TSTP Solution File: ITP144^1 by cvc5---1.0.5
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- Process Solution
%------------------------------------------------------------------------------
% File : cvc5---1.0.5
% Problem : ITP144^1 : TPTP v8.1.2. Released v7.5.0.
% Transfm : none
% Format : tptp
% Command : do_cvc5 %s %d
% Computer : n017.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 03:18:36 EDT 2023
% Result : Theorem 0.58s 0.77s
% Output : Proof 0.58s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.14 % Problem : ITP144^1 : TPTP v8.1.2. Released v7.5.0.
% 0.14/0.15 % Command : do_cvc5 %s %d
% 0.14/0.36 % Computer : n017.cluster.edu
% 0.14/0.36 % Model : x86_64 x86_64
% 0.14/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.36 % Memory : 8042.1875MB
% 0.14/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.36 % CPULimit : 300
% 0.14/0.36 % WCLimit : 300
% 0.14/0.36 % DateTime : Sun Aug 27 12:14:57 EDT 2023
% 0.14/0.36 % CPUTime :
% 0.21/0.51 %----Proving TH0
% 0.21/0.51 %------------------------------------------------------------------------------
% 0.21/0.51 % File : ITP144^1 : TPTP v8.1.2. Released v7.5.0.
% 0.21/0.51 % Domain : Interactive Theorem Proving
% 0.21/0.51 % Problem : Sledgehammer PHoareTotal problem prob_493__3263728_1
% 0.21/0.51 % Version : Especial.
% 0.21/0.51 % English :
% 0.21/0.51
% 0.21/0.51 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.21/0.51 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.21/0.51 % Source : [Des21]
% 0.21/0.51 % Names : PHoareTotal/prob_493__3263728_1 [Des21]
% 0.21/0.51
% 0.21/0.51 % Status : Theorem
% 0.21/0.51 % Rating : 0.23 v8.1.0, 0.27 v7.5.0
% 0.21/0.51 % Syntax : Number of formulae : 138 ( 54 unt; 26 typ; 0 def)
% 0.21/0.51 % Number of atoms : 265 ( 153 equ; 0 cnn)
% 0.21/0.51 % Maximal formula atoms : 7 ( 2 avg)
% 0.21/0.51 % Number of connectives : 728 ( 51 ~; 4 |; 26 &; 546 @)
% 0.21/0.51 % ( 0 <=>; 101 =>; 0 <=; 0 <~>)
% 0.21/0.51 % Maximal formula depth : 15 ( 5 avg)
% 0.21/0.51 % Number of types : 5 ( 4 usr)
% 0.21/0.51 % Number of type conns : 91 ( 91 >; 0 *; 0 +; 0 <<)
% 0.21/0.51 % Number of symbols : 24 ( 22 usr; 3 con; 0-3 aty)
% 0.21/0.51 % Number of variables : 286 ( 41 ^; 230 !; 15 ?; 286 :)
% 0.21/0.51 % SPC : TH0_THM_EQU_NAR
% 0.21/0.51
% 0.21/0.51 % Comments : This file was generated by Sledgehammer 2021-02-23 15:33:49.690
% 0.21/0.51 %------------------------------------------------------------------------------
% 0.21/0.51 % Could-be-implicit typings (4)
% 0.21/0.51 thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
% 0.21/0.51 set_set_b: $tType ).
% 0.21/0.51
% 0.21/0.51 thf(ty_n_t__Set__Oset_Itf__b_J,type,
% 0.21/0.51 set_b: $tType ).
% 0.21/0.51
% 0.21/0.51 thf(ty_n_tf__b,type,
% 0.21/0.51 b: $tType ).
% 0.21/0.51
% 0.21/0.51 thf(ty_n_tf__a,type,
% 0.21/0.51 a: $tType ).
% 0.21/0.51
% 0.21/0.51 % Explicit typings (22)
% 0.21/0.51 thf(sy_c_Fun_Obij__betw_001tf__b_001tf__b,type,
% 0.21/0.51 bij_betw_b_b: ( b > b ) > set_b > set_b > $o ).
% 0.21/0.51
% 0.21/0.51 thf(sy_c_Fun_Oinj__on_001tf__b_001t__Set__Oset_Itf__b_J,type,
% 0.21/0.51 inj_on_b_set_b: ( b > set_b ) > set_b > $o ).
% 0.21/0.51
% 0.21/0.51 thf(sy_c_Fun_Oinj__on_001tf__b_001tf__b,type,
% 0.21/0.51 inj_on_b_b: ( b > b ) > set_b > $o ).
% 0.21/0.51
% 0.21/0.51 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__b_J,type,
% 0.21/0.51 minus_minus_set_b: set_b > set_b > set_b ).
% 0.21/0.51
% 0.21/0.51 thf(sy_c_Hilbert__Choice_OEps_001tf__b,type,
% 0.21/0.51 hilbert_Eps_b: ( b > $o ) > b ).
% 0.21/0.51
% 0.21/0.51 thf(sy_c_Hilbert__Choice_Oinv__into_001tf__b_001tf__b,type,
% 0.21/0.51 hilbert_inv_into_b_b: set_b > ( b > b ) > b > b ).
% 0.21/0.51
% 0.21/0.51 thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__b_M_Eo_J,type,
% 0.21/0.51 bot_bot_b_o: b > $o ).
% 0.21/0.51
% 0.21/0.51 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__b_J,type,
% 0.21/0.51 bot_bot_set_b: set_b ).
% 0.21/0.51
% 0.21/0.51 thf(sy_c_Orderings_Otop__class_Otop_001_062_Itf__b_M_Eo_J,type,
% 0.21/0.51 top_top_b_o: b > $o ).
% 0.21/0.51
% 0.21/0.51 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__b_J,type,
% 0.21/0.51 top_top_set_b: set_b ).
% 0.21/0.51
% 0.21/0.51 thf(sy_c_Set_OCollect_001tf__b,type,
% 0.21/0.51 collect_b: ( b > $o ) > set_b ).
% 0.21/0.51
% 0.21/0.51 thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__b_J_001t__Set__Oset_Itf__b_J,type,
% 0.21/0.51 image_set_b_set_b: ( set_b > set_b ) > set_set_b > set_set_b ).
% 0.21/0.51
% 0.21/0.51 thf(sy_c_Set_Oimage_001tf__b_001tf__b,type,
% 0.21/0.51 image_b_b: ( b > b ) > set_b > set_b ).
% 0.21/0.51
% 0.21/0.51 thf(sy_c_Set_Oinsert_001tf__b,type,
% 0.21/0.51 insert_b: b > set_b > set_b ).
% 0.21/0.51
% 0.21/0.51 thf(sy_c_Set_Ois__empty_001tf__b,type,
% 0.21/0.51 is_empty_b: set_b > $o ).
% 0.21/0.51
% 0.21/0.51 thf(sy_c_Set_Ois__singleton_001tf__b,type,
% 0.21/0.51 is_singleton_b: set_b > $o ).
% 0.21/0.51
% 0.21/0.51 thf(sy_c_Set_Oremove_001tf__b,type,
% 0.21/0.51 remove_b: b > set_b > set_b ).
% 0.21/0.51
% 0.21/0.51 thf(sy_c_Set_Othe__elem_001tf__b,type,
% 0.21/0.51 the_elem_b: set_b > b ).
% 0.21/0.51
% 0.21/0.51 thf(sy_c_member_001t__Set__Oset_Itf__b_J,type,
% 0.21/0.51 member_set_b: set_b > set_set_b > $o ).
% 0.21/0.51
% 0.21/0.51 thf(sy_c_member_001tf__b,type,
% 0.21/0.51 member_b: b > set_b > $o ).
% 0.21/0.51
% 0.21/0.51 thf(sy_v_P,type,
% 0.21/0.51 p: a > $o ).
% 0.21/0.51
% 0.21/0.51 thf(sy_v_Q,type,
% 0.21/0.51 q: a > b > $o ).
% 0.21/0.51
% 0.21/0.51 % Relevant facts (110)
% 0.21/0.51 thf(fact_0_some__equality,axiom,
% 0.21/0.51 ! [P: b > $o,A: b] :
% 0.21/0.51 ( ( P @ A )
% 0.21/0.51 => ( ! [X: b] :
% 0.21/0.51 ( ( P @ X )
% 0.21/0.51 => ( X = A ) )
% 0.21/0.51 => ( ( hilbert_Eps_b @ P )
% 0.21/0.51 = A ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % some_equality
% 0.21/0.51 thf(fact_1_some__eq__trivial,axiom,
% 0.21/0.51 ! [X2: b] :
% 0.21/0.51 ( ( hilbert_Eps_b
% 0.21/0.51 @ ^ [Y: b] : ( Y = X2 ) )
% 0.21/0.51 = X2 ) ).
% 0.21/0.51
% 0.21/0.51 % some_eq_trivial
% 0.21/0.51 thf(fact_2_some__sym__eq__trivial,axiom,
% 0.21/0.51 ! [X2: b] :
% 0.21/0.51 ( ( hilbert_Eps_b
% 0.21/0.51 @ ( ^ [Y2: b,Z: b] : ( Y2 = Z )
% 0.21/0.51 @ X2 ) )
% 0.21/0.51 = X2 ) ).
% 0.21/0.51
% 0.21/0.51 % some_sym_eq_trivial
% 0.21/0.51 thf(fact_3_verit__sko__ex_H,axiom,
% 0.21/0.51 ! [P: b > $o,A2: $o] :
% 0.21/0.51 ( ( ( P @ ( hilbert_Eps_b @ P ) )
% 0.21/0.51 = A2 )
% 0.21/0.51 => ( ( ? [X3: b] : ( P @ X3 ) )
% 0.21/0.51 = A2 ) ) ).
% 0.21/0.51
% 0.21/0.51 % verit_sko_ex'
% 0.21/0.51 thf(fact_4_verit__sko__forall,axiom,
% 0.21/0.51 ( ( ^ [P2: b > $o] :
% 0.21/0.51 ! [X3: b] : ( P2 @ X3 ) )
% 0.21/0.51 = ( ^ [P3: b > $o] :
% 0.21/0.51 ( P3
% 0.21/0.51 @ ( hilbert_Eps_b
% 0.21/0.51 @ ^ [X4: b] :
% 0.21/0.51 ~ ( P3 @ X4 ) ) ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % verit_sko_forall
% 0.21/0.51 thf(fact_5_someI2,axiom,
% 0.21/0.51 ! [P: b > $o,A: b,Q: b > $o] :
% 0.21/0.51 ( ( P @ A )
% 0.21/0.51 => ( ! [X: b] :
% 0.21/0.51 ( ( P @ X )
% 0.21/0.51 => ( Q @ X ) )
% 0.21/0.51 => ( Q @ ( hilbert_Eps_b @ P ) ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % someI2
% 0.21/0.51 thf(fact_6_verit__sko__forall_H,axiom,
% 0.21/0.51 ! [P: b > $o,A2: $o] :
% 0.21/0.51 ( ( ( P
% 0.21/0.51 @ ( hilbert_Eps_b
% 0.21/0.51 @ ^ [X4: b] :
% 0.21/0.51 ~ ( P @ X4 ) ) )
% 0.21/0.51 = A2 )
% 0.21/0.51 => ( ( ! [X3: b] : ( P @ X3 ) )
% 0.21/0.51 = A2 ) ) ).
% 0.21/0.51
% 0.21/0.51 % verit_sko_forall'
% 0.21/0.51 thf(fact_7_verit__sko__forall_H_H,axiom,
% 0.21/0.51 ! [B: b,A2: b,P: b > $o] :
% 0.21/0.51 ( ( B = A2 )
% 0.21/0.51 => ( ( ( hilbert_Eps_b @ P )
% 0.21/0.51 = A2 )
% 0.21/0.51 = ( ( hilbert_Eps_b @ P )
% 0.21/0.51 = B ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % verit_sko_forall''
% 0.21/0.51 thf(fact_8_someI__ex,axiom,
% 0.21/0.51 ! [P: b > $o] :
% 0.21/0.51 ( ? [X_1: b] : ( P @ X_1 )
% 0.21/0.51 => ( P @ ( hilbert_Eps_b @ P ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % someI_ex
% 0.21/0.51 thf(fact_9_someI2__ex,axiom,
% 0.21/0.51 ! [P: b > $o,Q: b > $o] :
% 0.21/0.51 ( ? [X_1: b] : ( P @ X_1 )
% 0.21/0.51 => ( ! [X: b] :
% 0.21/0.51 ( ( P @ X )
% 0.21/0.51 => ( Q @ X ) )
% 0.21/0.51 => ( Q @ ( hilbert_Eps_b @ P ) ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % someI2_ex
% 0.21/0.51 thf(fact_10_someI2__bex,axiom,
% 0.21/0.51 ! [A2: set_b,P: b > $o,Q: b > $o] :
% 0.21/0.51 ( ? [X5: b] :
% 0.21/0.51 ( ( member_b @ X5 @ A2 )
% 0.21/0.51 & ( P @ X5 ) )
% 0.21/0.51 => ( ! [X: b] :
% 0.21/0.51 ( ( ( member_b @ X @ A2 )
% 0.21/0.51 & ( P @ X ) )
% 0.21/0.51 => ( Q @ X ) )
% 0.21/0.51 => ( Q
% 0.21/0.51 @ ( hilbert_Eps_b
% 0.21/0.51 @ ^ [X4: b] :
% 0.21/0.51 ( ( member_b @ X4 @ A2 )
% 0.21/0.51 & ( P @ X4 ) ) ) ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % someI2_bex
% 0.21/0.51 thf(fact_11_some__eq__ex,axiom,
% 0.21/0.51 ! [P: b > $o] :
% 0.21/0.51 ( ( P @ ( hilbert_Eps_b @ P ) )
% 0.21/0.51 = ( ? [X3: b] : ( P @ X3 ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % some_eq_ex
% 0.21/0.51 thf(fact_12_some__eq__imp,axiom,
% 0.21/0.51 ! [P: b > $o,A: b,B2: b] :
% 0.21/0.51 ( ( ( hilbert_Eps_b @ P )
% 0.21/0.51 = A )
% 0.21/0.51 => ( ( P @ B2 )
% 0.21/0.51 => ( P @ A ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % some_eq_imp
% 0.21/0.51 thf(fact_13_tfl__some,axiom,
% 0.21/0.51 ! [P4: b > $o,X5: b] :
% 0.21/0.51 ( ( P4 @ X5 )
% 0.21/0.51 => ( P4 @ ( hilbert_Eps_b @ P4 ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % tfl_some
% 0.21/0.51 thf(fact_14_Eps__cong,axiom,
% 0.21/0.51 ! [P: b > $o,Q: b > $o] :
% 0.21/0.51 ( ! [X: b] :
% 0.21/0.51 ( ( P @ X )
% 0.21/0.51 = ( Q @ X ) )
% 0.21/0.51 => ( ( hilbert_Eps_b @ P )
% 0.21/0.51 = ( hilbert_Eps_b @ Q ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % Eps_cong
% 0.21/0.51 thf(fact_15_someI,axiom,
% 0.21/0.51 ! [P: b > $o,X2: b] :
% 0.21/0.51 ( ( P @ X2 )
% 0.21/0.51 => ( P @ ( hilbert_Eps_b @ P ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % someI
% 0.21/0.51 thf(fact_16_verit__sko__forall__indirect2,axiom,
% 0.21/0.51 ! [X2: b,P: b > $o,P5: b > $o] :
% 0.21/0.51 ( ( X2
% 0.21/0.51 = ( hilbert_Eps_b
% 0.21/0.51 @ ^ [X4: b] :
% 0.21/0.51 ~ ( P @ X4 ) ) )
% 0.21/0.51 => ( ! [X: b] :
% 0.21/0.51 ( ( P @ X )
% 0.21/0.51 = ( P5 @ X ) )
% 0.21/0.51 => ( ( ! [X3: b] : ( P5 @ X3 ) )
% 0.21/0.51 = ( P @ X2 ) ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % verit_sko_forall_indirect2
% 0.21/0.51 thf(fact_17_verit__sko__forall__indirect,axiom,
% 0.21/0.51 ! [X2: b,P: b > $o] :
% 0.21/0.51 ( ( X2
% 0.21/0.51 = ( hilbert_Eps_b
% 0.21/0.51 @ ^ [X4: b] :
% 0.21/0.51 ~ ( P @ X4 ) ) )
% 0.21/0.51 => ( ( ! [X3: b] : ( P @ X3 ) )
% 0.21/0.51 = ( P @ X2 ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % verit_sko_forall_indirect
% 0.21/0.51 thf(fact_18_some1__equality,axiom,
% 0.21/0.51 ! [P: b > $o,A: b] :
% 0.21/0.51 ( ? [X5: b] :
% 0.21/0.51 ( ( P @ X5 )
% 0.21/0.51 & ! [Y3: b] :
% 0.21/0.51 ( ( P @ Y3 )
% 0.21/0.51 => ( Y3 = X5 ) ) )
% 0.21/0.51 => ( ( P @ A )
% 0.21/0.51 => ( ( hilbert_Eps_b @ P )
% 0.21/0.51 = A ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % some1_equality
% 0.21/0.51 thf(fact_19_verit__sko__ex__indirect2,axiom,
% 0.21/0.51 ! [X2: b,P: b > $o,P5: b > $o] :
% 0.21/0.51 ( ( X2
% 0.21/0.51 = ( hilbert_Eps_b @ P ) )
% 0.21/0.51 => ( ! [X: b] :
% 0.21/0.51 ( ( P @ X )
% 0.21/0.51 = ( P5 @ X ) )
% 0.21/0.51 => ( ( ? [X3: b] : ( P5 @ X3 ) )
% 0.21/0.51 = ( P @ X2 ) ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % verit_sko_ex_indirect2
% 0.21/0.51 thf(fact_20_verit__sko__ex__indirect,axiom,
% 0.21/0.51 ! [X2: b,P: b > $o] :
% 0.21/0.51 ( ( X2
% 0.21/0.51 = ( hilbert_Eps_b @ P ) )
% 0.21/0.51 => ( ( ? [X3: b] : ( P @ X3 ) )
% 0.21/0.51 = ( P @ X2 ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % verit_sko_ex_indirect
% 0.21/0.51 thf(fact_21_Nitpick_OEps__psimp,axiom,
% 0.21/0.51 ! [P: b > $o,X2: b,Y4: b] :
% 0.21/0.51 ( ( P @ X2 )
% 0.21/0.51 => ( ~ ( P @ Y4 )
% 0.21/0.51 => ( ( ( hilbert_Eps_b @ P )
% 0.21/0.51 = Y4 )
% 0.21/0.51 => ( ( hilbert_Eps_b @ P )
% 0.21/0.51 = X2 ) ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % Nitpick.Eps_psimp
% 0.21/0.51 thf(fact_22_some__in__eq,axiom,
% 0.21/0.51 ! [A2: set_b] :
% 0.21/0.51 ( ( member_b
% 0.21/0.51 @ ( hilbert_Eps_b
% 0.21/0.51 @ ^ [X4: b] : ( member_b @ X4 @ A2 ) )
% 0.21/0.51 @ A2 )
% 0.21/0.51 = ( A2 != bot_bot_set_b ) ) ).
% 0.21/0.51
% 0.21/0.51 % some_in_eq
% 0.21/0.51 thf(fact_23_exE__some,axiom,
% 0.21/0.51 ! [P: b > $o,C: b] :
% 0.21/0.51 ( ? [X_1: b] : ( P @ X_1 )
% 0.21/0.51 => ( ( C
% 0.21/0.51 = ( hilbert_Eps_b @ P ) )
% 0.21/0.51 => ( P @ C ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % exE_some
% 0.21/0.51 thf(fact_24_empty__Collect__eq,axiom,
% 0.21/0.51 ! [P: b > $o] :
% 0.21/0.51 ( ( bot_bot_set_b
% 0.21/0.51 = ( collect_b @ P ) )
% 0.21/0.51 = ( ! [X4: b] :
% 0.21/0.51 ~ ( P @ X4 ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % empty_Collect_eq
% 0.21/0.51 thf(fact_25_Collect__empty__eq,axiom,
% 0.21/0.51 ! [P: b > $o] :
% 0.21/0.51 ( ( ( collect_b @ P )
% 0.21/0.51 = bot_bot_set_b )
% 0.21/0.51 = ( ! [X4: b] :
% 0.21/0.51 ~ ( P @ X4 ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % Collect_empty_eq
% 0.21/0.51 thf(fact_26_all__not__in__conv,axiom,
% 0.21/0.51 ! [A2: set_b] :
% 0.21/0.51 ( ( ! [X4: b] :
% 0.21/0.51 ~ ( member_b @ X4 @ A2 ) )
% 0.21/0.51 = ( A2 = bot_bot_set_b ) ) ).
% 0.21/0.51
% 0.21/0.51 % all_not_in_conv
% 0.21/0.51 thf(fact_27_empty__iff,axiom,
% 0.21/0.51 ! [C: b] :
% 0.21/0.51 ~ ( member_b @ C @ bot_bot_set_b ) ).
% 0.21/0.51
% 0.21/0.51 % empty_iff
% 0.21/0.51 thf(fact_28_mem__Collect__eq,axiom,
% 0.21/0.51 ! [A: b,P: b > $o] :
% 0.21/0.51 ( ( member_b @ A @ ( collect_b @ P ) )
% 0.21/0.51 = ( P @ A ) ) ).
% 0.21/0.51
% 0.21/0.51 % mem_Collect_eq
% 0.21/0.51 thf(fact_29_Collect__mem__eq,axiom,
% 0.21/0.51 ! [A2: set_b] :
% 0.21/0.51 ( ( collect_b
% 0.21/0.51 @ ^ [X4: b] : ( member_b @ X4 @ A2 ) )
% 0.21/0.51 = A2 ) ).
% 0.21/0.51
% 0.21/0.51 % Collect_mem_eq
% 0.21/0.51 thf(fact_30_bot__set__def,axiom,
% 0.21/0.51 ( bot_bot_set_b
% 0.21/0.51 = ( collect_b @ bot_bot_b_o ) ) ).
% 0.21/0.51
% 0.21/0.51 % bot_set_def
% 0.21/0.51 thf(fact_31_emptyE,axiom,
% 0.21/0.51 ! [A: b] :
% 0.21/0.51 ~ ( member_b @ A @ bot_bot_set_b ) ).
% 0.21/0.51
% 0.21/0.51 % emptyE
% 0.21/0.51 thf(fact_32_equals0D,axiom,
% 0.21/0.51 ! [A2: set_b,A: b] :
% 0.21/0.51 ( ( A2 = bot_bot_set_b )
% 0.21/0.51 => ~ ( member_b @ A @ A2 ) ) ).
% 0.21/0.51
% 0.21/0.51 % equals0D
% 0.21/0.51 thf(fact_33_equals0I,axiom,
% 0.21/0.51 ! [A2: set_b] :
% 0.21/0.51 ( ! [Y3: b] :
% 0.21/0.51 ~ ( member_b @ Y3 @ A2 )
% 0.21/0.51 => ( A2 = bot_bot_set_b ) ) ).
% 0.21/0.51
% 0.21/0.51 % equals0I
% 0.21/0.51 thf(fact_34_ex__in__conv,axiom,
% 0.21/0.51 ! [A2: set_b] :
% 0.21/0.51 ( ( ? [X4: b] : ( member_b @ X4 @ A2 ) )
% 0.21/0.51 = ( A2 != bot_bot_set_b ) ) ).
% 0.21/0.51
% 0.21/0.51 % ex_in_conv
% 0.21/0.51 thf(fact_35_empty__def,axiom,
% 0.21/0.51 ( bot_bot_set_b
% 0.21/0.51 = ( collect_b
% 0.21/0.51 @ ^ [X4: b] : $false ) ) ).
% 0.21/0.51
% 0.21/0.51 % empty_def
% 0.21/0.51 thf(fact_36_Set_Ois__empty__def,axiom,
% 0.21/0.51 ( is_empty_b
% 0.21/0.51 = ( ^ [A3: set_b] : ( A3 = bot_bot_set_b ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % Set.is_empty_def
% 0.21/0.51 thf(fact_37_Collect__empty__eq__bot,axiom,
% 0.21/0.51 ! [P: b > $o] :
% 0.21/0.51 ( ( ( collect_b @ P )
% 0.21/0.51 = bot_bot_set_b )
% 0.21/0.51 = ( P = bot_bot_b_o ) ) ).
% 0.21/0.51
% 0.21/0.51 % Collect_empty_eq_bot
% 0.21/0.51 thf(fact_38_bot__empty__eq,axiom,
% 0.21/0.51 ( bot_bot_b_o
% 0.21/0.51 = ( ^ [X4: b] : ( member_b @ X4 @ bot_bot_set_b ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % bot_empty_eq
% 0.21/0.51 thf(fact_39_is__singletonI_H,axiom,
% 0.21/0.51 ! [A2: set_b] :
% 0.21/0.51 ( ( A2 != bot_bot_set_b )
% 0.21/0.51 => ( ! [X: b,Y3: b] :
% 0.21/0.51 ( ( member_b @ X @ A2 )
% 0.21/0.51 => ( ( member_b @ Y3 @ A2 )
% 0.21/0.51 => ( X = Y3 ) ) )
% 0.21/0.51 => ( is_singleton_b @ A2 ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % is_singletonI'
% 0.21/0.51 thf(fact_40_UNIV__I,axiom,
% 0.21/0.51 ! [X2: b] : ( member_b @ X2 @ top_top_set_b ) ).
% 0.21/0.51
% 0.21/0.51 % UNIV_I
% 0.21/0.51 thf(fact_41_Collect__const,axiom,
% 0.21/0.51 ! [P: $o] :
% 0.21/0.51 ( ( P
% 0.21/0.51 => ( ( collect_b
% 0.21/0.51 @ ^ [S: b] : P )
% 0.21/0.51 = top_top_set_b ) )
% 0.21/0.51 & ( ~ P
% 0.21/0.51 => ( ( collect_b
% 0.21/0.51 @ ^ [S: b] : P )
% 0.21/0.51 = bot_bot_set_b ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % Collect_const
% 0.21/0.51 thf(fact_42_UNIV__eq__I,axiom,
% 0.21/0.51 ! [A2: set_b] :
% 0.21/0.51 ( ! [X: b] : ( member_b @ X @ A2 )
% 0.21/0.51 => ( top_top_set_b = A2 ) ) ).
% 0.21/0.51
% 0.21/0.51 % UNIV_eq_I
% 0.21/0.51 thf(fact_43_UNIV__witness,axiom,
% 0.21/0.51 ? [X: b] : ( member_b @ X @ top_top_set_b ) ).
% 0.21/0.51
% 0.21/0.51 % UNIV_witness
% 0.21/0.51 thf(fact_44_empty__not__UNIV,axiom,
% 0.21/0.51 bot_bot_set_b != top_top_set_b ).
% 0.21/0.51
% 0.21/0.51 % empty_not_UNIV
% 0.21/0.51 thf(fact_45_iso__tuple__UNIV__I,axiom,
% 0.21/0.51 ! [X2: b] : ( member_b @ X2 @ top_top_set_b ) ).
% 0.21/0.51
% 0.21/0.51 % iso_tuple_UNIV_I
% 0.21/0.51 thf(fact_46_is__singletonI,axiom,
% 0.21/0.51 ! [X2: b] : ( is_singleton_b @ ( insert_b @ X2 @ bot_bot_set_b ) ) ).
% 0.21/0.51
% 0.21/0.51 % is_singletonI
% 0.21/0.51 thf(fact_47_insert__iff,axiom,
% 0.21/0.51 ! [A: b,B2: b,A2: set_b] :
% 0.21/0.51 ( ( member_b @ A @ ( insert_b @ B2 @ A2 ) )
% 0.21/0.51 = ( ( A = B2 )
% 0.21/0.51 | ( member_b @ A @ A2 ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % insert_iff
% 0.21/0.51 thf(fact_48_insertCI,axiom,
% 0.21/0.51 ! [A: b,B: set_b,B2: b] :
% 0.21/0.51 ( ( ~ ( member_b @ A @ B )
% 0.21/0.51 => ( A = B2 ) )
% 0.21/0.51 => ( member_b @ A @ ( insert_b @ B2 @ B ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % insertCI
% 0.21/0.51 thf(fact_49_singletonI,axiom,
% 0.21/0.51 ! [A: b] : ( member_b @ A @ ( insert_b @ A @ bot_bot_set_b ) ) ).
% 0.21/0.51
% 0.21/0.51 % singletonI
% 0.21/0.51 thf(fact_50_singleton__conv,axiom,
% 0.21/0.51 ! [A: b] :
% 0.21/0.51 ( ( collect_b
% 0.21/0.51 @ ^ [X4: b] : ( X4 = A ) )
% 0.21/0.51 = ( insert_b @ A @ bot_bot_set_b ) ) ).
% 0.21/0.51
% 0.21/0.51 % singleton_conv
% 0.21/0.51 thf(fact_51_singleton__conv2,axiom,
% 0.21/0.51 ! [A: b] :
% 0.21/0.51 ( ( collect_b
% 0.21/0.51 @ ( ^ [Y2: b,Z: b] : ( Y2 = Z )
% 0.21/0.51 @ A ) )
% 0.21/0.51 = ( insert_b @ A @ bot_bot_set_b ) ) ).
% 0.21/0.51
% 0.21/0.51 % singleton_conv2
% 0.21/0.51 thf(fact_52_mk__disjoint__insert,axiom,
% 0.21/0.51 ! [A: b,A2: set_b] :
% 0.21/0.51 ( ( member_b @ A @ A2 )
% 0.21/0.51 => ? [B3: set_b] :
% 0.21/0.51 ( ( A2
% 0.21/0.51 = ( insert_b @ A @ B3 ) )
% 0.21/0.51 & ~ ( member_b @ A @ B3 ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % mk_disjoint_insert
% 0.21/0.51 thf(fact_53_insert__eq__iff,axiom,
% 0.21/0.51 ! [A: b,A2: set_b,B2: b,B: set_b] :
% 0.21/0.51 ( ~ ( member_b @ A @ A2 )
% 0.21/0.51 => ( ~ ( member_b @ B2 @ B )
% 0.21/0.51 => ( ( ( insert_b @ A @ A2 )
% 0.21/0.51 = ( insert_b @ B2 @ B ) )
% 0.21/0.51 = ( ( ( A = B2 )
% 0.21/0.51 => ( A2 = B ) )
% 0.21/0.51 & ( ( A != B2 )
% 0.21/0.51 => ? [C2: set_b] :
% 0.21/0.51 ( ( A2
% 0.21/0.51 = ( insert_b @ B2 @ C2 ) )
% 0.21/0.51 & ~ ( member_b @ B2 @ C2 )
% 0.21/0.51 & ( B
% 0.21/0.51 = ( insert_b @ A @ C2 ) )
% 0.21/0.51 & ~ ( member_b @ A @ C2 ) ) ) ) ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % insert_eq_iff
% 0.21/0.51 thf(fact_54_insert__absorb,axiom,
% 0.21/0.51 ! [A: b,A2: set_b] :
% 0.21/0.51 ( ( member_b @ A @ A2 )
% 0.21/0.51 => ( ( insert_b @ A @ A2 )
% 0.21/0.51 = A2 ) ) ).
% 0.21/0.51
% 0.21/0.51 % insert_absorb
% 0.21/0.51 thf(fact_55_insert__ident,axiom,
% 0.21/0.51 ! [X2: b,A2: set_b,B: set_b] :
% 0.21/0.51 ( ~ ( member_b @ X2 @ A2 )
% 0.21/0.51 => ( ~ ( member_b @ X2 @ B )
% 0.21/0.51 => ( ( ( insert_b @ X2 @ A2 )
% 0.21/0.51 = ( insert_b @ X2 @ B ) )
% 0.21/0.51 = ( A2 = B ) ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % insert_ident
% 0.21/0.51 thf(fact_56_insert__compr,axiom,
% 0.21/0.51 ( insert_b
% 0.21/0.51 = ( ^ [A4: b,B4: set_b] :
% 0.21/0.51 ( collect_b
% 0.21/0.51 @ ^ [X4: b] :
% 0.21/0.51 ( ( X4 = A4 )
% 0.21/0.51 | ( member_b @ X4 @ B4 ) ) ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % insert_compr
% 0.21/0.51 thf(fact_57_Set_Oset__insert,axiom,
% 0.21/0.51 ! [X2: b,A2: set_b] :
% 0.21/0.51 ( ( member_b @ X2 @ A2 )
% 0.21/0.51 => ~ ! [B3: set_b] :
% 0.21/0.51 ( ( A2
% 0.21/0.51 = ( insert_b @ X2 @ B3 ) )
% 0.21/0.51 => ( member_b @ X2 @ B3 ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % Set.set_insert
% 0.21/0.51 thf(fact_58_insertI2,axiom,
% 0.21/0.51 ! [A: b,B: set_b,B2: b] :
% 0.21/0.51 ( ( member_b @ A @ B )
% 0.21/0.51 => ( member_b @ A @ ( insert_b @ B2 @ B ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % insertI2
% 0.21/0.51 thf(fact_59_insertI1,axiom,
% 0.21/0.51 ! [A: b,B: set_b] : ( member_b @ A @ ( insert_b @ A @ B ) ) ).
% 0.21/0.51
% 0.21/0.51 % insertI1
% 0.21/0.51 thf(fact_60_insertE,axiom,
% 0.21/0.51 ! [A: b,B2: b,A2: set_b] :
% 0.21/0.51 ( ( member_b @ A @ ( insert_b @ B2 @ A2 ) )
% 0.21/0.51 => ( ( A != B2 )
% 0.21/0.51 => ( member_b @ A @ A2 ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % insertE
% 0.21/0.51 thf(fact_61_top__empty__eq,axiom,
% 0.21/0.51 ( top_top_b_o
% 0.21/0.51 = ( ^ [X4: b] : ( member_b @ X4 @ top_top_set_b ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % top_empty_eq
% 0.21/0.51 thf(fact_62_singleton__inject,axiom,
% 0.21/0.51 ! [A: b,B2: b] :
% 0.21/0.51 ( ( ( insert_b @ A @ bot_bot_set_b )
% 0.21/0.51 = ( insert_b @ B2 @ bot_bot_set_b ) )
% 0.21/0.51 => ( A = B2 ) ) ).
% 0.21/0.51
% 0.21/0.51 % singleton_inject
% 0.21/0.51 thf(fact_63_insert__not__empty,axiom,
% 0.21/0.51 ! [A: b,A2: set_b] :
% 0.21/0.51 ( ( insert_b @ A @ A2 )
% 0.21/0.51 != bot_bot_set_b ) ).
% 0.21/0.51
% 0.21/0.51 % insert_not_empty
% 0.21/0.51 thf(fact_64_doubleton__eq__iff,axiom,
% 0.21/0.51 ! [A: b,B2: b,C: b,D: b] :
% 0.21/0.51 ( ( ( insert_b @ A @ ( insert_b @ B2 @ bot_bot_set_b ) )
% 0.21/0.51 = ( insert_b @ C @ ( insert_b @ D @ bot_bot_set_b ) ) )
% 0.21/0.51 = ( ( ( A = C )
% 0.21/0.51 & ( B2 = D ) )
% 0.21/0.51 | ( ( A = D )
% 0.21/0.51 & ( B2 = C ) ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % doubleton_eq_iff
% 0.21/0.51 thf(fact_65_singleton__iff,axiom,
% 0.21/0.51 ! [B2: b,A: b] :
% 0.21/0.51 ( ( member_b @ B2 @ ( insert_b @ A @ bot_bot_set_b ) )
% 0.21/0.51 = ( B2 = A ) ) ).
% 0.21/0.51
% 0.21/0.51 % singleton_iff
% 0.21/0.51 thf(fact_66_singletonD,axiom,
% 0.21/0.51 ! [B2: b,A: b] :
% 0.21/0.51 ( ( member_b @ B2 @ ( insert_b @ A @ bot_bot_set_b ) )
% 0.21/0.51 => ( B2 = A ) ) ).
% 0.21/0.51
% 0.21/0.51 % singletonD
% 0.21/0.51 thf(fact_67_Collect__conv__if,axiom,
% 0.21/0.51 ! [P: b > $o,A: b] :
% 0.21/0.51 ( ( ( P @ A )
% 0.21/0.51 => ( ( collect_b
% 0.21/0.51 @ ^ [X4: b] :
% 0.21/0.51 ( ( X4 = A )
% 0.21/0.51 & ( P @ X4 ) ) )
% 0.21/0.51 = ( insert_b @ A @ bot_bot_set_b ) ) )
% 0.21/0.51 & ( ~ ( P @ A )
% 0.21/0.51 => ( ( collect_b
% 0.21/0.51 @ ^ [X4: b] :
% 0.21/0.51 ( ( X4 = A )
% 0.21/0.51 & ( P @ X4 ) ) )
% 0.21/0.51 = bot_bot_set_b ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % Collect_conv_if
% 0.21/0.51 thf(fact_68_Collect__conv__if2,axiom,
% 0.21/0.51 ! [P: b > $o,A: b] :
% 0.21/0.51 ( ( ( P @ A )
% 0.21/0.51 => ( ( collect_b
% 0.21/0.51 @ ^ [X4: b] :
% 0.21/0.51 ( ( A = X4 )
% 0.21/0.51 & ( P @ X4 ) ) )
% 0.21/0.51 = ( insert_b @ A @ bot_bot_set_b ) ) )
% 0.21/0.51 & ( ~ ( P @ A )
% 0.21/0.51 => ( ( collect_b
% 0.21/0.51 @ ^ [X4: b] :
% 0.21/0.51 ( ( A = X4 )
% 0.21/0.51 & ( P @ X4 ) ) )
% 0.21/0.51 = bot_bot_set_b ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % Collect_conv_if2
% 0.21/0.51 thf(fact_69_is__singleton__def,axiom,
% 0.21/0.51 ( is_singleton_b
% 0.21/0.51 = ( ^ [A3: set_b] :
% 0.21/0.51 ? [X4: b] :
% 0.21/0.51 ( A3
% 0.21/0.51 = ( insert_b @ X4 @ bot_bot_set_b ) ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % is_singleton_def
% 0.21/0.51 thf(fact_70_is__singletonE,axiom,
% 0.21/0.51 ! [A2: set_b] :
% 0.21/0.51 ( ( is_singleton_b @ A2 )
% 0.21/0.51 => ~ ! [X: b] :
% 0.21/0.51 ( A2
% 0.21/0.51 != ( insert_b @ X @ bot_bot_set_b ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % is_singletonE
% 0.21/0.51 thf(fact_71_is__singleton__the__elem,axiom,
% 0.21/0.51 ( is_singleton_b
% 0.21/0.51 = ( ^ [A3: set_b] :
% 0.21/0.51 ( A3
% 0.21/0.51 = ( insert_b @ ( the_elem_b @ A3 ) @ bot_bot_set_b ) ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % is_singleton_the_elem
% 0.21/0.51 thf(fact_72_the__elem__eq,axiom,
% 0.21/0.51 ! [X2: b] :
% 0.21/0.51 ( ( the_elem_b @ ( insert_b @ X2 @ bot_bot_set_b ) )
% 0.21/0.51 = X2 ) ).
% 0.21/0.51
% 0.21/0.51 % the_elem_eq
% 0.21/0.51 thf(fact_73_image__eqI,axiom,
% 0.21/0.51 ! [B2: b,F: b > b,X2: b,A2: set_b] :
% 0.21/0.51 ( ( B2
% 0.21/0.51 = ( F @ X2 ) )
% 0.21/0.51 => ( ( member_b @ X2 @ A2 )
% 0.21/0.51 => ( member_b @ B2 @ ( image_b_b @ F @ A2 ) ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % image_eqI
% 0.21/0.51 thf(fact_74_image__is__empty,axiom,
% 0.21/0.51 ! [F: b > b,A2: set_b] :
% 0.21/0.51 ( ( ( image_b_b @ F @ A2 )
% 0.21/0.51 = bot_bot_set_b )
% 0.21/0.51 = ( A2 = bot_bot_set_b ) ) ).
% 0.21/0.51
% 0.21/0.51 % image_is_empty
% 0.21/0.51 thf(fact_75_empty__is__image,axiom,
% 0.21/0.51 ! [F: b > b,A2: set_b] :
% 0.21/0.51 ( ( bot_bot_set_b
% 0.21/0.51 = ( image_b_b @ F @ A2 ) )
% 0.21/0.51 = ( A2 = bot_bot_set_b ) ) ).
% 0.21/0.51
% 0.21/0.51 % empty_is_image
% 0.21/0.51 thf(fact_76_image__empty,axiom,
% 0.21/0.51 ! [F: b > b] :
% 0.21/0.51 ( ( image_b_b @ F @ bot_bot_set_b )
% 0.21/0.51 = bot_bot_set_b ) ).
% 0.21/0.51
% 0.21/0.51 % image_empty
% 0.21/0.51 thf(fact_77_imageE,axiom,
% 0.21/0.51 ! [B2: b,F: b > b,A2: set_b] :
% 0.21/0.51 ( ( member_b @ B2 @ ( image_b_b @ F @ A2 ) )
% 0.21/0.51 => ~ ! [X: b] :
% 0.21/0.51 ( ( B2
% 0.21/0.51 = ( F @ X ) )
% 0.21/0.51 => ~ ( member_b @ X @ A2 ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % imageE
% 0.21/0.51 thf(fact_78_imageI,axiom,
% 0.21/0.51 ! [X2: b,A2: set_b,F: b > b] :
% 0.21/0.51 ( ( member_b @ X2 @ A2 )
% 0.21/0.51 => ( member_b @ ( F @ X2 ) @ ( image_b_b @ F @ A2 ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % imageI
% 0.21/0.51 thf(fact_79_rev__image__eqI,axiom,
% 0.21/0.51 ! [X2: b,A2: set_b,B2: b,F: b > b] :
% 0.21/0.51 ( ( member_b @ X2 @ A2 )
% 0.21/0.51 => ( ( B2
% 0.21/0.51 = ( F @ X2 ) )
% 0.21/0.51 => ( member_b @ B2 @ ( image_b_b @ F @ A2 ) ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % rev_image_eqI
% 0.21/0.51 thf(fact_80_Compr__image__eq,axiom,
% 0.21/0.51 ! [F: b > b,A2: set_b,P: b > $o] :
% 0.21/0.51 ( ( collect_b
% 0.21/0.51 @ ^ [X4: b] :
% 0.21/0.51 ( ( member_b @ X4 @ ( image_b_b @ F @ A2 ) )
% 0.21/0.51 & ( P @ X4 ) ) )
% 0.21/0.51 = ( image_b_b @ F
% 0.21/0.51 @ ( collect_b
% 0.21/0.51 @ ^ [X4: b] :
% 0.21/0.51 ( ( member_b @ X4 @ A2 )
% 0.21/0.51 & ( P @ ( F @ X4 ) ) ) ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % Compr_image_eq
% 0.21/0.51 thf(fact_81_inv__into__into,axiom,
% 0.21/0.51 ! [X2: b,F: b > b,A2: set_b] :
% 0.21/0.51 ( ( member_b @ X2 @ ( image_b_b @ F @ A2 ) )
% 0.21/0.51 => ( member_b @ ( hilbert_inv_into_b_b @ A2 @ F @ X2 ) @ A2 ) ) ).
% 0.21/0.51
% 0.21/0.51 % inv_into_into
% 0.21/0.51 thf(fact_82_image__constant,axiom,
% 0.21/0.51 ! [X2: b,A2: set_b,C: b] :
% 0.21/0.51 ( ( member_b @ X2 @ A2 )
% 0.21/0.51 => ( ( image_b_b
% 0.21/0.51 @ ^ [X4: b] : C
% 0.21/0.51 @ A2 )
% 0.21/0.51 = ( insert_b @ C @ bot_bot_set_b ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % image_constant
% 0.21/0.51 thf(fact_83_image__constant__conv,axiom,
% 0.21/0.51 ! [A2: set_b,C: b] :
% 0.21/0.51 ( ( ( A2 = bot_bot_set_b )
% 0.21/0.51 => ( ( image_b_b
% 0.21/0.51 @ ^ [X4: b] : C
% 0.21/0.51 @ A2 )
% 0.21/0.51 = bot_bot_set_b ) )
% 0.21/0.51 & ( ( A2 != bot_bot_set_b )
% 0.21/0.51 => ( ( image_b_b
% 0.21/0.51 @ ^ [X4: b] : C
% 0.21/0.51 @ A2 )
% 0.21/0.51 = ( insert_b @ C @ bot_bot_set_b ) ) ) ) ).
% 0.21/0.51
% 0.21/0.51 % image_constant_conv
% 0.21/0.51 thf(fact_84_bij__betw__empty2,axiom,
% 0.21/0.51 ! [F: b > b,A2: set_b] :
% 0.21/0.52 ( ( bij_betw_b_b @ F @ A2 @ bot_bot_set_b )
% 0.21/0.52 => ( A2 = bot_bot_set_b ) ) ).
% 0.21/0.52
% 0.21/0.52 % bij_betw_empty2
% 0.21/0.52 thf(fact_85_bij__betw__empty1,axiom,
% 0.21/0.52 ! [F: b > b,A2: set_b] :
% 0.21/0.52 ( ( bij_betw_b_b @ F @ bot_bot_set_b @ A2 )
% 0.21/0.52 => ( A2 = bot_bot_set_b ) ) ).
% 0.21/0.52
% 0.21/0.52 % bij_betw_empty1
% 0.21/0.52 thf(fact_86_inj__singleton,axiom,
% 0.21/0.52 ! [A2: set_b] :
% 0.21/0.52 ( inj_on_b_set_b
% 0.21/0.52 @ ^ [X4: b] : ( insert_b @ X4 @ bot_bot_set_b )
% 0.21/0.52 @ A2 ) ).
% 0.21/0.52
% 0.21/0.52 % inj_singleton
% 0.21/0.52 thf(fact_87_inj__img__insertE,axiom,
% 0.21/0.52 ! [F: b > b,A2: set_b,X2: b,B: set_b] :
% 0.21/0.52 ( ( inj_on_b_b @ F @ A2 )
% 0.21/0.52 => ( ~ ( member_b @ X2 @ B )
% 0.21/0.52 => ( ( ( insert_b @ X2 @ B )
% 0.21/0.52 = ( image_b_b @ F @ A2 ) )
% 0.21/0.52 => ~ ! [X6: b,A5: set_b] :
% 0.21/0.52 ( ~ ( member_b @ X6 @ A5 )
% 0.21/0.52 => ( ( A2
% 0.21/0.52 = ( insert_b @ X6 @ A5 ) )
% 0.21/0.52 => ( ( X2
% 0.21/0.52 = ( F @ X6 ) )
% 0.21/0.52 => ( B
% 0.21/0.52 != ( image_b_b @ F @ A5 ) ) ) ) ) ) ) ) ).
% 0.21/0.52
% 0.21/0.52 % inj_img_insertE
% 0.21/0.52 thf(fact_88_inj__image__mem__iff,axiom,
% 0.21/0.52 ! [F: b > b,A: b,A2: set_b] :
% 0.21/0.52 ( ( inj_on_b_b @ F @ top_top_set_b )
% 0.21/0.52 => ( ( member_b @ ( F @ A ) @ ( image_b_b @ F @ A2 ) )
% 0.21/0.52 = ( member_b @ A @ A2 ) ) ) ).
% 0.21/0.52
% 0.21/0.52 % inj_image_mem_iff
% 0.21/0.52 thf(fact_89_inj__on__insert,axiom,
% 0.21/0.52 ! [F: b > b,A: b,A2: set_b] :
% 0.21/0.52 ( ( inj_on_b_b @ F @ ( insert_b @ A @ A2 ) )
% 0.21/0.52 = ( ( inj_on_b_b @ F @ A2 )
% 0.21/0.52 & ~ ( member_b @ ( F @ A ) @ ( image_b_b @ F @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) ) ) ) ) ).
% 0.21/0.52
% 0.21/0.52 % inj_on_insert
% 0.21/0.52 thf(fact_90_DiffI,axiom,
% 0.21/0.52 ! [C: b,A2: set_b,B: set_b] :
% 0.21/0.52 ( ( member_b @ C @ A2 )
% 0.21/0.52 => ( ~ ( member_b @ C @ B )
% 0.21/0.52 => ( member_b @ C @ ( minus_minus_set_b @ A2 @ B ) ) ) ) ).
% 0.21/0.52
% 0.21/0.52 % DiffI
% 0.21/0.52 thf(fact_91_Diff__iff,axiom,
% 0.21/0.52 ! [C: b,A2: set_b,B: set_b] :
% 0.21/0.52 ( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B ) )
% 0.21/0.52 = ( ( member_b @ C @ A2 )
% 0.21/0.52 & ~ ( member_b @ C @ B ) ) ) ).
% 0.21/0.52
% 0.21/0.52 % Diff_iff
% 0.21/0.52 thf(fact_92_Diff__empty,axiom,
% 0.21/0.52 ! [A2: set_b] :
% 0.21/0.52 ( ( minus_minus_set_b @ A2 @ bot_bot_set_b )
% 0.21/0.52 = A2 ) ).
% 0.21/0.52
% 0.21/0.52 % Diff_empty
% 0.21/0.52 thf(fact_93_empty__Diff,axiom,
% 0.21/0.52 ! [A2: set_b] :
% 0.21/0.52 ( ( minus_minus_set_b @ bot_bot_set_b @ A2 )
% 0.21/0.52 = bot_bot_set_b ) ).
% 0.21/0.52
% 0.21/0.52 % empty_Diff
% 0.21/0.52 thf(fact_94_Diff__cancel,axiom,
% 0.21/0.52 ! [A2: set_b] :
% 0.21/0.52 ( ( minus_minus_set_b @ A2 @ A2 )
% 0.21/0.52 = bot_bot_set_b ) ).
% 0.21/0.52
% 0.21/0.52 % Diff_cancel
% 0.21/0.52 thf(fact_95_insert__Diff1,axiom,
% 0.21/0.52 ! [X2: b,B: set_b,A2: set_b] :
% 0.21/0.52 ( ( member_b @ X2 @ B )
% 0.21/0.52 => ( ( minus_minus_set_b @ ( insert_b @ X2 @ A2 ) @ B )
% 0.21/0.52 = ( minus_minus_set_b @ A2 @ B ) ) ) ).
% 0.21/0.52
% 0.21/0.52 % insert_Diff1
% 0.21/0.52 thf(fact_96_Diff__insert0,axiom,
% 0.21/0.52 ! [X2: b,A2: set_b,B: set_b] :
% 0.21/0.52 ( ~ ( member_b @ X2 @ A2 )
% 0.21/0.52 => ( ( minus_minus_set_b @ A2 @ ( insert_b @ X2 @ B ) )
% 0.21/0.52 = ( minus_minus_set_b @ A2 @ B ) ) ) ).
% 0.21/0.52
% 0.21/0.52 % Diff_insert0
% 0.21/0.52 thf(fact_97_insert__Diff__single,axiom,
% 0.21/0.52 ! [A: b,A2: set_b] :
% 0.21/0.52 ( ( insert_b @ A @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) )
% 0.21/0.52 = ( insert_b @ A @ A2 ) ) ).
% 0.21/0.52
% 0.21/0.52 % insert_Diff_single
% 0.21/0.52 thf(fact_98_Diff__UNIV,axiom,
% 0.21/0.52 ! [A2: set_b] :
% 0.21/0.52 ( ( minus_minus_set_b @ A2 @ top_top_set_b )
% 0.21/0.52 = bot_bot_set_b ) ).
% 0.21/0.52
% 0.21/0.52 % Diff_UNIV
% 0.21/0.52 thf(fact_99_Diff__insert__absorb,axiom,
% 0.21/0.52 ! [X2: b,A2: set_b] :
% 0.21/0.52 ( ~ ( member_b @ X2 @ A2 )
% 0.21/0.52 => ( ( minus_minus_set_b @ ( insert_b @ X2 @ A2 ) @ ( insert_b @ X2 @ bot_bot_set_b ) )
% 0.21/0.52 = A2 ) ) ).
% 0.21/0.52
% 0.21/0.52 % Diff_insert_absorb
% 0.21/0.52 thf(fact_100_Diff__insert2,axiom,
% 0.21/0.52 ! [A2: set_b,A: b,B: set_b] :
% 0.21/0.52 ( ( minus_minus_set_b @ A2 @ ( insert_b @ A @ B ) )
% 0.21/0.52 = ( minus_minus_set_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) @ B ) ) ).
% 0.21/0.52
% 0.21/0.52 % Diff_insert2
% 0.21/0.52 thf(fact_101_insert__Diff,axiom,
% 0.21/0.52 ! [A: b,A2: set_b] :
% 0.21/0.52 ( ( member_b @ A @ A2 )
% 0.21/0.52 => ( ( insert_b @ A @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) )
% 0.21/0.52 = A2 ) ) ).
% 0.21/0.52
% 0.21/0.52 % insert_Diff
% 0.21/0.52 thf(fact_102_Diff__insert,axiom,
% 0.21/0.52 ! [A2: set_b,A: b,B: set_b] :
% 0.21/0.52 ( ( minus_minus_set_b @ A2 @ ( insert_b @ A @ B ) )
% 0.21/0.52 = ( minus_minus_set_b @ ( minus_minus_set_b @ A2 @ B ) @ ( insert_b @ A @ bot_bot_set_b ) ) ) ).
% 0.21/0.52
% 0.21/0.52 % Diff_insert
% 0.21/0.52 thf(fact_103_in__image__insert__iff,axiom,
% 0.21/0.54 ! [B: set_set_b,X2: b,A2: set_b] :
% 0.21/0.54 ( ! [C3: set_b] :
% 0.21/0.54 ( ( member_set_b @ C3 @ B )
% 0.21/0.54 => ~ ( member_b @ X2 @ C3 ) )
% 0.21/0.54 => ( ( member_set_b @ A2 @ ( image_set_b_set_b @ ( insert_b @ X2 ) @ B ) )
% 0.21/0.54 = ( ( member_b @ X2 @ A2 )
% 0.21/0.54 & ( member_set_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ X2 @ bot_bot_set_b ) ) @ B ) ) ) ) ).
% 0.21/0.54
% 0.21/0.54 % in_image_insert_iff
% 0.21/0.54 thf(fact_104_DiffE,axiom,
% 0.21/0.54 ! [C: b,A2: set_b,B: set_b] :
% 0.21/0.54 ( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B ) )
% 0.21/0.54 => ~ ( ( member_b @ C @ A2 )
% 0.21/0.54 => ( member_b @ C @ B ) ) ) ).
% 0.21/0.54
% 0.21/0.54 % DiffE
% 0.21/0.54 thf(fact_105_DiffD1,axiom,
% 0.21/0.54 ! [C: b,A2: set_b,B: set_b] :
% 0.21/0.54 ( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B ) )
% 0.21/0.54 => ( member_b @ C @ A2 ) ) ).
% 0.21/0.54
% 0.21/0.54 % DiffD1
% 0.21/0.54 thf(fact_106_DiffD2,axiom,
% 0.21/0.54 ! [C: b,A2: set_b,B: set_b] :
% 0.21/0.54 ( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B ) )
% 0.21/0.54 => ~ ( member_b @ C @ B ) ) ).
% 0.21/0.54
% 0.21/0.54 % DiffD2
% 0.21/0.54 thf(fact_107_set__diff__eq,axiom,
% 0.21/0.54 ( minus_minus_set_b
% 0.21/0.54 = ( ^ [A3: set_b,B4: set_b] :
% 0.21/0.54 ( collect_b
% 0.21/0.54 @ ^ [X4: b] :
% 0.21/0.54 ( ( member_b @ X4 @ A3 )
% 0.21/0.54 & ~ ( member_b @ X4 @ B4 ) ) ) ) ) ).
% 0.21/0.54
% 0.21/0.54 % set_diff_eq
% 0.21/0.54 thf(fact_108_insert__Diff__if,axiom,
% 0.21/0.54 ! [X2: b,B: set_b,A2: set_b] :
% 0.21/0.54 ( ( ( member_b @ X2 @ B )
% 0.21/0.54 => ( ( minus_minus_set_b @ ( insert_b @ X2 @ A2 ) @ B )
% 0.21/0.54 = ( minus_minus_set_b @ A2 @ B ) ) )
% 0.21/0.54 & ( ~ ( member_b @ X2 @ B )
% 0.21/0.54 => ( ( minus_minus_set_b @ ( insert_b @ X2 @ A2 ) @ B )
% 0.21/0.54 = ( insert_b @ X2 @ ( minus_minus_set_b @ A2 @ B ) ) ) ) ) ).
% 0.21/0.54
% 0.21/0.54 % insert_Diff_if
% 0.21/0.54 thf(fact_109_remove__def,axiom,
% 0.21/0.54 ( remove_b
% 0.21/0.54 = ( ^ [X4: b,A3: set_b] : ( minus_minus_set_b @ A3 @ ( insert_b @ X4 @ bot_bot_set_b ) ) ) ) ).
% 0.21/0.54
% 0.21/0.54 % remove_def
% 0.21/0.54
% 0.21/0.54 % Conjectures (2)
% 0.21/0.54 thf(conj_0,hypothesis,
% 0.21/0.54 ! [X5: a] :
% 0.21/0.54 ( ( p @ X5 )
% 0.21/0.54 => ? [X_12: b] : ( q @ X5 @ X_12 ) ) ).
% 0.21/0.54
% 0.21/0.54 thf(conj_1,conjecture,
% 0.21/0.54 ! [X: a] :
% 0.21/0.54 ( ~ ( p @ X )
% 0.21/0.54 | ( q @ X @ ( hilbert_Eps_b @ ( q @ X ) ) ) ) ).
% 0.21/0.54
% 0.21/0.54 %------------------------------------------------------------------------------
% 0.21/0.54 ------- convert to smt2 : /export/starexec/sandbox2/tmp/tmp.NNuyy0VoPK/cvc5---1.0.5_31926.p...
% 0.21/0.54 (declare-sort $$unsorted 0)
% 0.21/0.54 (declare-sort tptp.set_set_b 0)
% 0.21/0.54 (declare-sort tptp.set_b 0)
% 0.21/0.54 (declare-sort tptp.b 0)
% 0.21/0.54 (declare-sort tptp.a 0)
% 0.21/0.54 (declare-fun tptp.bij_betw_b_b ((-> tptp.b tptp.b) tptp.set_b tptp.set_b) Bool)
% 0.21/0.54 (declare-fun tptp.inj_on_b_set_b ((-> tptp.b tptp.set_b) tptp.set_b) Bool)
% 0.21/0.54 (declare-fun tptp.inj_on_b_b ((-> tptp.b tptp.b) tptp.set_b) Bool)
% 0.21/0.54 (declare-fun tptp.minus_minus_set_b (tptp.set_b tptp.set_b) tptp.set_b)
% 0.21/0.54 (declare-fun tptp.hilbert_Eps_b ((-> tptp.b Bool)) tptp.b)
% 0.21/0.54 (declare-fun tptp.hilbert_inv_into_b_b (tptp.set_b (-> tptp.b tptp.b) tptp.b) tptp.b)
% 0.21/0.54 (declare-fun tptp.bot_bot_b_o (tptp.b) Bool)
% 0.21/0.54 (declare-fun tptp.bot_bot_set_b () tptp.set_b)
% 0.21/0.54 (declare-fun tptp.top_top_b_o (tptp.b) Bool)
% 0.21/0.54 (declare-fun tptp.top_top_set_b () tptp.set_b)
% 0.21/0.54 (declare-fun tptp.collect_b ((-> tptp.b Bool)) tptp.set_b)
% 0.21/0.54 (declare-fun tptp.image_set_b_set_b ((-> tptp.set_b tptp.set_b) tptp.set_set_b) tptp.set_set_b)
% 0.21/0.54 (declare-fun tptp.image_b_b ((-> tptp.b tptp.b) tptp.set_b) tptp.set_b)
% 0.21/0.54 (declare-fun tptp.insert_b (tptp.b tptp.set_b) tptp.set_b)
% 0.21/0.54 (declare-fun tptp.is_empty_b (tptp.set_b) Bool)
% 0.21/0.54 (declare-fun tptp.is_singleton_b (tptp.set_b) Bool)
% 0.21/0.54 (declare-fun tptp.remove_b (tptp.b tptp.set_b) tptp.set_b)
% 0.21/0.54 (declare-fun tptp.the_elem_b (tptp.set_b) tptp.b)
% 0.21/0.54 (declare-fun tptp.member_set_b (tptp.set_b tptp.set_set_b) Bool)
% 0.21/0.54 (declare-fun tptp.member_b (tptp.b tptp.set_b) Bool)
% 0.21/0.54 (declare-fun tptp.p (tptp.a) Bool)
% 0.21/0.54 (declare-fun tptp.q (tptp.a tptp.b) Bool)
% 0.21/0.54 (assert (forall ((P (-> tptp.b Bool)) (A tptp.b)) (=> (@ P A) (=> (forall ((X tptp.b)) (=> (@ P X) (= X A))) (= (@ tptp.hilbert_Eps_b P) A)))))
% 0.21/0.54 (assert (forall ((X2 tptp.b)) (= (@ tptp.hilbert_Eps_b (lambda ((Y tptp.b)) (= Y X2))) X2)))
% 0.21/0.54 (assert (forall ((X2 tptp.b)) (= (@ tptp.hilbert_Eps_b (@ (lambda ((Y2 tptp.b) (Z tptp.b)) (= Y2 Z)) X2)) X2)))
% 0.21/0.54 (assert (forall ((P (-> tptp.b Bool)) (A2 Bool)) (=> (= (@ P (@ tptp.hilbert_Eps_b P)) A2) (= (exists ((X3 tptp.b)) (@ P X3)) A2))))
% 0.21/0.54 (assert (= (lambda ((P2 (-> tptp.b Bool))) (forall ((X3 tptp.b)) (@ P2 X3))) (lambda ((P3 (-> tptp.b Bool))) (@ P3 (@ tptp.hilbert_Eps_b (lambda ((X4 tptp.b)) (not (@ P3 X4))))))))
% 0.21/0.54 (assert (forall ((P (-> tptp.b Bool)) (A tptp.b) (Q (-> tptp.b Bool))) (=> (@ P A) (=> (forall ((X tptp.b)) (=> (@ P X) (@ Q X))) (@ Q (@ tptp.hilbert_Eps_b P))))))
% 0.21/0.54 (assert (forall ((P (-> tptp.b Bool)) (A2 Bool)) (=> (= (@ P (@ tptp.hilbert_Eps_b (lambda ((X4 tptp.b)) (not (@ P X4))))) A2) (= (forall ((X3 tptp.b)) (@ P X3)) A2))))
% 0.21/0.54 (assert (forall ((B tptp.b) (A2 tptp.b) (P (-> tptp.b Bool))) (let ((_let_1 (@ tptp.hilbert_Eps_b P))) (=> (= B A2) (= (= _let_1 A2) (= _let_1 B))))))
% 0.21/0.54 (assert (forall ((P (-> tptp.b Bool))) (=> (exists ((X_1 tptp.b)) (@ P X_1)) (@ P (@ tptp.hilbert_Eps_b P)))))
% 0.21/0.54 (assert (forall ((P (-> tptp.b Bool)) (Q (-> tptp.b Bool))) (=> (exists ((X_1 tptp.b)) (@ P X_1)) (=> (forall ((X tptp.b)) (=> (@ P X) (@ Q X))) (@ Q (@ tptp.hilbert_Eps_b P))))))
% 0.21/0.54 (assert (forall ((A2 tptp.set_b) (P (-> tptp.b Bool)) (Q (-> tptp.b Bool))) (=> (exists ((X5 tptp.b)) (and (@ (@ tptp.member_b X5) A2) (@ P X5))) (=> (forall ((X tptp.b)) (=> (and (@ (@ tptp.member_b X) A2) (@ P X)) (@ Q X))) (@ Q (@ tptp.hilbert_Eps_b (lambda ((X4 tptp.b)) (and (@ (@ tptp.member_b X4) A2) (@ P X4)))))))))
% 0.21/0.54 (assert (forall ((P (-> tptp.b Bool))) (= (@ P (@ tptp.hilbert_Eps_b P)) (exists ((X3 tptp.b)) (@ P X3)))))
% 0.21/0.54 (assert (forall ((P (-> tptp.b Bool)) (A tptp.b) (B2 tptp.b)) (=> (= (@ tptp.hilbert_Eps_b P) A) (=> (@ P B2) (@ P A)))))
% 0.21/0.54 (assert (forall ((P4 (-> tptp.b Bool)) (X5 tptp.b)) (=> (@ P4 X5) (@ P4 (@ tptp.hilbert_Eps_b P4)))))
% 0.21/0.54 (assert (forall ((P (-> tptp.b Bool)) (Q (-> tptp.b Bool))) (=> (forall ((X tptp.b)) (= (@ P X) (@ Q X))) (= (@ tptp.hilbert_Eps_b P) (@ tptp.hilbert_Eps_b Q)))))
% 0.21/0.54 (assert (forall ((P (-> tptp.b Bool)) (X2 tptp.b)) (=> (@ P X2) (@ P (@ tptp.hilbert_Eps_b P)))))
% 0.21/0.54 (assert (forall ((X2 tptp.b) (P (-> tptp.b Bool)) (P5 (-> tptp.b Bool))) (=> (= X2 (@ tptp.hilbert_Eps_b (lambda ((X4 tptp.b)) (not (@ P X4))))) (=> (forall ((X tptp.b)) (= (@ P X) (@ P5 X))) (= (forall ((X3 tptp.b)) (@ P5 X3)) (@ P X2))))))
% 0.21/0.54 (assert (forall ((X2 tptp.b) (P (-> tptp.b Bool))) (=> (= X2 (@ tptp.hilbert_Eps_b (lambda ((X4 tptp.b)) (not (@ P X4))))) (= (forall ((X3 tptp.b)) (@ P X3)) (@ P X2)))))
% 0.21/0.54 (assert (forall ((P (-> tptp.b Bool)) (A tptp.b)) (=> (exists ((X5 tptp.b)) (and (@ P X5) (forall ((Y3 tptp.b)) (=> (@ P Y3) (= Y3 X5))))) (=> (@ P A) (= (@ tptp.hilbert_Eps_b P) A)))))
% 0.21/0.54 (assert (forall ((X2 tptp.b) (P (-> tptp.b Bool)) (P5 (-> tptp.b Bool))) (=> (= X2 (@ tptp.hilbert_Eps_b P)) (=> (forall ((X tptp.b)) (= (@ P X) (@ P5 X))) (= (exists ((X3 tptp.b)) (@ P5 X3)) (@ P X2))))))
% 0.21/0.54 (assert (forall ((X2 tptp.b) (P (-> tptp.b Bool))) (=> (= X2 (@ tptp.hilbert_Eps_b P)) (= (exists ((X3 tptp.b)) (@ P X3)) (@ P X2)))))
% 0.21/0.54 (assert (forall ((P (-> tptp.b Bool)) (X2 tptp.b) (Y4 tptp.b)) (let ((_let_1 (@ tptp.hilbert_Eps_b P))) (=> (@ P X2) (=> (not (@ P Y4)) (=> (= _let_1 Y4) (= _let_1 X2)))))))
% 0.21/0.54 (assert (forall ((A2 tptp.set_b)) (= (@ (@ tptp.member_b (@ tptp.hilbert_Eps_b (lambda ((X4 tptp.b)) (@ (@ tptp.member_b X4) A2)))) A2) (not (= A2 tptp.bot_bot_set_b)))))
% 0.21/0.54 (assert (forall ((P (-> tptp.b Bool)) (C tptp.b)) (=> (exists ((X_1 tptp.b)) (@ P X_1)) (=> (= C (@ tptp.hilbert_Eps_b P)) (@ P C)))))
% 0.21/0.54 (assert (forall ((P (-> tptp.b Bool))) (= (= tptp.bot_bot_set_b (@ tptp.collect_b P)) (forall ((X4 tptp.b)) (not (@ P X4))))))
% 0.21/0.54 (assert (forall ((P (-> tptp.b Bool))) (= (= (@ tptp.collect_b P) tptp.bot_bot_set_b) (forall ((X4 tptp.b)) (not (@ P X4))))))
% 0.21/0.54 (assert (forall ((A2 tptp.set_b)) (= (forall ((X4 tptp.b)) (not (@ (@ tptp.member_b X4) A2))) (= A2 tptp.bot_bot_set_b))))
% 0.21/0.54 (assert (forall ((C tptp.b)) (not (@ (@ tptp.member_b C) tptp.bot_bot_set_b))))
% 0.21/0.54 (assert (forall ((A tptp.b) (P (-> tptp.b Bool))) (= (@ (@ tptp.member_b A) (@ tptp.collect_b P)) (@ P A))))
% 0.21/0.54 (assert (forall ((A2 tptp.set_b)) (= (@ tptp.collect_b (lambda ((X4 tptp.b)) (@ (@ tptp.member_b X4) A2))) A2)))
% 0.21/0.54 (assert (= tptp.bot_bot_set_b (@ tptp.collect_b tptp.bot_bot_b_o)))
% 0.21/0.54 (assert (forall ((A tptp.b)) (not (@ (@ tptp.member_b A) tptp.bot_bot_set_b))))
% 0.21/0.54 (assert (forall ((A2 tptp.set_b) (A tptp.b)) (=> (= A2 tptp.bot_bot_set_b) (not (@ (@ tptp.member_b A) A2)))))
% 0.21/0.54 (assert (forall ((A2 tptp.set_b)) (=> (forall ((Y3 tptp.b)) (not (@ (@ tptp.member_b Y3) A2))) (= A2 tptp.bot_bot_set_b))))
% 0.21/0.54 (assert (forall ((A2 tptp.set_b)) (= (exists ((X4 tptp.b)) (@ (@ tptp.member_b X4) A2)) (not (= A2 tptp.bot_bot_set_b)))))
% 0.21/0.54 (assert (= tptp.bot_bot_set_b (@ tptp.collect_b (lambda ((X4 tptp.b)) false))))
% 0.21/0.54 (assert (= tptp.is_empty_b (lambda ((A3 tptp.set_b)) (= A3 tptp.bot_bot_set_b))))
% 0.21/0.54 (assert (forall ((P (-> tptp.b Bool))) (= (= (@ tptp.collect_b P) tptp.bot_bot_set_b) (= P tptp.bot_bot_b_o))))
% 0.21/0.54 (assert (= tptp.bot_bot_b_o (lambda ((X4 tptp.b)) (@ (@ tptp.member_b X4) tptp.bot_bot_set_b))))
% 0.21/0.54 (assert (forall ((A2 tptp.set_b)) (=> (not (= A2 tptp.bot_bot_set_b)) (=> (forall ((X tptp.b) (Y3 tptp.b)) (=> (@ (@ tptp.member_b X) A2) (=> (@ (@ tptp.member_b Y3) A2) (= X Y3)))) (@ tptp.is_singleton_b A2)))))
% 0.21/0.54 (assert (forall ((X2 tptp.b)) (@ (@ tptp.member_b X2) tptp.top_top_set_b)))
% 0.21/0.54 (assert (forall ((P Bool)) (and (=> P (= (@ tptp.collect_b (lambda ((S tptp.b)) P)) tptp.top_top_set_b)) (=> (not P) (= (@ tptp.collect_b (lambda ((S tptp.b)) P)) tptp.bot_bot_set_b)))))
% 0.21/0.54 (assert (forall ((A2 tptp.set_b)) (=> (forall ((X tptp.b)) (@ (@ tptp.member_b X) A2)) (= tptp.top_top_set_b A2))))
% 0.21/0.54 (assert (exists ((X tptp.b)) (@ (@ tptp.member_b X) tptp.top_top_set_b)))
% 0.21/0.54 (assert (not (= tptp.bot_bot_set_b tptp.top_top_set_b)))
% 0.21/0.54 (assert (forall ((X2 tptp.b)) (@ (@ tptp.member_b X2) tptp.top_top_set_b)))
% 0.21/0.54 (assert (forall ((X2 tptp.b)) (@ tptp.is_singleton_b (@ (@ tptp.insert_b X2) tptp.bot_bot_set_b))))
% 0.21/0.54 (assert (forall ((A tptp.b) (B2 tptp.b) (A2 tptp.set_b)) (let ((_let_1 (@ tptp.member_b A))) (= (@ _let_1 (@ (@ tptp.insert_b B2) A2)) (or (= A B2) (@ _let_1 A2))))))
% 0.21/0.54 (assert (forall ((A tptp.b) (B tptp.set_b) (B2 tptp.b)) (let ((_let_1 (@ tptp.member_b A))) (=> (=> (not (@ _let_1 B)) (= A B2)) (@ _let_1 (@ (@ tptp.insert_b B2) B))))))
% 0.21/0.54 (assert (forall ((A tptp.b)) (@ (@ tptp.member_b A) (@ (@ tptp.insert_b A) tptp.bot_bot_set_b))))
% 0.21/0.54 (assert (forall ((A tptp.b)) (= (@ tptp.collect_b (lambda ((X4 tptp.b)) (= X4 A))) (@ (@ tptp.insert_b A) tptp.bot_bot_set_b))))
% 0.21/0.54 (assert (forall ((A tptp.b)) (= (@ tptp.collect_b (@ (lambda ((Y2 tptp.b) (Z tptp.b)) (= Y2 Z)) A)) (@ (@ tptp.insert_b A) tptp.bot_bot_set_b))))
% 0.21/0.54 (assert (forall ((A tptp.b) (A2 tptp.set_b)) (=> (@ (@ tptp.member_b A) A2) (exists ((B3 tptp.set_b)) (and (= A2 (@ (@ tptp.insert_b A) B3)) (not (@ (@ tptp.member_b A) B3)))))))
% 0.21/0.54 (assert (forall ((A tptp.b) (A2 tptp.set_b) (B2 tptp.b) (B tptp.set_b)) (let ((_let_1 (= A B2))) (=> (not (@ (@ tptp.member_b A) A2)) (=> (not (@ (@ tptp.member_b B2) B)) (= (= (@ (@ tptp.insert_b A) A2) (@ (@ tptp.insert_b B2) B)) (and (=> _let_1 (= A2 B)) (=> (not _let_1) (exists ((C2 tptp.set_b)) (and (= A2 (@ (@ tptp.insert_b B2) C2)) (not (@ (@ tptp.member_b B2) C2)) (= B (@ (@ tptp.insert_b A) C2)) (not (@ (@ tptp.member_b A) C2))))))))))))
% 0.21/0.54 (assert (forall ((A tptp.b) (A2 tptp.set_b)) (=> (@ (@ tptp.member_b A) A2) (= (@ (@ tptp.insert_b A) A2) A2))))
% 0.21/0.54 (assert (forall ((X2 tptp.b) (A2 tptp.set_b) (B tptp.set_b)) (let ((_let_1 (@ tptp.insert_b X2))) (let ((_let_2 (@ tptp.member_b X2))) (=> (not (@ _let_2 A2)) (=> (not (@ _let_2 B)) (= (= (@ _let_1 A2) (@ _let_1 B)) (= A2 B))))))))
% 0.21/0.54 (assert (= tptp.insert_b (lambda ((A4 tptp.b) (B4 tptp.set_b)) (@ tptp.collect_b (lambda ((X4 tptp.b)) (or (= X4 A4) (@ (@ tptp.member_b X4) B4)))))))
% 0.21/0.54 (assert (forall ((X2 tptp.b) (A2 tptp.set_b)) (=> (@ (@ tptp.member_b X2) A2) (not (forall ((B3 tptp.set_b)) (=> (= A2 (@ (@ tptp.insert_b X2) B3)) (@ (@ tptp.member_b X2) B3)))))))
% 0.21/0.54 (assert (forall ((A tptp.b) (B tptp.set_b) (B2 tptp.b)) (let ((_let_1 (@ tptp.member_b A))) (=> (@ _let_1 B) (@ _let_1 (@ (@ tptp.insert_b B2) B))))))
% 0.21/0.54 (assert (forall ((A tptp.b) (B tptp.set_b)) (@ (@ tptp.member_b A) (@ (@ tptp.insert_b A) B))))
% 0.21/0.54 (assert (forall ((A tptp.b) (B2 tptp.b) (A2 tptp.set_b)) (let ((_let_1 (@ tptp.member_b A))) (=> (@ _let_1 (@ (@ tptp.insert_b B2) A2)) (=> (not (= A B2)) (@ _let_1 A2))))))
% 0.21/0.54 (assert (= tptp.top_top_b_o (lambda ((X4 tptp.b)) (@ (@ tptp.member_b X4) tptp.top_top_set_b))))
% 0.21/0.54 (assert (forall ((A tptp.b) (B2 tptp.b)) (=> (= (@ (@ tptp.insert_b A) tptp.bot_bot_set_b) (@ (@ tptp.insert_b B2) tptp.bot_bot_set_b)) (= A B2))))
% 0.21/0.54 (assert (forall ((A tptp.b) (A2 tptp.set_b)) (not (= (@ (@ tptp.insert_b A) A2) tptp.bot_bot_set_b))))
% 0.21/0.54 (assert (forall ((A tptp.b) (B2 tptp.b) (C tptp.b) (D tptp.b)) (= (= (@ (@ tptp.insert_b A) (@ (@ tptp.insert_b B2) tptp.bot_bot_set_b)) (@ (@ tptp.insert_b C) (@ (@ tptp.insert_b D) tptp.bot_bot_set_b))) (or (and (= A C) (= B2 D)) (and (= A D) (= B2 C))))))
% 0.21/0.54 (assert (forall ((B2 tptp.b) (A tptp.b)) (= (@ (@ tptp.member_b B2) (@ (@ tptp.insert_b A) tptp.bot_bot_set_b)) (= B2 A))))
% 0.21/0.54 (assert (forall ((B2 tptp.b) (A tptp.b)) (=> (@ (@ tptp.member_b B2) (@ (@ tptp.insert_b A) tptp.bot_bot_set_b)) (= B2 A))))
% 0.21/0.54 (assert (forall ((P (-> tptp.b Bool)) (A tptp.b)) (let ((_let_1 (@ P A))) (and (=> _let_1 (= (@ tptp.collect_b (lambda ((X4 tptp.b)) (and (= X4 A) (@ P X4)))) (@ (@ tptp.insert_b A) tptp.bot_bot_set_b))) (=> (not _let_1) (= (@ tptp.collect_b (lambda ((X4 tptp.b)) (and (= X4 A) (@ P X4)))) tptp.bot_bot_set_b))))))
% 0.21/0.54 (assert (forall ((P (-> tptp.b Bool)) (A tptp.b)) (let ((_let_1 (@ P A))) (and (=> _let_1 (= (@ tptp.collect_b (lambda ((X4 tptp.b)) (and (= A X4) (@ P X4)))) (@ (@ tptp.insert_b A) tptp.bot_bot_set_b))) (=> (not _let_1) (= (@ tptp.collect_b (lambda ((X4 tptp.b)) (and (= A X4) (@ P X4)))) tptp.bot_bot_set_b))))))
% 0.21/0.54 (assert (= tptp.is_singleton_b (lambda ((A3 tptp.set_b)) (exists ((X4 tptp.b)) (= A3 (@ (@ tptp.insert_b X4) tptp.bot_bot_set_b))))))
% 0.21/0.54 (assert (forall ((A2 tptp.set_b)) (=> (@ tptp.is_singleton_b A2) (not (forall ((X tptp.b)) (not (= A2 (@ (@ tptp.insert_b X) tptp.bot_bot_set_b))))))))
% 0.21/0.54 (assert (= tptp.is_singleton_b (lambda ((A3 tptp.set_b)) (= A3 (@ (@ tptp.insert_b (@ tptp.the_elem_b A3)) tptp.bot_bot_set_b)))))
% 0.21/0.54 (assert (forall ((X2 tptp.b)) (= (@ tptp.the_elem_b (@ (@ tptp.insert_b X2) tptp.bot_bot_set_b)) X2)))
% 0.21/0.54 (assert (forall ((B2 tptp.b) (F (-> tptp.b tptp.b)) (X2 tptp.b) (A2 tptp.set_b)) (=> (= B2 (@ F X2)) (=> (@ (@ tptp.member_b X2) A2) (@ (@ tptp.member_b B2) (@ (@ tptp.image_b_b F) A2))))))
% 0.21/0.54 (assert (forall ((F (-> tptp.b tptp.b)) (A2 tptp.set_b)) (= (= (@ (@ tptp.image_b_b F) A2) tptp.bot_bot_set_b) (= A2 tptp.bot_bot_set_b))))
% 0.21/0.54 (assert (forall ((F (-> tptp.b tptp.b)) (A2 tptp.set_b)) (= (= tptp.bot_bot_set_b (@ (@ tptp.image_b_b F) A2)) (= A2 tptp.bot_bot_set_b))))
% 0.21/0.54 (assert (forall ((F (-> tptp.b tptp.b))) (= (@ (@ tptp.image_b_b F) tptp.bot_bot_set_b) tptp.bot_bot_set_b)))
% 0.21/0.54 (assert (forall ((B2 tptp.b) (F (-> tptp.b tptp.b)) (A2 tptp.set_b)) (=> (@ (@ tptp.member_b B2) (@ (@ tptp.image_b_b F) A2)) (not (forall ((X tptp.b)) (=> (= B2 (@ F X)) (not (@ (@ tptp.member_b X) A2))))))))
% 0.21/0.54 (assert (forall ((X2 tptp.b) (A2 tptp.set_b) (F (-> tptp.b tptp.b))) (=> (@ (@ tptp.member_b X2) A2) (@ (@ tptp.member_b (@ F X2)) (@ (@ tptp.image_b_b F) A2)))))
% 0.21/0.54 (assert (forall ((X2 tptp.b) (A2 tptp.set_b) (B2 tptp.b) (F (-> tptp.b tptp.b))) (=> (@ (@ tptp.member_b X2) A2) (=> (= B2 (@ F X2)) (@ (@ tptp.member_b B2) (@ (@ tptp.image_b_b F) A2))))))
% 0.21/0.54 (assert (forall ((F (-> tptp.b tptp.b)) (A2 tptp.set_b) (P (-> tptp.b Bool))) (= (@ tptp.collect_b (lambda ((X4 tptp.b)) (and (@ (@ tptp.member_b X4) (@ (@ tptp.image_b_b F) A2)) (@ P X4)))) (@ (@ tptp.image_b_b F) (@ tptp.collect_b (lambda ((X4 tptp.b)) (and (@ (@ tptp.member_b X4) A2) (@ P (@ F X4)))))))))
% 0.21/0.54 (assert (forall ((X2 tptp.b) (F (-> tptp.b tptp.b)) (A2 tptp.set_b)) (=> (@ (@ tptp.member_b X2) (@ (@ tptp.image_b_b F) A2)) (@ (@ tptp.member_b (@ (@ (@ tptp.hilbert_inv_into_b_b A2) F) X2)) A2))))
% 0.21/0.54 (assert (forall ((X2 tptp.b) (A2 tptp.set_b) (C tptp.b)) (=> (@ (@ tptp.member_b X2) A2) (= (@ (@ tptp.image_b_b (lambda ((X4 tptp.b)) C)) A2) (@ (@ tptp.insert_b C) tptp.bot_bot_set_b)))))
% 0.21/0.54 (assert (forall ((A2 tptp.set_b) (C tptp.b)) (let ((_let_1 (= A2 tptp.bot_bot_set_b))) (and (=> _let_1 (= (@ (@ tptp.image_b_b (lambda ((X4 tptp.b)) C)) A2) tptp.bot_bot_set_b)) (=> (not _let_1) (= (@ (@ tptp.image_b_b (lambda ((X4 tptp.b)) C)) A2) (@ (@ tptp.insert_b C) tptp.bot_bot_set_b)))))))
% 0.21/0.54 (assert (forall ((F (-> tptp.b tptp.b)) (A2 tptp.set_b)) (=> (@ (@ (@ tptp.bij_betw_b_b F) A2) tptp.bot_bot_set_b) (= A2 tptp.bot_bot_set_b))))
% 0.21/0.54 (assert (forall ((F (-> tptp.b tptp.b)) (A2 tptp.set_b)) (=> (@ (@ (@ tptp.bij_betw_b_b F) tptp.bot_bot_set_b) A2) (= A2 tptp.bot_bot_set_b))))
% 0.21/0.54 (assert (forall ((A2 tptp.set_b)) (@ (@ tptp.inj_on_b_set_b (lambda ((X4 tptp.b)) (@ (@ tptp.insert_b X4) tptp.bot_bot_set_b))) A2)))
% 0.21/0.54 (assert (forall ((F (-> tptp.b tptp.b)) (A2 tptp.set_b) (X2 tptp.b) (B tptp.set_b)) (=> (@ (@ tptp.inj_on_b_b F) A2) (=> (not (@ (@ tptp.member_b X2) B)) (=> (= (@ (@ tptp.insert_b X2) B) (@ (@ tptp.image_b_b F) A2)) (not (forall ((X6 tptp.b) (A5 tptp.set_b)) (=> (not (@ (@ tptp.member_b X6) A5)) (=> (= A2 (@ (@ tptp.insert_b X6) A5)) (=> (= X2 (@ F X6)) (not (= B (@ (@ tptp.image_b_b F) A5)))))))))))))
% 0.21/0.54 (assert (forall ((F (-> tptp.b tptp.b)) (A tptp.b) (A2 tptp.set_b)) (=> (@ (@ tptp.inj_on_b_b F) tptp.top_top_set_b) (= (@ (@ tptp.member_b (@ F A)) (@ (@ tptp.image_b_b F) A2)) (@ (@ tptp.member_b A) A2)))))
% 0.21/0.54 (assert (forall ((F (-> tptp.b tptp.b)) (A tptp.b) (A2 tptp.set_b)) (let ((_let_1 (@ tptp.insert_b A))) (let ((_let_2 (@ tptp.inj_on_b_b F))) (= (@ _let_2 (@ _let_1 A2)) (and (@ _let_2 A2) (not (@ (@ tptp.member_b (@ F A)) (@ (@ tptp.image_b_b F) (@ (@ tptp.minus_minus_set_b A2) (@ _let_1 tptp.bot_bot_set_b)))))))))))
% 0.21/0.54 (assert (forall ((C tptp.b) (A2 tptp.set_b) (B tptp.set_b)) (let ((_let_1 (@ tptp.member_b C))) (=> (@ _let_1 A2) (=> (not (@ _let_1 B)) (@ _let_1 (@ (@ tptp.minus_minus_set_b A2) B)))))))
% 0.21/0.54 (assert (forall ((C tptp.b) (A2 tptp.set_b) (B tptp.set_b)) (let ((_let_1 (@ tptp.member_b C))) (= (@ _let_1 (@ (@ tptp.minus_minus_set_b A2) B)) (and (@ _let_1 A2) (not (@ _let_1 B)))))))
% 0.21/0.54 (assert (forall ((A2 tptp.set_b)) (= (@ (@ tptp.minus_minus_set_b A2) tptp.bot_bot_set_b) A2)))
% 0.21/0.54 (assert (forall ((A2 tptp.set_b)) (= (@ (@ tptp.minus_minus_set_b tptp.bot_bot_set_b) A2) tptp.bot_bot_set_b)))
% 0.21/0.54 (assert (forall ((A2 tptp.set_b)) (= (@ (@ tptp.minus_minus_set_b A2) A2) tptp.bot_bot_set_b)))
% 0.21/0.54 (assert (forall ((X2 tptp.b) (B tptp.set_b) (A2 tptp.set_b)) (=> (@ (@ tptp.member_b X2) B) (= (@ (@ tptp.minus_minus_set_b (@ (@ tptp.insert_b X2) A2)) B) (@ (@ tptp.minus_minus_set_b A2) B)))))
% 0.21/0.54 (assert (forall ((X2 tptp.b) (A2 tptp.set_b) (B tptp.set_b)) (let ((_let_1 (@ tptp.minus_minus_set_b A2))) (=> (not (@ (@ tptp.member_b X2) A2)) (= (@ _let_1 (@ (@ tptp.insert_b X2) B)) (@ _let_1 B))))))
% 0.21/0.54 (assert (forall ((A tptp.b) (A2 tptp.set_b)) (let ((_let_1 (@ tptp.insert_b A))) (= (@ _let_1 (@ (@ tptp.minus_minus_set_b A2) (@ _let_1 tptp.bot_bot_set_b))) (@ _let_1 A2)))))
% 0.21/0.54 (assert (forall ((A2 tptp.set_b)) (= (@ (@ tptp.minus_minus_set_b A2) tptp.top_top_set_b) tptp.bot_bot_set_b)))
% 0.21/0.54 (assert (forall ((X2 tptp.b) (A2 tptp.set_b)) (let ((_let_1 (@ tptp.insert_b X2))) (=> (not (@ (@ tptp.member_b X2) A2)) (= (@ (@ tptp.minus_minus_set_b (@ _let_1 A2)) (@ _let_1 tptp.bot_bot_set_b)) A2)))))
% 0.21/0.54 (assert (forall ((A2 tptp.set_b) (A tptp.b) (B tptp.set_b)) (let ((_let_1 (@ tptp.insert_b A))) (let ((_let_2 (@ tptp.minus_minus_set_b A2))) (= (@ _let_2 (@ _let_1 B)) (@ (@ tptp.minus_minus_set_b (@ _let_2 (@ _let_1 tptp.bot_bot_set_b))) B))))))
% 0.21/0.54 (assert (forall ((A tptp.b) (A2 tptp.set_b)) (let ((_let_1 (@ tptp.insert_b A))) (=> (@ (@ tptp.member_b A) A2) (= (@ _let_1 (@ (@ tptp.minus_minus_set_b A2) (@ _let_1 tptp.bot_bot_set_b))) A2)))))
% 0.21/0.54 (assert (forall ((A2 tptp.set_b) (A tptp.b) (B tptp.set_b)) (let ((_let_1 (@ tptp.insert_b A))) (let ((_let_2 (@ tptp.minus_minus_set_b A2))) (= (@ _let_2 (@ _let_1 B)) (@ (@ tptp.minus_minus_set_b (@ _let_2 B)) (@ _let_1 tptp.bot_bot_set_b)))))))
% 0.21/0.54 (assert (forall ((B tptp.set_set_b) (X2 tptp.b) (A2 tptp.set_b)) (let ((_let_1 (@ tptp.insert_b X2))) (=> (forall ((C3 tptp.set_b)) (=> (@ (@ tptp.member_set_b C3) B) (not (@ (@ tptp.member_b X2) C3)))) (= (@ (@ tptp.member_set_b A2) (@ (@ tptp.image_set_b_set_b _let_1) B)) (and (@ (@ tptp.member_b X2) A2) (@ (@ tptp.member_set_b (@ (@ tptp.minus_minus_set_b A2) (@ _let_1 tptp.bot_bot_set_b))) B)))))))
% 0.21/0.54 (assert (forall ((C tptp.b) (A2 tptp.set_b) (B tptp.set_b)) (let ((_let_1 (@ tptp.member_b C))) (=> (@ _let_1 (@ (@ tptp.minus_minus_set_b A2) B)) (not (=> (@ _let_1 A2) (@ _let_1 B)))))))
% 0.58/0.77 (assert (forall ((C tptp.b) (A2 tptp.set_b) (B tptp.set_b)) (let ((_let_1 (@ tptp.member_b C))) (=> (@ _let_1 (@ (@ tptp.minus_minus_set_b A2) B)) (@ _let_1 A2)))))
% 0.58/0.77 (assert (forall ((C tptp.b) (A2 tptp.set_b) (B tptp.set_b)) (let ((_let_1 (@ tptp.member_b C))) (=> (@ _let_1 (@ (@ tptp.minus_minus_set_b A2) B)) (not (@ _let_1 B))))))
% 0.58/0.77 (assert (= tptp.minus_minus_set_b (lambda ((A3 tptp.set_b) (B4 tptp.set_b)) (@ tptp.collect_b (lambda ((X4 tptp.b)) (let ((_let_1 (@ tptp.member_b X4))) (and (@ _let_1 A3) (not (@ _let_1 B4)))))))))
% 0.58/0.77 (assert (forall ((X2 tptp.b) (B tptp.set_b) (A2 tptp.set_b)) (let ((_let_1 (@ (@ tptp.minus_minus_set_b A2) B))) (let ((_let_2 (@ tptp.insert_b X2))) (let ((_let_3 (@ (@ tptp.minus_minus_set_b (@ _let_2 A2)) B))) (let ((_let_4 (@ (@ tptp.member_b X2) B))) (and (=> _let_4 (= _let_3 _let_1)) (=> (not _let_4) (= _let_3 (@ _let_2 _let_1))))))))))
% 0.58/0.77 (assert (= tptp.remove_b (lambda ((X4 tptp.b) (A3 tptp.set_b)) (@ (@ tptp.minus_minus_set_b A3) (@ (@ tptp.insert_b X4) tptp.bot_bot_set_b)))))
% 0.58/0.77 (assert (forall ((X5 tptp.a)) (=> (@ tptp.p X5) (exists ((X_12 tptp.b)) (@ (@ tptp.q X5) X_12)))))
% 0.58/0.77 (assert (not (forall ((X tptp.a)) (let ((_let_1 (@ tptp.q X))) (or (not (@ tptp.p X)) (@ _let_1 (@ tptp.hilbert_Eps_b _let_1)))))))
% 0.58/0.77 (set-info :filename cvc5---1.0.5_31926)
% 0.58/0.77 (check-sat-assuming ( true ))
% 0.58/0.77 ------- get file name : TPTP file name is ITP144^1
% 0.58/0.77 ------- cvc5-thf : /export/starexec/sandbox2/solver/bin/cvc5---1.0.5_31926.smt2...
% 0.58/0.77 --- Run --ho-elim --full-saturate-quant at 10...
% 0.58/0.77 % SZS status Theorem for ITP144^1
% 0.58/0.77 % SZS output start Proof for ITP144^1
% 0.58/0.77 (
% 0.58/0.77 (let ((_let_1 (not (forall ((X tptp.a)) (let ((_let_1 (@ tptp.q X))) (or (not (@ tptp.p X)) (@ _let_1 (@ tptp.hilbert_Eps_b _let_1)))))))) (let ((_let_2 (forall ((X5 tptp.a)) (=> (@ tptp.p X5) (exists ((X_12 tptp.b)) (@ (@ tptp.q X5) X_12)))))) (let ((_let_3 (forall ((P (-> tptp.b Bool)) (A2 Bool)) (=> (= (@ P (@ tptp.hilbert_Eps_b P)) A2) (= (exists ((X3 tptp.b)) (@ P X3)) A2))))) (let ((_let_4 (forall ((X3 tptp.b)) (not (ho_121 (ho_288 k_287 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_292) X3))))) (let ((_let_5 (forall ((X_12 tptp.b)) (not (ho_121 (ho_288 k_287 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_292) X_12))))) (let ((_let_6 (not _let_4))) (let ((_let_7 (ho_288 k_287 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_292))) (let ((_let_8 (ho_121 _let_7 (ho_267 k_266 _let_7)))) (let ((_let_9 (= _let_8 _let_6))) (let ((_let_10 (forall ((BOUND_VARIABLE_10539 |u_(-> tptp.b Bool)|)) (= (not (forall ((X3 tptp.b)) (not (ho_121 BOUND_VARIABLE_10539 X3)))) (ho_121 BOUND_VARIABLE_10539 (ho_267 k_266 BOUND_VARIABLE_10539)))))) (let ((_let_11 (EQ_RESOLVE (ASSUME :args (_let_3)) (TRANS (MACRO_SR_EQ_INTRO :args (_let_3 SB_DEFAULT SBA_FIXPOINT)) (PREPROCESS :args ((= (forall ((P (-> tptp.b Bool))) (= (@ P (@ tptp.hilbert_Eps_b P)) (not (forall ((X3 tptp.b)) (not (@ P X3)))))) _let_10))))))) (let ((_let_12 (ho_290 k_289 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_292))) (let ((_let_13 (not _let_12))) (let ((_let_14 (or _let_13 _let_8))) (let ((_let_15 (forall ((X tptp.a)) (let ((_let_1 (ho_288 k_287 X))) (or (not (ho_290 k_289 X)) (ho_121 _let_1 (ho_267 k_266 _let_1))))))) (let ((_let_16 (not _let_14))) (let ((_let_17 (not _let_15))) (let ((_let_18 (EQ_RESOLVE (ASSUME :args (_let_1)) (PREPROCESS :args ((= _let_1 _let_17)))))) (let ((_let_19 (or))) (let ((_let_20 (MACRO_RESOLUTION_TRUST (EQ_RESOLVE (IMPLIES_ELIM (SCOPE (SKOLEMIZE _let_18) :args (_let_17))) (CONG (MACRO_SR_PRED_INTRO :args ((= (not _let_17) _let_15))) (REFL :args (_let_16)) :args _let_19)) _let_18 :args (_let_16 true _let_15)))) (let ((_let_21 (not _let_9))) (let ((_let_22 (not _let_5))) (let ((_let_23 (or _let_13 _let_22))) (let ((_let_24 (forall ((X5 tptp.a)) (or (not (ho_290 k_289 X5)) (not (forall ((X_12 tptp.b)) (not (ho_121 (ho_288 k_287 X5) X_12)))))))) (let ((_let_25 (EQ_RESOLVE (ASSUME :args (_let_2)) (TRANS (MACRO_SR_EQ_INTRO :args (_let_2 SB_DEFAULT SBA_FIXPOINT)) (PREPROCESS :args ((= (forall ((X5 tptp.a)) (or (not (@ tptp.p X5)) (not (forall ((X_12 tptp.b)) (not (@ (@ tptp.q X5) X_12)))))) _let_24))))))) (SCOPE (SCOPE (MACRO_RESOLUTION_TRUST (EQUIV_ELIM1 (ALPHA_EQUIV :args (_let_4 (= X3 X_12)))) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_OR_POS :args (_let_23)) :args ((or _let_13 _let_22 (not _let_23)))) (MACRO_RESOLUTION_TRUST (REORDERING (EQ_RESOLVE (CNF_OR_NEG :args (_let_14 0)) (CONG (REFL :args (_let_14)) (MACRO_SR_PRED_INTRO :args ((= (not _let_13) _let_12))) :args _let_19)) :args ((or _let_12 _let_14))) _let_20 :args (_let_12 true _let_14)) (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE _let_25 :args (SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_292 QUANTIFIERS_INST_E_MATCHING_SIMPLE ((not (= (ho_290 k_289 X5) false))))) :args (_let_24))) _let_25 :args (_let_23 false _let_24)) :args (_let_22 false _let_12 false _let_23)) (MACRO_RESOLUTION_TRUST (REORDERING (EQ_RESOLVE (CNF_EQUIV_POS2 :args (_let_9)) (CONG (REFL :args (_let_21)) (REFL :args (_let_8)) (MACRO_SR_PRED_INTRO :args ((= (not _let_6) _let_4))) :args _let_19)) :args ((or _let_8 _let_4 _let_21))) (MACRO_RESOLUTION_TRUST (CNF_OR_NEG :args (_let_14 1)) _let_20 :args ((not _let_8) true _let_14)) (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (MACRO_SR_PRED_ELIM (SCOPE (INSTANTIATE _let_11 :args (_let_7 QUANTIFIERS_INST_E_MATCHING_SIMPLE ((ho_267 k_266 BOUND_VARIABLE_10539)))) :args (_let_10)))) _let_11 :args (_let_9 false _let_10)) :args (_let_4 true _let_8 false _let_9)) :args (false true _let_5 false _let_4)) :args ((forall ((P (-> tptp.b Bool)) (A tptp.b)) (=> (@ P A) (=> (forall ((X tptp.b)) (=> (@ P X) (= X A))) (= (@ tptp.hilbert_Eps_b P) A)))) (forall ((X2 tptp.b)) (= (@ tptp.hilbert_Eps_b (lambda ((Y tptp.b)) (= Y X2))) X2)) (forall ((X2 tptp.b)) (= (@ tptp.hilbert_Eps_b (@ (lambda ((Y2 tptp.b) (Z tptp.b)) (= Y2 Z)) X2)) X2)) _let_3 (= (lambda ((P2 (-> tptp.b Bool))) (forall ((X3 tptp.b)) (@ P2 X3))) (lambda ((P3 (-> tptp.b Bool))) (@ P3 (@ tptp.hilbert_Eps_b (lambda ((X4 tptp.b)) (not (@ P3 X4))))))) (forall ((P (-> tptp.b Bool)) (A tptp.b) (Q (-> tptp.b Bool))) (=> (@ P A) (=> (forall ((X tptp.b)) (=> (@ P X) (@ Q X))) (@ Q (@ tptp.hilbert_Eps_b P))))) (forall ((P (-> tptp.b Bool)) (A2 Bool)) (=> (= (@ P (@ tptp.hilbert_Eps_b (lambda ((X4 tptp.b)) (not (@ P X4))))) A2) (= (forall ((X3 tptp.b)) (@ P X3)) A2))) (forall ((B tptp.b) (A2 tptp.b) (P (-> tptp.b Bool))) (let ((_let_1 (@ tptp.hilbert_Eps_b P))) (=> (= B A2) (= (= _let_1 A2) (= _let_1 B))))) (forall ((P (-> tptp.b Bool))) (=> (exists ((X_1 tptp.b)) (@ P X_1)) (@ P (@ tptp.hilbert_Eps_b P)))) (forall ((P (-> tptp.b Bool)) (Q (-> tptp.b Bool))) (=> (exists ((X_1 tptp.b)) (@ P X_1)) (=> (forall ((X tptp.b)) (=> (@ P X) (@ Q X))) (@ Q (@ tptp.hilbert_Eps_b P))))) (forall ((A2 tptp.set_b) (P (-> tptp.b Bool)) (Q (-> tptp.b Bool))) (=> (exists ((X5 tptp.b)) (and (@ (@ tptp.member_b X5) A2) (@ P X5))) (=> (forall ((X tptp.b)) (=> (and (@ (@ tptp.member_b X) A2) (@ P X)) (@ Q X))) (@ Q (@ tptp.hilbert_Eps_b (lambda ((X4 tptp.b)) (and (@ (@ tptp.member_b X4) A2) (@ P X4)))))))) (forall ((P (-> tptp.b Bool))) (= (@ P (@ tptp.hilbert_Eps_b P)) (exists ((X3 tptp.b)) (@ P X3)))) (forall ((P (-> tptp.b Bool)) (A tptp.b) (B2 tptp.b)) (=> (= (@ tptp.hilbert_Eps_b P) A) (=> (@ P B2) (@ P A)))) (forall ((P4 (-> tptp.b Bool)) (X5 tptp.b)) (=> (@ P4 X5) (@ P4 (@ tptp.hilbert_Eps_b P4)))) (forall ((P (-> tptp.b Bool)) (Q (-> tptp.b Bool))) (=> (forall ((X tptp.b)) (= (@ P X) (@ Q X))) (= (@ tptp.hilbert_Eps_b P) (@ tptp.hilbert_Eps_b Q)))) (forall ((P (-> tptp.b Bool)) (X2 tptp.b)) (=> (@ P X2) (@ P (@ tptp.hilbert_Eps_b P)))) (forall ((X2 tptp.b) (P (-> tptp.b Bool)) (P5 (-> tptp.b Bool))) (=> (= X2 (@ tptp.hilbert_Eps_b (lambda ((X4 tptp.b)) (not (@ P X4))))) (=> (forall ((X tptp.b)) (= (@ P X) (@ P5 X))) (= (forall ((X3 tptp.b)) (@ P5 X3)) (@ P X2))))) (forall ((X2 tptp.b) (P (-> tptp.b Bool))) (=> (= X2 (@ tptp.hilbert_Eps_b (lambda ((X4 tptp.b)) (not (@ P X4))))) (= (forall ((X3 tptp.b)) (@ P X3)) (@ P X2)))) (forall ((P (-> tptp.b Bool)) (A tptp.b)) (=> (exists ((X5 tptp.b)) (and (@ P X5) (forall ((Y3 tptp.b)) (=> (@ P Y3) (= Y3 X5))))) (=> (@ P A) (= (@ tptp.hilbert_Eps_b P) A)))) (forall ((X2 tptp.b) (P (-> tptp.b Bool)) (P5 (-> tptp.b Bool))) (=> (= X2 (@ tptp.hilbert_Eps_b P)) (=> (forall ((X tptp.b)) (= (@ P X) (@ P5 X))) (= (exists ((X3 tptp.b)) (@ P5 X3)) (@ P X2))))) (forall ((X2 tptp.b) (P (-> tptp.b Bool))) (=> (= X2 (@ tptp.hilbert_Eps_b P)) (= (exists ((X3 tptp.b)) (@ P X3)) (@ P X2)))) (forall ((P (-> tptp.b Bool)) (X2 tptp.b) (Y4 tptp.b)) (let ((_let_1 (@ tptp.hilbert_Eps_b P))) (=> (@ P X2) (=> (not (@ P Y4)) (=> (= _let_1 Y4) (= _let_1 X2)))))) (forall ((A2 tptp.set_b)) (= (@ (@ tptp.member_b (@ tptp.hilbert_Eps_b (lambda ((X4 tptp.b)) (@ (@ tptp.member_b X4) A2)))) A2) (not (= A2 tptp.bot_bot_set_b)))) (forall ((P (-> tptp.b Bool)) (C tptp.b)) (=> (exists ((X_1 tptp.b)) (@ P X_1)) (=> (= C (@ tptp.hilbert_Eps_b P)) (@ P C)))) (forall ((P (-> tptp.b Bool))) (= (= tptp.bot_bot_set_b (@ tptp.collect_b P)) (forall ((X4 tptp.b)) (not (@ P X4))))) (forall ((P (-> tptp.b Bool))) (= (= (@ tptp.collect_b P) tptp.bot_bot_set_b) (forall ((X4 tptp.b)) (not (@ P X4))))) (forall ((A2 tptp.set_b)) (= (forall ((X4 tptp.b)) (not (@ (@ tptp.member_b X4) A2))) (= A2 tptp.bot_bot_set_b))) (forall ((C tptp.b)) (not (@ (@ tptp.member_b C) tptp.bot_bot_set_b))) (forall ((A tptp.b) (P (-> tptp.b Bool))) (= (@ (@ tptp.member_b A) (@ tptp.collect_b P)) (@ P A))) (forall ((A2 tptp.set_b)) (= (@ tptp.collect_b (lambda ((X4 tptp.b)) (@ (@ tptp.member_b X4) A2))) A2)) (= tptp.bot_bot_set_b (@ tptp.collect_b tptp.bot_bot_b_o)) (forall ((A tptp.b)) (not (@ (@ tptp.member_b A) tptp.bot_bot_set_b))) (forall ((A2 tptp.set_b) (A tptp.b)) (=> (= A2 tptp.bot_bot_set_b) (not (@ (@ tptp.member_b A) A2)))) (forall ((A2 tptp.set_b)) (=> (forall ((Y3 tptp.b)) (not (@ (@ tptp.member_b Y3) A2))) (= A2 tptp.bot_bot_set_b))) (forall ((A2 tptp.set_b)) (= (exists ((X4 tptp.b)) (@ (@ tptp.member_b X4) A2)) (not (= A2 tptp.bot_bot_set_b)))) (= tptp.bot_bot_set_b (@ tptp.collect_b (lambda ((X4 tptp.b)) false))) (= tptp.is_empty_b (lambda ((A3 tptp.set_b)) (= A3 tptp.bot_bot_set_b))) (forall ((P (-> tptp.b Bool))) (= (= (@ tptp.collect_b P) tptp.bot_bot_set_b) (= P tptp.bot_bot_b_o))) (= tptp.bot_bot_b_o (lambda ((X4 tptp.b)) (@ (@ tptp.member_b X4) tptp.bot_bot_set_b))) (forall ((A2 tptp.set_b)) (=> (not (= A2 tptp.bot_bot_set_b)) (=> (forall ((X tptp.b) (Y3 tptp.b)) (=> (@ (@ tptp.member_b X) A2) (=> (@ (@ tptp.member_b Y3) A2) (= X Y3)))) (@ tptp.is_singleton_b A2)))) (forall ((X2 tptp.b)) (@ (@ tptp.member_b X2) tptp.top_top_set_b)) (forall ((P Bool)) (and (=> P (= (@ tptp.collect_b (lambda ((S tptp.b)) P)) tptp.top_top_set_b)) (=> (not P) (= (@ tptp.collect_b (lambda ((S tptp.b)) P)) tptp.bot_bot_set_b)))) (forall ((A2 tptp.set_b)) (=> (forall ((X tptp.b)) (@ (@ tptp.member_b X) A2)) (= tptp.top_top_set_b A2))) (exists ((X tptp.b)) (@ (@ tptp.member_b X) tptp.top_top_set_b)) (not (= tptp.bot_bot_set_b tptp.top_top_set_b)) (forall ((X2 tptp.b)) (@ (@ tptp.member_b X2) tptp.top_top_set_b)) (forall ((X2 tptp.b)) (@ tptp.is_singleton_b (@ (@ tptp.insert_b X2) tptp.bot_bot_set_b))) (forall ((A tptp.b) (B2 tptp.b) (A2 tptp.set_b)) (let ((_let_1 (@ tptp.member_b A))) (= (@ _let_1 (@ (@ tptp.insert_b B2) A2)) (or (= A B2) (@ _let_1 A2))))) (forall ((A tptp.b) (B tptp.set_b) (B2 tptp.b)) (let ((_let_1 (@ tptp.member_b A))) (=> (=> (not (@ _let_1 B)) (= A B2)) (@ _let_1 (@ (@ tptp.insert_b B2) B))))) (forall ((A tptp.b)) (@ (@ tptp.member_b A) (@ (@ tptp.insert_b A) tptp.bot_bot_set_b))) (forall ((A tptp.b)) (= (@ tptp.collect_b (lambda ((X4 tptp.b)) (= X4 A))) (@ (@ tptp.insert_b A) tptp.bot_bot_set_b))) (forall ((A tptp.b)) (= (@ tptp.collect_b (@ (lambda ((Y2 tptp.b) (Z tptp.b)) (= Y2 Z)) A)) (@ (@ tptp.insert_b A) tptp.bot_bot_set_b))) (forall ((A tptp.b) (A2 tptp.set_b)) (=> (@ (@ tptp.member_b A) A2) (exists ((B3 tptp.set_b)) (and (= A2 (@ (@ tptp.insert_b A) B3)) (not (@ (@ tptp.member_b A) B3)))))) (forall ((A tptp.b) (A2 tptp.set_b) (B2 tptp.b) (B tptp.set_b)) (let ((_let_1 (= A B2))) (=> (not (@ (@ tptp.member_b A) A2)) (=> (not (@ (@ tptp.member_b B2) B)) (= (= (@ (@ tptp.insert_b A) A2) (@ (@ tptp.insert_b B2) B)) (and (=> _let_1 (= A2 B)) (=> (not _let_1) (exists ((C2 tptp.set_b)) (and (= A2 (@ (@ tptp.insert_b B2) C2)) (not (@ (@ tptp.member_b B2) C2)) (= B (@ (@ tptp.insert_b A) C2)) (not (@ (@ tptp.member_b A) C2))))))))))) (forall ((A tptp.b) (A2 tptp.set_b)) (=> (@ (@ tptp.member_b A) A2) (= (@ (@ tptp.insert_b A) A2) A2))) (forall ((X2 tptp.b) (A2 tptp.set_b) (B tptp.set_b)) (let ((_let_1 (@ tptp.insert_b X2))) (let ((_let_2 (@ tptp.member_b X2))) (=> (not (@ _let_2 A2)) (=> (not (@ _let_2 B)) (= (= (@ _let_1 A2) (@ _let_1 B)) (= A2 B))))))) (= tptp.insert_b (lambda ((A4 tptp.b) (B4 tptp.set_b)) (@ tptp.collect_b (lambda ((X4 tptp.b)) (or (= X4 A4) (@ (@ tptp.member_b X4) B4)))))) (forall ((X2 tptp.b) (A2 tptp.set_b)) (=> (@ (@ tptp.member_b X2) A2) (not (forall ((B3 tptp.set_b)) (=> (= A2 (@ (@ tptp.insert_b X2) B3)) (@ (@ tptp.member_b X2) B3)))))) (forall ((A tptp.b) (B tptp.set_b) (B2 tptp.b)) (let ((_let_1 (@ tptp.member_b A))) (=> (@ _let_1 B) (@ _let_1 (@ (@ tptp.insert_b B2) B))))) (forall ((A tptp.b) (B tptp.set_b)) (@ (@ tptp.member_b A) (@ (@ tptp.insert_b A) B))) (forall ((A tptp.b) (B2 tptp.b) (A2 tptp.set_b)) (let ((_let_1 (@ tptp.member_b A))) (=> (@ _let_1 (@ (@ tptp.insert_b B2) A2)) (=> (not (= A B2)) (@ _let_1 A2))))) (= tptp.top_top_b_o (lambda ((X4 tptp.b)) (@ (@ tptp.member_b X4) tptp.top_top_set_b))) (forall ((A tptp.b) (B2 tptp.b)) (=> (= (@ (@ tptp.insert_b A) tptp.bot_bot_set_b) (@ (@ tptp.insert_b B2) tptp.bot_bot_set_b)) (= A B2))) (forall ((A tptp.b) (A2 tptp.set_b)) (not (= (@ (@ tptp.insert_b A) A2) tptp.bot_bot_set_b))) (forall ((A tptp.b) (B2 tptp.b) (C tptp.b) (D tptp.b)) (= (= (@ (@ tptp.insert_b A) (@ (@ tptp.insert_b B2) tptp.bot_bot_set_b)) (@ (@ tptp.insert_b C) (@ (@ tptp.insert_b D) tptp.bot_bot_set_b))) (or (and (= A C) (= B2 D)) (and (= A D) (= B2 C))))) (forall ((B2 tptp.b) (A tptp.b)) (= (@ (@ tptp.member_b B2) (@ (@ tptp.insert_b A) tptp.bot_bot_set_b)) (= B2 A))) (forall ((B2 tptp.b) (A tptp.b)) (=> (@ (@ tptp.member_b B2) (@ (@ tptp.insert_b A) tptp.bot_bot_set_b)) (= B2 A))) (forall ((P (-> tptp.b Bool)) (A tptp.b)) (let ((_let_1 (@ P A))) (and (=> _let_1 (= (@ tptp.collect_b (lambda ((X4 tptp.b)) (and (= X4 A) (@ P X4)))) (@ (@ tptp.insert_b A) tptp.bot_bot_set_b))) (=> (not _let_1) (= (@ tptp.collect_b (lambda ((X4 tptp.b)) (and (= X4 A) (@ P X4)))) tptp.bot_bot_set_b))))) (forall ((P (-> tptp.b Bool)) (A tptp.b)) (let ((_let_1 (@ P A))) (and (=> _let_1 (= (@ tptp.collect_b (lambda ((X4 tptp.b)) (and (= A X4) (@ P X4)))) (@ (@ tptp.insert_b A) tptp.bot_bot_set_b))) (=> (not _let_1) (= (@ tptp.collect_b (lambda ((X4 tptp.b)) (and (= A X4) (@ P X4)))) tptp.bot_bot_set_b))))) (= tptp.is_singleton_b (lambda ((A3 tptp.set_b)) (exists ((X4 tptp.b)) (= A3 (@ (@ tptp.insert_b X4) tptp.bot_bot_set_b))))) (forall ((A2 tptp.set_b)) (=> (@ tptp.is_singleton_b A2) (not (forall ((X tptp.b)) (not (= A2 (@ (@ tptp.insert_b X) tptp.bot_bot_set_b))))))) (= tptp.is_singleton_b (lambda ((A3 tptp.set_b)) (= A3 (@ (@ tptp.insert_b (@ tptp.the_elem_b A3)) tptp.bot_bot_set_b)))) (forall ((X2 tptp.b)) (= (@ tptp.the_elem_b (@ (@ tptp.insert_b X2) tptp.bot_bot_set_b)) X2)) (forall ((B2 tptp.b) (F (-> tptp.b tptp.b)) (X2 tptp.b) (A2 tptp.set_b)) (=> (= B2 (@ F X2)) (=> (@ (@ tptp.member_b X2) A2) (@ (@ tptp.member_b B2) (@ (@ tptp.image_b_b F) A2))))) (forall ((F (-> tptp.b tptp.b)) (A2 tptp.set_b)) (= (= (@ (@ tptp.image_b_b F) A2) tptp.bot_bot_set_b) (= A2 tptp.bot_bot_set_b))) (forall ((F (-> tptp.b tptp.b)) (A2 tptp.set_b)) (= (= tptp.bot_bot_set_b (@ (@ tptp.image_b_b F) A2)) (= A2 tptp.bot_bot_set_b))) (forall ((F (-> tptp.b tptp.b))) (= (@ (@ tptp.image_b_b F) tptp.bot_bot_set_b) tptp.bot_bot_set_b)) (forall ((B2 tptp.b) (F (-> tptp.b tptp.b)) (A2 tptp.set_b)) (=> (@ (@ tptp.member_b B2) (@ (@ tptp.image_b_b F) A2)) (not (forall ((X tptp.b)) (=> (= B2 (@ F X)) (not (@ (@ tptp.member_b X) A2))))))) (forall ((X2 tptp.b) (A2 tptp.set_b) (F (-> tptp.b tptp.b))) (=> (@ (@ tptp.member_b X2) A2) (@ (@ tptp.member_b (@ F X2)) (@ (@ tptp.image_b_b F) A2)))) (forall ((X2 tptp.b) (A2 tptp.set_b) (B2 tptp.b) (F (-> tptp.b tptp.b))) (=> (@ (@ tptp.member_b X2) A2) (=> (= B2 (@ F X2)) (@ (@ tptp.member_b B2) (@ (@ tptp.image_b_b F) A2))))) (forall ((F (-> tptp.b tptp.b)) (A2 tptp.set_b) (P (-> tptp.b Bool))) (= (@ tptp.collect_b (lambda ((X4 tptp.b)) (and (@ (@ tptp.member_b X4) (@ (@ tptp.image_b_b F) A2)) (@ P X4)))) (@ (@ tptp.image_b_b F) (@ tptp.collect_b (lambda ((X4 tptp.b)) (and (@ (@ tptp.member_b X4) A2) (@ P (@ F X4)))))))) (forall ((X2 tptp.b) (F (-> tptp.b tptp.b)) (A2 tptp.set_b)) (=> (@ (@ tptp.member_b X2) (@ (@ tptp.image_b_b F) A2)) (@ (@ tptp.member_b (@ (@ (@ tptp.hilbert_inv_into_b_b A2) F) X2)) A2))) (forall ((X2 tptp.b) (A2 tptp.set_b) (C tptp.b)) (=> (@ (@ tptp.member_b X2) A2) (= (@ (@ tptp.image_b_b (lambda ((X4 tptp.b)) C)) A2) (@ (@ tptp.insert_b C) tptp.bot_bot_set_b)))) (forall ((A2 tptp.set_b) (C tptp.b)) (let ((_let_1 (= A2 tptp.bot_bot_set_b))) (and (=> _let_1 (= (@ (@ tptp.image_b_b (lambda ((X4 tptp.b)) C)) A2) tptp.bot_bot_set_b)) (=> (not _let_1) (= (@ (@ tptp.image_b_b (lambda ((X4 tptp.b)) C)) A2) (@ (@ tptp.insert_b C) tptp.bot_bot_set_b)))))) (forall ((F (-> tptp.b tptp.b)) (A2 tptp.set_b)) (=> (@ (@ (@ tptp.bij_betw_b_b F) A2) tptp.bot_bot_set_b) (= A2 tptp.bot_bot_set_b))) (forall ((F (-> tptp.b tptp.b)) (A2 tptp.set_b)) (=> (@ (@ (@ tptp.bij_betw_b_b F) tptp.bot_bot_set_b) A2) (= A2 tptp.bot_bot_set_b))) (forall ((A2 tptp.set_b)) (@ (@ tptp.inj_on_b_set_b (lambda ((X4 tptp.b)) (@ (@ tptp.insert_b X4) tptp.bot_bot_set_b))) A2)) (forall ((F (-> tptp.b tptp.b)) (A2 tptp.set_b) (X2 tptp.b) (B tptp.set_b)) (=> (@ (@ tptp.inj_on_b_b F) A2) (=> (not (@ (@ tptp.member_b X2) B)) (=> (= (@ (@ tptp.insert_b X2) B) (@ (@ tptp.image_b_b F) A2)) (not (forall ((X6 tptp.b) (A5 tptp.set_b)) (=> (not (@ (@ tptp.member_b X6) A5)) (=> (= A2 (@ (@ tptp.insert_b X6) A5)) (=> (= X2 (@ F X6)) (not (= B (@ (@ tptp.image_b_b F) A5)))))))))))) (forall ((F (-> tptp.b tptp.b)) (A tptp.b) (A2 tptp.set_b)) (=> (@ (@ tptp.inj_on_b_b F) tptp.top_top_set_b) (= (@ (@ tptp.member_b (@ F A)) (@ (@ tptp.image_b_b F) A2)) (@ (@ tptp.member_b A) A2)))) (forall ((F (-> tptp.b tptp.b)) (A tptp.b) (A2 tptp.set_b)) (let ((_let_1 (@ tptp.insert_b A))) (let ((_let_2 (@ tptp.inj_on_b_b F))) (= (@ _let_2 (@ _let_1 A2)) (and (@ _let_2 A2) (not (@ (@ tptp.member_b (@ F A)) (@ (@ tptp.image_b_b F) (@ (@ tptp.minus_minus_set_b A2) (@ _let_1 tptp.bot_bot_set_b)))))))))) (forall ((C tptp.b) (A2 tptp.set_b) (B tptp.set_b)) (let ((_let_1 (@ tptp.member_b C))) (=> (@ _let_1 A2) (=> (not (@ _let_1 B)) (@ _let_1 (@ (@ tptp.minus_minus_set_b A2) B)))))) (forall ((C tptp.b) (A2 tptp.set_b) (B tptp.set_b)) (let ((_let_1 (@ tptp.member_b C))) (= (@ _let_1 (@ (@ tptp.minus_minus_set_b A2) B)) (and (@ _let_1 A2) (not (@ _let_1 B)))))) (forall ((A2 tptp.set_b)) (= (@ (@ tptp.minus_minus_set_b A2) tptp.bot_bot_set_b) A2)) (forall ((A2 tptp.set_b)) (= (@ (@ tptp.minus_minus_set_b tptp.bot_bot_set_b) A2) tptp.bot_bot_set_b)) (forall ((A2 tptp.set_b)) (= (@ (@ tptp.minus_minus_set_b A2) A2) tptp.bot_bot_set_b)) (forall ((X2 tptp.b) (B tptp.set_b) (A2 tptp.set_b)) (=> (@ (@ tptp.member_b X2) B) (= (@ (@ tptp.minus_minus_set_b (@ (@ tptp.insert_b X2) A2)) B) (@ (@ tptp.minus_minus_set_b A2) B)))) (forall ((X2 tptp.b) (A2 tptp.set_b) (B tptp.set_b)) (let ((_let_1 (@ tptp.minus_minus_set_b A2))) (=> (not (@ (@ tptp.member_b X2) A2)) (= (@ _let_1 (@ (@ tptp.insert_b X2) B)) (@ _let_1 B))))) (forall ((A tptp.b) (A2 tptp.set_b)) (let ((_let_1 (@ tptp.insert_b A))) (= (@ _let_1 (@ (@ tptp.minus_minus_set_b A2) (@ _let_1 tptp.bot_bot_set_b))) (@ _let_1 A2)))) (forall ((A2 tptp.set_b)) (= (@ (@ tptp.minus_minus_set_b A2) tptp.top_top_set_b) tptp.bot_bot_set_b)) (forall ((X2 tptp.b) (A2 tptp.set_b)) (let ((_let_1 (@ tptp.insert_b X2))) (=> (not (@ (@ tptp.member_b X2) A2)) (= (@ (@ tptp.minus_minus_set_b (@ _let_1 A2)) (@ _let_1 tptp.bot_bot_set_b)) A2)))) (forall ((A2 tptp.set_b) (A tptp.b) (B tptp.set_b)) (let ((_let_1 (@ tptp.insert_b A))) (let ((_let_2 (@ tptp.minus_minus_set_b A2))) (= (@ _let_2 (@ _let_1 B)) (@ (@ tptp.minus_minus_set_b (@ _let_2 (@ _let_1 tptp.bot_bot_set_b))) B))))) (forall ((A tptp.b) (A2 tptp.set_b)) (let ((_let_1 (@ tptp.insert_b A))) (=> (@ (@ tptp.member_b A) A2) (= (@ _let_1 (@ (@ tptp.minus_minus_set_b A2) (@ _let_1 tptp.bot_bot_set_b))) A2)))) (forall ((A2 tptp.set_b) (A tptp.b) (B tptp.set_b)) (let ((_let_1 (@ tptp.insert_b A))) (let ((_let_2 (@ tptp.minus_minus_set_b A2))) (= (@ _let_2 (@ _let_1 B)) (@ (@ tptp.minus_minus_set_b (@ _let_2 B)) (@ _let_1 tptp.bot_bot_set_b)))))) (forall ((B tptp.set_set_b) (X2 tptp.b) (A2 tptp.set_b)) (let ((_let_1 (@ tptp.insert_b X2))) (=> (forall ((C3 tptp.set_b)) (=> (@ (@ tptp.member_set_b C3) B) (not (@ (@ tptp.member_b X2) C3)))) (= (@ (@ tptp.member_set_b A2) (@ (@ tptp.image_set_b_set_b _let_1) B)) (and (@ (@ tptp.member_b X2) A2) (@ (@ tptp.member_set_b (@ (@ tptp.minus_minus_set_b A2) (@ _let_1 tptp.bot_bot_set_b))) B)))))) (forall ((C tptp.b) (A2 tptp.set_b) (B tptp.set_b)) (let ((_let_1 (@ tptp.member_b C))) (=> (@ _let_1 (@ (@ tptp.minus_minus_set_b A2) B)) (not (=> (@ _let_1 A2) (@ _let_1 B)))))) (forall ((C tptp.b) (A2 tptp.set_b) (B tptp.set_b)) (let ((_let_1 (@ tptp.member_b C))) (=> (@ _let_1 (@ (@ tptp.minus_minus_set_b A2) B)) (@ _let_1 A2)))) (forall ((C tptp.b) (A2 tptp.set_b) (B tptp.set_b)) (let ((_let_1 (@ tptp.member_b C))) (=> (@ _let_1 (@ (@ tptp.minus_minus_set_b A2) B)) (not (@ _let_1 B))))) (= tptp.minus_minus_set_b (lambda ((A3 tptp.set_b) (B4 tptp.set_b)) (@ tptp.collect_b (lambda ((X4 tptp.b)) (let ((_let_1 (@ tptp.member_b X4))) (and (@ _let_1 A3) (not (@ _let_1 B4)))))))) (forall ((X2 tptp.b) (B tptp.set_b) (A2 tptp.set_b)) (let ((_let_1 (@ (@ tptp.minus_minus_set_b A2) B))) (let ((_let_2 (@ tptp.insert_b X2))) (let ((_let_3 (@ (@ tptp.minus_minus_set_b (@ _let_2 A2)) B))) (let ((_let_4 (@ (@ tptp.member_b X2) B))) (and (=> _let_4 (= _let_3 _let_1)) (=> (not _let_4) (= _let_3 (@ _let_2 _let_1))))))))) (= tptp.remove_b (lambda ((X4 tptp.b) (A3 tptp.set_b)) (@ (@ tptp.minus_minus_set_b A3) (@ (@ tptp.insert_b X4) tptp.bot_bot_set_b)))) _let_2 _let_1 true))))))))))))))))))))))))))))
% 0.58/0.78 )
% 0.58/0.78 % SZS output end Proof for ITP144^1
% 0.58/0.78 % cvc5---1.0.5 exiting
% 0.58/0.78 % cvc5---1.0.5 exiting
%------------------------------------------------------------------------------