( union_7072582408929332827la_a_b @ ( restri4663030114664890836pl_a_b @ X @ Q1 ) @ ( restri4663030114664890836pl_a_b @ X @ ( relational_Neg_a_b @ V ) ) ) ) ).
% gen_impl.simps(18)
thf(fact_29_fmla_Oinject_I4_J,axiom,
! [X4: relational_fmla_a_b,Y4: relational_fmla_a_b] :
( ( ( relational_Neg_a_b @ X4 )
= ( relational_Neg_a_b @ Y4 ) )
= ( X4 = Y4 ) ) ).
% fmla.inject(4)
thf(fact_30_gen_Ointros_I3_J,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X @ Q @ G )
=> ( relational_gen_a_b @ X @ ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q ) ) @ G ) ) ).
% gen.intros(3)
thf(fact_31_fmla_Odistinct_I31_J,axiom,
! [X4: relational_fmla_a_b,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
( ( relational_Neg_a_b @ X4 )
!= ( relational_Conj_a_b @ X51 @ X52 ) ) ).
% fmla.distinct(31)
thf(fact_32_can__select__set__list__ex1,axiom,
! [P: relational_fmla_a_b > $o,A: list_R8263082107343818799la_a_b] :
( ( can_se1623252625881650914la_a_b @ P @ ( set_Re9104216502384355786la_a_b @ A ) )
= ( list_e1892966175763416342la_a_b @ P @ A ) ) ).
% can_select_set_list_ex1
thf(fact_33_can__select__set__list__ex1,axiom,
! [P: list_R8263082107343818799la_a_b > $o,A: list_l3216422893793986623la_a_b] :
( ( can_se4691468443538722802la_a_b @ P @ ( set_li8988895567068466906la_a_b @ A ) )
= ( list_e3575657358457426982la_a_b @ P @ A ) ) ).
% can_select_set_list_ex1
thf(fact_34_can__select__set__list__ex1,axiom,
! [P: list_s3671366213839843941la_a_b > $o,A: list_l1512381142989314933la_a_b] :
( ( can_se1196957838938351144la_a_b @ P @ ( set_li3253934503784678672la_a_b @ A ) )
= ( list_e7636442695929993820la_a_b @ P @ A ) ) ).
% can_select_set_list_ex1
thf(fact_35_can__select__set__list__ex1,axiom,
! [P: list_nat > $o,A: list_list_nat] :
( ( can_select_list_nat @ P @ ( set_list_nat2 @ A ) )
= ( list_ex1_list_nat @ P @ A ) ) ).
% can_select_set_list_ex1
thf(fact_36_can__select__set__list__ex1,axiom,
! [P: nat > $o,A: list_nat] :
( ( can_select_nat @ P @ ( set_nat2 @ A ) )
= ( list_ex1_nat @ P @ A ) ) ).
% can_select_set_list_ex1
thf(fact_37_can__select__set__list__ex1,axiom,
! [P: set_Re381260168593705685la_a_b > $o,A: list_s3671366213839843941la_a_b] :
( ( can_se982673022169791512la_a_b @ P @ ( set_se5111286776497155840la_a_b @ A ) )
= ( list_e6968285794140641868la_a_b @ P @ A ) ) ).
% can_select_set_list_ex1
thf(fact_38_gen__impl_Osimps_I20_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,V: relational_fmla_a_b,Va: relational_fmla_a_b] :
( ( restri4663030114664890836pl_a_b @ X @ ( relational_Conj_a_b @ Q1 @ ( relational_Disj_a_b @ V @ Va ) ) )
= ( union_7072582408929332827la_a_b @ ( restri4663030114664890836pl_a_b @ X @ Q1 ) @ ( restri4663030114664890836pl_a_b @ X @ ( relational_Disj_a_b @ V @ Va ) ) ) ) ).
% gen_impl.simps(20)
thf(fact_39_gen__impl_Osimps_I16_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,V: $o] :
( ( restri4663030114664890836pl_a_b @ X @ ( relational_Conj_a_b @ Q1 @ ( relational_Bool_a_b @ V ) ) )
= ( union_7072582408929332827la_a_b @ ( restri4663030114664890836pl_a_b @ X @ Q1 ) @ ( restri4663030114664890836pl_a_b @ X @ ( relational_Bool_a_b @ V ) ) ) ) ).
% gen_impl.simps(16)
thf(fact_40_gen__impl_Osimps_I21_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,V: nat,Va: relational_fmla_a_b] :
( ( restri4663030114664890836pl_a_b @ X @ ( relational_Conj_a_b @ Q1 @ ( relati591517084277583526ts_a_b @ V @ Va ) ) )
= ( union_7072582408929332827la_a_b @ ( restri4663030114664890836pl_a_b @ X @ Q1 ) @ ( restri4663030114664890836pl_a_b @ X @ ( relati591517084277583526ts_a_b @ V @ Va ) ) ) ) ).
% gen_impl.simps(21)
thf(fact_41_gen__impl_Osimps_I15_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,V: b,Va: list_R6823256787227418703term_a] :
( ( restri4663030114664890836pl_a_b @ X @ ( relational_Conj_a_b @ Q1 @ ( relational_Pred_b_a @ V @ Va ) ) )
= ( union_7072582408929332827la_a_b @ ( restri4663030114664890836pl_a_b @ X @ Q1 ) @ ( restri4663030114664890836pl_a_b @ X @ ( relational_Pred_b_a @ V @ Va ) ) ) ) ).
% gen_impl.simps(15)
thf(fact_42_gen_Ointros_I5_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,Q2: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q2 ) ) @ G )
=> ( relational_gen_a_b @ X @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) ) @ G ) ) ).
% gen.intros(5)
thf(fact_43_gen_Ointros_I4_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,Q2: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q2 ) ) @ G )
=> ( relational_gen_a_b @ X @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) ) @ G ) ) ).
% gen.intros(4)
thf(fact_44_gen__impl_Osimps_I7_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( restri4663030114664890836pl_a_b @ X @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) ) )
= ( restri4663030114664890836pl_a_b @ X @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q2 ) ) ) ) ).
% gen_impl.simps(7)
thf(fact_45_gen__impl_Osimps_I8_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( restri4663030114664890836pl_a_b @ X @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) ) )
= ( restri4663030114664890836pl_a_b @ X @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q2 ) ) ) ) ).
% gen_impl.simps(8)
thf(fact_46_set__union,axiom,
! [Xs2: list_R8263082107343818799la_a_b,Ys: list_R8263082107343818799la_a_b] :
( ( set_Re9104216502384355786la_a_b @ ( union_5669641164172981541la_a_b @ Xs2 @ Ys ) )
= ( sup_su5130108678486352897la_a_b @ ( set_Re9104216502384355786la_a_b @ Xs2 ) @ ( set_Re9104216502384355786la_a_b @ Ys ) ) ) ).
% set_union
thf(fact_47_set__union,axiom,
! [Xs2: list_l3216422893793986623la_a_b,Ys: list_l3216422893793986623la_a_b] :
( ( set_li8988895567068466906la_a_b @ ( union_2282237785388101813la_a_b @ Xs2 @ Ys ) )
= ( sup_su8392741853961112337la_a_b @ ( set_li8988895567068466906la_a_b @ Xs2 ) @ ( set_li8988895567068466906la_a_b @ Ys ) ) ) ).
% set_union
thf(fact_48_set__union,axiom,
! [Xs2: list_l1512381142989314933la_a_b,Ys: list_l1512381142989314933la_a_b] :
( ( set_li3253934503784678672la_a_b @ ( union_6544486529039940331la_a_b @ Xs2 @ Ys ) )
= ( sup_su2721708343125988935la_a_b @ ( set_li3253934503784678672la_a_b @ Xs2 ) @ ( set_li3253934503784678672la_a_b @ Ys ) ) ) ).
% set_union
thf(fact_49_set__union,axiom,
! [Xs2: list_list_nat,Ys: list_list_nat] :
( ( set_list_nat2 @ ( union_list_nat @ Xs2 @ Ys ) )
= ( sup_sup_set_list_nat @ ( set_list_nat2 @ Xs2 ) @ ( set_list_nat2 @ Ys ) ) ) ).
% set_union
thf(fact_50_set__union,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( set_nat2 @ ( union_nat @ Xs2 @ Ys ) )
= ( sup_sup_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ Ys ) ) ) ).
% set_union
thf(fact_51_set__union,axiom,
! [Xs2: list_b,Ys: list_b] :
( ( set_b2 @ ( union_b @ Xs2 @ Ys ) )
= ( sup_sup_set_b @ ( set_b2 @ Xs2 ) @ ( set_b2 @ Ys ) ) ) ).
% set_union
thf(fact_52_set__union,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( set_a2 @ ( union_a @ Xs2 @ Ys ) )
= ( sup_sup_set_a @ ( set_a2 @ Xs2 ) @ ( set_a2 @ Ys ) ) ) ).
% set_union
thf(fact_53_set__union,axiom,
! [Xs2: list_s3671366213839843941la_a_b,Ys: list_s3671366213839843941la_a_b] :
( ( set_se5111286776497155840la_a_b @ ( union_7072582408929332827la_a_b @ Xs2 @ Ys ) )
= ( sup_su4783144482993978935la_a_b @ ( set_se5111286776497155840la_a_b @ Xs2 ) @ ( set_se5111286776497155840la_a_b @ Ys ) ) ) ).
% set_union
thf(fact_54_in__set__insert,axiom,
! [X: set_se6865892389300016395la_a_b,Xs2: list_s5517785546295487899la_a_b] :
( ( member1693404252451860130la_a_b @ X @ ( set_se4873105269812759862la_a_b @ Xs2 ) )
=> ( ( insert3187468692655853807la_a_b @ X @ Xs2 )
= Xs2 ) ) ).
% in_set_insert
thf(fact_55_in__set__insert,axiom,
! [X: b,Xs2: list_b] :
( ( member_b @ X @ ( set_b2 @ Xs2 ) )
=> ( ( insert_b @ X @ Xs2 )
= Xs2 ) ) ).
% in_set_insert
thf(fact_56_in__set__insert,axiom,
! [X: relational_fmla_a_b,Xs2: list_R8263082107343818799la_a_b] :
( ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Xs2 ) )
=> ( ( insert5052544347734373763la_a_b @ X @ Xs2 )
= Xs2 ) ) ).
% in_set_insert
thf(fact_57_in__set__insert,axiom,
! [X: list_R8263082107343818799la_a_b,Xs2: list_l3216422893793986623la_a_b] :
( ( member8712135081189583430la_a_b @ X @ ( set_li8988895567068466906la_a_b @ Xs2 ) )
=> ( ( insert3209172818022656531la_a_b @ X @ Xs2 )
= Xs2 ) ) ).
% in_set_insert
thf(fact_58_in__set__insert,axiom,
! [X: list_s3671366213839843941la_a_b,Xs2: list_l1512381142989314933la_a_b] :
( ( member987187531357909116la_a_b @ X @ ( set_li3253934503784678672la_a_b @ Xs2 ) )
=> ( ( insert3122035991742315593la_a_b @ X @ Xs2 )
= Xs2 ) ) ).
% in_set_insert
thf(fact_59_in__set__insert,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
=> ( ( insert_list_nat @ X @ Xs2 )
= Xs2 ) ) ).
% in_set_insert
thf(fact_60_in__set__insert,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ( insert_nat @ X @ Xs2 )
= Xs2 ) ) ).
% in_set_insert
thf(fact_61_in__set__insert,axiom,
! [X: set_Re381260168593705685la_a_b,Xs2: list_s3671366213839843941la_a_b] :
( ( member3481406638322139244la_a_b @ X @ ( set_se5111286776497155840la_a_b @ Xs2 ) )
=> ( ( insert443281300761040569la_a_b @ X @ Xs2 )
= Xs2 ) ) ).
% in_set_insert
thf(fact_62_list__ex1__simps_I1_J,axiom,
! [P: list_nat > $o] :
~ ( list_ex1_list_nat @ P @ nil_list_nat ) ).
% list_ex1_simps(1)
thf(fact_63_list__ex1__simps_I1_J,axiom,
! [P: set_nat > $o] :
~ ( list_ex1_set_nat @ P @ nil_set_nat ) ).
% list_ex1_simps(1)
thf(fact_64_list__ex1__simps_I1_J,axiom,
! [P: set_Re381260168593705685la_a_b > $o] :
~ ( list_e6968285794140641868la_a_b @ P @ nil_se6596116450819887951la_a_b ) ).
% list_ex1_simps(1)
thf(fact_65_list__ex1__simps_I1_J,axiom,
! [P: nat > $o] :
~ ( list_ex1_nat @ P @ nil_nat ) ).
% list_ex1_simps(1)
thf(fact_66_fmla_Oinject_I6_J,axiom,
! [X61: relational_fmla_a_b,X62: relational_fmla_a_b,Y61: relational_fmla_a_b,Y62: relational_fmla_a_b] :
( ( ( relational_Disj_a_b @ X61 @ X62 )
= ( relational_Disj_a_b @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% fmla.inject(6)
thf(fact_67_fmla_Oinject_I2_J,axiom,
! [X22: $o,Y2: $o] :
( ( ( relational_Bool_a_b @ X22 )
= ( relational_Bool_a_b @ Y2 ) )
= ( X22 = Y2 ) ) ).
% fmla.inject(2)
thf(fact_68_fmla_Oinject_I7_J,axiom,
! [X71: nat,X72: relational_fmla_a_b,Y71: nat,Y72: relational_fmla_a_b] :
( ( ( relati591517084277583526ts_a_b @ X71 @ X72 )
= ( relati591517084277583526ts_a_b @ Y71 @ Y72 ) )
= ( ( X71 = Y71 )
& ( X72 = Y72 ) ) ) ).
% fmla.inject(7)
thf(fact_69_fmla_Oinject_I1_J,axiom,
! [X11: b,X12: list_R6823256787227418703term_a,Y11: b,Y12: list_R6823256787227418703term_a] :
( ( ( relational_Pred_b_a @ X11 @ X12 )
= ( relational_Pred_b_a @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 ) ) ) ).
% fmla.inject(1)
thf(fact_70_gen__Bool__True,axiom,
! [X: nat,G: set_Re381260168593705685la_a_b] :
~ ( relational_gen_a_b @ X @ ( relational_Bool_a_b @ $true ) @ G ) ).
% gen_Bool_True
thf(fact_71_fmla_Odistinct_I41_J,axiom,
! [X61: relational_fmla_a_b,X62: relational_fmla_a_b,X71: nat,X72: relational_fmla_a_b] :
( ( relational_Disj_a_b @ X61 @ X62 )
!= ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).
% fmla.distinct(41)
thf(fact_72_fmla_Odistinct_I21_J,axiom,
! [X22: $o,X71: nat,X72: relational_fmla_a_b] :
( ( relational_Bool_a_b @ X22 )
!= ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).
% fmla.distinct(21)
thf(fact_73_fmla_Odistinct_I19_J,axiom,
! [X22: $o,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
( ( relational_Bool_a_b @ X22 )
!= ( relational_Disj_a_b @ X61 @ X62 ) ) ).
% fmla.distinct(19)
thf(fact_74_fmla_Odistinct_I11_J,axiom,
! [X11: b,X12: list_R6823256787227418703term_a,X71: nat,X72: relational_fmla_a_b] :
( ( relational_Pred_b_a @ X11 @ X12 )
!= ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).
% fmla.distinct(11)
thf(fact_75_fmla_Odistinct_I9_J,axiom,
! [X11: b,X12: list_R6823256787227418703term_a,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
( ( relational_Pred_b_a @ X11 @ X12 )
!= ( relational_Disj_a_b @ X61 @ X62 ) ) ).
% fmla.distinct(9)
thf(fact_76_fmla_Odistinct_I1_J,axiom,
! [X11: b,X12: list_R6823256787227418703term_a,X22: $o] :
( ( relational_Pred_b_a @ X11 @ X12 )
!= ( relational_Bool_a_b @ X22 ) ) ).
% fmla.distinct(1)
thf(fact_77_gen_Ointros_I6_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,G1: set_Re381260168593705685la_a_b,Q2: relational_fmla_a_b,G2: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X @ Q1 @ G1 )
=> ( ( relational_gen_a_b @ X @ Q2 @ G2 )
=> ( relational_gen_a_b @ X @ ( relational_Disj_a_b @ Q1 @ Q2 ) @ ( sup_su5130108678486352897la_a_b @ G1 @ G2 ) ) ) ) ).
% gen.intros(6)
thf(fact_78_gen__impl_Osimps_I2_J,axiom,
! [X: nat] :
( ( restri4663030114664890836pl_a_b @ X @ ( relational_Bool_a_b @ $true ) )
= nil_se6596116450819887951la_a_b ) ).
% gen_impl.simps(2)
thf(fact_79_gen__impl_Osimps_I9_J,axiom,
! [X: nat,V: b,Va: list_R6823256787227418703term_a] :
( ( restri4663030114664890836pl_a_b @ X @ ( relational_Neg_a_b @ ( relational_Pred_b_a @ V @ Va ) ) )
= nil_se6596116450819887951la_a_b ) ).
% gen_impl.simps(9)
thf(fact_80_gen__impl_Osimps_I12_J,axiom,
! [X: nat,V: nat,Va: relational_fmla_a_b] :
( ( restri4663030114664890836pl_a_b @ X @ ( relational_Neg_a_b @ ( relati591517084277583526ts_a_b @ V @ Va ) ) )
= nil_se6596116450819887951la_a_b ) ).
% gen_impl.simps(12)
thf(fact_81_gen__impl_Osimps_I10_J,axiom,
! [X: nat,V: $o] :
( ( restri4663030114664890836pl_a_b @ X @ ( relational_Neg_a_b @ ( relational_Bool_a_b @ V ) ) )
= nil_se6596116450819887951la_a_b ) ).
% gen_impl.simps(10)
thf(fact_82_mem__Collect__eq,axiom,
! [A2: list_nat,P: list_nat > $o] :
( ( member_list_nat @ A2 @ ( collect_list_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_83_mem__Collect__eq,axiom,
! [A2: set_se6865892389300016395la_a_b,P: set_se6865892389300016395la_a_b > $o] :
( ( member1693404252451860130la_a_b @ A2 @ ( collec7641561117972753376la_a_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_84_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_85_mem__Collect__eq,axiom,
! [A2: b,P: b > $o] :
( ( member_b @ A2 @ ( collect_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_86_mem__Collect__eq,axiom,
! [A2: list_s3671366213839843941la_a_b,P: list_s3671366213839843941la_a_b > $o] :
( ( member987187531357909116la_a_b @ A2 @ ( collec5397960198663843514la_a_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_87_mem__Collect__eq,axiom,
! [A2: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( member3481406638322139244la_a_b @ A2 @ ( collec2099942116761351594la_a_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_88_Collect__mem__eq,axiom,
! [A: set_list_nat] :
( ( collect_list_nat
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_89_Collect__mem__eq,axiom,
! [A: set_se1626149436229564481la_a_b] :
( ( collec7641561117972753376la_a_b
@ ^ [X2: set_se6865892389300016395la_a_b] : ( member1693404252451860130la_a_b @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_90_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_91_Collect__mem__eq,axiom,
! [A: set_b] :
( ( collect_b
@ ^ [X2: b] : ( member_b @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_92_Collect__mem__eq,axiom,
! [A: set_li7574107037410898715la_a_b] :
( ( collec5397960198663843514la_a_b
@ ^ [X2: list_s3671366213839843941la_a_b] : ( member987187531357909116la_a_b @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_93_Collect__mem__eq,axiom,
! [A: set_se6865892389300016395la_a_b] :
( ( collec2099942116761351594la_a_b
@ ^ [X2: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_94_Collect__cong,axiom,
! [P: set_Re381260168593705685la_a_b > $o,Q: set_Re381260168593705685la_a_b > $o] :
( ! [X3: set_Re381260168593705685la_a_b] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec2099942116761351594la_a_b @ P )
= ( collec2099942116761351594la_a_b @ Q ) ) ) ).
% Collect_cong
thf(fact_95_Collect__cong,axiom,
! [P: list_s3671366213839843941la_a_b > $o,Q: list_s3671366213839843941la_a_b > $o] :
( ! [X3: list_s3671366213839843941la_a_b] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec5397960198663843514la_a_b @ P )
= ( collec5397960198663843514la_a_b @ Q ) ) ) ).
% Collect_cong
thf(fact_96_fmla_Odistinct_I5_J,axiom,
! [X11: b,X12: list_R6823256787227418703term_a,X4: relational_fmla_a_b] :
( ( relational_Pred_b_a @ X11 @ X12 )
!= ( relational_Neg_a_b @ X4 ) ) ).
% fmla.distinct(5)
thf(fact_97_fmla_Odistinct_I7_J,axiom,
! [X11: b,X12: list_R6823256787227418703term_a,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
( ( relational_Pred_b_a @ X11 @ X12 )
!= ( relational_Conj_a_b @ X51 @ X52 ) ) ).
% fmla.distinct(7)
thf(fact_98_fmla_Odistinct_I35_J,axiom,
! [X4: relational_fmla_a_b,X71: nat,X72: relational_fmla_a_b] :
( ( relational_Neg_a_b @ X4 )
!= ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).
% fmla.distinct(35)
thf(fact_99_fmla_Odistinct_I15_J,axiom,
! [X22: $o,X4: relational_fmla_a_b] :
( ( relational_Bool_a_b @ X22 )
!= ( relational_Neg_a_b @ X4 ) ) ).
% fmla.distinct(15)
thf(fact_100_fmla_Odistinct_I39_J,axiom,
! [X51: relational_fmla_a_b,X52: relational_fmla_a_b,X71: nat,X72: relational_fmla_a_b] :
( ( relational_Conj_a_b @ X51 @ X52 )
!= ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).
% fmla.distinct(39)
thf(fact_101_fmla_Odistinct_I33_J,axiom,
! [X4: relational_fmla_a_b,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
( ( relational_Neg_a_b @ X4 )
!= ( relational_Disj_a_b @ X61 @ X62 ) ) ).
% fmla.distinct(33)
thf(fact_102_fmla_Odistinct_I17_J,axiom,
! [X22: $o,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
( ( relational_Bool_a_b @ X22 )
!= ( relational_Conj_a_b @ X51 @ X52 ) ) ).
% fmla.distinct(17)
thf(fact_103_fmla_Odistinct_I37_J,axiom,
! [X51: relational_fmla_a_b,X52: relational_fmla_a_b,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
( ( relational_Conj_a_b @ X51 @ X52 )
!= ( relational_Disj_a_b @ X61 @ X62 ) ) ).
% fmla.distinct(37)
thf(fact_104_can__select__def,axiom,
( can_se1196957838938351144la_a_b
= ( ^ [P2: list_s3671366213839843941la_a_b > $o,A3: set_li7574107037410898715la_a_b] :
? [X2: list_s3671366213839843941la_a_b] :
( ( member987187531357909116la_a_b @ X2 @ A3 )
& ( P2 @ X2 )
& ! [Y: list_s3671366213839843941la_a_b] :
( ( ( member987187531357909116la_a_b @ Y @ A3 )
& ( P2 @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_105_can__select__def,axiom,
( can_select_list_nat
= ( ^ [P2: list_nat > $o,A3: set_list_nat] :
? [X2: list_nat] :
( ( member_list_nat @ X2 @ A3 )
& ( P2 @ X2 )
& ! [Y: list_nat] :
( ( ( member_list_nat @ Y @ A3 )
& ( P2 @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_106_can__select__def,axiom,
( can_se4503359670287350350la_a_b
= ( ^ [P2: set_se6865892389300016395la_a_b > $o,A3: set_se1626149436229564481la_a_b] :
? [X2: set_se6865892389300016395la_a_b] :
( ( member1693404252451860130la_a_b @ X2 @ A3 )
& ( P2 @ X2 )
& ! [Y: set_se6865892389300016395la_a_b] :
( ( ( member1693404252451860130la_a_b @ Y @ A3 )
& ( P2 @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_107_can__select__def,axiom,
( can_select_nat
= ( ^ [P2: nat > $o,A3: set_nat] :
? [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( P2 @ X2 )
& ! [Y: nat] :
( ( ( member_nat @ Y @ A3 )
& ( P2 @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_108_can__select__def,axiom,
( can_select_b
= ( ^ [P2: b > $o,A3: set_b] :
? [X2: b] :
( ( member_b @ X2 @ A3 )
& ( P2 @ X2 )
& ! [Y: b] :
( ( ( member_b @ Y @ A3 )
& ( P2 @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_109_can__select__def,axiom,
( can_se982673022169791512la_a_b
= ( ^ [P2: set_Re381260168593705685la_a_b > $o,A3: set_se6865892389300016395la_a_b] :
? [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ A3 )
& ( P2 @ X2 )
& ! [Y: set_Re381260168593705685la_a_b] :
( ( ( member3481406638322139244la_a_b @ Y @ A3 )
& ( P2 @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_110_list__all__rec_Osimps_I1_J,axiom,
! [P: list_nat > $o] : ( param_5680023212210265682st_nat @ P @ nil_list_nat ) ).
% list_all_rec.simps(1)
thf(fact_111_list__all__rec_Osimps_I1_J,axiom,
! [P: set_nat > $o] : ( param_9064775779286910968et_nat @ P @ nil_set_nat ) ).
% list_all_rec.simps(1)
thf(fact_112_list__all__rec_Osimps_I1_J,axiom,
! [P: nat > $o] : ( param_3668158194536031298ec_nat @ P @ nil_nat ) ).
% list_all_rec.simps(1)
thf(fact_113_list__all__rec_Osimps_I1_J,axiom,
! [P: set_Re381260168593705685la_a_b > $o] : ( param_8621423519915839815la_a_b @ P @ nil_se6596116450819887951la_a_b ) ).
% list_all_rec.simps(1)
thf(fact_114_list__ex__rec_Osimps_I1_J,axiom,
! [P: list_nat > $o] :
~ ( param_519148586836064220st_nat @ P @ nil_list_nat ) ).
% list_ex_rec.simps(1)
thf(fact_115_list__ex__rec_Osimps_I1_J,axiom,
! [P: set_nat > $o] :
~ ( param_1280123760176727682et_nat @ P @ nil_set_nat ) ).
% list_ex_rec.simps(1)
thf(fact_116_list__ex__rec_Osimps_I1_J,axiom,
! [P: nat > $o] :
~ ( param_7592452054083848908ec_nat @ P @ nil_nat ) ).
% list_ex_rec.simps(1)
thf(fact_117_list__ex__rec_Osimps_I1_J,axiom,
! [P: set_Re381260168593705685la_a_b > $o] :
~ ( param_2860366322937624017la_a_b @ P @ nil_se6596116450819887951la_a_b ) ).
% list_ex_rec.simps(1)
thf(fact_118_Un__iff,axiom,
! [C: list_s3671366213839843941la_a_b,A: set_li7574107037410898715la_a_b,B: set_li7574107037410898715la_a_b] :
( ( member987187531357909116la_a_b @ C @ ( sup_su2721708343125988935la_a_b @ A @ B ) )
= ( ( member987187531357909116la_a_b @ C @ A )
| ( member987187531357909116la_a_b @ C @ B ) ) ) ).
% Un_iff
thf(fact_119_Un__iff,axiom,
! [C: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat @ C @ ( sup_sup_set_list_nat @ A @ B ) )
= ( ( member_list_nat @ C @ A )
| ( member_list_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_120_Un__iff,axiom,
! [C: set_se6865892389300016395la_a_b,A: set_se1626149436229564481la_a_b,B: set_se1626149436229564481la_a_b] :
( ( member1693404252451860130la_a_b @ C @ ( sup_su4447918665525899629la_a_b @ A @ B ) )
= ( ( member1693404252451860130la_a_b @ C @ A )
| ( member1693404252451860130la_a_b @ C @ B ) ) ) ).
% Un_iff
thf(fact_121_Un__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
| ( member_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_122_Un__iff,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ ( sup_sup_set_b @ A @ B ) )
= ( ( member_b @ C @ A )
| ( member_b @ C @ B ) ) ) ).
% Un_iff
thf(fact_123_Un__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
| ( member_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_124_Un__iff,axiom,
! [C: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ C @ ( sup_su4783144482993978935la_a_b @ A @ B ) )
= ( ( member3481406638322139244la_a_b @ C @ A )
| ( member3481406638322139244la_a_b @ C @ B ) ) ) ).
% Un_iff
thf(fact_125_UnCI,axiom,
! [C: list_s3671366213839843941la_a_b,B: set_li7574107037410898715la_a_b,A: set_li7574107037410898715la_a_b] :
( ( ~ ( member987187531357909116la_a_b @ C @ B )
=> ( member987187531357909116la_a_b @ C @ A ) )
=> ( member987187531357909116la_a_b @ C @ ( sup_su2721708343125988935la_a_b @ A @ B ) ) ) ).
% UnCI
thf(fact_126_UnCI,axiom,
! [C: list_nat,B: set_list_nat,A: set_list_nat] :
( ( ~ ( member_list_nat @ C @ B )
=> ( member_list_nat @ C @ A ) )
=> ( member_list_nat @ C @ ( sup_sup_set_list_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_127_UnCI,axiom,
! [C: set_se6865892389300016395la_a_b,B: set_se1626149436229564481la_a_b,A: set_se1626149436229564481la_a_b] :
( ( ~ ( member1693404252451860130la_a_b @ C @ B )
=> ( member1693404252451860130la_a_b @ C @ A ) )
=> ( member1693404252451860130la_a_b @ C @ ( sup_su4447918665525899629la_a_b @ A @ B ) ) ) ).
% UnCI
thf(fact_128_UnCI,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ A ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_129_UnCI,axiom,
! [C: b,B: set_b,A: set_b] :
( ( ~ ( member_b @ C @ B )
=> ( member_b @ C @ A ) )
=> ( member_b @ C @ ( sup_sup_set_b @ A @ B ) ) ) ).
% UnCI
thf(fact_130_UnCI,axiom,
! [C: a,B: set_a,A: set_a] :
( ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ A ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnCI
thf(fact_131_UnCI,axiom,
! [C: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b,A: set_se6865892389300016395la_a_b] :
( ( ~ ( member3481406638322139244la_a_b @ C @ B )
=> ( member3481406638322139244la_a_b @ C @ A ) )
=> ( member3481406638322139244la_a_b @ C @ ( sup_su4783144482993978935la_a_b @ A @ B ) ) ) ).
% UnCI
thf(fact_132_sup__apply,axiom,
( sup_su2223990397388305166_a_b_o
= ( ^ [F: set_Re381260168593705685la_a_b > $o,G3: set_Re381260168593705685la_a_b > $o,X2: set_Re381260168593705685la_a_b] : ( sup_sup_o @ ( F @ X2 ) @ ( G3 @ X2 ) ) ) ) ).
% sup_apply
thf(fact_133_sup_Oright__idem,axiom,
! [A2: set_se6865892389300016395la_a_b,B2: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ ( sup_su4783144482993978935la_a_b @ A2 @ B2 ) @ B2 )
= ( sup_su4783144482993978935la_a_b @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_134_sup_Oright__idem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_135_sup_Oright__idem,axiom,
! [A2: set_b,B2: set_b] :
( ( sup_sup_set_b @ ( sup_sup_set_b @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_b @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_136_sup_Oright__idem,axiom,
! [A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_137_sup_Oright__idem,axiom,
! [A2: set_Re381260168593705685la_a_b > $o,B2: set_Re381260168593705685la_a_b > $o] :
( ( sup_su2223990397388305166_a_b_o @ ( sup_su2223990397388305166_a_b_o @ A2 @ B2 ) @ B2 )
= ( sup_su2223990397388305166_a_b_o @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_138_sup__left__idem,axiom,
! [X: set_se6865892389300016395la_a_b,Y3: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ X @ ( sup_su4783144482993978935la_a_b @ X @ Y3 ) )
= ( sup_su4783144482993978935la_a_b @ X @ Y3 ) ) ).
% sup_left_idem
thf(fact_139_sup__left__idem,axiom,
! [X: set_nat,Y3: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y3 ) )
= ( sup_sup_set_nat @ X @ Y3 ) ) ).
% sup_left_idem
thf(fact_140_sup__left__idem,axiom,
! [X: set_b,Y3: set_b] :
( ( sup_sup_set_b @ X @ ( sup_sup_set_b @ X @ Y3 ) )
= ( sup_sup_set_b @ X @ Y3 ) ) ).
% sup_left_idem
thf(fact_141_sup__left__idem,axiom,
! [X: set_a,Y3: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y3 ) )
= ( sup_sup_set_a @ X @ Y3 ) ) ).
% sup_left_idem
thf(fact_142_sup__left__idem,axiom,
! [X: set_Re381260168593705685la_a_b > $o,Y3: set_Re381260168593705685la_a_b > $o] :
( ( sup_su2223990397388305166_a_b_o @ X @ ( sup_su2223990397388305166_a_b_o @ X @ Y3 ) )
= ( sup_su2223990397388305166_a_b_o @ X @ Y3 ) ) ).
% sup_left_idem
thf(fact_143_sup_Oleft__idem,axiom,
! [A2: set_se6865892389300016395la_a_b,B2: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ A2 @ ( sup_su4783144482993978935la_a_b @ A2 @ B2 ) )
= ( sup_su4783144482993978935la_a_b @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_144_sup_Oleft__idem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_145_sup_Oleft__idem,axiom,
! [A2: set_b,B2: set_b] :
( ( sup_sup_set_b @ A2 @ ( sup_sup_set_b @ A2 @ B2 ) )
= ( sup_sup_set_b @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_146_sup_Oleft__idem,axiom,
! [A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) )
= ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_147_sup_Oleft__idem,axiom,
! [A2: set_Re381260168593705685la_a_b > $o,B2: set_Re381260168593705685la_a_b > $o] :
( ( sup_su2223990397388305166_a_b_o @ A2 @ ( sup_su2223990397388305166_a_b_o @ A2 @ B2 ) )
= ( sup_su2223990397388305166_a_b_o @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_148_sup__idem,axiom,
! [X: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ X @ X )
= X ) ).
% sup_idem
thf(fact_149_sup__idem,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_150_sup__idem,axiom,
! [X: set_b] :
( ( sup_sup_set_b @ X @ X )
= X ) ).
% sup_idem
thf(fact_151_sup__idem,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ X )
= X ) ).
% sup_idem
thf(fact_152_sup__idem,axiom,
! [X: set_Re381260168593705685la_a_b > $o] :
( ( sup_su2223990397388305166_a_b_o @ X @ X )
= X ) ).
% sup_idem
thf(fact_153_sup_Oidem,axiom,
! [A2: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_154_sup_Oidem,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_155_sup_Oidem,axiom,
! [A2: set_b] :
( ( sup_sup_set_b @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_156_sup_Oidem,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_157_sup_Oidem,axiom,
! [A2: set_Re381260168593705685la_a_b > $o] :
( ( sup_su2223990397388305166_a_b_o @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_158_in__set__simps_I3_J,axiom,
! [X: set_se6865892389300016395la_a_b] :
~ ( member1693404252451860130la_a_b @ X @ ( set_se4873105269812759862la_a_b @ nil_se4484975911477474693la_a_b ) ) ).
% in_set_simps(3)
thf(fact_159_in__set__simps_I3_J,axiom,
! [X: b] :
~ ( member_b @ X @ ( set_b2 @ nil_b ) ) ).
% in_set_simps(3)
thf(fact_160_in__set__simps_I3_J,axiom,
! [X: set_nat] :
~ ( member_set_nat @ X @ ( set_set_nat2 @ nil_set_nat ) ) ).
% in_set_simps(3)
thf(fact_161_in__set__simps_I3_J,axiom,
! [X: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ nil_Re6358386334527539737la_a_b ) ) ).
% in_set_simps(3)
thf(fact_162_in__set__simps_I3_J,axiom,
! [X: list_R8263082107343818799la_a_b] :
~ ( member8712135081189583430la_a_b @ X @ ( set_li8988895567068466906la_a_b @ nil_li4862165533606214569la_a_b ) ) ).
% in_set_simps(3)
thf(fact_163_in__set__simps_I3_J,axiom,
! [X: list_s3671366213839843941la_a_b] :
~ ( member987187531357909116la_a_b @ X @ ( set_li3253934503784678672la_a_b @ nil_li6858553494156228063la_a_b ) ) ).
% in_set_simps(3)
thf(fact_164_in__set__simps_I3_J,axiom,
! [X: list_nat] :
~ ( member_list_nat @ X @ ( set_list_nat2 @ nil_list_nat ) ) ).
% in_set_simps(3)
thf(fact_165_in__set__simps_I3_J,axiom,
! [X: nat] :
~ ( member_nat @ X @ ( set_nat2 @ nil_nat ) ) ).
% in_set_simps(3)
thf(fact_166_in__set__simps_I3_J,axiom,
! [X: set_Re381260168593705685la_a_b] :
~ ( member3481406638322139244la_a_b @ X @ ( set_se5111286776497155840la_a_b @ nil_se6596116450819887951la_a_b ) ) ).
% in_set_simps(3)
thf(fact_167_cp_Ocases,axiom,
! [X: relational_fmla_a_b] :
( ! [X3: nat,T: relational_term_a] :
( X
!= ( relational_Eq_a_b @ X3 @ T ) )
=> ( ! [Q3: relational_fmla_a_b] :
( X
!= ( relational_Neg_a_b @ Q3 ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( X
!= ( relational_Conj_a_b @ Q12 @ Q22 ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( X
!= ( relational_Disj_a_b @ Q12 @ Q22 ) )
=> ( ! [X3: nat,Q3: relational_fmla_a_b] :
( X
!= ( relati591517084277583526ts_a_b @ X3 @ Q3 ) )
=> ( ! [V2: b,Va2: list_R6823256787227418703term_a] :
( X
!= ( relational_Pred_b_a @ V2 @ Va2 ) )
=> ~ ! [V2: $o] :
( X
!= ( relational_Bool_a_b @ V2 ) ) ) ) ) ) ) ) ).
% cp.cases
thf(fact_168_fv_Ocases,axiom,
! [X: relational_fmla_a_b] :
( ! [Uu: b,Ts: list_R6823256787227418703term_a] :
( X
!= ( relational_Pred_b_a @ Uu @ Ts ) )
=> ( ! [B3: $o] :
( X
!= ( relational_Bool_a_b @ B3 ) )
=> ( ! [X3: nat,T2: relational_term_a] :
( X
!= ( relational_Eq_a_b @ X3 @ T2 ) )
=> ( ! [Phi: relational_fmla_a_b] :
( X
!= ( relational_Neg_a_b @ Phi ) )
=> ( ! [Phi: relational_fmla_a_b,Psi: relational_fmla_a_b] :
( X
!= ( relational_Conj_a_b @ Phi @ Psi ) )
=> ( ! [Phi: relational_fmla_a_b,Psi: relational_fmla_a_b] :
( X
!= ( relational_Disj_a_b @ Phi @ Psi ) )
=> ~ ! [Z: nat,Phi: relational_fmla_a_b] :
( X
!= ( relati591517084277583526ts_a_b @ Z @ Phi ) ) ) ) ) ) ) ) ).
% fv.cases
thf(fact_169_fmla_Oinject_I3_J,axiom,
! [X31: nat,X32: relational_term_a,Y31: nat,Y32: relational_term_a] :
( ( ( relational_Eq_a_b @ X31 @ X32 )
= ( relational_Eq_a_b @ Y31 @ Y32 ) )
= ( ( X31 = Y31 )
& ( X32 = Y32 ) ) ) ).
% fmla.inject(3)
thf(fact_170_fmla_Odistinct_I23_J,axiom,
! [X31: nat,X32: relational_term_a,X4: relational_fmla_a_b] :
( ( relational_Eq_a_b @ X31 @ X32 )
!= ( relational_Neg_a_b @ X4 ) ) ).
% fmla.distinct(23)
thf(fact_171_fmla_Odistinct_I25_J,axiom,
! [X31: nat,X32: relational_term_a,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
( ( relational_Eq_a_b @ X31 @ X32 )
!= ( relational_Conj_a_b @ X51 @ X52 ) ) ).
% fmla.distinct(25)
thf(fact_172_fmla_Odistinct_I27_J,axiom,
! [X31: nat,X32: relational_term_a,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
( ( relational_Eq_a_b @ X31 @ X32 )
!= ( relational_Disj_a_b @ X61 @ X62 ) ) ).
% fmla.distinct(27)
thf(fact_173_fmla_Odistinct_I13_J,axiom,
! [X22: $o,X31: nat,X32: relational_term_a] :
( ( relational_Bool_a_b @ X22 )
!= ( relational_Eq_a_b @ X31 @ X32 ) ) ).
% fmla.distinct(13)
thf(fact_174_fmla_Odistinct_I29_J,axiom,
! [X31: nat,X32: relational_term_a,X71: nat,X72: relational_fmla_a_b] :
( ( relational_Eq_a_b @ X31 @ X32 )
!= ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).
% fmla.distinct(29)
thf(fact_175_fmla_Odistinct_I3_J,axiom,
! [X11: b,X12: list_R6823256787227418703term_a,X31: nat,X32: relational_term_a] :
( ( relational_Pred_b_a @ X11 @ X12 )
!= ( relational_Eq_a_b @ X31 @ X32 ) ) ).
% fmla.distinct(3)
thf(fact_176_gen__impl_Osimps_I11_J,axiom,
! [X: nat,V: nat,Va: relational_term_a] :
( ( restri4663030114664890836pl_a_b @ X @ ( relational_Neg_a_b @ ( relational_Eq_a_b @ V @ Va ) ) )
= nil_se6596116450819887951la_a_b ) ).
% gen_impl.simps(11)
thf(fact_177_inf__sup__aci_I8_J,axiom,
! [X: set_se6865892389300016395la_a_b,Y3: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ X @ ( sup_su4783144482993978935la_a_b @ X @ Y3 ) )
= ( sup_su4783144482993978935la_a_b @ X @ Y3 ) ) ).
% inf_sup_aci(8)
thf(fact_178_inf__sup__aci_I8_J,axiom,
! [X: set_nat,Y3: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y3 ) )
= ( sup_sup_set_nat @ X @ Y3 ) ) ).
% inf_sup_aci(8)
thf(fact_179_inf__sup__aci_I8_J,axiom,
! [X: set_b,Y3: set_b] :
( ( sup_sup_set_b @ X @ ( sup_sup_set_b @ X @ Y3 ) )
= ( sup_sup_set_b @ X @ Y3 ) ) ).
% inf_sup_aci(8)
thf(fact_180_inf__sup__aci_I8_J,axiom,
! [X: set_a,Y3: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y3 ) )
= ( sup_sup_set_a @ X @ Y3 ) ) ).
% inf_sup_aci(8)
thf(fact_181_inf__sup__aci_I8_J,axiom,
! [X: set_Re381260168593705685la_a_b > $o,Y3: set_Re381260168593705685la_a_b > $o] :
( ( sup_su2223990397388305166_a_b_o @ X @ ( sup_su2223990397388305166_a_b_o @ X @ Y3 ) )
= ( sup_su2223990397388305166_a_b_o @ X @ Y3 ) ) ).
% inf_sup_aci(8)
thf(fact_182_inf__sup__aci_I7_J,axiom,
! [X: set_se6865892389300016395la_a_b,Y3: set_se6865892389300016395la_a_b,Z2: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ X @ ( sup_su4783144482993978935la_a_b @ Y3 @ Z2 ) )
= ( sup_su4783144482993978935la_a_b @ Y3 @ ( sup_su4783144482993978935la_a_b @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_183_inf__sup__aci_I7_J,axiom,
! [X: set_nat,Y3: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y3 @ Z2 ) )
= ( sup_sup_set_nat @ Y3 @ ( sup_sup_set_nat @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_184_inf__sup__aci_I7_J,axiom,
! [X: set_b,Y3: set_b,Z2: set_b] :
( ( sup_sup_set_b @ X @ ( sup_sup_set_b @ Y3 @ Z2 ) )
= ( sup_sup_set_b @ Y3 @ ( sup_sup_set_b @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_185_inf__sup__aci_I7_J,axiom,
! [X: set_a,Y3: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y3 @ Z2 ) )
= ( sup_sup_set_a @ Y3 @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_186_inf__sup__aci_I7_J,axiom,
! [X: set_Re381260168593705685la_a_b > $o,Y3: set_Re381260168593705685la_a_b > $o,Z2: set_Re381260168593705685la_a_b > $o] :
( ( sup_su2223990397388305166_a_b_o @ X @ ( sup_su2223990397388305166_a_b_o @ Y3 @ Z2 ) )
= ( sup_su2223990397388305166_a_b_o @ Y3 @ ( sup_su2223990397388305166_a_b_o @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_187_inf__sup__aci_I6_J,axiom,
! [X: set_se6865892389300016395la_a_b,Y3: set_se6865892389300016395la_a_b,Z2: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ ( sup_su4783144482993978935la_a_b @ X @ Y3 ) @ Z2 )
= ( sup_su4783144482993978935la_a_b @ X @ ( sup_su4783144482993978935la_a_b @ Y3 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_188_inf__sup__aci_I6_J,axiom,
! [X: set_nat,Y3: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y3 ) @ Z2 )
= ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y3 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_189_inf__sup__aci_I6_J,axiom,
! [X: set_b,Y3: set_b,Z2: set_b] :
( ( sup_sup_set_b @ ( sup_sup_set_b @ X @ Y3 ) @ Z2 )
= ( sup_sup_set_b @ X @ ( sup_sup_set_b @ Y3 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_190_inf__sup__aci_I6_J,axiom,
! [X: set_a,Y3: set_a,Z2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y3 ) @ Z2 )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y3 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_191_inf__sup__aci_I6_J,axiom,
! [X: set_Re381260168593705685la_a_b > $o,Y3: set_Re381260168593705685la_a_b > $o,Z2: set_Re381260168593705685la_a_b > $o] :
( ( sup_su2223990397388305166_a_b_o @ ( sup_su2223990397388305166_a_b_o @ X @ Y3 ) @ Z2 )
= ( sup_su2223990397388305166_a_b_o @ X @ ( sup_su2223990397388305166_a_b_o @ Y3 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_192_inf__sup__aci_I5_J,axiom,
( sup_su4783144482993978935la_a_b
= ( ^ [X2: set_se6865892389300016395la_a_b,Y: set_se6865892389300016395la_a_b] : ( sup_su4783144482993978935la_a_b @ Y @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_193_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X2: set_nat,Y: set_nat] : ( sup_sup_set_nat @ Y @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_194_inf__sup__aci_I5_J,axiom,
( sup_sup_set_b
= ( ^ [X2: set_b,Y: set_b] : ( sup_sup_set_b @ Y @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_195_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X2: set_a,Y: set_a] : ( sup_sup_set_a @ Y @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_196_inf__sup__aci_I5_J,axiom,
( sup_su2223990397388305166_a_b_o
= ( ^ [X2: set_Re381260168593705685la_a_b > $o,Y: set_Re381260168593705685la_a_b > $o] : ( sup_su2223990397388305166_a_b_o @ Y @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_197_sup_Oassoc,axiom,
! [A2: set_se6865892389300016395la_a_b,B2: set_se6865892389300016395la_a_b,C: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ ( sup_su4783144482993978935la_a_b @ A2 @ B2 ) @ C )
= ( sup_su4783144482993978935la_a_b @ A2 @ ( sup_su4783144482993978935la_a_b @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_198_sup_Oassoc,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_199_sup_Oassoc,axiom,
! [A2: set_b,B2: set_b,C: set_b] :
( ( sup_sup_set_b @ ( sup_sup_set_b @ A2 @ B2 ) @ C )
= ( sup_sup_set_b @ A2 @ ( sup_sup_set_b @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_200_sup_Oassoc,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_201_sup_Oassoc,axiom,
! [A2: set_Re381260168593705685la_a_b > $o,B2: set_Re381260168593705685la_a_b > $o,C: set_Re381260168593705685la_a_b > $o] :
( ( sup_su2223990397388305166_a_b_o @ ( sup_su2223990397388305166_a_b_o @ A2 @ B2 ) @ C )
= ( sup_su2223990397388305166_a_b_o @ A2 @ ( sup_su2223990397388305166_a_b_o @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_202_sup__assoc,axiom,
! [X: set_se6865892389300016395la_a_b,Y3: set_se6865892389300016395la_a_b,Z2: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ ( sup_su4783144482993978935la_a_b @ X @ Y3 ) @ Z2 )
= ( sup_su4783144482993978935la_a_b @ X @ ( sup_su4783144482993978935la_a_b @ Y3 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_203_sup__assoc,axiom,
! [X: set_nat,Y3: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y3 ) @ Z2 )
= ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y3 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_204_sup__assoc,axiom,
! [X: set_b,Y3: set_b,Z2: set_b] :
( ( sup_sup_set_b @ ( sup_sup_set_b @ X @ Y3 ) @ Z2 )
= ( sup_sup_set_b @ X @ ( sup_sup_set_b @ Y3 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_205_sup__assoc,axiom,
! [X: set_a,Y3: set_a,Z2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y3 ) @ Z2 )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y3 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_206_sup__assoc,axiom,
! [X: set_Re381260168593705685la_a_b > $o,Y3: set_Re381260168593705685la_a_b > $o,Z2: set_Re381260168593705685la_a_b > $o] :
( ( sup_su2223990397388305166_a_b_o @ ( sup_su2223990397388305166_a_b_o @ X @ Y3 ) @ Z2 )
= ( sup_su2223990397388305166_a_b_o @ X @ ( sup_su2223990397388305166_a_b_o @ Y3 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_207_sup_Ocommute,axiom,
( sup_su4783144482993978935la_a_b
= ( ^ [A4: set_se6865892389300016395la_a_b,B4: set_se6865892389300016395la_a_b] : ( sup_su4783144482993978935la_a_b @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_208_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( sup_sup_set_nat @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_209_sup_Ocommute,axiom,
( sup_sup_set_b
= ( ^ [A4: set_b,B4: set_b] : ( sup_sup_set_b @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_210_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B4: set_a] : ( sup_sup_set_a @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_211_sup_Ocommute,axiom,
( sup_su2223990397388305166_a_b_o
= ( ^ [A4: set_Re381260168593705685la_a_b > $o,B4: set_Re381260168593705685la_a_b > $o] : ( sup_su2223990397388305166_a_b_o @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_212_sup__commute,axiom,
( sup_su4783144482993978935la_a_b
= ( ^ [X2: set_se6865892389300016395la_a_b,Y: set_se6865892389300016395la_a_b] : ( sup_su4783144482993978935la_a_b @ Y @ X2 ) ) ) ).
% sup_commute
thf(fact_213_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X2: set_nat,Y: set_nat] : ( sup_sup_set_nat @ Y @ X2 ) ) ) ).
% sup_commute
thf(fact_214_sup__commute,axiom,
( sup_sup_set_b
= ( ^ [X2: set_b,Y: set_b] : ( sup_sup_set_b @ Y @ X2 ) ) ) ).
% sup_commute
thf(fact_215_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X2: set_a,Y: set_a] : ( sup_sup_set_a @ Y @ X2 ) ) ) ).
% sup_commute
thf(fact_216_sup__commute,axiom,
( sup_su2223990397388305166_a_b_o
= ( ^ [X2: set_Re381260168593705685la_a_b > $o,Y: set_Re381260168593705685la_a_b > $o] : ( sup_su2223990397388305166_a_b_o @ Y @ X2 ) ) ) ).
% sup_commute
thf(fact_217_sup_Oleft__commute,axiom,
! [B2: set_se6865892389300016395la_a_b,A2: set_se6865892389300016395la_a_b,C: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ B2 @ ( sup_su4783144482993978935la_a_b @ A2 @ C ) )
= ( sup_su4783144482993978935la_a_b @ A2 @ ( sup_su4783144482993978935la_a_b @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_218_sup_Oleft__commute,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ C ) )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_219_sup_Oleft__commute,axiom,
! [B2: set_b,A2: set_b,C: set_b] :
( ( sup_sup_set_b @ B2 @ ( sup_sup_set_b @ A2 @ C ) )
= ( sup_sup_set_b @ A2 @ ( sup_sup_set_b @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_220_sup_Oleft__commute,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( sup_sup_set_a @ B2 @ ( sup_sup_set_a @ A2 @ C ) )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_221_sup_Oleft__commute,axiom,
! [B2: set_Re381260168593705685la_a_b > $o,A2: set_Re381260168593705685la_a_b > $o,C: set_Re381260168593705685la_a_b > $o] :
( ( sup_su2223990397388305166_a_b_o @ B2 @ ( sup_su2223990397388305166_a_b_o @ A2 @ C ) )
= ( sup_su2223990397388305166_a_b_o @ A2 @ ( sup_su2223990397388305166_a_b_o @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_222_sup__left__commute,axiom,
! [X: set_se6865892389300016395la_a_b,Y3: set_se6865892389300016395la_a_b,Z2: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ X @ ( sup_su4783144482993978935la_a_b @ Y3 @ Z2 ) )
= ( sup_su4783144482993978935la_a_b @ Y3 @ ( sup_su4783144482993978935la_a_b @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_223_sup__left__commute,axiom,
! [X: set_nat,Y3: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y3 @ Z2 ) )
= ( sup_sup_set_nat @ Y3 @ ( sup_sup_set_nat @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_224_sup__left__commute,axiom,
! [X: set_b,Y3: set_b,Z2: set_b] :
( ( sup_sup_set_b @ X @ ( sup_sup_set_b @ Y3 @ Z2 ) )
= ( sup_sup_set_b @ Y3 @ ( sup_sup_set_b @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_225_sup__left__commute,axiom,
! [X: set_a,Y3: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y3 @ Z2 ) )
= ( sup_sup_set_a @ Y3 @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_226_sup__left__commute,axiom,
! [X: set_Re381260168593705685la_a_b > $o,Y3: set_Re381260168593705685la_a_b > $o,Z2: set_Re381260168593705685la_a_b > $o] :
( ( sup_su2223990397388305166_a_b_o @ X @ ( sup_su2223990397388305166_a_b_o @ Y3 @ Z2 ) )
= ( sup_su2223990397388305166_a_b_o @ Y3 @ ( sup_su2223990397388305166_a_b_o @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_227_sup__fun__def,axiom,
( sup_su2223990397388305166_a_b_o
= ( ^ [F: set_Re381260168593705685la_a_b > $o,G3: set_Re381260168593705685la_a_b > $o,X2: set_Re381260168593705685la_a_b] : ( sup_sup_o @ ( F @ X2 ) @ ( G3 @ X2 ) ) ) ) ).
% sup_fun_def
thf(fact_228_UnE,axiom,
! [C: list_s3671366213839843941la_a_b,A: set_li7574107037410898715la_a_b,B: set_li7574107037410898715la_a_b] :
( ( member987187531357909116la_a_b @ C @ ( sup_su2721708343125988935la_a_b @ A @ B ) )
=> ( ~ ( member987187531357909116la_a_b @ C @ A )
=> ( member987187531357909116la_a_b @ C @ B ) ) ) ).
% UnE
thf(fact_229_UnE,axiom,
! [C: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat @ C @ ( sup_sup_set_list_nat @ A @ B ) )
=> ( ~ ( member_list_nat @ C @ A )
=> ( member_list_nat @ C @ B ) ) ) ).
% UnE
thf(fact_230_UnE,axiom,
! [C: set_se6865892389300016395la_a_b,A: set_se1626149436229564481la_a_b,B: set_se1626149436229564481la_a_b] :
( ( member1693404252451860130la_a_b @ C @ ( sup_su4447918665525899629la_a_b @ A @ B ) )
=> ( ~ ( member1693404252451860130la_a_b @ C @ A )
=> ( member1693404252451860130la_a_b @ C @ B ) ) ) ).
% UnE
thf(fact_231_UnE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
=> ( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% UnE
thf(fact_232_UnE,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ ( sup_sup_set_b @ A @ B ) )
=> ( ~ ( member_b @ C @ A )
=> ( member_b @ C @ B ) ) ) ).
% UnE
thf(fact_233_UnE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B ) )
=> ( ~ ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% UnE
thf(fact_234_UnE,axiom,
! [C: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ C @ ( sup_su4783144482993978935la_a_b @ A @ B ) )
=> ( ~ ( member3481406638322139244la_a_b @ C @ A )
=> ( member3481406638322139244la_a_b @ C @ B ) ) ) ).
% UnE
thf(fact_235_UnI1,axiom,
! [C: list_s3671366213839843941la_a_b,A: set_li7574107037410898715la_a_b,B: set_li7574107037410898715la_a_b] :
( ( member987187531357909116la_a_b @ C @ A )
=> ( member987187531357909116la_a_b @ C @ ( sup_su2721708343125988935la_a_b @ A @ B ) ) ) ).
% UnI1
thf(fact_236_UnI1,axiom,
! [C: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat @ C @ A )
=> ( member_list_nat @ C @ ( sup_sup_set_list_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_237_UnI1,axiom,
! [C: set_se6865892389300016395la_a_b,A: set_se1626149436229564481la_a_b,B: set_se1626149436229564481la_a_b] :
( ( member1693404252451860130la_a_b @ C @ A )
=> ( member1693404252451860130la_a_b @ C @ ( sup_su4447918665525899629la_a_b @ A @ B ) ) ) ).
% UnI1
thf(fact_238_UnI1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_239_UnI1,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ A )
=> ( member_b @ C @ ( sup_sup_set_b @ A @ B ) ) ) ).
% UnI1
thf(fact_240_UnI1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI1
thf(fact_241_UnI1,axiom,
! [C: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ C @ A )
=> ( member3481406638322139244la_a_b @ C @ ( sup_su4783144482993978935la_a_b @ A @ B ) ) ) ).
% UnI1
thf(fact_242_UnI2,axiom,
! [C: list_s3671366213839843941la_a_b,B: set_li7574107037410898715la_a_b,A: set_li7574107037410898715la_a_b] :
( ( member987187531357909116la_a_b @ C @ B )
=> ( member987187531357909116la_a_b @ C @ ( sup_su2721708343125988935la_a_b @ A @ B ) ) ) ).
% UnI2
thf(fact_243_UnI2,axiom,
! [C: list_nat,B: set_list_nat,A: set_list_nat] :
( ( member_list_nat @ C @ B )
=> ( member_list_nat @ C @ ( sup_sup_set_list_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_244_UnI2,axiom,
! [C: set_se6865892389300016395la_a_b,B: set_se1626149436229564481la_a_b,A: set_se1626149436229564481la_a_b] :
( ( member1693404252451860130la_a_b @ C @ B )
=> ( member1693404252451860130la_a_b @ C @ ( sup_su4447918665525899629la_a_b @ A @ B ) ) ) ).
% UnI2
thf(fact_245_UnI2,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_246_UnI2,axiom,
! [C: b,B: set_b,A: set_b] :
( ( member_b @ C @ B )
=> ( member_b @ C @ ( sup_sup_set_b @ A @ B ) ) ) ).
% UnI2
thf(fact_247_UnI2,axiom,
! [C: a,B: set_a,A: set_a] :
( ( member_a @ C @ B )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI2
thf(fact_248_UnI2,axiom,
! [C: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b,A: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ C @ B )
=> ( member3481406638322139244la_a_b @ C @ ( sup_su4783144482993978935la_a_b @ A @ B ) ) ) ).
% UnI2
thf(fact_249_bex__Un,axiom,
! [A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( ? [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ ( sup_su4783144482993978935la_a_b @ A @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ A )
& ( P @ X2 ) )
| ? [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_250_bex__Un,axiom,
! [A: set_nat,B: set_nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( sup_sup_set_nat @ A @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ X2 ) )
| ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_251_bex__Un,axiom,
! [A: set_b,B: set_b,P: b > $o] :
( ( ? [X2: b] :
( ( member_b @ X2 @ ( sup_sup_set_b @ A @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: b] :
( ( member_b @ X2 @ A )
& ( P @ X2 ) )
| ? [X2: b] :
( ( member_b @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_252_bex__Un,axiom,
! [A: set_a,B: set_a,P: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( sup_sup_set_a @ A @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ( P @ X2 ) )
| ? [X2: a] :
( ( member_a @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_253_ball__Un,axiom,
! [A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( ! [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ ( sup_su4783144482993978935la_a_b @ A @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ A )
=> ( P @ X2 ) )
& ! [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_254_ball__Un,axiom,
! [A: set_nat,B: set_nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( sup_sup_set_nat @ A @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( P @ X2 ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_255_ball__Un,axiom,
! [A: set_b,B: set_b,P: b > $o] :
( ( ! [X2: b] :
( ( member_b @ X2 @ ( sup_sup_set_b @ A @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: b] :
( ( member_b @ X2 @ A )
=> ( P @ X2 ) )
& ! [X2: b] :
( ( member_b @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_256_ball__Un,axiom,
! [A: set_a,B: set_a,P: a > $o] :
( ( ! [X2: a] :
( ( member_a @ X2 @ ( sup_sup_set_a @ A @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( P @ X2 ) )
& ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_257_Un__assoc,axiom,
! [A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b,C2: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ ( sup_su4783144482993978935la_a_b @ A @ B ) @ C2 )
= ( sup_su4783144482993978935la_a_b @ A @ ( sup_su4783144482993978935la_a_b @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_258_Un__assoc,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_259_Un__assoc,axiom,
! [A: set_b,B: set_b,C2: set_b] :
( ( sup_sup_set_b @ ( sup_sup_set_b @ A @ B ) @ C2 )
= ( sup_sup_set_b @ A @ ( sup_sup_set_b @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_260_Un__assoc,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_261_Un__absorb,axiom,
! [A: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ A @ A )
= A ) ).
% Un_absorb
thf(fact_262_Un__absorb,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_263_Un__absorb,axiom,
! [A: set_b] :
( ( sup_sup_set_b @ A @ A )
= A ) ).
% Un_absorb
thf(fact_264_Un__absorb,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% Un_absorb
thf(fact_265_Un__commute,axiom,
( sup_su4783144482993978935la_a_b
= ( ^ [A3: set_se6865892389300016395la_a_b,B5: set_se6865892389300016395la_a_b] : ( sup_su4783144482993978935la_a_b @ B5 @ A3 ) ) ) ).
% Un_commute
thf(fact_266_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B5: set_nat] : ( sup_sup_set_nat @ B5 @ A3 ) ) ) ).
% Un_commute
thf(fact_267_Un__commute,axiom,
( sup_sup_set_b
= ( ^ [A3: set_b,B5: set_b] : ( sup_sup_set_b @ B5 @ A3 ) ) ) ).
% Un_commute
thf(fact_268_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A3: set_a,B5: set_a] : ( sup_sup_set_a @ B5 @ A3 ) ) ) ).
% Un_commute
thf(fact_269_Un__left__absorb,axiom,
! [A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ A @ ( sup_su4783144482993978935la_a_b @ A @ B ) )
= ( sup_su4783144482993978935la_a_b @ A @ B ) ) ).
% Un_left_absorb
thf(fact_270_Un__left__absorb,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ A @ B ) ) ).
% Un_left_absorb
thf(fact_271_Un__left__absorb,axiom,
! [A: set_b,B: set_b] :
( ( sup_sup_set_b @ A @ ( sup_sup_set_b @ A @ B ) )
= ( sup_sup_set_b @ A @ B ) ) ).
% Un_left_absorb
thf(fact_272_Un__left__absorb,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B ) )
= ( sup_sup_set_a @ A @ B ) ) ).
% Un_left_absorb
thf(fact_273_Un__left__commute,axiom,
! [A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b,C2: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ A @ ( sup_su4783144482993978935la_a_b @ B @ C2 ) )
= ( sup_su4783144482993978935la_a_b @ B @ ( sup_su4783144482993978935la_a_b @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_274_Un__left__commute,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) )
= ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_275_Un__left__commute,axiom,
! [A: set_b,B: set_b,C2: set_b] :
( ( sup_sup_set_b @ A @ ( sup_sup_set_b @ B @ C2 ) )
= ( sup_sup_set_b @ B @ ( sup_sup_set_b @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_276_Un__left__commute,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) )
= ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_277_gen__size_Ocases,axiom,
! [X: relational_fmla_a_b] :
( ! [B3: $o] :
( X
!= ( relational_Bool_a_b @ B3 ) )
=> ( ! [X3: nat,T: relational_term_a] :
( X
!= ( relational_Eq_a_b @ X3 @ T ) )
=> ( ! [P3: b,Ts: list_R6823256787227418703term_a] :
( X
!= ( relational_Pred_b_a @ P3 @ Ts ) )
=> ( ! [Q3: relational_fmla_a_b] :
( X
!= ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q3 ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( X
!= ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q12 @ Q22 ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( X
!= ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q12 @ Q22 ) ) )
=> ( ! [V2: b,Va2: list_R6823256787227418703term_a] :
( X
!= ( relational_Neg_a_b @ ( relational_Pred_b_a @ V2 @ Va2 ) ) )
=> ( ! [V2: $o] :
( X
!= ( relational_Neg_a_b @ ( relational_Bool_a_b @ V2 ) ) )
=> ( ! [V2: nat,Va2: relational_term_a] :
( X
!= ( relational_Neg_a_b @ ( relational_Eq_a_b @ V2 @ Va2 ) ) )
=> ( ! [V2: nat,Va2: relational_fmla_a_b] :
( X
!= ( relational_Neg_a_b @ ( relati591517084277583526ts_a_b @ V2 @ Va2 ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( X
!= ( relational_Conj_a_b @ Q12 @ Q22 ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( X
!= ( relational_Disj_a_b @ Q12 @ Q22 ) )
=> ~ ! [X3: nat,Q3: relational_fmla_a_b] :
( X
!= ( relati591517084277583526ts_a_b @ X3 @ Q3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% gen_size.cases
thf(fact_278_fmla_Oexhaust,axiom,
! [Y3: relational_fmla_a_b] :
( ! [X112: b,X122: list_R6823256787227418703term_a] :
( Y3
!= ( relational_Pred_b_a @ X112 @ X122 ) )
=> ( ! [X23: $o] :
( Y3
!= ( relational_Bool_a_b @ X23 ) )
=> ( ! [X312: nat,X322: relational_term_a] :
( Y3
!= ( relational_Eq_a_b @ X312 @ X322 ) )
=> ( ! [X42: relational_fmla_a_b] :
( Y3
!= ( relational_Neg_a_b @ X42 ) )
=> ( ! [X512: relational_fmla_a_b,X522: relational_fmla_a_b] :
( Y3
!= ( relational_Conj_a_b @ X512 @ X522 ) )
=> ( ! [X612: relational_fmla_a_b,X622: relational_fmla_a_b] :
( Y3
!= ( relational_Disj_a_b @ X612 @ X622 ) )
=> ~ ! [X712: nat,X722: relational_fmla_a_b] :
( Y3
!= ( relati591517084277583526ts_a_b @ X712 @ X722 ) ) ) ) ) ) ) ) ).
% fmla.exhaust
thf(fact_279_nocp_Ocases,axiom,
! [X: relational_fmla_a_b] :
( ! [B3: $o] :
( X
!= ( relational_Bool_a_b @ B3 ) )
=> ( ! [P3: b,Ts: list_R6823256787227418703term_a] :
( X
!= ( relational_Pred_b_a @ P3 @ Ts ) )
=> ( ! [X3: nat,T: relational_term_a] :
( X
!= ( relational_Eq_a_b @ X3 @ T ) )
=> ( ! [Q3: relational_fmla_a_b] :
( X
!= ( relational_Neg_a_b @ Q3 ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( X
!= ( relational_Conj_a_b @ Q12 @ Q22 ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( X
!= ( relational_Disj_a_b @ Q12 @ Q22 ) )
=> ~ ! [X3: nat,Q3: relational_fmla_a_b] :
( X
!= ( relati591517084277583526ts_a_b @ X3 @ Q3 ) ) ) ) ) ) ) ) ).
% nocp.cases
thf(fact_280_flat__Disj_Ocases,axiom,
! [X: relational_fmla_a_b] :
( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( X
!= ( relational_Disj_a_b @ Q12 @ Q22 ) )
=> ( ! [V2: b,Va2: list_R6823256787227418703term_a] :
( X
!= ( relational_Pred_b_a @ V2 @ Va2 ) )
=> ( ! [V2: $o] :
( X
!= ( relational_Bool_a_b @ V2 ) )
=> ( ! [V2: nat,Va2: relational_term_a] :
( X
!= ( relational_Eq_a_b @ V2 @ Va2 ) )
=> ( ! [V2: relational_fmla_a_b] :
( X
!= ( relational_Neg_a_b @ V2 ) )
=> ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( X
!= ( relational_Conj_a_b @ V2 @ Va2 ) )
=> ~ ! [V2: nat,Va2: relational_fmla_a_b] :
( X
!= ( relati591517084277583526ts_a_b @ V2 @ Va2 ) ) ) ) ) ) ) ) ).
% flat_Disj.cases
thf(fact_281_erase_Ocases,axiom,
! [X: produc7366699395886430672_b_nat] :
( ! [T: $o,X3: nat] :
( X
!= ( produc4282057684358614024_b_nat @ ( relational_Bool_a_b @ T ) @ X3 ) )
=> ( ! [P3: b,Ts: list_R6823256787227418703term_a,X3: nat] :
( X
!= ( produc4282057684358614024_b_nat @ ( relational_Pred_b_a @ P3 @ Ts ) @ X3 ) )
=> ( ! [Z: nat,T: relational_term_a,X3: nat] :
( X
!= ( produc4282057684358614024_b_nat @ ( relational_Eq_a_b @ Z @ T ) @ X3 ) )
=> ( ! [Q3: relational_fmla_a_b,X3: nat] :
( X
!= ( produc4282057684358614024_b_nat @ ( relational_Neg_a_b @ Q3 ) @ X3 ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b,X3: nat] :
( X
!= ( produc4282057684358614024_b_nat @ ( relational_Conj_a_b @ Q12 @ Q22 ) @ X3 ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b,X3: nat] :
( X
!= ( produc4282057684358614024_b_nat @ ( relational_Disj_a_b @ Q12 @ Q22 ) @ X3 ) )
=> ~ ! [Z: nat,Q3: relational_fmla_a_b,X3: nat] :
( X
!= ( produc4282057684358614024_b_nat @ ( relati591517084277583526ts_a_b @ Z @ Q3 ) @ X3 ) ) ) ) ) ) ) ) ).
% erase.cases
thf(fact_282_fmla_Oset__cases_I2_J,axiom,
! [E: b,A2: relational_fmla_a_b] :
( ( member_b @ E @ ( relati8924981150291758614la_a_b @ A2 ) )
=> ( ! [Z22: list_R6823256787227418703term_a] :
( A2
!= ( relational_Pred_b_a @ E @ Z22 ) )
=> ( ! [Z: relational_fmla_a_b] :
( ( A2
= ( relational_Neg_a_b @ Z ) )
=> ~ ( member_b @ E @ ( relati8924981150291758614la_a_b @ Z ) ) )
=> ( ! [Z1: relational_fmla_a_b] :
( ? [Z22: relational_fmla_a_b] :
( A2
= ( relational_Conj_a_b @ Z1 @ Z22 ) )
=> ~ ( member_b @ E @ ( relati8924981150291758614la_a_b @ Z1 ) ) )
=> ( ! [Z1: relational_fmla_a_b,Z22: relational_fmla_a_b] :
( ( A2
= ( relational_Conj_a_b @ Z1 @ Z22 ) )
=> ~ ( member_b @ E @ ( relati8924981150291758614la_a_b @ Z22 ) ) )
=> ( ! [Z1: relational_fmla_a_b] :
( ? [Z22: relational_fmla_a_b] :
( A2
= ( relational_Disj_a_b @ Z1 @ Z22 ) )
=> ~ ( member_b @ E @ ( relati8924981150291758614la_a_b @ Z1 ) ) )
=> ( ! [Z1: relational_fmla_a_b,Z22: relational_fmla_a_b] :
( ( A2
= ( relational_Disj_a_b @ Z1 @ Z22 ) )
=> ~ ( member_b @ E @ ( relati8924981150291758614la_a_b @ Z22 ) ) )
=> ~ ! [Z1: nat,Z22: relational_fmla_a_b] :
( ( A2
= ( relati591517084277583526ts_a_b @ Z1 @ Z22 ) )
=> ~ ( member_b @ E @ ( relati8924981150291758614la_a_b @ Z22 ) ) ) ) ) ) ) ) ) ) ).
% fmla.set_cases(2)
thf(fact_283_gen__impl_Osimps_I17_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,V: nat,Vb: a] :
( ( restri4663030114664890836pl_a_b @ X @ ( relational_Conj_a_b @ Q1 @ ( relational_Eq_a_b @ V @ ( relational_Const_a @ Vb ) ) ) )
= ( union_7072582408929332827la_a_b @ ( restri4663030114664890836pl_a_b @ X @ Q1 ) @ ( restri4663030114664890836pl_a_b @ X @ ( relational_Eq_a_b @ V @ ( relational_Const_a @ Vb ) ) ) ) ) ).
% gen_impl.simps(17)
thf(fact_284_term_Oinject_I2_J,axiom,
! [X22: nat,Y2: nat] :
( ( ( relational_Var_a @ X22 )
= ( relational_Var_a @ Y2 ) )
= ( X22 = Y2 ) ) ).
% term.inject(2)
thf(fact_285_cp__idem,axiom,
! [Q: relational_fmla_a_b] :
( ( relational_cp_a_b @ ( relational_cp_a_b @ Q ) )
= ( relational_cp_a_b @ Q ) ) ).
% cp_idem
thf(fact_286_term_Oinject_I1_J,axiom,
! [X1: a,Y1: a] :
( ( ( relational_Const_a @ X1 )
= ( relational_Const_a @ Y1 ) )
= ( X1 = Y1 ) ) ).
% term.inject(1)
thf(fact_287_term_Odistinct_I1_J,axiom,
! [X1: a,X22: nat] :
( ( relational_Const_a @ X1 )
!= ( relational_Var_a @ X22 ) ) ).
% term.distinct(1)
thf(fact_288_Relational__Calculus_Oterm_Oexhaust,axiom,
! [Y3: relational_term_a] :
( ! [X13: a] :
( Y3
!= ( relational_Const_a @ X13 ) )
=> ~ ! [X23: nat] :
( Y3
!= ( relational_Var_a @ X23 ) ) ) ).
% Relational_Calculus.term.exhaust
thf(fact_289_fv__fo__term__list_Ocases,axiom,
! [X: relational_term_a] :
( ! [N: nat] :
( X
!= ( relational_Var_a @ N ) )
=> ~ ! [V2: a] :
( X
!= ( relational_Const_a @ V2 ) ) ) ).
% fv_fo_term_list.cases
thf(fact_290_cp_Osimps_I7_J,axiom,
! [V: $o] :
( ( relational_cp_a_b @ ( relational_Bool_a_b @ V ) )
= ( relational_Bool_a_b @ V ) ) ).
% cp.simps(7)
thf(fact_291_cp_Osimps_I6_J,axiom,
! [V: b,Va: list_R6823256787227418703term_a] :
( ( relational_cp_a_b @ ( relational_Pred_b_a @ V @ Va ) )
= ( relational_Pred_b_a @ V @ Va ) ) ).
% cp.simps(6)
thf(fact_292_gen__Gen__cp,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X @ Q @ G )
=> ? [X_12: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ ( relational_cp_a_b @ Q ) @ X_12 ) ) ).
% gen_Gen_cp
thf(fact_293_Gen__cp,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ Q @ X_1 )
=> ? [X_12: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ ( relational_cp_a_b @ Q ) @ X_12 ) ) ).
% Gen_cp
thf(fact_294_fmla_Oset__intros_I10_J,axiom,
! [Ys: b,X4: relational_fmla_a_b] :
( ( member_b @ Ys @ ( relati8924981150291758614la_a_b @ X4 ) )
=> ( member_b @ Ys @ ( relati8924981150291758614la_a_b @ ( relational_Neg_a_b @ X4 ) ) ) ) ).
% fmla.set_intros(10)
thf(fact_295_fmla_Osimps_I130_J,axiom,
! [X4: relational_fmla_a_b] :
( ( relati8924981150291758614la_a_b @ ( relational_Neg_a_b @ X4 ) )
= ( relati8924981150291758614la_a_b @ X4 ) ) ).
% fmla.simps(130)
thf(fact_296_fmla_Oset__intros_I12_J,axiom,
! [Yw: b,X52: relational_fmla_a_b,X51: relational_fmla_a_b] :
( ( member_b @ Yw @ ( relati8924981150291758614la_a_b @ X52 ) )
=> ( member_b @ Yw @ ( relati8924981150291758614la_a_b @ ( relational_Conj_a_b @ X51 @ X52 ) ) ) ) ).
% fmla.set_intros(12)
thf(fact_297_fmla_Oset__intros_I11_J,axiom,
! [Yu: b,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
( ( member_b @ Yu @ ( relati8924981150291758614la_a_b @ X51 ) )
=> ( member_b @ Yu @ ( relati8924981150291758614la_a_b @ ( relational_Conj_a_b @ X51 @ X52 ) ) ) ) ).
% fmla.set_intros(11)
thf(fact_298_fmla_Oset__intros_I14_J,axiom,
! [Za: b,X62: relational_fmla_a_b,X61: relational_fmla_a_b] :
( ( member_b @ Za @ ( relati8924981150291758614la_a_b @ X62 ) )
=> ( member_b @ Za @ ( relati8924981150291758614la_a_b @ ( relational_Disj_a_b @ X61 @ X62 ) ) ) ) ).
% fmla.set_intros(14)
thf(fact_299_fmla_Oset__intros_I13_J,axiom,
! [Yy: b,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
( ( member_b @ Yy @ ( relati8924981150291758614la_a_b @ X61 ) )
=> ( member_b @ Yy @ ( relati8924981150291758614la_a_b @ ( relational_Disj_a_b @ X61 @ X62 ) ) ) ) ).
% fmla.set_intros(13)
thf(fact_300_fmla_Osimps_I133_J,axiom,
! [X71: nat,X72: relational_fmla_a_b] :
( ( relati8924981150291758614la_a_b @ ( relati591517084277583526ts_a_b @ X71 @ X72 ) )
= ( relati8924981150291758614la_a_b @ X72 ) ) ).
% fmla.simps(133)
thf(fact_301_fmla_Oset__intros_I15_J,axiom,
! [Zc: b,X72: relational_fmla_a_b,X71: nat] :
( ( member_b @ Zc @ ( relati8924981150291758614la_a_b @ X72 ) )
=> ( member_b @ Zc @ ( relati8924981150291758614la_a_b @ ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ) ) ).
% fmla.set_intros(15)
thf(fact_302_fmla_Oset__intros_I9_J,axiom,
! [X11: b,X12: list_R6823256787227418703term_a] : ( member_b @ X11 @ ( relati8924981150291758614la_a_b @ ( relational_Pred_b_a @ X11 @ X12 ) ) ) ).
% fmla.set_intros(9)
thf(fact_303_fmla_Osimps_I131_J,axiom,
! [X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
( ( relati8924981150291758614la_a_b @ ( relational_Conj_a_b @ X51 @ X52 ) )
= ( sup_sup_set_b @ ( relati8924981150291758614la_a_b @ X51 ) @ ( relati8924981150291758614la_a_b @ X52 ) ) ) ).
% fmla.simps(131)
thf(fact_304_fmla_Osimps_I132_J,axiom,
! [X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
( ( relati8924981150291758614la_a_b @ ( relational_Disj_a_b @ X61 @ X62 ) )
= ( sup_sup_set_b @ ( relati8924981150291758614la_a_b @ X61 ) @ ( relati8924981150291758614la_a_b @ X62 ) ) ) ).
% fmla.simps(132)
thf(fact_305_gen__impl_Osimps_I4_J,axiom,
! [X: nat,Y3: nat,Z2: nat] :
( ( restri4663030114664890836pl_a_b @ X @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ Z2 ) ) )
= nil_se6596116450819887951la_a_b ) ).
% gen_impl.simps(4)
thf(fact_306_qp__impl_Ocases,axiom,
! [X: relational_fmla_a_b] :
( ! [X3: nat,C3: a] :
( X
!= ( relational_Eq_a_b @ X3 @ ( relational_Const_a @ C3 ) ) )
=> ( ! [X3: b,Ts: list_R6823256787227418703term_a] :
( X
!= ( relational_Pred_b_a @ X3 @ Ts ) )
=> ( ! [X3: nat,Q3: relational_fmla_a_b] :
( X
!= ( relati591517084277583526ts_a_b @ X3 @ Q3 ) )
=> ( ! [V2: $o] :
( X
!= ( relational_Bool_a_b @ V2 ) )
=> ( ! [V2: nat,Vb2: nat] :
( X
!= ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb2 ) ) )
=> ( ! [V2: relational_fmla_a_b] :
( X
!= ( relational_Neg_a_b @ V2 ) )
=> ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( X
!= ( relational_Conj_a_b @ V2 @ Va2 ) )
=> ~ ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( X
!= ( relational_Disj_a_b @ V2 @ Va2 ) ) ) ) ) ) ) ) ) ).
% qp_impl.cases
thf(fact_307_gen__impl_Ocases,axiom,
! [X: produc2220957400383678800la_a_b] :
( ! [X3: nat] :
( X
!= ( produc8327306639710187272la_a_b @ X3 @ ( relational_Bool_a_b @ $false ) ) )
=> ( ! [X3: nat] :
( X
!= ( produc8327306639710187272la_a_b @ X3 @ ( relational_Bool_a_b @ $true ) ) )
=> ( ! [X3: nat,Y5: nat,C3: a] :
( X
!= ( produc8327306639710187272la_a_b @ X3 @ ( relational_Eq_a_b @ Y5 @ ( relational_Const_a @ C3 ) ) ) )
=> ( ! [X3: nat,Y5: nat,Z: nat] :
( X
!= ( produc8327306639710187272la_a_b @ X3 @ ( relational_Eq_a_b @ Y5 @ ( relational_Var_a @ Z ) ) ) )
=> ( ! [X3: nat,P3: b,Ts: list_R6823256787227418703term_a] :
( X
!= ( produc8327306639710187272la_a_b @ X3 @ ( relational_Pred_b_a @ P3 @ Ts ) ) )
=> ( ! [X3: nat,Q3: relational_fmla_a_b] :
( X
!= ( produc8327306639710187272la_a_b @ X3 @ ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q3 ) ) ) )
=> ( ! [X3: nat,Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( X
!= ( produc8327306639710187272la_a_b @ X3 @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q12 @ Q22 ) ) ) )
=> ( ! [X3: nat,Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( X
!= ( produc8327306639710187272la_a_b @ X3 @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q12 @ Q22 ) ) ) )
=> ( ! [X3: nat,V2: b,Va2: list_R6823256787227418703term_a] :
( X
!= ( produc8327306639710187272la_a_b @ X3 @ ( relational_Neg_a_b @ ( relational_Pred_b_a @ V2 @ Va2 ) ) ) )
=> ( ! [X3: nat,V2: $o] :
( X
!= ( produc8327306639710187272la_a_b @ X3 @ ( relational_Neg_a_b @ ( relational_Bool_a_b @ V2 ) ) ) )
=> ( ! [X3: nat,V2: nat,Va2: relational_term_a] :
( X
!= ( produc8327306639710187272la_a_b @ X3 @ ( relational_Neg_a_b @ ( relational_Eq_a_b @ V2 @ Va2 ) ) ) )
=> ( ! [X3: nat,V2: nat,Va2: relational_fmla_a_b] :
( X
!= ( produc8327306639710187272la_a_b @ X3 @ ( relational_Neg_a_b @ ( relati591517084277583526ts_a_b @ V2 @ Va2 ) ) ) )
=> ( ! [X3: nat,Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( X
!= ( produc8327306639710187272la_a_b @ X3 @ ( relational_Disj_a_b @ Q12 @ Q22 ) ) )
=> ( ! [X3: nat,Q12: relational_fmla_a_b,Y5: nat,Z: nat] :
( X
!= ( produc8327306639710187272la_a_b @ X3 @ ( relational_Conj_a_b @ Q12 @ ( relational_Eq_a_b @ Y5 @ ( relational_Var_a @ Z ) ) ) ) )
=> ( ! [X3: nat,Q12: relational_fmla_a_b,V2: b,Va2: list_R6823256787227418703term_a] :
( X
!= ( produc8327306639710187272la_a_b @ X3 @ ( relational_Conj_a_b @ Q12 @ ( relational_Pred_b_a @ V2 @ Va2 ) ) ) )
=> ( ! [X3: nat,Q12: relational_fmla_a_b,V2: $o] :
( X
!= ( produc8327306639710187272la_a_b @ X3 @ ( relational_Conj_a_b @ Q12 @ ( relational_Bool_a_b @ V2 ) ) ) )
=> ( ! [X3: nat,Q12: relational_fmla_a_b,V2: nat,Vb2: a] :
( X
!= ( produc8327306639710187272la_a_b @ X3 @ ( relational_Conj_a_b @ Q12 @ ( relational_Eq_a_b @ V2 @ ( relational_Const_a @ Vb2 ) ) ) ) )
=> ( ! [X3: nat,Q12: relational_fmla_a_b,V2: relational_fmla_a_b] :
( X
!= ( produc8327306639710187272la_a_b @ X3 @ ( relational_Conj_a_b @ Q12 @ ( relational_Neg_a_b @ V2 ) ) ) )
=> ( ! [X3: nat,Q12: relational_fmla_a_b,V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( X
!= ( produc8327306639710187272la_a_b @ X3 @ ( relational_Conj_a_b @ Q12 @ ( relational_Conj_a_b @ V2 @ Va2 ) ) ) )
=> ( ! [X3: nat,Q12: relational_fmla_a_b,V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( X
!= ( produc8327306639710187272la_a_b @ X3 @ ( relational_Conj_a_b @ Q12 @ ( relational_Disj_a_b @ V2 @ Va2 ) ) ) )
=> ( ! [X3: nat,Q12: relational_fmla_a_b,V2: nat,Va2: relational_fmla_a_b] :
( X
!= ( produc8327306639710187272la_a_b @ X3 @ ( relational_Conj_a_b @ Q12 @ ( relati591517084277583526ts_a_b @ V2 @ Va2 ) ) ) )
=> ~ ! [X3: nat,Y5: nat,Q3: relational_fmla_a_b] :
( X
!= ( produc8327306639710187272la_a_b @ X3 @ ( relati591517084277583526ts_a_b @ Y5 @ Q3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% gen_impl.cases
thf(fact_308_old_Oprod_Oinject,axiom,
! [A2: product_prod_b_nat > set_list_a,B2: nat > a,A5: product_prod_b_nat > set_list_a,B6: nat > a] :
( ( ( produc2895298938842563487_nat_a @ A2 @ B2 )
= ( produc2895298938842563487_nat_a @ A5 @ B6 ) )
= ( ( A2 = A5 )
& ( B2 = B6 ) ) ) ).
% old.prod.inject
thf(fact_309_old_Oprod_Oinject,axiom,
! [A2: nat > $o,B2: produc8368264342088597808st_nat,A5: nat > $o,B6: produc8368264342088597808st_nat] :
( ( ( produc2404789219858123352st_nat @ A2 @ B2 )
= ( produc2404789219858123352st_nat @ A5 @ B6 ) )
= ( ( A2 = A5 )
& ( B2 = B6 ) ) ) ).
% old.prod.inject
thf(fact_310_old_Oprod_Oinject,axiom,
! [A2: nat > nat > $o,B2: produc1828647624359046049st_nat,A5: nat > nat > $o,B6: produc1828647624359046049st_nat] :
( ( ( produc3127733452865184594st_nat @ A2 @ B2 )
= ( produc3127733452865184594st_nat @ A5 @ B6 ) )
= ( ( A2 = A5 )
& ( B2 = B6 ) ) ) ).
% old.prod.inject
thf(fact_311_old_Oprod_Oinject,axiom,
! [A2: nat > nat > $o,B2: list_nat,A5: nat > nat > $o,B6: list_nat] :
( ( ( produc4727192421694094319st_nat @ A2 @ B2 )
= ( produc4727192421694094319st_nat @ A5 @ B6 ) )
= ( ( A2 = A5 )
& ( B2 = B6 ) ) ) ).
% old.prod.inject
thf(fact_312_old_Oprod_Oinject,axiom,
! [A2: nat > nat > nat,B2: produc1540777390238407569at_nat,A5: nat > nat > nat,B6: produc1540777390238407569at_nat] :
( ( ( produc4302411105509421706at_nat @ A2 @ B2 )
= ( produc4302411105509421706at_nat @ A5 @ B6 ) )
= ( ( A2 = A5 )
& ( B2 = B6 ) ) ) ).
% old.prod.inject
thf(fact_313_old_Oprod_Oinject,axiom,
! [A2: diff_a789849492003309546t_unit,B2: list_nat,A5: diff_a789849492003309546t_unit,B6: list_nat] :
( ( ( produc6847244279022580211st_nat @ A2 @ B2 )
= ( produc6847244279022580211st_nat @ A5 @ B6 ) )
= ( ( A2 = A5 )
& ( B2 = B6 ) ) ) ).
% old.prod.inject
thf(fact_314_prod_Oinject,axiom,
! [X1: product_prod_b_nat > set_list_a,X22: nat > a,Y1: product_prod_b_nat > set_list_a,Y2: nat > a] :
( ( ( produc2895298938842563487_nat_a @ X1 @ X22 )
= ( produc2895298938842563487_nat_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_315_prod_Oinject,axiom,
! [X1: nat > $o,X22: produc8368264342088597808st_nat,Y1: nat > $o,Y2: produc8368264342088597808st_nat] :
( ( ( produc2404789219858123352st_nat @ X1 @ X22 )
= ( produc2404789219858123352st_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_316_prod_Oinject,axiom,
! [X1: nat > nat > $o,X22: produc1828647624359046049st_nat,Y1: nat > nat > $o,Y2: produc1828647624359046049st_nat] :
( ( ( produc3127733452865184594st_nat @ X1 @ X22 )
= ( produc3127733452865184594st_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_317_prod_Oinject,axiom,
! [X1: nat > nat > $o,X22: list_nat,Y1: nat > nat > $o,Y2: list_nat] :
( ( ( produc4727192421694094319st_nat @ X1 @ X22 )
= ( produc4727192421694094319st_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_318_prod_Oinject,axiom,
! [X1: nat > nat > nat,X22: produc1540777390238407569at_nat,Y1: nat > nat > nat,Y2: produc1540777390238407569at_nat] :
( ( ( produc4302411105509421706at_nat @ X1 @ X22 )
= ( produc4302411105509421706at_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_319_prod_Oinject,axiom,
! [X1: diff_a789849492003309546t_unit,X22: list_nat,Y1: diff_a789849492003309546t_unit,Y2: list_nat] :
( ( ( produc6847244279022580211st_nat @ X1 @ X22 )
= ( produc6847244279022580211st_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_320_genempty_Ocases,axiom,
! [A2: relational_fmla_a_b] :
( ( relati5999705594545617851ty_a_b @ A2 )
=> ( ( A2
!= ( relational_Bool_a_b @ $false ) )
=> ( ! [Q3: relational_fmla_a_b] :
( ( A2
= ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q3 ) ) )
=> ~ ( relati5999705594545617851ty_a_b @ Q3 ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( A2
= ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q12 @ Q22 ) ) )
=> ~ ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q22 ) ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( A2
= ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q12 @ Q22 ) ) )
=> ~ ( relati5999705594545617851ty_a_b @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q22 ) ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( A2
= ( relational_Disj_a_b @ Q12 @ Q22 ) )
=> ( ( relati5999705594545617851ty_a_b @ Q12 )
=> ~ ( relati5999705594545617851ty_a_b @ Q22 ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( A2
= ( relational_Conj_a_b @ Q12 @ Q22 ) )
=> ~ ( ( relati5999705594545617851ty_a_b @ Q12 )
| ( relati5999705594545617851ty_a_b @ Q22 ) ) )
=> ( ! [Q3: relational_fmla_a_b] :
( ? [X3: nat,Y5: nat] :
( A2
= ( relational_Conj_a_b @ Q3 @ ( relational_Eq_a_b @ X3 @ ( relational_Var_a @ Y5 ) ) ) )
=> ~ ( relati5999705594545617851ty_a_b @ Q3 ) )
=> ( ! [Q3: relational_fmla_a_b] :
( ? [Y5: nat,X3: nat] :
( A2
= ( relational_Conj_a_b @ Q3 @ ( relational_Eq_a_b @ Y5 @ ( relational_Var_a @ X3 ) ) ) )
=> ~ ( relati5999705594545617851ty_a_b @ Q3 ) )
=> ~ ! [Q3: relational_fmla_a_b] :
( ? [Y5: nat] :
( A2
= ( relati591517084277583526ts_a_b @ Y5 @ Q3 ) )
=> ~ ( relati5999705594545617851ty_a_b @ Q3 ) ) ) ) ) ) ) ) ) ) ) ).
% genempty.cases
thf(fact_321_genempty_Osimps,axiom,
( relati5999705594545617851ty_a_b
= ( ^ [A4: relational_fmla_a_b] :
( ( A4
= ( relational_Bool_a_b @ $false ) )
| ? [Q4: relational_fmla_a_b] :
( ( A4
= ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q4 ) ) )
& ( relati5999705594545617851ty_a_b @ Q4 ) )
| ? [Q13: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( A4
= ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q13 @ Q23 ) ) )
& ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q23 ) ) ) )
| ? [Q13: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( A4
= ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q13 @ Q23 ) ) )
& ( relati5999705594545617851ty_a_b @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q23 ) ) ) )
| ? [Q13: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( A4
= ( relational_Disj_a_b @ Q13 @ Q23 ) )
& ( relati5999705594545617851ty_a_b @ Q13 )
& ( relati5999705594545617851ty_a_b @ Q23 ) )
| ? [Q13: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( A4
= ( relational_Conj_a_b @ Q13 @ Q23 ) )
& ( ( relati5999705594545617851ty_a_b @ Q13 )
| ( relati5999705594545617851ty_a_b @ Q23 ) ) )
| ? [Q4: relational_fmla_a_b] :
( ? [X2: nat,Y: nat] :
( A4
= ( relational_Conj_a_b @ Q4 @ ( relational_Eq_a_b @ X2 @ ( relational_Var_a @ Y ) ) ) )
& ( relati5999705594545617851ty_a_b @ Q4 ) )
| ? [Q4: relational_fmla_a_b] :
( ? [Y: nat] :
( A4
= ( relati591517084277583526ts_a_b @ Y @ Q4 ) )
& ( relati5999705594545617851ty_a_b @ Q4 ) ) ) ) ) ).
% genempty.simps
thf(fact_322_subst__term_Ocases,axiom,
! [X: produc6058688428250151583at_nat] :
( ! [Z: nat,X3: nat,Y5: nat] :
( X
!= ( produc2180204704594896271at_nat @ ( relational_Var_a @ Z ) @ ( product_Pair_nat_nat @ X3 @ Y5 ) ) )
=> ~ ! [C3: a,X3: nat,Y5: nat] :
( X
!= ( produc2180204704594896271at_nat @ ( relational_Const_a @ C3 ) @ ( product_Pair_nat_nat @ X3 @ Y5 ) ) ) ) ).
% subst_term.cases
thf(fact_323_genempty_Ointros_I2_J,axiom,
! [Q: relational_fmla_a_b] :
( ( relati5999705594545617851ty_a_b @ Q )
=> ( relati5999705594545617851ty_a_b @ ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q ) ) ) ) ).
% genempty.intros(2)
thf(fact_324_genempty_Ointros_I6_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( ( relati5999705594545617851ty_a_b @ Q1 )
| ( relati5999705594545617851ty_a_b @ Q2 ) )
=> ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) ) ) ).
% genempty.intros(6)
thf(fact_325_genempty_Ointros_I5_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( relati5999705594545617851ty_a_b @ Q1 )
=> ( ( relati5999705594545617851ty_a_b @ Q2 )
=> ( relati5999705594545617851ty_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) ) ) ) ).
% genempty.intros(5)
thf(fact_326_genempty_Ointros_I1_J,axiom,
relati5999705594545617851ty_a_b @ ( relational_Bool_a_b @ $false ) ).
% genempty.intros(1)
thf(fact_327_genempty_Ointros_I9_J,axiom,
! [Q: relational_fmla_a_b,Y3: nat] :
( ( relati5999705594545617851ty_a_b @ Q )
=> ( relati5999705594545617851ty_a_b @ ( relati591517084277583526ts_a_b @ Y3 @ Q ) ) ) ).
% genempty.intros(9)
thf(fact_328_eval__term_Ocases,axiom,
! [X: produc8608687409264118859term_a] :
( ! [Sigma: nat > a,C3: a] :
( X
!= ( produc8917778089171359291term_a @ Sigma @ ( relational_Const_a @ C3 ) ) )
=> ~ ! [Sigma: nat > a,N: nat] :
( X
!= ( produc8917778089171359291term_a @ Sigma @ ( relational_Var_a @ N ) ) ) ) ).
% eval_term.cases
thf(fact_329_genempty__cp,axiom,
! [Q: relational_fmla_a_b] :
( ( relati5999705594545617851ty_a_b @ Q )
=> ( ( relational_cp_a_b @ Q )
= ( relational_Bool_a_b @ $false ) ) ) ).
% genempty_cp
thf(fact_330_old_Oprod_Oexhaust,axiom,
! [Y3: produc5835360497134304175_nat_a] :
~ ! [A6: product_prod_b_nat > set_list_a,B3: nat > a] :
( Y3
!= ( produc2895298938842563487_nat_a @ A6 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_331_old_Oprod_Oexhaust,axiom,
! [Y3: produc6155704871766690142st_nat] :
~ ! [A6: nat > $o,B3: produc8368264342088597808st_nat] :
( Y3
!= ( produc2404789219858123352st_nat @ A6 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_332_old_Oprod_Oexhaust,axiom,
! [Y3: produc4787317212837456354st_nat] :
~ ! [A6: nat > nat > $o,B3: produc1828647624359046049st_nat] :
( Y3
!= ( produc3127733452865184594st_nat @ A6 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_333_old_Oprod_Oexhaust,axiom,
! [Y3: produc254973753779126261st_nat] :
~ ! [A6: nat > nat > $o,B3: list_nat] :
( Y3
!= ( produc4727192421694094319st_nat @ A6 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_334_old_Oprod_Oexhaust,axiom,
! [Y3: produc7291604503153307608at_nat] :
~ ! [A6: nat > nat > nat,B3: produc1540777390238407569at_nat] :
( Y3
!= ( produc4302411105509421706at_nat @ A6 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_335_old_Oprod_Oexhaust,axiom,
! [Y3: produc2026718404927347961st_nat] :
~ ! [A6: diff_a789849492003309546t_unit,B3: list_nat] :
( Y3
!= ( produc6847244279022580211st_nat @ A6 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_336_surj__pair,axiom,
! [P4: produc5835360497134304175_nat_a] :
? [X3: product_prod_b_nat > set_list_a,Y5: nat > a] :
( P4
= ( produc2895298938842563487_nat_a @ X3 @ Y5 ) ) ).
% surj_pair
thf(fact_337_surj__pair,axiom,
! [P4: produc6155704871766690142st_nat] :
? [X3: nat > $o,Y5: produc8368264342088597808st_nat] :
( P4
= ( produc2404789219858123352st_nat @ X3 @ Y5 ) ) ).
% surj_pair
thf(fact_338_surj__pair,axiom,
! [P4: produc4787317212837456354st_nat] :
? [X3: nat > nat > $o,Y5: produc1828647624359046049st_nat] :
( P4
= ( produc3127733452865184594st_nat @ X3 @ Y5 ) ) ).
% surj_pair
thf(fact_339_surj__pair,axiom,
! [P4: produc254973753779126261st_nat] :
? [X3: nat > nat > $o,Y5: list_nat] :
( P4
= ( produc4727192421694094319st_nat @ X3 @ Y5 ) ) ).
% surj_pair
thf(fact_340_surj__pair,axiom,
! [P4: produc7291604503153307608at_nat] :
? [X3: nat > nat > nat,Y5: produc1540777390238407569at_nat] :
( P4
= ( produc4302411105509421706at_nat @ X3 @ Y5 ) ) ).
% surj_pair
thf(fact_341_surj__pair,axiom,
! [P4: produc2026718404927347961st_nat] :
? [X3: diff_a789849492003309546t_unit,Y5: list_nat] :
( P4
= ( produc6847244279022580211st_nat @ X3 @ Y5 ) ) ).
% surj_pair
thf(fact_342_prod__cases,axiom,
! [P: produc5835360497134304175_nat_a > $o,P4: produc5835360497134304175_nat_a] :
( ! [A6: product_prod_b_nat > set_list_a,B3: nat > a] : ( P @ ( produc2895298938842563487_nat_a @ A6 @ B3 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_343_prod__cases,axiom,
! [P: produc6155704871766690142st_nat > $o,P4: produc6155704871766690142st_nat] :
( ! [A6: nat > $o,B3: produc8368264342088597808st_nat] : ( P @ ( produc2404789219858123352st_nat @ A6 @ B3 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_344_prod__cases,axiom,
! [P: produc4787317212837456354st_nat > $o,P4: produc4787317212837456354st_nat] :
( ! [A6: nat > nat > $o,B3: produc1828647624359046049st_nat] : ( P @ ( produc3127733452865184594st_nat @ A6 @ B3 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_345_prod__cases,axiom,
! [P: produc254973753779126261st_nat > $o,P4: produc254973753779126261st_nat] :
( ! [A6: nat > nat > $o,B3: list_nat] : ( P @ ( produc4727192421694094319st_nat @ A6 @ B3 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_346_prod__cases,axiom,
! [P: produc7291604503153307608at_nat > $o,P4: produc7291604503153307608at_nat] :
( ! [A6: nat > nat > nat,B3: produc1540777390238407569at_nat] : ( P @ ( produc4302411105509421706at_nat @ A6 @ B3 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_347_prod__cases,axiom,
! [P: produc2026718404927347961st_nat > $o,P4: produc2026718404927347961st_nat] :
( ! [A6: diff_a789849492003309546t_unit,B3: list_nat] : ( P @ ( produc6847244279022580211st_nat @ A6 @ B3 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_348_Pair__inject,axiom,
! [A2: product_prod_b_nat > set_list_a,B2: nat > a,A5: product_prod_b_nat > set_list_a,B6: nat > a] :
( ( ( produc2895298938842563487_nat_a @ A2 @ B2 )
= ( produc2895298938842563487_nat_a @ A5 @ B6 ) )
=> ~ ( ( A2 = A5 )
=> ( B2 != B6 ) ) ) ).
% Pair_inject
thf(fact_349_Pair__inject,axiom,
! [A2: nat > $o,B2: produc8368264342088597808st_nat,A5: nat > $o,B6: produc8368264342088597808st_nat] :
( ( ( produc2404789219858123352st_nat @ A2 @ B2 )
= ( produc2404789219858123352st_nat @ A5 @ B6 ) )
=> ~ ( ( A2 = A5 )
=> ( B2 != B6 ) ) ) ).
% Pair_inject
thf(fact_350_Pair__inject,axiom,
! [A2: nat > nat > $o,B2: produc1828647624359046049st_nat,A5: nat > nat > $o,B6: produc1828647624359046049st_nat] :
( ( ( produc3127733452865184594st_nat @ A2 @ B2 )
= ( produc3127733452865184594st_nat @ A5 @ B6 ) )
=> ~ ( ( A2 = A5 )
=> ( B2 != B6 ) ) ) ).
% Pair_inject
thf(fact_351_Pair__inject,axiom,
! [A2: nat > nat > $o,B2: list_nat,A5: nat > nat > $o,B6: list_nat] :
( ( ( produc4727192421694094319st_nat @ A2 @ B2 )
= ( produc4727192421694094319st_nat @ A5 @ B6 ) )
=> ~ ( ( A2 = A5 )
=> ( B2 != B6 ) ) ) ).
% Pair_inject
thf(fact_352_Pair__inject,axiom,
! [A2: nat > nat > nat,B2: produc1540777390238407569at_nat,A5: nat > nat > nat,B6: produc1540777390238407569at_nat] :
( ( ( produc4302411105509421706at_nat @ A2 @ B2 )
= ( produc4302411105509421706at_nat @ A5 @ B6 ) )
=> ~ ( ( A2 = A5 )
=> ( B2 != B6 ) ) ) ).
% Pair_inject
thf(fact_353_Pair__inject,axiom,
! [A2: diff_a789849492003309546t_unit,B2: list_nat,A5: diff_a789849492003309546t_unit,B6: list_nat] :
( ( ( produc6847244279022580211st_nat @ A2 @ B2 )
= ( produc6847244279022580211st_nat @ A5 @ B6 ) )
=> ~ ( ( A2 = A5 )
=> ( B2 != B6 ) ) ) ).
% Pair_inject
thf(fact_354_prod__cases3,axiom,
! [Y3: produc6155704871766690142st_nat] :
~ ! [A6: nat > $o,B3: produc1828647624359046049st_nat,C3: list_nat] :
( Y3
!= ( produc2404789219858123352st_nat @ A6 @ ( produc1778712578675790378st_nat @ B3 @ C3 ) ) ) ).
% prod_cases3
thf(fact_355_prod__cases3,axiom,
! [Y3: produc4787317212837456354st_nat] :
~ ! [A6: nat > nat > $o,B3: list_nat,C3: list_nat] :
( Y3
!= ( produc3127733452865184594st_nat @ A6 @ ( produc2694037385005941721st_nat @ B3 @ C3 ) ) ) ).
% prod_cases3
thf(fact_356_prod__cases3,axiom,
! [Y3: produc7291604503153307608at_nat] :
~ ! [A6: nat > nat > nat,B3: list_nat,C3: nat] :
( Y3
!= ( produc4302411105509421706at_nat @ A6 @ ( produc1298395424260782409at_nat @ B3 @ C3 ) ) ) ).
% prod_cases3
thf(fact_357_prod__induct3,axiom,
! [P: produc6155704871766690142st_nat > $o,X: produc6155704871766690142st_nat] :
( ! [A6: nat > $o,B3: produc1828647624359046049st_nat,C3: list_nat] : ( P @ ( produc2404789219858123352st_nat @ A6 @ ( produc1778712578675790378st_nat @ B3 @ C3 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_358_prod__induct3,axiom,
! [P: produc4787317212837456354st_nat > $o,X: produc4787317212837456354st_nat] :
( ! [A6: nat > nat > $o,B3: list_nat,C3: list_nat] : ( P @ ( produc3127733452865184594st_nat @ A6 @ ( produc2694037385005941721st_nat @ B3 @ C3 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_359_prod__induct3,axiom,
! [P: produc7291604503153307608at_nat > $o,X: produc7291604503153307608at_nat] :
( ! [A6: nat > nat > nat,B3: list_nat,C3: nat] : ( P @ ( produc4302411105509421706at_nat @ A6 @ ( produc1298395424260782409at_nat @ B3 @ C3 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_360_genempty_Ointros_I4_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( relati5999705594545617851ty_a_b @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q2 ) ) )
=> ( relati5999705594545617851ty_a_b @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) ) ) ) ).
% genempty.intros(4)
thf(fact_361_genempty_Ointros_I3_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q2 ) ) )
=> ( relati5999705594545617851ty_a_b @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) ) ) ) ).
% genempty.intros(3)
thf(fact_362_genempty_Ointros_I7_J,axiom,
! [Q: relational_fmla_a_b,X: nat,Y3: nat] :
( ( relati5999705594545617851ty_a_b @ Q )
=> ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y3 ) ) ) ) ) ).
% genempty.intros(7)
thf(fact_363_genempty_Ointros_I8_J,axiom,
! [Q: relational_fmla_a_b,Y3: nat,X: nat] :
( ( relati5999705594545617851ty_a_b @ Q )
=> ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ X ) ) ) ) ) ).
% genempty.intros(8)
thf(fact_364_sat_Ocases,axiom,
! [X: produc1132964494702330949_nat_a] :
( ! [R: b,Ts: list_R6823256787227418703term_a,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( X
!= ( produc6598558901832717687_nat_a @ ( relational_Pred_b_a @ R @ Ts ) @ ( produc2895298938842563487_nat_a @ I @ Sigma ) ) )
=> ( ! [B3: $o,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( X
!= ( produc6598558901832717687_nat_a @ ( relational_Bool_a_b @ B3 ) @ ( produc2895298938842563487_nat_a @ I @ Sigma ) ) )
=> ( ! [X3: nat,T2: relational_term_a,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( X
!= ( produc6598558901832717687_nat_a @ ( relational_Eq_a_b @ X3 @ T2 ) @ ( produc2895298938842563487_nat_a @ I @ Sigma ) ) )
=> ( ! [Phi: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( X
!= ( produc6598558901832717687_nat_a @ ( relational_Neg_a_b @ Phi ) @ ( produc2895298938842563487_nat_a @ I @ Sigma ) ) )
=> ( ! [Phi: relational_fmla_a_b,Psi: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( X
!= ( produc6598558901832717687_nat_a @ ( relational_Conj_a_b @ Phi @ Psi ) @ ( produc2895298938842563487_nat_a @ I @ Sigma ) ) )
=> ( ! [Phi: relational_fmla_a_b,Psi: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( X
!= ( produc6598558901832717687_nat_a @ ( relational_Disj_a_b @ Phi @ Psi ) @ ( produc2895298938842563487_nat_a @ I @ Sigma ) ) )
=> ~ ! [Z: nat,Phi: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( X
!= ( produc6598558901832717687_nat_a @ ( relati591517084277583526ts_a_b @ Z @ Phi ) @ ( produc2895298938842563487_nat_a @ I @ Sigma ) ) ) ) ) ) ) ) ) ).
% sat.cases
thf(fact_365_subst_Ocases,axiom,
! [X: produc8867654947514737559at_nat] :
( ! [T: $o,X3: nat,Y5: nat] :
( X
!= ( produc6913411929637712585at_nat @ ( relational_Bool_a_b @ T ) @ ( product_Pair_nat_nat @ X3 @ Y5 ) ) )
=> ( ! [P3: b,Ts: list_R6823256787227418703term_a,X3: nat,Y5: nat] :
( X
!= ( produc6913411929637712585at_nat @ ( relational_Pred_b_a @ P3 @ Ts ) @ ( product_Pair_nat_nat @ X3 @ Y5 ) ) )
=> ( ! [Z: nat,T: relational_term_a,X3: nat,Y5: nat] :
( X
!= ( produc6913411929637712585at_nat @ ( relational_Eq_a_b @ Z @ T ) @ ( product_Pair_nat_nat @ X3 @ Y5 ) ) )
=> ( ! [Q3: relational_fmla_a_b,X3: nat,Y5: nat] :
( X
!= ( produc6913411929637712585at_nat @ ( relational_Neg_a_b @ Q3 ) @ ( product_Pair_nat_nat @ X3 @ Y5 ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b,X3: nat,Y5: nat] :
( X
!= ( produc6913411929637712585at_nat @ ( relational_Conj_a_b @ Q12 @ Q22 ) @ ( product_Pair_nat_nat @ X3 @ Y5 ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b,X3: nat,Y5: nat] :
( X
!= ( produc6913411929637712585at_nat @ ( relational_Disj_a_b @ Q12 @ Q22 ) @ ( product_Pair_nat_nat @ X3 @ Y5 ) ) )
=> ~ ! [Z: nat,Q3: relational_fmla_a_b,X3: nat,Y5: nat] :
( X
!= ( produc6913411929637712585at_nat @ ( relati591517084277583526ts_a_b @ Z @ Q3 ) @ ( product_Pair_nat_nat @ X3 @ Y5 ) ) ) ) ) ) ) ) ) ).
% subst.cases
thf(fact_366_gen__impl_Opinduct,axiom,
! [A0: nat,A1: relational_fmla_a_b,P: nat > relational_fmla_a_b > $o] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ A0 @ A1 ) )
=> ( ! [X3: nat] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X3 @ ( relational_Bool_a_b @ $false ) ) )
=> ( P @ X3 @ ( relational_Bool_a_b @ $false ) ) )
=> ( ! [X3: nat] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X3 @ ( relational_Bool_a_b @ $true ) ) )
=> ( P @ X3 @ ( relational_Bool_a_b @ $true ) ) )
=> ( ! [X3: nat,Y5: nat,C3: a] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X3 @ ( relational_Eq_a_b @ Y5 @ ( relational_Const_a @ C3 ) ) ) )
=> ( P @ X3 @ ( relational_Eq_a_b @ Y5 @ ( relational_Const_a @ C3 ) ) ) )
=> ( ! [X3: nat,Y5: nat,Z: nat] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X3 @ ( relational_Eq_a_b @ Y5 @ ( relational_Var_a @ Z ) ) ) )
=> ( P @ X3 @ ( relational_Eq_a_b @ Y5 @ ( relational_Var_a @ Z ) ) ) )
=> ( ! [X3: nat,P3: b,Ts: list_R6823256787227418703term_a] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X3 @ ( relational_Pred_b_a @ P3 @ Ts ) ) )
=> ( P @ X3 @ ( relational_Pred_b_a @ P3 @ Ts ) ) )
=> ( ! [X3: nat,Q3: relational_fmla_a_b] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X3 @ ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q3 ) ) ) )
=> ( ( P @ X3 @ Q3 )
=> ( P @ X3 @ ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q3 ) ) ) ) )
=> ( ! [X3: nat,Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X3 @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q12 @ Q22 ) ) ) )
=> ( ( P @ X3 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q22 ) ) )
=> ( P @ X3 @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q12 @ Q22 ) ) ) ) )
=> ( ! [X3: nat,Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X3 @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q12 @ Q22 ) ) ) )
=> ( ( P @ X3 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q22 ) ) )
=> ( P @ X3 @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q12 @ Q22 ) ) ) ) )
=> ( ! [X3: nat,V2: b,Va2: list_R6823256787227418703term_a] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X3 @ ( relational_Neg_a_b @ ( relational_Pred_b_a @ V2 @ Va2 ) ) ) )
=> ( P @ X3 @ ( relational_Neg_a_b @ ( relational_Pred_b_a @ V2 @ Va2 ) ) ) )
=> ( ! [X3: nat,V2: $o] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X3 @ ( relational_Neg_a_b @ ( relational_Bool_a_b @ V2 ) ) ) )
=> ( P @ X3 @ ( relational_Neg_a_b @ ( relational_Bool_a_b @ V2 ) ) ) )
=> ( ! [X3: nat,V2: nat,Va2: relational_term_a] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X3 @ ( relational_Neg_a_b @ ( relational_Eq_a_b @ V2 @ Va2 ) ) ) )
=> ( P @ X3 @ ( relational_Neg_a_b @ ( relational_Eq_a_b @ V2 @ Va2 ) ) ) )
=> ( ! [X3: nat,V2: nat,Va2: relational_fmla_a_b] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X3 @ ( relational_Neg_a_b @ ( relati591517084277583526ts_a_b @ V2 @ Va2 ) ) ) )
=> ( P @ X3 @ ( relational_Neg_a_b @ ( relati591517084277583526ts_a_b @ V2 @ Va2 ) ) ) )
=> ( ! [X3: nat,Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X3 @ ( relational_Disj_a_b @ Q12 @ Q22 ) ) )
=> ( ( P @ X3 @ Q12 )
=> ( ( ? [Xa: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ Xa @ ( set_se5111286776497155840la_a_b @ ( restri4663030114664890836pl_a_b @ X3 @ Q12 ) ) )
=> ( P @ X3 @ Q22 ) )
=> ( P @ X3 @ ( relational_Disj_a_b @ Q12 @ Q22 ) ) ) ) )
=> ( ! [X3: nat,Q12: relational_fmla_a_b,Y5: nat,Z: nat] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X3 @ ( relational_Conj_a_b @ Q12 @ ( relational_Eq_a_b @ Y5 @ ( relational_Var_a @ Z ) ) ) ) )
=> ( ( ( X3 = Y5 )
=> ( P @ X3 @ Q12 ) )
=> ( ( ( X3 = Y5 )
=> ( P @ Z @ Q12 ) )
=> ( ( ( X3 != Y5 )
=> ( ( X3 = Z )
=> ( P @ X3 @ Q12 ) ) )
=> ( ( ( X3 != Y5 )
=> ( ( X3 = Z )
=> ( P @ Y5 @ Q12 ) ) )
=> ( ( ( X3 != Y5 )
=> ( ( X3 != Z )
=> ( P @ X3 @ Q12 ) ) )
=> ( P @ X3 @ ( relational_Conj_a_b @ Q12 @ ( relational_Eq_a_b @ Y5 @ ( relational_Var_a @ Z ) ) ) ) ) ) ) ) ) )
=> ( ! [X3: nat,Q12: relational_fmla_a_b,V2: b,Va2: list_R6823256787227418703term_a] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X3 @ ( relational_Conj_a_b @ Q12 @ ( relational_Pred_b_a @ V2 @ Va2 ) ) ) )
=> ( ( P @ X3 @ Q12 )
=> ( ( P @ X3 @ ( relational_Pred_b_a @ V2 @ Va2 ) )
=> ( P @ X3 @ ( relational_Conj_a_b @ Q12 @ ( relational_Pred_b_a @ V2 @ Va2 ) ) ) ) ) )
=> ( ! [X3: nat,Q12: relational_fmla_a_b,V2: $o] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X3 @ ( relational_Conj_a_b @ Q12 @ ( relational_Bool_a_b @ V2 ) ) ) )
=> ( ( P @ X3 @ Q12 )
=> ( ( P @ X3 @ ( relational_Bool_a_b @ V2 ) )
=> ( P @ X3 @ ( relational_Conj_a_b @ Q12 @ ( relational_Bool_a_b @ V2 ) ) ) ) ) )
=> ( ! [X3: nat,Q12: relational_fmla_a_b,V2: nat,Vb2: a] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X3 @ ( relational_Conj_a_b @ Q12 @ ( relational_Eq_a_b @ V2 @ ( relational_Const_a @ Vb2 ) ) ) ) )
=> ( ( P @ X3 @ Q12 )
=> ( ( P @ X3 @ ( relational_Eq_a_b @ V2 @ ( relational_Const_a @ Vb2 ) ) )
=> ( P @ X3 @ ( relational_Conj_a_b @ Q12 @ ( relational_Eq_a_b @ V2 @ ( relational_Const_a @ Vb2 ) ) ) ) ) ) )
=> ( ! [X3: nat,Q12: relational_fmla_a_b,V2: relational_fmla_a_b] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X3 @ ( relational_Conj_a_b @ Q12 @ ( relational_Neg_a_b @ V2 ) ) ) )
=> ( ( P @ X3 @ Q12 )
=> ( ( P @ X3 @ ( relational_Neg_a_b @ V2 ) )
=> ( P @ X3 @ ( relational_Conj_a_b @ Q12 @ ( relational_Neg_a_b @ V2 ) ) ) ) ) )
=> ( ! [X3: nat,Q12: relational_fmla_a_b,V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X3 @ ( relational_Conj_a_b @ Q12 @ ( relational_Conj_a_b @ V2 @ Va2 ) ) ) )
=> ( ( P @ X3 @ Q12 )
=> ( ( P @ X3 @ ( relational_Conj_a_b @ V2 @ Va2 ) )
=> ( P @ X3 @ ( relational_Conj_a_b @ Q12 @ ( relational_Conj_a_b @ V2 @ Va2 ) ) ) ) ) )
=> ( ! [X3: nat,Q12: relational_fmla_a_b,V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X3 @ ( relational_Conj_a_b @ Q12 @ ( relational_Disj_a_b @ V2 @ Va2 ) ) ) )
=> ( ( P @ X3 @ Q12 )
=> ( ( P @ X3 @ ( relational_Disj_a_b @ V2 @ Va2 ) )
=> ( P @ X3 @ ( relational_Conj_a_b @ Q12 @ ( relational_Disj_a_b @ V2 @ Va2 ) ) ) ) ) )
=> ( ! [X3: nat,Q12: relational_fmla_a_b,V2: nat,Va2: relational_fmla_a_b] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X3 @ ( relational_Conj_a_b @ Q12 @ ( relati591517084277583526ts_a_b @ V2 @ Va2 ) ) ) )
=> ( ( P @ X3 @ Q12 )
=> ( ( P @ X3 @ ( relati591517084277583526ts_a_b @ V2 @ Va2 ) )
=> ( P @ X3 @ ( relational_Conj_a_b @ Q12 @ ( relati591517084277583526ts_a_b @ V2 @ Va2 ) ) ) ) ) )
=> ( ! [X3: nat,Y5: nat,Q3: relational_fmla_a_b] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X3 @ ( relati591517084277583526ts_a_b @ Y5 @ Q3 ) ) )
=> ( ( ( X3 != Y5 )
=> ( P @ X3 @ Q3 ) )
=> ( P @ X3 @ ( relati591517084277583526ts_a_b @ Y5 @ Q3 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% gen_impl.pinduct
thf(fact_367_cpropagated__simps_I3_J,axiom,
! [X: nat,T3: relational_term_a] :
( ( relati1591879772219623554ed_a_b @ ( relational_Eq_a_b @ X @ T3 ) )
= ( T3
!= ( relational_Var_a @ X ) ) ) ).
% cpropagated_simps(3)
thf(fact_368_nocp_Oelims_I1_J,axiom,
! [X: relational_fmla_a_b,Y3: $o] :
( ( ( relational_nocp_a_b @ X )
= Y3 )
=> ( ( ? [B3: $o] :
( X
= ( relational_Bool_a_b @ B3 ) )
=> Y3 )
=> ( ( ? [P3: b,Ts: list_R6823256787227418703term_a] :
( X
= ( relational_Pred_b_a @ P3 @ Ts ) )
=> ~ Y3 )
=> ( ! [X3: nat,T: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X3 @ T ) )
=> ( Y3
= ( T
= ( relational_Var_a @ X3 ) ) ) )
=> ( ! [Q3: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q3 ) )
=> ( Y3
= ( ~ ( relational_nocp_a_b @ Q3 ) ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q12 @ Q22 ) )
=> ( Y3
= ( ~ ( ( relational_nocp_a_b @ Q12 )
& ( relational_nocp_a_b @ Q22 ) ) ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q12 @ Q22 ) )
=> ( Y3
= ( ~ ( ( relational_nocp_a_b @ Q12 )
& ( relational_nocp_a_b @ Q22 ) ) ) ) )
=> ~ ! [X3: nat,Q3: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X3 @ Q3 ) )
=> ( Y3
= ( ~ ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q3 ) )
& ( relational_nocp_a_b @ Q3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% nocp.elims(1)
thf(fact_369_ap_Ocases,axiom,
! [A2: relational_fmla_a_b] :
( ( relational_ap_a_b @ A2 )
=> ( ! [P3: b,Ts: list_R6823256787227418703term_a] :
( A2
!= ( relational_Pred_b_a @ P3 @ Ts ) )
=> ~ ! [X3: nat,C3: a] :
( A2
!= ( relational_Eq_a_b @ X3 @ ( relational_Const_a @ C3 ) ) ) ) ) ).
% ap.cases
thf(fact_370_ap_Osimps,axiom,
( relational_ap_a_b
= ( ^ [A4: relational_fmla_a_b] :
( ? [P5: b,Ts2: list_R6823256787227418703term_a] :
( A4
= ( relational_Pred_b_a @ P5 @ Ts2 ) )
| ? [X2: nat,C4: a] :
( A4
= ( relational_Eq_a_b @ X2 @ ( relational_Const_a @ C4 ) ) ) ) ) ) ).
% ap.simps
thf(fact_371_cpropagated__simps_I1_J,axiom,
! [B2: $o] : ( relati1591879772219623554ed_a_b @ ( relational_Bool_a_b @ B2 ) ) ).
% cpropagated_simps(1)
thf(fact_372_cpropagated__simps_I2_J,axiom,
! [P4: b,Ts3: list_R6823256787227418703term_a] : ( relati1591879772219623554ed_a_b @ ( relational_Pred_b_a @ P4 @ Ts3 ) ) ).
% cpropagated_simps(2)
thf(fact_373_cpropagated__simps_I4_J,axiom,
! [Q: relational_fmla_a_b] :
( ( relati1591879772219623554ed_a_b @ ( relational_Neg_a_b @ Q ) )
= ( relational_nocp_a_b @ Q ) ) ).
% cpropagated_simps(4)
thf(fact_374_cpropagated__simps_I5_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( relati1591879772219623554ed_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) )
= ( ( relational_nocp_a_b @ Q1 )
& ( relational_nocp_a_b @ Q2 ) ) ) ).
% cpropagated_simps(5)
thf(fact_375_cpropagated__simps_I6_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( relati1591879772219623554ed_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) )
= ( ( relational_nocp_a_b @ Q1 )
& ( relational_nocp_a_b @ Q2 ) ) ) ).
% cpropagated_simps(6)
thf(fact_376_cpropagated__simps_I7_J,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ( relati1591879772219623554ed_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) )
= ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
& ( relational_nocp_a_b @ Q ) ) ) ).
% cpropagated_simps(7)
thf(fact_377_cpropagated__nocp,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relati1591879772219623554ed_a_b @ Q )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( relational_nocp_a_b @ Q ) ) ) ).
% cpropagated_nocp
thf(fact_378_nocp__cpropagated,axiom,
! [Q: relational_fmla_a_b] :
( ( relational_nocp_a_b @ Q )
=> ( relati1591879772219623554ed_a_b @ Q ) ) ).
% nocp_cpropagated
thf(fact_379_nocp_Osimps_I7_J,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ( relational_nocp_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) )
= ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
& ( relational_nocp_a_b @ Q ) ) ) ).
% nocp.simps(7)
thf(fact_380_fv_Osimps_I4_J,axiom,
! [Phi2: relational_fmla_a_b] :
( ( relational_fv_a_b @ ( relational_Neg_a_b @ Phi2 ) )
= ( relational_fv_a_b @ Phi2 ) ) ).
% fv.simps(4)
thf(fact_381_fv_Osimps_I5_J,axiom,
! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
( ( relational_fv_a_b @ ( relational_Conj_a_b @ Phi2 @ Psi2 ) )
= ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( relational_fv_a_b @ Psi2 ) ) ) ).
% fv.simps(5)
thf(fact_382_fv_Osimps_I6_J,axiom,
! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
( ( relational_fv_a_b @ ( relational_Disj_a_b @ Phi2 @ Psi2 ) )
= ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( relational_fv_a_b @ Psi2 ) ) ) ).
% fv.simps(6)
thf(fact_383_nocp_Osimps_I4_J,axiom,
! [Q: relational_fmla_a_b] :
( ( relational_nocp_a_b @ ( relational_Neg_a_b @ Q ) )
= ( relational_nocp_a_b @ Q ) ) ).
% nocp.simps(4)
thf(fact_384_nocp_Osimps_I5_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( relational_nocp_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) )
= ( ( relational_nocp_a_b @ Q1 )
& ( relational_nocp_a_b @ Q2 ) ) ) ).
% nocp.simps(5)
thf(fact_385_nocp_Osimps_I6_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( relational_nocp_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) )
= ( ( relational_nocp_a_b @ Q1 )
& ( relational_nocp_a_b @ Q2 ) ) ) ).
% nocp.simps(6)
thf(fact_386_nocp_Osimps_I1_J,axiom,
! [B2: $o] :
~ ( relational_nocp_a_b @ ( relational_Bool_a_b @ B2 ) ) ).
% nocp.simps(1)
thf(fact_387_nocp_Osimps_I2_J,axiom,
! [P4: b,Ts3: list_R6823256787227418703term_a] : ( relational_nocp_a_b @ ( relational_Pred_b_a @ P4 @ Ts3 ) ) ).
% nocp.simps(2)
thf(fact_388_nocp__cp__triv,axiom,
! [Q: relational_fmla_a_b] :
( ( relational_nocp_a_b @ Q )
=> ( ( relational_cp_a_b @ Q )
= Q ) ) ).
% nocp_cp_triv
thf(fact_389_Pred,axiom,
! [P4: b,Ts3: list_R6823256787227418703term_a] : ( relational_ap_a_b @ ( relational_Pred_b_a @ P4 @ Ts3 ) ) ).
% Pred
thf(fact_390_gen__impl_Opsimps_I6_J,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X @ ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q ) ) ) )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q ) ) )
= ( restri4663030114664890836pl_a_b @ X @ Q ) ) ) ).
% gen_impl.psimps(6)
thf(fact_391_ap__cp,axiom,
! [Q: relational_fmla_a_b] :
( ( relational_ap_a_b @ Q )
=> ( relational_ap_a_b @ ( relational_cp_a_b @ Q ) ) ) ).
% ap_cp
thf(fact_392_ap__cp__triv,axiom,
! [Q: relational_fmla_a_b] :
( ( relational_ap_a_b @ Q )
=> ( ( relational_cp_a_b @ Q )
= Q ) ) ).
% ap_cp_triv
thf(fact_393_cpropagated__cp,axiom,
! [Q: relational_fmla_a_b] : ( relati1591879772219623554ed_a_b @ ( relational_cp_a_b @ Q ) ) ).
% cpropagated_cp
thf(fact_394_cpropagated__cp__triv,axiom,
! [Q: relational_fmla_a_b] :
( ( relati1591879772219623554ed_a_b @ Q )
=> ( ( relational_cp_a_b @ Q )
= Q ) ) ).
% cpropagated_cp_triv
thf(fact_395_gen__impl_Opsimps_I2_J,axiom,
! [X: nat] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X @ ( relational_Bool_a_b @ $true ) ) )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relational_Bool_a_b @ $true ) )
= nil_se6596116450819887951la_a_b ) ) ).
% gen_impl.psimps(2)
thf(fact_396_gen__impl_Opsimps_I19_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,V: relational_fmla_a_b,Va: relational_fmla_a_b] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X @ ( relational_Conj_a_b @ Q1 @ ( relational_Conj_a_b @ V @ Va ) ) ) )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relational_Conj_a_b @ Q1 @ ( relational_Conj_a_b @ V @ Va ) ) )
= ( union_7072582408929332827la_a_b @ ( restri4663030114664890836pl_a_b @ X @ Q1 ) @ ( restri4663030114664890836pl_a_b @ X @ ( relational_Conj_a_b @ V @ Va ) ) ) ) ) ).
% gen_impl.psimps(19)
thf(fact_397_nocp_Osimps_I3_J,axiom,
! [X: nat,T3: relational_term_a] :
( ( relational_nocp_a_b @ ( relational_Eq_a_b @ X @ T3 ) )
= ( T3
!= ( relational_Var_a @ X ) ) ) ).
% nocp.simps(3)
thf(fact_398_gen__impl_Opsimps_I7_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) ) ) )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) ) )
= ( restri4663030114664890836pl_a_b @ X @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q2 ) ) ) ) ) ).
% gen_impl.psimps(7)
thf(fact_399_gen__impl_Opsimps_I8_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) ) ) )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) ) )
= ( restri4663030114664890836pl_a_b @ X @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q2 ) ) ) ) ) ).
% gen_impl.psimps(8)
thf(fact_400_gen__impl_Opsimps_I11_J,axiom,
! [X: nat,V: nat,Va: relational_term_a] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X @ ( relational_Neg_a_b @ ( relational_Eq_a_b @ V @ Va ) ) ) )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relational_Neg_a_b @ ( relational_Eq_a_b @ V @ Va ) ) )
= nil_se6596116450819887951la_a_b ) ) ).
% gen_impl.psimps(11)
thf(fact_401_gen__impl_Opsimps_I10_J,axiom,
! [X: nat,V: $o] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X @ ( relational_Neg_a_b @ ( relational_Bool_a_b @ V ) ) ) )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relational_Neg_a_b @ ( relational_Bool_a_b @ V ) ) )
= nil_se6596116450819887951la_a_b ) ) ).
% gen_impl.psimps(10)
thf(fact_402_gen__impl_Opsimps_I12_J,axiom,
! [X: nat,V: nat,Va: relational_fmla_a_b] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X @ ( relational_Neg_a_b @ ( relati591517084277583526ts_a_b @ V @ Va ) ) ) )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relational_Neg_a_b @ ( relati591517084277583526ts_a_b @ V @ Va ) ) )
= nil_se6596116450819887951la_a_b ) ) ).
% gen_impl.psimps(12)
thf(fact_403_gen__impl_Opsimps_I9_J,axiom,
! [X: nat,V: b,Va: list_R6823256787227418703term_a] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X @ ( relational_Neg_a_b @ ( relational_Pred_b_a @ V @ Va ) ) ) )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relational_Neg_a_b @ ( relational_Pred_b_a @ V @ Va ) ) )
= nil_se6596116450819887951la_a_b ) ) ).
% gen_impl.psimps(9)
thf(fact_404_gen__impl_Opsimps_I18_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,V: relational_fmla_a_b] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X @ ( relational_Conj_a_b @ Q1 @ ( relational_Neg_a_b @ V ) ) ) )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relational_Conj_a_b @ Q1 @ ( relational_Neg_a_b @ V ) ) )
= ( union_7072582408929332827la_a_b @ ( restri4663030114664890836pl_a_b @ X @ Q1 ) @ ( restri4663030114664890836pl_a_b @ X @ ( relational_Neg_a_b @ V ) ) ) ) ) ).
% gen_impl.psimps(18)
thf(fact_405_gen__impl_Opsimps_I4_J,axiom,
! [X: nat,Y3: nat,Z2: nat] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ Z2 ) ) ) )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ Z2 ) ) )
= nil_se6596116450819887951la_a_b ) ) ).
% gen_impl.psimps(4)
thf(fact_406_gen__impl_Opsimps_I20_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,V: relational_fmla_a_b,Va: relational_fmla_a_b] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X @ ( relational_Conj_a_b @ Q1 @ ( relational_Disj_a_b @ V @ Va ) ) ) )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relational_Conj_a_b @ Q1 @ ( relational_Disj_a_b @ V @ Va ) ) )
= ( union_7072582408929332827la_a_b @ ( restri4663030114664890836pl_a_b @ X @ Q1 ) @ ( restri4663030114664890836pl_a_b @ X @ ( relational_Disj_a_b @ V @ Va ) ) ) ) ) ).
% gen_impl.psimps(20)
thf(fact_407_gen__impl_Opsimps_I16_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,V: $o] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X @ ( relational_Conj_a_b @ Q1 @ ( relational_Bool_a_b @ V ) ) ) )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relational_Conj_a_b @ Q1 @ ( relational_Bool_a_b @ V ) ) )
= ( union_7072582408929332827la_a_b @ ( restri4663030114664890836pl_a_b @ X @ Q1 ) @ ( restri4663030114664890836pl_a_b @ X @ ( relational_Bool_a_b @ V ) ) ) ) ) ).
% gen_impl.psimps(16)
thf(fact_408_gen__impl_Opsimps_I21_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,V: nat,Va: relational_fmla_a_b] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X @ ( relational_Conj_a_b @ Q1 @ ( relati591517084277583526ts_a_b @ V @ Va ) ) ) )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relational_Conj_a_b @ Q1 @ ( relati591517084277583526ts_a_b @ V @ Va ) ) )
= ( union_7072582408929332827la_a_b @ ( restri4663030114664890836pl_a_b @ X @ Q1 ) @ ( restri4663030114664890836pl_a_b @ X @ ( relati591517084277583526ts_a_b @ V @ Va ) ) ) ) ) ).
% gen_impl.psimps(21)
thf(fact_409_gen__impl_Opsimps_I15_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,V: b,Va: list_R6823256787227418703term_a] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X @ ( relational_Conj_a_b @ Q1 @ ( relational_Pred_b_a @ V @ Va ) ) ) )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relational_Conj_a_b @ Q1 @ ( relational_Pred_b_a @ V @ Va ) ) )
= ( union_7072582408929332827la_a_b @ ( restri4663030114664890836pl_a_b @ X @ Q1 ) @ ( restri4663030114664890836pl_a_b @ X @ ( relational_Pred_b_a @ V @ Va ) ) ) ) ) ).
% gen_impl.psimps(15)
thf(fact_410_Eqc,axiom,
! [X: nat,C: a] : ( relational_ap_a_b @ ( relational_Eq_a_b @ X @ ( relational_Const_a @ C ) ) ) ).
% Eqc
thf(fact_411_gen__impl_Opsimps_I17_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,V: nat,Vb: a] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X @ ( relational_Conj_a_b @ Q1 @ ( relational_Eq_a_b @ V @ ( relational_Const_a @ Vb ) ) ) ) )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relational_Conj_a_b @ Q1 @ ( relational_Eq_a_b @ V @ ( relational_Const_a @ Vb ) ) ) )
= ( union_7072582408929332827la_a_b @ ( restri4663030114664890836pl_a_b @ X @ Q1 ) @ ( restri4663030114664890836pl_a_b @ X @ ( relational_Eq_a_b @ V @ ( relational_Const_a @ Vb ) ) ) ) ) ) ).
% gen_impl.psimps(17)
thf(fact_412_nocp_Oelims_I3_J,axiom,
! [X: relational_fmla_a_b] :
( ~ ( relational_nocp_a_b @ X )
=> ( ! [B3: $o] :
( X
!= ( relational_Bool_a_b @ B3 ) )
=> ( ! [X3: nat,T: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X3 @ T ) )
=> ( T
!= ( relational_Var_a @ X3 ) ) )
=> ( ! [Q3: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q3 ) )
=> ( relational_nocp_a_b @ Q3 ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q12 @ Q22 ) )
=> ( ( relational_nocp_a_b @ Q12 )
& ( relational_nocp_a_b @ Q22 ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q12 @ Q22 ) )
=> ( ( relational_nocp_a_b @ Q12 )
& ( relational_nocp_a_b @ Q22 ) ) )
=> ~ ! [X3: nat,Q3: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X3 @ Q3 ) )
=> ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q3 ) )
& ( relational_nocp_a_b @ Q3 ) ) ) ) ) ) ) ) ) ).
% nocp.elims(3)
thf(fact_413_nocp_Oelims_I2_J,axiom,
! [X: relational_fmla_a_b] :
( ( relational_nocp_a_b @ X )
=> ( ! [P3: b,Ts: list_R6823256787227418703term_a] :
( X
!= ( relational_Pred_b_a @ P3 @ Ts ) )
=> ( ! [X3: nat,T: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X3 @ T ) )
=> ( T
= ( relational_Var_a @ X3 ) ) )
=> ( ! [Q3: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q3 ) )
=> ~ ( relational_nocp_a_b @ Q3 ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q12 @ Q22 ) )
=> ~ ( ( relational_nocp_a_b @ Q12 )
& ( relational_nocp_a_b @ Q22 ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q12 @ Q22 ) )
=> ~ ( ( relational_nocp_a_b @ Q12 )
& ( relational_nocp_a_b @ Q22 ) ) )
=> ~ ! [X3: nat,Q3: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X3 @ Q3 ) )
=> ~ ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q3 ) )
& ( relational_nocp_a_b @ Q3 ) ) ) ) ) ) ) ) ) ).
% nocp.elims(2)
thf(fact_414_fv__cp__foldr1__Disj,axiom,
! [Qs: list_R8263082107343818799la_a_b,A: set_nat] :
( ( ( relational_cp_a_b @ ( relati1724936043867567613la_a_b @ relational_Disj_a_b @ Qs @ ( relational_Bool_a_b @ $false ) ) )
!= ( relational_Bool_a_b @ $false ) )
=> ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( set_Re9104216502384355786la_a_b @ Qs ) )
=> ( ( relati1591879772219623554ed_a_b @ X3 )
& ( ( relational_fv_a_b @ X3 )
= A ) ) )
=> ( ( relational_fv_a_b @ ( relational_cp_a_b @ ( relati1724936043867567613la_a_b @ relational_Disj_a_b @ Qs @ ( relational_Bool_a_b @ $false ) ) ) )
= A ) ) ) ).
% fv_cp_foldr1_Disj
thf(fact_415_qp__impl_Oelims_I1_J,axiom,
! [X: relational_fmla_a_b,Y3: $o] :
( ( ( relati3725921752842749053pl_a_b @ X )
= Y3 )
=> ( ( ? [X3: nat,C3: a] :
( X
= ( relational_Eq_a_b @ X3 @ ( relational_Const_a @ C3 ) ) )
=> ~ Y3 )
=> ( ( ? [X3: b,Ts: list_R6823256787227418703term_a] :
( X
= ( relational_Pred_b_a @ X3 @ Ts ) )
=> ~ Y3 )
=> ( ! [X3: nat,Q3: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X3 @ Q3 ) )
=> ( Y3
= ( ~ ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q3 ) )
& ( relational_qp_a_b @ Q3 ) ) ) ) )
=> ( ( ? [V2: $o] :
( X
= ( relational_Bool_a_b @ V2 ) )
=> Y3 )
=> ( ( ? [V2: nat,Vb2: nat] :
( X
= ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb2 ) ) )
=> Y3 )
=> ( ( ? [V2: relational_fmla_a_b] :
( X
= ( relational_Neg_a_b @ V2 ) )
=> Y3 )
=> ( ( ? [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( X
= ( relational_Conj_a_b @ V2 @ Va2 ) )
=> Y3 )
=> ~ ( ? [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( X
= ( relational_Disj_a_b @ V2 @ Va2 ) )
=> Y3 ) ) ) ) ) ) ) ) ) ).
% qp_impl.elims(1)
thf(fact_416_sr__Conj__eq,axiom,
! [Q: relational_fmla_a_b,X: nat,Y3: nat] :
( ( relational_sr_a_b @ Q )
=> ( ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
| ( member_nat @ Y3 @ ( relational_fv_a_b @ Q ) ) )
=> ( relational_sr_a_b @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y3 ) ) ) ) ) ) ).
% sr_Conj_eq
thf(fact_417_nocp_Opelims_I1_J,axiom,
! [X: relational_fmla_a_b,Y3: $o] :
( ( ( relational_nocp_a_b @ X )
= Y3 )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ X )
=> ( ! [B3: $o] :
( ( X
= ( relational_Bool_a_b @ B3 ) )
=> ( ~ Y3
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Bool_a_b @ B3 ) ) ) )
=> ( ! [P3: b,Ts: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ P3 @ Ts ) )
=> ( Y3
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Pred_b_a @ P3 @ Ts ) ) ) )
=> ( ! [X3: nat,T: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X3 @ T ) )
=> ( ( Y3
= ( T
!= ( relational_Var_a @ X3 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Eq_a_b @ X3 @ T ) ) ) )
=> ( ! [Q3: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q3 ) )
=> ( ( Y3
= ( relational_nocp_a_b @ Q3 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Neg_a_b @ Q3 ) ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q12 @ Q22 ) )
=> ( ( Y3
= ( ( relational_nocp_a_b @ Q12 )
& ( relational_nocp_a_b @ Q22 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Conj_a_b @ Q12 @ Q22 ) ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q12 @ Q22 ) )
=> ( ( Y3
= ( ( relational_nocp_a_b @ Q12 )
& ( relational_nocp_a_b @ Q22 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Disj_a_b @ Q12 @ Q22 ) ) ) )
=> ~ ! [X3: nat,Q3: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X3 @ Q3 ) )
=> ( ( Y3
= ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q3 ) )
& ( relational_nocp_a_b @ Q3 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relati591517084277583526ts_a_b @ X3 @ Q3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% nocp.pelims(1)
thf(fact_418_qp__impl_Oelims_I3_J,axiom,
! [X: relational_fmla_a_b] :
( ~ ( relati3725921752842749053pl_a_b @ X )
=> ( ! [X3: nat,Q3: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X3 @ Q3 ) )
=> ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q3 ) )
& ( relational_qp_a_b @ Q3 ) ) )
=> ( ! [V2: $o] :
( X
!= ( relational_Bool_a_b @ V2 ) )
=> ( ! [V2: nat,Vb2: nat] :
( X
!= ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb2 ) ) )
=> ( ! [V2: relational_fmla_a_b] :
( X
!= ( relational_Neg_a_b @ V2 ) )
=> ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( X
!= ( relational_Conj_a_b @ V2 @ Va2 ) )
=> ~ ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( X
!= ( relational_Disj_a_b @ V2 @ Va2 ) ) ) ) ) ) ) ) ).
% qp_impl.elims(3)
thf(fact_419_nocp_Opelims_I2_J,axiom,
! [X: relational_fmla_a_b] :
( ( relational_nocp_a_b @ X )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ X )
=> ( ! [P3: b,Ts: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ P3 @ Ts ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Pred_b_a @ P3 @ Ts ) ) )
=> ( ! [X3: nat,T: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X3 @ T ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Eq_a_b @ X3 @ T ) )
=> ( T
= ( relational_Var_a @ X3 ) ) ) )
=> ( ! [Q3: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q3 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Neg_a_b @ Q3 ) )
=> ~ ( relational_nocp_a_b @ Q3 ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q12 @ Q22 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Conj_a_b @ Q12 @ Q22 ) )
=> ~ ( ( relational_nocp_a_b @ Q12 )
& ( relational_nocp_a_b @ Q22 ) ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q12 @ Q22 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Disj_a_b @ Q12 @ Q22 ) )
=> ~ ( ( relational_nocp_a_b @ Q12 )
& ( relational_nocp_a_b @ Q22 ) ) ) )
=> ~ ! [X3: nat,Q3: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X3 @ Q3 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relati591517084277583526ts_a_b @ X3 @ Q3 ) )
=> ~ ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q3 ) )
& ( relational_nocp_a_b @ Q3 ) ) ) ) ) ) ) ) ) ) ) ).
% nocp.pelims(2)
thf(fact_420_nocp_Opelims_I3_J,axiom,
! [X: relational_fmla_a_b] :
( ~ ( relational_nocp_a_b @ X )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ X )
=> ( ! [B3: $o] :
( ( X
= ( relational_Bool_a_b @ B3 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Bool_a_b @ B3 ) ) )
=> ( ! [X3: nat,T: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X3 @ T ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Eq_a_b @ X3 @ T ) )
=> ( T
!= ( relational_Var_a @ X3 ) ) ) )
=> ( ! [Q3: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q3 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Neg_a_b @ Q3 ) )
=> ( relational_nocp_a_b @ Q3 ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q12 @ Q22 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Conj_a_b @ Q12 @ Q22 ) )
=> ( ( relational_nocp_a_b @ Q12 )
& ( relational_nocp_a_b @ Q22 ) ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q12 @ Q22 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Disj_a_b @ Q12 @ Q22 ) )
=> ( ( relational_nocp_a_b @ Q12 )
& ( relational_nocp_a_b @ Q22 ) ) ) )
=> ~ ! [X3: nat,Q3: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X3 @ Q3 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relati591517084277583526ts_a_b @ X3 @ Q3 ) )
=> ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q3 ) )
& ( relational_nocp_a_b @ Q3 ) ) ) ) ) ) ) ) ) ) ) ).
% nocp.pelims(3)
thf(fact_421_qp__impl__imp__qp,axiom,
! [Q: relational_fmla_a_b] :
( ( relati3725921752842749053pl_a_b @ Q )
=> ( relational_qp_a_b @ Q ) ) ).
% qp_impl_imp_qp
thf(fact_422_qp__imp__qp__impl,axiom,
! [Q: relational_fmla_a_b] :
( ( relational_qp_a_b @ Q )
=> ( relati3725921752842749053pl_a_b @ Q ) ) ).
% qp_imp_qp_impl
thf(fact_423_qp__code,axiom,
relational_qp_a_b = relati3725921752842749053pl_a_b ).
% qp_code
thf(fact_424_qp__Neg,axiom,
! [Q: relational_fmla_a_b] :
~ ( relational_qp_a_b @ ( relational_Neg_a_b @ Q ) ) ).
% qp_Neg
thf(fact_425_qp__Conj,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
~ ( relational_qp_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) ) ).
% qp_Conj
thf(fact_426_qp__Disj,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
~ ( relational_qp_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) ) ).
% qp_Disj
thf(fact_427_qp__impl_Osimps_I3_J,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ( relati3725921752842749053pl_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) )
= ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
& ( relational_qp_a_b @ Q ) ) ) ).
% qp_impl.simps(3)
thf(fact_428_qp__cp__triv,axiom,
! [Q: relational_fmla_a_b] :
( ( relational_qp_a_b @ Q )
=> ( ( relational_cp_a_b @ Q )
= Q ) ) ).
% qp_cp_triv
thf(fact_429_qp__cp,axiom,
! [Q: relational_fmla_a_b] :
( ( relational_qp_a_b @ Q )
=> ( relational_qp_a_b @ ( relational_cp_a_b @ Q ) ) ) ).
% qp_cp
thf(fact_430_gen__qp,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
( ( relational_gen_a_b @ X @ Q @ G )
=> ( ( member4680049679412964150la_a_b @ Qqp @ G )
=> ( relational_qp_a_b @ Qqp ) ) ) ).
% gen_qp
thf(fact_431_sr__foldr1__Disj,axiom,
! [Qs: list_R8263082107343818799la_a_b,X5: set_nat,Q: relational_fmla_a_b] :
( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( set_Re9104216502384355786la_a_b @ Qs ) )
=> ( ( relational_fv_a_b @ X3 )
= X5 ) )
=> ( ( relational_sr_a_b @ ( relati1724936043867567613la_a_b @ relational_Disj_a_b @ Qs @ Q ) )
= ( ( ( Qs = nil_Re6358386334527539737la_a_b )
=> ( relational_sr_a_b @ Q ) )
& ( ( Qs != nil_Re6358386334527539737la_a_b )
=> ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ ( set_Re9104216502384355786la_a_b @ Qs ) )
=> ( relational_sr_a_b @ X2 ) ) ) ) ) ) ).
% sr_foldr1_Disj
thf(fact_432_qp_Oap,axiom,
! [Q: relational_fmla_a_b] :
( ( relational_ap_a_b @ Q )
=> ( relational_qp_a_b @ Q ) ) ).
% qp.ap
thf(fact_433_sr__False,axiom,
relational_sr_a_b @ ( relational_Bool_a_b @ $false ) ).
% sr_False
thf(fact_434_qp__impl_Osimps_I6_J,axiom,
! [V: relational_fmla_a_b] :
~ ( relati3725921752842749053pl_a_b @ ( relational_Neg_a_b @ V ) ) ).
% qp_impl.simps(6)
thf(fact_435_sr__cp,axiom,
! [Q: relational_fmla_a_b] :
( ( relational_sr_a_b @ Q )
=> ( relational_sr_a_b @ ( relational_cp_a_b @ Q ) ) ) ).
% sr_cp
thf(fact_436_qp__impl_Osimps_I7_J,axiom,
! [V: relational_fmla_a_b,Va: relational_fmla_a_b] :
~ ( relati3725921752842749053pl_a_b @ ( relational_Conj_a_b @ V @ Va ) ) ).
% qp_impl.simps(7)
thf(fact_437_qp__impl_Osimps_I8_J,axiom,
! [V: relational_fmla_a_b,Va: relational_fmla_a_b] :
~ ( relati3725921752842749053pl_a_b @ ( relational_Disj_a_b @ V @ Va ) ) ).
% qp_impl.simps(8)
thf(fact_438_qp__impl_Osimps_I4_J,axiom,
! [V: $o] :
~ ( relati3725921752842749053pl_a_b @ ( relational_Bool_a_b @ V ) ) ).
% qp_impl.simps(4)
thf(fact_439_qp__impl_Osimps_I2_J,axiom,
! [X: b,Ts3: list_R6823256787227418703term_a] : ( relati3725921752842749053pl_a_b @ ( relational_Pred_b_a @ X @ Ts3 ) ) ).
% qp_impl.simps(2)
thf(fact_440_qp__eq,axiom,
! [X: nat,Y3: nat] :
~ ( relational_qp_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y3 ) ) ) ).
% qp_eq
thf(fact_441_qp__Exists,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relational_qp_a_b @ Q )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( relational_qp_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) ) ) ) ).
% qp_Exists
thf(fact_442_qp__ExistsE,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ( relational_qp_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) )
=> ~ ( ( relational_qp_a_b @ Q )
=> ~ ( member_nat @ X @ ( relational_fv_a_b @ Q ) ) ) ) ).
% qp_ExistsE
thf(fact_443_qp__Gen,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relational_qp_a_b @ Q )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ? [X_12: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ Q @ X_12 ) ) ) ).
% qp_Gen
thf(fact_444_foldr1_Osimps_I2_J,axiom,
! [F2: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,Z2: set_Re381260168593705685la_a_b] :
( ( relati7835315018957680435la_a_b @ F2 @ nil_se6596116450819887951la_a_b @ Z2 )
= Z2 ) ).
% foldr1.simps(2)
thf(fact_445_foldr1_Osimps_I2_J,axiom,
! [F2: list_nat > list_nat > list_nat,Z2: list_nat] :
( ( relati7377577593470194238st_nat @ F2 @ nil_list_nat @ Z2 )
= Z2 ) ).
% foldr1.simps(2)
thf(fact_446_foldr1_Osimps_I2_J,axiom,
! [F2: set_nat > set_nat > set_nat,Z2: set_nat] :
( ( relati7114214420354183652et_nat @ F2 @ nil_set_nat @ Z2 )
= Z2 ) ).
% foldr1.simps(2)
thf(fact_447_foldr1_Osimps_I2_J,axiom,
! [F2: nat > nat > nat,Z2: nat] :
( ( relati2262546126839610926r1_nat @ F2 @ nil_nat @ Z2 )
= Z2 ) ).
% foldr1.simps(2)
thf(fact_448_sr__Disj,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( ( relational_fv_a_b @ Q1 )
= ( relational_fv_a_b @ Q2 ) )
=> ( ( relational_sr_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) )
= ( ( relational_sr_a_b @ Q1 )
& ( relational_sr_a_b @ Q2 ) ) ) ) ).
% sr_Disj
thf(fact_449_qp__impl_Oelims_I2_J,axiom,
! [X: relational_fmla_a_b] :
( ( relati3725921752842749053pl_a_b @ X )
=> ( ! [X3: nat,C3: a] :
( X
!= ( relational_Eq_a_b @ X3 @ ( relational_Const_a @ C3 ) ) )
=> ( ! [X3: b,Ts: list_R6823256787227418703term_a] :
( X
!= ( relational_Pred_b_a @ X3 @ Ts ) )
=> ~ ! [X3: nat,Q3: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X3 @ Q3 ) )
=> ~ ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q3 ) )
& ( relational_qp_a_b @ Q3 ) ) ) ) ) ) ).
% qp_impl.elims(2)
thf(fact_450_qp__impl_Osimps_I5_J,axiom,
! [V: nat,Vb: nat] :
~ ( relati3725921752842749053pl_a_b @ ( relational_Eq_a_b @ V @ ( relational_Var_a @ Vb ) ) ) ).
% qp_impl.simps(5)
thf(fact_451_qp__impl_Osimps_I1_J,axiom,
! [X: nat,C: a] : ( relati3725921752842749053pl_a_b @ ( relational_Eq_a_b @ X @ ( relational_Const_a @ C ) ) ) ).
% qp_impl.simps(1)
thf(fact_452_qp__impl_Opelims_I1_J,axiom,
! [X: relational_fmla_a_b,Y3: $o] :
( ( ( relati3725921752842749053pl_a_b @ X )
= Y3 )
=> ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ X )
=> ( ! [X3: nat,C3: a] :
( ( X
= ( relational_Eq_a_b @ X3 @ ( relational_Const_a @ C3 ) ) )
=> ( Y3
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Eq_a_b @ X3 @ ( relational_Const_a @ C3 ) ) ) ) )
=> ( ! [X3: b,Ts: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ X3 @ Ts ) )
=> ( Y3
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Pred_b_a @ X3 @ Ts ) ) ) )
=> ( ! [X3: nat,Q3: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X3 @ Q3 ) )
=> ( ( Y3
= ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q3 ) )
& ( relational_qp_a_b @ Q3 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relati591517084277583526ts_a_b @ X3 @ Q3 ) ) ) )
=> ( ! [V2: $o] :
( ( X
= ( relational_Bool_a_b @ V2 ) )
=> ( ~ Y3
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Bool_a_b @ V2 ) ) ) )
=> ( ! [V2: nat,Vb2: nat] :
( ( X
= ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb2 ) ) )
=> ( ~ Y3
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb2 ) ) ) ) )
=> ( ! [V2: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ V2 ) )
=> ( ~ Y3
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Neg_a_b @ V2 ) ) ) )
=> ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ V2 @ Va2 ) )
=> ( ~ Y3
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Conj_a_b @ V2 @ Va2 ) ) ) )
=> ~ ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ V2 @ Va2 ) )
=> ( ~ Y3
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Disj_a_b @ V2 @ Va2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% qp_impl.pelims(1)
thf(fact_453_qp__impl_Opelims_I3_J,axiom,
! [X: relational_fmla_a_b] :
( ~ ( relati3725921752842749053pl_a_b @ X )
=> ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ X )
=> ( ! [X3: nat,Q3: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X3 @ Q3 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relati591517084277583526ts_a_b @ X3 @ Q3 ) )
=> ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q3 ) )
& ( relational_qp_a_b @ Q3 ) ) ) )
=> ( ! [V2: $o] :
( ( X
= ( relational_Bool_a_b @ V2 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Bool_a_b @ V2 ) ) )
=> ( ! [V2: nat,Vb2: nat] :
( ( X
= ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb2 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb2 ) ) ) )
=> ( ! [V2: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ V2 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Neg_a_b @ V2 ) ) )
=> ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ V2 @ Va2 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Conj_a_b @ V2 @ Va2 ) ) )
=> ~ ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ V2 @ Va2 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Disj_a_b @ V2 @ Va2 ) ) ) ) ) ) ) ) ) ) ).
% qp_impl.pelims(3)
thf(fact_454_qp__impl_Opelims_I2_J,axiom,
! [X: relational_fmla_a_b] :
( ( relati3725921752842749053pl_a_b @ X )
=> ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ X )
=> ( ! [X3: nat,C3: a] :
( ( X
= ( relational_Eq_a_b @ X3 @ ( relational_Const_a @ C3 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Eq_a_b @ X3 @ ( relational_Const_a @ C3 ) ) ) )
=> ( ! [X3: b,Ts: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ X3 @ Ts ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Pred_b_a @ X3 @ Ts ) ) )
=> ~ ! [X3: nat,Q3: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X3 @ Q3 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relati591517084277583526ts_a_b @ X3 @ Q3 ) )
=> ~ ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q3 ) )
& ( relational_qp_a_b @ Q3 ) ) ) ) ) ) ) ) ).
% qp_impl.pelims(2)
thf(fact_455_Exists__in__sub__cp__foldr1__Disj,axiom,
! [X: nat,Q5: relational_fmla_a_b,Qs: list_R8263082107343818799la_a_b,Q: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X @ Q5 ) @ ( relational_sub_a_b @ ( relational_cp_a_b @ ( relati1724936043867567613la_a_b @ relational_Disj_a_b @ Qs @ Q ) ) ) )
=> ( ( ( Qs = nil_Re6358386334527539737la_a_b )
& ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X @ Q5 ) @ ( relational_sub_a_b @ ( relational_cp_a_b @ Q ) ) ) )
| ? [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( set_Re9104216502384355786la_a_b @ Qs ) )
& ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X @ Q5 ) @ ( relational_sub_a_b @ ( relational_cp_a_b @ X3 ) ) ) ) ) ) ).
% Exists_in_sub_cp_foldr1_Disj
thf(fact_456_Exists__in__sub__foldr1__Disj,axiom,
! [X: nat,Q5: relational_fmla_a_b,Qs: list_R8263082107343818799la_a_b,Q: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X @ Q5 ) @ ( relational_sub_a_b @ ( relati1724936043867567613la_a_b @ relational_Disj_a_b @ Qs @ Q ) ) )
=> ( ( ( Qs = nil_Re6358386334527539737la_a_b )
& ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X @ Q5 ) @ ( relational_sub_a_b @ Q ) ) )
| ? [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( set_Re9104216502384355786la_a_b @ Qs ) )
& ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X @ Q5 ) @ ( relational_sub_a_b @ X3 ) ) ) ) ) ).
% Exists_in_sub_foldr1_Disj
thf(fact_457_foldr1__Disj__equiv,axiom,
! [Qs: list_R8263082107343818799la_a_b,Qs2: list_R8263082107343818799la_a_b] :
( ( ( set_Re9104216502384355786la_a_b @ Qs )
= ( set_Re9104216502384355786la_a_b @ Qs2 ) )
=> ( relational_equiv_a_b @ ( relati1724936043867567613la_a_b @ relational_Disj_a_b @ Qs @ ( relational_Bool_a_b @ $false ) ) @ ( relati1724936043867567613la_a_b @ relational_Disj_a_b @ Qs2 @ ( relational_Bool_a_b @ $false ) ) ) ) ).
% foldr1_Disj_equiv
thf(fact_458_sr__foldr__Disj,axiom,
! [Qs: list_R8263082107343818799la_a_b,Q: relational_fmla_a_b] :
( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( set_Re9104216502384355786la_a_b @ Qs ) )
=> ( ( relational_fv_a_b @ X3 )
= ( relational_fv_a_b @ Q ) ) )
=> ( ( relational_sr_a_b @ ( foldr_789212930732525799la_a_b @ relational_Disj_a_b @ Qs @ Q ) )
= ( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ ( set_Re9104216502384355786la_a_b @ Qs ) )
=> ( relational_sr_a_b @ X2 ) )
& ( relational_sr_a_b @ Q ) ) ) ) ).
% sr_foldr_Disj
thf(fact_459_equiv__refl,axiom,
! [Q: relational_fmla_a_b] : ( relational_equiv_a_b @ Q @ Q ) ).
% equiv_refl
thf(fact_460_equiv__sym,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( relational_equiv_a_b @ Q1 @ Q2 )
=> ( relational_equiv_a_b @ Q2 @ Q1 ) ) ).
% equiv_sym
thf(fact_461_equiv__trans,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b,Q32: relational_fmla_a_b] :
( ( relational_equiv_a_b @ Q1 @ Q2 )
=> ( ( relational_equiv_a_b @ Q2 @ Q32 )
=> ( relational_equiv_a_b @ Q1 @ Q32 ) ) ) ).
% equiv_trans
thf(fact_462_equiv__Neg__cong,axiom,
! [Q: relational_fmla_a_b,Q5: relational_fmla_a_b] :
( ( relational_equiv_a_b @ Q @ Q5 )
=> ( relational_equiv_a_b @ ( relational_Neg_a_b @ Q ) @ ( relational_Neg_a_b @ Q5 ) ) ) ).
% equiv_Neg_cong
thf(fact_463_equiv__Conj__cong,axiom,
! [Q1: relational_fmla_a_b,Q14: relational_fmla_a_b,Q2: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( relational_equiv_a_b @ Q1 @ Q14 )
=> ( ( relational_equiv_a_b @ Q2 @ Q24 )
=> ( relational_equiv_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ ( relational_Conj_a_b @ Q14 @ Q24 ) ) ) ) ).
% equiv_Conj_cong
thf(fact_464_equiv__Disj__cong,axiom,
! [Q1: relational_fmla_a_b,Q14: relational_fmla_a_b,Q2: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( relational_equiv_a_b @ Q1 @ Q14 )
=> ( ( relational_equiv_a_b @ Q2 @ Q24 )
=> ( relational_equiv_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) @ ( relational_Disj_a_b @ Q14 @ Q24 ) ) ) ) ).
% equiv_Disj_cong
thf(fact_465_equiv__Disj__Assoc,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b,Q32: relational_fmla_a_b] : ( relational_equiv_a_b @ ( relational_Disj_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) @ Q32 ) @ ( relational_Disj_a_b @ Q1 @ ( relational_Disj_a_b @ Q2 @ Q32 ) ) ) ).
% equiv_Disj_Assoc
thf(fact_466_equiv__Exists__cong,axiom,
! [Q: relational_fmla_a_b,Q5: relational_fmla_a_b,X: nat] :
( ( relational_equiv_a_b @ Q @ Q5 )
=> ( relational_equiv_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) @ ( relati591517084277583526ts_a_b @ X @ Q5 ) ) ) ).
% equiv_Exists_cong
thf(fact_467_equiv__cp,axiom,
! [Q: relational_fmla_a_b] : ( relational_equiv_a_b @ ( relational_cp_a_b @ Q ) @ Q ) ).
% equiv_cp
thf(fact_468_equiv__cp__cong,axiom,
! [Q: relational_fmla_a_b,Q5: relational_fmla_a_b] :
( ( relational_equiv_a_b @ Q @ Q5 )
=> ( relational_equiv_a_b @ ( relational_cp_a_b @ Q ) @ ( relational_cp_a_b @ Q5 ) ) ) ).
% equiv_cp_cong
thf(fact_469_foldr__cong,axiom,
! [A2: nat,B2: nat,L: list_nat,K: list_nat,F2: nat > nat > nat,G4: nat > nat > nat] :
( ( A2 = B2 )
=> ( ( L = K )
=> ( ! [A6: nat,X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ L ) )
=> ( ( F2 @ X3 @ A6 )
= ( G4 @ X3 @ A6 ) ) )
=> ( ( foldr_nat_nat @ F2 @ L @ A2 )
= ( foldr_nat_nat @ G4 @ K @ B2 ) ) ) ) ) ).
% foldr_cong
thf(fact_470_foldr__cong,axiom,
! [A2: list_nat,B2: list_nat,L: list_nat,K: list_nat,F2: nat > list_nat > list_nat,G4: nat > list_nat > list_nat] :
( ( A2 = B2 )
=> ( ( L = K )
=> ( ! [A6: list_nat,X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ L ) )
=> ( ( F2 @ X3 @ A6 )
= ( G4 @ X3 @ A6 ) ) )
=> ( ( foldr_nat_list_nat @ F2 @ L @ A2 )
= ( foldr_nat_list_nat @ G4 @ K @ B2 ) ) ) ) ) ).
% foldr_cong
thf(fact_471_cpropagated__sub,axiom,
! [Q: relational_fmla_a_b,Q5: relational_fmla_a_b] :
( ( relati1591879772219623554ed_a_b @ Q )
=> ( ( member4680049679412964150la_a_b @ Q5 @ ( relational_sub_a_b @ Q ) )
=> ( relati1591879772219623554ed_a_b @ Q5 ) ) ) ).
% cpropagated_sub
thf(fact_472_Exists__in__sub__foldr__Disj,axiom,
! [X: nat,Q5: relational_fmla_a_b,Qs: list_R8263082107343818799la_a_b,Q: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X @ Q5 ) @ ( relational_sub_a_b @ ( foldr_789212930732525799la_a_b @ relational_Disj_a_b @ Qs @ Q ) ) )
=> ( ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X @ Q5 ) @ ( relational_sub_a_b @ Q ) )
| ? [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( set_Re9104216502384355786la_a_b @ Qs ) )
& ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X @ Q5 ) @ ( relational_sub_a_b @ X3 ) ) ) ) ) ).
% Exists_in_sub_foldr_Disj
thf(fact_473_Exists__in__sub__cp__foldr__Disj,axiom,
! [X: nat,Q5: relational_fmla_a_b,Qs: list_R8263082107343818799la_a_b,Q: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X @ Q5 ) @ ( relational_sub_a_b @ ( relational_cp_a_b @ ( foldr_789212930732525799la_a_b @ relational_Disj_a_b @ Qs @ Q ) ) ) )
=> ( ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X @ Q5 ) @ ( relational_sub_a_b @ ( relational_cp_a_b @ Q ) ) )
| ? [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( set_Re9104216502384355786la_a_b @ Qs ) )
& ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X @ Q5 ) @ ( relational_sub_a_b @ ( relational_cp_a_b @ X3 ) ) ) ) ) ) ).
% Exists_in_sub_cp_foldr_Disj
thf(fact_474_equiv__Exists__Disj,axiom,
! [X: nat,Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] : ( relational_equiv_a_b @ ( relati591517084277583526ts_a_b @ X @ ( relational_Disj_a_b @ Q1 @ Q2 ) ) @ ( relational_Disj_a_b @ ( relati591517084277583526ts_a_b @ X @ Q1 ) @ ( relati591517084277583526ts_a_b @ X @ Q2 ) ) ) ).
% equiv_Exists_Disj
thf(fact_475_Exists__nonfree__equiv,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ~ ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( relational_equiv_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) @ Q ) ) ).
% Exists_nonfree_equiv
thf(fact_476_sat__foldr1__Disj,axiom,
! [Xs2: list_R8263082107343818799la_a_b,Q: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma2: nat > a] :
( ( relational_sat_a_b @ ( relati1724936043867567613la_a_b @ relational_Disj_a_b @ Xs2 @ Q ) @ I2 @ Sigma2 )
= ( ( ( Xs2 = nil_Re6358386334527539737la_a_b )
=> ( relational_sat_a_b @ Q @ I2 @ Sigma2 ) )
& ( ( Xs2 != nil_Re6358386334527539737la_a_b )
=> ? [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ ( set_Re9104216502384355786la_a_b @ Xs2 ) )
& ( relational_sat_a_b @ X2 @ I2 @ Sigma2 ) ) ) ) ) ).
% sat_foldr1_Disj
thf(fact_477_foldr__Disj__equiv,axiom,
! [Q: relational_fmla_a_b,Qs: list_R8263082107343818799la_a_b,Q5: relational_fmla_a_b,Qs2: list_R8263082107343818799la_a_b] :
( ( ( insert7010464514620295119la_a_b @ Q @ ( set_Re9104216502384355786la_a_b @ Qs ) )
= ( insert7010464514620295119la_a_b @ Q5 @ ( set_Re9104216502384355786la_a_b @ Qs2 ) ) )
=> ( relational_equiv_a_b @ ( foldr_789212930732525799la_a_b @ relational_Disj_a_b @ Qs @ Q ) @ ( foldr_789212930732525799la_a_b @ relational_Disj_a_b @ Qs2 @ Q5 ) ) ) ).
% foldr_Disj_equiv
thf(fact_478_sub_Oelims,axiom,
! [X: relational_fmla_a_b,Y3: set_Re381260168593705685la_a_b] :
( ( ( relational_sub_a_b @ X )
= Y3 )
=> ( ! [T: $o] :
( ( X
= ( relational_Bool_a_b @ T ) )
=> ( Y3
!= ( insert7010464514620295119la_a_b @ ( relational_Bool_a_b @ T ) @ bot_bo4495933725496725865la_a_b ) ) )
=> ( ! [P3: b,Ts: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ P3 @ Ts ) )
=> ( Y3
!= ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ P3 @ Ts ) @ bot_bo4495933725496725865la_a_b ) ) )
=> ( ! [X3: nat,T: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X3 @ T ) )
=> ( Y3
!= ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X3 @ T ) @ bot_bo4495933725496725865la_a_b ) ) )
=> ( ! [Q3: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q3 ) )
=> ( Y3
!= ( insert7010464514620295119la_a_b @ ( relational_Neg_a_b @ Q3 ) @ ( relational_sub_a_b @ Q3 ) ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q12 @ Q22 ) )
=> ( Y3
!= ( insert7010464514620295119la_a_b @ ( relational_Conj_a_b @ Q12 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q12 ) @ ( relational_sub_a_b @ Q22 ) ) ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q12 @ Q22 ) )
=> ( Y3
!= ( insert7010464514620295119la_a_b @ ( relational_Disj_a_b @ Q12 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q12 ) @ ( relational_sub_a_b @ Q22 ) ) ) ) )
=> ~ ! [Z: nat,Q3: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ Z @ Q3 ) )
=> ( Y3
!= ( insert7010464514620295119la_a_b @ ( relati591517084277583526ts_a_b @ Z @ Q3 ) @ ( relational_sub_a_b @ Q3 ) ) ) ) ) ) ) ) ) ) ) ).
% sub.elims
thf(fact_479_sub_Opelims,axiom,
! [X: relational_fmla_a_b,Y3: set_Re381260168593705685la_a_b] :
( ( ( relational_sub_a_b @ X )
= Y3 )
=> ( ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ X )
=> ( ! [T: $o] :
( ( X
= ( relational_Bool_a_b @ T ) )
=> ( ( Y3
= ( insert7010464514620295119la_a_b @ ( relational_Bool_a_b @ T ) @ bot_bo4495933725496725865la_a_b ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Bool_a_b @ T ) ) ) )
=> ( ! [P3: b,Ts: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ P3 @ Ts ) )
=> ( ( Y3
= ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ P3 @ Ts ) @ bot_bo4495933725496725865la_a_b ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Pred_b_a @ P3 @ Ts ) ) ) )
=> ( ! [X3: nat,T: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X3 @ T ) )
=> ( ( Y3
= ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X3 @ T ) @ bot_bo4495933725496725865la_a_b ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Eq_a_b @ X3 @ T ) ) ) )
=> ( ! [Q3: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q3 ) )
=> ( ( Y3
= ( insert7010464514620295119la_a_b @ ( relational_Neg_a_b @ Q3 ) @ ( relational_sub_a_b @ Q3 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Neg_a_b @ Q3 ) ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q12 @ Q22 ) )
=> ( ( Y3
= ( insert7010464514620295119la_a_b @ ( relational_Conj_a_b @ Q12 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q12 ) @ ( relational_sub_a_b @ Q22 ) ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Conj_a_b @ Q12 @ Q22 ) ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q12 @ Q22 ) )
=> ( ( Y3
= ( insert7010464514620295119la_a_b @ ( relational_Disj_a_b @ Q12 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q12 ) @ ( relational_sub_a_b @ Q22 ) ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Disj_a_b @ Q12 @ Q22 ) ) ) )
=> ~ ! [Z: nat,Q3: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ Z @ Q3 ) )
=> ( ( Y3
= ( insert7010464514620295119la_a_b @ ( relati591517084277583526ts_a_b @ Z @ Q3 ) @ ( relational_sub_a_b @ Q3 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relati591517084277583526ts_a_b @ Z @ Q3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% sub.pelims
thf(fact_480_fv__cp__foldr__Disj,axiom,
! [Qs: list_R8263082107343818799la_a_b,Q: relational_fmla_a_b,A: set_nat] :
( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( sup_su5130108678486352897la_a_b @ ( set_Re9104216502384355786la_a_b @ Qs ) @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) )
=> ( ( relati1591879772219623554ed_a_b @ X3 )
& ( ( relational_fv_a_b @ X3 )
= A ) ) )
=> ( ( relational_fv_a_b @ ( relational_cp_a_b @ ( foldr_789212930732525799la_a_b @ relational_Disj_a_b @ Qs @ Q ) ) )
= A ) ) ).
% fv_cp_foldr_Disj
thf(fact_481_ap__cp__erase,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relational_ap_a_b @ Q )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q @ X ) )
= ( relational_Bool_a_b @ $false ) ) ) ) ).
% ap_cp_erase
thf(fact_482_empty__Collect__eq,axiom,
! [P: list_s3671366213839843941la_a_b > $o] :
( ( bot_bo7655846380748197807la_a_b
= ( collec5397960198663843514la_a_b @ P ) )
= ( ! [X2: list_s3671366213839843941la_a_b] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_483_empty__Collect__eq,axiom,
! [P: relational_fmla_a_b > $o] :
( ( bot_bo4495933725496725865la_a_b
= ( collec3419995626248312948la_a_b @ P ) )
= ( ! [X2: relational_fmla_a_b] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_484_empty__Collect__eq,axiom,
! [P: set_Re381260168593705685la_a_b > $o] :
( ( bot_bo2891247006866115487la_a_b
= ( collec2099942116761351594la_a_b @ P ) )
= ( ! [X2: set_Re381260168593705685la_a_b] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_485_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_486_empty__Collect__eq,axiom,
! [P: list_nat > $o] :
( ( bot_bot_set_list_nat
= ( collect_list_nat @ P ) )
= ( ! [X2: list_nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_487_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_488_Collect__empty__eq,axiom,
! [P: list_s3671366213839843941la_a_b > $o] :
( ( ( collec5397960198663843514la_a_b @ P )
= bot_bo7655846380748197807la_a_b )
= ( ! [X2: list_s3671366213839843941la_a_b] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_489_Collect__empty__eq,axiom,
! [P: relational_fmla_a_b > $o] :
( ( ( collec3419995626248312948la_a_b @ P )
= bot_bo4495933725496725865la_a_b )
= ( ! [X2: relational_fmla_a_b] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_490_Collect__empty__eq,axiom,
! [P: set_Re381260168593705685la_a_b > $o] :
( ( ( collec2099942116761351594la_a_b @ P )
= bot_bo2891247006866115487la_a_b )
= ( ! [X2: set_Re381260168593705685la_a_b] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_491_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_492_Collect__empty__eq,axiom,
! [P: list_nat > $o] :
( ( ( collect_list_nat @ P )
= bot_bot_set_list_nat )
= ( ! [X2: list_nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_493_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_494_all__not__in__conv,axiom,
! [A: set_li7574107037410898715la_a_b] :
( ( ! [X2: list_s3671366213839843941la_a_b] :
~ ( member987187531357909116la_a_b @ X2 @ A ) )
= ( A = bot_bo7655846380748197807la_a_b ) ) ).
% all_not_in_conv
thf(fact_495_all__not__in__conv,axiom,
! [A: set_se1626149436229564481la_a_b] :
( ( ! [X2: set_se6865892389300016395la_a_b] :
~ ( member1693404252451860130la_a_b @ X2 @ A ) )
= ( A = bot_bo1069171165257752277la_a_b ) ) ).
% all_not_in_conv
thf(fact_496_all__not__in__conv,axiom,
! [A: set_b] :
( ( ! [X2: b] :
~ ( member_b @ X2 @ A ) )
= ( A = bot_bot_set_b ) ) ).
% all_not_in_conv
thf(fact_497_all__not__in__conv,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( ! [X2: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ X2 @ A ) )
= ( A = bot_bo4495933725496725865la_a_b ) ) ).
% all_not_in_conv
thf(fact_498_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_499_all__not__in__conv,axiom,
! [A: set_list_nat] :
( ( ! [X2: list_nat] :
~ ( member_list_nat @ X2 @ A ) )
= ( A = bot_bot_set_list_nat ) ) ).
% all_not_in_conv
thf(fact_500_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_501_all__not__in__conv,axiom,
! [A: set_se6865892389300016395la_a_b] :
( ( ! [X2: set_Re381260168593705685la_a_b] :
~ ( member3481406638322139244la_a_b @ X2 @ A ) )
= ( A = bot_bo2891247006866115487la_a_b ) ) ).
% all_not_in_conv
thf(fact_502_empty__iff,axiom,
! [C: list_s3671366213839843941la_a_b] :
~ ( member987187531357909116la_a_b @ C @ bot_bo7655846380748197807la_a_b ) ).
% empty_iff
thf(fact_503_empty__iff,axiom,
! [C: set_se6865892389300016395la_a_b] :
~ ( member1693404252451860130la_a_b @ C @ bot_bo1069171165257752277la_a_b ) ).
% empty_iff
thf(fact_504_empty__iff,axiom,
! [C: b] :
~ ( member_b @ C @ bot_bot_set_b ) ).
% empty_iff
thf(fact_505_empty__iff,axiom,
! [C: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ C @ bot_bo4495933725496725865la_a_b ) ).
% empty_iff
thf(fact_506_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_507_empty__iff,axiom,
! [C: list_nat] :
~ ( member_list_nat @ C @ bot_bot_set_list_nat ) ).
% empty_iff
thf(fact_508_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_509_empty__iff,axiom,
! [C: set_Re381260168593705685la_a_b] :
~ ( member3481406638322139244la_a_b @ C @ bot_bo2891247006866115487la_a_b ) ).
% empty_iff
thf(fact_510_insert__absorb2,axiom,
! [X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( insert7010464514620295119la_a_b @ X @ ( insert7010464514620295119la_a_b @ X @ A ) )
= ( insert7010464514620295119la_a_b @ X @ A ) ) ).
% insert_absorb2
thf(fact_511_insert__absorb2,axiom,
! [X: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ( insert2023870700798818565la_a_b @ X @ ( insert2023870700798818565la_a_b @ X @ A ) )
= ( insert2023870700798818565la_a_b @ X @ A ) ) ).
% insert_absorb2
thf(fact_512_insert__absorb2,axiom,
! [X: nat,A: set_nat] :
( ( insert_nat2 @ X @ ( insert_nat2 @ X @ A ) )
= ( insert_nat2 @ X @ A ) ) ).
% insert_absorb2
thf(fact_513_insert__absorb2,axiom,
! [X: list_nat,A: set_list_nat] :
( ( insert_list_nat2 @ X @ ( insert_list_nat2 @ X @ A ) )
= ( insert_list_nat2 @ X @ A ) ) ).
% insert_absorb2
thf(fact_514_insert__iff,axiom,
! [A2: relational_fmla_a_b,B2: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member4680049679412964150la_a_b @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_515_insert__iff,axiom,
! [A2: list_s3671366213839843941la_a_b,B2: list_s3671366213839843941la_a_b,A: set_li7574107037410898715la_a_b] :
( ( member987187531357909116la_a_b @ A2 @ ( insert6225987573742651541la_a_b @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member987187531357909116la_a_b @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_516_insert__iff,axiom,
! [A2: list_nat,B2: list_nat,A: set_list_nat] :
( ( member_list_nat @ A2 @ ( insert_list_nat2 @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_list_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_517_insert__iff,axiom,
! [A2: set_se6865892389300016395la_a_b,B2: set_se6865892389300016395la_a_b,A: set_se1626149436229564481la_a_b] :
( ( member1693404252451860130la_a_b @ A2 @ ( insert1549238021488837435la_a_b @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member1693404252451860130la_a_b @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_518_insert__iff,axiom,
! [A2: nat,B2: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat2 @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_519_insert__iff,axiom,
! [A2: b,B2: b,A: set_b] :
( ( member_b @ A2 @ ( insert_b2 @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_b @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_520_insert__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member3481406638322139244la_a_b @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_521_insertCI,axiom,
! [A2: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,B2: relational_fmla_a_b] :
( ( ~ ( member4680049679412964150la_a_b @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member4680049679412964150la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ B2 @ B ) ) ) ).
% insertCI
thf(fact_522_insertCI,axiom,
! [A2: list_s3671366213839843941la_a_b,B: set_li7574107037410898715la_a_b,B2: list_s3671366213839843941la_a_b] :
( ( ~ ( member987187531357909116la_a_b @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member987187531357909116la_a_b @ A2 @ ( insert6225987573742651541la_a_b @ B2 @ B ) ) ) ).
% insertCI
thf(fact_523_insertCI,axiom,
! [A2: list_nat,B: set_list_nat,B2: list_nat] :
( ( ~ ( member_list_nat @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_list_nat @ A2 @ ( insert_list_nat2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_524_insertCI,axiom,
! [A2: set_se6865892389300016395la_a_b,B: set_se1626149436229564481la_a_b,B2: set_se6865892389300016395la_a_b] :
( ( ~ ( member1693404252451860130la_a_b @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member1693404252451860130la_a_b @ A2 @ ( insert1549238021488837435la_a_b @ B2 @ B ) ) ) ).
% insertCI
thf(fact_525_insertCI,axiom,
! [A2: nat,B: set_nat,B2: nat] :
( ( ~ ( member_nat @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_nat @ A2 @ ( insert_nat2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_526_insertCI,axiom,
! [A2: b,B: set_b,B2: b] :
( ( ~ ( member_b @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_b @ A2 @ ( insert_b2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_527_insertCI,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ~ ( member3481406638322139244la_a_b @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member3481406638322139244la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ B2 @ B ) ) ) ).
% insertCI
thf(fact_528_singletonI,axiom,
! [A2: list_s3671366213839843941la_a_b] : ( member987187531357909116la_a_b @ A2 @ ( insert6225987573742651541la_a_b @ A2 @ bot_bo7655846380748197807la_a_b ) ) ).
% singletonI
thf(fact_529_singletonI,axiom,
! [A2: set_se6865892389300016395la_a_b] : ( member1693404252451860130la_a_b @ A2 @ ( insert1549238021488837435la_a_b @ A2 @ bot_bo1069171165257752277la_a_b ) ) ).
% singletonI
thf(fact_530_singletonI,axiom,
! [A2: b] : ( member_b @ A2 @ ( insert_b2 @ A2 @ bot_bot_set_b ) ) ).
% singletonI
thf(fact_531_singletonI,axiom,
! [A2: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ) ).
% singletonI
thf(fact_532_singletonI,axiom,
! [A2: nat] : ( member_nat @ A2 @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_533_singletonI,axiom,
! [A2: list_nat] : ( member_list_nat @ A2 @ ( insert_list_nat2 @ A2 @ bot_bot_set_list_nat ) ) ).
% singletonI
thf(fact_534_singletonI,axiom,
! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_535_singletonI,axiom,
! [A2: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ A2 @ bot_bo2891247006866115487la_a_b ) ) ).
% singletonI
thf(fact_536_sup__bot__left,axiom,
! [X: set_b] :
( ( sup_sup_set_b @ bot_bot_set_b @ X )
= X ) ).
% sup_bot_left
thf(fact_537_sup__bot__left,axiom,
! [X: set_Re381260168593705685la_a_b > $o] :
( ( sup_su2223990397388305166_a_b_o @ bot_bo8397419132826146726_a_b_o @ X )
= X ) ).
% sup_bot_left
thf(fact_538_sup__bot__left,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ bot_bo4495933725496725865la_a_b @ X )
= X ) ).
% sup_bot_left
thf(fact_539_sup__bot__left,axiom,
! [X: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ bot_bo2891247006866115487la_a_b @ X )
= X ) ).
% sup_bot_left
thf(fact_540_sup__bot__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_541_sup__bot__left,axiom,
! [X: set_list_nat] :
( ( sup_sup_set_list_nat @ bot_bot_set_list_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_542_sup__bot__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_543_sup__bot__right,axiom,
! [X: set_b] :
( ( sup_sup_set_b @ X @ bot_bot_set_b )
= X ) ).
% sup_bot_right
thf(fact_544_sup__bot__right,axiom,
! [X: set_Re381260168593705685la_a_b > $o] :
( ( sup_su2223990397388305166_a_b_o @ X @ bot_bo8397419132826146726_a_b_o )
= X ) ).
% sup_bot_right
thf(fact_545_sup__bot__right,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ bot_bo4495933725496725865la_a_b )
= X ) ).
% sup_bot_right
thf(fact_546_sup__bot__right,axiom,
! [X: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ X @ bot_bo2891247006866115487la_a_b )
= X ) ).
% sup_bot_right
thf(fact_547_sup__bot__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% sup_bot_right
thf(fact_548_sup__bot__right,axiom,
! [X: set_list_nat] :
( ( sup_sup_set_list_nat @ X @ bot_bot_set_list_nat )
= X ) ).
% sup_bot_right
thf(fact_549_sup__bot__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% sup_bot_right
thf(fact_550_bot__eq__sup__iff,axiom,
! [X: set_b,Y3: set_b] :
( ( bot_bot_set_b
= ( sup_sup_set_b @ X @ Y3 ) )
= ( ( X = bot_bot_set_b )
& ( Y3 = bot_bot_set_b ) ) ) ).
% bot_eq_sup_iff
thf(fact_551_bot__eq__sup__iff,axiom,
! [X: set_Re381260168593705685la_a_b > $o,Y3: set_Re381260168593705685la_a_b > $o] :
( ( bot_bo8397419132826146726_a_b_o
= ( sup_su2223990397388305166_a_b_o @ X @ Y3 ) )
= ( ( X = bot_bo8397419132826146726_a_b_o )
& ( Y3 = bot_bo8397419132826146726_a_b_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_552_bot__eq__sup__iff,axiom,
! [X: set_Re381260168593705685la_a_b,Y3: set_Re381260168593705685la_a_b] :
( ( bot_bo4495933725496725865la_a_b
= ( sup_su5130108678486352897la_a_b @ X @ Y3 ) )
= ( ( X = bot_bo4495933725496725865la_a_b )
& ( Y3 = bot_bo4495933725496725865la_a_b ) ) ) ).
% bot_eq_sup_iff
thf(fact_553_bot__eq__sup__iff,axiom,
! [X: set_se6865892389300016395la_a_b,Y3: set_se6865892389300016395la_a_b] :
( ( bot_bo2891247006866115487la_a_b
= ( sup_su4783144482993978935la_a_b @ X @ Y3 ) )
= ( ( X = bot_bo2891247006866115487la_a_b )
& ( Y3 = bot_bo2891247006866115487la_a_b ) ) ) ).
% bot_eq_sup_iff
thf(fact_554_bot__eq__sup__iff,axiom,
! [X: set_nat,Y3: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X @ Y3 ) )
= ( ( X = bot_bot_set_nat )
& ( Y3 = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_555_bot__eq__sup__iff,axiom,
! [X: set_list_nat,Y3: set_list_nat] :
( ( bot_bot_set_list_nat
= ( sup_sup_set_list_nat @ X @ Y3 ) )
= ( ( X = bot_bot_set_list_nat )
& ( Y3 = bot_bot_set_list_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_556_bot__eq__sup__iff,axiom,
! [X: set_a,Y3: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X @ Y3 ) )
= ( ( X = bot_bot_set_a )
& ( Y3 = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_557_sup__eq__bot__iff,axiom,
! [X: set_b,Y3: set_b] :
( ( ( sup_sup_set_b @ X @ Y3 )
= bot_bot_set_b )
= ( ( X = bot_bot_set_b )
& ( Y3 = bot_bot_set_b ) ) ) ).
% sup_eq_bot_iff
thf(fact_558_sup__eq__bot__iff,axiom,
! [X: set_Re381260168593705685la_a_b > $o,Y3: set_Re381260168593705685la_a_b > $o] :
( ( ( sup_su2223990397388305166_a_b_o @ X @ Y3 )
= bot_bo8397419132826146726_a_b_o )
= ( ( X = bot_bo8397419132826146726_a_b_o )
& ( Y3 = bot_bo8397419132826146726_a_b_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_559_sup__eq__bot__iff,axiom,
! [X: set_Re381260168593705685la_a_b,Y3: set_Re381260168593705685la_a_b] :
( ( ( sup_su5130108678486352897la_a_b @ X @ Y3 )
= bot_bo4495933725496725865la_a_b )
= ( ( X = bot_bo4495933725496725865la_a_b )
& ( Y3 = bot_bo4495933725496725865la_a_b ) ) ) ).
% sup_eq_bot_iff
thf(fact_560_sup__eq__bot__iff,axiom,
! [X: set_se6865892389300016395la_a_b,Y3: set_se6865892389300016395la_a_b] :
( ( ( sup_su4783144482993978935la_a_b @ X @ Y3 )
= bot_bo2891247006866115487la_a_b )
= ( ( X = bot_bo2891247006866115487la_a_b )
& ( Y3 = bot_bo2891247006866115487la_a_b ) ) ) ).
% sup_eq_bot_iff
thf(fact_561_sup__eq__bot__iff,axiom,
! [X: set_nat,Y3: set_nat] :
( ( ( sup_sup_set_nat @ X @ Y3 )
= bot_bot_set_nat )
= ( ( X = bot_bot_set_nat )
& ( Y3 = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_562_sup__eq__bot__iff,axiom,
! [X: set_list_nat,Y3: set_list_nat] :
( ( ( sup_sup_set_list_nat @ X @ Y3 )
= bot_bot_set_list_nat )
= ( ( X = bot_bot_set_list_nat )
& ( Y3 = bot_bot_set_list_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_563_sup__eq__bot__iff,axiom,
! [X: set_a,Y3: set_a] :
( ( ( sup_sup_set_a @ X @ Y3 )
= bot_bot_set_a )
= ( ( X = bot_bot_set_a )
& ( Y3 = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_564_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_b,B2: set_b] :
( ( ( sup_sup_set_b @ A2 @ B2 )
= bot_bot_set_b )
= ( ( A2 = bot_bot_set_b )
& ( B2 = bot_bot_set_b ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_565_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_Re381260168593705685la_a_b > $o,B2: set_Re381260168593705685la_a_b > $o] :
( ( ( sup_su2223990397388305166_a_b_o @ A2 @ B2 )
= bot_bo8397419132826146726_a_b_o )
= ( ( A2 = bot_bo8397419132826146726_a_b_o )
& ( B2 = bot_bo8397419132826146726_a_b_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_566_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ( sup_su5130108678486352897la_a_b @ A2 @ B2 )
= bot_bo4495933725496725865la_a_b )
= ( ( A2 = bot_bo4495933725496725865la_a_b )
& ( B2 = bot_bo4495933725496725865la_a_b ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_567_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_se6865892389300016395la_a_b,B2: set_se6865892389300016395la_a_b] :
( ( ( sup_su4783144482993978935la_a_b @ A2 @ B2 )
= bot_bo2891247006866115487la_a_b )
= ( ( A2 = bot_bo2891247006866115487la_a_b )
& ( B2 = bot_bo2891247006866115487la_a_b ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_568_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_569_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_list_nat,B2: set_list_nat] :
( ( ( sup_sup_set_list_nat @ A2 @ B2 )
= bot_bot_set_list_nat )
= ( ( A2 = bot_bot_set_list_nat )
& ( B2 = bot_bot_set_list_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_570_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( sup_sup_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_571_sup__bot_Oleft__neutral,axiom,
! [A2: set_b] :
( ( sup_sup_set_b @ bot_bot_set_b @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_572_sup__bot_Oleft__neutral,axiom,
! [A2: set_Re381260168593705685la_a_b > $o] :
( ( sup_su2223990397388305166_a_b_o @ bot_bo8397419132826146726_a_b_o @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_573_sup__bot_Oleft__neutral,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ bot_bo4495933725496725865la_a_b @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_574_sup__bot_Oleft__neutral,axiom,
! [A2: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ bot_bo2891247006866115487la_a_b @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_575_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_576_sup__bot_Oleft__neutral,axiom,
! [A2: set_list_nat] :
( ( sup_sup_set_list_nat @ bot_bot_set_list_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_577_sup__bot_Oleft__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_578_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_b,B2: set_b] :
( ( bot_bot_set_b
= ( sup_sup_set_b @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_b )
& ( B2 = bot_bot_set_b ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_579_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_Re381260168593705685la_a_b > $o,B2: set_Re381260168593705685la_a_b > $o] :
( ( bot_bo8397419132826146726_a_b_o
= ( sup_su2223990397388305166_a_b_o @ A2 @ B2 ) )
= ( ( A2 = bot_bo8397419132826146726_a_b_o )
& ( B2 = bot_bo8397419132826146726_a_b_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_580_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( bot_bo4495933725496725865la_a_b
= ( sup_su5130108678486352897la_a_b @ A2 @ B2 ) )
= ( ( A2 = bot_bo4495933725496725865la_a_b )
& ( B2 = bot_bo4495933725496725865la_a_b ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_581_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_se6865892389300016395la_a_b,B2: set_se6865892389300016395la_a_b] :
( ( bot_bo2891247006866115487la_a_b
= ( sup_su4783144482993978935la_a_b @ A2 @ B2 ) )
= ( ( A2 = bot_bo2891247006866115487la_a_b )
& ( B2 = bot_bo2891247006866115487la_a_b ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_582_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_583_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_list_nat,B2: set_list_nat] :
( ( bot_bot_set_list_nat
= ( sup_sup_set_list_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_list_nat )
& ( B2 = bot_bot_set_list_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_584_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_585_sup__bot_Oright__neutral,axiom,
! [A2: set_b] :
( ( sup_sup_set_b @ A2 @ bot_bot_set_b )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_586_sup__bot_Oright__neutral,axiom,
! [A2: set_Re381260168593705685la_a_b > $o] :
( ( sup_su2223990397388305166_a_b_o @ A2 @ bot_bo8397419132826146726_a_b_o )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_587_sup__bot_Oright__neutral,axiom,
! [A2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A2 @ bot_bo4495933725496725865la_a_b )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_588_sup__bot_Oright__neutral,axiom,
! [A2: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ A2 @ bot_bo2891247006866115487la_a_b )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_589_sup__bot_Oright__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_590_sup__bot_Oright__neutral,axiom,
! [A2: set_list_nat] :
( ( sup_sup_set_list_nat @ A2 @ bot_bot_set_list_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_591_sup__bot_Oright__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_592_Un__empty,axiom,
! [A: set_b,B: set_b] :
( ( ( sup_sup_set_b @ A @ B )
= bot_bot_set_b )
= ( ( A = bot_bot_set_b )
& ( B = bot_bot_set_b ) ) ) ).
% Un_empty
thf(fact_593_Un__empty,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( sup_su5130108678486352897la_a_b @ A @ B )
= bot_bo4495933725496725865la_a_b )
= ( ( A = bot_bo4495933725496725865la_a_b )
& ( B = bot_bo4495933725496725865la_a_b ) ) ) ).
% Un_empty
thf(fact_594_Un__empty,axiom,
! [A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( ( sup_su4783144482993978935la_a_b @ A @ B )
= bot_bo2891247006866115487la_a_b )
= ( ( A = bot_bo2891247006866115487la_a_b )
& ( B = bot_bo2891247006866115487la_a_b ) ) ) ).
% Un_empty
thf(fact_595_Un__empty,axiom,
! [A: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_596_Un__empty,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( ( sup_sup_set_list_nat @ A @ B )
= bot_bot_set_list_nat )
= ( ( A = bot_bot_set_list_nat )
& ( B = bot_bot_set_list_nat ) ) ) ).
% Un_empty
thf(fact_597_Un__empty,axiom,
! [A: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A @ B )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_598_Un__insert__left,axiom,
! [A2: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,C2: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ ( insert7010464514620295119la_a_b @ A2 @ B ) @ C2 )
= ( insert7010464514620295119la_a_b @ A2 @ ( sup_su5130108678486352897la_a_b @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_599_Un__insert__left,axiom,
! [A2: list_nat,B: set_list_nat,C2: set_list_nat] :
( ( sup_sup_set_list_nat @ ( insert_list_nat2 @ A2 @ B ) @ C2 )
= ( insert_list_nat2 @ A2 @ ( sup_sup_set_list_nat @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_600_Un__insert__left,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b,C2: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ ( insert2023870700798818565la_a_b @ A2 @ B ) @ C2 )
= ( insert2023870700798818565la_a_b @ A2 @ ( sup_su4783144482993978935la_a_b @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_601_Un__insert__left,axiom,
! [A2: nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat2 @ A2 @ B ) @ C2 )
= ( insert_nat2 @ A2 @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_602_Un__insert__left,axiom,
! [A2: b,B: set_b,C2: set_b] :
( ( sup_sup_set_b @ ( insert_b2 @ A2 @ B ) @ C2 )
= ( insert_b2 @ A2 @ ( sup_sup_set_b @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_603_Un__insert__left,axiom,
! [A2: a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( insert_a @ A2 @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_604_Un__insert__right,axiom,
! [A: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A @ ( insert7010464514620295119la_a_b @ A2 @ B ) )
= ( insert7010464514620295119la_a_b @ A2 @ ( sup_su5130108678486352897la_a_b @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_605_Un__insert__right,axiom,
! [A: set_list_nat,A2: list_nat,B: set_list_nat] :
( ( sup_sup_set_list_nat @ A @ ( insert_list_nat2 @ A2 @ B ) )
= ( insert_list_nat2 @ A2 @ ( sup_sup_set_list_nat @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_606_Un__insert__right,axiom,
! [A: set_se6865892389300016395la_a_b,A2: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ A @ ( insert2023870700798818565la_a_b @ A2 @ B ) )
= ( insert2023870700798818565la_a_b @ A2 @ ( sup_su4783144482993978935la_a_b @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_607_Un__insert__right,axiom,
! [A: set_nat,A2: nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( insert_nat2 @ A2 @ B ) )
= ( insert_nat2 @ A2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_608_Un__insert__right,axiom,
! [A: set_b,A2: b,B: set_b] :
( ( sup_sup_set_b @ A @ ( insert_b2 @ A2 @ B ) )
= ( insert_b2 @ A2 @ ( sup_sup_set_b @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_609_Un__insert__right,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( sup_sup_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_610_sat__cp,axiom,
! [Q: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma2: nat > a] :
( ( relational_sat_a_b @ ( relational_cp_a_b @ Q ) @ I2 @ Sigma2 )
= ( relational_sat_a_b @ Q @ I2 @ Sigma2 ) ) ).
% sat_cp
thf(fact_611_List_Oset__empty,axiom,
! [Xs2: list_set_nat] :
( ( ( set_set_nat2 @ Xs2 )
= bot_bot_set_set_nat )
= ( Xs2 = nil_set_nat ) ) ).
% List.set_empty
thf(fact_612_List_Oset__empty,axiom,
! [Xs2: list_l3216422893793986623la_a_b] :
( ( ( set_li8988895567068466906la_a_b @ Xs2 )
= bot_bo3230846185698255993la_a_b )
= ( Xs2 = nil_li4862165533606214569la_a_b ) ) ).
% List.set_empty
thf(fact_613_List_Oset__empty,axiom,
! [Xs2: list_l1512381142989314933la_a_b] :
( ( ( set_li3253934503784678672la_a_b @ Xs2 )
= bot_bo7655846380748197807la_a_b )
= ( Xs2 = nil_li6858553494156228063la_a_b ) ) ).
% List.set_empty
thf(fact_614_List_Oset__empty,axiom,
! [Xs2: list_R8263082107343818799la_a_b] :
( ( ( set_Re9104216502384355786la_a_b @ Xs2 )
= bot_bo4495933725496725865la_a_b )
= ( Xs2 = nil_Re6358386334527539737la_a_b ) ) ).
% List.set_empty
thf(fact_615_List_Oset__empty,axiom,
! [Xs2: list_list_nat] :
( ( ( set_list_nat2 @ Xs2 )
= bot_bot_set_list_nat )
= ( Xs2 = nil_list_nat ) ) ).
% List.set_empty
thf(fact_616_List_Oset__empty,axiom,
! [Xs2: list_a] :
( ( ( set_a2 @ Xs2 )
= bot_bot_set_a )
= ( Xs2 = nil_a ) ) ).
% List.set_empty
thf(fact_617_List_Oset__empty,axiom,
! [Xs2: list_nat] :
( ( ( set_nat2 @ Xs2 )
= bot_bot_set_nat )
= ( Xs2 = nil_nat ) ) ).
% List.set_empty
thf(fact_618_List_Oset__empty,axiom,
! [Xs2: list_s3671366213839843941la_a_b] :
( ( ( set_se5111286776497155840la_a_b @ Xs2 )
= bot_bo2891247006866115487la_a_b )
= ( Xs2 = nil_se6596116450819887951la_a_b ) ) ).
% List.set_empty
thf(fact_619_set__empty2,axiom,
! [Xs2: list_set_nat] :
( ( bot_bot_set_set_nat
= ( set_set_nat2 @ Xs2 ) )
= ( Xs2 = nil_set_nat ) ) ).
% set_empty2
thf(fact_620_set__empty2,axiom,
! [Xs2: list_l3216422893793986623la_a_b] :
( ( bot_bo3230846185698255993la_a_b
= ( set_li8988895567068466906la_a_b @ Xs2 ) )
= ( Xs2 = nil_li4862165533606214569la_a_b ) ) ).
% set_empty2
thf(fact_621_set__empty2,axiom,
! [Xs2: list_l1512381142989314933la_a_b] :
( ( bot_bo7655846380748197807la_a_b
= ( set_li3253934503784678672la_a_b @ Xs2 ) )
= ( Xs2 = nil_li6858553494156228063la_a_b ) ) ).
% set_empty2
thf(fact_622_set__empty2,axiom,
! [Xs2: list_R8263082107343818799la_a_b] :
( ( bot_bo4495933725496725865la_a_b
= ( set_Re9104216502384355786la_a_b @ Xs2 ) )
= ( Xs2 = nil_Re6358386334527539737la_a_b ) ) ).
% set_empty2
thf(fact_623_set__empty2,axiom,
! [Xs2: list_list_nat] :
( ( bot_bot_set_list_nat
= ( set_list_nat2 @ Xs2 ) )
= ( Xs2 = nil_list_nat ) ) ).
% set_empty2
thf(fact_624_set__empty2,axiom,
! [Xs2: list_a] :
( ( bot_bot_set_a
= ( set_a2 @ Xs2 ) )
= ( Xs2 = nil_a ) ) ).
% set_empty2
thf(fact_625_set__empty2,axiom,
! [Xs2: list_nat] :
( ( bot_bot_set_nat
= ( set_nat2 @ Xs2 ) )
= ( Xs2 = nil_nat ) ) ).
% set_empty2
thf(fact_626_set__empty2,axiom,
! [Xs2: list_s3671366213839843941la_a_b] :
( ( bot_bo2891247006866115487la_a_b
= ( set_se5111286776497155840la_a_b @ Xs2 ) )
= ( Xs2 = nil_se6596116450819887951la_a_b ) ) ).
% set_empty2
thf(fact_627_gen__Bool__False,axiom,
! [X: nat,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X @ ( relational_Bool_a_b @ $false ) @ G )
= ( G = bot_bo4495933725496725865la_a_b ) ) ).
% gen_Bool_False
thf(fact_628_List_Oset__insert,axiom,
! [X: relational_fmla_a_b,Xs2: list_R8263082107343818799la_a_b] :
( ( set_Re9104216502384355786la_a_b @ ( insert5052544347734373763la_a_b @ X @ Xs2 ) )
= ( insert7010464514620295119la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Xs2 ) ) ) ).
% List.set_insert
thf(fact_629_List_Oset__insert,axiom,
! [X: list_R8263082107343818799la_a_b,Xs2: list_l3216422893793986623la_a_b] :
( ( set_li8988895567068466906la_a_b @ ( insert3209172818022656531la_a_b @ X @ Xs2 ) )
= ( insert3617074100887873119la_a_b @ X @ ( set_li8988895567068466906la_a_b @ Xs2 ) ) ) ).
% List.set_insert
thf(fact_630_List_Oset__insert,axiom,
! [X: list_s3671366213839843941la_a_b,Xs2: list_l1512381142989314933la_a_b] :
( ( set_li3253934503784678672la_a_b @ ( insert3122035991742315593la_a_b @ X @ Xs2 ) )
= ( insert6225987573742651541la_a_b @ X @ ( set_li3253934503784678672la_a_b @ Xs2 ) ) ) ).
% List.set_insert
thf(fact_631_List_Oset__insert,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( set_list_nat2 @ ( insert_list_nat @ X @ Xs2 ) )
= ( insert_list_nat2 @ X @ ( set_list_nat2 @ Xs2 ) ) ) ).
% List.set_insert
thf(fact_632_List_Oset__insert,axiom,
! [X: nat,Xs2: list_nat] :
( ( set_nat2 @ ( insert_nat @ X @ Xs2 ) )
= ( insert_nat2 @ X @ ( set_nat2 @ Xs2 ) ) ) ).
% List.set_insert
thf(fact_633_List_Oset__insert,axiom,
! [X: set_Re381260168593705685la_a_b,Xs2: list_s3671366213839843941la_a_b] :
( ( set_se5111286776497155840la_a_b @ ( insert443281300761040569la_a_b @ X @ Xs2 ) )
= ( insert2023870700798818565la_a_b @ X @ ( set_se5111286776497155840la_a_b @ Xs2 ) ) ) ).
% List.set_insert
thf(fact_634_gen__Eq,axiom,
! [Z2: nat,A2: nat,T3: relational_term_a,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ Z2 @ ( relational_Eq_a_b @ A2 @ T3 ) @ G )
= ( ( Z2 = A2 )
& ? [C4: a] :
( ( T3
= ( relational_Const_a @ C4 ) )
& ( G
= ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ A2 @ T3 ) @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ).
% gen_Eq
thf(fact_635_sat__foldr__Disj,axiom,
! [Xs2: list_R8263082107343818799la_a_b,Q: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma2: nat > a] :
( ( relational_sat_a_b @ ( foldr_789212930732525799la_a_b @ relational_Disj_a_b @ Xs2 @ Q ) @ I2 @ Sigma2 )
= ( ? [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ ( sup_su5130108678486352897la_a_b @ ( set_Re9104216502384355786la_a_b @ Xs2 ) @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) )
& ( relational_sat_a_b @ X2 @ I2 @ Sigma2 ) ) ) ) ).
% sat_foldr_Disj
thf(fact_636_gen__sat__erase,axiom,
! [Y3: nat,Q: relational_fmla_a_b,Gy: set_Re381260168593705685la_a_b,X: nat,I2: product_prod_b_nat > set_list_a,Sigma2: nat > a] :
( ( relational_gen_a_b @ Y3 @ Q @ Gy )
=> ( ( relational_sat_a_b @ ( relational_erase_a_b @ Q @ X ) @ I2 @ Sigma2 )
=> ? [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ Gy )
& ( relational_sat_a_b @ X3 @ I2 @ Sigma2 ) ) ) ) ).
% gen_sat_erase
thf(fact_637_mk__disjoint__insert,axiom,
! [A2: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ A )
=> ? [B7: set_Re381260168593705685la_a_b] :
( ( A
= ( insert7010464514620295119la_a_b @ A2 @ B7 ) )
& ~ ( member4680049679412964150la_a_b @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_638_mk__disjoint__insert,axiom,
! [A2: list_s3671366213839843941la_a_b,A: set_li7574107037410898715la_a_b] :
( ( member987187531357909116la_a_b @ A2 @ A )
=> ? [B7: set_li7574107037410898715la_a_b] :
( ( A
= ( insert6225987573742651541la_a_b @ A2 @ B7 ) )
& ~ ( member987187531357909116la_a_b @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_639_mk__disjoint__insert,axiom,
! [A2: list_nat,A: set_list_nat] :
( ( member_list_nat @ A2 @ A )
=> ? [B7: set_list_nat] :
( ( A
= ( insert_list_nat2 @ A2 @ B7 ) )
& ~ ( member_list_nat @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_640_mk__disjoint__insert,axiom,
! [A2: set_se6865892389300016395la_a_b,A: set_se1626149436229564481la_a_b] :
( ( member1693404252451860130la_a_b @ A2 @ A )
=> ? [B7: set_se1626149436229564481la_a_b] :
( ( A
= ( insert1549238021488837435la_a_b @ A2 @ B7 ) )
& ~ ( member1693404252451860130la_a_b @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_641_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ? [B7: set_nat] :
( ( A
= ( insert_nat2 @ A2 @ B7 ) )
& ~ ( member_nat @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_642_mk__disjoint__insert,axiom,
! [A2: b,A: set_b] :
( ( member_b @ A2 @ A )
=> ? [B7: set_b] :
( ( A
= ( insert_b2 @ A2 @ B7 ) )
& ~ ( member_b @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_643_mk__disjoint__insert,axiom,
! [A2: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ A2 @ A )
=> ? [B7: set_se6865892389300016395la_a_b] :
( ( A
= ( insert2023870700798818565la_a_b @ A2 @ B7 ) )
& ~ ( member3481406638322139244la_a_b @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_644_singleton__inject,axiom,
! [A2: relational_fmla_a_b,B2: relational_fmla_a_b] :
( ( ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b )
= ( insert7010464514620295119la_a_b @ B2 @ bot_bo4495933725496725865la_a_b ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_645_singleton__inject,axiom,
! [A2: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( ( insert2023870700798818565la_a_b @ A2 @ bot_bo2891247006866115487la_a_b )
= ( insert2023870700798818565la_a_b @ B2 @ bot_bo2891247006866115487la_a_b ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_646_singleton__inject,axiom,
! [A2: nat,B2: nat] :
( ( ( insert_nat2 @ A2 @ bot_bot_set_nat )
= ( insert_nat2 @ B2 @ bot_bot_set_nat ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_647_singleton__inject,axiom,
! [A2: list_nat,B2: list_nat] :
( ( ( insert_list_nat2 @ A2 @ bot_bot_set_list_nat )
= ( insert_list_nat2 @ B2 @ bot_bot_set_list_nat ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_648_singleton__inject,axiom,
! [A2: a,B2: a] :
( ( ( insert_a @ A2 @ bot_bot_set_a )
= ( insert_a @ B2 @ bot_bot_set_a ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_649_insert__not__empty,axiom,
! [A2: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( insert7010464514620295119la_a_b @ A2 @ A )
!= bot_bo4495933725496725865la_a_b ) ).
% insert_not_empty
thf(fact_650_insert__not__empty,axiom,
! [A2: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ( insert2023870700798818565la_a_b @ A2 @ A )
!= bot_bo2891247006866115487la_a_b ) ).
% insert_not_empty
thf(fact_651_insert__not__empty,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat2 @ A2 @ A )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_652_insert__not__empty,axiom,
! [A2: list_nat,A: set_list_nat] :
( ( insert_list_nat2 @ A2 @ A )
!= bot_bot_set_list_nat ) ).
% insert_not_empty
thf(fact_653_insert__not__empty,axiom,
! [A2: a,A: set_a] :
( ( insert_a @ A2 @ A )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_654_doubleton__eq__iff,axiom,
! [A2: relational_fmla_a_b,B2: relational_fmla_a_b,C: relational_fmla_a_b,D: relational_fmla_a_b] :
( ( ( insert7010464514620295119la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ B2 @ bot_bo4495933725496725865la_a_b ) )
= ( insert7010464514620295119la_a_b @ C @ ( insert7010464514620295119la_a_b @ D @ bot_bo4495933725496725865la_a_b ) ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_655_doubleton__eq__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,C: set_Re381260168593705685la_a_b,D: set_Re381260168593705685la_a_b] :
( ( ( insert2023870700798818565la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ B2 @ bot_bo2891247006866115487la_a_b ) )
= ( insert2023870700798818565la_a_b @ C @ ( insert2023870700798818565la_a_b @ D @ bot_bo2891247006866115487la_a_b ) ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_656_doubleton__eq__iff,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ( insert_nat2 @ A2 @ ( insert_nat2 @ B2 @ bot_bot_set_nat ) )
= ( insert_nat2 @ C @ ( insert_nat2 @ D @ bot_bot_set_nat ) ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_657_doubleton__eq__iff,axiom,
! [A2: list_nat,B2: list_nat,C: list_nat,D: list_nat] :
( ( ( insert_list_nat2 @ A2 @ ( insert_list_nat2 @ B2 @ bot_bot_set_list_nat ) )
= ( insert_list_nat2 @ C @ ( insert_list_nat2 @ D @ bot_bot_set_list_nat ) ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_658_doubleton__eq__iff,axiom,
! [A2: a,B2: a,C: a,D: a] :
( ( ( insert_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_659_insert__commute,axiom,
! [X: relational_fmla_a_b,Y3: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( insert7010464514620295119la_a_b @ X @ ( insert7010464514620295119la_a_b @ Y3 @ A ) )
= ( insert7010464514620295119la_a_b @ Y3 @ ( insert7010464514620295119la_a_b @ X @ A ) ) ) ).
% insert_commute
thf(fact_660_insert__commute,axiom,
! [X: set_Re381260168593705685la_a_b,Y3: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ( insert2023870700798818565la_a_b @ X @ ( insert2023870700798818565la_a_b @ Y3 @ A ) )
= ( insert2023870700798818565la_a_b @ Y3 @ ( insert2023870700798818565la_a_b @ X @ A ) ) ) ).
% insert_commute
thf(fact_661_insert__commute,axiom,
! [X: nat,Y3: nat,A: set_nat] :
( ( insert_nat2 @ X @ ( insert_nat2 @ Y3 @ A ) )
= ( insert_nat2 @ Y3 @ ( insert_nat2 @ X @ A ) ) ) ).
% insert_commute
thf(fact_662_insert__commute,axiom,
! [X: list_nat,Y3: list_nat,A: set_list_nat] :
( ( insert_list_nat2 @ X @ ( insert_list_nat2 @ Y3 @ A ) )
= ( insert_list_nat2 @ Y3 @ ( insert_list_nat2 @ X @ A ) ) ) ).
% insert_commute
thf(fact_663_singleton__iff,axiom,
! [B2: list_s3671366213839843941la_a_b,A2: list_s3671366213839843941la_a_b] :
( ( member987187531357909116la_a_b @ B2 @ ( insert6225987573742651541la_a_b @ A2 @ bot_bo7655846380748197807la_a_b ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_664_singleton__iff,axiom,
! [B2: set_se6865892389300016395la_a_b,A2: set_se6865892389300016395la_a_b] :
( ( member1693404252451860130la_a_b @ B2 @ ( insert1549238021488837435la_a_b @ A2 @ bot_bo1069171165257752277la_a_b ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_665_singleton__iff,axiom,
! [B2: b,A2: b] :
( ( member_b @ B2 @ ( insert_b2 @ A2 @ bot_bot_set_b ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_666_singleton__iff,axiom,
! [B2: relational_fmla_a_b,A2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ B2 @ ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_667_singleton__iff,axiom,
! [B2: nat,A2: nat] :
( ( member_nat @ B2 @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_668_singleton__iff,axiom,
! [B2: list_nat,A2: list_nat] :
( ( member_list_nat @ B2 @ ( insert_list_nat2 @ A2 @ bot_bot_set_list_nat ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_669_singleton__iff,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_670_singleton__iff,axiom,
! [B2: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ B2 @ ( insert2023870700798818565la_a_b @ A2 @ bot_bo2891247006866115487la_a_b ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_671_insert__eq__iff,axiom,
! [A2: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B2: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ A2 @ A )
=> ( ~ ( member4680049679412964150la_a_b @ B2 @ B )
=> ( ( ( insert7010464514620295119la_a_b @ A2 @ A )
= ( insert7010464514620295119la_a_b @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C5: set_Re381260168593705685la_a_b] :
( ( A
= ( insert7010464514620295119la_a_b @ B2 @ C5 ) )
& ~ ( member4680049679412964150la_a_b @ B2 @ C5 )
& ( B
= ( insert7010464514620295119la_a_b @ A2 @ C5 ) )
& ~ ( member4680049679412964150la_a_b @ A2 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_672_insert__eq__iff,axiom,
! [A2: list_s3671366213839843941la_a_b,A: set_li7574107037410898715la_a_b,B2: list_s3671366213839843941la_a_b,B: set_li7574107037410898715la_a_b] :
( ~ ( member987187531357909116la_a_b @ A2 @ A )
=> ( ~ ( member987187531357909116la_a_b @ B2 @ B )
=> ( ( ( insert6225987573742651541la_a_b @ A2 @ A )
= ( insert6225987573742651541la_a_b @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C5: set_li7574107037410898715la_a_b] :
( ( A
= ( insert6225987573742651541la_a_b @ B2 @ C5 ) )
& ~ ( member987187531357909116la_a_b @ B2 @ C5 )
& ( B
= ( insert6225987573742651541la_a_b @ A2 @ C5 ) )
& ~ ( member987187531357909116la_a_b @ A2 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_673_insert__eq__iff,axiom,
! [A2: list_nat,A: set_list_nat,B2: list_nat,B: set_list_nat] :
( ~ ( member_list_nat @ A2 @ A )
=> ( ~ ( member_list_nat @ B2 @ B )
=> ( ( ( insert_list_nat2 @ A2 @ A )
= ( insert_list_nat2 @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C5: set_list_nat] :
( ( A
= ( insert_list_nat2 @ B2 @ C5 ) )
& ~ ( member_list_nat @ B2 @ C5 )
& ( B
= ( insert_list_nat2 @ A2 @ C5 ) )
& ~ ( member_list_nat @ A2 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_674_insert__eq__iff,axiom,
! [A2: set_se6865892389300016395la_a_b,A: set_se1626149436229564481la_a_b,B2: set_se6865892389300016395la_a_b,B: set_se1626149436229564481la_a_b] :
( ~ ( member1693404252451860130la_a_b @ A2 @ A )
=> ( ~ ( member1693404252451860130la_a_b @ B2 @ B )
=> ( ( ( insert1549238021488837435la_a_b @ A2 @ A )
= ( insert1549238021488837435la_a_b @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C5: set_se1626149436229564481la_a_b] :
( ( A
= ( insert1549238021488837435la_a_b @ B2 @ C5 ) )
& ~ ( member1693404252451860130la_a_b @ B2 @ C5 )
& ( B
= ( insert1549238021488837435la_a_b @ A2 @ C5 ) )
& ~ ( member1693404252451860130la_a_b @ A2 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_675_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B2: nat,B: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ~ ( member_nat @ B2 @ B )
=> ( ( ( insert_nat2 @ A2 @ A )
= ( insert_nat2 @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C5: set_nat] :
( ( A
= ( insert_nat2 @ B2 @ C5 ) )
& ~ ( member_nat @ B2 @ C5 )
& ( B
= ( insert_nat2 @ A2 @ C5 ) )
& ~ ( member_nat @ A2 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_676_insert__eq__iff,axiom,
! [A2: b,A: set_b,B2: b,B: set_b] :
( ~ ( member_b @ A2 @ A )
=> ( ~ ( member_b @ B2 @ B )
=> ( ( ( insert_b2 @ A2 @ A )
= ( insert_b2 @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C5: set_b] :
( ( A
= ( insert_b2 @ B2 @ C5 ) )
& ~ ( member_b @ B2 @ C5 )
& ( B
= ( insert_b2 @ A2 @ C5 ) )
& ~ ( member_b @ A2 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_677_insert__eq__iff,axiom,
! [A2: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,B2: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b] :
( ~ ( member3481406638322139244la_a_b @ A2 @ A )
=> ( ~ ( member3481406638322139244la_a_b @ B2 @ B )
=> ( ( ( insert2023870700798818565la_a_b @ A2 @ A )
= ( insert2023870700798818565la_a_b @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C5: set_se6865892389300016395la_a_b] :
( ( A
= ( insert2023870700798818565la_a_b @ B2 @ C5 ) )
& ~ ( member3481406638322139244la_a_b @ B2 @ C5 )
& ( B
= ( insert2023870700798818565la_a_b @ A2 @ C5 ) )
& ~ ( member3481406638322139244la_a_b @ A2 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_678_insert__absorb,axiom,
! [A2: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ A )
=> ( ( insert7010464514620295119la_a_b @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_679_insert__absorb,axiom,
! [A2: list_s3671366213839843941la_a_b,A: set_li7574107037410898715la_a_b] :
( ( member987187531357909116la_a_b @ A2 @ A )
=> ( ( insert6225987573742651541la_a_b @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_680_insert__absorb,axiom,
! [A2: list_nat,A: set_list_nat] :
( ( member_list_nat @ A2 @ A )
=> ( ( insert_list_nat2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_681_insert__absorb,axiom,
! [A2: set_se6865892389300016395la_a_b,A: set_se1626149436229564481la_a_b] :
( ( member1693404252451860130la_a_b @ A2 @ A )
=> ( ( insert1549238021488837435la_a_b @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_682_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_683_insert__absorb,axiom,
! [A2: b,A: set_b] :
( ( member_b @ A2 @ A )
=> ( ( insert_b2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_684_insert__absorb,axiom,
! [A2: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ A2 @ A )
=> ( ( insert2023870700798818565la_a_b @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_685_insert__ident,axiom,
! [X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X @ A )
=> ( ~ ( member4680049679412964150la_a_b @ X @ B )
=> ( ( ( insert7010464514620295119la_a_b @ X @ A )
= ( insert7010464514620295119la_a_b @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_686_insert__ident,axiom,
! [X: list_s3671366213839843941la_a_b,A: set_li7574107037410898715la_a_b,B: set_li7574107037410898715la_a_b] :
( ~ ( member987187531357909116la_a_b @ X @ A )
=> ( ~ ( member987187531357909116la_a_b @ X @ B )
=> ( ( ( insert6225987573742651541la_a_b @ X @ A )
= ( insert6225987573742651541la_a_b @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_687_insert__ident,axiom,
! [X: list_nat,A: set_list_nat,B: set_list_nat] :
( ~ ( member_list_nat @ X @ A )
=> ( ~ ( member_list_nat @ X @ B )
=> ( ( ( insert_list_nat2 @ X @ A )
= ( insert_list_nat2 @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_688_insert__ident,axiom,
! [X: set_se6865892389300016395la_a_b,A: set_se1626149436229564481la_a_b,B: set_se1626149436229564481la_a_b] :
( ~ ( member1693404252451860130la_a_b @ X @ A )
=> ( ~ ( member1693404252451860130la_a_b @ X @ B )
=> ( ( ( insert1549238021488837435la_a_b @ X @ A )
= ( insert1549238021488837435la_a_b @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_689_insert__ident,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ~ ( member_nat @ X @ B )
=> ( ( ( insert_nat2 @ X @ A )
= ( insert_nat2 @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_690_insert__ident,axiom,
! [X: b,A: set_b,B: set_b] :
( ~ ( member_b @ X @ A )
=> ( ~ ( member_b @ X @ B )
=> ( ( ( insert_b2 @ X @ A )
= ( insert_b2 @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_691_insert__ident,axiom,
! [X: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ~ ( member3481406638322139244la_a_b @ X @ A )
=> ( ~ ( member3481406638322139244la_a_b @ X @ B )
=> ( ( ( insert2023870700798818565la_a_b @ X @ A )
= ( insert2023870700798818565la_a_b @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_692_singletonD,axiom,
! [B2: list_s3671366213839843941la_a_b,A2: list_s3671366213839843941la_a_b] :
( ( member987187531357909116la_a_b @ B2 @ ( insert6225987573742651541la_a_b @ A2 @ bot_bo7655846380748197807la_a_b ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_693_singletonD,axiom,
! [B2: set_se6865892389300016395la_a_b,A2: set_se6865892389300016395la_a_b] :
( ( member1693404252451860130la_a_b @ B2 @ ( insert1549238021488837435la_a_b @ A2 @ bot_bo1069171165257752277la_a_b ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_694_singletonD,axiom,
! [B2: b,A2: b] :
( ( member_b @ B2 @ ( insert_b2 @ A2 @ bot_bot_set_b ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_695_singletonD,axiom,
! [B2: relational_fmla_a_b,A2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ B2 @ ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_696_singletonD,axiom,
! [B2: nat,A2: nat] :
( ( member_nat @ B2 @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_697_singletonD,axiom,
! [B2: list_nat,A2: list_nat] :
( ( member_list_nat @ B2 @ ( insert_list_nat2 @ A2 @ bot_bot_set_list_nat ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_698_singletonD,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_699_singletonD,axiom,
! [B2: set_Re381260168593705685la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ B2 @ ( insert2023870700798818565la_a_b @ A2 @ bot_bo2891247006866115487la_a_b ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_700_Set_Oset__insert,axiom,
! [X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X @ A )
=> ~ ! [B7: set_Re381260168593705685la_a_b] :
( ( A
= ( insert7010464514620295119la_a_b @ X @ B7 ) )
=> ( member4680049679412964150la_a_b @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_701_Set_Oset__insert,axiom,
! [X: list_s3671366213839843941la_a_b,A: set_li7574107037410898715la_a_b] :
( ( member987187531357909116la_a_b @ X @ A )
=> ~ ! [B7: set_li7574107037410898715la_a_b] :
( ( A
= ( insert6225987573742651541la_a_b @ X @ B7 ) )
=> ( member987187531357909116la_a_b @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_702_Set_Oset__insert,axiom,
! [X: list_nat,A: set_list_nat] :
( ( member_list_nat @ X @ A )
=> ~ ! [B7: set_list_nat] :
( ( A
= ( insert_list_nat2 @ X @ B7 ) )
=> ( member_list_nat @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_703_Set_Oset__insert,axiom,
! [X: set_se6865892389300016395la_a_b,A: set_se1626149436229564481la_a_b] :
( ( member1693404252451860130la_a_b @ X @ A )
=> ~ ! [B7: set_se1626149436229564481la_a_b] :
( ( A
= ( insert1549238021488837435la_a_b @ X @ B7 ) )
=> ( member1693404252451860130la_a_b @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_704_Set_Oset__insert,axiom,
! [X: nat,A: set_nat] :
( ( member_nat @ X @ A )
=> ~ ! [B7: set_nat] :
( ( A
= ( insert_nat2 @ X @ B7 ) )
=> ( member_nat @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_705_Set_Oset__insert,axiom,
! [X: b,A: set_b] :
( ( member_b @ X @ A )
=> ~ ! [B7: set_b] :
( ( A
= ( insert_b2 @ X @ B7 ) )
=> ( member_b @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_706_Set_Oset__insert,axiom,
! [X: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ X @ A )
=> ~ ! [B7: set_se6865892389300016395la_a_b] :
( ( A
= ( insert2023870700798818565la_a_b @ X @ B7 ) )
=> ( member3481406638322139244la_a_b @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_707_ex__in__conv,axiom,
! [A: set_li7574107037410898715la_a_b] :
( ( ? [X2: list_s3671366213839843941la_a_b] : ( member987187531357909116la_a_b @ X2 @ A ) )
= ( A != bot_bo7655846380748197807la_a_b ) ) ).
% ex_in_conv
thf(fact_708_ex__in__conv,axiom,
! [A: set_se1626149436229564481la_a_b] :
( ( ? [X2: set_se6865892389300016395la_a_b] : ( member1693404252451860130la_a_b @ X2 @ A ) )
= ( A != bot_bo1069171165257752277la_a_b ) ) ).
% ex_in_conv
thf(fact_709_ex__in__conv,axiom,
! [A: set_b] :
( ( ? [X2: b] : ( member_b @ X2 @ A ) )
= ( A != bot_bot_set_b ) ) ).
% ex_in_conv
thf(fact_710_ex__in__conv,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( ? [X2: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X2 @ A ) )
= ( A != bot_bo4495933725496725865la_a_b ) ) ).
% ex_in_conv
thf(fact_711_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_712_ex__in__conv,axiom,
! [A: set_list_nat] :
( ( ? [X2: list_nat] : ( member_list_nat @ X2 @ A ) )
= ( A != bot_bot_set_list_nat ) ) ).
% ex_in_conv
thf(fact_713_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_714_ex__in__conv,axiom,
! [A: set_se6865892389300016395la_a_b] :
( ( ? [X2: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ X2 @ A ) )
= ( A != bot_bo2891247006866115487la_a_b ) ) ).
% ex_in_conv
thf(fact_715_insertI2,axiom,
! [A2: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,B2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ B )
=> ( member4680049679412964150la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ B2 @ B ) ) ) ).
% insertI2
thf(fact_716_insertI2,axiom,
! [A2: list_s3671366213839843941la_a_b,B: set_li7574107037410898715la_a_b,B2: list_s3671366213839843941la_a_b] :
( ( member987187531357909116la_a_b @ A2 @ B )
=> ( member987187531357909116la_a_b @ A2 @ ( insert6225987573742651541la_a_b @ B2 @ B ) ) ) ).
% insertI2
thf(fact_717_insertI2,axiom,
! [A2: list_nat,B: set_list_nat,B2: list_nat] :
( ( member_list_nat @ A2 @ B )
=> ( member_list_nat @ A2 @ ( insert_list_nat2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_718_insertI2,axiom,
! [A2: set_se6865892389300016395la_a_b,B: set_se1626149436229564481la_a_b,B2: set_se6865892389300016395la_a_b] :
( ( member1693404252451860130la_a_b @ A2 @ B )
=> ( member1693404252451860130la_a_b @ A2 @ ( insert1549238021488837435la_a_b @ B2 @ B ) ) ) ).
% insertI2
thf(fact_719_insertI2,axiom,
! [A2: nat,B: set_nat,B2: nat] :
( ( member_nat @ A2 @ B )
=> ( member_nat @ A2 @ ( insert_nat2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_720_insertI2,axiom,
! [A2: b,B: set_b,B2: b] :
( ( member_b @ A2 @ B )
=> ( member_b @ A2 @ ( insert_b2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_721_insertI2,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b,B2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ A2 @ B )
=> ( member3481406638322139244la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ B2 @ B ) ) ) ).
% insertI2
thf(fact_722_insertI1,axiom,
! [A2: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] : ( member4680049679412964150la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A2 @ B ) ) ).
% insertI1
thf(fact_723_insertI1,axiom,
! [A2: list_s3671366213839843941la_a_b,B: set_li7574107037410898715la_a_b] : ( member987187531357909116la_a_b @ A2 @ ( insert6225987573742651541la_a_b @ A2 @ B ) ) ).
% insertI1
thf(fact_724_insertI1,axiom,
! [A2: list_nat,B: set_list_nat] : ( member_list_nat @ A2 @ ( insert_list_nat2 @ A2 @ B ) ) ).
% insertI1
thf(fact_725_insertI1,axiom,
! [A2: set_se6865892389300016395la_a_b,B: set_se1626149436229564481la_a_b] : ( member1693404252451860130la_a_b @ A2 @ ( insert1549238021488837435la_a_b @ A2 @ B ) ) ).
% insertI1
thf(fact_726_insertI1,axiom,
! [A2: nat,B: set_nat] : ( member_nat @ A2 @ ( insert_nat2 @ A2 @ B ) ) ).
% insertI1
thf(fact_727_insertI1,axiom,
! [A2: b,B: set_b] : ( member_b @ A2 @ ( insert_b2 @ A2 @ B ) ) ).
% insertI1
thf(fact_728_insertI1,axiom,
! [A2: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b] : ( member3481406638322139244la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ A2 @ B ) ) ).
% insertI1
thf(fact_729_equals0I,axiom,
! [A: set_li7574107037410898715la_a_b] :
( ! [Y5: list_s3671366213839843941la_a_b] :
~ ( member987187531357909116la_a_b @ Y5 @ A )
=> ( A = bot_bo7655846380748197807la_a_b ) ) ).
% equals0I
thf(fact_730_equals0I,axiom,
! [A: set_se1626149436229564481la_a_b] :
( ! [Y5: set_se6865892389300016395la_a_b] :
~ ( member1693404252451860130la_a_b @ Y5 @ A )
=> ( A = bot_bo1069171165257752277la_a_b ) ) ).
% equals0I
thf(fact_731_equals0I,axiom,
! [A: set_b] :
( ! [Y5: b] :
~ ( member_b @ Y5 @ A )
=> ( A = bot_bot_set_b ) ) ).
% equals0I
thf(fact_732_equals0I,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ! [Y5: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ Y5 @ A )
=> ( A = bot_bo4495933725496725865la_a_b ) ) ).
% equals0I
thf(fact_733_equals0I,axiom,
! [A: set_nat] :
( ! [Y5: nat] :
~ ( member_nat @ Y5 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_734_equals0I,axiom,
! [A: set_list_nat] :
( ! [Y5: list_nat] :
~ ( member_list_nat @ Y5 @ A )
=> ( A = bot_bot_set_list_nat ) ) ).
% equals0I
thf(fact_735_equals0I,axiom,
! [A: set_a] :
( ! [Y5: a] :
~ ( member_a @ Y5 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_736_equals0I,axiom,
! [A: set_se6865892389300016395la_a_b] :
( ! [Y5: set_Re381260168593705685la_a_b] :
~ ( member3481406638322139244la_a_b @ Y5 @ A )
=> ( A = bot_bo2891247006866115487la_a_b ) ) ).
% equals0I
thf(fact_737_equals0D,axiom,
! [A: set_li7574107037410898715la_a_b,A2: list_s3671366213839843941la_a_b] :
( ( A = bot_bo7655846380748197807la_a_b )
=> ~ ( member987187531357909116la_a_b @ A2 @ A ) ) ).
% equals0D
thf(fact_738_equals0D,axiom,
! [A: set_se1626149436229564481la_a_b,A2: set_se6865892389300016395la_a_b] :
( ( A = bot_bo1069171165257752277la_a_b )
=> ~ ( member1693404252451860130la_a_b @ A2 @ A ) ) ).
% equals0D
thf(fact_739_equals0D,axiom,
! [A: set_b,A2: b] :
( ( A = bot_bot_set_b )
=> ~ ( member_b @ A2 @ A ) ) ).
% equals0D
thf(fact_740_equals0D,axiom,
! [A: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b] :
( ( A = bot_bo4495933725496725865la_a_b )
=> ~ ( member4680049679412964150la_a_b @ A2 @ A ) ) ).
% equals0D
thf(fact_741_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_742_equals0D,axiom,
! [A: set_list_nat,A2: list_nat] :
( ( A = bot_bot_set_list_nat )
=> ~ ( member_list_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_743_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_744_equals0D,axiom,
! [A: set_se6865892389300016395la_a_b,A2: set_Re381260168593705685la_a_b] :
( ( A = bot_bo2891247006866115487la_a_b )
=> ~ ( member3481406638322139244la_a_b @ A2 @ A ) ) ).
% equals0D
thf(fact_745_insertE,axiom,
! [A2: relational_fmla_a_b,B2: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member4680049679412964150la_a_b @ A2 @ A ) ) ) ).
% insertE
thf(fact_746_insertE,axiom,
! [A2: list_s3671366213839843941la_a_b,B2: list_s3671366213839843941la_a_b,A: set_li7574107037410898715la_a_b] :
( ( member987187531357909116la_a_b @ A2 @ ( insert6225987573742651541la_a_b @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member987187531357909116la_a_b @ A2 @ A ) ) ) ).
% insertE
thf(fact_747_insertE,axiom,
! [A2: list_nat,B2: list_nat,A: set_list_nat] :
( ( member_list_nat @ A2 @ ( insert_list_nat2 @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_list_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_748_insertE,axiom,
! [A2: set_se6865892389300016395la_a_b,B2: set_se6865892389300016395la_a_b,A: set_se1626149436229564481la_a_b] :
( ( member1693404252451860130la_a_b @ A2 @ ( insert1549238021488837435la_a_b @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member1693404252451860130la_a_b @ A2 @ A ) ) ) ).
% insertE
thf(fact_749_insertE,axiom,
! [A2: nat,B2: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat2 @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_750_insertE,axiom,
! [A2: b,B2: b,A: set_b] :
( ( member_b @ A2 @ ( insert_b2 @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_b @ A2 @ A ) ) ) ).
% insertE
thf(fact_751_insertE,axiom,
! [A2: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ A2 @ ( insert2023870700798818565la_a_b @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member3481406638322139244la_a_b @ A2 @ A ) ) ) ).
% insertE
thf(fact_752_emptyE,axiom,
! [A2: list_s3671366213839843941la_a_b] :
~ ( member987187531357909116la_a_b @ A2 @ bot_bo7655846380748197807la_a_b ) ).
% emptyE
thf(fact_753_emptyE,axiom,
! [A2: set_se6865892389300016395la_a_b] :
~ ( member1693404252451860130la_a_b @ A2 @ bot_bo1069171165257752277la_a_b ) ).
% emptyE
thf(fact_754_emptyE,axiom,
! [A2: b] :
~ ( member_b @ A2 @ bot_bot_set_b ) ).
% emptyE
thf(fact_755_emptyE,axiom,
! [A2: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ).
% emptyE
thf(fact_756_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_757_emptyE,axiom,
! [A2: list_nat] :
~ ( member_list_nat @ A2 @ bot_bot_set_list_nat ) ).
% emptyE
thf(fact_758_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_759_emptyE,axiom,
! [A2: set_Re381260168593705685la_a_b] :
~ ( member3481406638322139244la_a_b @ A2 @ bot_bo2891247006866115487la_a_b ) ).
% emptyE
thf(fact_760_sat__erase,axiom,
! [Q: relational_fmla_a_b,X: nat,I2: product_prod_b_nat > set_list_a,Sigma2: nat > a,Z2: a] :
( ( relational_sat_a_b @ ( relational_erase_a_b @ Q @ X ) @ I2 @ ( fun_upd_nat_a @ Sigma2 @ X @ Z2 ) )
= ( relational_sat_a_b @ ( relational_erase_a_b @ Q @ X ) @ I2 @ Sigma2 ) ) ).
% sat_erase
thf(fact_761_singleton__Un__iff,axiom,
! [X: b,A: set_b,B: set_b] :
( ( ( insert_b2 @ X @ bot_bot_set_b )
= ( sup_sup_set_b @ A @ B ) )
= ( ( ( A = bot_bot_set_b )
& ( B
= ( insert_b2 @ X @ bot_bot_set_b ) ) )
| ( ( A
= ( insert_b2 @ X @ bot_bot_set_b ) )
& ( B = bot_bot_set_b ) )
| ( ( A
= ( insert_b2 @ X @ bot_bot_set_b ) )
& ( B
= ( insert_b2 @ X @ bot_bot_set_b ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_762_singleton__Un__iff,axiom,
! [X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b )
= ( sup_su5130108678486352897la_a_b @ A @ B ) )
= ( ( ( A = bot_bo4495933725496725865la_a_b )
& ( B
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) )
| ( ( A
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
& ( B = bot_bo4495933725496725865la_a_b ) )
| ( ( A
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
& ( B
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_763_singleton__Un__iff,axiom,
! [X: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b )
= ( sup_su4783144482993978935la_a_b @ A @ B ) )
= ( ( ( A = bot_bo2891247006866115487la_a_b )
& ( B
= ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) ) )
| ( ( A
= ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) )
& ( B = bot_bo2891247006866115487la_a_b ) )
| ( ( A
= ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) )
& ( B
= ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_764_singleton__Un__iff,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ( ( insert_nat2 @ X @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A @ B ) )
= ( ( ( A = bot_bot_set_nat )
& ( B
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) )
| ( ( A
= ( insert_nat2 @ X @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A
= ( insert_nat2 @ X @ bot_bot_set_nat ) )
& ( B
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_765_singleton__Un__iff,axiom,
! [X: list_nat,A: set_list_nat,B: set_list_nat] :
( ( ( insert_list_nat2 @ X @ bot_bot_set_list_nat )
= ( sup_sup_set_list_nat @ A @ B ) )
= ( ( ( A = bot_bot_set_list_nat )
& ( B
= ( insert_list_nat2 @ X @ bot_bot_set_list_nat ) ) )
| ( ( A
= ( insert_list_nat2 @ X @ bot_bot_set_list_nat ) )
& ( B = bot_bot_set_list_nat ) )
| ( ( A
= ( insert_list_nat2 @ X @ bot_bot_set_list_nat ) )
& ( B
= ( insert_list_nat2 @ X @ bot_bot_set_list_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_766_singleton__Un__iff,axiom,
! [X: a,A: set_a,B: set_a] :
( ( ( insert_a @ X @ bot_bot_set_a )
= ( sup_sup_set_a @ A @ B ) )
= ( ( ( A = bot_bot_set_a )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_767_Un__singleton__iff,axiom,
! [A: set_b,B: set_b,X: b] :
( ( ( sup_sup_set_b @ A @ B )
= ( insert_b2 @ X @ bot_bot_set_b ) )
= ( ( ( A = bot_bot_set_b )
& ( B
= ( insert_b2 @ X @ bot_bot_set_b ) ) )
| ( ( A
= ( insert_b2 @ X @ bot_bot_set_b ) )
& ( B = bot_bot_set_b ) )
| ( ( A
= ( insert_b2 @ X @ bot_bot_set_b ) )
& ( B
= ( insert_b2 @ X @ bot_bot_set_b ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_768_Un__singleton__iff,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,X: relational_fmla_a_b] :
( ( ( sup_su5130108678486352897la_a_b @ A @ B )
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
= ( ( ( A = bot_bo4495933725496725865la_a_b )
& ( B
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) )
| ( ( A
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
& ( B = bot_bo4495933725496725865la_a_b ) )
| ( ( A
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
& ( B
= ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_769_Un__singleton__iff,axiom,
! [A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b,X: set_Re381260168593705685la_a_b] :
( ( ( sup_su4783144482993978935la_a_b @ A @ B )
= ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) )
= ( ( ( A = bot_bo2891247006866115487la_a_b )
& ( B
= ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) ) )
| ( ( A
= ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) )
& ( B = bot_bo2891247006866115487la_a_b ) )
| ( ( A
= ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) )
& ( B
= ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_770_Un__singleton__iff,axiom,
! [A: set_nat,B: set_nat,X: nat] :
( ( ( sup_sup_set_nat @ A @ B )
= ( insert_nat2 @ X @ bot_bot_set_nat ) )
= ( ( ( A = bot_bot_set_nat )
& ( B
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) )
| ( ( A
= ( insert_nat2 @ X @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A
= ( insert_nat2 @ X @ bot_bot_set_nat ) )
& ( B
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_771_Un__singleton__iff,axiom,
! [A: set_list_nat,B: set_list_nat,X: list_nat] :
( ( ( sup_sup_set_list_nat @ A @ B )
= ( insert_list_nat2 @ X @ bot_bot_set_list_nat ) )
= ( ( ( A = bot_bot_set_list_nat )
& ( B
= ( insert_list_nat2 @ X @ bot_bot_set_list_nat ) ) )
| ( ( A
= ( insert_list_nat2 @ X @ bot_bot_set_list_nat ) )
& ( B = bot_bot_set_list_nat ) )
| ( ( A
= ( insert_list_nat2 @ X @ bot_bot_set_list_nat ) )
& ( B
= ( insert_list_nat2 @ X @ bot_bot_set_list_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_772_Un__singleton__iff,axiom,
! [A: set_a,B: set_a,X: a] :
( ( ( sup_sup_set_a @ A @ B )
= ( insert_a @ X @ bot_bot_set_a ) )
= ( ( ( A = bot_bot_set_a )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_773_insert__is__Un,axiom,
( insert_b2
= ( ^ [A4: b] : ( sup_sup_set_b @ ( insert_b2 @ A4 @ bot_bot_set_b ) ) ) ) ).
% insert_is_Un
thf(fact_774_insert__is__Un,axiom,
( insert7010464514620295119la_a_b
= ( ^ [A4: relational_fmla_a_b] : ( sup_su5130108678486352897la_a_b @ ( insert7010464514620295119la_a_b @ A4 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% insert_is_Un
thf(fact_775_insert__is__Un,axiom,
( insert2023870700798818565la_a_b
= ( ^ [A4: set_Re381260168593705685la_a_b] : ( sup_su4783144482993978935la_a_b @ ( insert2023870700798818565la_a_b @ A4 @ bot_bo2891247006866115487la_a_b ) ) ) ) ).
% insert_is_Un
thf(fact_776_insert__is__Un,axiom,
( insert_nat2
= ( ^ [A4: nat] : ( sup_sup_set_nat @ ( insert_nat2 @ A4 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_777_insert__is__Un,axiom,
( insert_list_nat2
= ( ^ [A4: list_nat] : ( sup_sup_set_list_nat @ ( insert_list_nat2 @ A4 @ bot_bot_set_list_nat ) ) ) ) ).
% insert_is_Un
thf(fact_778_insert__is__Un,axiom,
( insert_a
= ( ^ [A4: a] : ( sup_sup_set_a @ ( insert_a @ A4 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_779_sat_Osimps_I4_J,axiom,
! [Phi2: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma2: nat > a] :
( ( relational_sat_a_b @ ( relational_Neg_a_b @ Phi2 ) @ I2 @ Sigma2 )
= ( ~ ( relational_sat_a_b @ Phi2 @ I2 @ Sigma2 ) ) ) ).
% sat.simps(4)
thf(fact_780_sat_Osimps_I5_J,axiom,
! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma2: nat > a] :
( ( relational_sat_a_b @ ( relational_Conj_a_b @ Phi2 @ Psi2 ) @ I2 @ Sigma2 )
= ( ( relational_sat_a_b @ Phi2 @ I2 @ Sigma2 )
& ( relational_sat_a_b @ Psi2 @ I2 @ Sigma2 ) ) ) ).
% sat.simps(5)
thf(fact_781_sat_Osimps_I6_J,axiom,
! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma2: nat > a] :
( ( relational_sat_a_b @ ( relational_Disj_a_b @ Phi2 @ Psi2 ) @ I2 @ Sigma2 )
= ( ( relational_sat_a_b @ Phi2 @ I2 @ Sigma2 )
| ( relational_sat_a_b @ Psi2 @ I2 @ Sigma2 ) ) ) ).
% sat.simps(6)
thf(fact_782_sat_Osimps_I2_J,axiom,
! [B2: $o,I2: product_prod_b_nat > set_list_a,Sigma2: nat > a] :
( ( relational_sat_a_b @ ( relational_Bool_a_b @ B2 ) @ I2 @ Sigma2 )
= B2 ) ).
% sat.simps(2)
thf(fact_783_sat__fv__cong,axiom,
! [Phi2: relational_fmla_a_b,Sigma2: nat > a,Sigma3: nat > a,I2: product_prod_b_nat > set_list_a] :
( ! [N: nat] :
( ( member_nat @ N @ ( relational_fv_a_b @ Phi2 ) )
=> ( ( Sigma2 @ N )
= ( Sigma3 @ N ) ) )
=> ( ( relational_sat_a_b @ Phi2 @ I2 @ Sigma2 )
= ( relational_sat_a_b @ Phi2 @ I2 @ Sigma3 ) ) ) ).
% sat_fv_cong
thf(fact_784_gen__sat,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,I2: product_prod_b_nat > set_list_a,Sigma2: nat > a] :
( ( relational_gen_a_b @ X @ Q @ G )
=> ( ( relational_sat_a_b @ Q @ I2 @ Sigma2 )
=> ? [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ G )
& ( relational_sat_a_b @ X3 @ I2 @ Sigma2 ) ) ) ) ).
% gen_sat
thf(fact_785_erase_Osimps_I4_J,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relational_erase_a_b @ ( relational_Neg_a_b @ Q ) @ X )
= ( relational_Neg_a_b @ ( relational_erase_a_b @ Q @ X ) ) ) ).
% erase.simps(4)
thf(fact_786_erase_Osimps_I5_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b,X: nat] :
( ( relational_erase_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ X )
= ( relational_Conj_a_b @ ( relational_erase_a_b @ Q1 @ X ) @ ( relational_erase_a_b @ Q2 @ X ) ) ) ).
% erase.simps(5)
thf(fact_787_erase_Osimps_I6_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b,X: nat] :
( ( relational_erase_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) @ X )
= ( relational_Disj_a_b @ ( relational_erase_a_b @ Q1 @ X ) @ ( relational_erase_a_b @ Q2 @ X ) ) ) ).
% erase.simps(6)
thf(fact_788_erase_Osimps_I1_J,axiom,
! [T3: $o,X: nat] :
( ( relational_erase_a_b @ ( relational_Bool_a_b @ T3 ) @ X )
= ( relational_Bool_a_b @ T3 ) ) ).
% erase.simps(1)
thf(fact_789_erase_Osimps_I7_J,axiom,
! [X: nat,Z2: nat,Q: relational_fmla_a_b] :
( ( ( X = Z2 )
=> ( ( relational_erase_a_b @ ( relati591517084277583526ts_a_b @ Z2 @ Q ) @ X )
= ( relati591517084277583526ts_a_b @ X @ Q ) ) )
& ( ( X != Z2 )
=> ( ( relational_erase_a_b @ ( relati591517084277583526ts_a_b @ Z2 @ Q ) @ X )
= ( relati591517084277583526ts_a_b @ Z2 @ ( relational_erase_a_b @ Q @ X ) ) ) ) ) ).
% erase.simps(7)
thf(fact_790_fmla_Osimps_I127_J,axiom,
! [X11: b,X12: list_R6823256787227418703term_a] :
( ( relati8924981150291758614la_a_b @ ( relational_Pred_b_a @ X11 @ X12 ) )
= ( insert_b2 @ X11 @ bot_bot_set_b ) ) ).
% fmla.simps(127)
thf(fact_791_sub_Osimps_I3_J,axiom,
! [X: nat,T3: relational_term_a] :
( ( relational_sub_a_b @ ( relational_Eq_a_b @ X @ T3 ) )
= ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ T3 ) @ bot_bo4495933725496725865la_a_b ) ) ).
% sub.simps(3)
thf(fact_792_sub_Osimps_I1_J,axiom,
! [T3: $o] :
( ( relational_sub_a_b @ ( relational_Bool_a_b @ T3 ) )
= ( insert7010464514620295119la_a_b @ ( relational_Bool_a_b @ T3 ) @ bot_bo4495933725496725865la_a_b ) ) ).
% sub.simps(1)
thf(fact_793_sub_Osimps_I2_J,axiom,
! [P4: b,Ts3: list_R6823256787227418703term_a] :
( ( relational_sub_a_b @ ( relational_Pred_b_a @ P4 @ Ts3 ) )
= ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ P4 @ Ts3 ) @ bot_bo4495933725496725865la_a_b ) ) ).
% sub.simps(2)
thf(fact_794_Gen__erase,axiom,
! [X: nat,Q: relational_fmla_a_b,Z2: nat] :
( ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ Q @ X_1 )
=> ? [X_12: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ ( relational_erase_a_b @ Q @ Z2 ) @ X_12 ) ) ).
% Gen_erase
thf(fact_795_gen__Gen__erase,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Z2: nat] :
( ( relational_gen_a_b @ X @ Q @ G )
=> ? [X_12: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ ( relational_erase_a_b @ Q @ Z2 ) @ X_12 ) ) ).
% gen_Gen_erase
thf(fact_796_Un__empty__left,axiom,
! [B: set_b] :
( ( sup_sup_set_b @ bot_bot_set_b @ B )
= B ) ).
% Un_empty_left
thf(fact_797_Un__empty__left,axiom,
! [B: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ bot_bo4495933725496725865la_a_b @ B )
= B ) ).
% Un_empty_left
thf(fact_798_Un__empty__left,axiom,
! [B: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ bot_bo2891247006866115487la_a_b @ B )
= B ) ).
% Un_empty_left
thf(fact_799_Un__empty__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_800_Un__empty__left,axiom,
! [B: set_list_nat] :
( ( sup_sup_set_list_nat @ bot_bot_set_list_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_801_Un__empty__left,axiom,
! [B: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B )
= B ) ).
% Un_empty_left
thf(fact_802_Un__empty__right,axiom,
! [A: set_b] :
( ( sup_sup_set_b @ A @ bot_bot_set_b )
= A ) ).
% Un_empty_right
thf(fact_803_Un__empty__right,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ A @ bot_bo4495933725496725865la_a_b )
= A ) ).
% Un_empty_right
thf(fact_804_Un__empty__right,axiom,
! [A: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ A @ bot_bo2891247006866115487la_a_b )
= A ) ).
% Un_empty_right
thf(fact_805_Un__empty__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Un_empty_right
thf(fact_806_Un__empty__right,axiom,
! [A: set_list_nat] :
( ( sup_sup_set_list_nat @ A @ bot_bot_set_list_nat )
= A ) ).
% Un_empty_right
thf(fact_807_Un__empty__right,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% Un_empty_right
thf(fact_808_qp__gen,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relational_qp_a_b @ Q )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( relational_gen_a_b @ X @ Q @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% qp_gen
thf(fact_809_gen_Ointros_I2_J,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relational_ap_a_b @ Q )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( relational_gen_a_b @ X @ Q @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% gen.intros(2)
thf(fact_810_sat_Osimps_I7_J,axiom,
! [Z2: nat,Phi2: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma2: nat > a] :
( ( relational_sat_a_b @ ( relati591517084277583526ts_a_b @ Z2 @ Phi2 ) @ I2 @ Sigma2 )
= ( ? [X2: a] : ( relational_sat_a_b @ Phi2 @ I2 @ ( fun_upd_nat_a @ Sigma2 @ Z2 @ X2 ) ) ) ) ).
% sat.simps(7)
thf(fact_811_sat__fun__upd,axiom,
! [N2: nat,Q: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma2: nat > a,Z2: a] :
( ~ ( member_nat @ N2 @ ( relational_fv_a_b @ Q ) )
=> ( ( relational_sat_a_b @ Q @ I2 @ ( fun_upd_nat_a @ Sigma2 @ N2 @ Z2 ) )
= ( relational_sat_a_b @ Q @ I2 @ Sigma2 ) ) ) ).
% sat_fun_upd
thf(fact_812_empty__set,axiom,
( bot_bot_set_set_nat
= ( set_set_nat2 @ nil_set_nat ) ) ).
% empty_set
thf(fact_813_empty__set,axiom,
( bot_bo3230846185698255993la_a_b
= ( set_li8988895567068466906la_a_b @ nil_li4862165533606214569la_a_b ) ) ).
% empty_set
thf(fact_814_empty__set,axiom,
( bot_bo7655846380748197807la_a_b
= ( set_li3253934503784678672la_a_b @ nil_li6858553494156228063la_a_b ) ) ).
% empty_set
thf(fact_815_empty__set,axiom,
( bot_bo4495933725496725865la_a_b
= ( set_Re9104216502384355786la_a_b @ nil_Re6358386334527539737la_a_b ) ) ).
% empty_set
thf(fact_816_empty__set,axiom,
( bot_bot_set_list_nat
= ( set_list_nat2 @ nil_list_nat ) ) ).
% empty_set
thf(fact_817_empty__set,axiom,
( bot_bot_set_a
= ( set_a2 @ nil_a ) ) ).
% empty_set
thf(fact_818_empty__set,axiom,
( bot_bot_set_nat
= ( set_nat2 @ nil_nat ) ) ).
% empty_set
thf(fact_819_empty__set,axiom,
( bot_bo2891247006866115487la_a_b
= ( set_se5111286776497155840la_a_b @ nil_se6596116450819887951la_a_b ) ) ).
% empty_set
thf(fact_820_sub_Osimps_I4_J,axiom,
! [Q: relational_fmla_a_b] :
( ( relational_sub_a_b @ ( relational_Neg_a_b @ Q ) )
= ( insert7010464514620295119la_a_b @ ( relational_Neg_a_b @ Q ) @ ( relational_sub_a_b @ Q ) ) ) ).
% sub.simps(4)
thf(fact_821_sub_Osimps_I7_J,axiom,
! [Z2: nat,Q: relational_fmla_a_b] :
( ( relational_sub_a_b @ ( relati591517084277583526ts_a_b @ Z2 @ Q ) )
= ( insert7010464514620295119la_a_b @ ( relati591517084277583526ts_a_b @ Z2 @ Q ) @ ( relational_sub_a_b @ Q ) ) ) ).
% sub.simps(7)
thf(fact_822_gen_Ointros_I1_J,axiom,
! [X: nat] : ( relational_gen_a_b @ X @ ( relational_Bool_a_b @ $false ) @ bot_bo4495933725496725865la_a_b ) ).
% gen.intros(1)
thf(fact_823_fmla_Osimps_I129_J,axiom,
! [X31: nat,X32: relational_term_a] :
( ( relati8924981150291758614la_a_b @ ( relational_Eq_a_b @ X31 @ X32 ) )
= bot_bot_set_b ) ).
% fmla.simps(129)
thf(fact_824_fmla_Osimps_I128_J,axiom,
! [X22: $o] :
( ( relati8924981150291758614la_a_b @ ( relational_Bool_a_b @ X22 ) )
= bot_bot_set_b ) ).
% fmla.simps(128)
thf(fact_825_gen__genempty,axiom,
! [Z2: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ Z2 @ Q @ G )
=> ( ( G = bot_bo4495933725496725865la_a_b )
=> ( relati5999705594545617851ty_a_b @ Q ) ) ) ).
% gen_genempty
thf(fact_826_gen__cp__erase,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
( ( relational_gen_a_b @ X @ Q @ G )
=> ( ( member4680049679412964150la_a_b @ Qqp @ G )
=> ( ( relational_cp_a_b @ ( relational_erase_a_b @ Qqp @ X ) )
= ( relational_Bool_a_b @ $false ) ) ) ) ).
% gen_cp_erase
thf(fact_827_sub_Osimps_I5_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( relational_sub_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) )
= ( insert7010464514620295119la_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q1 ) @ ( relational_sub_a_b @ Q2 ) ) ) ) ).
% sub.simps(5)
thf(fact_828_sub_Osimps_I6_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( relational_sub_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) )
= ( insert7010464514620295119la_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q1 ) @ ( relational_sub_a_b @ Q2 ) ) ) ) ).
% sub.simps(6)
thf(fact_829_gen__empty__cp,axiom,
! [Z2: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ Z2 @ Q @ G )
=> ( ( G = bot_bo4495933725496725865la_a_b )
=> ( ( relational_cp_a_b @ Q )
= ( relational_Bool_a_b @ $false ) ) ) ) ).
% gen_empty_cp
thf(fact_830_qp__cp__erase,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relational_qp_a_b @ Q )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q @ X ) )
= ( relational_Bool_a_b @ $false ) ) ) ) ).
% qp_cp_erase
thf(fact_831_flat__Disj_Oelims,axiom,
! [X: relational_fmla_a_b,Y3: set_Re381260168593705685la_a_b] :
( ( ( restri569617705344514291sj_a_b @ X )
= Y3 )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q12 @ Q22 ) )
=> ( Y3
!= ( sup_su5130108678486352897la_a_b @ ( restri569617705344514291sj_a_b @ Q12 ) @ ( restri569617705344514291sj_a_b @ Q22 ) ) ) )
=> ( ! [V2: b,Va2: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ V2 @ Va2 ) )
=> ( Y3
!= ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ V2 @ Va2 ) @ bot_bo4495933725496725865la_a_b ) ) )
=> ( ! [V2: $o] :
( ( X
= ( relational_Bool_a_b @ V2 ) )
=> ( Y3
!= ( insert7010464514620295119la_a_b @ ( relational_Bool_a_b @ V2 ) @ bot_bo4495933725496725865la_a_b ) ) )
=> ( ! [V2: nat,Va2: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ V2 @ Va2 ) )
=> ( Y3
!= ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ V2 @ Va2 ) @ bot_bo4495933725496725865la_a_b ) ) )
=> ( ! [V2: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ V2 ) )
=> ( Y3
!= ( insert7010464514620295119la_a_b @ ( relational_Neg_a_b @ V2 ) @ bot_bo4495933725496725865la_a_b ) ) )
=> ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ V2 @ Va2 ) )
=> ( Y3
!= ( insert7010464514620295119la_a_b @ ( relational_Conj_a_b @ V2 @ Va2 ) @ bot_bo4495933725496725865la_a_b ) ) )
=> ~ ! [V2: nat,Va2: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ V2 @ Va2 ) )
=> ( Y3
!= ( insert7010464514620295119la_a_b @ ( relati591517084277583526ts_a_b @ V2 @ Va2 ) @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ) ) ) ) ) ).
% flat_Disj.elims
thf(fact_832_gen__impl_Opsimps_I3_J,axiom,
! [X: nat,Y3: nat,C: a] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X @ ( relational_Eq_a_b @ Y3 @ ( relational_Const_a @ C ) ) ) )
=> ( ( ( X = Y3 )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relational_Eq_a_b @ Y3 @ ( relational_Const_a @ C ) ) )
= ( cons_s2104095074707585951la_a_b @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ Y3 @ ( relational_Const_a @ C ) ) @ bot_bo4495933725496725865la_a_b ) @ nil_se6596116450819887951la_a_b ) ) )
& ( ( X != Y3 )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relational_Eq_a_b @ Y3 @ ( relational_Const_a @ C ) ) )
= nil_se6596116450819887951la_a_b ) ) ) ) ).
% gen_impl.psimps(3)
thf(fact_833_flat__Disj_Opelims,axiom,
! [X: relational_fmla_a_b,Y3: set_Re381260168593705685la_a_b] :
( ( ( restri569617705344514291sj_a_b @ X )
= Y3 )
=> ( ( accp_R989495437599811158la_a_b @ restri7773364413411414152el_a_b @ X )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q12 @ Q22 ) )
=> ( ( Y3
= ( sup_su5130108678486352897la_a_b @ ( restri569617705344514291sj_a_b @ Q12 ) @ ( restri569617705344514291sj_a_b @ Q22 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ restri7773364413411414152el_a_b @ ( relational_Disj_a_b @ Q12 @ Q22 ) ) ) )
=> ( ! [V2: b,Va2: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ V2 @ Va2 ) )
=> ( ( Y3
= ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ V2 @ Va2 ) @ bot_bo4495933725496725865la_a_b ) )
=> ~ ( accp_R989495437599811158la_a_b @ restri7773364413411414152el_a_b @ ( relational_Pred_b_a @ V2 @ Va2 ) ) ) )
=> ( ! [V2: $o] :
( ( X
= ( relational_Bool_a_b @ V2 ) )
=> ( ( Y3
= ( insert7010464514620295119la_a_b @ ( relational_Bool_a_b @ V2 ) @ bot_bo4495933725496725865la_a_b ) )
=> ~ ( accp_R989495437599811158la_a_b @ restri7773364413411414152el_a_b @ ( relational_Bool_a_b @ V2 ) ) ) )
=> ( ! [V2: nat,Va2: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ V2 @ Va2 ) )
=> ( ( Y3
= ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ V2 @ Va2 ) @ bot_bo4495933725496725865la_a_b ) )
=> ~ ( accp_R989495437599811158la_a_b @ restri7773364413411414152el_a_b @ ( relational_Eq_a_b @ V2 @ Va2 ) ) ) )
=> ( ! [V2: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ V2 ) )
=> ( ( Y3
= ( insert7010464514620295119la_a_b @ ( relational_Neg_a_b @ V2 ) @ bot_bo4495933725496725865la_a_b ) )
=> ~ ( accp_R989495437599811158la_a_b @ restri7773364413411414152el_a_b @ ( relational_Neg_a_b @ V2 ) ) ) )
=> ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ V2 @ Va2 ) )
=> ( ( Y3
= ( insert7010464514620295119la_a_b @ ( relational_Conj_a_b @ V2 @ Va2 ) @ bot_bo4495933725496725865la_a_b ) )
=> ~ ( accp_R989495437599811158la_a_b @ restri7773364413411414152el_a_b @ ( relational_Conj_a_b @ V2 @ Va2 ) ) ) )
=> ~ ! [V2: nat,Va2: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ V2 @ Va2 ) )
=> ( ( Y3
= ( insert7010464514620295119la_a_b @ ( relati591517084277583526ts_a_b @ V2 @ Va2 ) @ bot_bo4495933725496725865la_a_b ) )
=> ~ ( accp_R989495437599811158la_a_b @ restri7773364413411414152el_a_b @ ( relati591517084277583526ts_a_b @ V2 @ Va2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% flat_Disj.pelims
thf(fact_834_the__elem__eq,axiom,
! [X: relational_fmla_a_b] :
( ( the_el6350558617753882986la_a_b @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
= X ) ).
% the_elem_eq
thf(fact_835_the__elem__eq,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( the_el7486773796875720352la_a_b @ ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) )
= X ) ).
% the_elem_eq
thf(fact_836_the__elem__eq,axiom,
! [X: nat] :
( ( the_elem_nat @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= X ) ).
% the_elem_eq
thf(fact_837_the__elem__eq,axiom,
! [X: list_nat] :
( ( the_elem_list_nat @ ( insert_list_nat2 @ X @ bot_bot_set_list_nat ) )
= X ) ).
% the_elem_eq
thf(fact_838_the__elem__eq,axiom,
! [X: a] :
( ( the_elem_a @ ( insert_a @ X @ bot_bot_set_a ) )
= X ) ).
% the_elem_eq
thf(fact_839_gen__Pred,axiom,
! [Z2: nat,P4: b,Ts3: list_R6823256787227418703term_a,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ Z2 @ ( relational_Pred_b_a @ P4 @ Ts3 ) @ G )
= ( ( member_nat @ Z2 @ ( relati4569515538964159125_set_a @ Ts3 ) )
& ( G
= ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ P4 @ Ts3 ) @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% gen_Pred
thf(fact_840_gen__impl_Opsimps_I1_J,axiom,
! [X: nat] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X @ ( relational_Bool_a_b @ $false ) ) )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relational_Bool_a_b @ $false ) )
= ( cons_s2104095074707585951la_a_b @ bot_bo4495933725496725865la_a_b @ nil_se6596116450819887951la_a_b ) ) ) ).
% gen_impl.psimps(1)
thf(fact_841_list_Oinject,axiom,
! [X21: set_Re381260168593705685la_a_b,X222: list_s3671366213839843941la_a_b,Y21: set_Re381260168593705685la_a_b,Y22: list_s3671366213839843941la_a_b] :
( ( ( cons_s2104095074707585951la_a_b @ X21 @ X222 )
= ( cons_s2104095074707585951la_a_b @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_842_list_Oinject,axiom,
! [X21: list_nat,X222: list_list_nat,Y21: list_nat,Y22: list_list_nat] :
( ( ( cons_list_nat @ X21 @ X222 )
= ( cons_list_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_843_list_Oinject,axiom,
! [X21: nat,X222: list_nat,Y21: nat,Y22: list_nat] :
( ( ( cons_nat @ X21 @ X222 )
= ( cons_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_844_list_Osimps_I15_J,axiom,
! [X21: relational_fmla_a_b,X222: list_R8263082107343818799la_a_b] :
( ( set_Re9104216502384355786la_a_b @ ( cons_R4161027894713053801la_a_b @ X21 @ X222 ) )
= ( insert7010464514620295119la_a_b @ X21 @ ( set_Re9104216502384355786la_a_b @ X222 ) ) ) ).
% list.simps(15)
thf(fact_845_list_Osimps_I15_J,axiom,
! [X21: list_R8263082107343818799la_a_b,X222: list_l3216422893793986623la_a_b] :
( ( set_li8988895567068466906la_a_b @ ( cons_l6716632503304403961la_a_b @ X21 @ X222 ) )
= ( insert3617074100887873119la_a_b @ X21 @ ( set_li8988895567068466906la_a_b @ X222 ) ) ) ).
% list.simps(15)
thf(fact_846_list_Osimps_I15_J,axiom,
! [X21: list_s3671366213839843941la_a_b,X222: list_l1512381142989314933la_a_b] :
( ( set_li3253934503784678672la_a_b @ ( cons_l1224881792622212655la_a_b @ X21 @ X222 ) )
= ( insert6225987573742651541la_a_b @ X21 @ ( set_li3253934503784678672la_a_b @ X222 ) ) ) ).
% list.simps(15)
thf(fact_847_list_Osimps_I15_J,axiom,
! [X21: list_nat,X222: list_list_nat] :
( ( set_list_nat2 @ ( cons_list_nat @ X21 @ X222 ) )
= ( insert_list_nat2 @ X21 @ ( set_list_nat2 @ X222 ) ) ) ).
% list.simps(15)
thf(fact_848_list_Osimps_I15_J,axiom,
! [X21: set_Re381260168593705685la_a_b,X222: list_s3671366213839843941la_a_b] :
( ( set_se5111286776497155840la_a_b @ ( cons_s2104095074707585951la_a_b @ X21 @ X222 ) )
= ( insert2023870700798818565la_a_b @ X21 @ ( set_se5111286776497155840la_a_b @ X222 ) ) ) ).
% list.simps(15)
thf(fact_849_list_Osimps_I15_J,axiom,
! [X21: nat,X222: list_nat] :
( ( set_nat2 @ ( cons_nat @ X21 @ X222 ) )
= ( insert_nat2 @ X21 @ ( set_nat2 @ X222 ) ) ) ).
% list.simps(15)
thf(fact_850_insert__Nil,axiom,
! [X: set_nat] :
( ( insert_set_nat @ X @ nil_set_nat )
= ( cons_set_nat @ X @ nil_set_nat ) ) ).
% insert_Nil
thf(fact_851_insert__Nil,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( insert443281300761040569la_a_b @ X @ nil_se6596116450819887951la_a_b )
= ( cons_s2104095074707585951la_a_b @ X @ nil_se6596116450819887951la_a_b ) ) ).
% insert_Nil
thf(fact_852_insert__Nil,axiom,
! [X: list_nat] :
( ( insert_list_nat @ X @ nil_list_nat )
= ( cons_list_nat @ X @ nil_list_nat ) ) ).
% insert_Nil
thf(fact_853_insert__Nil,axiom,
! [X: nat] :
( ( insert_nat @ X @ nil_nat )
= ( cons_nat @ X @ nil_nat ) ) ).
% insert_Nil
thf(fact_854_not__in__set__insert,axiom,
! [X: set_se6865892389300016395la_a_b,Xs2: list_s5517785546295487899la_a_b] :
( ~ ( member1693404252451860130la_a_b @ X @ ( set_se4873105269812759862la_a_b @ Xs2 ) )
=> ( ( insert3187468692655853807la_a_b @ X @ Xs2 )
= ( cons_s4528938952058510805la_a_b @ X @ Xs2 ) ) ) ).
% not_in_set_insert
thf(fact_855_not__in__set__insert,axiom,
! [X: b,Xs2: list_b] :
( ~ ( member_b @ X @ ( set_b2 @ Xs2 ) )
=> ( ( insert_b @ X @ Xs2 )
= ( cons_b @ X @ Xs2 ) ) ) ).
% not_in_set_insert
thf(fact_856_not__in__set__insert,axiom,
! [X: relational_fmla_a_b,Xs2: list_R8263082107343818799la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Xs2 ) )
=> ( ( insert5052544347734373763la_a_b @ X @ Xs2 )
= ( cons_R4161027894713053801la_a_b @ X @ Xs2 ) ) ) ).
% not_in_set_insert
thf(fact_857_not__in__set__insert,axiom,
! [X: list_R8263082107343818799la_a_b,Xs2: list_l3216422893793986623la_a_b] :
( ~ ( member8712135081189583430la_a_b @ X @ ( set_li8988895567068466906la_a_b @ Xs2 ) )
=> ( ( insert3209172818022656531la_a_b @ X @ Xs2 )
= ( cons_l6716632503304403961la_a_b @ X @ Xs2 ) ) ) ).
% not_in_set_insert
thf(fact_858_not__in__set__insert,axiom,
! [X: list_s3671366213839843941la_a_b,Xs2: list_l1512381142989314933la_a_b] :
( ~ ( member987187531357909116la_a_b @ X @ ( set_li3253934503784678672la_a_b @ Xs2 ) )
=> ( ( insert3122035991742315593la_a_b @ X @ Xs2 )
= ( cons_l1224881792622212655la_a_b @ X @ Xs2 ) ) ) ).
% not_in_set_insert
thf(fact_859_not__in__set__insert,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ~ ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
=> ( ( insert_list_nat @ X @ Xs2 )
= ( cons_list_nat @ X @ Xs2 ) ) ) ).
% not_in_set_insert
thf(fact_860_not__in__set__insert,axiom,
! [X: set_Re381260168593705685la_a_b,Xs2: list_s3671366213839843941la_a_b] :
( ~ ( member3481406638322139244la_a_b @ X @ ( set_se5111286776497155840la_a_b @ Xs2 ) )
=> ( ( insert443281300761040569la_a_b @ X @ Xs2 )
= ( cons_s2104095074707585951la_a_b @ X @ Xs2 ) ) ) ).
% not_in_set_insert
thf(fact_861_not__in__set__insert,axiom,
! [X: nat,Xs2: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ( insert_nat @ X @ Xs2 )
= ( cons_nat @ X @ Xs2 ) ) ) ).
% not_in_set_insert
thf(fact_862_not__Cons__self2,axiom,
! [X: set_Re381260168593705685la_a_b,Xs2: list_s3671366213839843941la_a_b] :
( ( cons_s2104095074707585951la_a_b @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_863_not__Cons__self2,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( cons_list_nat @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_864_not__Cons__self2,axiom,
! [X: nat,Xs2: list_nat] :
( ( cons_nat @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_865_transpose_Ocases,axiom,
! [X: list_list_set_nat] :
( ( X != nil_list_set_nat )
=> ( ! [Xss: list_list_set_nat] :
( X
!= ( cons_list_set_nat @ nil_set_nat @ Xss ) )
=> ~ ! [X3: set_nat,Xs3: list_set_nat,Xss: list_list_set_nat] :
( X
!= ( cons_list_set_nat @ ( cons_set_nat @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_866_transpose_Ocases,axiom,
! [X: list_l1512381142989314933la_a_b] :
( ( X != nil_li6858553494156228063la_a_b )
=> ( ! [Xss: list_l1512381142989314933la_a_b] :
( X
!= ( cons_l1224881792622212655la_a_b @ nil_se6596116450819887951la_a_b @ Xss ) )
=> ~ ! [X3: set_Re381260168593705685la_a_b,Xs3: list_s3671366213839843941la_a_b,Xss: list_l1512381142989314933la_a_b] :
( X
!= ( cons_l1224881792622212655la_a_b @ ( cons_s2104095074707585951la_a_b @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_867_transpose_Ocases,axiom,
! [X: list_list_list_nat] :
( ( X != nil_list_list_nat )
=> ( ! [Xss: list_list_list_nat] :
( X
!= ( cons_list_list_nat @ nil_list_nat @ Xss ) )
=> ~ ! [X3: list_nat,Xs3: list_list_nat,Xss: list_list_list_nat] :
( X
!= ( cons_list_list_nat @ ( cons_list_nat @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_868_transpose_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X3: nat,Xs3: list_nat,Xss: list_list_nat] :
( X
!= ( cons_list_nat @ ( cons_nat @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_869_bot__set__def,axiom,
( bot_bo7655846380748197807la_a_b
= ( collec5397960198663843514la_a_b @ bot_bo4327600545544596886_a_b_o ) ) ).
% bot_set_def
thf(fact_870_bot__set__def,axiom,
( bot_bo4495933725496725865la_a_b
= ( collec3419995626248312948la_a_b @ bot_bo8852203127187332700_a_b_o ) ) ).
% bot_set_def
thf(fact_871_bot__set__def,axiom,
( bot_bo2891247006866115487la_a_b
= ( collec2099942116761351594la_a_b @ bot_bo8397419132826146726_a_b_o ) ) ).
% bot_set_def
thf(fact_872_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_873_bot__set__def,axiom,
( bot_bot_set_list_nat
= ( collect_list_nat @ bot_bot_list_nat_o ) ) ).
% bot_set_def
thf(fact_874_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_875_list_Odistinct_I1_J,axiom,
! [X21: set_nat,X222: list_set_nat] :
( nil_set_nat
!= ( cons_set_nat @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_876_list_Odistinct_I1_J,axiom,
! [X21: set_Re381260168593705685la_a_b,X222: list_s3671366213839843941la_a_b] :
( nil_se6596116450819887951la_a_b
!= ( cons_s2104095074707585951la_a_b @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_877_list_Odistinct_I1_J,axiom,
! [X21: list_nat,X222: list_list_nat] :
( nil_list_nat
!= ( cons_list_nat @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_878_list_Odistinct_I1_J,axiom,
! [X21: nat,X222: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_879_list_OdiscI,axiom,
! [List: list_set_nat,X21: set_nat,X222: list_set_nat] :
( ( List
= ( cons_set_nat @ X21 @ X222 ) )
=> ( List != nil_set_nat ) ) ).
% list.discI
thf(fact_880_list_OdiscI,axiom,
! [List: list_s3671366213839843941la_a_b,X21: set_Re381260168593705685la_a_b,X222: list_s3671366213839843941la_a_b] :
( ( List
= ( cons_s2104095074707585951la_a_b @ X21 @ X222 ) )
=> ( List != nil_se6596116450819887951la_a_b ) ) ).
% list.discI
thf(fact_881_list_OdiscI,axiom,
! [List: list_list_nat,X21: list_nat,X222: list_list_nat] :
( ( List
= ( cons_list_nat @ X21 @ X222 ) )
=> ( List != nil_list_nat ) ) ).
% list.discI
thf(fact_882_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X222: list_nat] :
( ( List
= ( cons_nat @ X21 @ X222 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_883_list_Oexhaust,axiom,
! [Y3: list_set_nat] :
( ( Y3 != nil_set_nat )
=> ~ ! [X212: set_nat,X223: list_set_nat] :
( Y3
!= ( cons_set_nat @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_884_list_Oexhaust,axiom,
! [Y3: list_s3671366213839843941la_a_b] :
( ( Y3 != nil_se6596116450819887951la_a_b )
=> ~ ! [X212: set_Re381260168593705685la_a_b,X223: list_s3671366213839843941la_a_b] :
( Y3
!= ( cons_s2104095074707585951la_a_b @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_885_list_Oexhaust,axiom,
! [Y3: list_list_nat] :
( ( Y3 != nil_list_nat )
=> ~ ! [X212: list_nat,X223: list_list_nat] :
( Y3
!= ( cons_list_nat @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_886_list_Oexhaust,axiom,
! [Y3: list_nat] :
( ( Y3 != nil_nat )
=> ~ ! [X212: nat,X223: list_nat] :
( Y3
!= ( cons_nat @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_887_min__list_Ocases,axiom,
! [X: list_set_nat] :
( ! [X3: set_nat,Xs3: list_set_nat] :
( X
!= ( cons_set_nat @ X3 @ Xs3 ) )
=> ( X = nil_set_nat ) ) ).
% min_list.cases
thf(fact_888_min__list_Ocases,axiom,
! [X: list_s3671366213839843941la_a_b] :
( ! [X3: set_Re381260168593705685la_a_b,Xs3: list_s3671366213839843941la_a_b] :
( X
!= ( cons_s2104095074707585951la_a_b @ X3 @ Xs3 ) )
=> ( X = nil_se6596116450819887951la_a_b ) ) ).
% min_list.cases
thf(fact_889_min__list_Ocases,axiom,
! [X: list_nat] :
( ! [X3: nat,Xs3: list_nat] :
( X
!= ( cons_nat @ X3 @ Xs3 ) )
=> ( X = nil_nat ) ) ).
% min_list.cases
thf(fact_890_remdups__adj_Ocases,axiom,
! [X: list_set_nat] :
( ( X != nil_set_nat )
=> ( ! [X3: set_nat] :
( X
!= ( cons_set_nat @ X3 @ nil_set_nat ) )
=> ~ ! [X3: set_nat,Y5: set_nat,Xs3: list_set_nat] :
( X
!= ( cons_set_nat @ X3 @ ( cons_set_nat @ Y5 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_891_remdups__adj_Ocases,axiom,
! [X: list_s3671366213839843941la_a_b] :
( ( X != nil_se6596116450819887951la_a_b )
=> ( ! [X3: set_Re381260168593705685la_a_b] :
( X
!= ( cons_s2104095074707585951la_a_b @ X3 @ nil_se6596116450819887951la_a_b ) )
=> ~ ! [X3: set_Re381260168593705685la_a_b,Y5: set_Re381260168593705685la_a_b,Xs3: list_s3671366213839843941la_a_b] :
( X
!= ( cons_s2104095074707585951la_a_b @ X3 @ ( cons_s2104095074707585951la_a_b @ Y5 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_892_remdups__adj_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [X3: list_nat] :
( X
!= ( cons_list_nat @ X3 @ nil_list_nat ) )
=> ~ ! [X3: list_nat,Y5: list_nat,Xs3: list_list_nat] :
( X
!= ( cons_list_nat @ X3 @ ( cons_list_nat @ Y5 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_893_remdups__adj_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ( ! [X3: nat] :
( X
!= ( cons_nat @ X3 @ nil_nat ) )
=> ~ ! [X3: nat,Y5: nat,Xs3: list_nat] :
( X
!= ( cons_nat @ X3 @ ( cons_nat @ Y5 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_894_neq__Nil__conv,axiom,
! [Xs2: list_set_nat] :
( ( Xs2 != nil_set_nat )
= ( ? [Y: set_nat,Ys2: list_set_nat] :
( Xs2
= ( cons_set_nat @ Y @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_895_neq__Nil__conv,axiom,
! [Xs2: list_s3671366213839843941la_a_b] :
( ( Xs2 != nil_se6596116450819887951la_a_b )
= ( ? [Y: set_Re381260168593705685la_a_b,Ys2: list_s3671366213839843941la_a_b] :
( Xs2
= ( cons_s2104095074707585951la_a_b @ Y @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_896_neq__Nil__conv,axiom,
! [Xs2: list_list_nat] :
( ( Xs2 != nil_list_nat )
= ( ? [Y: list_nat,Ys2: list_list_nat] :
( Xs2
= ( cons_list_nat @ Y @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_897_neq__Nil__conv,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
= ( ? [Y: nat,Ys2: list_nat] :
( Xs2
= ( cons_nat @ Y @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_898_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs2: list_nat,Ys: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X3 @ Xs3 ) @ nil_nat )
=> ( ! [Y5: nat,Ys3: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y5 @ Ys3 ) )
=> ( ! [X3: nat,Xs3: list_nat,Y5: nat,Ys3: list_nat] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y5 @ Ys3 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_899_list__induct2_H,axiom,
! [P: list_set_nat > list_nat > $o,Xs2: list_set_nat,Ys: list_nat] :
( ( P @ nil_set_nat @ nil_nat )
=> ( ! [X3: set_nat,Xs3: list_set_nat] : ( P @ ( cons_set_nat @ X3 @ Xs3 ) @ nil_nat )
=> ( ! [Y5: nat,Ys3: list_nat] : ( P @ nil_set_nat @ ( cons_nat @ Y5 @ Ys3 ) )
=> ( ! [X3: set_nat,Xs3: list_set_nat,Y5: nat,Ys3: list_nat] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_set_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y5 @ Ys3 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_900_list__induct2_H,axiom,
! [P: list_nat > list_set_nat > $o,Xs2: list_nat,Ys: list_set_nat] :
( ( P @ nil_nat @ nil_set_nat )
=> ( ! [X3: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X3 @ Xs3 ) @ nil_set_nat )
=> ( ! [Y5: set_nat,Ys3: list_set_nat] : ( P @ nil_nat @ ( cons_set_nat @ Y5 @ Ys3 ) )
=> ( ! [X3: nat,Xs3: list_nat,Y5: set_nat,Ys3: list_set_nat] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_set_nat @ Y5 @ Ys3 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_901_list__induct2_H,axiom,
! [P: list_nat > list_list_nat > $o,Xs2: list_nat,Ys: list_list_nat] :
( ( P @ nil_nat @ nil_list_nat )
=> ( ! [X3: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X3 @ Xs3 ) @ nil_list_nat )
=> ( ! [Y5: list_nat,Ys3: list_list_nat] : ( P @ nil_nat @ ( cons_list_nat @ Y5 @ Ys3 ) )
=> ( ! [X3: nat,Xs3: list_nat,Y5: list_nat,Ys3: list_list_nat] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_list_nat @ Y5 @ Ys3 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_902_list__induct2_H,axiom,
! [P: list_list_nat > list_nat > $o,Xs2: list_list_nat,Ys: list_nat] :
( ( P @ nil_list_nat @ nil_nat )
=> ( ! [X3: list_nat,Xs3: list_list_nat] : ( P @ ( cons_list_nat @ X3 @ Xs3 ) @ nil_nat )
=> ( ! [Y5: nat,Ys3: list_nat] : ( P @ nil_list_nat @ ( cons_nat @ Y5 @ Ys3 ) )
=> ( ! [X3: list_nat,Xs3: list_list_nat,Y5: nat,Ys3: list_nat] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_list_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y5 @ Ys3 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_903_list__induct2_H,axiom,
! [P: list_set_nat > list_set_nat > $o,Xs2: list_set_nat,Ys: list_set_nat] :
( ( P @ nil_set_nat @ nil_set_nat )
=> ( ! [X3: set_nat,Xs3: list_set_nat] : ( P @ ( cons_set_nat @ X3 @ Xs3 ) @ nil_set_nat )
=> ( ! [Y5: set_nat,Ys3: list_set_nat] : ( P @ nil_set_nat @ ( cons_set_nat @ Y5 @ Ys3 ) )
=> ( ! [X3: set_nat,Xs3: list_set_nat,Y5: set_nat,Ys3: list_set_nat] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_set_nat @ X3 @ Xs3 ) @ ( cons_set_nat @ Y5 @ Ys3 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_904_list__induct2_H,axiom,
! [P: list_set_nat > list_list_nat > $o,Xs2: list_set_nat,Ys: list_list_nat] :
( ( P @ nil_set_nat @ nil_list_nat )
=> ( ! [X3: set_nat,Xs3: list_set_nat] : ( P @ ( cons_set_nat @ X3 @ Xs3 ) @ nil_list_nat )
=> ( ! [Y5: list_nat,Ys3: list_list_nat] : ( P @ nil_set_nat @ ( cons_list_nat @ Y5 @ Ys3 ) )
=> ( ! [X3: set_nat,Xs3: list_set_nat,Y5: list_nat,Ys3: list_list_nat] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_set_nat @ X3 @ Xs3 ) @ ( cons_list_nat @ Y5 @ Ys3 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_905_list__induct2_H,axiom,
! [P: list_list_nat > list_set_nat > $o,Xs2: list_list_nat,Ys: list_set_nat] :
( ( P @ nil_list_nat @ nil_set_nat )
=> ( ! [X3: list_nat,Xs3: list_list_nat] : ( P @ ( cons_list_nat @ X3 @ Xs3 ) @ nil_set_nat )
=> ( ! [Y5: set_nat,Ys3: list_set_nat] : ( P @ nil_list_nat @ ( cons_set_nat @ Y5 @ Ys3 ) )
=> ( ! [X3: list_nat,Xs3: list_list_nat,Y5: set_nat,Ys3: list_set_nat] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_list_nat @ X3 @ Xs3 ) @ ( cons_set_nat @ Y5 @ Ys3 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_906_list__induct2_H,axiom,
! [P: list_list_nat > list_list_nat > $o,Xs2: list_list_nat,Ys: list_list_nat] :
( ( P @ nil_list_nat @ nil_list_nat )
=> ( ! [X3: list_nat,Xs3: list_list_nat] : ( P @ ( cons_list_nat @ X3 @ Xs3 ) @ nil_list_nat )
=> ( ! [Y5: list_nat,Ys3: list_list_nat] : ( P @ nil_list_nat @ ( cons_list_nat @ Y5 @ Ys3 ) )
=> ( ! [X3: list_nat,Xs3: list_list_nat,Y5: list_nat,Ys3: list_list_nat] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_list_nat @ X3 @ Xs3 ) @ ( cons_list_nat @ Y5 @ Ys3 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_907_list__induct2_H,axiom,
! [P: list_nat > list_s3671366213839843941la_a_b > $o,Xs2: list_nat,Ys: list_s3671366213839843941la_a_b] :
( ( P @ nil_nat @ nil_se6596116450819887951la_a_b )
=> ( ! [X3: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X3 @ Xs3 ) @ nil_se6596116450819887951la_a_b )
=> ( ! [Y5: set_Re381260168593705685la_a_b,Ys3: list_s3671366213839843941la_a_b] : ( P @ nil_nat @ ( cons_s2104095074707585951la_a_b @ Y5 @ Ys3 ) )
=> ( ! [X3: nat,Xs3: list_nat,Y5: set_Re381260168593705685la_a_b,Ys3: list_s3671366213839843941la_a_b] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_s2104095074707585951la_a_b @ Y5 @ Ys3 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_908_list__nonempty__induct,axiom,
! [Xs2: list_set_nat,P: list_set_nat > $o] :
( ( Xs2 != nil_set_nat )
=> ( ! [X3: set_nat] : ( P @ ( cons_set_nat @ X3 @ nil_set_nat ) )
=> ( ! [X3: set_nat,Xs3: list_set_nat] :
( ( Xs3 != nil_set_nat )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_set_nat @ X3 @ Xs3 ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_909_list__nonempty__induct,axiom,
! [Xs2: list_s3671366213839843941la_a_b,P: list_s3671366213839843941la_a_b > $o] :
( ( Xs2 != nil_se6596116450819887951la_a_b )
=> ( ! [X3: set_Re381260168593705685la_a_b] : ( P @ ( cons_s2104095074707585951la_a_b @ X3 @ nil_se6596116450819887951la_a_b ) )
=> ( ! [X3: set_Re381260168593705685la_a_b,Xs3: list_s3671366213839843941la_a_b] :
( ( Xs3 != nil_se6596116450819887951la_a_b )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_s2104095074707585951la_a_b @ X3 @ Xs3 ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_910_list__nonempty__induct,axiom,
! [Xs2: list_list_nat,P: list_list_nat > $o] :
( ( Xs2 != nil_list_nat )
=> ( ! [X3: list_nat] : ( P @ ( cons_list_nat @ X3 @ nil_list_nat ) )
=> ( ! [X3: list_nat,Xs3: list_list_nat] :
( ( Xs3 != nil_list_nat )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_list_nat @ X3 @ Xs3 ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_911_list__nonempty__induct,axiom,
! [Xs2: list_nat,P: list_nat > $o] :
( ( Xs2 != nil_nat )
=> ( ! [X3: nat] : ( P @ ( cons_nat @ X3 @ nil_nat ) )
=> ( ! [X3: nat,Xs3: list_nat] :
( ( Xs3 != nil_nat )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_nat @ X3 @ Xs3 ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_912_shuffles_Ocases,axiom,
! [X: produc1024980458911918861et_nat] :
( ! [Ys3: list_set_nat] :
( X
!= ( produc2728781817031571013et_nat @ nil_set_nat @ Ys3 ) )
=> ( ! [Xs3: list_set_nat] :
( X
!= ( produc2728781817031571013et_nat @ Xs3 @ nil_set_nat ) )
=> ~ ! [X3: set_nat,Xs3: list_set_nat,Y5: set_nat,Ys3: list_set_nat] :
( X
!= ( produc2728781817031571013et_nat @ ( cons_set_nat @ X3 @ Xs3 ) @ ( cons_set_nat @ Y5 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_913_shuffles_Ocases,axiom,
! [X: produc2432081345522950571la_a_b] :
( ! [Ys3: list_s3671366213839843941la_a_b] :
( X
!= ( produc6301070565816922723la_a_b @ nil_se6596116450819887951la_a_b @ Ys3 ) )
=> ( ! [Xs3: list_s3671366213839843941la_a_b] :
( X
!= ( produc6301070565816922723la_a_b @ Xs3 @ nil_se6596116450819887951la_a_b ) )
=> ~ ! [X3: set_Re381260168593705685la_a_b,Xs3: list_s3671366213839843941la_a_b,Y5: set_Re381260168593705685la_a_b,Ys3: list_s3671366213839843941la_a_b] :
( X
!= ( produc6301070565816922723la_a_b @ ( cons_s2104095074707585951la_a_b @ X3 @ Xs3 ) @ ( cons_s2104095074707585951la_a_b @ Y5 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_914_shuffles_Ocases,axiom,
! [X: produc4326814125627636033st_nat] :
( ! [Ys3: list_list_nat] :
( X
!= ( produc7129799990162260089st_nat @ nil_list_nat @ Ys3 ) )
=> ( ! [Xs3: list_list_nat] :
( X
!= ( produc7129799990162260089st_nat @ Xs3 @ nil_list_nat ) )
=> ~ ! [X3: list_nat,Xs3: list_list_nat,Y5: list_nat,Ys3: list_list_nat] :
( X
!= ( produc7129799990162260089st_nat @ ( cons_list_nat @ X3 @ Xs3 ) @ ( cons_list_nat @ Y5 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_915_shuffles_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [Ys3: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys3 ) )
=> ( ! [Xs3: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ Xs3 @ nil_nat ) )
=> ~ ! [X3: nat,Xs3: list_nat,Y5: nat,Ys3: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y5 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_916_sorted__wrt_Ocases,axiom,
! [X: produc1700580196396568895et_nat] :
( ! [P6: set_nat > set_nat > $o] :
( X
!= ( produc7645457933056093241et_nat @ P6 @ nil_set_nat ) )
=> ~ ! [P6: set_nat > set_nat > $o,X3: set_nat,Ys3: list_set_nat] :
( X
!= ( produc7645457933056093241et_nat @ P6 @ ( cons_set_nat @ X3 @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_917_sorted__wrt_Ocases,axiom,
! [X: produc4286288314677443696la_a_b] :
( ! [P6: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > $o] :
( X
!= ( produc3269847925915422698la_a_b @ P6 @ nil_se6596116450819887951la_a_b ) )
=> ~ ! [P6: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > $o,X3: set_Re381260168593705685la_a_b,Ys3: list_s3671366213839843941la_a_b] :
( X
!= ( produc3269847925915422698la_a_b @ P6 @ ( cons_s2104095074707585951la_a_b @ X3 @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_918_sorted__wrt_Ocases,axiom,
! [X: produc2068713617857708901st_nat] :
( ! [P6: list_nat > list_nat > $o] :
( X
!= ( produc6344517630863869919st_nat @ P6 @ nil_list_nat ) )
=> ~ ! [P6: list_nat > list_nat > $o,X3: list_nat,Ys3: list_list_nat] :
( X
!= ( produc6344517630863869919st_nat @ P6 @ ( cons_list_nat @ X3 @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_919_sorted__wrt_Ocases,axiom,
! [X: produc254973753779126261st_nat] :
( ! [P6: nat > nat > $o] :
( X
!= ( produc4727192421694094319st_nat @ P6 @ nil_nat ) )
=> ~ ! [P6: nat > nat > $o,X3: nat,Ys3: list_nat] :
( X
!= ( produc4727192421694094319st_nat @ P6 @ ( cons_nat @ X3 @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_920_successively_Ocases,axiom,
! [X: produc1700580196396568895et_nat] :
( ! [P6: set_nat > set_nat > $o] :
( X
!= ( produc7645457933056093241et_nat @ P6 @ nil_set_nat ) )
=> ( ! [P6: set_nat > set_nat > $o,X3: set_nat] :
( X
!= ( produc7645457933056093241et_nat @ P6 @ ( cons_set_nat @ X3 @ nil_set_nat ) ) )
=> ~ ! [P6: set_nat > set_nat > $o,X3: set_nat,Y5: set_nat,Xs3: list_set_nat] :
( X
!= ( produc7645457933056093241et_nat @ P6 @ ( cons_set_nat @ X3 @ ( cons_set_nat @ Y5 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_921_successively_Ocases,axiom,
! [X: produc4286288314677443696la_a_b] :
( ! [P6: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > $o] :
( X
!= ( produc3269847925915422698la_a_b @ P6 @ nil_se6596116450819887951la_a_b ) )
=> ( ! [P6: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > $o,X3: set_Re381260168593705685la_a_b] :
( X
!= ( produc3269847925915422698la_a_b @ P6 @ ( cons_s2104095074707585951la_a_b @ X3 @ nil_se6596116450819887951la_a_b ) ) )
=> ~ ! [P6: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > $o,X3: set_Re381260168593705685la_a_b,Y5: set_Re381260168593705685la_a_b,Xs3: list_s3671366213839843941la_a_b] :
( X
!= ( produc3269847925915422698la_a_b @ P6 @ ( cons_s2104095074707585951la_a_b @ X3 @ ( cons_s2104095074707585951la_a_b @ Y5 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_922_successively_Ocases,axiom,
! [X: produc2068713617857708901st_nat] :
( ! [P6: list_nat > list_nat > $o] :
( X
!= ( produc6344517630863869919st_nat @ P6 @ nil_list_nat ) )
=> ( ! [P6: list_nat > list_nat > $o,X3: list_nat] :
( X
!= ( produc6344517630863869919st_nat @ P6 @ ( cons_list_nat @ X3 @ nil_list_nat ) ) )
=> ~ ! [P6: list_nat > list_nat > $o,X3: list_nat,Y5: list_nat,Xs3: list_list_nat] :
( X
!= ( produc6344517630863869919st_nat @ P6 @ ( cons_list_nat @ X3 @ ( cons_list_nat @ Y5 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_923_successively_Ocases,axiom,
! [X: produc254973753779126261st_nat] :
( ! [P6: nat > nat > $o] :
( X
!= ( produc4727192421694094319st_nat @ P6 @ nil_nat ) )
=> ( ! [P6: nat > nat > $o,X3: nat] :
( X
!= ( produc4727192421694094319st_nat @ P6 @ ( cons_nat @ X3 @ nil_nat ) ) )
=> ~ ! [P6: nat > nat > $o,X3: nat,Y5: nat,Xs3: list_nat] :
( X
!= ( produc4727192421694094319st_nat @ P6 @ ( cons_nat @ X3 @ ( cons_nat @ Y5 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_924_subst__var_Ocases,axiom,
! [X: produc3830279506226590812et_nat] :
( ! [X3: set_nat,Xs3: list_set_nat,Y5: set_nat,Ys3: list_set_nat,Z: set_nat] :
( X
!= ( produc7721406734314559374et_nat @ ( cons_set_nat @ X3 @ Xs3 ) @ ( produc2800461512284786101et_nat @ ( cons_set_nat @ Y5 @ Ys3 ) @ Z ) ) )
=> ( ! [Uv: list_set_nat,Z: set_nat] :
( X
!= ( produc7721406734314559374et_nat @ nil_set_nat @ ( produc2800461512284786101et_nat @ Uv @ Z ) ) )
=> ~ ! [Uu: list_set_nat,Z: set_nat] :
( X
!= ( produc7721406734314559374et_nat @ Uu @ ( produc2800461512284786101et_nat @ nil_set_nat @ Z ) ) ) ) ) ).
% subst_var.cases
thf(fact_925_subst__var_Ocases,axiom,
! [X: produc4815183511976467691la_a_b] :
( ! [X3: set_Re381260168593705685la_a_b,Xs3: list_s3671366213839843941la_a_b,Y5: set_Re381260168593705685la_a_b,Ys3: list_s3671366213839843941la_a_b,Z: set_Re381260168593705685la_a_b] :
( X
!= ( produc3775367036610755613la_a_b @ ( cons_s2104095074707585951la_a_b @ X3 @ Xs3 ) @ ( produc6029123889615053267la_a_b @ ( cons_s2104095074707585951la_a_b @ Y5 @ Ys3 ) @ Z ) ) )
=> ( ! [Uv: list_s3671366213839843941la_a_b,Z: set_Re381260168593705685la_a_b] :
( X
!= ( produc3775367036610755613la_a_b @ nil_se6596116450819887951la_a_b @ ( produc6029123889615053267la_a_b @ Uv @ Z ) ) )
=> ~ ! [Uu: list_s3671366213839843941la_a_b,Z: set_Re381260168593705685la_a_b] :
( X
!= ( produc3775367036610755613la_a_b @ Uu @ ( produc6029123889615053267la_a_b @ nil_se6596116450819887951la_a_b @ Z ) ) ) ) ) ).
% subst_var.cases
thf(fact_926_subst__var_Ocases,axiom,
! [X: produc7745895538441958454st_nat] :
( ! [X3: list_nat,Xs3: list_list_nat,Y5: list_nat,Ys3: list_list_nat,Z: list_nat] :
( X
!= ( produc5136709007628231656st_nat @ ( cons_list_nat @ X3 @ Xs3 ) @ ( produc8303078633012690025st_nat @ ( cons_list_nat @ Y5 @ Ys3 ) @ Z ) ) )
=> ( ! [Uv: list_list_nat,Z: list_nat] :
( X
!= ( produc5136709007628231656st_nat @ nil_list_nat @ ( produc8303078633012690025st_nat @ Uv @ Z ) ) )
=> ~ ! [Uu: list_list_nat,Z: list_nat] :
( X
!= ( produc5136709007628231656st_nat @ Uu @ ( produc8303078633012690025st_nat @ nil_list_nat @ Z ) ) ) ) ) ).
% subst_var.cases
thf(fact_927_subst__var_Ocases,axiom,
! [X: produc5824815472048091686at_nat] :
( ! [X3: nat,Xs3: list_nat,Y5: nat,Ys3: list_nat,Z: nat] :
( X
!= ( produc4483744321870515032at_nat @ ( cons_nat @ X3 @ Xs3 ) @ ( produc1298395424260782409at_nat @ ( cons_nat @ Y5 @ Ys3 ) @ Z ) ) )
=> ( ! [Uv: list_nat,Z: nat] :
( X
!= ( produc4483744321870515032at_nat @ nil_nat @ ( produc1298395424260782409at_nat @ Uv @ Z ) ) )
=> ~ ! [Uu: list_nat,Z: nat] :
( X
!= ( produc4483744321870515032at_nat @ Uu @ ( produc1298395424260782409at_nat @ nil_nat @ Z ) ) ) ) ) ).
% subst_var.cases
thf(fact_928_foldr1_Ocases,axiom,
! [X: produc2766782161934916770et_nat] :
( ! [F3: set_nat > set_nat > set_nat,X3: set_nat,Xs3: list_set_nat,Z: set_nat] :
( X
!= ( produc8556424306328756564et_nat @ F3 @ ( produc2800461512284786101et_nat @ ( cons_set_nat @ X3 @ Xs3 ) @ Z ) ) )
=> ~ ! [F3: set_nat > set_nat > set_nat,Z: set_nat] :
( X
!= ( produc8556424306328756564et_nat @ F3 @ ( produc2800461512284786101et_nat @ nil_set_nat @ Z ) ) ) ) ).
% foldr1.cases
thf(fact_929_foldr1_Ocases,axiom,
! [X: produc6723062099890885523la_a_b] :
( ! [F3: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,X3: set_Re381260168593705685la_a_b,Xs3: list_s3671366213839843941la_a_b,Z: set_Re381260168593705685la_a_b] :
( X
!= ( produc9177147133837083205la_a_b @ F3 @ ( produc6029123889615053267la_a_b @ ( cons_s2104095074707585951la_a_b @ X3 @ Xs3 ) @ Z ) ) )
=> ~ ! [F3: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,Z: set_Re381260168593705685la_a_b] :
( X
!= ( produc9177147133837083205la_a_b @ F3 @ ( produc6029123889615053267la_a_b @ nil_se6596116450819887951la_a_b @ Z ) ) ) ) ).
% foldr1.cases
thf(fact_930_foldr1_Ocases,axiom,
! [X: produc2412404074066063688st_nat] :
( ! [F3: list_nat > list_nat > list_nat,X3: list_nat,Xs3: list_list_nat,Z: list_nat] :
( X
!= ( produc6545234417207796090st_nat @ F3 @ ( produc8303078633012690025st_nat @ ( cons_list_nat @ X3 @ Xs3 ) @ Z ) ) )
=> ~ ! [F3: list_nat > list_nat > list_nat,Z: list_nat] :
( X
!= ( produc6545234417207796090st_nat @ F3 @ ( produc8303078633012690025st_nat @ nil_list_nat @ Z ) ) ) ) ).
% foldr1.cases
thf(fact_931_foldr1_Ocases,axiom,
! [X: produc7291604503153307608at_nat] :
( ! [F3: nat > nat > nat,X3: nat,Xs3: list_nat,Z: nat] :
( X
!= ( produc4302411105509421706at_nat @ F3 @ ( produc1298395424260782409at_nat @ ( cons_nat @ X3 @ Xs3 ) @ Z ) ) )
=> ~ ! [F3: nat > nat > nat,Z: nat] :
( X
!= ( produc4302411105509421706at_nat @ F3 @ ( produc1298395424260782409at_nat @ nil_nat @ Z ) ) ) ) ).
% foldr1.cases
thf(fact_932_list_Oset__intros_I2_J,axiom,
! [Y3: set_se6865892389300016395la_a_b,X222: list_s5517785546295487899la_a_b,X21: set_se6865892389300016395la_a_b] :
( ( member1693404252451860130la_a_b @ Y3 @ ( set_se4873105269812759862la_a_b @ X222 ) )
=> ( member1693404252451860130la_a_b @ Y3 @ ( set_se4873105269812759862la_a_b @ ( cons_s4528938952058510805la_a_b @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_933_list_Oset__intros_I2_J,axiom,
! [Y3: b,X222: list_b,X21: b] :
( ( member_b @ Y3 @ ( set_b2 @ X222 ) )
=> ( member_b @ Y3 @ ( set_b2 @ ( cons_b @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_934_list_Oset__intros_I2_J,axiom,
! [Y3: relational_fmla_a_b,X222: list_R8263082107343818799la_a_b,X21: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y3 @ ( set_Re9104216502384355786la_a_b @ X222 ) )
=> ( member4680049679412964150la_a_b @ Y3 @ ( set_Re9104216502384355786la_a_b @ ( cons_R4161027894713053801la_a_b @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_935_list_Oset__intros_I2_J,axiom,
! [Y3: list_R8263082107343818799la_a_b,X222: list_l3216422893793986623la_a_b,X21: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ Y3 @ ( set_li8988895567068466906la_a_b @ X222 ) )
=> ( member8712135081189583430la_a_b @ Y3 @ ( set_li8988895567068466906la_a_b @ ( cons_l6716632503304403961la_a_b @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_936_list_Oset__intros_I2_J,axiom,
! [Y3: list_s3671366213839843941la_a_b,X222: list_l1512381142989314933la_a_b,X21: list_s3671366213839843941la_a_b] :
( ( member987187531357909116la_a_b @ Y3 @ ( set_li3253934503784678672la_a_b @ X222 ) )
=> ( member987187531357909116la_a_b @ Y3 @ ( set_li3253934503784678672la_a_b @ ( cons_l1224881792622212655la_a_b @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_937_list_Oset__intros_I2_J,axiom,
! [Y3: list_nat,X222: list_list_nat,X21: list_nat] :
( ( member_list_nat @ Y3 @ ( set_list_nat2 @ X222 ) )
=> ( member_list_nat @ Y3 @ ( set_list_nat2 @ ( cons_list_nat @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_938_list_Oset__intros_I2_J,axiom,
! [Y3: set_Re381260168593705685la_a_b,X222: list_s3671366213839843941la_a_b,X21: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ Y3 @ ( set_se5111286776497155840la_a_b @ X222 ) )
=> ( member3481406638322139244la_a_b @ Y3 @ ( set_se5111286776497155840la_a_b @ ( cons_s2104095074707585951la_a_b @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_939_list_Oset__intros_I2_J,axiom,
! [Y3: nat,X222: list_nat,X21: nat] :
( ( member_nat @ Y3 @ ( set_nat2 @ X222 ) )
=> ( member_nat @ Y3 @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_940_list_Oset__intros_I1_J,axiom,
! [X21: set_se6865892389300016395la_a_b,X222: list_s5517785546295487899la_a_b] : ( member1693404252451860130la_a_b @ X21 @ ( set_se4873105269812759862la_a_b @ ( cons_s4528938952058510805la_a_b @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_941_list_Oset__intros_I1_J,axiom,
! [X21: b,X222: list_b] : ( member_b @ X21 @ ( set_b2 @ ( cons_b @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_942_list_Oset__intros_I1_J,axiom,
! [X21: relational_fmla_a_b,X222: list_R8263082107343818799la_a_b] : ( member4680049679412964150la_a_b @ X21 @ ( set_Re9104216502384355786la_a_b @ ( cons_R4161027894713053801la_a_b @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_943_list_Oset__intros_I1_J,axiom,
! [X21: list_R8263082107343818799la_a_b,X222: list_l3216422893793986623la_a_b] : ( member8712135081189583430la_a_b @ X21 @ ( set_li8988895567068466906la_a_b @ ( cons_l6716632503304403961la_a_b @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_944_list_Oset__intros_I1_J,axiom,
! [X21: list_s3671366213839843941la_a_b,X222: list_l1512381142989314933la_a_b] : ( member987187531357909116la_a_b @ X21 @ ( set_li3253934503784678672la_a_b @ ( cons_l1224881792622212655la_a_b @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_945_list_Oset__intros_I1_J,axiom,
! [X21: list_nat,X222: list_list_nat] : ( member_list_nat @ X21 @ ( set_list_nat2 @ ( cons_list_nat @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_946_list_Oset__intros_I1_J,axiom,
! [X21: set_Re381260168593705685la_a_b,X222: list_s3671366213839843941la_a_b] : ( member3481406638322139244la_a_b @ X21 @ ( set_se5111286776497155840la_a_b @ ( cons_s2104095074707585951la_a_b @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_947_list_Oset__intros_I1_J,axiom,
! [X21: nat,X222: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_948_list_Oset__cases,axiom,
! [E: set_se6865892389300016395la_a_b,A2: list_s5517785546295487899la_a_b] :
( ( member1693404252451860130la_a_b @ E @ ( set_se4873105269812759862la_a_b @ A2 ) )
=> ( ! [Z22: list_s5517785546295487899la_a_b] :
( A2
!= ( cons_s4528938952058510805la_a_b @ E @ Z22 ) )
=> ~ ! [Z1: set_se6865892389300016395la_a_b,Z22: list_s5517785546295487899la_a_b] :
( ( A2
= ( cons_s4528938952058510805la_a_b @ Z1 @ Z22 ) )
=> ~ ( member1693404252451860130la_a_b @ E @ ( set_se4873105269812759862la_a_b @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_949_list_Oset__cases,axiom,
! [E: b,A2: list_b] :
( ( member_b @ E @ ( set_b2 @ A2 ) )
=> ( ! [Z22: list_b] :
( A2
!= ( cons_b @ E @ Z22 ) )
=> ~ ! [Z1: b,Z22: list_b] :
( ( A2
= ( cons_b @ Z1 @ Z22 ) )
=> ~ ( member_b @ E @ ( set_b2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_950_list_Oset__cases,axiom,
! [E: relational_fmla_a_b,A2: list_R8263082107343818799la_a_b] :
( ( member4680049679412964150la_a_b @ E @ ( set_Re9104216502384355786la_a_b @ A2 ) )
=> ( ! [Z22: list_R8263082107343818799la_a_b] :
( A2
!= ( cons_R4161027894713053801la_a_b @ E @ Z22 ) )
=> ~ ! [Z1: relational_fmla_a_b,Z22: list_R8263082107343818799la_a_b] :
( ( A2
= ( cons_R4161027894713053801la_a_b @ Z1 @ Z22 ) )
=> ~ ( member4680049679412964150la_a_b @ E @ ( set_Re9104216502384355786la_a_b @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_951_list_Oset__cases,axiom,
! [E: list_R8263082107343818799la_a_b,A2: list_l3216422893793986623la_a_b] :
( ( member8712135081189583430la_a_b @ E @ ( set_li8988895567068466906la_a_b @ A2 ) )
=> ( ! [Z22: list_l3216422893793986623la_a_b] :
( A2
!= ( cons_l6716632503304403961la_a_b @ E @ Z22 ) )
=> ~ ! [Z1: list_R8263082107343818799la_a_b,Z22: list_l3216422893793986623la_a_b] :
( ( A2
= ( cons_l6716632503304403961la_a_b @ Z1 @ Z22 ) )
=> ~ ( member8712135081189583430la_a_b @ E @ ( set_li8988895567068466906la_a_b @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_952_list_Oset__cases,axiom,
! [E: list_s3671366213839843941la_a_b,A2: list_l1512381142989314933la_a_b] :
( ( member987187531357909116la_a_b @ E @ ( set_li3253934503784678672la_a_b @ A2 ) )
=> ( ! [Z22: list_l1512381142989314933la_a_b] :
( A2
!= ( cons_l1224881792622212655la_a_b @ E @ Z22 ) )
=> ~ ! [Z1: list_s3671366213839843941la_a_b,Z22: list_l1512381142989314933la_a_b] :
( ( A2
= ( cons_l1224881792622212655la_a_b @ Z1 @ Z22 ) )
=> ~ ( member987187531357909116la_a_b @ E @ ( set_li3253934503784678672la_a_b @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_953_list_Oset__cases,axiom,
! [E: list_nat,A2: list_list_nat] :
( ( member_list_nat @ E @ ( set_list_nat2 @ A2 ) )
=> ( ! [Z22: list_list_nat] :
( A2
!= ( cons_list_nat @ E @ Z22 ) )
=> ~ ! [Z1: list_nat,Z22: list_list_nat] :
( ( A2
= ( cons_list_nat @ Z1 @ Z22 ) )
=> ~ ( member_list_nat @ E @ ( set_list_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_954_list_Oset__cases,axiom,
! [E: set_Re381260168593705685la_a_b,A2: list_s3671366213839843941la_a_b] :
( ( member3481406638322139244la_a_b @ E @ ( set_se5111286776497155840la_a_b @ A2 ) )
=> ( ! [Z22: list_s3671366213839843941la_a_b] :
( A2
!= ( cons_s2104095074707585951la_a_b @ E @ Z22 ) )
=> ~ ! [Z1: set_Re381260168593705685la_a_b,Z22: list_s3671366213839843941la_a_b] :
( ( A2
= ( cons_s2104095074707585951la_a_b @ Z1 @ Z22 ) )
=> ~ ( member3481406638322139244la_a_b @ E @ ( set_se5111286776497155840la_a_b @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_955_list_Oset__cases,axiom,
! [E: nat,A2: list_nat] :
( ( member_nat @ E @ ( set_nat2 @ A2 ) )
=> ( ! [Z22: list_nat] :
( A2
!= ( cons_nat @ E @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A2
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_956_set__ConsD,axiom,
! [Y3: set_se6865892389300016395la_a_b,X: set_se6865892389300016395la_a_b,Xs2: list_s5517785546295487899la_a_b] :
( ( member1693404252451860130la_a_b @ Y3 @ ( set_se4873105269812759862la_a_b @ ( cons_s4528938952058510805la_a_b @ X @ Xs2 ) ) )
=> ( ( Y3 = X )
| ( member1693404252451860130la_a_b @ Y3 @ ( set_se4873105269812759862la_a_b @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_957_set__ConsD,axiom,
! [Y3: b,X: b,Xs2: list_b] :
( ( member_b @ Y3 @ ( set_b2 @ ( cons_b @ X @ Xs2 ) ) )
=> ( ( Y3 = X )
| ( member_b @ Y3 @ ( set_b2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_958_set__ConsD,axiom,
! [Y3: relational_fmla_a_b,X: relational_fmla_a_b,Xs2: list_R8263082107343818799la_a_b] :
( ( member4680049679412964150la_a_b @ Y3 @ ( set_Re9104216502384355786la_a_b @ ( cons_R4161027894713053801la_a_b @ X @ Xs2 ) ) )
=> ( ( Y3 = X )
| ( member4680049679412964150la_a_b @ Y3 @ ( set_Re9104216502384355786la_a_b @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_959_set__ConsD,axiom,
! [Y3: list_R8263082107343818799la_a_b,X: list_R8263082107343818799la_a_b,Xs2: list_l3216422893793986623la_a_b] :
( ( member8712135081189583430la_a_b @ Y3 @ ( set_li8988895567068466906la_a_b @ ( cons_l6716632503304403961la_a_b @ X @ Xs2 ) ) )
=> ( ( Y3 = X )
| ( member8712135081189583430la_a_b @ Y3 @ ( set_li8988895567068466906la_a_b @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_960_set__ConsD,axiom,
! [Y3: list_s3671366213839843941la_a_b,X: list_s3671366213839843941la_a_b,Xs2: list_l1512381142989314933la_a_b] :
( ( member987187531357909116la_a_b @ Y3 @ ( set_li3253934503784678672la_a_b @ ( cons_l1224881792622212655la_a_b @ X @ Xs2 ) ) )
=> ( ( Y3 = X )
| ( member987187531357909116la_a_b @ Y3 @ ( set_li3253934503784678672la_a_b @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_961_set__ConsD,axiom,
! [Y3: list_nat,X: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ Y3 @ ( set_list_nat2 @ ( cons_list_nat @ X @ Xs2 ) ) )
=> ( ( Y3 = X )
| ( member_list_nat @ Y3 @ ( set_list_nat2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_962_set__ConsD,axiom,
! [Y3: set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b,Xs2: list_s3671366213839843941la_a_b] :
( ( member3481406638322139244la_a_b @ Y3 @ ( set_se5111286776497155840la_a_b @ ( cons_s2104095074707585951la_a_b @ X @ Xs2 ) ) )
=> ( ( Y3 = X )
| ( member3481406638322139244la_a_b @ Y3 @ ( set_se5111286776497155840la_a_b @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_963_set__ConsD,axiom,
! [Y3: nat,X: nat,Xs2: list_nat] :
( ( member_nat @ Y3 @ ( set_nat2 @ ( cons_nat @ X @ Xs2 ) ) )
=> ( ( Y3 = X )
| ( member_nat @ Y3 @ ( set_nat2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_964_in__set__simps_I1_J,axiom,
! [X: set_se6865892389300016395la_a_b,Y3: set_se6865892389300016395la_a_b,Z2: set_se6865892389300016395la_a_b,Ys: list_s5517785546295487899la_a_b] :
( ( member1693404252451860130la_a_b @ X @ ( set_se4873105269812759862la_a_b @ ( cons_s4528938952058510805la_a_b @ Y3 @ ( cons_s4528938952058510805la_a_b @ Z2 @ Ys ) ) ) )
= ( ( X = Y3 )
| ( member1693404252451860130la_a_b @ X @ ( set_se4873105269812759862la_a_b @ ( cons_s4528938952058510805la_a_b @ Z2 @ Ys ) ) ) ) ) ).
% in_set_simps(1)
thf(fact_965_in__set__simps_I1_J,axiom,
! [X: b,Y3: b,Z2: b,Ys: list_b] :
( ( member_b @ X @ ( set_b2 @ ( cons_b @ Y3 @ ( cons_b @ Z2 @ Ys ) ) ) )
= ( ( X = Y3 )
| ( member_b @ X @ ( set_b2 @ ( cons_b @ Z2 @ Ys ) ) ) ) ) ).
% in_set_simps(1)
thf(fact_966_in__set__simps_I1_J,axiom,
! [X: relational_fmla_a_b,Y3: relational_fmla_a_b,Z2: relational_fmla_a_b,Ys: list_R8263082107343818799la_a_b] :
( ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ ( cons_R4161027894713053801la_a_b @ Y3 @ ( cons_R4161027894713053801la_a_b @ Z2 @ Ys ) ) ) )
= ( ( X = Y3 )
| ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ ( cons_R4161027894713053801la_a_b @ Z2 @ Ys ) ) ) ) ) ).
% in_set_simps(1)
thf(fact_967_in__set__simps_I1_J,axiom,
! [X: list_R8263082107343818799la_a_b,Y3: list_R8263082107343818799la_a_b,Z2: list_R8263082107343818799la_a_b,Ys: list_l3216422893793986623la_a_b] :
( ( member8712135081189583430la_a_b @ X @ ( set_li8988895567068466906la_a_b @ ( cons_l6716632503304403961la_a_b @ Y3 @ ( cons_l6716632503304403961la_a_b @ Z2 @ Ys ) ) ) )
= ( ( X = Y3 )
| ( member8712135081189583430la_a_b @ X @ ( set_li8988895567068466906la_a_b @ ( cons_l6716632503304403961la_a_b @ Z2 @ Ys ) ) ) ) ) ).
% in_set_simps(1)
thf(fact_968_in__set__simps_I1_J,axiom,
! [X: list_s3671366213839843941la_a_b,Y3: list_s3671366213839843941la_a_b,Z2: list_s3671366213839843941la_a_b,Ys: list_l1512381142989314933la_a_b] :
( ( member987187531357909116la_a_b @ X @ ( set_li3253934503784678672la_a_b @ ( cons_l1224881792622212655la_a_b @ Y3 @ ( cons_l1224881792622212655la_a_b @ Z2 @ Ys ) ) ) )
= ( ( X = Y3 )
| ( member987187531357909116la_a_b @ X @ ( set_li3253934503784678672la_a_b @ ( cons_l1224881792622212655la_a_b @ Z2 @ Ys ) ) ) ) ) ).
% in_set_simps(1)
thf(fact_969_in__set__simps_I1_J,axiom,
! [X: list_nat,Y3: list_nat,Z2: list_nat,Ys: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ ( cons_list_nat @ Y3 @ ( cons_list_nat @ Z2 @ Ys ) ) ) )
= ( ( X = Y3 )
| ( member_list_nat @ X @ ( set_list_nat2 @ ( cons_list_nat @ Z2 @ Ys ) ) ) ) ) ).
% in_set_simps(1)
thf(fact_970_in__set__simps_I1_J,axiom,
! [X: set_Re381260168593705685la_a_b,Y3: set_Re381260168593705685la_a_b,Z2: set_Re381260168593705685la_a_b,Ys: list_s3671366213839843941la_a_b] :
( ( member3481406638322139244la_a_b @ X @ ( set_se5111286776497155840la_a_b @ ( cons_s2104095074707585951la_a_b @ Y3 @ ( cons_s2104095074707585951la_a_b @ Z2 @ Ys ) ) ) )
= ( ( X = Y3 )
| ( member3481406638322139244la_a_b @ X @ ( set_se5111286776497155840la_a_b @ ( cons_s2104095074707585951la_a_b @ Z2 @ Ys ) ) ) ) ) ).
% in_set_simps(1)
thf(fact_971_in__set__simps_I1_J,axiom,
! [X: nat,Y3: nat,Z2: nat,Ys: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ ( cons_nat @ Y3 @ ( cons_nat @ Z2 @ Ys ) ) ) )
= ( ( X = Y3 )
| ( member_nat @ X @ ( set_nat2 @ ( cons_nat @ Z2 @ Ys ) ) ) ) ) ).
% in_set_simps(1)
thf(fact_972_the__elem__set,axiom,
! [X: set_nat] :
( ( the_elem_set_nat @ ( set_set_nat2 @ ( cons_set_nat @ X @ nil_set_nat ) ) )
= X ) ).
% the_elem_set
thf(fact_973_the__elem__set,axiom,
! [X: relational_fmla_a_b] :
( ( the_el6350558617753882986la_a_b @ ( set_Re9104216502384355786la_a_b @ ( cons_R4161027894713053801la_a_b @ X @ nil_Re6358386334527539737la_a_b ) ) )
= X ) ).
% the_elem_set
thf(fact_974_the__elem__set,axiom,
! [X: list_R8263082107343818799la_a_b] :
( ( the_el2253512510949844602la_a_b @ ( set_li8988895567068466906la_a_b @ ( cons_l6716632503304403961la_a_b @ X @ nil_li4862165533606214569la_a_b ) ) )
= X ) ).
% the_elem_set
thf(fact_975_the__elem__set,axiom,
! [X: list_s3671366213839843941la_a_b] :
( ( the_el1595598750643379376la_a_b @ ( set_li3253934503784678672la_a_b @ ( cons_l1224881792622212655la_a_b @ X @ nil_li6858553494156228063la_a_b ) ) )
= X ) ).
% the_elem_set
thf(fact_976_the__elem__set,axiom,
! [X: list_nat] :
( ( the_elem_list_nat @ ( set_list_nat2 @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= X ) ).
% the_elem_set
thf(fact_977_the__elem__set,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( the_el7486773796875720352la_a_b @ ( set_se5111286776497155840la_a_b @ ( cons_s2104095074707585951la_a_b @ X @ nil_se6596116450819887951la_a_b ) ) )
= X ) ).
% the_elem_set
thf(fact_978_the__elem__set,axiom,
! [X: nat] :
( ( the_elem_nat @ ( set_nat2 @ ( cons_nat @ X @ nil_nat ) ) )
= X ) ).
% the_elem_set
thf(fact_979_fv__flat__DisjD,axiom,
! [Q5: relational_fmla_a_b,Q: relational_fmla_a_b,X: nat] :
( ( member4680049679412964150la_a_b @ Q5 @ ( restri569617705344514291sj_a_b @ Q ) )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q5 ) )
=> ( member_nat @ X @ ( relational_fv_a_b @ Q ) ) ) ) ).
% fv_flat_DisjD
thf(fact_980_fv__terms__setD,axiom,
! [M: nat,Ts3: list_R6823256787227418703term_a] :
( ( member_nat @ M @ ( relati4569515538964159125_set_a @ Ts3 ) )
=> ( member5466445112152945266term_a @ ( relational_Var_a @ M ) @ ( set_Re3569617851344498910term_a @ Ts3 ) ) ) ).
% fv_terms_setD
thf(fact_981_fv__terms__setI,axiom,
! [M: nat,Ts3: list_R6823256787227418703term_a] :
( ( member5466445112152945266term_a @ ( relational_Var_a @ M ) @ ( set_Re3569617851344498910term_a @ Ts3 ) )
=> ( member_nat @ M @ ( relati4569515538964159125_set_a @ Ts3 ) ) ) ).
% fv_terms_setI
thf(fact_982_cpropagated__flat__DisjD,axiom,
! [Q5: relational_fmla_a_b,Q: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Q5 @ ( restri569617705344514291sj_a_b @ Q ) )
=> ( ( relati1591879772219623554ed_a_b @ Q )
=> ( relati1591879772219623554ed_a_b @ Q5 ) ) ) ).
% cpropagated_flat_DisjD
thf(fact_983_list__all__rec_Osimps_I2_J,axiom,
! [P: list_nat > $o,A2: list_nat,L: list_list_nat] :
( ( param_5680023212210265682st_nat @ P @ ( cons_list_nat @ A2 @ L ) )
= ( ( P @ A2 )
& ( param_5680023212210265682st_nat @ P @ L ) ) ) ).
% list_all_rec.simps(2)
thf(fact_984_list__all__rec_Osimps_I2_J,axiom,
! [P: nat > $o,A2: nat,L: list_nat] :
( ( param_3668158194536031298ec_nat @ P @ ( cons_nat @ A2 @ L ) )
= ( ( P @ A2 )
& ( param_3668158194536031298ec_nat @ P @ L ) ) ) ).
% list_all_rec.simps(2)
thf(fact_985_list__all__rec_Osimps_I2_J,axiom,
! [P: set_Re381260168593705685la_a_b > $o,A2: set_Re381260168593705685la_a_b,L: list_s3671366213839843941la_a_b] :
( ( param_8621423519915839815la_a_b @ P @ ( cons_s2104095074707585951la_a_b @ A2 @ L ) )
= ( ( P @ A2 )
& ( param_8621423519915839815la_a_b @ P @ L ) ) ) ).
% list_all_rec.simps(2)
thf(fact_986_list__ex__rec_Osimps_I2_J,axiom,
! [P: list_nat > $o,A2: list_nat,L: list_list_nat] :
( ( param_519148586836064220st_nat @ P @ ( cons_list_nat @ A2 @ L ) )
= ( ( P @ A2 )
| ( param_519148586836064220st_nat @ P @ L ) ) ) ).
% list_ex_rec.simps(2)
thf(fact_987_list__ex__rec_Osimps_I2_J,axiom,
! [P: nat > $o,A2: nat,L: list_nat] :
( ( param_7592452054083848908ec_nat @ P @ ( cons_nat @ A2 @ L ) )
= ( ( P @ A2 )
| ( param_7592452054083848908ec_nat @ P @ L ) ) ) ).
% list_ex_rec.simps(2)
thf(fact_988_list__ex__rec_Osimps_I2_J,axiom,
! [P: set_Re381260168593705685la_a_b > $o,A2: set_Re381260168593705685la_a_b,L: list_s3671366213839843941la_a_b] :
( ( param_2860366322937624017la_a_b @ P @ ( cons_s2104095074707585951la_a_b @ A2 @ L ) )
= ( ( P @ A2 )
| ( param_2860366322937624017la_a_b @ P @ L ) ) ) ).
% list_ex_rec.simps(2)
thf(fact_989_in__set__simps_I2_J,axiom,
! [X: set_se6865892389300016395la_a_b,Y3: set_se6865892389300016395la_a_b] :
( ( member1693404252451860130la_a_b @ X @ ( set_se4873105269812759862la_a_b @ ( cons_s4528938952058510805la_a_b @ Y3 @ nil_se4484975911477474693la_a_b ) ) )
= ( X = Y3 ) ) ).
% in_set_simps(2)
thf(fact_990_in__set__simps_I2_J,axiom,
! [X: b,Y3: b] :
( ( member_b @ X @ ( set_b2 @ ( cons_b @ Y3 @ nil_b ) ) )
= ( X = Y3 ) ) ).
% in_set_simps(2)
thf(fact_991_in__set__simps_I2_J,axiom,
! [X: set_nat,Y3: set_nat] :
( ( member_set_nat @ X @ ( set_set_nat2 @ ( cons_set_nat @ Y3 @ nil_set_nat ) ) )
= ( X = Y3 ) ) ).
% in_set_simps(2)
thf(fact_992_in__set__simps_I2_J,axiom,
! [X: relational_fmla_a_b,Y3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ ( cons_R4161027894713053801la_a_b @ Y3 @ nil_Re6358386334527539737la_a_b ) ) )
= ( X = Y3 ) ) ).
% in_set_simps(2)
thf(fact_993_in__set__simps_I2_J,axiom,
! [X: list_R8263082107343818799la_a_b,Y3: list_R8263082107343818799la_a_b] :
( ( member8712135081189583430la_a_b @ X @ ( set_li8988895567068466906la_a_b @ ( cons_l6716632503304403961la_a_b @ Y3 @ nil_li4862165533606214569la_a_b ) ) )
= ( X = Y3 ) ) ).
% in_set_simps(2)
thf(fact_994_in__set__simps_I2_J,axiom,
! [X: list_s3671366213839843941la_a_b,Y3: list_s3671366213839843941la_a_b] :
( ( member987187531357909116la_a_b @ X @ ( set_li3253934503784678672la_a_b @ ( cons_l1224881792622212655la_a_b @ Y3 @ nil_li6858553494156228063la_a_b ) ) )
= ( X = Y3 ) ) ).
% in_set_simps(2)
thf(fact_995_in__set__simps_I2_J,axiom,
! [X: list_nat,Y3: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ ( cons_list_nat @ Y3 @ nil_list_nat ) ) )
= ( X = Y3 ) ) ).
% in_set_simps(2)
thf(fact_996_in__set__simps_I2_J,axiom,
! [X: set_Re381260168593705685la_a_b,Y3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X @ ( set_se5111286776497155840la_a_b @ ( cons_s2104095074707585951la_a_b @ Y3 @ nil_se6596116450819887951la_a_b ) ) )
= ( X = Y3 ) ) ).
% in_set_simps(2)
thf(fact_997_in__set__simps_I2_J,axiom,
! [X: nat,Y3: nat] :
( ( member_nat @ X @ ( set_nat2 @ ( cons_nat @ Y3 @ nil_nat ) ) )
= ( X = Y3 ) ) ).
% in_set_simps(2)
thf(fact_998_fv_Osimps_I2_J,axiom,
! [B2: $o] :
( ( relational_fv_a_b @ ( relational_Bool_a_b @ B2 ) )
= bot_bot_set_nat ) ).
% fv.simps(2)
thf(fact_999_List_Oinsert__def,axiom,
( insert3187468692655853807la_a_b
= ( ^ [X2: set_se6865892389300016395la_a_b,Xs: list_s5517785546295487899la_a_b] : ( if_lis8471649527330808917la_a_b @ ( member1693404252451860130la_a_b @ X2 @ ( set_se4873105269812759862la_a_b @ Xs ) ) @ Xs @ ( cons_s4528938952058510805la_a_b @ X2 @ Xs ) ) ) ) ).
% List.insert_def
thf(fact_1000_List_Oinsert__def,axiom,
( insert_b
= ( ^ [X2: b,Xs: list_b] : ( if_list_b @ ( member_b @ X2 @ ( set_b2 @ Xs ) ) @ Xs @ ( cons_b @ X2 @ Xs ) ) ) ) ).
% List.insert_def
thf(fact_1001_List_Oinsert__def,axiom,
( insert5052544347734373763la_a_b
= ( ^ [X2: relational_fmla_a_b,Xs: list_R8263082107343818799la_a_b] : ( if_lis1478040310087653097la_a_b @ ( member4680049679412964150la_a_b @ X2 @ ( set_Re9104216502384355786la_a_b @ Xs ) ) @ Xs @ ( cons_R4161027894713053801la_a_b @ X2 @ Xs ) ) ) ) ).
% List.insert_def
thf(fact_1002_List_Oinsert__def,axiom,
( insert3209172818022656531la_a_b
= ( ^ [X2: list_R8263082107343818799la_a_b,Xs: list_l3216422893793986623la_a_b] : ( if_lis5916486426012477945la_a_b @ ( member8712135081189583430la_a_b @ X2 @ ( set_li8988895567068466906la_a_b @ Xs ) ) @ Xs @ ( cons_l6716632503304403961la_a_b @ X2 @ Xs ) ) ) ) ).
% List.insert_def
thf(fact_1003_List_Oinsert__def,axiom,
( insert3122035991742315593la_a_b
= ( ^ [X2: list_s3671366213839843941la_a_b,Xs: list_l1512381142989314933la_a_b] : ( if_lis8483221396762875183la_a_b @ ( member987187531357909116la_a_b @ X2 @ ( set_li3253934503784678672la_a_b @ Xs ) ) @ Xs @ ( cons_l1224881792622212655la_a_b @ X2 @ Xs ) ) ) ) ).
% List.insert_def
thf(fact_1004_List_Oinsert__def,axiom,
( insert_list_nat
= ( ^ [X2: list_nat,Xs: list_list_nat] : ( if_list_list_nat @ ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) ) @ Xs @ ( cons_list_nat @ X2 @ Xs ) ) ) ) ).
% List.insert_def
thf(fact_1005_List_Oinsert__def,axiom,
( insert443281300761040569la_a_b
= ( ^ [X2: set_Re381260168593705685la_a_b,Xs: list_s3671366213839843941la_a_b] : ( if_lis5716551597998330143la_a_b @ ( member3481406638322139244la_a_b @ X2 @ ( set_se5111286776497155840la_a_b @ Xs ) ) @ Xs @ ( cons_s2104095074707585951la_a_b @ X2 @ Xs ) ) ) ) ).
% List.insert_def
thf(fact_1006_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X2: nat,Xs: list_nat] : ( if_list_nat @ ( member_nat @ X2 @ ( set_nat2 @ Xs ) ) @ Xs @ ( cons_nat @ X2 @ Xs ) ) ) ) ).
% List.insert_def
thf(fact_1007_foldr1_Osimps_I1_J,axiom,
! [F2: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b,Xs2: list_s3671366213839843941la_a_b,Z2: set_Re381260168593705685la_a_b] :
( ( relati7835315018957680435la_a_b @ F2 @ ( cons_s2104095074707585951la_a_b @ X @ Xs2 ) @ Z2 )
= ( foldr_5041855395594655059la_a_b @ F2 @ Xs2 @ X ) ) ).
% foldr1.simps(1)
thf(fact_1008_foldr1_Osimps_I1_J,axiom,
! [F2: list_nat > list_nat > list_nat,X: list_nat,Xs2: list_list_nat,Z2: list_nat] :
( ( relati7377577593470194238st_nat @ F2 @ ( cons_list_nat @ X @ Xs2 ) @ Z2 )
= ( foldr_6871341030409798377st_nat @ F2 @ Xs2 @ X ) ) ).
% foldr1.simps(1)
thf(fact_1009_foldr1_Osimps_I1_J,axiom,
! [F2: nat > nat > nat,X: nat,Xs2: list_nat,Z2: nat] :
( ( relati2262546126839610926r1_nat @ F2 @ ( cons_nat @ X @ Xs2 ) @ Z2 )
= ( foldr_nat_nat @ F2 @ Xs2 @ X ) ) ).
% foldr1.simps(1)
thf(fact_1010_flat__Disj_Osimps_I1_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( restri569617705344514291sj_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) )
= ( sup_su5130108678486352897la_a_b @ ( restri569617705344514291sj_a_b @ Q1 ) @ ( restri569617705344514291sj_a_b @ Q2 ) ) ) ).
% flat_Disj.simps(1)
thf(fact_1011_flat__Disj__Exists__sub,axiom,
! [Q5: relational_fmla_a_b,Q: relational_fmla_a_b,Y3: nat,Qy: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Q5 @ ( restri569617705344514291sj_a_b @ Q ) )
=> ( ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ Y3 @ Qy ) @ ( relational_sub_a_b @ Q5 ) )
=> ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ Y3 @ Qy ) @ ( relational_sub_a_b @ Q ) ) ) ) ).
% flat_Disj_Exists_sub
thf(fact_1012_gen__impl_Osimps_I5_J,axiom,
! [X: nat,Ts3: list_R6823256787227418703term_a,P4: b] :
( ( ( member_nat @ X @ ( relati4569515538964159125_set_a @ Ts3 ) )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relational_Pred_b_a @ P4 @ Ts3 ) )
= ( cons_s2104095074707585951la_a_b @ ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ P4 @ Ts3 ) @ bot_bo4495933725496725865la_a_b ) @ nil_se6596116450819887951la_a_b ) ) )
& ( ~ ( member_nat @ X @ ( relati4569515538964159125_set_a @ Ts3 ) )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relational_Pred_b_a @ P4 @ Ts3 ) )
= nil_se6596116450819887951la_a_b ) ) ) ).
% gen_impl.simps(5)
thf(fact_1013_fv_Osimps_I1_J,axiom,
! [Uu2: b,Ts3: list_R6823256787227418703term_a] :
( ( relational_fv_a_b @ ( relational_Pred_b_a @ Uu2 @ Ts3 ) )
= ( relati4569515538964159125_set_a @ Ts3 ) ) ).
% fv.simps(1)
thf(fact_1014_foldr1_Oelims,axiom,
! [X: set_nat > set_nat > set_nat,Xa2: list_set_nat,Xb: set_nat,Y3: set_nat] :
( ( ( relati7114214420354183652et_nat @ X @ Xa2 @ Xb )
= Y3 )
=> ( ! [X3: set_nat,Xs3: list_set_nat] :
( ( Xa2
= ( cons_set_nat @ X3 @ Xs3 ) )
=> ( Y3
!= ( foldr_8664167560355425461et_nat @ X @ Xs3 @ X3 ) ) )
=> ~ ( ( Xa2 = nil_set_nat )
=> ( Y3 != Xb ) ) ) ) ).
% foldr1.elims
thf(fact_1015_foldr1_Oelims,axiom,
! [X: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,Xa2: list_s3671366213839843941la_a_b,Xb: set_Re381260168593705685la_a_b,Y3: set_Re381260168593705685la_a_b] :
( ( ( relati7835315018957680435la_a_b @ X @ Xa2 @ Xb )
= Y3 )
=> ( ! [X3: set_Re381260168593705685la_a_b,Xs3: list_s3671366213839843941la_a_b] :
( ( Xa2
= ( cons_s2104095074707585951la_a_b @ X3 @ Xs3 ) )
=> ( Y3
!= ( foldr_5041855395594655059la_a_b @ X @ Xs3 @ X3 ) ) )
=> ~ ( ( Xa2 = nil_se6596116450819887951la_a_b )
=> ( Y3 != Xb ) ) ) ) ).
% foldr1.elims
thf(fact_1016_foldr1_Oelims,axiom,
! [X: list_nat > list_nat > list_nat,Xa2: list_list_nat,Xb: list_nat,Y3: list_nat] :
( ( ( relati7377577593470194238st_nat @ X @ Xa2 @ Xb )
= Y3 )
=> ( ! [X3: list_nat,Xs3: list_list_nat] :
( ( Xa2
= ( cons_list_nat @ X3 @ Xs3 ) )
=> ( Y3
!= ( foldr_6871341030409798377st_nat @ X @ Xs3 @ X3 ) ) )
=> ~ ( ( Xa2 = nil_list_nat )
=> ( Y3 != Xb ) ) ) ) ).
% foldr1.elims
thf(fact_1017_foldr1_Oelims,axiom,
! [X: nat > nat > nat,Xa2: list_nat,Xb: nat,Y3: nat] :
( ( ( relati2262546126839610926r1_nat @ X @ Xa2 @ Xb )
= Y3 )
=> ( ! [X3: nat,Xs3: list_nat] :
( ( Xa2
= ( cons_nat @ X3 @ Xs3 ) )
=> ( Y3
!= ( foldr_nat_nat @ X @ Xs3 @ X3 ) ) )
=> ~ ( ( Xa2 = nil_nat )
=> ( Y3 != Xb ) ) ) ) ).
% foldr1.elims
thf(fact_1018_flat__Disj_Osimps_I5_J,axiom,
! [V: relational_fmla_a_b] :
( ( restri569617705344514291sj_a_b @ ( relational_Neg_a_b @ V ) )
= ( insert7010464514620295119la_a_b @ ( relational_Neg_a_b @ V ) @ bot_bo4495933725496725865la_a_b ) ) ).
% flat_Disj.simps(5)
thf(fact_1019_flat__Disj_Osimps_I6_J,axiom,
! [V: relational_fmla_a_b,Va: relational_fmla_a_b] :
( ( restri569617705344514291sj_a_b @ ( relational_Conj_a_b @ V @ Va ) )
= ( insert7010464514620295119la_a_b @ ( relational_Conj_a_b @ V @ Va ) @ bot_bo4495933725496725865la_a_b ) ) ).
% flat_Disj.simps(6)
thf(fact_1020_flat__Disj_Osimps_I4_J,axiom,
! [V: nat,Va: relational_term_a] :
( ( restri569617705344514291sj_a_b @ ( relational_Eq_a_b @ V @ Va ) )
= ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ V @ Va ) @ bot_bo4495933725496725865la_a_b ) ) ).
% flat_Disj.simps(4)
thf(fact_1021_flat__Disj_Osimps_I3_J,axiom,
! [V: $o] :
( ( restri569617705344514291sj_a_b @ ( relational_Bool_a_b @ V ) )
= ( insert7010464514620295119la_a_b @ ( relational_Bool_a_b @ V ) @ bot_bo4495933725496725865la_a_b ) ) ).
% flat_Disj.simps(3)
thf(fact_1022_flat__Disj_Osimps_I7_J,axiom,
! [V: nat,Va: relational_fmla_a_b] :
( ( restri569617705344514291sj_a_b @ ( relati591517084277583526ts_a_b @ V @ Va ) )
= ( insert7010464514620295119la_a_b @ ( relati591517084277583526ts_a_b @ V @ Va ) @ bot_bo4495933725496725865la_a_b ) ) ).
% flat_Disj.simps(7)
thf(fact_1023_flat__Disj_Osimps_I2_J,axiom,
! [V: b,Va: list_R6823256787227418703term_a] :
( ( restri569617705344514291sj_a_b @ ( relational_Pred_b_a @ V @ Va ) )
= ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ V @ Va ) @ bot_bo4495933725496725865la_a_b ) ) ).
% flat_Disj.simps(2)
thf(fact_1024_gen__impl_Osimps_I1_J,axiom,
! [X: nat] :
( ( restri4663030114664890836pl_a_b @ X @ ( relational_Bool_a_b @ $false ) )
= ( cons_s2104095074707585951la_a_b @ bot_bo4495933725496725865la_a_b @ nil_se6596116450819887951la_a_b ) ) ).
% gen_impl.simps(1)
thf(fact_1025_erase_Osimps_I2_J,axiom,
! [X: nat,Ts3: list_R6823256787227418703term_a,P4: b] :
( ( ( member_nat @ X @ ( relati4569515538964159125_set_a @ Ts3 ) )
=> ( ( relational_erase_a_b @ ( relational_Pred_b_a @ P4 @ Ts3 ) @ X )
= ( relational_Bool_a_b @ $false ) ) )
& ( ~ ( member_nat @ X @ ( relati4569515538964159125_set_a @ Ts3 ) )
=> ( ( relational_erase_a_b @ ( relational_Pred_b_a @ P4 @ Ts3 ) @ X )
= ( relational_Pred_b_a @ P4 @ Ts3 ) ) ) ) ).
% erase.simps(2)
thf(fact_1026_gen__impl_Opsimps_I5_J,axiom,
! [X: nat,P4: b,Ts3: list_R6823256787227418703term_a] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X @ ( relational_Pred_b_a @ P4 @ Ts3 ) ) )
=> ( ( ( member_nat @ X @ ( relati4569515538964159125_set_a @ Ts3 ) )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relational_Pred_b_a @ P4 @ Ts3 ) )
= ( cons_s2104095074707585951la_a_b @ ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ P4 @ Ts3 ) @ bot_bo4495933725496725865la_a_b ) @ nil_se6596116450819887951la_a_b ) ) )
& ( ~ ( member_nat @ X @ ( relati4569515538964159125_set_a @ Ts3 ) )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relational_Pred_b_a @ P4 @ Ts3 ) )
= nil_se6596116450819887951la_a_b ) ) ) ) ).
% gen_impl.psimps(5)
thf(fact_1027_gen__impl_Osimps_I3_J,axiom,
! [X: nat,Y3: nat,C: a] :
( ( ( X = Y3 )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relational_Eq_a_b @ Y3 @ ( relational_Const_a @ C ) ) )
= ( cons_s2104095074707585951la_a_b @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ Y3 @ ( relational_Const_a @ C ) ) @ bot_bo4495933725496725865la_a_b ) @ nil_se6596116450819887951la_a_b ) ) )
& ( ( X != Y3 )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relational_Eq_a_b @ Y3 @ ( relational_Const_a @ C ) ) )
= nil_se6596116450819887951la_a_b ) ) ) ).
% gen_impl.simps(3)
thf(fact_1028_erase_Oelims,axiom,
! [X: relational_fmla_a_b,Xa2: nat,Y3: relational_fmla_a_b] :
( ( ( relational_erase_a_b @ X @ Xa2 )
= Y3 )
=> ( ! [T: $o] :
( ( X
= ( relational_Bool_a_b @ T ) )
=> ( Y3
!= ( relational_Bool_a_b @ T ) ) )
=> ( ! [P3: b,Ts: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ P3 @ Ts ) )
=> ~ ( ( ( member_nat @ Xa2 @ ( relati4569515538964159125_set_a @ Ts ) )
=> ( Y3
= ( relational_Bool_a_b @ $false ) ) )
& ( ~ ( member_nat @ Xa2 @ ( relati4569515538964159125_set_a @ Ts ) )
=> ( Y3
= ( relational_Pred_b_a @ P3 @ Ts ) ) ) ) )
=> ( ! [Z: nat,T: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ Z @ T ) )
=> ~ ( ( ( T
= ( relational_Var_a @ Z ) )
=> ( Y3
= ( relational_Bool_a_b @ $true ) ) )
& ( ( T
!= ( relational_Var_a @ Z ) )
=> ( ( ( ( Xa2 = Z )
| ( member_nat @ Xa2 @ ( relati6004689760767320788_set_a @ T ) ) )
=> ( Y3
= ( relational_Bool_a_b @ $false ) ) )
& ( ~ ( ( Xa2 = Z )
| ( member_nat @ Xa2 @ ( relati6004689760767320788_set_a @ T ) ) )
=> ( Y3
= ( relational_Eq_a_b @ Z @ T ) ) ) ) ) ) )
=> ( ! [Q3: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q3 ) )
=> ( Y3
!= ( relational_Neg_a_b @ ( relational_erase_a_b @ Q3 @ Xa2 ) ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q12 @ Q22 ) )
=> ( Y3
!= ( relational_Conj_a_b @ ( relational_erase_a_b @ Q12 @ Xa2 ) @ ( relational_erase_a_b @ Q22 @ Xa2 ) ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q12 @ Q22 ) )
=> ( Y3
!= ( relational_Disj_a_b @ ( relational_erase_a_b @ Q12 @ Xa2 ) @ ( relational_erase_a_b @ Q22 @ Xa2 ) ) ) )
=> ~ ! [Z: nat,Q3: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ Z @ Q3 ) )
=> ~ ( ( ( Xa2 = Z )
=> ( Y3
= ( relati591517084277583526ts_a_b @ Xa2 @ Q3 ) ) )
& ( ( Xa2 != Z )
=> ( Y3
= ( relati591517084277583526ts_a_b @ Z @ ( relational_erase_a_b @ Q3 @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% erase.elims
thf(fact_1029_is__singleton__the__elem,axiom,
( is_sin6594375743535830443la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b] :
( A3
= ( insert7010464514620295119la_a_b @ ( the_el6350558617753882986la_a_b @ A3 ) @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% is_singleton_the_elem
thf(fact_1030_is__singleton__the__elem,axiom,
( is_sin1114528679824004833la_a_b
= ( ^ [A3: set_se6865892389300016395la_a_b] :
( A3
= ( insert2023870700798818565la_a_b @ ( the_el7486773796875720352la_a_b @ A3 ) @ bot_bo2891247006866115487la_a_b ) ) ) ) ).
% is_singleton_the_elem
thf(fact_1031_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
( A3
= ( insert_nat2 @ ( the_elem_nat @ A3 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_1032_is__singleton__the__elem,axiom,
( is_sin2641923865335537900st_nat
= ( ^ [A3: set_list_nat] :
( A3
= ( insert_list_nat2 @ ( the_elem_list_nat @ A3 ) @ bot_bot_set_list_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_1033_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A3: set_a] :
( A3
= ( insert_a @ ( the_elem_a @ A3 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_1034_is__singletonI,axiom,
! [X: relational_fmla_a_b] : ( is_sin6594375743535830443la_a_b @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ).
% is_singletonI
thf(fact_1035_is__singletonI,axiom,
! [X: set_Re381260168593705685la_a_b] : ( is_sin1114528679824004833la_a_b @ ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) ) ).
% is_singletonI
thf(fact_1036_is__singletonI,axiom,
! [X: nat] : ( is_singleton_nat @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_1037_is__singletonI,axiom,
! [X: list_nat] : ( is_sin2641923865335537900st_nat @ ( insert_list_nat2 @ X @ bot_bot_set_list_nat ) ) ).
% is_singletonI
thf(fact_1038_is__singletonI,axiom,
! [X: a] : ( is_singleton_a @ ( insert_a @ X @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_1039_csts__term_Oelims,axiom,
! [X: relati112041753218324778la_a_b,Y3: set_Re381260168593705685la_a_b] :
( ( ( relati1926769566493843000la_a_b @ X )
= Y3 )
=> ( ( ? [X3: nat] :
( X
= ( relati2528351936964124430la_a_b @ X3 ) )
=> ( Y3 != bot_bo4495933725496725865la_a_b ) )
=> ~ ! [C3: relational_fmla_a_b] :
( ( X
= ( relati3340789340644911146la_a_b @ C3 ) )
=> ( Y3
!= ( insert7010464514620295119la_a_b @ C3 @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ).
% csts_term.elims
thf(fact_1040_csts__term_Oelims,axiom,
! [X: relati7801739318661992288la_a_b,Y3: set_se6865892389300016395la_a_b] :
( ( ( relati9001609190041627502la_a_b @ X )
= Y3 )
=> ( ( ? [X3: nat] :
( X
= ( relati4371369962149319748la_a_b @ X3 ) )
=> ( Y3 != bot_bo2891247006866115487la_a_b ) )
=> ~ ! [C3: set_Re381260168593705685la_a_b] :
( ( X
= ( relati411545593891581280la_a_b @ C3 ) )
=> ( Y3
!= ( insert2023870700798818565la_a_b @ C3 @ bot_bo2891247006866115487la_a_b ) ) ) ) ) ).
% csts_term.elims
thf(fact_1041_csts__term_Oelims,axiom,
! [X: relational_term_nat,Y3: set_nat] :
( ( ( relati694035416245573993rm_nat @ X )
= Y3 )
=> ( ( ? [X3: nat] :
( X
= ( relational_Var_nat @ X3 ) )
=> ( Y3 != bot_bot_set_nat ) )
=> ~ ! [C3: nat] :
( ( X
= ( relational_Const_nat @ C3 ) )
=> ( Y3
!= ( insert_nat2 @ C3 @ bot_bot_set_nat ) ) ) ) ) ).
% csts_term.elims
thf(fact_1042_csts__term_Oelims,axiom,
! [X: relati643601439615553003st_nat,Y3: set_list_nat] :
( ( ( relati97657567636182009st_nat @ X )
= Y3 )
=> ( ( ? [X3: nat] :
( X
= ( relati8710011843476256719st_nat @ X3 ) )
=> ( Y3 != bot_bot_set_list_nat ) )
=> ~ ! [C3: list_nat] :
( ( X
= ( relati1027843696874716907st_nat @ C3 ) )
=> ( Y3
!= ( insert_list_nat2 @ C3 @ bot_bot_set_list_nat ) ) ) ) ) ).
% csts_term.elims
thf(fact_1043_csts__term_Oelims,axiom,
! [X: relational_term_a,Y3: set_a] :
( ( ( relati6638259395341848997term_a @ X )
= Y3 )
=> ( ( ? [X3: nat] :
( X
= ( relational_Var_a @ X3 ) )
=> ( Y3 != bot_bot_set_a ) )
=> ~ ! [C3: a] :
( ( X
= ( relational_Const_a @ C3 ) )
=> ( Y3
!= ( insert_a @ C3 @ bot_bot_set_a ) ) ) ) ) ).
% csts_term.elims
thf(fact_1044_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_b] :
( ( sup_sup_set_b @ X @ bot_bot_set_b )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_1045_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_Re381260168593705685la_a_b > $o] :
( ( sup_su2223990397388305166_a_b_o @ X @ bot_bo8397419132826146726_a_b_o )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_1046_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_Re381260168593705685la_a_b] :
( ( sup_su5130108678486352897la_a_b @ X @ bot_bo4495933725496725865la_a_b )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_1047_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_se6865892389300016395la_a_b] :
( ( sup_su4783144482993978935la_a_b @ X @ bot_bo2891247006866115487la_a_b )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_1048_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_1049_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_list_nat] :
( ( sup_sup_set_list_nat @ X @ bot_bot_set_list_nat )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_1050_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_1051_fv__term__setD,axiom,
! [N2: nat,T3: relational_term_a] :
( ( member_nat @ N2 @ ( relati6004689760767320788_set_a @ T3 ) )
=> ( T3
= ( relational_Var_a @ N2 ) ) ) ).
% fv_term_setD
thf(fact_1052_fv__term__set_Osimps_I1_J,axiom,
! [N2: nat] :
( ( relati6004689760767320788_set_a @ ( relational_Var_a @ N2 ) )
= ( insert_nat2 @ N2 @ bot_bot_set_nat ) ) ).
% fv_term_set.simps(1)
thf(fact_1053_fv__term__set_Osimps_I2_J,axiom,
! [V: a] :
( ( relati6004689760767320788_set_a @ ( relational_Const_a @ V ) )
= bot_bot_set_nat ) ).
% fv_term_set.simps(2)
thf(fact_1054_is__singletonI_H,axiom,
! [A: set_li7574107037410898715la_a_b] :
( ( A != bot_bo7655846380748197807la_a_b )
=> ( ! [X3: list_s3671366213839843941la_a_b,Y5: list_s3671366213839843941la_a_b] :
( ( member987187531357909116la_a_b @ X3 @ A )
=> ( ( member987187531357909116la_a_b @ Y5 @ A )
=> ( X3 = Y5 ) ) )
=> ( is_sin5582224557664405617la_a_b @ A ) ) ) ).
% is_singletonI'
thf(fact_1055_is__singletonI_H,axiom,
! [A: set_se1626149436229564481la_a_b] :
( ( A != bot_bo1069171165257752277la_a_b )
=> ( ! [X3: set_se6865892389300016395la_a_b,Y5: set_se6865892389300016395la_a_b] :
( ( member1693404252451860130la_a_b @ X3 @ A )
=> ( ( member1693404252451860130la_a_b @ Y5 @ A )
=> ( X3 = Y5 ) ) )
=> ( is_sin4154956242601603351la_a_b @ A ) ) ) ).
% is_singletonI'
thf(fact_1056_is__singletonI_H,axiom,
! [A: set_b] :
( ( A != bot_bot_set_b )
=> ( ! [X3: b,Y5: b] :
( ( member_b @ X3 @ A )
=> ( ( member_b @ Y5 @ A )
=> ( X3 = Y5 ) ) )
=> ( is_singleton_b @ A ) ) ) ).
% is_singletonI'
thf(fact_1057_is__singletonI_H,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( A != bot_bo4495933725496725865la_a_b )
=> ( ! [X3: relational_fmla_a_b,Y5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A )
=> ( ( member4680049679412964150la_a_b @ Y5 @ A )
=> ( X3 = Y5 ) ) )
=> ( is_sin6594375743535830443la_a_b @ A ) ) ) ).
% is_singletonI'
thf(fact_1058_is__singletonI_H,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X3: nat,Y5: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y5 @ A )
=> ( X3 = Y5 ) ) )
=> ( is_singleton_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_1059_is__singletonI_H,axiom,
! [A: set_list_nat] :
( ( A != bot_bot_set_list_nat )
=> ( ! [X3: list_nat,Y5: list_nat] :
( ( member_list_nat @ X3 @ A )
=> ( ( member_list_nat @ Y5 @ A )
=> ( X3 = Y5 ) ) )
=> ( is_sin2641923865335537900st_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_1060_is__singletonI_H,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
=> ( ! [X3: a,Y5: a] :
( ( member_a @ X3 @ A )
=> ( ( member_a @ Y5 @ A )
=> ( X3 = Y5 ) ) )
=> ( is_singleton_a @ A ) ) ) ).
% is_singletonI'
thf(fact_1061_is__singletonI_H,axiom,
! [A: set_se6865892389300016395la_a_b] :
( ( A != bot_bo2891247006866115487la_a_b )
=> ( ! [X3: set_Re381260168593705685la_a_b,Y5: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ A )
=> ( ( member3481406638322139244la_a_b @ Y5 @ A )
=> ( X3 = Y5 ) ) )
=> ( is_sin1114528679824004833la_a_b @ A ) ) ) ).
% is_singletonI'
thf(fact_1062_fv__term__set_Oelims,axiom,
! [X: relational_term_a,Y3: set_nat] :
( ( ( relati6004689760767320788_set_a @ X )
= Y3 )
=> ( ! [N: nat] :
( ( X
= ( relational_Var_a @ N ) )
=> ( Y3
!= ( insert_nat2 @ N @ bot_bot_set_nat ) ) )
=> ~ ( ? [V2: a] :
( X
= ( relational_Const_a @ V2 ) )
=> ( Y3 != bot_bot_set_nat ) ) ) ) ).
% fv_term_set.elims
thf(fact_1063_boolean__algebra__cancel_Osup1,axiom,
! [A: set_a,K: set_a,A2: set_a,B2: set_a] :
( ( A
= ( sup_sup_set_a @ K @ A2 ) )
=> ( ( sup_sup_set_a @ A @ B2 )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_1064_boolean__algebra__cancel_Osup1,axiom,
! [A: set_Re381260168593705685la_a_b > $o,K: set_Re381260168593705685la_a_b > $o,A2: set_Re381260168593705685la_a_b > $o,B2: set_Re381260168593705685la_a_b > $o] :
( ( A
= ( sup_su2223990397388305166_a_b_o @ K @ A2 ) )
=> ( ( sup_su2223990397388305166_a_b_o @ A @ B2 )
= ( sup_su2223990397388305166_a_b_o @ K @ ( sup_su2223990397388305166_a_b_o @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_1065_lookup_Ocases,axiom,
! [X: produc5824815472048091686at_nat] :
( ! [X3: nat,Xs3: list_nat,Y5: nat,Ys3: list_nat,Z: nat] :
( X
!= ( produc4483744321870515032at_nat @ ( cons_nat @ X3 @ Xs3 ) @ ( produc1298395424260782409at_nat @ ( cons_nat @ Y5 @ Ys3 ) @ Z ) ) )
=> ( ! [Uv: list_nat,Uw: nat] :
( X
!= ( produc4483744321870515032at_nat @ nil_nat @ ( produc1298395424260782409at_nat @ Uv @ Uw ) ) )
=> ~ ! [Uu: list_nat,Uw: nat] :
( X
!= ( produc4483744321870515032at_nat @ Uu @ ( produc1298395424260782409at_nat @ nil_nat @ Uw ) ) ) ) ) ).
% lookup.cases
thf(fact_1066_fv_Oelims,axiom,
! [X: relational_fmla_a_b,Y3: set_nat] :
( ( ( relational_fv_a_b @ X )
= Y3 )
=> ( ! [Uu: b,Ts: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ Uu @ Ts ) )
=> ( Y3
!= ( relati4569515538964159125_set_a @ Ts ) ) )
=> ( ( ? [B3: $o] :
( X
= ( relational_Bool_a_b @ B3 ) )
=> ( Y3 != bot_bot_set_nat ) )
=> ( ! [X3: nat,T2: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X3 @ T2 ) )
=> ( Y3
!= ( sup_sup_set_nat @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) @ ( relati6004689760767320788_set_a @ T2 ) ) ) )
=> ( ! [Phi: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Phi ) )
=> ( Y3
!= ( relational_fv_a_b @ Phi ) ) )
=> ( ! [Phi: relational_fmla_a_b,Psi: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Phi @ Psi ) )
=> ( Y3
!= ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi ) @ ( relational_fv_a_b @ Psi ) ) ) )
=> ( ! [Phi: relational_fmla_a_b,Psi: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Phi @ Psi ) )
=> ( Y3
!= ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi ) @ ( relational_fv_a_b @ Psi ) ) ) )
=> ~ ! [Z: nat,Phi: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ Z @ Phi ) )
=> ( Y3
!= ( minus_minus_set_nat @ ( relational_fv_a_b @ Phi ) @ ( insert_nat2 @ Z @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ) ) ).
% fv.elims
thf(fact_1067_list__all__zip_Ocases,axiom,
! [X: produc4787317212837456354st_nat] :
( ! [P6: nat > nat > $o] :
( X
!= ( produc3127733452865184594st_nat @ P6 @ ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) ) )
=> ( ! [P6: nat > nat > $o,A6: nat,As: list_nat,B3: nat,Bs: list_nat] :
( X
!= ( produc3127733452865184594st_nat @ P6 @ ( produc2694037385005941721st_nat @ ( cons_nat @ A6 @ As ) @ ( cons_nat @ B3 @ Bs ) ) ) )
=> ( ! [P6: nat > nat > $o,V2: nat,Va2: list_nat] :
( X
!= ( produc3127733452865184594st_nat @ P6 @ ( produc2694037385005941721st_nat @ ( cons_nat @ V2 @ Va2 ) @ nil_nat ) ) )
=> ~ ! [P6: nat > nat > $o,V2: nat,Va2: list_nat] :
( X
!= ( produc3127733452865184594st_nat @ P6 @ ( produc2694037385005941721st_nat @ nil_nat @ ( cons_nat @ V2 @ Va2 ) ) ) ) ) ) ) ).
% list_all_zip.cases
thf(fact_1068_DiffI,axiom,
! [C: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ C @ A )
=> ( ~ ( member3481406638322139244la_a_b @ C @ B )
=> ( member3481406638322139244la_a_b @ C @ ( minus_4705846553145473764la_a_b @ A @ B ) ) ) ) ).
% DiffI
thf(fact_1069_Diff__iff,axiom,
! [C: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ C @ ( minus_4705846553145473764la_a_b @ A @ B ) )
= ( ( member3481406638322139244la_a_b @ C @ A )
& ~ ( member3481406638322139244la_a_b @ C @ B ) ) ) ).
% Diff_iff
thf(fact_1070_Diff__insert0,axiom,
! [X: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ~ ( member3481406638322139244la_a_b @ X @ A )
=> ( ( minus_4705846553145473764la_a_b @ A @ ( insert2023870700798818565la_a_b @ X @ B ) )
= ( minus_4705846553145473764la_a_b @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_1071_insert__Diff1,axiom,
! [X: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b,A: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ X @ B )
=> ( ( minus_4705846553145473764la_a_b @ ( insert2023870700798818565la_a_b @ X @ A ) @ B )
= ( minus_4705846553145473764la_a_b @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_1072_DiffE,axiom,
! [C: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ C @ ( minus_4705846553145473764la_a_b @ A @ B ) )
=> ~ ( ( member3481406638322139244la_a_b @ C @ A )
=> ( member3481406638322139244la_a_b @ C @ B ) ) ) ).
% DiffE
thf(fact_1073_DiffD1,axiom,
! [C: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ C @ ( minus_4705846553145473764la_a_b @ A @ B ) )
=> ( member3481406638322139244la_a_b @ C @ A ) ) ).
% DiffD1
thf(fact_1074_DiffD2,axiom,
! [C: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ C @ ( minus_4705846553145473764la_a_b @ A @ B ) )
=> ~ ( member3481406638322139244la_a_b @ C @ B ) ) ).
% DiffD2
thf(fact_1075_insert__Diff__if,axiom,
! [X: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b,A: set_se6865892389300016395la_a_b] :
( ( ( member3481406638322139244la_a_b @ X @ B )
=> ( ( minus_4705846553145473764la_a_b @ ( insert2023870700798818565la_a_b @ X @ A ) @ B )
= ( minus_4705846553145473764la_a_b @ A @ B ) ) )
& ( ~ ( member3481406638322139244la_a_b @ X @ B )
=> ( ( minus_4705846553145473764la_a_b @ ( insert2023870700798818565la_a_b @ X @ A ) @ B )
= ( insert2023870700798818565la_a_b @ X @ ( minus_4705846553145473764la_a_b @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1076_insert__Diff,axiom,
! [A2: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ A2 @ A )
=> ( ( insert2023870700798818565la_a_b @ A2 @ ( minus_4705846553145473764la_a_b @ A @ ( insert2023870700798818565la_a_b @ A2 @ bot_bo2891247006866115487la_a_b ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1077_Diff__insert__absorb,axiom,
! [X: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ~ ( member3481406638322139244la_a_b @ X @ A )
=> ( ( minus_4705846553145473764la_a_b @ ( insert2023870700798818565la_a_b @ X @ A ) @ ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1078_bex2I,axiom,
! [A2: diff_a789849492003309546t_unit,B2: list_nat,S: set_Pr5509071407266205487st_nat,P: diff_a789849492003309546t_unit > list_nat > $o] :
( ( member3320643659109938320st_nat @ ( produc6847244279022580211st_nat @ A2 @ B2 ) @ S )
=> ( ( ( member3320643659109938320st_nat @ ( produc6847244279022580211st_nat @ A2 @ B2 ) @ S )
=> ( P @ A2 @ B2 ) )
=> ? [A6: diff_a789849492003309546t_unit,B3: list_nat] :
( ( member3320643659109938320st_nat @ ( produc6847244279022580211st_nat @ A6 @ B3 ) @ S )
& ( P @ A6 @ B3 ) ) ) ) ).
% bex2I
thf(fact_1079_Misc_Ofoldr__Cons,axiom,
! [Xs2: list_nat] :
( ( foldr_nat_list_nat @ cons_nat @ Xs2 @ nil_nat )
= Xs2 ) ).
% Misc.foldr_Cons
thf(fact_1080_mergesort__by__rel__merge__induct,axiom,
! [P: list_nat > list_nat > $o,R2: nat > nat > $o,Xs2: list_nat,Ys: list_nat] :
( ! [Xs3: list_nat] : ( P @ Xs3 @ nil_nat )
=> ( ! [X_12: list_nat] : ( P @ nil_nat @ X_12 )
=> ( ! [X3: nat,Xs3: list_nat,Y5: nat,Ys3: list_nat] :
( ( R2 @ X3 @ Y5 )
=> ( ( P @ Xs3 @ ( cons_nat @ Y5 @ Ys3 ) )
=> ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y5 @ Ys3 ) ) ) )
=> ( ! [X3: nat,Xs3: list_nat,Y5: nat,Ys3: list_nat] :
( ~ ( R2 @ X3 @ Y5 )
=> ( ( P @ ( cons_nat @ X3 @ Xs3 ) @ Ys3 )
=> ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y5 @ Ys3 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% mergesort_by_rel_merge_induct
thf(fact_1081_list__induct__first2,axiom,
! [P: list_nat > $o,Xs2: list_nat] :
( ( P @ nil_nat )
=> ( ! [X3: nat] : ( P @ ( cons_nat @ X3 @ nil_nat ) )
=> ( ! [X13: nat,X23: nat,Xs3: list_nat] :
( ( P @ Xs3 )
=> ( P @ ( cons_nat @ X13 @ ( cons_nat @ X23 @ Xs3 ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_induct_first2
thf(fact_1082_list__2pre__induct,axiom,
! [P: list_nat > list_nat > $o,W1: list_nat,W2: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [E2: nat,W12: list_nat,W22: list_nat] :
( ( P @ W12 @ W22 )
=> ( P @ ( cons_nat @ E2 @ W12 ) @ W22 ) )
=> ( ! [E2: nat,W13: list_nat,W23: list_nat] :
( ( P @ W13 @ W23 )
=> ( P @ W13 @ ( cons_nat @ E2 @ W23 ) ) )
=> ( P @ W1 @ W2 ) ) ) ) ).
% list_2pre_induct
thf(fact_1083_neq__NilE,axiom,
! [L: list_nat] :
( ( L != nil_nat )
=> ~ ! [X3: nat,Xs3: list_nat] :
( L
!= ( cons_nat @ X3 @ Xs3 ) ) ) ).
% neq_NilE
thf(fact_1084_mergesort__by__rel__split_Ocases,axiom,
! [X: produc8368264342088597808st_nat] :
( ! [Xs1: list_nat,Xs22: list_nat] :
( X
!= ( produc1778712578675790378st_nat @ ( produc2694037385005941721st_nat @ Xs1 @ Xs22 ) @ nil_nat ) )
=> ( ! [Xs1: list_nat,Xs22: list_nat,X3: nat] :
( X
!= ( produc1778712578675790378st_nat @ ( produc2694037385005941721st_nat @ Xs1 @ Xs22 ) @ ( cons_nat @ X3 @ nil_nat ) ) )
=> ~ ! [Xs1: list_nat,Xs22: list_nat,X13: nat,X23: nat,Xs3: list_nat] :
( X
!= ( produc1778712578675790378st_nat @ ( produc2694037385005941721st_nat @ Xs1 @ Xs22 ) @ ( cons_nat @ X13 @ ( cons_nat @ X23 @ Xs3 ) ) ) ) ) ) ).
% mergesort_by_rel_split.cases
thf(fact_1085_mergesort__by__rel__merge_Ocases,axiom,
! [X: produc4787317212837456354st_nat] :
( ! [R3: nat > nat > $o,X3: nat,Xs3: list_nat,Y5: nat,Ys3: list_nat] :
( X
!= ( produc3127733452865184594st_nat @ R3 @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y5 @ Ys3 ) ) ) )
=> ( ! [R3: nat > nat > $o,Xs3: list_nat] :
( X
!= ( produc3127733452865184594st_nat @ R3 @ ( produc2694037385005941721st_nat @ Xs3 @ nil_nat ) ) )
=> ~ ! [R3: nat > nat > $o,V2: nat,Va2: list_nat] :
( X
!= ( produc3127733452865184594st_nat @ R3 @ ( produc2694037385005941721st_nat @ nil_nat @ ( cons_nat @ V2 @ Va2 ) ) ) ) ) ) ).
% mergesort_by_rel_merge.cases
thf(fact_1086_restrict_Ocases,axiom,
! [X: produc2715289222843590288st_nat] :
( ! [A7: set_nat,X3: nat,Xs3: list_nat,Y5: nat,Ys3: list_nat] :
( X
!= ( produc7873415857651533250st_nat @ A7 @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y5 @ Ys3 ) ) ) )
=> ( ! [A7: set_nat,Uv: list_nat] :
( X
!= ( produc7873415857651533250st_nat @ A7 @ ( produc2694037385005941721st_nat @ nil_nat @ Uv ) ) )
=> ~ ! [A7: set_nat,Uu: list_nat] :
( X
!= ( produc7873415857651533250st_nat @ A7 @ ( produc2694037385005941721st_nat @ Uu @ nil_nat ) ) ) ) ) ).
% restrict.cases
thf(fact_1087_quicksort__by__rel_Ocases,axiom,
! [X: produc4787317212837456354st_nat] :
( ! [R3: nat > nat > $o,Sl: list_nat] :
( X
!= ( produc3127733452865184594st_nat @ R3 @ ( produc2694037385005941721st_nat @ Sl @ nil_nat ) ) )
=> ~ ! [R3: nat > nat > $o,Sl: list_nat,X3: nat,Xs3: list_nat] :
( X
!= ( produc3127733452865184594st_nat @ R3 @ ( produc2694037385005941721st_nat @ Sl @ ( cons_nat @ X3 @ Xs3 ) ) ) ) ) ).
% quicksort_by_rel.cases
thf(fact_1088_partition__rev_Ocases,axiom,
! [X: produc6155704871766690142st_nat] :
( ! [P6: nat > $o,Yes: list_nat,No: list_nat] :
( X
!= ( produc2404789219858123352st_nat @ P6 @ ( produc1778712578675790378st_nat @ ( produc2694037385005941721st_nat @ Yes @ No ) @ nil_nat ) ) )
=> ~ ! [P6: nat > $o,Yes: list_nat,No: list_nat,X3: nat,Xs3: list_nat] :
( X
!= ( produc2404789219858123352st_nat @ P6 @ ( produc1778712578675790378st_nat @ ( produc2694037385005941721st_nat @ Yes @ No ) @ ( cons_nat @ X3 @ Xs3 ) ) ) ) ) ).
% partition_rev.cases
thf(fact_1089_fv_Opelims,axiom,
! [X: relational_fmla_a_b,Y3: set_nat] :
( ( ( relational_fv_a_b @ X )
= Y3 )
=> ( ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ X )
=> ( ! [Uu: b,Ts: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ Uu @ Ts ) )
=> ( ( Y3
= ( relati4569515538964159125_set_a @ Ts ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Pred_b_a @ Uu @ Ts ) ) ) )
=> ( ! [B3: $o] :
( ( X
= ( relational_Bool_a_b @ B3 ) )
=> ( ( Y3 = bot_bot_set_nat )
=> ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Bool_a_b @ B3 ) ) ) )
=> ( ! [X3: nat,T2: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X3 @ T2 ) )
=> ( ( Y3
= ( sup_sup_set_nat @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) @ ( relati6004689760767320788_set_a @ T2 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Eq_a_b @ X3 @ T2 ) ) ) )
=> ( ! [Phi: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Phi ) )
=> ( ( Y3
= ( relational_fv_a_b @ Phi ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Neg_a_b @ Phi ) ) ) )
=> ( ! [Phi: relational_fmla_a_b,Psi: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Phi @ Psi ) )
=> ( ( Y3
= ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi ) @ ( relational_fv_a_b @ Psi ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Conj_a_b @ Phi @ Psi ) ) ) )
=> ( ! [Phi: relational_fmla_a_b,Psi: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Phi @ Psi ) )
=> ( ( Y3
= ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi ) @ ( relational_fv_a_b @ Psi ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Disj_a_b @ Phi @ Psi ) ) ) )
=> ~ ! [Z: nat,Phi: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ Z @ Phi ) )
=> ( ( Y3
= ( minus_minus_set_nat @ ( relational_fv_a_b @ Phi ) @ ( insert_nat2 @ Z @ bot_bot_set_nat ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relati591517084277583526ts_a_b @ Z @ Phi ) ) ) ) ) ) ) ) ) ) ) ) ).
% fv.pelims
thf(fact_1090_Autoref__Bindings__HOL_Olist__eq_Ocases,axiom,
! [X: produc4787317212837456354st_nat] :
( ! [Eq: nat > nat > $o] :
( X
!= ( produc3127733452865184594st_nat @ Eq @ ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) ) )
=> ( ! [Eq: nat > nat > $o,A6: nat,L2: list_nat,A8: nat,L3: list_nat] :
( X
!= ( produc3127733452865184594st_nat @ Eq @ ( produc2694037385005941721st_nat @ ( cons_nat @ A6 @ L2 ) @ ( cons_nat @ A8 @ L3 ) ) ) )
=> ( ! [Uu: nat > nat > $o,V2: nat,Va2: list_nat] :
( X
!= ( produc3127733452865184594st_nat @ Uu @ ( produc2694037385005941721st_nat @ ( cons_nat @ V2 @ Va2 ) @ nil_nat ) ) )
=> ~ ! [Uu: nat > nat > $o,V2: nat,Va2: list_nat] :
( X
!= ( produc3127733452865184594st_nat @ Uu @ ( produc2694037385005941721st_nat @ nil_nat @ ( cons_nat @ V2 @ Va2 ) ) ) ) ) ) ) ).
% Autoref_Bindings_HOL.list_eq.cases
thf(fact_1091_lsnd__basic_Og__from__list__aux_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [Accs: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ Accs @ nil_nat ) )
=> ~ ! [Accs: list_nat,X3: nat,L2: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ Accs @ ( cons_nat @ X3 @ L2 ) ) ) ) ).
% lsnd_basic.g_from_list_aux.cases
thf(fact_1092_lss__basic_Og__from__list__aux_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [Accs: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ Accs @ nil_nat ) )
=> ~ ! [Accs: list_nat,X3: nat,L2: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ Accs @ ( cons_nat @ X3 @ L2 ) ) ) ) ).
% lss_basic.g_from_list_aux.cases
thf(fact_1093_subset__eq__mset__impl_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [Ys3: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys3 ) )
=> ~ ! [X3: nat,Xs3: list_nat,Ys3: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs3 ) @ Ys3 ) ) ) ).
% subset_eq_mset_impl.cases
thf(fact_1094_subset__sorted_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [L22: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ L22 ) )
=> ( ! [X13: nat,L1: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X13 @ L1 ) @ nil_nat ) )
=> ~ ! [X13: nat,L1: list_nat,X23: nat,L22: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X13 @ L1 ) @ ( cons_nat @ X23 @ L22 ) ) ) ) ) ).
% subset_sorted.cases
thf(fact_1095_StdBasicSetDefs_Og__from__list__aux_Ocases,axiom,
! [X: produc2026718404927347961st_nat] :
( ! [Accs: diff_a789849492003309546t_unit] :
( X
!= ( produc6847244279022580211st_nat @ Accs @ nil_nat ) )
=> ~ ! [Accs: diff_a789849492003309546t_unit,X3: nat,L2: list_nat] :
( X
!= ( produc6847244279022580211st_nat @ Accs @ ( cons_nat @ X3 @ L2 ) ) ) ) ).
% StdBasicSetDefs.g_from_list_aux.cases
thf(fact_1096_gen__eqs,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Z2: nat] :
( ( relational_gen_a_b @ X @ Q @ G )
=> ( ( relational_eqs_a_b @ Z2 @ G )
= bot_bot_set_nat ) ) ).
% gen_eqs
thf(fact_1097_memb__sorted_Ocases,axiom,
! [X: produc1540777390238407569at_nat] :
( ! [X3: nat] :
( X
!= ( produc1298395424260782409at_nat @ nil_nat @ X3 ) )
=> ~ ! [Y5: nat,Xs3: list_nat,X3: nat] :
( X
!= ( produc1298395424260782409at_nat @ ( cons_nat @ Y5 @ Xs3 ) @ X3 ) ) ) ).
% memb_sorted.cases
thf(fact_1098_insertion__sort_Ocases,axiom,
! [X: produc4575160907756185873st_nat] :
( ! [X3: nat] :
( X
!= ( produc8282810413953273033st_nat @ X3 @ nil_nat ) )
=> ~ ! [X3: nat,Y5: nat,Xs3: list_nat] :
( X
!= ( produc8282810413953273033st_nat @ X3 @ ( cons_nat @ Y5 @ Xs3 ) ) ) ) ).
% insertion_sort.cases
thf(fact_1099_erase_Opelims,axiom,
! [X: relational_fmla_a_b,Xa2: nat,Y3: relational_fmla_a_b] :
( ( ( relational_erase_a_b @ X @ Xa2 )
= Y3 )
=> ( ( accp_P4351966040938400857_b_nat @ relati5987653437628155313el_a_b @ ( produc4282057684358614024_b_nat @ X @ Xa2 ) )
=> ( ! [T: $o] :
( ( X
= ( relational_Bool_a_b @ T ) )
=> ( ( Y3
= ( relational_Bool_a_b @ T ) )
=> ~ ( accp_P4351966040938400857_b_nat @ relati5987653437628155313el_a_b @ ( produc4282057684358614024_b_nat @ ( relational_Bool_a_b @ T ) @ Xa2 ) ) ) )
=> ( ! [P3: b,Ts: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ P3 @ Ts ) )
=> ( ( ( ( member_nat @ Xa2 @ ( relati4569515538964159125_set_a @ Ts ) )
=> ( Y3
= ( relational_Bool_a_b @ $false ) ) )
& ( ~ ( member_nat @ Xa2 @ ( relati4569515538964159125_set_a @ Ts ) )
=> ( Y3
= ( relational_Pred_b_a @ P3 @ Ts ) ) ) )
=> ~ ( accp_P4351966040938400857_b_nat @ relati5987653437628155313el_a_b @ ( produc4282057684358614024_b_nat @ ( relational_Pred_b_a @ P3 @ Ts ) @ Xa2 ) ) ) )
=> ( ! [Z: nat,T: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ Z @ T ) )
=> ( ( ( ( T
= ( relational_Var_a @ Z ) )
=> ( Y3
= ( relational_Bool_a_b @ $true ) ) )
& ( ( T
!= ( relational_Var_a @ Z ) )
=> ( ( ( ( Xa2 = Z )
| ( member_nat @ Xa2 @ ( relati6004689760767320788_set_a @ T ) ) )
=> ( Y3
= ( relational_Bool_a_b @ $false ) ) )
& ( ~ ( ( Xa2 = Z )
| ( member_nat @ Xa2 @ ( relati6004689760767320788_set_a @ T ) ) )
=> ( Y3
= ( relational_Eq_a_b @ Z @ T ) ) ) ) ) )
=> ~ ( accp_P4351966040938400857_b_nat @ relati5987653437628155313el_a_b @ ( produc4282057684358614024_b_nat @ ( relational_Eq_a_b @ Z @ T ) @ Xa2 ) ) ) )
=> ( ! [Q3: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q3 ) )
=> ( ( Y3
= ( relational_Neg_a_b @ ( relational_erase_a_b @ Q3 @ Xa2 ) ) )
=> ~ ( accp_P4351966040938400857_b_nat @ relati5987653437628155313el_a_b @ ( produc4282057684358614024_b_nat @ ( relational_Neg_a_b @ Q3 ) @ Xa2 ) ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q12 @ Q22 ) )
=> ( ( Y3
= ( relational_Conj_a_b @ ( relational_erase_a_b @ Q12 @ Xa2 ) @ ( relational_erase_a_b @ Q22 @ Xa2 ) ) )
=> ~ ( accp_P4351966040938400857_b_nat @ relati5987653437628155313el_a_b @ ( produc4282057684358614024_b_nat @ ( relational_Conj_a_b @ Q12 @ Q22 ) @ Xa2 ) ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q12 @ Q22 ) )
=> ( ( Y3
= ( relational_Disj_a_b @ ( relational_erase_a_b @ Q12 @ Xa2 ) @ ( relational_erase_a_b @ Q22 @ Xa2 ) ) )
=> ~ ( accp_P4351966040938400857_b_nat @ relati5987653437628155313el_a_b @ ( produc4282057684358614024_b_nat @ ( relational_Disj_a_b @ Q12 @ Q22 ) @ Xa2 ) ) ) )
=> ~ ! [Z: nat,Q3: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ Z @ Q3 ) )
=> ( ( ( ( Xa2 = Z )
=> ( Y3
= ( relati591517084277583526ts_a_b @ Xa2 @ Q3 ) ) )
& ( ( Xa2 != Z )
=> ( Y3
= ( relati591517084277583526ts_a_b @ Z @ ( relational_erase_a_b @ Q3 @ Xa2 ) ) ) ) )
=> ~ ( accp_P4351966040938400857_b_nat @ relati5987653437628155313el_a_b @ ( produc4282057684358614024_b_nat @ ( relati591517084277583526ts_a_b @ Z @ Q3 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% erase.pelims
thf(fact_1100_restrict_Oelims,axiom,
! [X: set_se6865892389300016395la_a_b,Xa2: list_s3671366213839843941la_a_b,Xb: list_nat,Y3: list_nat] :
( ( ( restri9062294100618341515_b_nat @ X @ Xa2 @ Xb )
= Y3 )
=> ( ! [X3: set_Re381260168593705685la_a_b,Xs3: list_s3671366213839843941la_a_b] :
( ( Xa2
= ( cons_s2104095074707585951la_a_b @ X3 @ Xs3 ) )
=> ! [Y5: nat,Ys3: list_nat] :
( ( Xb
= ( cons_nat @ Y5 @ Ys3 ) )
=> ~ ( ( ( member3481406638322139244la_a_b @ X3 @ X )
=> ( Y3
= ( cons_nat @ Y5 @ ( restri9062294100618341515_b_nat @ ( minus_4705846553145473764la_a_b @ X @ ( insert2023870700798818565la_a_b @ X3 @ bot_bo2891247006866115487la_a_b ) ) @ Xs3 @ Ys3 ) ) ) )
& ( ~ ( member3481406638322139244la_a_b @ X3 @ X )
=> ( Y3
= ( restri9062294100618341515_b_nat @ X @ Xs3 @ Ys3 ) ) ) ) ) )
=> ( ( ( Xa2 = nil_se6596116450819887951la_a_b )
=> ( Y3 != nil_nat ) )
=> ~ ( ( Xb = nil_nat )
=> ( Y3 != nil_nat ) ) ) ) ) ).
% restrict.elims
thf(fact_1101_restrict_Oelims,axiom,
! [X: set_nat,Xa2: list_nat,Xb: list_nat,Y3: list_nat] :
( ( ( restrict_nat_nat @ X @ Xa2 @ Xb )
= Y3 )
=> ( ! [X3: nat,Xs3: list_nat] :
( ( Xa2
= ( cons_nat @ X3 @ Xs3 ) )
=> ! [Y5: nat,Ys3: list_nat] :
( ( Xb
= ( cons_nat @ Y5 @ Ys3 ) )
=> ~ ( ( ( member_nat @ X3 @ X )
=> ( Y3
= ( cons_nat @ Y5 @ ( restrict_nat_nat @ ( minus_minus_set_nat @ X @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) @ Xs3 @ Ys3 ) ) ) )
& ( ~ ( member_nat @ X3 @ X )
=> ( Y3
= ( restrict_nat_nat @ X @ Xs3 @ Ys3 ) ) ) ) ) )
=> ( ( ( Xa2 = nil_nat )
=> ( Y3 != nil_nat ) )
=> ~ ( ( Xb = nil_nat )
=> ( Y3 != nil_nat ) ) ) ) ) ).
% restrict.elims
thf(fact_1102_gen__qps,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X @ Q @ G )
=> ( ( relational_qps_a_b @ G )
= G ) ) ).
% gen_qps
thf(fact_1103_restrict_Osimps_I2_J,axiom,
! [A: set_nat,Uv2: list_nat] :
( ( restrict_nat_nat @ A @ nil_nat @ Uv2 )
= nil_nat ) ).
% restrict.simps(2)
thf(fact_1104_set__removeAll,axiom,
! [X: set_Re381260168593705685la_a_b,Xs2: list_s3671366213839843941la_a_b] :
( ( set_se5111286776497155840la_a_b @ ( remove1701277439453780425la_a_b @ X @ Xs2 ) )
= ( minus_4705846553145473764la_a_b @ ( set_se5111286776497155840la_a_b @ Xs2 ) @ ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) ) ) ).
% set_removeAll
thf(fact_1105_restrict_Opelims,axiom,
! [X: set_se6865892389300016395la_a_b,Xa2: list_s3671366213839843941la_a_b,Xb: list_nat,Y3: list_nat] :
( ( ( restri9062294100618341515_b_nat @ X @ Xa2 @ Xb )
= Y3 )
=> ( ( accp_P6432500237705381767st_nat @ restri8460832097250669116_b_nat @ ( produc9195735010658895618st_nat @ X @ ( produc4035358903851717854st_nat @ Xa2 @ Xb ) ) )
=> ( ! [X3: set_Re381260168593705685la_a_b,Xs3: list_s3671366213839843941la_a_b] :
( ( Xa2
= ( cons_s2104095074707585951la_a_b @ X3 @ Xs3 ) )
=> ! [Y5: nat,Ys3: list_nat] :
( ( Xb
= ( cons_nat @ Y5 @ Ys3 ) )
=> ( ( ( ( member3481406638322139244la_a_b @ X3 @ X )
=> ( Y3
= ( cons_nat @ Y5 @ ( restri9062294100618341515_b_nat @ ( minus_4705846553145473764la_a_b @ X @ ( insert2023870700798818565la_a_b @ X3 @ bot_bo2891247006866115487la_a_b ) ) @ Xs3 @ Ys3 ) ) ) )
& ( ~ ( member3481406638322139244la_a_b @ X3 @ X )
=> ( Y3
= ( restri9062294100618341515_b_nat @ X @ Xs3 @ Ys3 ) ) ) )
=> ~ ( accp_P6432500237705381767st_nat @ restri8460832097250669116_b_nat @ ( produc9195735010658895618st_nat @ X @ ( produc4035358903851717854st_nat @ ( cons_s2104095074707585951la_a_b @ X3 @ Xs3 ) @ ( cons_nat @ Y5 @ Ys3 ) ) ) ) ) ) )
=> ( ( ( Xa2 = nil_se6596116450819887951la_a_b )
=> ( ( Y3 = nil_nat )
=> ~ ( accp_P6432500237705381767st_nat @ restri8460832097250669116_b_nat @ ( produc9195735010658895618st_nat @ X @ ( produc4035358903851717854st_nat @ nil_se6596116450819887951la_a_b @ Xb ) ) ) ) )
=> ~ ( ( Xb = nil_nat )
=> ( ( Y3 = nil_nat )
=> ~ ( accp_P6432500237705381767st_nat @ restri8460832097250669116_b_nat @ ( produc9195735010658895618st_nat @ X @ ( produc4035358903851717854st_nat @ Xa2 @ nil_nat ) ) ) ) ) ) ) ) ) ).
% restrict.pelims
thf(fact_1106_restrict_Opelims,axiom,
! [X: set_nat,Xa2: list_nat,Xb: list_nat,Y3: list_nat] :
( ( ( restrict_nat_nat @ X @ Xa2 @ Xb )
= Y3 )
=> ( ( accp_P4977102363926644935st_nat @ restrict_rel_nat_nat @ ( produc7873415857651533250st_nat @ X @ ( produc2694037385005941721st_nat @ Xa2 @ Xb ) ) )
=> ( ! [X3: nat,Xs3: list_nat] :
( ( Xa2
= ( cons_nat @ X3 @ Xs3 ) )
=> ! [Y5: nat,Ys3: list_nat] :
( ( Xb
= ( cons_nat @ Y5 @ Ys3 ) )
=> ( ( ( ( member_nat @ X3 @ X )
=> ( Y3
= ( cons_nat @ Y5 @ ( restrict_nat_nat @ ( minus_minus_set_nat @ X @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) @ Xs3 @ Ys3 ) ) ) )
& ( ~ ( member_nat @ X3 @ X )
=> ( Y3
= ( restrict_nat_nat @ X @ Xs3 @ Ys3 ) ) ) )
=> ~ ( accp_P4977102363926644935st_nat @ restrict_rel_nat_nat @ ( produc7873415857651533250st_nat @ X @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y5 @ Ys3 ) ) ) ) ) ) )
=> ( ( ( Xa2 = nil_nat )
=> ( ( Y3 = nil_nat )
=> ~ ( accp_P4977102363926644935st_nat @ restrict_rel_nat_nat @ ( produc7873415857651533250st_nat @ X @ ( produc2694037385005941721st_nat @ nil_nat @ Xb ) ) ) ) )
=> ~ ( ( Xb = nil_nat )
=> ( ( Y3 = nil_nat )
=> ~ ( accp_P4977102363926644935st_nat @ restrict_rel_nat_nat @ ( produc7873415857651533250st_nat @ X @ ( produc2694037385005941721st_nat @ Xa2 @ nil_nat ) ) ) ) ) ) ) ) ) ).
% restrict.pelims
thf(fact_1107_quicksort_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ~ ! [X3: nat,Xs3: list_nat] :
( X
!= ( cons_nat @ X3 @ Xs3 ) ) ) ).
% quicksort.cases
thf(fact_1108_removeAll__id,axiom,
! [X: set_Re381260168593705685la_a_b,Xs2: list_s3671366213839843941la_a_b] :
( ~ ( member3481406638322139244la_a_b @ X @ ( set_se5111286776497155840la_a_b @ Xs2 ) )
=> ( ( remove1701277439453780425la_a_b @ X @ Xs2 )
= Xs2 ) ) ).
% removeAll_id
thf(fact_1109_removeAll_Osimps_I2_J,axiom,
! [X: nat,Y3: nat,Xs2: list_nat] :
( ( ( X = Y3 )
=> ( ( removeAll_nat @ X @ ( cons_nat @ Y3 @ Xs2 ) )
= ( removeAll_nat @ X @ Xs2 ) ) )
& ( ( X != Y3 )
=> ( ( removeAll_nat @ X @ ( cons_nat @ Y3 @ Xs2 ) )
= ( cons_nat @ Y3 @ ( removeAll_nat @ X @ Xs2 ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_1110_removeAll_Osimps_I1_J,axiom,
! [X: nat] :
( ( removeAll_nat @ X @ nil_nat )
= nil_nat ) ).
% removeAll.simps(1)
thf(fact_1111_ahs__sbm_Obasic_Og__from__list__aux_Ocases,axiom,
! [X: produc8744035831292522300st_nat] :
( ! [Accs: hashma6156956020035564923t_unit] :
( X
!= ( produc6155107168611786996st_nat @ Accs @ nil_nat ) )
=> ~ ! [Accs: hashma6156956020035564923t_unit,X3: nat,L2: list_nat] :
( X
!= ( produc6155107168611786996st_nat @ Accs @ ( cons_nat @ X3 @ L2 ) ) ) ) ).
% ahs_sbm.basic.g_from_list_aux.cases
thf(fact_1112_rs__sbm_Oobasic_Og__from__list__aux_Ocases,axiom,
! [X: produc4729492286241681067st_nat] :
( ! [Accs: rbt_nat_Product_unit] :
( X
!= ( produc3872122648785725605st_nat @ Accs @ nil_nat ) )
=> ~ ! [Accs: rbt_nat_Product_unit,X3: nat,L2: list_nat] :
( X
!= ( produc3872122648785725605st_nat @ Accs @ ( cons_nat @ X3 @ L2 ) ) ) ) ).
% rs_sbm.obasic.g_from_list_aux.cases
thf(fact_1113_ls__basic_Og__from__list__aux_Ocases,axiom,
! [X: produc2330689195435815175st_nat] :
( ! [Accs: dlist_nat] :
( X
!= ( produc6862011220015528383st_nat @ Accs @ nil_nat ) )
=> ~ ! [Accs: dlist_nat,X3: nat,L2: list_nat] :
( X
!= ( produc6862011220015528383st_nat @ Accs @ ( cons_nat @ X3 @ L2 ) ) ) ) ).
% ls_basic.g_from_list_aux.cases
thf(fact_1114_hs__sbm_Obasic_Og__from__list__aux_Ocases,axiom,
! [X: produc2578004470728393911st_nat] :
( ! [Accs: hashma4580772665052883368t_unit] :
( X
!= ( produc137455288653200561st_nat @ Accs @ nil_nat ) )
=> ~ ! [Accs: hashma4580772665052883368t_unit,X3: nat,L2: list_nat] :
( X
!= ( produc137455288653200561st_nat @ Accs @ ( cons_nat @ X3 @ L2 ) ) ) ) ).
% hs_sbm.basic.g_from_list_aux.cases
thf(fact_1115_op__list__singleton__def,axiom,
( autore7800175773597574974on_nat
= ( ^ [X2: nat] : ( cons_nat @ X2 @ nil_nat ) ) ) ).
% op_list_singleton_def
thf(fact_1116_ias__sbm_Oobasic_Og__from__list__aux_Ocases,axiom,
! [X: produc2026718404927347961st_nat] :
( ! [Accs: diff_a789849492003309546t_unit] :
( X
!= ( produc6847244279022580211st_nat @ Accs @ nil_nat ) )
=> ~ ! [Accs: diff_a789849492003309546t_unit,X3: nat,L2: list_nat] :
( X
!= ( produc6847244279022580211st_nat @ Accs @ ( cons_nat @ X3 @ L2 ) ) ) ) ).
% ias_sbm.obasic.g_from_list_aux.cases
thf(fact_1117_subsetI,axiom,
! [A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ! [X3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ A )
=> ( member3481406638322139244la_a_b @ X3 @ B ) )
=> ( ord_le1577343677690852715la_a_b @ A @ B ) ) ).
% subsetI
thf(fact_1118_insert__subset,axiom,
! [X: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ( ord_le1577343677690852715la_a_b @ ( insert2023870700798818565la_a_b @ X @ A ) @ B )
= ( ( member3481406638322139244la_a_b @ X @ B )
& ( ord_le1577343677690852715la_a_b @ A @ B ) ) ) ).
% insert_subset
thf(fact_1119_rrb__simps_I4_J,axiom,
! [Q: relational_fmla_a_b] :
( ( relational_rrb_a_b @ ( relational_Neg_a_b @ Q ) )
= ( relational_rrb_a_b @ Q ) ) ).
% rrb_simps(4)
thf(fact_1120_rrb__simps_I6_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( relational_rrb_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) )
= ( ( relational_rrb_a_b @ Q1 )
& ( relational_rrb_a_b @ Q2 ) ) ) ).
% rrb_simps(6)
thf(fact_1121_rrb__simps_I7_J,axiom,
! [Y3: nat,Qy: relational_fmla_a_b] :
( ( relational_rrb_a_b @ ( relati591517084277583526ts_a_b @ Y3 @ Qy ) )
= ( ? [X6: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ Y3 @ Qy @ X6 )
& ( relational_rrb_a_b @ Qy ) ) ) ).
% rrb_simps(7)
thf(fact_1122_subset__insert,axiom,
! [X: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ~ ( member3481406638322139244la_a_b @ X @ A )
=> ( ( ord_le1577343677690852715la_a_b @ A @ ( insert2023870700798818565la_a_b @ X @ B ) )
= ( ord_le1577343677690852715la_a_b @ A @ B ) ) ) ).
% subset_insert
thf(fact_1123_subset__iff,axiom,
( ord_le1577343677690852715la_a_b
= ( ^ [A3: set_se6865892389300016395la_a_b,B5: set_se6865892389300016395la_a_b] :
! [T4: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ T4 @ A3 )
=> ( member3481406638322139244la_a_b @ T4 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_1124_subset__eq,axiom,
( ord_le1577343677690852715la_a_b
= ( ^ [A3: set_se6865892389300016395la_a_b,B5: set_se6865892389300016395la_a_b] :
! [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ A3 )
=> ( member3481406638322139244la_a_b @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_1125_subsetD,axiom,
! [A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b,C: set_Re381260168593705685la_a_b] :
( ( ord_le1577343677690852715la_a_b @ A @ B )
=> ( ( member3481406638322139244la_a_b @ C @ A )
=> ( member3481406638322139244la_a_b @ C @ B ) ) ) ).
% subsetD
thf(fact_1126_in__mono,axiom,
! [A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b,X: set_Re381260168593705685la_a_b] :
( ( ord_le1577343677690852715la_a_b @ A @ B )
=> ( ( member3481406638322139244la_a_b @ X @ A )
=> ( member3481406638322139244la_a_b @ X @ B ) ) ) ).
% in_mono
thf(fact_1127_subset__code_I1_J,axiom,
! [Xs2: list_s3671366213839843941la_a_b,B: set_se6865892389300016395la_a_b] :
( ( ord_le1577343677690852715la_a_b @ ( set_se5111286776497155840la_a_b @ Xs2 ) @ B )
= ( ! [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ ( set_se5111286776497155840la_a_b @ Xs2 ) )
=> ( member3481406638322139244la_a_b @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_1128_set__subset__Cons,axiom,
! [Xs2: list_s3671366213839843941la_a_b,X: set_Re381260168593705685la_a_b] : ( ord_le1577343677690852715la_a_b @ ( set_se5111286776497155840la_a_b @ Xs2 ) @ ( set_se5111286776497155840la_a_b @ ( cons_s2104095074707585951la_a_b @ X @ Xs2 ) ) ) ).
% set_subset_Cons
thf(fact_1129_set__subset__Cons,axiom,
! [Xs2: list_nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ ( cons_nat @ X @ Xs2 ) ) ) ).
% set_subset_Cons
thf(fact_1130_subset__Diff__insert,axiom,
! [A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b,X: set_Re381260168593705685la_a_b,C2: set_se6865892389300016395la_a_b] :
( ( ord_le1577343677690852715la_a_b @ A @ ( minus_4705846553145473764la_a_b @ B @ ( insert2023870700798818565la_a_b @ X @ C2 ) ) )
= ( ( ord_le1577343677690852715la_a_b @ A @ ( minus_4705846553145473764la_a_b @ B @ C2 ) )
& ~ ( member3481406638322139244la_a_b @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1131_gen__fv,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
( ( relational_gen_a_b @ X @ Q @ G )
=> ( ( member4680049679412964150la_a_b @ Qqp @ G )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Qqp ) )
& ( ord_less_eq_set_nat @ ( relational_fv_a_b @ Qqp ) @ ( relational_fv_a_b @ Q ) ) ) ) ) ).
% gen_fv
thf(fact_1132_subset__insert__iff,axiom,
! [A: set_se6865892389300016395la_a_b,X: set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b] :
( ( ord_le1577343677690852715la_a_b @ A @ ( insert2023870700798818565la_a_b @ X @ B ) )
= ( ( ( member3481406638322139244la_a_b @ X @ A )
=> ( ord_le1577343677690852715la_a_b @ ( minus_4705846553145473764la_a_b @ A @ ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) ) @ B ) )
& ( ~ ( member3481406638322139244la_a_b @ X @ A )
=> ( ord_le1577343677690852715la_a_b @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1133_rrb__def,axiom,
( relational_rrb_a_b
= ( ^ [Q4: relational_fmla_a_b] :
! [Y: nat,Qy2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ Y @ Qy2 ) @ ( relational_sub_a_b @ Q4 ) )
=> ? [X6: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ Y @ Qy2 @ X6 ) ) ) ) ).
% rrb_def
thf(fact_1134_product__lists_Osimps_I1_J,axiom,
( ( product_lists_nat @ nil_list_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% product_lists.simps(1)
thf(fact_1135_member__remove,axiom,
! [X: set_Re381260168593705685la_a_b,Y3: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ X @ ( remove6344774115224635824la_a_b @ Y3 @ A ) )
= ( ( member3481406638322139244la_a_b @ X @ A )
& ( X != Y3 ) ) ) ).
% member_remove
thf(fact_1136_remove__code_I1_J,axiom,
! [X: set_Re381260168593705685la_a_b,Xs2: list_s3671366213839843941la_a_b] :
( ( remove6344774115224635824la_a_b @ X @ ( set_se5111286776497155840la_a_b @ Xs2 ) )
= ( set_se5111286776497155840la_a_b @ ( remove1701277439453780425la_a_b @ X @ Xs2 ) ) ) ).
% remove_code(1)
thf(fact_1137_listset_Osimps_I1_J,axiom,
( ( listset_nat @ nil_set_nat )
= ( insert_list_nat2 @ nil_nat @ bot_bot_set_list_nat ) ) ).
% listset.simps(1)
thf(fact_1138_subseqs_Osimps_I1_J,axiom,
( ( subseqs_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% subseqs.simps(1)
thf(fact_1139_Cons__in__subseqsD,axiom,
! [Y3: nat,Ys: list_nat,Xs2: list_nat] :
( ( member_list_nat @ ( cons_nat @ Y3 @ Ys ) @ ( set_list_nat2 @ ( subseqs_nat @ Xs2 ) ) )
=> ( member_list_nat @ Ys @ ( set_list_nat2 @ ( subseqs_nat @ Xs2 ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_1140_extend_Osimps_I2_J,axiom,
! [A: set_nat,Uv2: list_nat] :
( ( extend_nat @ A @ nil_nat @ Uv2 )
= ( insert_list_nat2 @ nil_nat @ bot_bot_set_list_nat ) ) ).
% extend.simps(2)
thf(fact_1141_lists__empty,axiom,
( ( lists_nat @ bot_bot_set_nat )
= ( insert_list_nat2 @ nil_nat @ bot_bot_set_list_nat ) ) ).
% lists_empty
thf(fact_1142_Cons__in__lists__iff,axiom,
! [X: set_Re381260168593705685la_a_b,Xs2: list_s3671366213839843941la_a_b,A: set_se6865892389300016395la_a_b] :
( ( member987187531357909116la_a_b @ ( cons_s2104095074707585951la_a_b @ X @ Xs2 ) @ ( lists_8421378239952148961la_a_b @ A ) )
= ( ( member3481406638322139244la_a_b @ X @ A )
& ( member987187531357909116la_a_b @ Xs2 @ ( lists_8421378239952148961la_a_b @ A ) ) ) ) ).
% Cons_in_lists_iff
thf(fact_1143_Cons__in__lists__iff,axiom,
! [X: nat,Xs2: list_nat,A: set_nat] :
( ( member_list_nat @ ( cons_nat @ X @ Xs2 ) @ ( lists_nat @ A ) )
= ( ( member_nat @ X @ A )
& ( member_list_nat @ Xs2 @ ( lists_nat @ A ) ) ) ) ).
% Cons_in_lists_iff
thf(fact_1144_in__listsI,axiom,
! [Xs2: list_s3671366213839843941la_a_b,A: set_se6865892389300016395la_a_b] :
( ! [X3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ ( set_se5111286776497155840la_a_b @ Xs2 ) )
=> ( member3481406638322139244la_a_b @ X3 @ A ) )
=> ( member987187531357909116la_a_b @ Xs2 @ ( lists_8421378239952148961la_a_b @ A ) ) ) ).
% in_listsI
thf(fact_1145_lists_ONil,axiom,
! [A: set_nat] : ( member_list_nat @ nil_nat @ ( lists_nat @ A ) ) ).
% lists.Nil
thf(fact_1146_in__lists__conv__set,axiom,
! [Xs2: list_s3671366213839843941la_a_b,A: set_se6865892389300016395la_a_b] :
( ( member987187531357909116la_a_b @ Xs2 @ ( lists_8421378239952148961la_a_b @ A ) )
= ( ! [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ ( set_se5111286776497155840la_a_b @ Xs2 ) )
=> ( member3481406638322139244la_a_b @ X2 @ A ) ) ) ) ).
% in_lists_conv_set
thf(fact_1147_in__listsD,axiom,
! [Xs2: list_s3671366213839843941la_a_b,A: set_se6865892389300016395la_a_b] :
( ( member987187531357909116la_a_b @ Xs2 @ ( lists_8421378239952148961la_a_b @ A ) )
=> ! [X7: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X7 @ ( set_se5111286776497155840la_a_b @ Xs2 ) )
=> ( member3481406638322139244la_a_b @ X7 @ A ) ) ) ).
% in_listsD
thf(fact_1148_lists_OCons,axiom,
! [A2: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,L: list_s3671366213839843941la_a_b] :
( ( member3481406638322139244la_a_b @ A2 @ A )
=> ( ( member987187531357909116la_a_b @ L @ ( lists_8421378239952148961la_a_b @ A ) )
=> ( member987187531357909116la_a_b @ ( cons_s2104095074707585951la_a_b @ A2 @ L ) @ ( lists_8421378239952148961la_a_b @ A ) ) ) ) ).
% lists.Cons
thf(fact_1149_lists_OCons,axiom,
! [A2: nat,A: set_nat,L: list_nat] :
( ( member_nat @ A2 @ A )
=> ( ( member_list_nat @ L @ ( lists_nat @ A ) )
=> ( member_list_nat @ ( cons_nat @ A2 @ L ) @ ( lists_nat @ A ) ) ) ) ).
% lists.Cons
thf(fact_1150_listsE,axiom,
! [X: set_Re381260168593705685la_a_b,L: list_s3671366213839843941la_a_b,A: set_se6865892389300016395la_a_b] :
( ( member987187531357909116la_a_b @ ( cons_s2104095074707585951la_a_b @ X @ L ) @ ( lists_8421378239952148961la_a_b @ A ) )
=> ~ ( ( member3481406638322139244la_a_b @ X @ A )
=> ~ ( member987187531357909116la_a_b @ L @ ( lists_8421378239952148961la_a_b @ A ) ) ) ) ).
% listsE
thf(fact_1151_listsE,axiom,
! [X: nat,L: list_nat,A: set_nat] :
( ( member_list_nat @ ( cons_nat @ X @ L ) @ ( lists_nat @ A ) )
=> ~ ( ( member_nat @ X @ A )
=> ~ ( member_list_nat @ L @ ( lists_nat @ A ) ) ) ) ).
% listsE
thf(fact_1152_lists_Ocases,axiom,
! [A2: list_s3671366213839843941la_a_b,A: set_se6865892389300016395la_a_b] :
( ( member987187531357909116la_a_b @ A2 @ ( lists_8421378239952148961la_a_b @ A ) )
=> ( ( A2 != nil_se6596116450819887951la_a_b )
=> ~ ! [A6: set_Re381260168593705685la_a_b,L2: list_s3671366213839843941la_a_b] :
( ( A2
= ( cons_s2104095074707585951la_a_b @ A6 @ L2 ) )
=> ( ( member3481406638322139244la_a_b @ A6 @ A )
=> ~ ( member987187531357909116la_a_b @ L2 @ ( lists_8421378239952148961la_a_b @ A ) ) ) ) ) ) ).
% lists.cases
thf(fact_1153_lists_Ocases,axiom,
! [A2: list_nat,A: set_nat] :
( ( member_list_nat @ A2 @ ( lists_nat @ A ) )
=> ( ( A2 != nil_nat )
=> ~ ! [A6: nat,L2: list_nat] :
( ( A2
= ( cons_nat @ A6 @ L2 ) )
=> ( ( member_nat @ A6 @ A )
=> ~ ( member_list_nat @ L2 @ ( lists_nat @ A ) ) ) ) ) ) ).
% lists.cases
thf(fact_1154_lists_Osimps,axiom,
! [A2: list_s3671366213839843941la_a_b,A: set_se6865892389300016395la_a_b] :
( ( member987187531357909116la_a_b @ A2 @ ( lists_8421378239952148961la_a_b @ A ) )
= ( ( A2 = nil_se6596116450819887951la_a_b )
| ? [A4: set_Re381260168593705685la_a_b,L4: list_s3671366213839843941la_a_b] :
( ( A2
= ( cons_s2104095074707585951la_a_b @ A4 @ L4 ) )
& ( member3481406638322139244la_a_b @ A4 @ A )
& ( member987187531357909116la_a_b @ L4 @ ( lists_8421378239952148961la_a_b @ A ) ) ) ) ) ).
% lists.simps
thf(fact_1155_lists_Osimps,axiom,
! [A2: list_nat,A: set_nat] :
( ( member_list_nat @ A2 @ ( lists_nat @ A ) )
= ( ( A2 = nil_nat )
| ? [A4: nat,L4: list_nat] :
( ( A2
= ( cons_nat @ A4 @ L4 ) )
& ( member_nat @ A4 @ A )
& ( member_list_nat @ L4 @ ( lists_nat @ A ) ) ) ) ) ).
% lists.simps
thf(fact_1156_partition__set,axiom,
! [P: set_Re381260168593705685la_a_b > $o,Xs2: list_s3671366213839843941la_a_b,Yes2: list_s3671366213839843941la_a_b,No2: list_s3671366213839843941la_a_b] :
( ( ( partit7571582890193829270la_a_b @ P @ Xs2 )
= ( produc6301070565816922723la_a_b @ Yes2 @ No2 ) )
=> ( ( sup_su4783144482993978935la_a_b @ ( set_se5111286776497155840la_a_b @ Yes2 ) @ ( set_se5111286776497155840la_a_b @ No2 ) )
= ( set_se5111286776497155840la_a_b @ Xs2 ) ) ) ).
% partition_set
thf(fact_1157_rrb__simps_I8_J,axiom,
! [Y3: nat,Qy: relational_fmla_a_b] :
( ( relational_rrb_a_b @ ( relati3989891337220013914ts_a_b @ Y3 @ Qy ) )
= ( ( ( member_nat @ Y3 @ ( relational_fv_a_b @ Qy ) )
=> ? [X6: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ Y3 @ Qy @ X6 ) )
& ( relational_rrb_a_b @ Qy ) ) ) ).
% rrb_simps(8)
thf(fact_1158_partition_Osimps_I1_J,axiom,
! [P: nat > $o] :
( ( partition_nat @ P @ nil_nat )
= ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) ) ).
% partition.simps(1)
thf(fact_1159_partition__P,axiom,
! [P: set_Re381260168593705685la_a_b > $o,Xs2: list_s3671366213839843941la_a_b,Yes2: list_s3671366213839843941la_a_b,No2: list_s3671366213839843941la_a_b] :
( ( ( partit7571582890193829270la_a_b @ P @ Xs2 )
= ( produc6301070565816922723la_a_b @ Yes2 @ No2 ) )
=> ( ! [X7: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X7 @ ( set_se5111286776497155840la_a_b @ Yes2 ) )
=> ( P @ X7 ) )
& ! [X7: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X7 @ ( set_se5111286776497155840la_a_b @ No2 ) )
=> ~ ( P @ X7 ) ) ) ) ).
% partition_P
thf(fact_1160_image__eqI,axiom,
! [B2: set_Re381260168593705685la_a_b,F2: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,X: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ( B2
= ( F2 @ X ) )
=> ( ( member3481406638322139244la_a_b @ X @ A )
=> ( member3481406638322139244la_a_b @ B2 @ ( image_7051608999182166449la_a_b @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_1161_map__is__Nil__conv,axiom,
! [F2: nat > nat,Xs2: list_nat] :
( ( ( map_nat_nat @ F2 @ Xs2 )
= nil_nat )
= ( Xs2 = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_1162_Nil__is__map__conv,axiom,
! [F2: nat > nat,Xs2: list_nat] :
( ( nil_nat
= ( map_nat_nat @ F2 @ Xs2 ) )
= ( Xs2 = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_1163_list_Omap__disc__iff,axiom,
! [F2: nat > nat,A2: list_nat] :
( ( ( map_nat_nat @ F2 @ A2 )
= nil_nat )
= ( A2 = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_1164_list_Oset__map,axiom,
! [F2: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,V: list_s3671366213839843941la_a_b] :
( ( set_se5111286776497155840la_a_b @ ( map_se8100028827536680816la_a_b @ F2 @ V ) )
= ( image_7051608999182166449la_a_b @ F2 @ ( set_se5111286776497155840la_a_b @ V ) ) ) ).
% list.set_map
thf(fact_1165_list_Omap__ident__strong,axiom,
! [T3: list_s3671366213839843941la_a_b,F2: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b] :
( ! [Z: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ Z @ ( set_se5111286776497155840la_a_b @ T3 ) )
=> ( ( F2 @ Z )
= Z ) )
=> ( ( map_se8100028827536680816la_a_b @ F2 @ T3 )
= T3 ) ) ).
% list.map_ident_strong
thf(fact_1166_map__idI,axiom,
! [Xs2: list_s3671366213839843941la_a_b,F2: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b] :
( ! [X3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ ( set_se5111286776497155840la_a_b @ Xs2 ) )
=> ( ( F2 @ X3 )
= X3 ) )
=> ( ( map_se8100028827536680816la_a_b @ F2 @ Xs2 )
= Xs2 ) ) ).
% map_idI
thf(fact_1167_image__set,axiom,
! [F2: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,Xs2: list_s3671366213839843941la_a_b] :
( ( image_7051608999182166449la_a_b @ F2 @ ( set_se5111286776497155840la_a_b @ Xs2 ) )
= ( set_se5111286776497155840la_a_b @ ( map_se8100028827536680816la_a_b @ F2 @ Xs2 ) ) ) ).
% image_set
thf(fact_1168_image__subsetI,axiom,
! [A: set_se6865892389300016395la_a_b,F2: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,B: set_se6865892389300016395la_a_b] :
( ! [X3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ A )
=> ( member3481406638322139244la_a_b @ ( F2 @ X3 ) @ B ) )
=> ( ord_le1577343677690852715la_a_b @ ( image_7051608999182166449la_a_b @ F2 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1169_list_Osimps_I8_J,axiom,
! [F2: nat > nat] :
( ( map_nat_nat @ F2 @ nil_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_1170_imageI,axiom,
! [X: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,F2: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X @ A )
=> ( member3481406638322139244la_a_b @ ( F2 @ X ) @ ( image_7051608999182166449la_a_b @ F2 @ A ) ) ) ).
% imageI
thf(fact_1171_rev__image__eqI,axiom,
! [X: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,B2: set_Re381260168593705685la_a_b,F2: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X @ A )
=> ( ( B2
= ( F2 @ X ) )
=> ( member3481406638322139244la_a_b @ B2 @ ( image_7051608999182166449la_a_b @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1172_list_Osimps_I9_J,axiom,
! [F2: nat > nat,X21: nat,X222: list_nat] :
( ( map_nat_nat @ F2 @ ( cons_nat @ X21 @ X222 ) )
= ( cons_nat @ ( F2 @ X21 ) @ ( map_nat_nat @ F2 @ X222 ) ) ) ).
% list.simps(9)
thf(fact_1173_Cons__eq__map__D,axiom,
! [X: nat,Xs2: list_nat,F2: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X @ Xs2 )
= ( map_nat_nat @ F2 @ Ys ) )
=> ? [Z: nat,Zs: list_nat] :
( ( Ys
= ( cons_nat @ Z @ Zs ) )
& ( X
= ( F2 @ Z ) )
& ( Xs2
= ( map_nat_nat @ F2 @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1174_map__eq__Cons__D,axiom,
! [F2: nat > nat,Xs2: list_nat,Y3: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F2 @ Xs2 )
= ( cons_nat @ Y3 @ Ys ) )
=> ? [Z: nat,Zs: list_nat] :
( ( Xs2
= ( cons_nat @ Z @ Zs ) )
& ( ( F2 @ Z )
= Y3 )
& ( ( map_nat_nat @ F2 @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1175_Cons__eq__map__conv,axiom,
! [X: nat,Xs2: list_nat,F2: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X @ Xs2 )
= ( map_nat_nat @ F2 @ Ys ) )
= ( ? [Z3: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z3 @ Zs2 ) )
& ( X
= ( F2 @ Z3 ) )
& ( Xs2
= ( map_nat_nat @ F2 @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1176_map__eq__Cons__conv,axiom,
! [F2: nat > nat,Xs2: list_nat,Y3: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F2 @ Xs2 )
= ( cons_nat @ Y3 @ Ys ) )
= ( ? [Z3: nat,Zs2: list_nat] :
( ( Xs2
= ( cons_nat @ Z3 @ Zs2 ) )
& ( ( F2 @ Z3 )
= Y3 )
& ( ( map_nat_nat @ F2 @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1177_gen_Ointros_I10_J,axiom,
! [X: nat,Y3: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( X != Y3 )
=> ( ( relational_gen_a_b @ X @ Q @ G )
=> ( relational_gen_a_b @ X @ ( relati591517084277583526ts_a_b @ Y3 @ Q ) @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y3 ) @ G ) ) ) ) ).
% gen.intros(10)
thf(fact_1178_gen__impl_Opsimps_I22_J,axiom,
! [X: nat,Y3: nat,Q: relational_fmla_a_b] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X @ ( relati591517084277583526ts_a_b @ Y3 @ Q ) ) )
=> ( ( ( X = Y3 )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relati591517084277583526ts_a_b @ Y3 @ Q ) )
= nil_se6596116450819887951la_a_b ) )
& ( ( X != Y3 )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relati591517084277583526ts_a_b @ Y3 @ Q ) )
= ( map_se8100028827536680816la_a_b @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y3 ) ) @ ( restri4663030114664890836pl_a_b @ X @ Q ) ) ) ) ) ) ).
% gen_impl.psimps(22)
thf(fact_1179_in__image__insert__iff,axiom,
! [B: set_se1626149436229564481la_a_b,X: set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ! [C6: set_se6865892389300016395la_a_b] :
( ( member1693404252451860130la_a_b @ C6 @ B )
=> ~ ( member3481406638322139244la_a_b @ X @ C6 ) )
=> ( ( member1693404252451860130la_a_b @ A @ ( image_5985962561035126813la_a_b @ ( insert2023870700798818565la_a_b @ X ) @ B ) )
= ( ( member3481406638322139244la_a_b @ X @ A )
& ( member1693404252451860130la_a_b @ ( minus_4705846553145473764la_a_b @ A @ ( insert2023870700798818565la_a_b @ X @ bot_bo2891247006866115487la_a_b ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1180_in__image__insert__iff,axiom,
! [B: set_se6865892389300016395la_a_b,X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ! [C6: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ C6 @ B )
=> ~ ( member4680049679412964150la_a_b @ X @ C6 ) )
=> ( ( member3481406638322139244la_a_b @ A @ ( image_7051608999182166449la_a_b @ ( insert7010464514620295119la_a_b @ X ) @ B ) )
= ( ( member4680049679412964150la_a_b @ X @ A )
& ( member3481406638322139244la_a_b @ ( minus_4077726661957047470la_a_b @ A @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1181_subset__subseqs,axiom,
! [X5: set_Re381260168593705685la_a_b,Xs2: list_R8263082107343818799la_a_b] :
( ( ord_le4112832032246704949la_a_b @ X5 @ ( set_Re9104216502384355786la_a_b @ Xs2 ) )
=> ( member3481406638322139244la_a_b @ X5 @ ( image_1636186708463624715la_a_b @ set_Re9104216502384355786la_a_b @ ( set_li8988895567068466906la_a_b @ ( subseq4216486041226450954la_a_b @ Xs2 ) ) ) ) ) ).
% subset_subseqs
thf(fact_1182_subset__subseqs,axiom,
! [X5: set_se6865892389300016395la_a_b,Xs2: list_s3671366213839843941la_a_b] :
( ( ord_le1577343677690852715la_a_b @ X5 @ ( set_se5111286776497155840la_a_b @ Xs2 ) )
=> ( member1693404252451860130la_a_b @ X5 @ ( image_6556082427161849975la_a_b @ set_se5111286776497155840la_a_b @ ( set_li3253934503784678672la_a_b @ ( subseq7006534010780147008la_a_b @ Xs2 ) ) ) ) ) ).
% subset_subseqs
thf(fact_1183_gen__impl_Osimps_I22_J,axiom,
! [X: nat,Y3: nat,Q: relational_fmla_a_b] :
( ( ( X = Y3 )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relati591517084277583526ts_a_b @ Y3 @ Q ) )
= nil_se6596116450819887951la_a_b ) )
& ( ( X != Y3 )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relati591517084277583526ts_a_b @ Y3 @ Q ) )
= ( map_se8100028827536680816la_a_b @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y3 ) ) @ ( restri4663030114664890836pl_a_b @ X @ Q ) ) ) ) ) ).
% gen_impl.simps(22)
thf(fact_1184_List_Ofinite__set,axiom,
! [Xs2: list_s3671366213839843941la_a_b] : ( finite5238674622262875500la_a_b @ ( set_se5111286776497155840la_a_b @ Xs2 ) ) ).
% List.finite_set
thf(fact_1185_finite__list,axiom,
! [A: set_se6865892389300016395la_a_b] :
( ( finite5238674622262875500la_a_b @ A )
=> ? [Xs3: list_s3671366213839843941la_a_b] :
( ( set_se5111286776497155840la_a_b @ Xs3 )
= A ) ) ).
% finite_list
thf(fact_1186_gen__finite,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X @ Q @ G )
=> ( finite5600759454172676150la_a_b @ G ) ) ).
% gen_finite
thf(fact_1187_infinite__Implies__mono__on,axiom,
! [X5: set_nat,Q: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a,Q5: relational_fmla_a_b] :
( ~ ( finite_finite_list_a @ ( relati8814510239606734169on_a_b @ X5 @ Q @ I2 ) )
=> ( ( finite_finite_nat @ X5 )
=> ( ! [X_12: nat > a] : ( relational_sat_a_b @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q ) @ Q5 ) @ I2 @ X_12 )
=> ~ ( finite_finite_list_a @ ( relati8814510239606734169on_a_b @ X5 @ Q5 @ I2 ) ) ) ) ) ).
% infinite_Implies_mono_on
thf(fact_1188_csts_Oelims,axiom,
! [X: relational_fmla_a_b,Y3: set_a] :
( ( ( relational_csts_a_b @ X )
= Y3 )
=> ( ( ? [B3: $o] :
( X
= ( relational_Bool_a_b @ B3 ) )
=> ( Y3 != bot_bot_set_a ) )
=> ( ! [P3: b,Ts: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ P3 @ Ts ) )
=> ( Y3
!= ( comple2307003609928055243_set_a @ ( image_1080223610614215258_set_a @ relati6638259395341848997term_a @ ( set_Re3569617851344498910term_a @ Ts ) ) ) ) )
=> ( ! [X3: nat,T: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X3 @ T ) )
=> ( Y3
!= ( relati6638259395341848997term_a @ T ) ) )
=> ( ! [Q3: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q3 ) )
=> ( Y3
!= ( relational_csts_a_b @ Q3 ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q12 @ Q22 ) )
=> ( Y3
!= ( sup_sup_set_a @ ( relational_csts_a_b @ Q12 ) @ ( relational_csts_a_b @ Q22 ) ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q12 @ Q22 ) )
=> ( Y3
!= ( sup_sup_set_a @ ( relational_csts_a_b @ Q12 ) @ ( relational_csts_a_b @ Q22 ) ) ) )
=> ~ ! [X3: nat,Q3: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X3 @ Q3 ) )
=> ( Y3
!= ( relational_csts_a_b @ Q3 ) ) ) ) ) ) ) ) ) ) ).
% csts.elims
thf(fact_1189_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_nat] :
( ( ( concat_nat @ Xss2 )
= nil_nat )
= ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xss2 ) )
=> ( X2 = nil_nat ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_1190_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_nat] :
( ( nil_nat
= ( concat_nat @ Xss2 ) )
= ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xss2 ) )
=> ( X2 = nil_nat ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_1191_set__concat,axiom,
! [Xs2: list_l1512381142989314933la_a_b] :
( ( set_se5111286776497155840la_a_b @ ( concat678213573965515860la_a_b @ Xs2 ) )
= ( comple7774258788699406174la_a_b @ ( image_6556082427161849975la_a_b @ set_se5111286776497155840la_a_b @ ( set_li3253934503784678672la_a_b @ Xs2 ) ) ) ) ).
% set_concat
thf(fact_1192_set__Cons__sing__Nil,axiom,
! [A: set_nat] :
( ( set_Cons_nat @ A @ ( insert_list_nat2 @ nil_nat @ bot_bot_set_list_nat ) )
= ( image_nat_list_nat
@ ^ [X2: nat] : ( cons_nat @ X2 @ nil_nat )
@ A ) ) ).
% set_Cons_sing_Nil
thf(fact_1193_gen__impl_Osimps_I14_J,axiom,
! [X: nat,Y3: nat,Q1: relational_fmla_a_b,Z2: nat] :
( ( ( X = Y3 )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relational_Conj_a_b @ Q1 @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ Z2 ) ) ) )
= ( union_7072582408929332827la_a_b @ ( restri4663030114664890836pl_a_b @ X @ Q1 )
@ ( map_se8100028827536680816la_a_b
@ ( image_6790371041703824709la_a_b
@ ^ [Q4: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q4 @ Z2 @ X ) ) )
@ ( restri4663030114664890836pl_a_b @ Z2 @ Q1 ) ) ) ) )
& ( ( X != Y3 )
=> ( ( ( X = Z2 )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relational_Conj_a_b @ Q1 @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ Z2 ) ) ) )
= ( union_7072582408929332827la_a_b @ ( restri4663030114664890836pl_a_b @ X @ Q1 )
@ ( map_se8100028827536680816la_a_b
@ ( image_6790371041703824709la_a_b
@ ^ [Q4: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q4 @ Y3 @ X ) ) )
@ ( restri4663030114664890836pl_a_b @ Y3 @ Q1 ) ) ) ) )
& ( ( X != Z2 )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relational_Conj_a_b @ Q1 @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ Z2 ) ) ) )
= ( restri4663030114664890836pl_a_b @ X @ Q1 ) ) ) ) ) ) ).
% gen_impl.simps(14)
thf(fact_1194_gen__impl_Opsimps_I13_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X @ ( relational_Disj_a_b @ Q1 @ Q2 ) ) )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relational_Disj_a_b @ Q1 @ Q2 ) )
= ( concat678213573965515860la_a_b
@ ( map_se4895820801628292480la_a_b
@ ^ [G12: set_Re381260168593705685la_a_b] : ( map_se8100028827536680816la_a_b @ ( sup_su5130108678486352897la_a_b @ G12 ) @ ( restri4663030114664890836pl_a_b @ X @ Q2 ) )
@ ( restri4663030114664890836pl_a_b @ X @ Q1 ) ) ) ) ) ).
% gen_impl.psimps(13)
thf(fact_1195_gen__impl_Osimps_I13_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( restri4663030114664890836pl_a_b @ X @ ( relational_Disj_a_b @ Q1 @ Q2 ) )
= ( concat678213573965515860la_a_b
@ ( map_se4895820801628292480la_a_b
@ ^ [G12: set_Re381260168593705685la_a_b] : ( map_se8100028827536680816la_a_b @ ( sup_su5130108678486352897la_a_b @ G12 ) @ ( restri4663030114664890836pl_a_b @ X @ Q2 ) )
@ ( restri4663030114664890836pl_a_b @ X @ Q1 ) ) ) ) ).
% gen_impl.simps(13)
thf(fact_1196_csts_Osimps_I5_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( relational_csts_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) )
= ( sup_sup_set_a @ ( relational_csts_a_b @ Q1 ) @ ( relational_csts_a_b @ Q2 ) ) ) ).
% csts.simps(5)
thf(fact_1197_lists__eq__set,axiom,
( lists_8421378239952148961la_a_b
= ( ^ [A3: set_se6865892389300016395la_a_b] :
( collec5397960198663843514la_a_b
@ ^ [Xs: list_s3671366213839843941la_a_b] : ( ord_le1577343677690852715la_a_b @ ( set_se5111286776497155840la_a_b @ Xs ) @ A3 ) ) ) ) ).
% lists_eq_set
thf(fact_1198_Collect__subset,axiom,
! [A: set_se6865892389300016395la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ord_le1577343677690852715la_a_b
@ ( collec2099942116761351594la_a_b
@ ^ [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_1199_less__eq__set__def,axiom,
( ord_le1577343677690852715la_a_b
= ( ^ [A3: set_se6865892389300016395la_a_b,B5: set_se6865892389300016395la_a_b] :
( ord_le8511361905821865178_a_b_o
@ ^ [X2: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ X2 @ A3 )
@ ^ [X2: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_1200_insert__compr,axiom,
( insert2023870700798818565la_a_b
= ( ^ [A4: set_Re381260168593705685la_a_b,B5: set_se6865892389300016395la_a_b] :
( collec2099942116761351594la_a_b
@ ^ [X2: set_Re381260168593705685la_a_b] :
( ( X2 = A4 )
| ( member3481406638322139244la_a_b @ X2 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_1201_csts_Osimps_I4_J,axiom,
! [Q: relational_fmla_a_b] :
( ( relational_csts_a_b @ ( relational_Neg_a_b @ Q ) )
= ( relational_csts_a_b @ Q ) ) ).
% csts.simps(4)
thf(fact_1202_Un__def,axiom,
( sup_su4783144482993978935la_a_b
= ( ^ [A3: set_se6865892389300016395la_a_b,B5: set_se6865892389300016395la_a_b] :
( collec2099942116761351594la_a_b
@ ^ [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ A3 )
| ( member3481406638322139244la_a_b @ X2 @ B5 ) ) ) ) ) ).
% Un_def
thf(fact_1203_sup__set__def,axiom,
( sup_su4783144482993978935la_a_b
= ( ^ [A3: set_se6865892389300016395la_a_b,B5: set_se6865892389300016395la_a_b] :
( collec2099942116761351594la_a_b
@ ( sup_su2223990397388305166_a_b_o
@ ^ [X2: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ X2 @ A3 )
@ ^ [X2: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ X2 @ B5 ) ) ) ) ) ).
% sup_set_def
thf(fact_1204_subst_Osimps_I5_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b,X: nat,Y3: nat] :
( ( relational_subst_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ X @ Y3 )
= ( relational_Conj_a_b @ ( relational_subst_a_b @ Q1 @ X @ Y3 ) @ ( relational_subst_a_b @ Q2 @ X @ Y3 ) ) ) ).
% subst.simps(5)
thf(fact_1205_subst_Osimps_I4_J,axiom,
! [Q: relational_fmla_a_b,X: nat,Y3: nat] :
( ( relational_subst_a_b @ ( relational_Neg_a_b @ Q ) @ X @ Y3 )
= ( relational_Neg_a_b @ ( relational_subst_a_b @ Q @ X @ Y3 ) ) ) ).
% subst.simps(4)
thf(fact_1206_minus__set__def,axiom,
( minus_4705846553145473764la_a_b
= ( ^ [A3: set_se6865892389300016395la_a_b,B5: set_se6865892389300016395la_a_b] :
( collec2099942116761351594la_a_b
@ ( minus_5536082475972203681_a_b_o
@ ^ [X2: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ X2 @ A3 )
@ ^ [X2: set_Re381260168593705685la_a_b] : ( member3481406638322139244la_a_b @ X2 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_1207_set__diff__eq,axiom,
( minus_4705846553145473764la_a_b
= ( ^ [A3: set_se6865892389300016395la_a_b,B5: set_se6865892389300016395la_a_b] :
( collec2099942116761351594la_a_b
@ ^ [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ A3 )
& ~ ( member3481406638322139244la_a_b @ X2 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1208_Gen__cp__subst,axiom,
! [Z2: nat,Q: relational_fmla_a_b,X: nat,Y3: nat] :
( ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ Z2 @ Q @ X_1 )
=> ( ( Z2 != X )
=> ? [X_12: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ Z2 @ ( relational_cp_a_b @ ( relational_subst_a_b @ Q @ X @ Y3 ) ) @ X_12 ) ) ) ).
% Gen_cp_subst
thf(fact_1209_concat_Osimps_I1_J,axiom,
( ( concat_nat @ nil_list_nat )
= nil_nat ) ).
% concat.simps(1)
thf(fact_1210_gen__csts,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
( ( relational_gen_a_b @ X @ Q @ G )
=> ( ( member4680049679412964150la_a_b @ Qqp @ G )
=> ( ord_less_eq_set_a @ ( relational_csts_a_b @ Qqp ) @ ( relational_csts_a_b @ Q ) ) ) ) ).
% gen_csts
thf(fact_1211_gen_Ointros_I8_J,axiom,
! [Y3: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
( ( relational_gen_a_b @ Y3 @ Q @ G )
=> ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y3 ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q4: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q4 @ Y3 @ X ) )
@ G ) ) ) ).
% gen.intros(8)
thf(fact_1212_gen_Ointros_I9_J,axiom,
! [Y3: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
( ( relational_gen_a_b @ Y3 @ Q @ G )
=> ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ X ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q4: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q4 @ Y3 @ X ) )
@ G ) ) ) ).
% gen.intros(9)
thf(fact_1213_gen__nocp__intros_I1_J,axiom,
! [Y3: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
( ( relational_gen_a_b @ Y3 @ Q @ G )
=> ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y3 ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q4: relational_fmla_a_b] : ( relational_subst_a_b @ Q4 @ Y3 @ X )
@ G ) ) ) ).
% gen_nocp_intros(1)
thf(fact_1214_gen__nocp__intros_I2_J,axiom,
! [Y3: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
( ( relational_gen_a_b @ Y3 @ Q @ G )
=> ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ X ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q4: relational_fmla_a_b] : ( relational_subst_a_b @ Q4 @ Y3 @ X )
@ G ) ) ) ).
% gen_nocp_intros(2)
thf(fact_1215_Compr__image__eq,axiom,
! [F2: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( collec2099942116761351594la_a_b
@ ^ [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ ( image_7051608999182166449la_a_b @ F2 @ A ) )
& ( P @ X2 ) ) )
= ( image_7051608999182166449la_a_b @ F2
@ ( collec2099942116761351594la_a_b
@ ^ [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ A )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1216_imageE,axiom,
! [B2: set_Re381260168593705685la_a_b,F2: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b,A: set_se6865892389300016395la_a_b] :
( ( member3481406638322139244la_a_b @ B2 @ ( image_7051608999182166449la_a_b @ F2 @ A ) )
=> ~ ! [X3: set_Re381260168593705685la_a_b] :
( ( B2
= ( F2 @ X3 ) )
=> ~ ( member3481406638322139244la_a_b @ X3 @ A ) ) ) ).
% imageE
thf(fact_1217_finite__Sup__in,axiom,
! [A: set_se6865892389300016395la_a_b] :
( ( finite5238674622262875500la_a_b @ A )
=> ( ( A != bot_bo2891247006866115487la_a_b )
=> ( ! [X3: set_Re381260168593705685la_a_b,Y5: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ A )
=> ( ( member3481406638322139244la_a_b @ Y5 @ A )
=> ( member3481406638322139244la_a_b @ ( sup_su5130108678486352897la_a_b @ X3 @ Y5 ) @ A ) ) )
=> ( member3481406638322139244la_a_b @ ( comple8442120529048846632la_a_b @ A ) @ A ) ) ) ) ).
% finite_Sup_in
thf(fact_1218_gen__impl_Opsimps_I14_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,Y3: nat,Z2: nat] :
( ( accp_P8429596082290424793la_a_b @ restri6037960503177822439el_a_b @ ( produc8327306639710187272la_a_b @ X @ ( relational_Conj_a_b @ Q1 @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ Z2 ) ) ) ) )
=> ( ( ( X = Y3 )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relational_Conj_a_b @ Q1 @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ Z2 ) ) ) )
= ( union_7072582408929332827la_a_b @ ( restri4663030114664890836pl_a_b @ X @ Q1 )
@ ( map_se8100028827536680816la_a_b
@ ( image_6790371041703824709la_a_b
@ ^ [Q4: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q4 @ Z2 @ X ) ) )
@ ( restri4663030114664890836pl_a_b @ Z2 @ Q1 ) ) ) ) )
& ( ( X != Y3 )
=> ( ( ( X = Z2 )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relational_Conj_a_b @ Q1 @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ Z2 ) ) ) )
= ( union_7072582408929332827la_a_b @ ( restri4663030114664890836pl_a_b @ X @ Q1 )
@ ( map_se8100028827536680816la_a_b
@ ( image_6790371041703824709la_a_b
@ ^ [Q4: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q4 @ Y3 @ X ) ) )
@ ( restri4663030114664890836pl_a_b @ Y3 @ Q1 ) ) ) ) )
& ( ( X != Z2 )
=> ( ( restri4663030114664890836pl_a_b @ X @ ( relational_Conj_a_b @ Q1 @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ Z2 ) ) ) )
= ( restri4663030114664890836pl_a_b @ X @ Q1 ) ) ) ) ) ) ) ).
% gen_impl.psimps(14)
thf(fact_1219_gen__induct,axiom,
! [X1: nat,X22: relational_fmla_a_b,X33: set_Re381260168593705685la_a_b,P: nat > relational_fmla_a_b > set_Re381260168593705685la_a_b > $o] :
( ( relational_gen_a_b @ X1 @ X22 @ X33 )
=> ( ! [X3: nat] : ( P @ X3 @ ( relational_Bool_a_b @ $false ) @ bot_bo4495933725496725865la_a_b )
=> ( ! [Q3: relational_fmla_a_b] :
( ( relational_ap_a_b @ Q3 )
=> ! [X3: nat] :
( ( member_nat @ X3 @ ( relational_fv_a_b @ Q3 ) )
=> ( P @ X3 @ Q3 @ ( insert7010464514620295119la_a_b @ Q3 @ bot_bo4495933725496725865la_a_b ) ) ) )
=> ( ! [X3: nat,Q3: relational_fmla_a_b,G5: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X3 @ Q3 @ G5 )
=> ( ( P @ X3 @ Q3 @ G5 )
=> ( P @ X3 @ ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q3 ) ) @ G5 ) ) )
=> ( ! [X3: nat,Q12: relational_fmla_a_b,Q22: relational_fmla_a_b,G5: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X3 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q22 ) ) @ G5 )
=> ( ( P @ X3 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q22 ) ) @ G5 )
=> ( P @ X3 @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q12 @ Q22 ) ) @ G5 ) ) )
=> ( ! [X3: nat,Q12: relational_fmla_a_b,Q22: relational_fmla_a_b,G5: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X3 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q22 ) ) @ G5 )
=> ( ( P @ X3 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q22 ) ) @ G5 )
=> ( P @ X3 @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q12 @ Q22 ) ) @ G5 ) ) )
=> ( ! [X3: nat,Q12: relational_fmla_a_b,G13: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X3 @ Q12 @ G13 )
=> ( ( P @ X3 @ Q12 @ G13 )
=> ! [Q22: relational_fmla_a_b,G22: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X3 @ Q22 @ G22 )
=> ( ( P @ X3 @ Q22 @ G22 )
=> ( P @ X3 @ ( relational_Disj_a_b @ Q12 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ G13 @ G22 ) ) ) ) ) )
=> ( ! [X3: nat,Q12: relational_fmla_a_b,G5: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b] :
( ( ( ( relational_gen_a_b @ X3 @ Q12 @ G5 )
& ( P @ X3 @ Q12 @ G5 ) )
| ( ( relational_gen_a_b @ X3 @ Q22 @ G5 )
& ( P @ X3 @ Q22 @ G5 ) ) )
=> ( P @ X3 @ ( relational_Conj_a_b @ Q12 @ Q22 ) @ G5 ) )
=> ( ! [Y5: nat,Q3: relational_fmla_a_b,G5: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ Y5 @ Q3 @ G5 )
=> ( ( P @ Y5 @ Q3 @ G5 )
=> ! [X3: nat] :
( P @ X3 @ ( relational_Conj_a_b @ Q3 @ ( relational_Eq_a_b @ X3 @ ( relational_Var_a @ Y5 ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y5 @ X3 )
@ G5 ) ) ) )
=> ( ! [Y5: nat,Q3: relational_fmla_a_b,G5: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ Y5 @ Q3 @ G5 )
=> ( ( P @ Y5 @ Q3 @ G5 )
=> ! [X3: nat] :
( P @ X3 @ ( relational_Conj_a_b @ Q3 @ ( relational_Eq_a_b @ Y5 @ ( relational_Var_a @ X3 ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y5 @ X3 )
@ G5 ) ) ) )
=> ( ! [X3: nat,Y5: nat] :
( ( X3 != Y5 )
=> ! [Q3: relational_fmla_a_b,G5: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X3 @ Q3 @ G5 )
=> ( ( P @ X3 @ Q3 @ G5 )
=> ( P @ X3 @ ( relati591517084277583526ts_a_b @ Y5 @ Q3 ) @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y5 ) @ G5 ) ) ) ) )
=> ( P @ X1 @ X22 @ X33 ) ) ) ) ) ) ) ) ) ) ) ) ).
% gen_induct
thf(fact_1220_gen_Osimps,axiom,
( relational_gen_a_b
= ( ^ [A12: nat,A22: relational_fmla_a_b,A32: set_Re381260168593705685la_a_b] :
( ( ( A22
= ( relational_Bool_a_b @ $false ) )
& ( A32 = bot_bo4495933725496725865la_a_b ) )
| ( ( A32
= ( insert7010464514620295119la_a_b @ A22 @ bot_bo4495933725496725865la_a_b ) )
& ( relational_ap_a_b @ A22 )
& ( member_nat @ A12 @ ( relational_fv_a_b @ A22 ) ) )
| ? [Q4: relational_fmla_a_b] :
( ( A22
= ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q4 ) ) )
& ( relational_gen_a_b @ A12 @ Q4 @ A32 ) )
| ? [Q13: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( A22
= ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q13 @ Q23 ) ) )
& ( relational_gen_a_b @ A12 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q23 ) ) @ A32 ) )
| ? [Q13: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( A22
= ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q13 @ Q23 ) ) )
& ( relational_gen_a_b @ A12 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q23 ) ) @ A32 ) )
| ? [Q13: relational_fmla_a_b,G12: set_Re381260168593705685la_a_b,Q23: relational_fmla_a_b] :
( ( A22
= ( relational_Disj_a_b @ Q13 @ Q23 ) )
& ? [G23: set_Re381260168593705685la_a_b] :
( ( A32
= ( sup_su5130108678486352897la_a_b @ G12 @ G23 ) )
& ( relational_gen_a_b @ A12 @ Q13 @ G12 )
& ( relational_gen_a_b @ A12 @ Q23 @ G23 ) ) )
| ? [Q13: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( A22
= ( relational_Conj_a_b @ Q13 @ Q23 ) )
& ( ( relational_gen_a_b @ A12 @ Q13 @ A32 )
| ( relational_gen_a_b @ A12 @ Q23 @ A32 ) ) )
| ? [Y: nat,Q4: relational_fmla_a_b] :
( ( A22
= ( relational_Conj_a_b @ Q4 @ ( relational_Eq_a_b @ A12 @ ( relational_Var_a @ Y ) ) ) )
& ? [G6: set_Re381260168593705685la_a_b] :
( ( A32
= ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Qa @ Y @ A12 ) )
@ G6 ) )
& ( relational_gen_a_b @ Y @ Q4 @ G6 ) ) )
| ? [Y: nat,Q4: relational_fmla_a_b] :
( ( A22
= ( relational_Conj_a_b @ Q4 @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ A12 ) ) ) )
& ? [G6: set_Re381260168593705685la_a_b] :
( ( A32
= ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Qa @ Y @ A12 ) )
@ G6 ) )
& ( relational_gen_a_b @ Y @ Q4 @ G6 ) ) )
| ? [Y: nat,Q4: relational_fmla_a_b] :
( ( A22
= ( relati591517084277583526ts_a_b @ Y @ Q4 ) )
& ? [G6: set_Re381260168593705685la_a_b] :
( ( A32
= ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ G6 ) )
& ( A12 != Y )
& ( relational_gen_a_b @ A12 @ Q4 @ G6 ) ) ) ) ) ) ).
% gen.simps
thf(fact_1221_gen_Ocases,axiom,
! [A1: nat,A23: relational_fmla_a_b,A33: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ A1 @ A23 @ A33 )
=> ( ( ( A23
= ( relational_Bool_a_b @ $false ) )
=> ( A33 != bot_bo4495933725496725865la_a_b ) )
=> ( ( ( A33
= ( insert7010464514620295119la_a_b @ A23 @ bot_bo4495933725496725865la_a_b ) )
=> ( ( relational_ap_a_b @ A23 )
=> ~ ( member_nat @ A1 @ ( relational_fv_a_b @ A23 ) ) ) )
=> ( ! [Q3: relational_fmla_a_b] :
( ( A23
= ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q3 ) ) )
=> ~ ( relational_gen_a_b @ A1 @ Q3 @ A33 ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( A23
= ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q12 @ Q22 ) ) )
=> ~ ( relational_gen_a_b @ A1 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q22 ) ) @ A33 ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( A23
= ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q12 @ Q22 ) ) )
=> ~ ( relational_gen_a_b @ A1 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q22 ) ) @ A33 ) )
=> ( ! [Q12: relational_fmla_a_b,G13: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b] :
( ( A23
= ( relational_Disj_a_b @ Q12 @ Q22 ) )
=> ! [G22: set_Re381260168593705685la_a_b] :
( ( A33
= ( sup_su5130108678486352897la_a_b @ G13 @ G22 ) )
=> ( ( relational_gen_a_b @ A1 @ Q12 @ G13 )
=> ~ ( relational_gen_a_b @ A1 @ Q22 @ G22 ) ) ) )
=> ( ! [Q12: relational_fmla_a_b,G5: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b] :
( ( A23
= ( relational_Conj_a_b @ Q12 @ Q22 ) )
=> ( ( A33 = G5 )
=> ~ ( ( relational_gen_a_b @ A1 @ Q12 @ G5 )
| ( relational_gen_a_b @ A1 @ Q22 @ G5 ) ) ) )
=> ( ! [Y5: nat,Q3: relational_fmla_a_b] :
( ( A23
= ( relational_Conj_a_b @ Q3 @ ( relational_Eq_a_b @ A1 @ ( relational_Var_a @ Y5 ) ) ) )
=> ! [G5: set_Re381260168593705685la_a_b] :
( ( A33
= ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Qa @ Y5 @ A1 ) )
@ G5 ) )
=> ~ ( relational_gen_a_b @ Y5 @ Q3 @ G5 ) ) )
=> ( ! [Y5: nat,Q3: relational_fmla_a_b] :
( ( A23
= ( relational_Conj_a_b @ Q3 @ ( relational_Eq_a_b @ Y5 @ ( relational_Var_a @ A1 ) ) ) )
=> ! [G5: set_Re381260168593705685la_a_b] :
( ( A33
= ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Qa @ Y5 @ A1 ) )
@ G5 ) )
=> ~ ( relational_gen_a_b @ Y5 @ Q3 @ G5 ) ) )
=> ~ ! [Y5: nat,Q3: relational_fmla_a_b] :
( ( A23
= ( relati591517084277583526ts_a_b @ Y5 @ Q3 ) )
=> ! [G5: set_Re381260168593705685la_a_b] :
( ( A33
= ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y5 ) @ G5 ) )
=> ( ( A1 != Y5 )
=> ~ ( relational_gen_a_b @ A1 @ Q3 @ G5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% gen.cases
thf(fact_1222_product__lists_Osimps_I2_J,axiom,
! [Xs2: list_nat,Xss2: list_list_nat] :
( ( product_lists_nat @ ( cons_list_nat @ Xs2 @ Xss2 ) )
= ( concat_list_nat
@ ( map_na6205611841492582150st_nat
@ ^ [X2: nat] : ( map_li7225945977422193158st_nat @ ( cons_nat @ X2 ) @ ( product_lists_nat @ Xss2 ) )
@ Xs2 ) ) ) ).
% product_lists.simps(2)
thf(fact_1223_csts_Opelims,axiom,
! [X: relational_fmla_a_b,Y3: set_a] :
( ( ( relational_csts_a_b @ X )
= Y3 )
=> ( ( accp_R989495437599811158la_a_b @ relati7137348651719826542el_a_b @ X )
=> ( ! [B3: $o] :
( ( X
= ( relational_Bool_a_b @ B3 ) )
=> ( ( Y3 = bot_bot_set_a )
=> ~ ( accp_R989495437599811158la_a_b @ relati7137348651719826542el_a_b @ ( relational_Bool_a_b @ B3 ) ) ) )
=> ( ! [P3: b,Ts: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ P3 @ Ts ) )
=> ( ( Y3
= ( comple2307003609928055243_set_a @ ( image_1080223610614215258_set_a @ relati6638259395341848997term_a @ ( set_Re3569617851344498910term_a @ Ts ) ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7137348651719826542el_a_b @ ( relational_Pred_b_a @ P3 @ Ts ) ) ) )
=> ( ! [X3: nat,T: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X3 @ T ) )
=> ( ( Y3
= ( relati6638259395341848997term_a @ T ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7137348651719826542el_a_b @ ( relational_Eq_a_b @ X3 @ T ) ) ) )
=> ( ! [Q3: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q3 ) )
=> ( ( Y3
= ( relational_csts_a_b @ Q3 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7137348651719826542el_a_b @ ( relational_Neg_a_b @ Q3 ) ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q12 @ Q22 ) )
=> ( ( Y3
= ( sup_sup_set_a @ ( relational_csts_a_b @ Q12 ) @ ( relational_csts_a_b @ Q22 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7137348651719826542el_a_b @ ( relational_Conj_a_b @ Q12 @ Q22 ) ) ) )
=> ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q12 @ Q22 ) )
=> ( ( Y3
= ( sup_sup_set_a @ ( relational_csts_a_b @ Q12 ) @ ( relational_csts_a_b @ Q22 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7137348651719826542el_a_b @ ( relational_Disj_a_b @ Q12 @ Q22 ) ) ) )
=> ~ ! [X3: nat,Q3: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X3 @ Q3 ) )
=> ( ( Y3
= ( relational_csts_a_b @ Q3 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7137348651719826542el_a_b @ ( relati591517084277583526ts_a_b @ X3 @ Q3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% csts.pelims
thf(fact_1224_cov_H_OConj,axiom,
! [X: nat,Q1: relational_fmla_a_b,G1: set_Re381260168593705685la_a_b,Q2: relational_fmla_a_b,G2: set_Re381260168593705685la_a_b] :
( ( relational_cov_a_b2 @ X @ Q1 @ G1 )
=> ( ( relational_cov_a_b2 @ X @ Q2 @ G2 )
=> ( relational_cov_a_b2 @ X @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ ( sup_su5130108678486352897la_a_b @ G1 @ G2 ) ) ) ) ).
% cov'.Conj
thf(fact_1225_cov_H_ONeg,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_cov_a_b2 @ X @ Q @ G )
=> ( relational_cov_a_b2 @ X @ ( relational_Neg_a_b @ Q ) @ G ) ) ).
% cov'.Neg
thf(fact_1226_cov_H_OConjL,axiom,
! [X: nat,Q1: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q2: relational_fmla_a_b] :
( ( relational_cov_a_b2 @ X @ Q1 @ G )
=> ( ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q1 @ X ) )
= ( relational_Bool_a_b @ $false ) )
=> ( relational_cov_a_b2 @ X @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ G ) ) ) ).
% cov'.ConjL
thf(fact_1227_cov_H_OConjR,axiom,
! [X: nat,Q2: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q1: relational_fmla_a_b] :
( ( relational_cov_a_b2 @ X @ Q2 @ G )
=> ( ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q2 @ X ) )
= ( relational_Bool_a_b @ $false ) )
=> ( relational_cov_a_b2 @ X @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ G ) ) ) ).
% cov'.ConjR
thf(fact_1228_cov_H_OExists__gen,axiom,
! [X: nat,Y3: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Gy: set_Re381260168593705685la_a_b] :
( ( X != Y3 )
=> ( ( relational_cov_a_b2 @ X @ Q @ G )
=> ( ( relational_gen_a_b @ Y3 @ Q @ Gy )
=> ( relational_cov_a_b2 @ X @ ( relati591517084277583526ts_a_b @ Y3 @ Q )
@ ( sup_su5130108678486352897la_a_b @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y3 ) @ ( minus_4077726661957047470la_a_b @ G @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y3 ) ) @ bot_bo4495933725496725865la_a_b ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q4: relational_fmla_a_b] : ( relational_subst_a_b @ Q4 @ Y3 @ X )
@ Gy ) ) ) ) ) ) ).
% cov'.Exists_gen
thf(fact_1229_cov_H__cp__intros,axiom,
! [X: nat,Y3: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Gy: set_Re381260168593705685la_a_b] :
( ( X != Y3 )
=> ( ( relational_cov_a_b2 @ X @ Q @ G )
=> ( ( relational_gen_a_b @ Y3 @ Q @ Gy )
=> ( relational_cov_a_b2 @ X @ ( relati591517084277583526ts_a_b @ Y3 @ Q )
@ ( sup_su5130108678486352897la_a_b @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y3 ) @ ( minus_4077726661957047470la_a_b @ G @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y3 ) ) @ bot_bo4495933725496725865la_a_b ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q4: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q4 @ Y3 @ X ) )
@ Gy ) ) ) ) ) ) ).
% cov'_cp_intros
thf(fact_1230_cov_OConj,axiom,
! [X: nat,Q1: relational_fmla_a_b,G1: set_Re381260168593705685la_a_b,Q2: relational_fmla_a_b,G2: set_Re381260168593705685la_a_b] :
( ( relational_cov_a_b @ X @ Q1 @ G1 )
=> ( ( relational_cov_a_b @ X @ Q2 @ G2 )
=> ( relational_cov_a_b @ X @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ ( sup_su5130108678486352897la_a_b @ G1 @ G2 ) ) ) ) ).
% cov.Conj
thf(fact_1231_cov_ONeg,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_cov_a_b @ X @ Q @ G )
=> ( relational_cov_a_b @ X @ ( relational_Neg_a_b @ Q ) @ G ) ) ).
% cov.Neg
thf(fact_1232_cov_OConjR,axiom,
! [X: nat,Q2: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q1: relational_fmla_a_b] :
( ( relational_cov_a_b @ X @ Q2 @ G )
=> ( ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q2 @ X ) )
= ( relational_Bool_a_b @ $false ) )
=> ( relational_cov_a_b @ X @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ G ) ) ) ).
% cov.ConjR
thf(fact_1233_cov_OConjL,axiom,
! [X: nat,Q1: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q2: relational_fmla_a_b] :
( ( relational_cov_a_b @ X @ Q1 @ G )
=> ( ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q1 @ X ) )
= ( relational_Bool_a_b @ $false ) )
=> ( relational_cov_a_b @ X @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ G ) ) ) ).
% cov.ConjL
thf(fact_1234_cov__nocp__intros,axiom,
! [X: nat,Y3: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Gy: set_Re381260168593705685la_a_b] :
( ( X != Y3 )
=> ( ( relational_cov_a_b @ X @ Q @ G )
=> ( ( relational_gen_a_b @ Y3 @ Q @ Gy )
=> ( relational_cov_a_b @ X @ ( relati591517084277583526ts_a_b @ Y3 @ Q )
@ ( sup_su5130108678486352897la_a_b @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y3 ) @ ( minus_4077726661957047470la_a_b @ G @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y3 ) ) @ bot_bo4495933725496725865la_a_b ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q4: relational_fmla_a_b] : ( relational_subst_a_b @ Q4 @ Y3 @ X )
@ Gy ) ) ) ) ) ) ).
% cov_nocp_intros
thf(fact_1235_cov_OExists__gen,axiom,
! [X: nat,Y3: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Gy: set_Re381260168593705685la_a_b] :
( ( X != Y3 )
=> ( ( relational_cov_a_b @ X @ Q @ G )
=> ( ( relational_gen_a_b @ Y3 @ Q @ Gy )
=> ( relational_cov_a_b @ X @ ( relati591517084277583526ts_a_b @ Y3 @ Q )
@ ( sup_su5130108678486352897la_a_b @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y3 ) @ ( minus_4077726661957047470la_a_b @ G @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y3 ) ) @ bot_bo4495933725496725865la_a_b ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q4: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q4 @ Y3 @ X ) )
@ Gy ) ) ) ) ) ) ).
% cov.Exists_gen
thf(fact_1236_fixbound__empty__Gen,axiom,
! [Q6: set_Re381260168593705685la_a_b,X: nat,Q: relational_fmla_a_b] :
( ( ( restri6210327484173102631nd_a_b @ Q6 @ X )
= bot_bo4495933725496725865la_a_b )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( ( member4680049679412964150la_a_b @ Q @ Q6 )
=> ? [X_12: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ Q @ X_12 ) ) ) ) ).
% fixbound_empty_Gen
thf(fact_1237_gen__eq__gen_H,axiom,
relational_gen_a_b = relational_gen_a_b2 ).
% gen_eq_gen'
thf(fact_1238_gen__gen_H,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X @ Q @ G )
=> ( relational_gen_a_b2 @ X @ Q @ G ) ) ).
% gen_gen'
thf(fact_1239_gen_H__gen,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b2 @ X @ Q @ G )
=> ( relational_gen_a_b @ X @ Q @ G ) ) ).
% gen'_gen
thf(fact_1240_gen_H_Ointros_I7_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q2: relational_fmla_a_b] :
( ( ( relational_gen_a_b2 @ X @ Q1 @ G )
| ( relational_gen_a_b2 @ X @ Q2 @ G ) )
=> ( relational_gen_a_b2 @ X @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ G ) ) ).
% gen'.intros(7)
thf(fact_1241_gen_H_Ointros_I3_J,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b2 @ X @ Q @ G )
=> ( relational_gen_a_b2 @ X @ ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q ) ) @ G ) ) ).
% gen'.intros(3)
thf(fact_1242_gen_H_Ointros_I4_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,Q2: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b2 @ X @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q2 ) ) @ G )
=> ( relational_gen_a_b2 @ X @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) ) @ G ) ) ).
% gen'.intros(4)
thf(fact_1243_gen_H_Ointros_I5_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,Q2: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b2 @ X @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q2 ) ) @ G )
=> ( relational_gen_a_b2 @ X @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) ) @ G ) ) ).
% gen'.intros(5)
thf(fact_1244_gen_H_Ointros_I9_J,axiom,
! [Y3: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
( ( relational_gen_a_b2 @ Y3 @ Q @ G )
=> ( relational_gen_a_b2 @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ X ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q4: relational_fmla_a_b] : ( relational_subst_a_b @ Q4 @ Y3 @ X )
@ G ) ) ) ).
% gen'.intros(9)
thf(fact_1245_gen_H_Ointros_I8_J,axiom,
! [Y3: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
( ( relational_gen_a_b2 @ Y3 @ Q @ G )
=> ( relational_gen_a_b2 @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y3 ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q4: relational_fmla_a_b] : ( relational_subst_a_b @ Q4 @ Y3 @ X )
@ G ) ) ) ).
% gen'.intros(8)
thf(fact_1246_infinite__eval__Conj,axiom,
! [X: nat,Q: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a,Y3: nat] :
( ~ ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( ~ ( finite_finite_list_a @ ( relational_eval_a_b @ Q @ I2 ) )
=> ~ ( finite_finite_list_a @ ( relational_eval_a_b @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y3 ) ) ) @ I2 ) ) ) ) ).
% infinite_eval_Conj
thf(fact_1247_gen_H__cp__intros_I2_J,axiom,
! [Y3: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
( ( relational_gen_a_b2 @ Y3 @ Q @ G )
=> ( relational_gen_a_b2 @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ X ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q4: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q4 @ Y3 @ X ) )
@ G ) ) ) ).
% gen'_cp_intros(2)
thf(fact_1248_gen_H__cp__intros_I1_J,axiom,
! [Y3: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
( ( relational_gen_a_b2 @ Y3 @ Q @ G )
=> ( relational_gen_a_b2 @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y3 ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q4: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q4 @ Y3 @ X ) )
@ G ) ) ) ).
% gen'_cp_intros(1)
% Helper facts (17)
thf(help_If_2_1_If_001t__List__Olist_Itf__b_J_T,axiom,
! [X: list_b,Y3: list_b] :
( ( if_list_b @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__b_J_T,axiom,
! [X: list_b,Y3: list_b] :
( ( if_list_b @ $true @ X @ Y3 )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y3: list_nat] :
( ( if_list_nat @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y3: list_nat] :
( ( if_list_nat @ $true @ X @ Y3 )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_T,axiom,
! [X: list_list_nat,Y3: list_list_nat] :
( ( if_list_list_nat @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_T,axiom,
! [X: list_list_nat,Y3: list_list_nat] :
( ( if_list_list_nat @ $true @ X @ Y3 )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_T,axiom,
! [X: list_R8263082107343818799la_a_b,Y3: list_R8263082107343818799la_a_b] :
( ( if_lis1478040310087653097la_a_b @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_T,axiom,
! [X: list_R8263082107343818799la_a_b,Y3: list_R8263082107343818799la_a_b] :
( ( if_lis1478040310087653097la_a_b @ $true @ X @ Y3 )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J_T,axiom,
! [X: list_s3671366213839843941la_a_b,Y3: list_s3671366213839843941la_a_b] :
( ( if_lis5716551597998330143la_a_b @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J_T,axiom,
! [X: list_s3671366213839843941la_a_b,Y3: list_s3671366213839843941la_a_b] :
( ( if_lis5716551597998330143la_a_b @ $true @ X @ Y3 )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J_T,axiom,
! [X: list_l3216422893793986623la_a_b,Y3: list_l3216422893793986623la_a_b] :
( ( if_lis5916486426012477945la_a_b @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J_T,axiom,
! [X: list_l3216422893793986623la_a_b,Y3: list_l3216422893793986623la_a_b] :
( ( if_lis5916486426012477945la_a_b @ $true @ X @ Y3 )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Set__Oset_It__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J_J_T,axiom,
! [X: list_s5517785546295487899la_a_b,Y3: list_s5517785546295487899la_a_b] :
( ( if_lis8471649527330808917la_a_b @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Set__Oset_It__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J_J_T,axiom,
! [X: list_s5517785546295487899la_a_b,Y3: list_s5517785546295487899la_a_b] :
( ( if_lis8471649527330808917la_a_b @ $true @ X @ Y3 )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J_J_T,axiom,
! [X: list_l1512381142989314933la_a_b,Y3: list_l1512381142989314933la_a_b] :
( ( if_lis8483221396762875183la_a_b @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J_J_T,axiom,
! [X: list_l1512381142989314933la_a_b,Y3: list_l1512381142989314933la_a_b] :
( ( if_lis8483221396762875183la_a_b @ $true @ X @ Y3 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
member3481406638322139244la_a_b @ ga @ ( set_se5111286776497155840la_a_b @ ( restri4663030114664890836pl_a_b @ xa @ ( relational_Conj_a_b @ q1 @ q2 ) ) ) ).
%------------------------------------------------------------------------------