%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : COM209^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n009.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:41:58 AM UTC 2026
% Result : Theorem 2.87s 1.60s
% Output : Refutation 2.87s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : COM209^1 : TPTP v9.2.1. Released v7.0.0.
% 0.12/0.12 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.33 % Computer : n009.cluster.edu
% 0.15/0.33 % Model : x86_64 x86_64
% 0.15/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.33 % Memory : 8042.1875MB
% 0.15/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.33 % CPULimit : 300
% 0.15/0.33 % WCLimit : 300
% 0.15/0.33 % DateTime : Tue Feb 24 04:16:40 EST 2026
% 0.15/0.33 % CPUTime :
% 0.15/0.33 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.18/0.45 ---- Original DTF file ---
% 0.18/0.45 thf(spec,logic,$$dhol).
% 0.18/0.45 %------------------------------------------------------------------------------
% 0.18/0.45 % File : COM209^1 : TPTP v9.2.1. Released v7.0.0.
% 0.18/0.45 % Domain : Computing Theory
% 0.18/0.45 % Problem : Parallel extension to grammars and languages 133
% 0.18/0.45 % Version : [Bla16] axioms : Especial.
% 0.18/0.45 % English :
% 0.18/0.45
% 0.18/0.45 % Refs : [BH+14] Blanchette et al. (2014), Truly Modular (Co)datatypes
% 0.18/0.45 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.18/0.45 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.18/0.45 % Source : [Bla16]
% 0.18/0.45 % Names : parallel__133.p [Bla16]
% 0.18/0.45
% 0.18/0.45 % Status : Theorem
% 0.18/0.45 % Rating : 0.33 v9.1.0, 0.00 v7.2.0, 0.25 v7.1.0
% 0.18/0.45 % Syntax : Number of formulae : 337 ( 135 unt; 52 typ; 0 def)
% 0.18/0.45 % Number of atoms : 514 ( 320 equ; 0 cnn)
% 0.18/0.45 % Maximal formula atoms : 6 ( 1 avg)
% 0.18/0.45 % Number of connectives : 4631 ( 53 ~; 5 |; 18 &;4347 @)
% 0.18/0.45 % ( 0 <=>; 208 =>; 0 <=; 0 <~>)
% 0.18/0.45 % Maximal formula depth : 28 ( 8 avg)
% 0.18/0.45 % Number of types : 3 ( 2 usr)
% 0.18/0.45 % Number of type conns : 398 ( 398 >; 0 *; 0 +; 0 <<)
% 0.18/0.45 % Number of symbols : 53 ( 50 usr; 11 con; 0-7 aty)
% 0.18/0.45 % Number of variables : 1438 ( 50 ^;1299 !; 20 ?;1438 :)
% 0.18/0.45 % ( 69 !>; 0 ?*; 0 @-; 0 @+)
% 0.18/0.45 % SPC : TH1_THM_EQU_NAR
% 0.18/0.45
% 0.18/0.45 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.18/0.45 % 2016-07-13 14:39:20.562
% 0.18/0.45 %------------------------------------------------------------------------------
% 0.18/0.45 %----Could-be-implicit typings (6)
% 0.18/0.45 thf(ty_t_Product__Type_Oprod,type,
% 0.18/0.45 product_prod: $tType > $tType > $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_t_Sum__Type_Osum,type,
% 0.18/0.45 sum_sum: $tType > $tType > $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_t_DTree_Odtree,type,
% 0.18/0.45 dtree: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_t_Set_Oset,type,
% 0.18/0.45 set: $tType > $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_t_DTree_OT,type,
% 0.18/0.45 t: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_t_itself,type,
% 0.18/0.45 itself: $tType > $tType ).
% 0.18/0.45
% 0.18/0.45 %----Explicit typings (46)
% 0.18/0.45 thf(sy_cl_Lattices_Obounded__lattice,type,
% 0.18/0.45 bounded_lattice:
% 0.18/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_HOL_Otype,type,
% 0.18/0.45 type:
% 0.18/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Lattices_Olattice,type,
% 0.18/0.45 lattice:
% 0.18/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Lattices_Odistrib__lattice,type,
% 0.18/0.45 distrib_lattice:
% 0.18/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Lattices_Osemilattice__inf,type,
% 0.18/0.45 semilattice_inf:
% 0.18/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 0.18/0.45 semilattice_sup:
% 0.18/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Lattices_Obounded__lattice__top,type,
% 0.18/0.45 bounded_lattice_top:
% 0.18/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Lattices_Obounded__semilattice__inf__top,type,
% 0.18/0.45 bounde1561333602nf_top:
% 0.18/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
% 0.18/0.45 bNF_Greatest_image2:
% 0.18/0.45 !>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_DTree_Ocont,type,
% 0.29/0.45 cont: dtree > ( set @ ( sum_sum @ t @ dtree ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Fun_Ocomp,type,
% 0.29/0.45 comp:
% 0.29/0.45 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_If,type,
% 0.29/0.45 if:
% 0.29/0.45 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 0.29/0.45 inf_inf:
% 0.29/0.45 !>[A: $tType] : ( A > A > A ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Lattices_Osup__class_Osup,type,
% 0.29/0.45 sup_sup:
% 0.29/0.45 !>[A: $tType] : ( A > A > A ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.29/0.45 top_top:
% 0.29/0.45 !>[A: $tType] : A ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar,type,
% 0.29/0.45 parall1899940088le_par: ( product_prod @ dtree @ dtree ) > dtree ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Product__Type_OPair,type,
% 0.29/0.45 product_Pair:
% 0.29/0.45 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Product__Type_Ocurry,type,
% 0.29/0.45 product_curry:
% 0.29/0.45 !>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.29/0.45 produc2004651681e_prod:
% 0.29/0.45 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Product__Type_Omap__prod,type,
% 0.29/0.45 product_map_prod:
% 0.29/0.45 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.29/0.45 product_rec_prod:
% 0.29/0.45 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
% 0.29/0.45 product_case_prod:
% 0.29/0.45 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 0.29/0.45 product_swap:
% 0.29/0.45 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Product__Type_Oscomp,type,
% 0.29/0.45 product_scomp:
% 0.29/0.45 !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( B > C > D ) > A > D ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Pure_Otype,type,
% 0.29/0.45 type2:
% 0.29/0.45 !>[A: $tType] : ( itself @ A ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_OCollect,type,
% 0.29/0.45 collect:
% 0.29/0.45 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Oimage,type,
% 0.29/0.45 image:
% 0.29/0.45 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Ovimage,type,
% 0.29/0.45 vimage:
% 0.29/0.45 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ B ) > ( set @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Sum__Type_OInl,type,
% 0.29/0.45 sum_Inl:
% 0.29/0.45 !>[A: $tType,B: $tType] : ( A > ( sum_sum @ A @ B ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Sum__Type_OInr,type,
% 0.29/0.45 sum_Inr:
% 0.29/0.45 !>[B: $tType,A: $tType] : ( B > ( sum_sum @ A @ B ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Sum__Type_OPlus,type,
% 0.29/0.45 sum_Plus:
% 0.29/0.45 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( sum_sum @ A @ B ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Sum__Type_OSuml,type,
% 0.29/0.45 sum_Suml:
% 0.29/0.45 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( sum_sum @ A @ B ) > C ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Sum__Type_OSumr,type,
% 0.29/0.45 sum_Sumr:
% 0.29/0.45 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( sum_sum @ A @ B ) > C ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Sum__Type_Oold_Osum_Orec__sum,type,
% 0.29/0.45 sum_rec_sum:
% 0.29/0.45 !>[A: $tType,T: $tType,B: $tType] : ( ( A > T ) > ( B > T ) > ( sum_sum @ A @ B ) > T ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Sum__Type_Osum_Ocase__sum,type,
% 0.29/0.45 sum_case_sum:
% 0.29/0.45 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( B > C ) > ( sum_sum @ A @ B ) > C ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Typedef_Otype__definition,type,
% 0.29/0.45 type_definition:
% 0.29/0.45 !>[B: $tType,A: $tType] : ( ( B > A ) > ( A > B ) > ( set @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_member,type,
% 0.29/0.45 member:
% 0.29/0.45 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_tr1_H____,type,
% 0.29/0.45 tr1: dtree ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_tr1a____,type,
% 0.29/0.45 tr1a: dtree ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_tr2_H____,type,
% 0.29/0.45 tr2: dtree ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_tr2a____,type,
% 0.29/0.45 tr2a: dtree ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_tr3_H____,type,
% 0.29/0.45 tr3: dtree ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_tr3a____,type,
% 0.29/0.45 tr3a: dtree ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_trA_H____,type,
% 0.29/0.45 trA: dtree ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_trA____,type,
% 0.29/0.45 trA2: dtree ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_trB____,type,
% 0.29/0.45 trB: dtree ).
% 0.29/0.45
% 0.29/0.45 %----Relevant facts (256)
% 0.29/0.45 thf(fact_0__092_060open_062_092_060exists_062tr1_Atr2_Atr3_O_AtrA_A_061_A_Itr1_A_092_060parallel_062_Atr2_J_A_092_060parallel_062_Atr3_A_092_060and_062_AtrB_A_061_Atr1_A_092_060parallel_062_Atr2_A_092_060parallel_062_Atr3_092_060close_062,axiom,
% 0.29/0.45 ? [Tr1: dtree,Tr2: dtree,Tr3: dtree] :
% 0.29/0.45 ( ( trA2
% 0.29/0.45 = ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) ) @ Tr3 ) ) )
% 0.29/0.45 & ( trB
% 0.29/0.45 = ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr1 @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr2 @ Tr3 ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % \<open>\<exists>tr1 tr2 tr3. trA = (tr1 \<parallel> tr2) \<parallel> tr3 \<and> trB = tr1 \<parallel> tr2 \<parallel> tr3\<close>
% 0.29/0.45 thf(fact_1__092_060open_062Inr_Atr1_H_A_092_060in_062_Acont_Atr1_092_060close_062,axiom,
% 0.29/0.45 member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ tr1 ) @ ( cont @ tr1a ) ).
% 0.29/0.45
% 0.29/0.45 % \<open>Inr tr1' \<in> cont tr1\<close>
% 0.29/0.45 thf(fact_2__092_060open_062Inr_Atr2_H_A_092_060in_062_Acont_Atr2_092_060close_062,axiom,
% 0.29/0.45 member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ tr2 ) @ ( cont @ tr2a ) ).
% 0.29/0.45
% 0.29/0.45 % \<open>Inr tr2' \<in> cont tr2\<close>
% 0.29/0.45 thf(fact_3__092_060open_062Inr_Atr3_H_A_092_060in_062_Acont_Atr3_092_060close_062,axiom,
% 0.29/0.45 member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ tr3 ) @ ( cont @ tr3a ) ).
% 0.29/0.45
% 0.29/0.45 % \<open>Inr tr3' \<in> cont tr3\<close>
% 0.29/0.45 thf(fact_4__092_060open_062trA_H_A_061_A_Itr1_H_A_092_060parallel_062_Atr2_H_J_A_092_060parallel_062_Atr3_H_092_060close_062,axiom,
% 0.29/0.45 ( trA
% 0.29/0.45 = ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ tr1 @ tr2 ) ) @ tr3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % \<open>trA' = (tr1' \<parallel> tr2') \<parallel> tr3'\<close>
% 0.29/0.45 thf(fact_5__092_060open_062Inr_AtrA_H_A_092_060in_062_Acont_A_I_Itr1_A_092_060parallel_062_Atr2_J_A_092_060parallel_062_Atr3_J_092_060close_062,axiom,
% 0.29/0.45 member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ trA ) @ ( cont @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ tr1a @ tr2a ) ) @ tr3a ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % \<open>Inr trA' \<in> cont ((tr1 \<parallel> tr2) \<parallel> tr3)\<close>
% 0.29/0.45 thf(fact_6__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062tr1_H_Atr2_H_Atr3_H_O_A_092_060lbrakk_062trA_H_A_061_A_Itr1_H_A_092_060parallel_062_Atr2_H_J_A_092_060parallel_062_Atr3_H_059_AInr_Atr1_H_A_092_060in_062_Acont_Atr1_059_AInr_Atr2_H_A_092_060in_062_Acont_Atr2_059_AInr_Atr3_H_A_092_060in_062_Acont_Atr3_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.29/0.45 ~ ! [Tr12: dtree,Tr22: dtree,Tr32: dtree] :
% 0.29/0.45 ( ( trA
% 0.29/0.45 = ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) ) @ Tr32 ) ) )
% 0.29/0.45 => ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ tr1a ) )
% 0.29/0.45 => ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr22 ) @ ( cont @ tr2a ) )
% 0.29/0.45 => ~ ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr32 ) @ ( cont @ tr3a ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % \<open>\<And>thesis. (\<And>tr1' tr2' tr3'. \<lbrakk>trA' = (tr1' \<parallel> tr2') \<parallel> tr3'; Inr tr1' \<in> cont tr1; Inr tr2' \<in> cont tr2; Inr tr3' \<in> cont tr3\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.29/0.45 thf(fact_7_par__com,axiom,
% 0.29/0.45 ! [Tr13: dtree,Tr23: dtree] :
% 0.29/0.45 ( ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr13 @ Tr23 ) )
% 0.29/0.45 = ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr23 @ Tr13 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % par_com
% 0.29/0.45 thf(fact_8_par__r_Ocases,axiom,
% 0.29/0.45 ! [X: product_prod @ dtree @ dtree] :
% 0.29/0.45 ~ ! [Tr1: dtree,Tr2: dtree] :
% 0.29/0.45 ( X
% 0.29/0.45 != ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % par_r.cases
% 0.29/0.45 thf(fact_9_par__r_Oinduct,axiom,
% 0.29/0.45 ! [P: ( product_prod @ dtree @ dtree ) > $o,A0: product_prod @ dtree @ dtree] :
% 0.29/0.45 ( ! [Tr1: dtree,Tr2: dtree] : ( P @ ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) )
% 0.29/0.45 => ( P @ A0 ) ) ).
% 0.29/0.45
% 0.29/0.45 % par_r.induct
% 0.29/0.45 thf(fact_10_sum_Oinject_I2_J,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,X2: B,Y2: B] :
% 0.29/0.45 ( ( ( sum_Inr @ B @ A @ X2 )
% 0.29/0.45 = ( sum_Inr @ B @ A @ Y2 ) )
% 0.29/0.45 = ( X2 = Y2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % sum.inject(2)
% 0.29/0.45 thf(fact_11_old_Osum_Oinject_I2_J,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,B2: B,B3: B] :
% 0.29/0.45 ( ( ( sum_Inr @ B @ A @ B2 )
% 0.29/0.45 = ( sum_Inr @ B @ A @ B3 ) )
% 0.29/0.45 = ( B2 = B3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % old.sum.inject(2)
% 0.29/0.45 thf(fact_12_prod_Oinject,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
% 0.29/0.45 ( ( ( product_Pair @ A @ B @ X1 @ X2 )
% 0.29/0.45 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 0.29/0.45 = ( ( X1 = Y1 )
% 0.29/0.45 & ( X2 = Y2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % prod.inject
% 0.29/0.45 thf(fact_13_old_Oprod_Oinject,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 0.29/0.45 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.29/0.45 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.29/0.45 = ( ( A2 = A3 )
% 0.29/0.45 & ( B2 = B3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % old.prod.inject
% 0.29/0.45 thf(fact_14_Inr__inject,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,X: B,Y: B] :
% 0.29/0.45 ( ( ( sum_Inr @ B @ A @ X )
% 0.29/0.45 = ( sum_Inr @ B @ A @ Y ) )
% 0.29/0.45 => ( X = Y ) ) ).
% 0.29/0.45
% 0.29/0.45 % Inr_inject
% 0.29/0.45 thf(fact_15_not__arg__cong__Inr,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,X: A,Y: A] :
% 0.29/0.45 ( ( X != Y )
% 0.29/0.45 => ( ( sum_Inr @ A @ B @ X )
% 0.29/0.45 != ( sum_Inr @ A @ B @ Y ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % not_arg_cong_Inr
% 0.29/0.45 thf(fact_16_Inl__in__cont__par,axiom,
% 0.29/0.45 ! [T2: t,Tr13: dtree,Tr23: dtree] :
% 0.29/0.45 ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr13 @ Tr23 ) ) ) )
% 0.29/0.45 = ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ Tr13 ) )
% 0.29/0.45 | ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ Tr23 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Inl_in_cont_par
% 0.29/0.45 thf(fact_17_surj__pair,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 0.29/0.45 ? [X3: A,Y3: B] :
% 0.29/0.45 ( P2
% 0.29/0.45 = ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % surj_pair
% 0.29/0.45 thf(fact_18_prod__cases,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 0.29/0.45 ( ! [A4: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 0.29/0.45 => ( P @ P2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % prod_cases
% 0.29/0.45 thf(fact_19_Pair__inject,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 0.29/0.45 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.29/0.45 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.29/0.45 => ~ ( ( A2 = A3 )
% 0.29/0.45 => ( B2 != B3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Pair_inject
% 0.29/0.45 thf(fact_20_prod__cases3,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.29/0.45 ~ ! [A4: A,B4: B,C2: C] :
% 0.29/0.45 ( Y
% 0.29/0.45 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % prod_cases3
% 0.29/0.45 thf(fact_21_prod__cases4,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.29/0.45 ~ ! [A4: A,B4: B,C2: C,D2: D] :
% 0.29/0.45 ( Y
% 0.29/0.45 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % prod_cases4
% 0.29/0.45 thf(fact_22_sum_Oinject_I1_J,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,X1: A,Y1: A] :
% 0.29/0.45 ( ( ( sum_Inl @ A @ B @ X1 )
% 0.29/0.45 = ( sum_Inl @ A @ B @ Y1 ) )
% 0.29/0.45 = ( X1 = Y1 ) ) ).
% 0.29/0.45
% 0.29/0.45 % sum.inject(1)
% 0.29/0.45 thf(fact_23_old_Osum_Oinject_I1_J,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,A2: A,A3: A] :
% 0.29/0.45 ( ( ( sum_Inl @ A @ B @ A2 )
% 0.29/0.45 = ( sum_Inl @ A @ B @ A3 ) )
% 0.29/0.45 = ( A2 = A3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % old.sum.inject(1)
% 0.29/0.45 thf(fact_24_Inl__inject,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,X: A,Y: A] :
% 0.29/0.45 ( ( ( sum_Inl @ A @ B @ X )
% 0.29/0.45 = ( sum_Inl @ A @ B @ Y ) )
% 0.29/0.45 => ( X = Y ) ) ).
% 0.29/0.45
% 0.29/0.45 % Inl_inject
% 0.29/0.45 thf(fact_25_obj__sumE,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,S: sum_sum @ A @ B] :
% 0.29/0.45 ( ! [X3: A] :
% 0.29/0.45 ( S
% 0.29/0.45 != ( sum_Inl @ A @ B @ X3 ) )
% 0.29/0.45 => ~ ! [X3: B] :
% 0.29/0.45 ( S
% 0.29/0.45 != ( sum_Inr @ B @ A @ X3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % obj_sumE
% 0.29/0.45 thf(fact_26_old_Osum_Oinducts,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,P: ( sum_sum @ A @ B ) > $o,Sum: sum_sum @ A @ B] :
% 0.29/0.45 ( ! [A4: A] : ( P @ ( sum_Inl @ A @ B @ A4 ) )
% 0.29/0.45 => ( ! [B4: B] : ( P @ ( sum_Inr @ B @ A @ B4 ) )
% 0.29/0.45 => ( P @ Sum ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % old.sum.inducts
% 0.29/0.45 thf(fact_27_old_Osum_Oexhaust,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,Y: sum_sum @ A @ B] :
% 0.29/0.45 ( ! [A4: A] :
% 0.29/0.45 ( Y
% 0.29/0.45 != ( sum_Inl @ A @ B @ A4 ) )
% 0.29/0.45 => ~ ! [B4: B] :
% 0.29/0.45 ( Y
% 0.29/0.45 != ( sum_Inr @ B @ A @ B4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % old.sum.exhaust
% 0.29/0.45 thf(fact_28_split__sum__all,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType] :
% 0.29/0.45 ( ( ^ [P3: ( sum_sum @ A @ B ) > $o] :
% 0.29/0.45 ! [X4: sum_sum @ A @ B] : ( P3 @ X4 ) )
% 0.29/0.45 = ( ^ [P4: ( sum_sum @ A @ B ) > $o] :
% 0.29/0.45 ( ! [X5: A] : ( P4 @ ( sum_Inl @ A @ B @ X5 ) )
% 0.29/0.45 & ! [X5: B] : ( P4 @ ( sum_Inr @ B @ A @ X5 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % split_sum_all
% 0.29/0.45 thf(fact_29_split__sum__ex,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType] :
% 0.29/0.45 ( ( ^ [P3: ( sum_sum @ A @ B ) > $o] :
% 0.29/0.45 ? [X4: sum_sum @ A @ B] : ( P3 @ X4 ) )
% 0.29/0.45 = ( ^ [P4: ( sum_sum @ A @ B ) > $o] :
% 0.29/0.45 ( ? [X5: A] : ( P4 @ ( sum_Inl @ A @ B @ X5 ) )
% 0.29/0.45 | ? [X5: B] : ( P4 @ ( sum_Inr @ B @ A @ X5 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % split_sum_ex
% 0.29/0.45 thf(fact_30_Inr__not__Inl,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,B2: B,A2: A] :
% 0.29/0.45 ( ( sum_Inr @ B @ A @ B2 )
% 0.29/0.45 != ( sum_Inl @ A @ B @ A2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % Inr_not_Inl
% 0.29/0.45 thf(fact_31_sumE,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,S: sum_sum @ A @ B] :
% 0.29/0.45 ( ! [X3: A] :
% 0.29/0.45 ( S
% 0.29/0.45 != ( sum_Inl @ A @ B @ X3 ) )
% 0.29/0.45 => ~ ! [Y3: B] :
% 0.29/0.45 ( S
% 0.29/0.45 != ( sum_Inr @ B @ A @ Y3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % sumE
% 0.29/0.45 thf(fact_32_old_Osum_Odistinct_I1_J,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,A2: A,B3: B] :
% 0.29/0.45 ( ( sum_Inl @ A @ B @ A2 )
% 0.29/0.45 != ( sum_Inr @ B @ A @ B3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % old.sum.distinct(1)
% 0.29/0.45 thf(fact_33_old_Osum_Odistinct_I2_J,axiom,
% 0.29/0.45 ! [B5: $tType,A5: $tType,B6: B5,A6: A5] :
% 0.29/0.45 ( ( sum_Inr @ B5 @ A5 @ B6 )
% 0.29/0.45 != ( sum_Inl @ A5 @ B5 @ A6 ) ) ).
% 0.29/0.45
% 0.29/0.45 % old.sum.distinct(2)
% 0.29/0.45 thf(fact_34_sum_Odistinct_I1_J,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,X1: A,X2: B] :
% 0.29/0.45 ( ( sum_Inl @ A @ B @ X1 )
% 0.29/0.45 != ( sum_Inr @ B @ A @ X2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % sum.distinct(1)
% 0.29/0.45 thf(fact_35_old_Oprod_Oinducts,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.29/0.45 ( ! [A4: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 0.29/0.45 => ( P @ Prod ) ) ).
% 0.29/0.45
% 0.29/0.45 % old.prod.inducts
% 0.29/0.45 thf(fact_36_old_Oprod_Oexhaust,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 0.29/0.45 ~ ! [A4: A,B4: B] :
% 0.29/0.45 ( Y
% 0.29/0.45 != ( product_Pair @ A @ B @ A4 @ B4 ) ) ).
% 0.29/0.45
% 0.29/0.45 % old.prod.exhaust
% 0.29/0.45 thf(fact_37_prod__induct7,axiom,
% 0.29/0.45 ! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 0.29/0.45 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
% 0.29/0.45 => ( P @ X ) ) ).
% 0.29/0.45
% 0.29/0.45 % prod_induct7
% 0.29/0.45 thf(fact_38_prod__induct6,axiom,
% 0.29/0.45 ! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 0.29/0.45 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
% 0.29/0.45 => ( P @ X ) ) ).
% 0.29/0.45
% 0.29/0.45 % prod_induct6
% 0.29/0.45 thf(fact_39_prod__induct5,axiom,
% 0.29/0.45 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.29/0.45 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 0.29/0.45 => ( P @ X ) ) ).
% 0.29/0.45
% 0.29/0.45 % prod_induct5
% 0.29/0.45 thf(fact_40_prod__induct4,axiom,
% 0.29/0.45 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.29/0.45 ( ! [A4: A,B4: B,C2: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
% 0.29/0.45 => ( P @ X ) ) ).
% 0.29/0.45
% 0.29/0.45 % prod_induct4
% 0.29/0.45 thf(fact_41_prod__induct3,axiom,
% 0.29/0.45 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.29/0.45 ( ! [A4: A,B4: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
% 0.29/0.45 => ( P @ X ) ) ).
% 0.29/0.45
% 0.29/0.45 % prod_induct3
% 0.29/0.45 thf(fact_42_prod__cases7,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 0.29/0.45 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] :
% 0.29/0.45 ( Y
% 0.29/0.45 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % prod_cases7
% 0.29/0.45 thf(fact_43_prod__cases6,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 0.29/0.45 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] :
% 0.29/0.45 ( Y
% 0.29/0.45 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % prod_cases6
% 0.29/0.45 thf(fact_44_prod__cases5,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.29/0.45 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E] :
% 0.29/0.45 ( Y
% 0.29/0.45 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % prod_cases5
% 0.29/0.45 thf(fact_45_mem__Collect__eq,axiom,
% 0.29/0.45 ! [A: $tType,A2: A,P: A > $o] :
% 0.29/0.45 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.29/0.45 = ( P @ A2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % mem_Collect_eq
% 0.29/0.45 thf(fact_46_Collect__mem__eq,axiom,
% 0.29/0.45 ! [A: $tType,A7: set @ A] :
% 0.29/0.45 ( ( collect @ A
% 0.29/0.45 @ ^ [X5: A] : ( member @ A @ X5 @ A7 ) )
% 0.29/0.45 = A7 ) ).
% 0.29/0.45
% 0.29/0.45 % Collect_mem_eq
% 0.29/0.45 thf(fact_47_Collect__cong,axiom,
% 0.29/0.45 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.29/0.45 ( ! [X3: A] :
% 0.29/0.45 ( ( P @ X3 )
% 0.29/0.45 = ( Q @ X3 ) )
% 0.29/0.45 => ( ( collect @ A @ P )
% 0.29/0.45 = ( collect @ A @ Q ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Collect_cong
% 0.29/0.45 thf(fact_48_ext,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,F3: A > B,G3: A > B] :
% 0.29/0.45 ( ! [X3: A] :
% 0.29/0.45 ( ( F3 @ X3 )
% 0.29/0.45 = ( G3 @ X3 ) )
% 0.29/0.45 => ( F3 = G3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % ext
% 0.29/0.45 thf(fact_49_Inr__Inl__False,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,X: B,Y: A] :
% 0.29/0.45 ( ( sum_Inr @ B @ A @ X )
% 0.29/0.45 != ( sum_Inl @ A @ B @ Y ) ) ).
% 0.29/0.45
% 0.29/0.45 % Inr_Inl_False
% 0.29/0.45 thf(fact_50_Inl__Inr__False,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,X: A,Y: B] :
% 0.29/0.45 ( ( sum_Inl @ A @ B @ X )
% 0.29/0.45 != ( sum_Inr @ B @ A @ Y ) ) ).
% 0.29/0.45
% 0.29/0.45 % Inl_Inr_False
% 0.29/0.45 thf(fact_51_old_Oprod_Orec,axiom,
% 0.29/0.45 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 0.29/0.45 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.29/0.45 = ( F1 @ A2 @ B2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % old.prod.rec
% 0.29/0.45 thf(fact_52_old_Osum_Osimps_I7_J,axiom,
% 0.29/0.45 ! [B: $tType,T: $tType,A: $tType,F1: A > T,F22: B > T,A2: A] :
% 0.29/0.45 ( ( sum_rec_sum @ A @ T @ B @ F1 @ F22 @ ( sum_Inl @ A @ B @ A2 ) )
% 0.29/0.45 = ( F1 @ A2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % old.sum.simps(7)
% 0.29/0.45 thf(fact_53_old_Osum_Osimps_I8_J,axiom,
% 0.29/0.45 ! [A: $tType,T: $tType,B: $tType,F1: A > T,F22: B > T,B2: B] :
% 0.29/0.45 ( ( sum_rec_sum @ A @ T @ B @ F1 @ F22 @ ( sum_Inr @ B @ A @ B2 ) )
% 0.29/0.45 = ( F22 @ B2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % old.sum.simps(8)
% 0.29/0.45 thf(fact_54_obj__sumE__f,axiom,
% 0.29/0.45 ! [A: $tType,C: $tType,B: $tType,S: B,F3: ( sum_sum @ A @ C ) > B,P: $o] :
% 0.29/0.45 ( ! [X3: A] :
% 0.29/0.45 ( ( S
% 0.29/0.45 = ( F3 @ ( sum_Inl @ A @ C @ X3 ) ) )
% 0.29/0.45 => P )
% 0.29/0.45 => ( ! [X3: C] :
% 0.29/0.45 ( ( S
% 0.29/0.45 = ( F3 @ ( sum_Inr @ C @ A @ X3 ) ) )
% 0.29/0.45 => P )
% 0.29/0.45 => ! [X6: sum_sum @ A @ C] :
% 0.29/0.45 ( ( S
% 0.29/0.45 = ( F3 @ X6 ) )
% 0.29/0.45 => P ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % obj_sumE_f
% 0.29/0.45 thf(fact_55_internal__case__prod__conv,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B2: C] :
% 0.29/0.45 ( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 0.29/0.45 = ( C3 @ A2 @ B2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % internal_case_prod_conv
% 0.29/0.45 thf(fact_56_Suml_Osimps,axiom,
% 0.29/0.45 ! [B: $tType,C: $tType,A: $tType,F3: A > C,X: A] :
% 0.29/0.45 ( ( sum_Suml @ A @ C @ B @ F3 @ ( sum_Inl @ A @ B @ X ) )
% 0.29/0.45 = ( F3 @ X ) ) ).
% 0.29/0.45
% 0.29/0.45 % Suml.simps
% 0.29/0.45 thf(fact_57_Sumr_Osimps,axiom,
% 0.29/0.45 ! [A: $tType,C: $tType,B: $tType,F3: B > C,X: B] :
% 0.29/0.45 ( ( sum_Sumr @ B @ C @ A @ F3 @ ( sum_Inr @ B @ A @ X ) )
% 0.29/0.45 = ( F3 @ X ) ) ).
% 0.29/0.45
% 0.29/0.45 % Sumr.simps
% 0.29/0.45 thf(fact_58_image2__eqI,axiom,
% 0.29/0.45 ! [A: $tType,C: $tType,B: $tType,B2: A,F3: B > A,X: B,C3: C,G3: B > C,A7: set @ B] :
% 0.29/0.45 ( ( B2
% 0.29/0.45 = ( F3 @ X ) )
% 0.29/0.45 => ( ( C3
% 0.29/0.45 = ( G3 @ X ) )
% 0.29/0.45 => ( ( member @ B @ X @ A7 )
% 0.29/0.45 => ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B2 @ C3 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A7 @ F3 @ G3 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % image2_eqI
% 0.29/0.45 thf(fact_59_Sumr__inject,axiom,
% 0.29/0.45 ! [A: $tType,C: $tType,B: $tType,F3: B > C,G3: B > C] :
% 0.29/0.45 ( ( ( sum_Sumr @ B @ C @ A @ F3 )
% 0.29/0.45 = ( sum_Sumr @ B @ C @ A @ G3 ) )
% 0.29/0.45 => ( F3 = G3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % Sumr_inject
% 0.29/0.45 thf(fact_60_Suml__inject,axiom,
% 0.29/0.45 ! [B: $tType,C: $tType,A: $tType,F3: A > C,G3: A > C] :
% 0.29/0.45 ( ( ( sum_Suml @ A @ C @ B @ F3 )
% 0.29/0.45 = ( sum_Suml @ A @ C @ B @ G3 ) )
% 0.29/0.45 => ( F3 = G3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % Suml_inject
% 0.29/0.45 thf(fact_61_ssubst__Pair__rhs,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,R: A,S: B,R2: set @ ( product_prod @ A @ B ),S2: B] :
% 0.29/0.45 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S ) @ R2 )
% 0.29/0.45 => ( ( S2 = S )
% 0.29/0.45 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S2 ) @ R2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ssubst_Pair_rhs
% 0.29/0.45 thf(fact_62_curry__conv,axiom,
% 0.29/0.45 ! [A: $tType,C: $tType,B: $tType] :
% 0.29/0.45 ( ( product_curry @ B @ C @ A )
% 0.29/0.45 = ( ^ [F4: ( product_prod @ B @ C ) > A,A8: B,B7: C] : ( F4 @ ( product_Pair @ B @ C @ A8 @ B7 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % curry_conv
% 0.29/0.45 thf(fact_63_curryI,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 0.29/0.45 ( ( F3 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.29/0.45 => ( product_curry @ A @ B @ $o @ F3 @ A2 @ B2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % curryI
% 0.29/0.45 thf(fact_64_case__sum__if,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,C: $tType,P2: $o,F3: B > A,G3: C > A,X: B,Y: C] :
% 0.29/0.45 ( ( P2
% 0.29/0.45 => ( ( sum_case_sum @ B @ A @ C @ F3 @ G3 @ ( if @ ( sum_sum @ B @ C ) @ P2 @ ( sum_Inl @ B @ C @ X ) @ ( sum_Inr @ C @ B @ Y ) ) )
% 0.29/0.45 = ( F3 @ X ) ) )
% 0.29/0.45 & ( ~ P2
% 0.29/0.45 => ( ( sum_case_sum @ B @ A @ C @ F3 @ G3 @ ( if @ ( sum_sum @ B @ C ) @ P2 @ ( sum_Inl @ B @ C @ X ) @ ( sum_Inr @ C @ B @ Y ) ) )
% 0.29/0.45 = ( G3 @ Y ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % case_sum_if
% 0.29/0.45 thf(fact_65_swap__simp,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,X: B,Y: A] :
% 0.29/0.45 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 0.29/0.45 = ( product_Pair @ A @ B @ Y @ X ) ) ).
% 0.29/0.45
% 0.29/0.45 % swap_simp
% 0.29/0.45 thf(fact_66_Inl__cont__par,axiom,
% 0.29/0.45 ! [Tr13: dtree,Tr23: dtree] :
% 0.29/0.45 ( ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( cont @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr13 @ Tr23 ) ) ) )
% 0.29/0.45 = ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( sup_sup @ ( set @ ( sum_sum @ t @ dtree ) ) @ ( cont @ Tr13 ) @ ( cont @ Tr23 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Inl_cont_par
% 0.29/0.45 thf(fact_67_internal__case__prod__def,axiom,
% 0.29/0.45 ! [C: $tType,B: $tType,A: $tType] :
% 0.29/0.45 ( ( produc2004651681e_prod @ A @ B @ C )
% 0.29/0.45 = ( product_case_prod @ A @ B @ C ) ) ).
% 0.29/0.45
% 0.29/0.45 % internal_case_prod_def
% 0.29/0.45 thf(fact_68_swap__swap,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 0.29/0.45 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
% 0.29/0.45 = P2 ) ).
% 0.29/0.45
% 0.29/0.45 % swap_swap
% 0.29/0.45 thf(fact_69_curry__case__prod,axiom,
% 0.29/0.45 ! [C: $tType,B: $tType,A: $tType,F3: A > B > C] :
% 0.29/0.45 ( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F3 ) )
% 0.29/0.45 = F3 ) ).
% 0.29/0.45
% 0.29/0.45 % curry_case_prod
% 0.29/0.45 thf(fact_70_case__prod__curry,axiom,
% 0.29/0.45 ! [C: $tType,B: $tType,A: $tType,F3: ( product_prod @ A @ B ) > C] :
% 0.29/0.45 ( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F3 ) )
% 0.29/0.45 = F3 ) ).
% 0.29/0.45
% 0.29/0.45 % case_prod_curry
% 0.29/0.45 thf(fact_71_case__sum__step_I2_J,axiom,
% 0.29/0.45 ! [E: $tType,A: $tType,C: $tType,B: $tType,F3: E > A,F5: B > A,G4: C > A,P2: sum_sum @ B @ C] :
% 0.29/0.45 ( ( sum_case_sum @ E @ A @ ( sum_sum @ B @ C ) @ F3 @ ( sum_case_sum @ B @ A @ C @ F5 @ G4 ) @ ( sum_Inr @ ( sum_sum @ B @ C ) @ E @ P2 ) )
% 0.29/0.45 = ( sum_case_sum @ B @ A @ C @ F5 @ G4 @ P2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % case_sum_step(2)
% 0.29/0.45 thf(fact_72_case__sum__step_I1_J,axiom,
% 0.29/0.45 ! [D: $tType,A: $tType,C: $tType,B: $tType,F5: B > A,G4: C > A,G3: D > A,P2: sum_sum @ B @ C] :
% 0.29/0.45 ( ( sum_case_sum @ ( sum_sum @ B @ C ) @ A @ D @ ( sum_case_sum @ B @ A @ C @ F5 @ G4 ) @ G3 @ ( sum_Inl @ ( sum_sum @ B @ C ) @ D @ P2 ) )
% 0.29/0.45 = ( sum_case_sum @ B @ A @ C @ F5 @ G4 @ P2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % case_sum_step(1)
% 0.29/0.45 thf(fact_73_case__sum__inject,axiom,
% 0.29/0.45 ! [A: $tType,C: $tType,B: $tType,F1: A > C,F22: B > C,G1: A > C,G22: B > C] :
% 0.29/0.45 ( ( ( sum_case_sum @ A @ C @ B @ F1 @ F22 )
% 0.29/0.45 = ( sum_case_sum @ A @ C @ B @ G1 @ G22 ) )
% 0.29/0.45 => ~ ( ( F1 = G1 )
% 0.29/0.45 => ( F22 != G22 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % case_sum_inject
% 0.29/0.45 thf(fact_74_mem__case__prodE,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,C: $tType,Z: A,C3: B > C > ( set @ A ),P2: product_prod @ B @ C] :
% 0.29/0.45 ( ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ P2 ) )
% 0.29/0.45 => ~ ! [X3: B,Y3: C] :
% 0.29/0.45 ( ( P2
% 0.29/0.45 = ( product_Pair @ B @ C @ X3 @ Y3 ) )
% 0.29/0.45 => ~ ( member @ A @ Z @ ( C3 @ X3 @ Y3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % mem_case_prodE
% 0.29/0.45 thf(fact_75_old_Oprod_Ocase,axiom,
% 0.29/0.45 ! [A: $tType,C: $tType,B: $tType,F3: A > B > C,X1: A,X2: B] :
% 0.29/0.45 ( ( product_case_prod @ A @ B @ C @ F3 @ ( product_Pair @ A @ B @ X1 @ X2 ) )
% 0.29/0.45 = ( F3 @ X1 @ X2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % old.prod.case
% 0.29/0.45 thf(fact_76_old_Osum_Osimps_I6_J,axiom,
% 0.29/0.45 ! [A: $tType,C: $tType,B: $tType,F1: A > C,F22: B > C,X2: B] :
% 0.29/0.45 ( ( sum_case_sum @ A @ C @ B @ F1 @ F22 @ ( sum_Inr @ B @ A @ X2 ) )
% 0.29/0.45 = ( F22 @ X2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % old.sum.simps(6)
% 0.29/0.45 thf(fact_77_old_Osum_Osimps_I5_J,axiom,
% 0.29/0.45 ! [B: $tType,C: $tType,A: $tType,F1: A > C,F22: B > C,X1: A] :
% 0.29/0.45 ( ( sum_case_sum @ A @ C @ B @ F1 @ F22 @ ( sum_Inl @ A @ B @ X1 ) )
% 0.29/0.45 = ( F1 @ X1 ) ) ).
% 0.29/0.45
% 0.29/0.45 % old.sum.simps(5)
% 0.29/0.45 thf(fact_78_curryE,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 0.29/0.45 ( ( product_curry @ A @ B @ $o @ F3 @ A2 @ B2 )
% 0.29/0.45 => ( F3 @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % curryE
% 0.29/0.45 thf(fact_79_curryD,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 0.29/0.45 ( ( product_curry @ A @ B @ $o @ F3 @ A2 @ B2 )
% 0.29/0.45 => ( F3 @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % curryD
% 0.29/0.45 thf(fact_80_vimage__Un,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,F3: A > B,A7: set @ B,B8: set @ B] :
% 0.29/0.45 ( ( vimage @ A @ B @ F3 @ ( sup_sup @ ( set @ B ) @ A7 @ B8 ) )
% 0.29/0.45 = ( sup_sup @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ A7 ) @ ( vimage @ A @ B @ F3 @ B8 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % vimage_Un
% 0.29/0.45 thf(fact_81_Inl__Inr__image__cong,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,A7: set @ ( sum_sum @ A @ B ),B8: set @ ( sum_sum @ A @ B )] :
% 0.29/0.45 ( ( ( vimage @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ A7 )
% 0.29/0.45 = ( vimage @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ B8 ) )
% 0.29/0.45 => ( ( ( vimage @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ A7 )
% 0.29/0.45 = ( vimage @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ B8 ) )
% 0.29/0.45 => ( A7 = B8 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Inl_Inr_image_cong
% 0.29/0.45 thf(fact_82_vimageI,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,F3: B > A,A2: B,B2: A,B8: set @ A] :
% 0.29/0.45 ( ( ( F3 @ A2 )
% 0.29/0.45 = B2 )
% 0.29/0.45 => ( ( member @ A @ B2 @ B8 )
% 0.29/0.45 => ( member @ B @ A2 @ ( vimage @ B @ A @ F3 @ B8 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % vimageI
% 0.29/0.45 thf(fact_83_vimage__eq,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,A2: A,F3: A > B,B8: set @ B] :
% 0.29/0.45 ( ( member @ A @ A2 @ ( vimage @ A @ B @ F3 @ B8 ) )
% 0.29/0.45 = ( member @ B @ ( F3 @ A2 ) @ B8 ) ) ).
% 0.29/0.45
% 0.29/0.45 % vimage_eq
% 0.29/0.45 thf(fact_84_UnCI,axiom,
% 0.29/0.45 ! [A: $tType,C3: A,B8: set @ A,A7: set @ A] :
% 0.29/0.45 ( ( ~ ( member @ A @ C3 @ B8 )
% 0.29/0.45 => ( member @ A @ C3 @ A7 ) )
% 0.29/0.45 => ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A7 @ B8 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % UnCI
% 0.29/0.45 thf(fact_85_Un__iff,axiom,
% 0.29/0.45 ! [A: $tType,C3: A,A7: set @ A,B8: set @ A] :
% 0.29/0.45 ( ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A7 @ B8 ) )
% 0.29/0.45 = ( ( member @ A @ C3 @ A7 )
% 0.29/0.45 | ( member @ A @ C3 @ B8 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Un_iff
% 0.29/0.45 thf(fact_86_sup__apply,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType] :
% 0.29/0.45 ( ( semilattice_sup @ B @ ( type2 @ B ) )
% 0.29/0.45 => ( ( sup_sup @ ( A > B ) )
% 0.29/0.45 = ( ^ [F4: A > B,G5: A > B,X5: A] : ( sup_sup @ B @ ( F4 @ X5 ) @ ( G5 @ X5 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % sup_apply
% 0.29/0.45 thf(fact_87_sup_Oright__idem,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B2: A] :
% 0.29/0.45 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ B2 )
% 0.29/0.45 = ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % sup.right_idem
% 0.29/0.45 thf(fact_88_sup__left__idem,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 0.29/0.45 = ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % sup_left_idem
% 0.29/0.45 thf(fact_89_sup_Oleft__idem,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B2: A] :
% 0.29/0.45 ( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) )
% 0.29/0.45 = ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % sup.left_idem
% 0.29/0.45 thf(fact_90_sup__idem,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A] :
% 0.29/0.45 ( ( sup_sup @ A @ X @ X )
% 0.29/0.45 = X ) ) ).
% 0.29/0.45
% 0.29/0.45 % sup_idem
% 0.29/0.45 thf(fact_91_sup_Oidem,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ( ( sup_sup @ A @ A2 @ A2 )
% 0.29/0.45 = A2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % sup.idem
% 0.29/0.45 thf(fact_92_sup__left__commute,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A,Z: A] :
% 0.29/0.45 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
% 0.29/0.45 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % sup_left_commute
% 0.29/0.45 thf(fact_93_sup_Oleft__commute,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [B2: A,A2: A,C3: A] :
% 0.29/0.45 ( ( sup_sup @ A @ B2 @ ( sup_sup @ A @ A2 @ C3 ) )
% 0.29/0.45 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % sup.left_commute
% 0.29/0.45 thf(fact_94_sup__commute,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.45 => ( ( sup_sup @ A )
% 0.29/0.45 = ( ^ [X5: A,Y4: A] : ( sup_sup @ A @ Y4 @ X5 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % sup_commute
% 0.29/0.45 thf(fact_95_sup_Ocommute,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.45 => ( ( sup_sup @ A )
% 0.29/0.45 = ( ^ [A8: A,B7: A] : ( sup_sup @ A @ B7 @ A8 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % sup.commute
% 0.29/0.45 thf(fact_96_sup__assoc,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A,Z: A] :
% 0.29/0.45 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
% 0.29/0.45 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % sup_assoc
% 0.29/0.45 thf(fact_97_sup_Oassoc,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B2: A,C3: A] :
% 0.29/0.45 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ C3 )
% 0.29/0.45 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % sup.assoc
% 0.29/0.45 thf(fact_98_sup__fun__def,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType] :
% 0.29/0.45 ( ( semilattice_sup @ B @ ( type2 @ B ) )
% 0.29/0.45 => ( ( sup_sup @ ( A > B ) )
% 0.29/0.45 = ( ^ [F4: A > B,G5: A > B,X5: A] : ( sup_sup @ B @ ( F4 @ X5 ) @ ( G5 @ X5 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % sup_fun_def
% 0.29/0.45 thf(fact_99_inf__sup__aci_I5_J,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( lattice @ A @ ( type2 @ A ) )
% 0.29/0.45 => ( ( sup_sup @ A )
% 0.29/0.45 = ( ^ [X5: A,Y4: A] : ( sup_sup @ A @ Y4 @ X5 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf_sup_aci(5)
% 0.29/0.45 thf(fact_100_inf__sup__aci_I6_J,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( lattice @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A,Z: A] :
% 0.29/0.45 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
% 0.29/0.45 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf_sup_aci(6)
% 0.29/0.45 thf(fact_101_inf__sup__aci_I7_J,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( lattice @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A,Z: A] :
% 0.29/0.45 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
% 0.29/0.45 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf_sup_aci(7)
% 0.29/0.45 thf(fact_102_inf__sup__aci_I8_J,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( lattice @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 0.29/0.45 = ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf_sup_aci(8)
% 0.29/0.45 thf(fact_103_Un__left__commute,axiom,
% 0.29/0.45 ! [A: $tType,A7: set @ A,B8: set @ A,C4: set @ A] :
% 0.29/0.45 ( ( sup_sup @ ( set @ A ) @ A7 @ ( sup_sup @ ( set @ A ) @ B8 @ C4 ) )
% 0.29/0.45 = ( sup_sup @ ( set @ A ) @ B8 @ ( sup_sup @ ( set @ A ) @ A7 @ C4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Un_left_commute
% 0.29/0.45 thf(fact_104_Un__left__absorb,axiom,
% 0.29/0.45 ! [A: $tType,A7: set @ A,B8: set @ A] :
% 0.29/0.45 ( ( sup_sup @ ( set @ A ) @ A7 @ ( sup_sup @ ( set @ A ) @ A7 @ B8 ) )
% 0.29/0.45 = ( sup_sup @ ( set @ A ) @ A7 @ B8 ) ) ).
% 0.29/0.45
% 0.29/0.45 % Un_left_absorb
% 0.29/0.45 thf(fact_105_Un__commute,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( sup_sup @ ( set @ A ) )
% 0.29/0.45 = ( ^ [A9: set @ A,B9: set @ A] : ( sup_sup @ ( set @ A ) @ B9 @ A9 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Un_commute
% 0.29/0.45 thf(fact_106_Un__absorb,axiom,
% 0.29/0.45 ! [A: $tType,A7: set @ A] :
% 0.29/0.45 ( ( sup_sup @ ( set @ A ) @ A7 @ A7 )
% 0.29/0.45 = A7 ) ).
% 0.29/0.45
% 0.29/0.45 % Un_absorb
% 0.29/0.45 thf(fact_107_Un__assoc,axiom,
% 0.29/0.45 ! [A: $tType,A7: set @ A,B8: set @ A,C4: set @ A] :
% 0.29/0.45 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A7 @ B8 ) @ C4 )
% 0.29/0.45 = ( sup_sup @ ( set @ A ) @ A7 @ ( sup_sup @ ( set @ A ) @ B8 @ C4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Un_assoc
% 0.29/0.45 thf(fact_108_ball__Un,axiom,
% 0.29/0.45 ! [A: $tType,A7: set @ A,B8: set @ A,P: A > $o] :
% 0.29/0.45 ( ( ! [X5: A] :
% 0.29/0.45 ( ( member @ A @ X5 @ ( sup_sup @ ( set @ A ) @ A7 @ B8 ) )
% 0.29/0.45 => ( P @ X5 ) ) )
% 0.29/0.45 = ( ! [X5: A] :
% 0.29/0.45 ( ( member @ A @ X5 @ A7 )
% 0.29/0.45 => ( P @ X5 ) )
% 0.29/0.45 & ! [X5: A] :
% 0.29/0.45 ( ( member @ A @ X5 @ B8 )
% 0.29/0.45 => ( P @ X5 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ball_Un
% 0.29/0.45 thf(fact_109_bex__Un,axiom,
% 0.29/0.45 ! [A: $tType,A7: set @ A,B8: set @ A,P: A > $o] :
% 0.29/0.45 ( ( ? [X5: A] :
% 0.29/0.45 ( ( member @ A @ X5 @ ( sup_sup @ ( set @ A ) @ A7 @ B8 ) )
% 0.29/0.45 & ( P @ X5 ) ) )
% 0.29/0.45 = ( ? [X5: A] :
% 0.29/0.45 ( ( member @ A @ X5 @ A7 )
% 0.29/0.45 & ( P @ X5 ) )
% 0.29/0.45 | ? [X5: A] :
% 0.29/0.45 ( ( member @ A @ X5 @ B8 )
% 0.29/0.45 & ( P @ X5 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % bex_Un
% 0.29/0.45 thf(fact_110_UnI2,axiom,
% 0.29/0.45 ! [A: $tType,C3: A,B8: set @ A,A7: set @ A] :
% 0.29/0.45 ( ( member @ A @ C3 @ B8 )
% 0.29/0.45 => ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A7 @ B8 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % UnI2
% 0.29/0.45 thf(fact_111_UnI1,axiom,
% 0.29/0.45 ! [A: $tType,C3: A,A7: set @ A,B8: set @ A] :
% 0.29/0.45 ( ( member @ A @ C3 @ A7 )
% 0.29/0.45 => ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A7 @ B8 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % UnI1
% 0.29/0.45 thf(fact_112_UnE,axiom,
% 0.29/0.45 ! [A: $tType,C3: A,A7: set @ A,B8: set @ A] :
% 0.29/0.45 ( ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A7 @ B8 ) )
% 0.29/0.45 => ( ~ ( member @ A @ C3 @ A7 )
% 0.29/0.45 => ( member @ A @ C3 @ B8 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % UnE
% 0.29/0.45 thf(fact_113_vimage__Collect,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,P: B > $o,F3: A > B,Q: A > $o] :
% 0.29/0.45 ( ! [X3: A] :
% 0.29/0.45 ( ( P @ ( F3 @ X3 ) )
% 0.29/0.45 = ( Q @ X3 ) )
% 0.29/0.45 => ( ( vimage @ A @ B @ F3 @ ( collect @ B @ P ) )
% 0.29/0.45 = ( collect @ A @ Q ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % vimage_Collect
% 0.29/0.45 thf(fact_114_vimageI2,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,F3: B > A,A2: B,A7: set @ A] :
% 0.29/0.45 ( ( member @ A @ ( F3 @ A2 ) @ A7 )
% 0.29/0.45 => ( member @ B @ A2 @ ( vimage @ B @ A @ F3 @ A7 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % vimageI2
% 0.29/0.45 thf(fact_115_vimageE,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,A2: A,F3: A > B,B8: set @ B] :
% 0.29/0.45 ( ( member @ A @ A2 @ ( vimage @ A @ B @ F3 @ B8 ) )
% 0.29/0.45 => ( member @ B @ ( F3 @ A2 ) @ B8 ) ) ).
% 0.29/0.45
% 0.29/0.45 % vimageE
% 0.29/0.45 thf(fact_116_vimageD,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,A2: A,F3: A > B,A7: set @ B] :
% 0.29/0.45 ( ( member @ A @ A2 @ ( vimage @ A @ B @ F3 @ A7 ) )
% 0.29/0.45 => ( member @ B @ ( F3 @ A2 ) @ A7 ) ) ).
% 0.29/0.45
% 0.29/0.45 % vimageD
% 0.29/0.45 thf(fact_117_split__cong,axiom,
% 0.29/0.45 ! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F3: A > B > C,G3: A > B > C,P2: product_prod @ A @ B] :
% 0.29/0.45 ( ! [X3: A,Y3: B] :
% 0.29/0.45 ( ( ( product_Pair @ A @ B @ X3 @ Y3 )
% 0.29/0.45 = Q2 )
% 0.29/0.45 => ( ( F3 @ X3 @ Y3 )
% 0.29/0.45 = ( G3 @ X3 @ Y3 ) ) )
% 0.29/0.45 => ( ( P2 = Q2 )
% 0.29/0.45 => ( ( product_case_prod @ A @ B @ C @ F3 @ P2 )
% 0.29/0.45 = ( product_case_prod @ A @ B @ C @ G3 @ Q2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % split_cong
% 0.29/0.45 thf(fact_118_case__sum__expand__Inr__pointfree,axiom,
% 0.29/0.45 ! [B: $tType,C: $tType,A: $tType,F3: ( sum_sum @ A @ C ) > B,G3: A > B] :
% 0.29/0.45 ( ( ( comp @ ( sum_sum @ A @ C ) @ B @ A @ F3 @ ( sum_Inl @ A @ C ) )
% 0.29/0.45 = G3 )
% 0.29/0.45 => ( ( sum_case_sum @ A @ B @ C @ G3 @ ( comp @ ( sum_sum @ A @ C ) @ B @ C @ F3 @ ( sum_Inr @ C @ A ) ) )
% 0.29/0.45 = F3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % case_sum_expand_Inr_pointfree
% 0.29/0.45 thf(fact_119_case__sum__expand__Inr_H,axiom,
% 0.29/0.45 ! [B: $tType,C: $tType,A: $tType,F3: ( sum_sum @ A @ C ) > B,G3: A > B,H: C > B] :
% 0.29/0.45 ( ( ( comp @ ( sum_sum @ A @ C ) @ B @ A @ F3 @ ( sum_Inl @ A @ C ) )
% 0.29/0.45 = G3 )
% 0.29/0.45 => ( ( H
% 0.29/0.45 = ( comp @ ( sum_sum @ A @ C ) @ B @ C @ F3 @ ( sum_Inr @ C @ A ) ) )
% 0.29/0.45 = ( ( sum_case_sum @ A @ B @ C @ G3 @ H )
% 0.29/0.45 = F3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % case_sum_expand_Inr'
% 0.29/0.45 thf(fact_120_case__sum__expand__Inr,axiom,
% 0.29/0.45 ! [B: $tType,C: $tType,A: $tType,F3: ( sum_sum @ A @ C ) > B,G3: A > B,X: sum_sum @ A @ C] :
% 0.29/0.45 ( ( ( comp @ ( sum_sum @ A @ C ) @ B @ A @ F3 @ ( sum_Inl @ A @ C ) )
% 0.29/0.45 = G3 )
% 0.29/0.45 => ( ( F3 @ X )
% 0.29/0.45 = ( sum_case_sum @ A @ B @ C @ G3 @ ( comp @ ( sum_sum @ A @ C ) @ B @ C @ F3 @ ( sum_Inr @ C @ A ) ) @ X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % case_sum_expand_Inr
% 0.29/0.45 thf(fact_121_scomp__apply,axiom,
% 0.29/0.45 ! [A: $tType,D: $tType,C: $tType,B: $tType] :
% 0.29/0.45 ( ( product_scomp @ B @ C @ D @ A )
% 0.29/0.45 = ( ^ [F4: B > ( product_prod @ C @ D ),G5: C > D > A,X5: B] : ( product_case_prod @ C @ D @ A @ G5 @ ( F4 @ X5 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % scomp_apply
% 0.29/0.45 thf(fact_122_pair__in__swap__image,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,Y: A,X: B,A7: set @ ( product_prod @ B @ A )] :
% 0.29/0.45 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A7 ) )
% 0.29/0.45 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A7 ) ) ).
% 0.29/0.45
% 0.29/0.45 % pair_in_swap_image
% 0.29/0.45 thf(fact_123_image__eqI,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,B2: A,F3: B > A,X: B,A7: set @ B] :
% 0.29/0.45 ( ( B2
% 0.29/0.45 = ( F3 @ X ) )
% 0.29/0.45 => ( ( member @ B @ X @ A7 )
% 0.29/0.45 => ( member @ A @ B2 @ ( image @ B @ A @ F3 @ A7 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % image_eqI
% 0.29/0.45 thf(fact_124_o__case__sum,axiom,
% 0.29/0.45 ! [A: $tType,C: $tType,D: $tType,B: $tType,H: D > C,F3: A > D,G3: B > D] :
% 0.29/0.45 ( ( comp @ D @ C @ ( sum_sum @ A @ B ) @ H @ ( sum_case_sum @ A @ D @ B @ F3 @ G3 ) )
% 0.29/0.45 = ( sum_case_sum @ A @ C @ B @ ( comp @ D @ C @ A @ H @ F3 ) @ ( comp @ D @ C @ B @ H @ G3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % o_case_sum
% 0.29/0.45 thf(fact_125_rev__image__eqI,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,X: A,A7: set @ A,B2: B,F3: A > B] :
% 0.29/0.45 ( ( member @ A @ X @ A7 )
% 0.29/0.45 => ( ( B2
% 0.29/0.45 = ( F3 @ X ) )
% 0.29/0.45 => ( member @ B @ B2 @ ( image @ A @ B @ F3 @ A7 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % rev_image_eqI
% 0.29/0.45 thf(fact_126_ball__imageD,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,F3: B > A,A7: set @ B,P: A > $o] :
% 0.29/0.45 ( ! [X3: A] :
% 0.29/0.45 ( ( member @ A @ X3 @ ( image @ B @ A @ F3 @ A7 ) )
% 0.29/0.45 => ( P @ X3 ) )
% 0.29/0.45 => ! [X6: B] :
% 0.29/0.45 ( ( member @ B @ X6 @ A7 )
% 0.29/0.45 => ( P @ ( F3 @ X6 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ball_imageD
% 0.29/0.45 thf(fact_127_image__cong,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,M: set @ A,N: set @ A,F3: A > B,G3: A > B] :
% 0.29/0.45 ( ( M = N )
% 0.29/0.45 => ( ! [X3: A] :
% 0.29/0.45 ( ( member @ A @ X3 @ N )
% 0.29/0.45 => ( ( F3 @ X3 )
% 0.29/0.45 = ( G3 @ X3 ) ) )
% 0.29/0.45 => ( ( image @ A @ B @ F3 @ M )
% 0.29/0.45 = ( image @ A @ B @ G3 @ N ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % image_cong
% 0.29/0.45 thf(fact_128_bex__imageD,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,F3: B > A,A7: set @ B,P: A > $o] :
% 0.29/0.45 ( ? [X6: A] :
% 0.29/0.45 ( ( member @ A @ X6 @ ( image @ B @ A @ F3 @ A7 ) )
% 0.29/0.45 & ( P @ X6 ) )
% 0.29/0.45 => ? [X3: B] :
% 0.29/0.45 ( ( member @ B @ X3 @ A7 )
% 0.29/0.45 & ( P @ ( F3 @ X3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % bex_imageD
% 0.29/0.45 thf(fact_129_image__iff,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,Z: A,F3: B > A,A7: set @ B] :
% 0.29/0.45 ( ( member @ A @ Z @ ( image @ B @ A @ F3 @ A7 ) )
% 0.29/0.45 = ( ? [X5: B] :
% 0.29/0.45 ( ( member @ B @ X5 @ A7 )
% 0.29/0.45 & ( Z
% 0.29/0.45 = ( F3 @ X5 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % image_iff
% 0.29/0.45 thf(fact_130_imageI,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,X: A,A7: set @ A,F3: A > B] :
% 0.29/0.45 ( ( member @ A @ X @ A7 )
% 0.29/0.45 => ( member @ B @ ( F3 @ X ) @ ( image @ A @ B @ F3 @ A7 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % imageI
% 0.29/0.45 thf(fact_131_rewriteR__comp__comp2,axiom,
% 0.29/0.45 ! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G3: C > B,H: A > C,R1: D > B,R22: A > D,F3: B > E,L: D > E] :
% 0.29/0.45 ( ( ( comp @ C @ B @ A @ G3 @ H )
% 0.29/0.45 = ( comp @ D @ B @ A @ R1 @ R22 ) )
% 0.29/0.45 => ( ( ( comp @ B @ E @ D @ F3 @ R1 )
% 0.29/0.45 = L )
% 0.29/0.45 => ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F3 @ G3 ) @ H )
% 0.29/0.45 = ( comp @ D @ E @ A @ L @ R22 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % rewriteR_comp_comp2
% 0.29/0.45 thf(fact_132_rewriteL__comp__comp2,axiom,
% 0.29/0.45 ! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F3: C > B,G3: A > C,L1: D > B,L2: A > D,H: E > A,R: E > D] :
% 0.29/0.45 ( ( ( comp @ C @ B @ A @ F3 @ G3 )
% 0.29/0.45 = ( comp @ D @ B @ A @ L1 @ L2 ) )
% 0.29/0.45 => ( ( ( comp @ A @ D @ E @ L2 @ H )
% 0.29/0.45 = R )
% 0.29/0.45 => ( ( comp @ C @ B @ E @ F3 @ ( comp @ A @ C @ E @ G3 @ H ) )
% 0.29/0.45 = ( comp @ D @ B @ E @ L1 @ R ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % rewriteL_comp_comp2
% 0.29/0.45 thf(fact_133_rewriteR__comp__comp,axiom,
% 0.29/0.45 ! [C: $tType,D: $tType,B: $tType,A: $tType,G3: C > B,H: A > C,R: A > B,F3: B > D] :
% 0.29/0.45 ( ( ( comp @ C @ B @ A @ G3 @ H )
% 0.29/0.45 = R )
% 0.29/0.45 => ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F3 @ G3 ) @ H )
% 0.29/0.45 = ( comp @ B @ D @ A @ F3 @ R ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % rewriteR_comp_comp
% 0.29/0.45 thf(fact_134_rewriteL__comp__comp,axiom,
% 0.29/0.45 ! [C: $tType,B: $tType,A: $tType,D: $tType,F3: C > B,G3: A > C,L: A > B,H: D > A] :
% 0.29/0.45 ( ( ( comp @ C @ B @ A @ F3 @ G3 )
% 0.29/0.45 = L )
% 0.29/0.45 => ( ( comp @ C @ B @ D @ F3 @ ( comp @ A @ C @ D @ G3 @ H ) )
% 0.29/0.45 = ( comp @ A @ B @ D @ L @ H ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % rewriteL_comp_comp
% 0.29/0.45 thf(fact_135_comp__cong,axiom,
% 0.29/0.45 ! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F3: B > A,G3: C > B,X: C,F5: D > A,G4: E > D,X7: E] :
% 0.29/0.45 ( ( ( F3 @ ( G3 @ X ) )
% 0.29/0.45 = ( F5 @ ( G4 @ X7 ) ) )
% 0.29/0.45 => ( ( comp @ B @ A @ C @ F3 @ G3 @ X )
% 0.29/0.45 = ( comp @ D @ A @ E @ F5 @ G4 @ X7 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % comp_cong
% 0.29/0.45 thf(fact_136_image__Un,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,F3: B > A,A7: set @ B,B8: set @ B] :
% 0.29/0.45 ( ( image @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ A7 @ B8 ) )
% 0.29/0.45 = ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ F3 @ A7 ) @ ( image @ B @ A @ F3 @ B8 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % image_Un
% 0.29/0.45 thf(fact_137_scomp__Pair,axiom,
% 0.29/0.45 ! [C: $tType,B: $tType,A: $tType,X: A > ( product_prod @ B @ C )] :
% 0.29/0.45 ( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X @ ( product_Pair @ B @ C ) )
% 0.29/0.45 = X ) ).
% 0.29/0.45
% 0.29/0.45 % scomp_Pair
% 0.29/0.45 thf(fact_138_Pair__scomp,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,C: $tType,X: C,F3: C > A > B] :
% 0.29/0.45 ( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X ) @ F3 )
% 0.29/0.45 = ( F3 @ X ) ) ).
% 0.29/0.45
% 0.29/0.45 % Pair_scomp
% 0.29/0.45 thf(fact_139_scomp__def,axiom,
% 0.29/0.45 ! [D: $tType,C: $tType,B: $tType,A: $tType] :
% 0.29/0.45 ( ( product_scomp @ A @ B @ C @ D )
% 0.29/0.45 = ( ^ [F4: A > ( product_prod @ B @ C ),G5: B > C > D,X5: A] : ( product_case_prod @ B @ C @ D @ G5 @ ( F4 @ X5 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % scomp_def
% 0.29/0.45 thf(fact_140_case__sum__o__inj_I1_J,axiom,
% 0.29/0.45 ! [C: $tType,B: $tType,A: $tType,F3: A > B,G3: C > B] :
% 0.29/0.45 ( ( comp @ ( sum_sum @ A @ C ) @ B @ A @ ( sum_case_sum @ A @ B @ C @ F3 @ G3 ) @ ( sum_Inl @ A @ C ) )
% 0.29/0.45 = F3 ) ).
% 0.29/0.45
% 0.29/0.45 % case_sum_o_inj(1)
% 0.29/0.45 thf(fact_141_case__sum__o__inj_I2_J,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,C: $tType,F3: A > B,G3: C > B] :
% 0.29/0.45 ( ( comp @ ( sum_sum @ A @ C ) @ B @ C @ ( sum_case_sum @ A @ B @ C @ F3 @ G3 ) @ ( sum_Inr @ C @ A ) )
% 0.29/0.45 = G3 ) ).
% 0.29/0.45
% 0.29/0.45 % case_sum_o_inj(2)
% 0.29/0.45 thf(fact_142_comp__apply,axiom,
% 0.29/0.45 ! [C: $tType,A: $tType,B: $tType] :
% 0.29/0.45 ( ( comp @ B @ A @ C )
% 0.29/0.45 = ( ^ [F4: B > A,G5: C > B,X5: C] : ( F4 @ ( G5 @ X5 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % comp_apply
% 0.29/0.45 thf(fact_143_set_Ocomp,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,C: $tType,F3: C > B,G3: B > A] :
% 0.29/0.45 ( ( comp @ ( set @ B ) @ ( set @ C ) @ ( set @ A ) @ ( vimage @ C @ B @ F3 ) @ ( vimage @ B @ A @ G3 ) )
% 0.29/0.45 = ( vimage @ C @ A @ ( comp @ B @ A @ C @ G3 @ F3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % set.comp
% 0.29/0.45 thf(fact_144_comp__apply__eq,axiom,
% 0.29/0.45 ! [B: $tType,D: $tType,A: $tType,C: $tType,F3: B > A,G3: C > B,X: C,H: D > A,K: C > D] :
% 0.29/0.45 ( ( ( F3 @ ( G3 @ X ) )
% 0.29/0.45 = ( H @ ( K @ X ) ) )
% 0.29/0.45 => ( ( comp @ B @ A @ C @ F3 @ G3 @ X )
% 0.29/0.45 = ( comp @ D @ A @ C @ H @ K @ X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % comp_apply_eq
% 0.29/0.45 thf(fact_145_comp__eq__dest__lhs,axiom,
% 0.29/0.45 ! [C: $tType,B: $tType,A: $tType,A2: C > B,B2: A > C,C3: A > B,V: A] :
% 0.29/0.45 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 0.29/0.45 = C3 )
% 0.29/0.45 => ( ( A2 @ ( B2 @ V ) )
% 0.29/0.45 = ( C3 @ V ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % comp_eq_dest_lhs
% 0.29/0.45 thf(fact_146_comp__eq__elim,axiom,
% 0.29/0.45 ! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C3: D > B,D3: A > D] :
% 0.29/0.45 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 0.29/0.45 = ( comp @ D @ B @ A @ C3 @ D3 ) )
% 0.29/0.45 => ! [V2: A] :
% 0.29/0.45 ( ( A2 @ ( B2 @ V2 ) )
% 0.29/0.45 = ( C3 @ ( D3 @ V2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % comp_eq_elim
% 0.29/0.45 thf(fact_147_comp__eq__dest,axiom,
% 0.29/0.45 ! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C3: D > B,D3: A > D,V: A] :
% 0.29/0.45 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 0.29/0.45 = ( comp @ D @ B @ A @ C3 @ D3 ) )
% 0.29/0.45 => ( ( A2 @ ( B2 @ V ) )
% 0.29/0.45 = ( C3 @ ( D3 @ V ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % comp_eq_dest
% 0.29/0.45 thf(fact_148_comp__assoc,axiom,
% 0.29/0.45 ! [B: $tType,D: $tType,C: $tType,A: $tType,F3: D > B,G3: C > D,H: A > C] :
% 0.29/0.45 ( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F3 @ G3 ) @ H )
% 0.29/0.45 = ( comp @ D @ B @ A @ F3 @ ( comp @ C @ D @ A @ G3 @ H ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % comp_assoc
% 0.29/0.45 thf(fact_149_comp__def,axiom,
% 0.29/0.45 ! [A: $tType,C: $tType,B: $tType] :
% 0.29/0.45 ( ( comp @ B @ C @ A )
% 0.29/0.45 = ( ^ [F4: B > C,G5: A > B,X5: A] : ( F4 @ ( G5 @ X5 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % comp_def
% 0.29/0.45 thf(fact_150_image__eq__imp__comp,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,D: $tType,C: $tType,F3: B > A,A7: set @ B,G3: C > A,B8: set @ C,H: A > D] :
% 0.29/0.45 ( ( ( image @ B @ A @ F3 @ A7 )
% 0.29/0.45 = ( image @ C @ A @ G3 @ B8 ) )
% 0.29/0.45 => ( ( image @ B @ D @ ( comp @ A @ D @ B @ H @ F3 ) @ A7 )
% 0.29/0.45 = ( image @ C @ D @ ( comp @ A @ D @ C @ H @ G3 ) @ B8 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % image_eq_imp_comp
% 0.29/0.45 thf(fact_151_image__comp,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,C: $tType,F3: B > A,G3: C > B,R: set @ C] :
% 0.29/0.45 ( ( image @ B @ A @ F3 @ ( image @ C @ B @ G3 @ R ) )
% 0.29/0.45 = ( image @ C @ A @ ( comp @ B @ A @ C @ F3 @ G3 ) @ R ) ) ).
% 0.29/0.45
% 0.29/0.45 % image_comp
% 0.29/0.45 thf(fact_152_set_Ocompositionality,axiom,
% 0.29/0.45 ! [C: $tType,B: $tType,A: $tType,F3: C > B,G3: B > A,Set: set @ A] :
% 0.29/0.45 ( ( vimage @ C @ B @ F3 @ ( vimage @ B @ A @ G3 @ Set ) )
% 0.29/0.45 = ( vimage @ C @ A @ ( comp @ B @ A @ C @ G3 @ F3 ) @ Set ) ) ).
% 0.29/0.45
% 0.29/0.45 % set.compositionality
% 0.29/0.45 thf(fact_153_vimage__comp,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,C: $tType,F3: A > B,G3: B > C,X: set @ C] :
% 0.29/0.45 ( ( vimage @ A @ B @ F3 @ ( vimage @ B @ C @ G3 @ X ) )
% 0.29/0.45 = ( vimage @ A @ C @ ( comp @ B @ C @ A @ G3 @ F3 ) @ X ) ) ).
% 0.29/0.45
% 0.29/0.45 % vimage_comp
% 0.29/0.45 thf(fact_154_Sup_OSUP__image,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,C: $tType,Sup: ( set @ A ) > A,G3: B > A,F3: C > B,A7: set @ C] :
% 0.29/0.45 ( ( Sup @ ( image @ B @ A @ G3 @ ( image @ C @ B @ F3 @ A7 ) ) )
% 0.29/0.45 = ( Sup @ ( image @ C @ A @ ( comp @ B @ A @ C @ G3 @ F3 ) @ A7 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Sup.SUP_image
% 0.29/0.45 thf(fact_155_Inf_OINF__image,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,C: $tType,Inf: ( set @ A ) > A,G3: B > A,F3: C > B,A7: set @ C] :
% 0.29/0.45 ( ( Inf @ ( image @ B @ A @ G3 @ ( image @ C @ B @ F3 @ A7 ) ) )
% 0.29/0.45 = ( Inf @ ( image @ C @ A @ ( comp @ B @ A @ C @ G3 @ F3 ) @ A7 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Inf.INF_image
% 0.29/0.45 thf(fact_156_Plus__def,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType] :
% 0.29/0.45 ( ( sum_Plus @ A @ B )
% 0.29/0.45 = ( ^ [A9: set @ A,B9: set @ B] : ( sup_sup @ ( set @ ( sum_sum @ A @ B ) ) @ ( image @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ A9 ) @ ( image @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ B9 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Plus_def
% 0.29/0.45 thf(fact_157_surj__swap,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType] :
% 0.29/0.45 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 0.29/0.45 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % surj_swap
% 0.29/0.45 thf(fact_158_UNIV__I,axiom,
% 0.29/0.45 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % UNIV_I
% 0.29/0.45 thf(fact_159_sup__top__left,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( bounded_lattice_top @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A] :
% 0.29/0.45 ( ( sup_sup @ A @ ( top_top @ A ) @ X )
% 0.29/0.45 = ( top_top @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % sup_top_left
% 0.29/0.45 thf(fact_160_sup__top__right,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( bounded_lattice_top @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A] :
% 0.29/0.45 ( ( sup_sup @ A @ X @ ( top_top @ A ) )
% 0.29/0.45 = ( top_top @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % sup_top_right
% 0.29/0.45 thf(fact_161_vimage__UNIV,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,F3: A > B] :
% 0.29/0.45 ( ( vimage @ A @ B @ F3 @ ( top_top @ ( set @ B ) ) )
% 0.29/0.45 = ( top_top @ ( set @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % vimage_UNIV
% 0.29/0.45 thf(fact_162_UNIV__Plus__UNIV,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType] :
% 0.29/0.45 ( ( sum_Plus @ A @ B @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.29/0.45 = ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % UNIV_Plus_UNIV
% 0.29/0.45 thf(fact_163_InrI,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,B2: A,B8: set @ A,A7: set @ B] :
% 0.29/0.45 ( ( member @ A @ B2 @ B8 )
% 0.29/0.45 => ( member @ ( sum_sum @ B @ A ) @ ( sum_Inr @ A @ B @ B2 ) @ ( sum_Plus @ B @ A @ A7 @ B8 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % InrI
% 0.29/0.45 thf(fact_164_InlI,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,A2: A,A7: set @ A,B8: set @ B] :
% 0.29/0.45 ( ( member @ A @ A2 @ A7 )
% 0.29/0.45 => ( member @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B @ A2 ) @ ( sum_Plus @ A @ B @ A7 @ B8 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % InlI
% 0.29/0.45 thf(fact_165_range__eqI,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,B2: A,F3: B > A,X: B] :
% 0.29/0.45 ( ( B2
% 0.29/0.45 = ( F3 @ X ) )
% 0.29/0.45 => ( member @ A @ B2 @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % range_eqI
% 0.29/0.45 thf(fact_166_rangeI,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,F3: B > A,X: B] : ( member @ A @ ( F3 @ X ) @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % rangeI
% 0.29/0.45 thf(fact_167_surjD,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,F3: B > A,Y: A] :
% 0.29/0.45 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 0.29/0.45 = ( top_top @ ( set @ A ) ) )
% 0.29/0.45 => ? [X3: B] :
% 0.29/0.45 ( Y
% 0.29/0.45 = ( F3 @ X3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % surjD
% 0.29/0.45 thf(fact_168_surjE,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,F3: B > A,Y: A] :
% 0.29/0.45 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 0.29/0.45 = ( top_top @ ( set @ A ) ) )
% 0.29/0.45 => ~ ! [X3: B] :
% 0.29/0.45 ( Y
% 0.29/0.45 != ( F3 @ X3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % surjE
% 0.29/0.45 thf(fact_169_surjI,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,G3: B > A,F3: A > B] :
% 0.29/0.45 ( ! [X3: A] :
% 0.29/0.45 ( ( G3 @ ( F3 @ X3 ) )
% 0.29/0.45 = X3 )
% 0.29/0.45 => ( ( image @ B @ A @ G3 @ ( top_top @ ( set @ B ) ) )
% 0.29/0.45 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % surjI
% 0.29/0.45 thf(fact_170_surj__def,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,F3: B > A] :
% 0.29/0.45 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 0.29/0.45 = ( top_top @ ( set @ A ) ) )
% 0.29/0.45 = ( ! [Y4: A] :
% 0.29/0.45 ? [X5: B] :
% 0.29/0.45 ( Y4
% 0.29/0.45 = ( F3 @ X5 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % surj_def
% 0.29/0.45 thf(fact_171_UNIV__witness,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % UNIV_witness
% 0.29/0.45 thf(fact_172_UNIV__eq__I,axiom,
% 0.29/0.45 ! [A: $tType,A7: set @ A] :
% 0.29/0.45 ( ! [X3: A] : ( member @ A @ X3 @ A7 )
% 0.29/0.45 => ( ( top_top @ ( set @ A ) )
% 0.29/0.45 = A7 ) ) ).
% 0.29/0.45
% 0.29/0.45 % UNIV_eq_I
% 0.29/0.45 thf(fact_173_Un__UNIV__left,axiom,
% 0.29/0.45 ! [A: $tType,B8: set @ A] :
% 0.29/0.45 ( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B8 )
% 0.29/0.45 = ( top_top @ ( set @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Un_UNIV_left
% 0.29/0.45 thf(fact_174_Un__UNIV__right,axiom,
% 0.29/0.45 ! [A: $tType,A7: set @ A] :
% 0.29/0.45 ( ( sup_sup @ ( set @ A ) @ A7 @ ( top_top @ ( set @ A ) ) )
% 0.29/0.45 = ( top_top @ ( set @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Un_UNIV_right
% 0.29/0.45 thf(fact_175_UNIV__sum,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType] :
% 0.29/0.45 ( ( top_top @ ( set @ ( sum_sum @ A @ B ) ) )
% 0.29/0.45 = ( sup_sup @ ( set @ ( sum_sum @ A @ B ) ) @ ( image @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ ( top_top @ ( set @ A ) ) ) @ ( image @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % UNIV_sum
% 0.29/0.45 thf(fact_176_surj__fun__eq,axiom,
% 0.29/0.45 ! [B: $tType,C: $tType,A: $tType,F3: B > A,X8: set @ B,G1: A > C,G22: A > C] :
% 0.29/0.45 ( ( ( image @ B @ A @ F3 @ X8 )
% 0.29/0.45 = ( top_top @ ( set @ A ) ) )
% 0.29/0.45 => ( ! [X3: B] :
% 0.29/0.45 ( ( member @ B @ X3 @ X8 )
% 0.29/0.45 => ( ( comp @ A @ C @ B @ G1 @ F3 @ X3 )
% 0.29/0.45 = ( comp @ A @ C @ B @ G22 @ F3 @ X3 ) ) )
% 0.29/0.45 => ( G1 = G22 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % surj_fun_eq
% 0.29/0.45 thf(fact_177_comp__surj,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,C: $tType,F3: B > A,G3: A > C] :
% 0.29/0.45 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 0.29/0.45 = ( top_top @ ( set @ A ) ) )
% 0.29/0.45 => ( ( ( image @ A @ C @ G3 @ ( top_top @ ( set @ A ) ) )
% 0.29/0.45 = ( top_top @ ( set @ C ) ) )
% 0.29/0.45 => ( ( image @ B @ C @ ( comp @ A @ C @ B @ G3 @ F3 ) @ ( top_top @ ( set @ B ) ) )
% 0.29/0.45 = ( top_top @ ( set @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % comp_surj
% 0.29/0.45 thf(fact_178_surj__image__vimage__eq,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,F3: B > A,A7: set @ A] :
% 0.29/0.45 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 0.29/0.45 = ( top_top @ ( set @ A ) ) )
% 0.29/0.45 => ( ( image @ B @ A @ F3 @ ( vimage @ B @ A @ F3 @ A7 ) )
% 0.29/0.45 = A7 ) ) ).
% 0.29/0.45
% 0.29/0.45 % surj_image_vimage_eq
% 0.29/0.45 thf(fact_179_PlusE,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,U: sum_sum @ A @ B,A7: set @ A,B8: set @ B] :
% 0.29/0.45 ( ( member @ ( sum_sum @ A @ B ) @ U @ ( sum_Plus @ A @ B @ A7 @ B8 ) )
% 0.29/0.45 => ( ! [X3: A] :
% 0.29/0.45 ( ( member @ A @ X3 @ A7 )
% 0.29/0.45 => ( U
% 0.29/0.45 != ( sum_Inl @ A @ B @ X3 ) ) )
% 0.29/0.45 => ~ ! [Y3: B] :
% 0.29/0.45 ( ( member @ B @ Y3 @ B8 )
% 0.29/0.45 => ( U
% 0.29/0.45 != ( sum_Inr @ B @ A @ Y3 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % PlusE
% 0.29/0.45 thf(fact_180_Inf_OINF__cong,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,A7: set @ B,B8: set @ B,C4: B > A,D4: B > A,Inf: ( set @ A ) > A] :
% 0.29/0.45 ( ( A7 = B8 )
% 0.29/0.45 => ( ! [X3: B] :
% 0.29/0.45 ( ( member @ B @ X3 @ B8 )
% 0.29/0.45 => ( ( C4 @ X3 )
% 0.29/0.45 = ( D4 @ X3 ) ) )
% 0.29/0.45 => ( ( Inf @ ( image @ B @ A @ C4 @ A7 ) )
% 0.29/0.45 = ( Inf @ ( image @ B @ A @ D4 @ B8 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Inf.INF_cong
% 0.29/0.45 thf(fact_181_Sup_OSUP__cong,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,A7: set @ B,B8: set @ B,C4: B > A,D4: B > A,Sup: ( set @ A ) > A] :
% 0.29/0.45 ( ( A7 = B8 )
% 0.29/0.45 => ( ! [X3: B] :
% 0.29/0.45 ( ( member @ B @ X3 @ B8 )
% 0.29/0.45 => ( ( C4 @ X3 )
% 0.29/0.45 = ( D4 @ X3 ) ) )
% 0.29/0.45 => ( ( Sup @ ( image @ B @ A @ C4 @ A7 ) )
% 0.29/0.45 = ( Sup @ ( image @ B @ A @ D4 @ B8 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Sup.SUP_cong
% 0.29/0.45 thf(fact_182_fun_Oset__map,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,D: $tType,F3: A > B,V: D > A] :
% 0.29/0.45 ( ( image @ D @ B @ ( comp @ A @ B @ D @ F3 @ V ) @ ( top_top @ ( set @ D ) ) )
% 0.29/0.45 = ( image @ A @ B @ F3 @ ( image @ D @ A @ V @ ( top_top @ ( set @ D ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % fun.set_map
% 0.29/0.45 thf(fact_183_fun_Omap__cong,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,D: $tType,X: D > A,Ya: D > A,F3: A > B,G3: A > B] :
% 0.29/0.45 ( ( X = Ya )
% 0.29/0.45 => ( ! [Z2: A] :
% 0.29/0.45 ( ( member @ A @ Z2 @ ( image @ D @ A @ Ya @ ( top_top @ ( set @ D ) ) ) )
% 0.29/0.45 => ( ( F3 @ Z2 )
% 0.29/0.45 = ( G3 @ Z2 ) ) )
% 0.29/0.45 => ( ( comp @ A @ B @ D @ F3 @ X )
% 0.29/0.45 = ( comp @ A @ B @ D @ G3 @ Ya ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % fun.map_cong
% 0.29/0.45 thf(fact_184_fun_Omap__cong0,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,D: $tType,X: D > A,F3: A > B,G3: A > B] :
% 0.29/0.45 ( ! [Z2: A] :
% 0.29/0.45 ( ( member @ A @ Z2 @ ( image @ D @ A @ X @ ( top_top @ ( set @ D ) ) ) )
% 0.29/0.45 => ( ( F3 @ Z2 )
% 0.29/0.45 = ( G3 @ Z2 ) ) )
% 0.29/0.45 => ( ( comp @ A @ B @ D @ F3 @ X )
% 0.29/0.45 = ( comp @ A @ B @ D @ G3 @ X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % fun.map_cong0
% 0.29/0.45 thf(fact_185_top__set__def,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( top_top @ ( set @ A ) )
% 0.29/0.45 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % top_set_def
% 0.29/0.45 thf(fact_186_fun_Omap__comp,axiom,
% 0.29/0.45 ! [B: $tType,C: $tType,A: $tType,D: $tType,G3: B > C,F3: A > B,V: D > A] :
% 0.29/0.45 ( ( comp @ B @ C @ D @ G3 @ ( comp @ A @ B @ D @ F3 @ V ) )
% 0.29/0.45 = ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G3 @ F3 ) @ V ) ) ).
% 0.29/0.45
% 0.29/0.45 % fun.map_comp
% 0.29/0.45 thf(fact_187_fun_Oinj__map__strong,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,D: $tType,X: D > A,Xa: D > A,F3: A > B,Fa: A > B] :
% 0.29/0.45 ( ! [Z2: A,Za: A] :
% 0.29/0.45 ( ( member @ A @ Z2 @ ( image @ D @ A @ X @ ( top_top @ ( set @ D ) ) ) )
% 0.29/0.45 => ( ( member @ A @ Za @ ( image @ D @ A @ Xa @ ( top_top @ ( set @ D ) ) ) )
% 0.29/0.45 => ( ( ( F3 @ Z2 )
% 0.29/0.45 = ( Fa @ Za ) )
% 0.29/0.45 => ( Z2 = Za ) ) ) )
% 0.29/0.45 => ( ( ( comp @ A @ B @ D @ F3 @ X )
% 0.29/0.45 = ( comp @ A @ B @ D @ Fa @ Xa ) )
% 0.29/0.45 => ( X = Xa ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % fun.inj_map_strong
% 0.29/0.45 thf(fact_188_map__prod__surj,axiom,
% 0.29/0.45 ! [A: $tType,C: $tType,D: $tType,B: $tType,F3: A > B,G3: C > D] :
% 0.29/0.45 ( ( ( image @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.29/0.45 = ( top_top @ ( set @ B ) ) )
% 0.29/0.45 => ( ( ( image @ C @ D @ G3 @ ( top_top @ ( set @ C ) ) )
% 0.29/0.45 = ( top_top @ ( set @ D ) ) )
% 0.29/0.45 => ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F3 @ G3 ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
% 0.29/0.45 = ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % map_prod_surj
% 0.29/0.45 thf(fact_189_type__copy__set__map0,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,D: $tType,E: $tType,C: $tType,F: $tType,Rep: A > B,Abs: B > A,S3: B > ( set @ D ),M: C > B,F3: E > D,S4: C > ( set @ E ),G3: F > C] :
% 0.29/0.45 ( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
% 0.29/0.45 => ( ( ( comp @ B @ ( set @ D ) @ C @ S3 @ M )
% 0.29/0.45 = ( comp @ ( set @ E ) @ ( set @ D ) @ C @ ( image @ E @ D @ F3 ) @ S4 ) )
% 0.29/0.45 => ( ( comp @ A @ ( set @ D ) @ F @ ( comp @ B @ ( set @ D ) @ A @ S3 @ Rep ) @ ( comp @ C @ A @ F @ ( comp @ B @ A @ C @ Abs @ M ) @ G3 ) )
% 0.29/0.45 = ( comp @ ( set @ E ) @ ( set @ D ) @ F @ ( image @ E @ D @ F3 ) @ ( comp @ C @ ( set @ E ) @ F @ S4 @ G3 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % type_copy_set_map0
% 0.29/0.45 thf(fact_190_image__vimage__eq,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,F3: B > A,A7: set @ A] :
% 0.29/0.45 ( ( image @ B @ A @ F3 @ ( vimage @ B @ A @ F3 @ A7 ) )
% 0.29/0.45 = ( inf_inf @ ( set @ A ) @ A7 @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % image_vimage_eq
% 0.29/0.45 thf(fact_191_inf__right__idem,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Y )
% 0.29/0.45 = ( inf_inf @ A @ X @ Y ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf_right_idem
% 0.29/0.45 thf(fact_192_inf_Oright__idem,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B2: A] :
% 0.29/0.45 ( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ B2 )
% 0.29/0.45 = ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf.right_idem
% 0.29/0.45 thf(fact_193_inf__left__idem,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y ) )
% 0.29/0.45 = ( inf_inf @ A @ X @ Y ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf_left_idem
% 0.29/0.45 thf(fact_194_inf_Oleft__idem,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B2: A] :
% 0.29/0.45 ( ( inf_inf @ A @ A2 @ ( inf_inf @ A @ A2 @ B2 ) )
% 0.29/0.45 = ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf.left_idem
% 0.29/0.45 thf(fact_195_inf__idem,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A] :
% 0.29/0.45 ( ( inf_inf @ A @ X @ X )
% 0.29/0.45 = X ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf_idem
% 0.29/0.45 thf(fact_196_inf_Oidem,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ( ( inf_inf @ A @ A2 @ A2 )
% 0.29/0.45 = A2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf.idem
% 0.29/0.45 thf(fact_197_inf__apply,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType] :
% 0.29/0.45 ( ( semilattice_inf @ B @ ( type2 @ B ) )
% 0.29/0.45 => ( ( inf_inf @ ( A > B ) )
% 0.29/0.45 = ( ^ [F4: A > B,G5: A > B,X5: A] : ( inf_inf @ B @ ( F4 @ X5 ) @ ( G5 @ X5 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf_apply
% 0.29/0.45 thf(fact_198_Int__iff,axiom,
% 0.29/0.45 ! [A: $tType,C3: A,A7: set @ A,B8: set @ A] :
% 0.29/0.45 ( ( member @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A7 @ B8 ) )
% 0.29/0.45 = ( ( member @ A @ C3 @ A7 )
% 0.29/0.45 & ( member @ A @ C3 @ B8 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Int_iff
% 0.29/0.45 thf(fact_199_IntI,axiom,
% 0.29/0.45 ! [A: $tType,C3: A,A7: set @ A,B8: set @ A] :
% 0.29/0.45 ( ( member @ A @ C3 @ A7 )
% 0.29/0.45 => ( ( member @ A @ C3 @ B8 )
% 0.29/0.45 => ( member @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A7 @ B8 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % IntI
% 0.29/0.45 thf(fact_200_sup__inf__absorb,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( lattice @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( sup_sup @ A @ X @ ( inf_inf @ A @ X @ Y ) )
% 0.29/0.45 = X ) ) ).
% 0.29/0.45
% 0.29/0.45 % sup_inf_absorb
% 0.29/0.45 thf(fact_201_inf__sup__absorb,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( lattice @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( inf_inf @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 0.29/0.45 = X ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf_sup_absorb
% 0.29/0.45 thf(fact_202_inf__top_Oright__neutral,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( bounde1561333602nf_top @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ( ( inf_inf @ A @ A2 @ ( top_top @ A ) )
% 0.29/0.45 = A2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf_top.right_neutral
% 0.29/0.45 thf(fact_203_inf__top_Oleft__neutral,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( bounde1561333602nf_top @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ( ( inf_inf @ A @ ( top_top @ A ) @ A2 )
% 0.29/0.45 = A2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf_top.left_neutral
% 0.29/0.45 thf(fact_204_inf__eq__top__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( bounded_lattice_top @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ( inf_inf @ A @ X @ Y )
% 0.29/0.45 = ( top_top @ A ) )
% 0.29/0.45 = ( ( X
% 0.29/0.45 = ( top_top @ A ) )
% 0.29/0.45 & ( Y
% 0.29/0.45 = ( top_top @ A ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf_eq_top_iff
% 0.29/0.45 thf(fact_205_Int__UNIV,axiom,
% 0.29/0.45 ! [A: $tType,A7: set @ A,B8: set @ A] :
% 0.29/0.45 ( ( ( inf_inf @ ( set @ A ) @ A7 @ B8 )
% 0.29/0.45 = ( top_top @ ( set @ A ) ) )
% 0.29/0.45 = ( ( A7
% 0.29/0.45 = ( top_top @ ( set @ A ) ) )
% 0.29/0.45 & ( B8
% 0.29/0.45 = ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Int_UNIV
% 0.29/0.45 thf(fact_206_vimage__Int,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,F3: A > B,A7: set @ B,B8: set @ B] :
% 0.29/0.45 ( ( vimage @ A @ B @ F3 @ ( inf_inf @ ( set @ B ) @ A7 @ B8 ) )
% 0.29/0.45 = ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ A7 ) @ ( vimage @ A @ B @ F3 @ B8 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % vimage_Int
% 0.29/0.45 thf(fact_207_map__prod__simp,axiom,
% 0.29/0.45 ! [C: $tType,A: $tType,B: $tType,D: $tType,F3: C > A,G3: D > B,A2: C,B2: D] :
% 0.29/0.45 ( ( product_map_prod @ C @ A @ D @ B @ F3 @ G3 @ ( product_Pair @ C @ D @ A2 @ B2 ) )
% 0.29/0.45 = ( product_Pair @ A @ B @ ( F3 @ A2 ) @ ( G3 @ B2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % map_prod_simp
% 0.29/0.45 thf(fact_208_map__prod__imageI,axiom,
% 0.29/0.45 ! [D: $tType,C: $tType,B: $tType,A: $tType,A2: A,B2: B,R2: set @ ( product_prod @ A @ B ),F3: A > C,G3: B > D] :
% 0.29/0.45 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R2 )
% 0.29/0.45 => ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F3 @ A2 ) @ ( G3 @ B2 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F3 @ G3 ) @ R2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % map_prod_imageI
% 0.29/0.45 thf(fact_209_type__copy__wit,axiom,
% 0.29/0.45 ! [A: $tType,C: $tType,B: $tType,Rep: A > B,Abs: B > A,X: C,S3: B > ( set @ C ),Y: B] :
% 0.29/0.45 ( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
% 0.29/0.45 => ( ( member @ C @ X @ ( comp @ B @ ( set @ C ) @ A @ S3 @ Rep @ ( Abs @ Y ) ) )
% 0.29/0.45 => ( member @ C @ X @ ( S3 @ Y ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % type_copy_wit
% 0.29/0.45 thf(fact_210_type__copy__map__comp0,axiom,
% 0.29/0.45 ! [F: $tType,D: $tType,B: $tType,A: $tType,C: $tType,E: $tType,Rep: A > B,Abs: B > A,M: C > D,M1: B > D,M2: C > B,F3: D > F,G3: E > C] :
% 0.29/0.45 ( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
% 0.29/0.45 => ( ( M
% 0.29/0.45 = ( comp @ B @ D @ C @ M1 @ M2 ) )
% 0.29/0.45 => ( ( comp @ C @ F @ E @ ( comp @ D @ F @ C @ F3 @ M ) @ G3 )
% 0.29/0.45 = ( comp @ A @ F @ E @ ( comp @ B @ F @ A @ ( comp @ D @ F @ B @ F3 @ M1 ) @ Rep ) @ ( comp @ C @ A @ E @ ( comp @ B @ A @ C @ Abs @ M2 ) @ G3 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % type_copy_map_comp0
% 0.29/0.45 thf(fact_211_inf__top__right,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( bounded_lattice_top @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A] :
% 0.29/0.45 ( ( inf_inf @ A @ X @ ( top_top @ A ) )
% 0.29/0.45 = X ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf_top_right
% 0.29/0.45 thf(fact_212_inf__top__left,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( bounded_lattice_top @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A] :
% 0.29/0.45 ( ( inf_inf @ A @ ( top_top @ A ) @ X )
% 0.29/0.45 = X ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf_top_left
% 0.29/0.45 thf(fact_213_Int__UNIV__right,axiom,
% 0.29/0.45 ! [A: $tType,A7: set @ A] :
% 0.29/0.45 ( ( inf_inf @ ( set @ A ) @ A7 @ ( top_top @ ( set @ A ) ) )
% 0.29/0.45 = A7 ) ).
% 0.29/0.45
% 0.29/0.45 % Int_UNIV_right
% 0.29/0.45 thf(fact_214_Int__UNIV__left,axiom,
% 0.29/0.45 ! [A: $tType,B8: set @ A] :
% 0.29/0.45 ( ( inf_inf @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B8 )
% 0.29/0.45 = B8 ) ).
% 0.29/0.45
% 0.29/0.45 % Int_UNIV_left
% 0.29/0.45 thf(fact_215_type__copy__obj__one__point__absE,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,S: A] :
% 0.29/0.45 ( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
% 0.29/0.45 => ~ ! [X3: B] :
% 0.29/0.45 ( S
% 0.29/0.45 != ( Abs @ X3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % type_copy_obj_one_point_absE
% 0.29/0.45 thf(fact_216_top__conj_I1_J,axiom,
% 0.29/0.45 ! [A: $tType,X: A,P: $o] :
% 0.29/0.45 ( ( ( top_top @ ( A > $o ) @ X )
% 0.29/0.45 & P )
% 0.29/0.45 = P ) ).
% 0.29/0.45
% 0.29/0.45 % top_conj(1)
% 0.29/0.45 thf(fact_217_top__conj_I2_J,axiom,
% 0.29/0.45 ! [A: $tType,P: $o,X: A] :
% 0.29/0.45 ( ( P
% 0.29/0.45 & ( top_top @ ( A > $o ) @ X ) )
% 0.29/0.45 = P ) ).
% 0.29/0.45
% 0.29/0.45 % top_conj(2)
% 0.29/0.45 thf(fact_218_prod_Omap__comp,axiom,
% 0.29/0.45 ! [D: $tType,F: $tType,E: $tType,C: $tType,B: $tType,A: $tType,G1: C > E,G22: D > F,F1: A > C,F22: B > D,V: product_prod @ A @ B] :
% 0.29/0.45 ( ( product_map_prod @ C @ E @ D @ F @ G1 @ G22 @ ( product_map_prod @ A @ C @ B @ D @ F1 @ F22 @ V ) )
% 0.29/0.45 = ( product_map_prod @ A @ E @ B @ F @ ( comp @ C @ E @ A @ G1 @ F1 ) @ ( comp @ D @ F @ B @ G22 @ F22 ) @ V ) ) ).
% 0.29/0.45
% 0.29/0.45 % prod.map_comp
% 0.29/0.45 thf(fact_219_map__prod_Ocompositionality,axiom,
% 0.29/0.45 ! [D: $tType,F: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F3: C > E,G3: D > F,H: A > C,I: B > D,Prod: product_prod @ A @ B] :
% 0.29/0.45 ( ( product_map_prod @ C @ E @ D @ F @ F3 @ G3 @ ( product_map_prod @ A @ C @ B @ D @ H @ I @ Prod ) )
% 0.29/0.45 = ( product_map_prod @ A @ E @ B @ F @ ( comp @ C @ E @ A @ F3 @ H ) @ ( comp @ D @ F @ B @ G3 @ I ) @ Prod ) ) ).
% 0.29/0.45
% 0.29/0.45 % map_prod.compositionality
% 0.29/0.45 thf(fact_220_map__prod__compose,axiom,
% 0.29/0.45 ! [D: $tType,C: $tType,A: $tType,E: $tType,F: $tType,B: $tType,F1: E > C,F22: A > E,G1: F > D,G22: B > F] :
% 0.29/0.45 ( ( product_map_prod @ A @ C @ B @ D @ ( comp @ E @ C @ A @ F1 @ F22 ) @ ( comp @ F @ D @ B @ G1 @ G22 ) )
% 0.29/0.45 = ( comp @ ( product_prod @ E @ F ) @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ E @ C @ F @ D @ F1 @ G1 ) @ ( product_map_prod @ A @ E @ B @ F @ F22 @ G22 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % map_prod_compose
% 0.29/0.45 thf(fact_221_map__prod_Ocomp,axiom,
% 0.29/0.45 ! [A: $tType,C: $tType,E: $tType,F: $tType,D: $tType,B: $tType,F3: C > E,G3: D > F,H: A > C,I: B > D] :
% 0.29/0.45 ( ( comp @ ( product_prod @ C @ D ) @ ( product_prod @ E @ F ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ E @ D @ F @ F3 @ G3 ) @ ( product_map_prod @ A @ C @ B @ D @ H @ I ) )
% 0.29/0.45 = ( product_map_prod @ A @ E @ B @ F @ ( comp @ C @ E @ A @ F3 @ H ) @ ( comp @ D @ F @ B @ G3 @ I ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % map_prod.comp
% 0.29/0.45 thf(fact_222_inf__left__commute,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A,Z: A] :
% 0.29/0.45 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
% 0.29/0.45 = ( inf_inf @ A @ Y @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf_left_commute
% 0.29/0.45 thf(fact_223_inf_Oleft__commute,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [B2: A,A2: A,C3: A] :
% 0.29/0.45 ( ( inf_inf @ A @ B2 @ ( inf_inf @ A @ A2 @ C3 ) )
% 0.29/0.45 = ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B2 @ C3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf.left_commute
% 0.29/0.45 thf(fact_224_inf__commute,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.29/0.45 => ( ( inf_inf @ A )
% 0.29/0.45 = ( ^ [X5: A,Y4: A] : ( inf_inf @ A @ Y4 @ X5 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf_commute
% 0.29/0.45 thf(fact_225_inf_Ocommute,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.29/0.45 => ( ( inf_inf @ A )
% 0.29/0.45 = ( ^ [A8: A,B7: A] : ( inf_inf @ A @ B7 @ A8 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf.commute
% 0.29/0.45 thf(fact_226_inf__assoc,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A,Z: A] :
% 0.29/0.45 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Z )
% 0.29/0.45 = ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf_assoc
% 0.29/0.45 thf(fact_227_inf_Oassoc,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B2: A,C3: A] :
% 0.29/0.45 ( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C3 )
% 0.29/0.45 = ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B2 @ C3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf.assoc
% 0.29/0.45 thf(fact_228_Int__left__commute,axiom,
% 0.29/0.45 ! [A: $tType,A7: set @ A,B8: set @ A,C4: set @ A] :
% 0.29/0.45 ( ( inf_inf @ ( set @ A ) @ A7 @ ( inf_inf @ ( set @ A ) @ B8 @ C4 ) )
% 0.29/0.45 = ( inf_inf @ ( set @ A ) @ B8 @ ( inf_inf @ ( set @ A ) @ A7 @ C4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Int_left_commute
% 0.29/0.45 thf(fact_229_inf__fun__def,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType] :
% 0.29/0.45 ( ( semilattice_inf @ B @ ( type2 @ B ) )
% 0.29/0.45 => ( ( inf_inf @ ( A > B ) )
% 0.29/0.45 = ( ^ [F4: A > B,G5: A > B,X5: A] : ( inf_inf @ B @ ( F4 @ X5 ) @ ( G5 @ X5 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf_fun_def
% 0.29/0.45 thf(fact_230_Int__left__absorb,axiom,
% 0.29/0.45 ! [A: $tType,A7: set @ A,B8: set @ A] :
% 0.29/0.45 ( ( inf_inf @ ( set @ A ) @ A7 @ ( inf_inf @ ( set @ A ) @ A7 @ B8 ) )
% 0.29/0.45 = ( inf_inf @ ( set @ A ) @ A7 @ B8 ) ) ).
% 0.29/0.45
% 0.29/0.45 % Int_left_absorb
% 0.29/0.45 thf(fact_231_Int__commute,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( inf_inf @ ( set @ A ) )
% 0.29/0.45 = ( ^ [A9: set @ A,B9: set @ A] : ( inf_inf @ ( set @ A ) @ B9 @ A9 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Int_commute
% 0.29/0.45 thf(fact_232_Int__absorb,axiom,
% 0.29/0.45 ! [A: $tType,A7: set @ A] :
% 0.29/0.45 ( ( inf_inf @ ( set @ A ) @ A7 @ A7 )
% 0.29/0.45 = A7 ) ).
% 0.29/0.45
% 0.29/0.45 % Int_absorb
% 0.29/0.45 thf(fact_233_Int__assoc,axiom,
% 0.29/0.45 ! [A: $tType,A7: set @ A,B8: set @ A,C4: set @ A] :
% 0.29/0.45 ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A7 @ B8 ) @ C4 )
% 0.29/0.45 = ( inf_inf @ ( set @ A ) @ A7 @ ( inf_inf @ ( set @ A ) @ B8 @ C4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Int_assoc
% 0.29/0.45 thf(fact_234_IntD2,axiom,
% 0.29/0.45 ! [A: $tType,C3: A,A7: set @ A,B8: set @ A] :
% 0.29/0.45 ( ( member @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A7 @ B8 ) )
% 0.29/0.45 => ( member @ A @ C3 @ B8 ) ) ).
% 0.29/0.45
% 0.29/0.45 % IntD2
% 0.29/0.45 thf(fact_235_IntD1,axiom,
% 0.29/0.45 ! [A: $tType,C3: A,A7: set @ A,B8: set @ A] :
% 0.29/0.45 ( ( member @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A7 @ B8 ) )
% 0.29/0.45 => ( member @ A @ C3 @ A7 ) ) ).
% 0.29/0.45
% 0.29/0.45 % IntD1
% 0.29/0.45 thf(fact_236_IntE,axiom,
% 0.29/0.45 ! [A: $tType,C3: A,A7: set @ A,B8: set @ A] :
% 0.29/0.45 ( ( member @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A7 @ B8 ) )
% 0.29/0.45 => ~ ( ( member @ A @ C3 @ A7 )
% 0.29/0.45 => ~ ( member @ A @ C3 @ B8 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % IntE
% 0.29/0.45 thf(fact_237_inf__sup__aci_I1_J,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( lattice @ A @ ( type2 @ A ) )
% 0.29/0.45 => ( ( inf_inf @ A )
% 0.29/0.45 = ( ^ [X5: A,Y4: A] : ( inf_inf @ A @ Y4 @ X5 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf_sup_aci(1)
% 0.29/0.45 thf(fact_238_inf__sup__aci_I2_J,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( lattice @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A,Z: A] :
% 0.29/0.45 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Z )
% 0.29/0.45 = ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf_sup_aci(2)
% 0.29/0.45 thf(fact_239_inf__sup__aci_I3_J,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( lattice @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A,Z: A] :
% 0.29/0.45 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
% 0.29/0.45 = ( inf_inf @ A @ Y @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf_sup_aci(3)
% 0.29/0.45 thf(fact_240_inf__sup__aci_I4_J,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( lattice @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y ) )
% 0.29/0.45 = ( inf_inf @ A @ X @ Y ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf_sup_aci(4)
% 0.29/0.45 thf(fact_241_vimage__inter__cong,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,S3: set @ A,F3: A > B,G3: A > B,Y: set @ B] :
% 0.29/0.45 ( ! [W: A] :
% 0.29/0.45 ( ( member @ A @ W @ S3 )
% 0.29/0.45 => ( ( F3 @ W )
% 0.29/0.45 = ( G3 @ W ) ) )
% 0.29/0.45 => ( ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ Y ) @ S3 )
% 0.29/0.45 = ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ G3 @ Y ) @ S3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % vimage_inter_cong
% 0.29/0.45 thf(fact_242_sup__inf__distrib2,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( distrib_lattice @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [Y: A,Z: A,X: A] :
% 0.29/0.45 ( ( sup_sup @ A @ ( inf_inf @ A @ Y @ Z ) @ X )
% 0.29/0.45 = ( inf_inf @ A @ ( sup_sup @ A @ Y @ X ) @ ( sup_sup @ A @ Z @ X ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % sup_inf_distrib2
% 0.29/0.45 thf(fact_243_sup__inf__distrib1,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( distrib_lattice @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A,Z: A] :
% 0.29/0.45 ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
% 0.29/0.45 = ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % sup_inf_distrib1
% 0.29/0.45 thf(fact_244_inf__sup__distrib2,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( distrib_lattice @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [Y: A,Z: A,X: A] :
% 0.29/0.45 ( ( inf_inf @ A @ ( sup_sup @ A @ Y @ Z ) @ X )
% 0.29/0.45 = ( sup_sup @ A @ ( inf_inf @ A @ Y @ X ) @ ( inf_inf @ A @ Z @ X ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf_sup_distrib2
% 0.29/0.45 thf(fact_245_inf__sup__distrib1,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( distrib_lattice @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A,Z: A] :
% 0.29/0.45 ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
% 0.29/0.45 = ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % inf_sup_distrib1
% 0.29/0.45 thf(fact_246_distrib__imp2,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( lattice @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A,Z: A] :
% 0.29/0.45 ( ! [X3: A,Y3: A,Z2: A] :
% 0.29/0.45 ( ( sup_sup @ A @ X3 @ ( inf_inf @ A @ Y3 @ Z2 ) )
% 0.29/0.45 = ( inf_inf @ A @ ( sup_sup @ A @ X3 @ Y3 ) @ ( sup_sup @ A @ X3 @ Z2 ) ) )
% 0.29/0.45 => ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
% 0.29/0.45 = ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % distrib_imp2
% 0.29/0.45 thf(fact_247_distrib__imp1,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( lattice @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A,Z: A] :
% 0.29/0.45 ( ! [X3: A,Y3: A,Z2: A] :
% 0.29/0.45 ( ( inf_inf @ A @ X3 @ ( sup_sup @ A @ Y3 @ Z2 ) )
% 0.29/0.45 = ( sup_sup @ A @ ( inf_inf @ A @ X3 @ Y3 ) @ ( inf_inf @ A @ X3 @ Z2 ) ) )
% 0.29/0.45 => ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
% 0.29/0.45 = ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % distrib_imp1
% 0.29/0.45 thf(fact_248_Un__Int__distrib2,axiom,
% 0.29/0.45 ! [A: $tType,B8: set @ A,C4: set @ A,A7: set @ A] :
% 0.29/0.45 ( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B8 @ C4 ) @ A7 )
% 0.29/0.45 = ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B8 @ A7 ) @ ( sup_sup @ ( set @ A ) @ C4 @ A7 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Un_Int_distrib2
% 0.29/0.45 thf(fact_249_Int__Un__distrib2,axiom,
% 0.29/0.45 ! [A: $tType,B8: set @ A,C4: set @ A,A7: set @ A] :
% 0.29/0.45 ( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B8 @ C4 ) @ A7 )
% 0.29/0.45 = ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B8 @ A7 ) @ ( inf_inf @ ( set @ A ) @ C4 @ A7 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Int_Un_distrib2
% 0.29/0.45 thf(fact_250_Un__Int__distrib,axiom,
% 0.29/0.45 ! [A: $tType,A7: set @ A,B8: set @ A,C4: set @ A] :
% 0.29/0.45 ( ( sup_sup @ ( set @ A ) @ A7 @ ( inf_inf @ ( set @ A ) @ B8 @ C4 ) )
% 0.29/0.45 = ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A7 @ B8 ) @ ( sup_sup @ ( set @ A ) @ A7 @ C4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Un_Int_distrib
% 0.29/0.45 thf(fact_251_Int__Un__distrib,axiom,
% 0.29/0.45 ! [A: $tType,A7: set @ A,B8: set @ A,C4: set @ A] :
% 0.29/0.45 ( ( inf_inf @ ( set @ A ) @ A7 @ ( sup_sup @ ( set @ A ) @ B8 @ C4 ) )
% 0.29/0.45 = ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A7 @ B8 ) @ ( inf_inf @ ( set @ A ) @ A7 @ C4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Int_Un_distrib
% 0.29/0.45 thf(fact_252_Un__Int__crazy,axiom,
% 0.29/0.45 ! [A: $tType,A7: set @ A,B8: set @ A,C4: set @ A] :
% 0.29/0.45 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A7 @ B8 ) @ ( inf_inf @ ( set @ A ) @ B8 @ C4 ) ) @ ( inf_inf @ ( set @ A ) @ C4 @ A7 ) )
% 0.29/0.45 = ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A7 @ B8 ) @ ( sup_sup @ ( set @ A ) @ B8 @ C4 ) ) @ ( sup_sup @ ( set @ A ) @ C4 @ A7 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Un_Int_crazy
% 0.29/0.45 thf(fact_253_type__copy__map__comp0__undo,axiom,
% 0.29/0.45 ! [E: $tType,A: $tType,C: $tType,B: $tType,D: $tType,F: $tType,Rep: A > B,Abs: B > A,Rep2: C > D,Abs2: D > C,Rep3: E > F,Abs3: F > E,M: F > D,M1: B > D,M2: F > B] :
% 0.29/0.45 ( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
% 0.29/0.45 => ( ( type_definition @ C @ D @ Rep2 @ Abs2 @ ( top_top @ ( set @ D ) ) )
% 0.29/0.45 => ( ( type_definition @ E @ F @ Rep3 @ Abs3 @ ( top_top @ ( set @ F ) ) )
% 0.29/0.45 => ( ( ( comp @ F @ C @ E @ ( comp @ D @ C @ F @ Abs2 @ M ) @ Rep3 )
% 0.29/0.45 = ( comp @ A @ C @ E @ ( comp @ B @ C @ A @ ( comp @ D @ C @ B @ Abs2 @ M1 ) @ Rep ) @ ( comp @ F @ A @ E @ ( comp @ B @ A @ F @ Abs @ M2 ) @ Rep3 ) ) )
% 0.29/0.45 => ( ( comp @ B @ D @ F @ M1 @ M2 )
% 0.29/0.45 = M ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % type_copy_map_comp0_undo
% 0.29/0.45 thf(fact_254_prod__fun__imageE,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,D: $tType,C: $tType,C3: product_prod @ A @ B,F3: C > A,G3: D > B,R2: set @ ( product_prod @ C @ D )] :
% 0.29/0.45 ( ( member @ ( product_prod @ A @ B ) @ C3 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F3 @ G3 ) @ R2 ) )
% 0.29/0.45 => ~ ! [X3: C,Y3: D] :
% 0.29/0.45 ( ( C3
% 0.29/0.45 = ( product_Pair @ A @ B @ ( F3 @ X3 ) @ ( G3 @ Y3 ) ) )
% 0.29/0.45 => ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X3 @ Y3 ) @ R2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % prod_fun_imageE
% 0.29/0.45 thf(fact_255_type__copy__map__cong0,axiom,
% 0.29/0.45 ! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M: B > A,G3: C > B,X: C,N: D > A,H: C > D,F3: A > E] :
% 0.29/0.45 ( ( ( M @ ( G3 @ X ) )
% 0.29/0.45 = ( N @ ( H @ X ) ) )
% 0.29/0.45 => ( ( comp @ B @ E @ C @ ( comp @ A @ E @ B @ F3 @ M ) @ G3 @ X )
% 0.29/0.45 = ( comp @ D @ E @ C @ ( comp @ A @ E @ D @ F3 @ N ) @ H @ X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % type_copy_map_cong0
% 0.29/0.45
% 0.29/0.45 %----Type constructors (21)
% 0.29/0.45 thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
% 0.29/0.45 bounded_lattice @ $o @ ( type2 @ $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
% 0.29/0.45 ! [A5: $tType] : ( bounded_lattice @ ( set @ A5 ) @ ( type2 @ ( set @ A5 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
% 0.29/0.45 ! [A5: $tType,A10: $tType] :
% 0.29/0.45 ( ( bounded_lattice @ A10 @ ( type2 @ A10 ) )
% 0.29/0.45 => ( bounded_lattice @ ( A5 > A10 ) @ ( type2 @ ( A5 > A10 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
% 0.29/0.45 ! [A5: $tType,A10: $tType] :
% 0.29/0.45 ( ( bounded_lattice @ A10 @ ( type2 @ A10 ) )
% 0.29/0.45 => ( bounde1561333602nf_top @ ( A5 > A10 ) @ ( type2 @ ( A5 > A10 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
% 0.29/0.45 ! [A5: $tType,A10: $tType] :
% 0.29/0.45 ( ( bounded_lattice @ A10 @ ( type2 @ A10 ) )
% 0.29/0.45 => ( bounded_lattice_top @ ( A5 > A10 ) @ ( type2 @ ( A5 > A10 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 0.29/0.45 ! [A5: $tType,A10: $tType] :
% 0.29/0.45 ( ( semilattice_sup @ A10 @ ( type2 @ A10 ) )
% 0.29/0.46 => ( semilattice_sup @ ( A5 > A10 ) @ ( type2 @ ( A5 > A10 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
% 0.29/0.46 ! [A5: $tType,A10: $tType] :
% 0.29/0.46 ( ( semilattice_inf @ A10 @ ( type2 @ A10 ) )
% 0.29/0.46 => ( semilattice_inf @ ( A5 > A10 ) @ ( type2 @ ( A5 > A10 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Lattices_Odistrib__lattice,axiom,
% 0.29/0.46 ! [A5: $tType,A10: $tType] :
% 0.29/0.46 ( ( distrib_lattice @ A10 @ ( type2 @ A10 ) )
% 0.29/0.46 => ( distrib_lattice @ ( A5 > A10 ) @ ( type2 @ ( A5 > A10 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Lattices_Olattice,axiom,
% 0.29/0.46 ! [A5: $tType,A10: $tType] :
% 0.29/0.46 ( ( lattice @ A10 @ ( type2 @ A10 ) )
% 0.29/0.46 => ( lattice @ ( A5 > A10 ) @ ( type2 @ ( A5 > A10 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_3,axiom,
% 0.29/0.46 ! [A5: $tType] : ( bounde1561333602nf_top @ ( set @ A5 ) @ ( type2 @ ( set @ A5 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_4,axiom,
% 0.29/0.46 ! [A5: $tType] : ( bounded_lattice_top @ ( set @ A5 ) @ ( type2 @ ( set @ A5 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_5,axiom,
% 0.29/0.46 ! [A5: $tType] : ( semilattice_sup @ ( set @ A5 ) @ ( type2 @ ( set @ A5 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_6,axiom,
% 0.29/0.46 ! [A5: $tType] : ( semilattice_inf @ ( set @ A5 ) @ ( type2 @ ( set @ A5 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Lattices_Odistrib__lattice_7,axiom,
% 0.29/0.46 ! [A5: $tType] : ( distrib_lattice @ ( set @ A5 ) @ ( type2 @ ( set @ A5 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Lattices_Olattice_8,axiom,
% 0.29/0.46 ! [A5: $tType] : ( lattice @ ( set @ A5 ) @ ( type2 @ ( set @ A5 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_9,axiom,
% 0.29/0.46 bounde1561333602nf_top @ $o @ ( type2 @ $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_10,axiom,
% 0.29/0.46 bounded_lattice_top @ $o @ ( type2 @ $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_11,axiom,
% 0.29/0.46 semilattice_sup @ $o @ ( type2 @ $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_12,axiom,
% 0.29/0.46 semilattice_inf @ $o @ ( type2 @ $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_13,axiom,
% 0.29/0.46 distrib_lattice @ $o @ ( type2 @ $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Lattices_Olattice_14,axiom,
% 0.29/0.46 lattice @ $o @ ( type2 @ $o ) ).
% 0.29/0.46
% 0.29/0.46 %----Helper facts (3)
% 0.29/0.46 thf(help_If_3_1_T,axiom,
% 0.29/0.46 ! [P: $o] :
% 0.29/0.46 ( ( P = $true )
% 0.29/0.46 | ( P = $false ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(help_If_2_1_T,axiom,
% 0.29/0.46 ! [A: $tType,X: A,Y: A] :
% 0.29/0.46 ( ( if @ A @ $false @ X @ Y )
% 0.29/0.46 = Y ) ).
% 0.29/0.46
% 0.29/0.46 thf(help_If_1_1_T,axiom,
% 0.29/0.46 ! [A: $tType,X: A,Y: A] :
% 0.29/0.46 ( ( if @ A @ $true @ X @ Y )
% 0.29/0.46 = X ) ).
% 0.29/0.46
% 0.29/0.46 %----Conjectures (5)
% 0.29/0.46 thf(conj_0,hypothesis,
% 0.29/0.46 $true ).
% 0.29/0.46
% 0.29/0.46 thf(conj_1,hypothesis,
% 0.29/0.46 $true ).
% 0.29/0.46
% 0.29/0.46 thf(conj_2,hypothesis,
% 0.29/0.46 $true ).
% 0.29/0.46
% 0.29/0.46 thf(conj_3,hypothesis,
% 0.29/0.46 $true ).
% 0.29/0.46
% 0.29/0.46 thf(conj_4,conjecture,
% 0.29/0.46 ? [Tr14: dtree,Tr24: dtree,Tr33: dtree] :
% 0.29/0.46 ( ( trA
% 0.29/0.46 = ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr14 @ Tr24 ) ) @ Tr33 ) ) )
% 0.29/0.46 & ( ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ tr1 @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ tr2 @ tr3 ) ) ) )
% 0.29/0.46 = ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr14 @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr24 @ Tr33 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 ------------------------
% 2.54/1.49 ---- Embedded in THF ---
% 2.54/1.49 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.54/1.49 %%% Generated on Tue Feb 24 04:16:41 EST 2026
% 2.54/1.49 %%% using '$$dhol' embedding, version 1.3.0.
% 2.54/1.49 %%% Logic specification used:
% 2.54/1.49 %%% thf(spec, logic, $$dhol).
% 2.54/1.49
% 2.54/1.49 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.sn5SmONUSK/DTF2DTF10309.p
% 2.54/1.49 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 2.54/1.49 thf(ty_t_Sum__Type_Osum, type, sum_sum: ($tType > ($tType > $tType))).
% 2.54/1.49 thf(ty_t_DTree_Odtree, type, dtree: $tType).
% 2.54/1.49 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.54/1.49 thf(ty_t_DTree_OT, type, t: $tType).
% 2.54/1.49 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.54/1.49 thf(sy_cl_Lattices_Obounded__lattice, type, bounded_lattice: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.54/1.49 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.54/1.49 thf(sy_cl_Lattices_Olattice, type, lattice: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.54/1.49 thf(sy_cl_Lattices_Odistrib__lattice, type, distrib_lattice: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.54/1.49 thf(sy_cl_Lattices_Osemilattice__inf, type, semilattice_inf: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.54/1.49 thf(sy_cl_Lattices_Osemilattice__sup, type, semilattice_sup: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.54/1.49 thf(sy_cl_Lattices_Obounded__lattice__top, type, bounded_lattice_top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.54/1.49 thf(sy_cl_Lattices_Obounded__semilattice__inf__top, type, bounde1561333602nf_top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.54/1.49 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2, type, bNF_Greatest_image2: (!> [C:$tType,A:$tType,B:$tType]: (((set @ C) > ((C > A) > ((C > B) > (set @ ((product_prod @ A) @ B)))))))).
% 2.54/1.49 thf(sy_c_DTree_Ocont, type, cont: (dtree > (set @ ((sum_sum @ t) @ dtree)))).
% 2.54/1.49 thf(sy_c_Fun_Ocomp, type, comp: (!> [B:$tType,C:$tType,A:$tType]: (((B > C) > ((A > B) > (A > C)))))).
% 2.54/1.49 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.54/1.49 thf(sy_c_Lattices_Oinf__class_Oinf, type, inf_inf: (!> [A:$tType]: ((A > (A > A))))).
% 2.54/1.49 thf(sy_c_Lattices_Osup__class_Osup, type, sup_sup: (!> [A:$tType]: ((A > (A > A))))).
% 2.54/1.49 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 2.54/1.49 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar, type, parall1899940088le_par: (((product_prod @ dtree) @ dtree) > dtree)).
% 2.54/1.49 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 2.54/1.49 thf(sy_c_Product__Type_Ocurry, type, product_curry: (!> [A:$tType,B:$tType,C:$tType]: (((((product_prod @ A) @ B) > C) > (A > (B > C)))))).
% 2.54/1.49 thf(sy_c_Product__Type_Ointernal__case__prod, type, produc2004651681e_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 2.54/1.49 thf(sy_c_Product__Type_Omap__prod, type, product_map_prod: (!> [A:$tType,C:$tType,B:$tType,D:$tType]: (((A > C) > ((B > D) > (((product_prod @ A) @ B) > ((product_prod @ C) @ D))))))).
% 2.54/1.49 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 2.54/1.49 thf(sy_c_Product__Type_Oprod_Ocase__prod, type, product_case_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 2.54/1.49 thf(sy_c_Product__Type_Oprod_Oswap, type, product_swap: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > ((product_prod @ B) @ A))))).
% 2.54/1.49 thf(sy_c_Product__Type_Oscomp, type, product_scomp: (!> [A:$tType,B:$tType,C:$tType,D:$tType]: (((A > ((product_prod @ B) @ C)) > ((B > (C > D)) > (A > D)))))).
% 2.54/1.49 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 2.54/1.49 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.54/1.49 thf(sy_c_Set_Oimage, type, image: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (set @ B)))))).
% 2.54/1.49 thf(sy_c_Set_Ovimage, type, vimage: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ B) > (set @ A)))))).
% 2.54/1.49 thf(sy_c_Sum__Type_OInl, type, sum_Inl: (!> [A:$tType,B:$tType]: ((A > ((sum_sum @ A) @ B))))).
% 2.54/1.49 thf(sy_c_Sum__Type_OInr, type, sum_Inr: (!> [B:$tType,A:$tType]: ((B > ((sum_sum @ A) @ B))))).
% 2.54/1.49 thf(sy_c_Sum__Type_OPlus, type, sum_Plus: (!> [A:$tType,B:$tType]: (((set @ A) > ((set @ B) > (set @ ((sum_sum @ A) @ B))))))).
% 2.54/1.49 thf(sy_c_Sum__Type_OSuml, type, sum_Suml: (!> [A:$tType,C:$tType,B:$tType]: (((A > C) > (((sum_sum @ A) @ B) > C))))).
% 2.54/1.49 thf(sy_c_Sum__Type_OSumr, type, sum_Sumr: (!> [B:$tType,C:$tType,A:$tType]: (((B > C) > (((sum_sum @ A) @ B) > C))))).
% 2.54/1.49 thf(sy_c_Sum__Type_Oold_Osum_Orec__sum, type, sum_rec_sum: (!> [A:$tType,T:$tType,B:$tType]: (((A > T) > ((B > T) > (((sum_sum @ A) @ B) > T)))))).
% 2.54/1.49 thf(sy_c_Sum__Type_Osum_Ocase__sum, type, sum_case_sum: (!> [A:$tType,C:$tType,B:$tType]: (((A > C) > ((B > C) > (((sum_sum @ A) @ B) > C)))))).
% 2.54/1.49 thf(sy_c_Typedef_Otype__definition, type, type_definition: (!> [B:$tType,A:$tType]: (((B > A) > ((A > B) > ((set @ A) > $o)))))).
% 2.54/1.49 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.54/1.49 thf(sy_v_tr1_H____, type, tr1: dtree).
% 2.54/1.49 thf(sy_v_tr1a____, type, tr1a: dtree).
% 2.54/1.49 thf(sy_v_tr2_H____, type, tr2: dtree).
% 2.54/1.49 thf(sy_v_tr2a____, type, tr2a: dtree).
% 2.54/1.49 thf(sy_v_tr3_H____, type, tr3: dtree).
% 2.54/1.49 thf(sy_v_tr3a____, type, tr3a: dtree).
% 2.54/1.49 thf(sy_v_trA_H____, type, trA: dtree).
% 2.54/1.49 thf(sy_v_trA____, type, trA2: dtree).
% 2.54/1.49 thf(sy_v_trB____, type, trB: dtree).
% 2.54/1.49 thf(fact_0__092_060open_062_092_060exists_062tr1_Atr2_Atr3_O_AtrA_A_061_A_Itr1_A_092_060parallel_062_Atr2_J_A_092_060parallel_062_Atr3_A_092_060and_062_AtrB_A_061_Atr1_A_092_060parallel_062_Atr2_A_092_060parallel_062_Atr3_092_060close_062, axiom, (? [Tr1:dtree,Tr2:dtree,Tr3:dtree]: (((trA2 = (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ Tr1) @ Tr2))) @ Tr3))) & (trB = (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ Tr1) @ (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ Tr2) @ Tr3))))))))).
% 2.54/1.49 thf(fact_1__092_060open_062Inr_Atr1_H_A_092_060in_062_Acont_Atr1_092_060close_062, axiom, (((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inr @ dtree) @ t) @ tr1)) @ (cont @ tr1a))).
% 2.54/1.49 thf(fact_2__092_060open_062Inr_Atr2_H_A_092_060in_062_Acont_Atr2_092_060close_062, axiom, (((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inr @ dtree) @ t) @ tr2)) @ (cont @ tr2a))).
% 2.54/1.49 thf(fact_3__092_060open_062Inr_Atr3_H_A_092_060in_062_Acont_Atr3_092_060close_062, axiom, (((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inr @ dtree) @ t) @ tr3)) @ (cont @ tr3a))).
% 2.54/1.49 thf(fact_4__092_060open_062trA_H_A_061_A_Itr1_H_A_092_060parallel_062_Atr2_H_J_A_092_060parallel_062_Atr3_H_092_060close_062, axiom, (trA = (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ tr1) @ tr2))) @ tr3)))).
% 2.54/1.49 thf(fact_5__092_060open_062Inr_AtrA_H_A_092_060in_062_Acont_A_I_Itr1_A_092_060parallel_062_Atr2_J_A_092_060parallel_062_Atr3_J_092_060close_062, axiom, (((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inr @ dtree) @ t) @ trA)) @ (cont @ (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ tr1a) @ tr2a))) @ tr3a))))).
% 2.54/1.49 thf(fact_6__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062tr1_H_Atr2_H_Atr3_H_O_A_092_060lbrakk_062trA_H_A_061_A_Itr1_H_A_092_060parallel_062_Atr2_H_J_A_092_060parallel_062_Atr3_H_059_AInr_Atr1_H_A_092_060in_062_Acont_Atr1_059_AInr_Atr2_H_A_092_060in_062_Acont_Atr2_059_AInr_Atr3_H_A_092_060in_062_Acont_Atr3_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom, (~ ((! [Tr12:dtree,Tr22:dtree,Tr32:dtree]: (((trA = (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ Tr12) @ Tr22))) @ Tr32))) => ((((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inr @ dtree) @ t) @ Tr12)) @ (cont @ tr1a)) => ((((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inr @ dtree) @ t) @ Tr22)) @ (cont @ tr2a)) => (~ ((((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inr @ dtree) @ t) @ Tr32)) @ (cont @ tr3a)))))))))))).
% 2.54/1.49 thf(fact_7_par__com, axiom, (! [Tr13:dtree,Tr23:dtree]: (((parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ Tr13) @ Tr23)) = (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ Tr23) @ Tr13)))))).
% 2.54/1.49 thf(fact_8_par__r_Ocases, axiom, (! [X:((product_prod @ dtree) @ dtree)]: ((~ ((! [Tr1:dtree,Tr2:dtree]: ((X != ((((product_Pair @ dtree) @ dtree) @ Tr1) @ Tr2))))))))).
% 2.54/1.49 thf(fact_9_par__r_Oinduct, axiom, (! [P:(((product_prod @ dtree) @ dtree) > $o),A0:((product_prod @ dtree) @ dtree)]: (((! [Tr1:dtree,Tr2:dtree]: ((P @ ((((product_Pair @ dtree) @ dtree) @ Tr1) @ Tr2)))) => (P @ A0))))).
% 2.54/1.49 thf(fact_10_sum_Oinject_I2_J, axiom, (! [A:$tType,B:$tType,X2:B,Y2:B]: ((((((sum_Inr @ B) @ A) @ X2) = (((sum_Inr @ B) @ A) @ Y2)) = (X2 = Y2))))).
% 2.54/1.49 thf(fact_11_old_Osum_Oinject_I2_J, axiom, (! [A:$tType,B:$tType,B2:B,B3:B]: ((((((sum_Inr @ B) @ A) @ B2) = (((sum_Inr @ B) @ A) @ B3)) = (B2 = B3))))).
% 2.54/1.49 thf(fact_12_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X2:B,Y1:A,Y2:B]: (((((((product_Pair @ A) @ B) @ X1) @ X2) = ((((product_Pair @ A) @ B) @ Y1) @ Y2)) = ((X1 = Y1) & (X2 = Y2)))))).
% 2.54/1.49 thf(fact_13_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A3:A,B3:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A3) @ B3)) = ((A2 = A3) & (B2 = B3)))))).
% 2.54/1.49 thf(fact_14_Inr__inject, axiom, (! [A:$tType,B:$tType,X:B,Y:B]: ((((((sum_Inr @ B) @ A) @ X) = (((sum_Inr @ B) @ A) @ Y)) => (X = Y))))).
% 2.54/1.49 thf(fact_15_not__arg__cong__Inr, axiom, (! [B:$tType,A:$tType,X:A,Y:A]: (((X != Y) => ((((sum_Inr @ A) @ B) @ X) != (((sum_Inr @ A) @ B) @ Y)))))).
% 2.54/1.49 thf(fact_16_Inl__in__cont__par, axiom, (! [T2:t,Tr13:dtree,Tr23:dtree]: (((((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inl @ t) @ dtree) @ T2)) @ (cont @ (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ Tr13) @ Tr23)))) = ((((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inl @ t) @ dtree) @ T2)) @ (cont @ Tr13)) | (((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inl @ t) @ dtree) @ T2)) @ (cont @ Tr23))))))).
% 2.54/1.49 thf(fact_17_surj__pair, axiom, (! [A:$tType,B:$tType,P2:((product_prod @ A) @ B)]: ((? [X3:A,Y3:B]: ((P2 = ((((product_Pair @ A) @ B) @ X3) @ Y3))))))).
% 2.54/1.49 thf(fact_18_prod__cases, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),P2:((product_prod @ A) @ B)]: (((! [A4:A,B4:B]: ((P @ ((((product_Pair @ A) @ B) @ A4) @ B4)))) => (P @ P2))))).
% 2.54/1.49 thf(fact_19_Pair__inject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A3:A,B3:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A3) @ B3)) => (~ (((A2 = A3) => (B2 != B3)))))))).
% 2.54/1.49 thf(fact_20_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A4:A,B4:B,C2:C]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A4) @ ((((product_Pair @ B) @ C) @ B4) @ C2)))))))))).
% 2.54/1.49 thf(fact_21_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: ((~ ((! [A4:A,B4:B,C2:C,D2:D]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B4) @ ((((product_Pair @ C) @ D) @ C2) @ D2))))))))))).
% 2.54/1.49 thf(fact_22_sum_Oinject_I1_J, axiom, (! [B:$tType,A:$tType,X1:A,Y1:A]: ((((((sum_Inl @ A) @ B) @ X1) = (((sum_Inl @ A) @ B) @ Y1)) = (X1 = Y1))))).
% 2.54/1.49 thf(fact_23_old_Osum_Oinject_I1_J, axiom, (! [B:$tType,A:$tType,A2:A,A3:A]: ((((((sum_Inl @ A) @ B) @ A2) = (((sum_Inl @ A) @ B) @ A3)) = (A2 = A3))))).
% 2.54/1.49 thf(fact_24_Inl__inject, axiom, (! [B:$tType,A:$tType,X:A,Y:A]: ((((((sum_Inl @ A) @ B) @ X) = (((sum_Inl @ A) @ B) @ Y)) => (X = Y))))).
% 2.54/1.49 thf(fact_25_obj__sumE, axiom, (! [A:$tType,B:$tType,S:((sum_sum @ A) @ B)]: (((! [X3:A]: ((S != (((sum_Inl @ A) @ B) @ X3)))) => (~ ((! [X3:B]: ((S != (((sum_Inr @ B) @ A) @ X3)))))))))).
% 2.54/1.49 thf(fact_26_old_Osum_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((sum_sum @ A) @ B) > $o),Sum:((sum_sum @ A) @ B)]: (((! [A4:A]: ((P @ (((sum_Inl @ A) @ B) @ A4)))) => ((! [B4:B]: ((P @ (((sum_Inr @ B) @ A) @ B4)))) => (P @ Sum)))))).
% 2.54/1.49 thf(fact_27_old_Osum_Oexhaust, axiom, (! [A:$tType,B:$tType,Y:((sum_sum @ A) @ B)]: (((! [A4:A]: ((Y != (((sum_Inl @ A) @ B) @ A4)))) => (~ ((! [B4:B]: ((Y != (((sum_Inr @ B) @ A) @ B4)))))))))).
% 2.54/1.49 thf(fact_28_split__sum__all, axiom, (! [B:$tType,A:$tType]: (((^ [P3:(((sum_sum @ A) @ B) > $o)]: ((! [X4:((sum_sum @ A) @ B)]: ((P3 @ X4))))) = (^ [P4:(((sum_sum @ A) @ B) > $o)]: (((! [X5:A]: ((P4 @ (((sum_Inl @ A) @ B) @ X5)))) & (! [X5:B]: ((P4 @ (((sum_Inr @ B) @ A) @ X5))))))))))).
% 2.54/1.49 thf(fact_29_split__sum__ex, axiom, (! [B:$tType,A:$tType]: (((^ [P3:(((sum_sum @ A) @ B) > $o)]: ((? [X4:((sum_sum @ A) @ B)]: ((P3 @ X4))))) = (^ [P4:(((sum_sum @ A) @ B) > $o)]: (((? [X5:A]: ((P4 @ (((sum_Inl @ A) @ B) @ X5)))) | (? [X5:B]: ((P4 @ (((sum_Inr @ B) @ A) @ X5))))))))))).
% 2.54/1.49 thf(fact_30_Inr__not__Inl, axiom, (! [B:$tType,A:$tType,B2:B,A2:A]: (((((sum_Inr @ B) @ A) @ B2) != (((sum_Inl @ A) @ B) @ A2))))).
% 2.54/1.49 thf(fact_31_sumE, axiom, (! [A:$tType,B:$tType,S:((sum_sum @ A) @ B)]: (((! [X3:A]: ((S != (((sum_Inl @ A) @ B) @ X3)))) => (~ ((! [Y3:B]: ((S != (((sum_Inr @ B) @ A) @ Y3)))))))))).
% 2.54/1.49 thf(fact_32_old_Osum_Odistinct_I1_J, axiom, (! [A:$tType,B:$tType,A2:A,B3:B]: (((((sum_Inl @ A) @ B) @ A2) != (((sum_Inr @ B) @ A) @ B3))))).
% 2.54/1.49 thf(fact_33_old_Osum_Odistinct_I2_J, axiom, (! [B5:$tType,A5:$tType,B6:B5,A6:A5]: (((((sum_Inr @ B5) @ A5) @ B6) != (((sum_Inl @ A5) @ B5) @ A6))))).
% 2.54/1.49 thf(fact_34_sum_Odistinct_I1_J, axiom, (! [A:$tType,B:$tType,X1:A,X2:B]: (((((sum_Inl @ A) @ B) @ X1) != (((sum_Inr @ B) @ A) @ X2))))).
% 2.54/1.49 thf(fact_35_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A4:A,B4:B]: ((P @ ((((product_Pair @ A) @ B) @ A4) @ B4)))) => (P @ Prod))))).
% 2.54/1.49 thf(fact_36_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y:((product_prod @ A) @ B)]: ((~ ((! [A4:A,B4:B]: ((Y != ((((product_Pair @ A) @ B) @ A4) @ B4))))))))).
% 2.54/1.49 thf(fact_37_prod__induct7, axiom, (! [G:$tType,F:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))))]: (((! [A4:A,B4:B,C2:C,D2:D,E2:E,F2:F,G2:G]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F) @ G)) @ E2) @ ((((product_Pair @ F) @ G) @ F2) @ G2))))))))) => (P @ X))))).
% 2.54/1.49 thf(fact_38_prod__induct6, axiom, (! [F:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))))]: (((! [A4:A,B4:B,C2:C,D2:D,E2:E,F2:F]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F)) @ D2) @ ((((product_Pair @ E) @ F) @ E2) @ F2)))))))) => (P @ X))))).
% 2.54/1.49 thf(fact_39_prod__induct5, axiom, (! [E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: (((! [A4:A,B4:B,C2:C,D2:D,E2:E]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C2) @ ((((product_Pair @ D) @ E) @ D2) @ E2))))))) => (P @ X))))).
% 2.54/1.49 thf(fact_40_prod__induct4, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: (((! [A4:A,B4:B,C2:C,D2:D]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B4) @ ((((product_Pair @ C) @ D) @ C2) @ D2)))))) => (P @ X))))).
% 2.54/1.49 thf(fact_41_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A4:A,B4:B,C2:C]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A4) @ ((((product_Pair @ B) @ C) @ B4) @ C2))))) => (P @ X))))).
% 2.54/1.49 thf(fact_42_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F:$tType,G:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))))]: ((~ ((! [A4:A,B4:B,C2:C,D2:D,E2:E,F2:F,G2:G]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F) @ G)) @ E2) @ ((((product_Pair @ F) @ G) @ F2) @ G2)))))))))))))).
% 2.54/1.49 thf(fact_43_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))))]: ((~ ((! [A4:A,B4:B,C2:C,D2:D,E2:E,F2:F]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F)) @ D2) @ ((((product_Pair @ E) @ F) @ E2) @ F2))))))))))))).
% 2.54/1.49 thf(fact_44_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: ((~ ((! [A4:A,B4:B,C2:C,D2:D,E2:E]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C2) @ ((((product_Pair @ D) @ E) @ D2) @ E2)))))))))))).
% 2.54/1.49 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.54/1.49 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A7:(set @ A)]: ((((collect @ A) @ (^ [X5:A]: ((((member @ A) @ X5) @ A7)))) = A7)))).
% 2.54/1.49 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.54/1.49 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F3:(A > B),G3:(A > B)]: (((! [X3:A]: (((F3 @ X3) = (G3 @ X3)))) => (F3 = G3))))).
% 2.54/1.49 thf(fact_49_Inr__Inl__False, axiom, (! [B:$tType,A:$tType,X:B,Y:A]: (((((sum_Inr @ B) @ A) @ X) != (((sum_Inl @ A) @ B) @ Y))))).
% 2.54/1.49 thf(fact_50_Inl__Inr__False, axiom, (! [A:$tType,B:$tType,X:A,Y:B]: (((((sum_Inl @ A) @ B) @ X) != (((sum_Inr @ B) @ A) @ Y))))).
% 2.54/1.49 thf(fact_51_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A2:A,B2:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) = ((F1 @ A2) @ B2))))).
% 2.54/1.49 thf(fact_52_old_Osum_Osimps_I7_J, axiom, (! [B:$tType,T:$tType,A:$tType,F1:(A > T),F22:(B > T),A2:A]: ((((((((sum_rec_sum @ A) @ T) @ B) @ F1) @ F22) @ (((sum_Inl @ A) @ B) @ A2)) = (F1 @ A2))))).
% 2.54/1.49 thf(fact_53_old_Osum_Osimps_I8_J, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > T),F22:(B > T),B2:B]: ((((((((sum_rec_sum @ A) @ T) @ B) @ F1) @ F22) @ (((sum_Inr @ B) @ A) @ B2)) = (F22 @ B2))))).
% 2.54/1.49 thf(fact_54_obj__sumE__f, axiom, (! [A:$tType,C:$tType,B:$tType,S:B,F3:(((sum_sum @ A) @ C) > B),P:$o]: (((! [X3:A]: (((S = (F3 @ (((sum_Inl @ A) @ C) @ X3))) => P))) => ((! [X3:C]: (((S = (F3 @ (((sum_Inr @ C) @ A) @ X3))) => P))) => (! [X6:((sum_sum @ A) @ C)]: (((S = (F3 @ X6)) => P)))))))).
% 2.54/1.49 thf(fact_55_internal__case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,C3:(B > (C > A)),A2:B,B2:C]: (((((((produc2004651681e_prod @ B) @ C) @ A) @ C3) @ ((((product_Pair @ B) @ C) @ A2) @ B2)) = ((C3 @ A2) @ B2))))).
% 2.54/1.49 thf(fact_56_Suml_Osimps, axiom, (! [B:$tType,C:$tType,A:$tType,F3:(A > C),X:A]: (((((((sum_Suml @ A) @ C) @ B) @ F3) @ (((sum_Inl @ A) @ B) @ X)) = (F3 @ X))))).
% 2.54/1.49 thf(fact_57_Sumr_Osimps, axiom, (! [A:$tType,C:$tType,B:$tType,F3:(B > C),X:B]: (((((((sum_Sumr @ B) @ C) @ A) @ F3) @ (((sum_Inr @ B) @ A) @ X)) = (F3 @ X))))).
% 2.54/1.49 thf(fact_58_image2__eqI, axiom, (! [A:$tType,C:$tType,B:$tType,B2:A,F3:(B > A),X:B,C3:C,G3:(B > C),A7:(set @ B)]: (((B2 = (F3 @ X)) => ((C3 = (G3 @ X)) => ((((member @ B) @ X) @ A7) => (((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ B2) @ C3)) @ ((((((bNF_Greatest_image2 @ B) @ A) @ C) @ A7) @ F3) @ G3)))))))).
% 2.54/1.49 thf(fact_59_Sumr__inject, axiom, (! [A:$tType,C:$tType,B:$tType,F3:(B > C),G3:(B > C)]: (((((((sum_Sumr @ B) @ C) @ A) @ F3) = ((((sum_Sumr @ B) @ C) @ A) @ G3)) => (F3 = G3))))).
% 2.54/1.49 thf(fact_60_Suml__inject, axiom, (! [B:$tType,C:$tType,A:$tType,F3:(A > C),G3:(A > C)]: (((((((sum_Suml @ A) @ C) @ B) @ F3) = ((((sum_Suml @ A) @ C) @ B) @ G3)) => (F3 = G3))))).
% 2.54/1.49 thf(fact_61_ssubst__Pair__rhs, axiom, (! [B:$tType,A:$tType,R:A,S:B,R2:(set @ ((product_prod @ A) @ B)),S2:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R) @ S)) @ R2) => ((S2 = S) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R) @ S2)) @ R2)))))).
% 2.54/1.49 thf(fact_62_curry__conv, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((product_curry @ B) @ C) @ A) = (^ [F4:(((product_prod @ B) @ C) > A),A8:B,B7:C]: ((F4 @ ((((product_Pair @ B) @ C) @ A8) @ B7)))))))).
% 2.54/1.49 thf(fact_63_curryI, axiom, (! [A:$tType,B:$tType,F3:(((product_prod @ A) @ B) > $o),A2:A,B2:B]: (((F3 @ ((((product_Pair @ A) @ B) @ A2) @ B2)) => ((((((product_curry @ A) @ B) @ $o) @ F3) @ A2) @ B2))))).
% 2.54/1.49 thf(fact_64_case__sum__if, axiom, (! [B:$tType,A:$tType,C:$tType,P2:$o,F3:(B > A),G3:(C > A),X:B,Y:C]: (((P2 => (((((((sum_case_sum @ B) @ A) @ C) @ F3) @ G3) @ ((((if @ ((sum_sum @ B) @ C)) @ P2) @ (((sum_Inl @ B) @ C) @ X)) @ (((sum_Inr @ C) @ B) @ Y))) = (F3 @ X))) & ((~ (P2)) => (((((((sum_case_sum @ B) @ A) @ C) @ F3) @ G3) @ ((((if @ ((sum_sum @ B) @ C)) @ P2) @ (((sum_Inl @ B) @ C) @ X)) @ (((sum_Inr @ C) @ B) @ Y))) = (G3 @ Y))))))).
% 2.54/1.49 thf(fact_65_swap__simp, axiom, (! [A:$tType,B:$tType,X:B,Y:A]: (((((product_swap @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y)) = ((((product_Pair @ A) @ B) @ Y) @ X))))).
% 2.54/1.49 thf(fact_66_Inl__cont__par, axiom, (! [Tr13:dtree,Tr23:dtree]: ((((((vimage @ t) @ ((sum_sum @ t) @ dtree)) @ ((sum_Inl @ t) @ dtree)) @ (cont @ (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ Tr13) @ Tr23)))) = ((((vimage @ t) @ ((sum_sum @ t) @ dtree)) @ ((sum_Inl @ t) @ dtree)) @ (((sup_sup @ (set @ ((sum_sum @ t) @ dtree))) @ (cont @ Tr13)) @ (cont @ Tr23))))))).
% 2.54/1.49 thf(fact_67_internal__case__prod__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((produc2004651681e_prod @ A) @ B) @ C) = (((product_case_prod @ A) @ B) @ C))))).
% 2.54/1.49 thf(fact_68_swap__swap, axiom, (! [B:$tType,A:$tType,P2:((product_prod @ A) @ B)]: (((((product_swap @ B) @ A) @ (((product_swap @ A) @ B) @ P2)) = P2)))).
% 2.54/1.49 thf(fact_69_curry__case__prod, axiom, (! [C:$tType,B:$tType,A:$tType,F3:(A > (B > C))]: ((((((product_curry @ A) @ B) @ C) @ ((((product_case_prod @ A) @ B) @ C) @ F3)) = F3)))).
% 2.54/1.49 thf(fact_70_case__prod__curry, axiom, (! [C:$tType,B:$tType,A:$tType,F3:(((product_prod @ A) @ B) > C)]: ((((((product_case_prod @ A) @ B) @ C) @ ((((product_curry @ A) @ B) @ C) @ F3)) = F3)))).
% 2.54/1.49 thf(fact_71_case__sum__step_I2_J, axiom, (! [E:$tType,A:$tType,C:$tType,B:$tType,F3:(E > A),F5:(B > A),G4:(C > A),P2:((sum_sum @ B) @ C)]: ((((((((sum_case_sum @ E) @ A) @ ((sum_sum @ B) @ C)) @ F3) @ (((((sum_case_sum @ B) @ A) @ C) @ F5) @ G4)) @ (((sum_Inr @ ((sum_sum @ B) @ C)) @ E) @ P2)) = ((((((sum_case_sum @ B) @ A) @ C) @ F5) @ G4) @ P2))))).
% 2.54/1.49 thf(fact_72_case__sum__step_I1_J, axiom, (! [D:$tType,A:$tType,C:$tType,B:$tType,F5:(B > A),G4:(C > A),G3:(D > A),P2:((sum_sum @ B) @ C)]: ((((((((sum_case_sum @ ((sum_sum @ B) @ C)) @ A) @ D) @ (((((sum_case_sum @ B) @ A) @ C) @ F5) @ G4)) @ G3) @ (((sum_Inl @ ((sum_sum @ B) @ C)) @ D) @ P2)) = ((((((sum_case_sum @ B) @ A) @ C) @ F5) @ G4) @ P2))))).
% 2.54/1.49 thf(fact_73_case__sum__inject, axiom, (! [A:$tType,C:$tType,B:$tType,F1:(A > C),F22:(B > C),G1:(A > C),G22:(B > C)]: ((((((((sum_case_sum @ A) @ C) @ B) @ F1) @ F22) = (((((sum_case_sum @ A) @ C) @ B) @ G1) @ G22)) => (~ (((F1 = G1) => (F22 != G22)))))))).
% 2.54/1.49 thf(fact_74_mem__case__prodE, axiom, (! [B:$tType,A:$tType,C:$tType,Z:A,C3:(B > (C > (set @ A))),P2:((product_prod @ B) @ C)]: (((((member @ A) @ Z) @ (((((product_case_prod @ B) @ C) @ (set @ A)) @ C3) @ P2)) => (~ ((! [X3:B,Y3:C]: (((P2 = ((((product_Pair @ B) @ C) @ X3) @ Y3)) => (~ ((((member @ A) @ Z) @ ((C3 @ X3) @ Y3))))))))))))).
% 2.54/1.49 thf(fact_75_old_Oprod_Ocase, axiom, (! [A:$tType,C:$tType,B:$tType,F3:(A > (B > C)),X1:A,X2:B]: (((((((product_case_prod @ A) @ B) @ C) @ F3) @ ((((product_Pair @ A) @ B) @ X1) @ X2)) = ((F3 @ X1) @ X2))))).
% 2.54/1.49 thf(fact_76_old_Osum_Osimps_I6_J, axiom, (! [A:$tType,C:$tType,B:$tType,F1:(A > C),F22:(B > C),X2:B]: ((((((((sum_case_sum @ A) @ C) @ B) @ F1) @ F22) @ (((sum_Inr @ B) @ A) @ X2)) = (F22 @ X2))))).
% 2.54/1.49 thf(fact_77_old_Osum_Osimps_I5_J, axiom, (! [B:$tType,C:$tType,A:$tType,F1:(A > C),F22:(B > C),X1:A]: ((((((((sum_case_sum @ A) @ C) @ B) @ F1) @ F22) @ (((sum_Inl @ A) @ B) @ X1)) = (F1 @ X1))))).
% 2.54/1.49 thf(fact_78_curryE, axiom, (! [A:$tType,B:$tType,F3:(((product_prod @ A) @ B) > $o),A2:A,B2:B]: ((((((((product_curry @ A) @ B) @ $o) @ F3) @ A2) @ B2) => (F3 @ ((((product_Pair @ A) @ B) @ A2) @ B2)))))).
% 2.54/1.49 thf(fact_79_curryD, axiom, (! [A:$tType,B:$tType,F3:(((product_prod @ A) @ B) > $o),A2:A,B2:B]: ((((((((product_curry @ A) @ B) @ $o) @ F3) @ A2) @ B2) => (F3 @ ((((product_Pair @ A) @ B) @ A2) @ B2)))))).
% 2.54/1.49 thf(fact_80_vimage__Un, axiom, (! [A:$tType,B:$tType,F3:(A > B),A7:(set @ B),B8:(set @ B)]: ((((((vimage @ A) @ B) @ F3) @ (((sup_sup @ (set @ B)) @ A7) @ B8)) = (((sup_sup @ (set @ A)) @ ((((vimage @ A) @ B) @ F3) @ A7)) @ ((((vimage @ A) @ B) @ F3) @ B8)))))).
% 2.54/1.49 thf(fact_81_Inl__Inr__image__cong, axiom, (! [B:$tType,A:$tType,A7:(set @ ((sum_sum @ A) @ B)),B8:(set @ ((sum_sum @ A) @ B))]: (((((((vimage @ A) @ ((sum_sum @ A) @ B)) @ ((sum_Inl @ A) @ B)) @ A7) = ((((vimage @ A) @ ((sum_sum @ A) @ B)) @ ((sum_Inl @ A) @ B)) @ B8)) => ((((((vimage @ B) @ ((sum_sum @ A) @ B)) @ ((sum_Inr @ B) @ A)) @ A7) = ((((vimage @ B) @ ((sum_sum @ A) @ B)) @ ((sum_Inr @ B) @ A)) @ B8)) => (A7 = B8)))))).
% 2.54/1.49 thf(fact_82_vimageI, axiom, (! [B:$tType,A:$tType,F3:(B > A),A2:B,B2:A,B8:(set @ A)]: ((((F3 @ A2) = B2) => ((((member @ A) @ B2) @ B8) => (((member @ B) @ A2) @ ((((vimage @ B) @ A) @ F3) @ B8))))))).
% 2.54/1.49 thf(fact_83_vimage__eq, axiom, (! [A:$tType,B:$tType,A2:A,F3:(A > B),B8:(set @ B)]: (((((member @ A) @ A2) @ ((((vimage @ A) @ B) @ F3) @ B8)) = (((member @ B) @ (F3 @ A2)) @ B8))))).
% 2.54/1.49 thf(fact_84_UnCI, axiom, (! [A:$tType,C3:A,B8:(set @ A),A7:(set @ A)]: ((((~ ((((member @ A) @ C3) @ B8))) => (((member @ A) @ C3) @ A7)) => (((member @ A) @ C3) @ (((sup_sup @ (set @ A)) @ A7) @ B8)))))).
% 2.54/1.49 thf(fact_85_Un__iff, axiom, (! [A:$tType,C3:A,A7:(set @ A),B8:(set @ A)]: (((((member @ A) @ C3) @ (((sup_sup @ (set @ A)) @ A7) @ B8)) = ((((member @ A) @ C3) @ A7) | (((member @ A) @ C3) @ B8)))))).
% 2.54/1.49 thf(fact_86_sup__apply, axiom, (! [B:$tType,A:$tType]: ((((semilattice_sup @ B) @ (type2 @ B)) => ((sup_sup @ (A > B)) = (^ [F4:(A > B),G5:(A > B),X5:A]: ((((sup_sup @ B) @ (F4 @ X5)) @ (G5 @ X5))))))))).
% 2.54/1.49 thf(fact_87_sup_Oright__idem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A2) @ B2)) @ B2) = (((sup_sup @ A) @ A2) @ B2)))))))).
% 2.54/1.49 thf(fact_88_sup__left__idem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ X) @ Y)) = (((sup_sup @ A) @ X) @ Y)))))))).
% 2.54/1.49 thf(fact_89_sup_Oleft__idem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((sup_sup @ A) @ A2) @ (((sup_sup @ A) @ A2) @ B2)) = (((sup_sup @ A) @ A2) @ B2)))))))).
% 2.54/1.49 thf(fact_90_sup__idem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A]: (((((sup_sup @ A) @ X) @ X) = X))))))).
% 2.54/1.49 thf(fact_91_sup_Oidem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A2:A]: (((((sup_sup @ A) @ A2) @ A2) = A2))))))).
% 2.54/1.49 thf(fact_92_sup__left__commute, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z)) = (((sup_sup @ A) @ Y) @ (((sup_sup @ A) @ X) @ Z))))))))).
% 2.54/1.49 thf(fact_93_sup_Oleft__commute, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C3:A]: (((((sup_sup @ A) @ B2) @ (((sup_sup @ A) @ A2) @ C3)) = (((sup_sup @ A) @ A2) @ (((sup_sup @ A) @ B2) @ C3))))))))).
% 2.54/1.49 thf(fact_94_sup__commute, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => ((sup_sup @ A) = (^ [X5:A,Y4:A]: ((((sup_sup @ A) @ Y4) @ X5)))))))).
% 2.54/1.49 thf(fact_95_sup_Ocommute, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => ((sup_sup @ A) = (^ [A8:A,B7:A]: ((((sup_sup @ A) @ B7) @ A8)))))))).
% 2.54/1.49 thf(fact_96_sup__assoc, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X) @ Y)) @ Z) = (((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z))))))))).
% 2.54/1.49 thf(fact_97_sup_Oassoc, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C3:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A2) @ B2)) @ C3) = (((sup_sup @ A) @ A2) @ (((sup_sup @ A) @ B2) @ C3))))))))).
% 2.54/1.49 thf(fact_98_sup__fun__def, axiom, (! [B:$tType,A:$tType]: ((((semilattice_sup @ B) @ (type2 @ B)) => ((sup_sup @ (A > B)) = (^ [F4:(A > B),G5:(A > B),X5:A]: ((((sup_sup @ B) @ (F4 @ X5)) @ (G5 @ X5))))))))).
% 2.54/1.49 thf(fact_99_inf__sup__aci_I5_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => ((sup_sup @ A) = (^ [X5:A,Y4:A]: ((((sup_sup @ A) @ Y4) @ X5)))))))).
% 2.54/1.49 thf(fact_100_inf__sup__aci_I6_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X) @ Y)) @ Z) = (((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z))))))))).
% 2.54/1.49 thf(fact_101_inf__sup__aci_I7_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z)) = (((sup_sup @ A) @ Y) @ (((sup_sup @ A) @ X) @ Z))))))))).
% 2.54/1.49 thf(fact_102_inf__sup__aci_I8_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ X) @ Y)) = (((sup_sup @ A) @ X) @ Y)))))))).
% 2.54/1.49 thf(fact_103_Un__left__commute, axiom, (! [A:$tType,A7:(set @ A),B8:(set @ A),C4:(set @ A)]: (((((sup_sup @ (set @ A)) @ A7) @ (((sup_sup @ (set @ A)) @ B8) @ C4)) = (((sup_sup @ (set @ A)) @ B8) @ (((sup_sup @ (set @ A)) @ A7) @ C4)))))).
% 2.54/1.49 thf(fact_104_Un__left__absorb, axiom, (! [A:$tType,A7:(set @ A),B8:(set @ A)]: (((((sup_sup @ (set @ A)) @ A7) @ (((sup_sup @ (set @ A)) @ A7) @ B8)) = (((sup_sup @ (set @ A)) @ A7) @ B8))))).
% 2.54/1.49 thf(fact_105_Un__commute, axiom, (! [A:$tType]: (((sup_sup @ (set @ A)) = (^ [A9:(set @ A),B9:(set @ A)]: ((((sup_sup @ (set @ A)) @ B9) @ A9))))))).
% 2.54/1.49 thf(fact_106_Un__absorb, axiom, (! [A:$tType,A7:(set @ A)]: (((((sup_sup @ (set @ A)) @ A7) @ A7) = A7)))).
% 2.54/1.49 thf(fact_107_Un__assoc, axiom, (! [A:$tType,A7:(set @ A),B8:(set @ A),C4:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A7) @ B8)) @ C4) = (((sup_sup @ (set @ A)) @ A7) @ (((sup_sup @ (set @ A)) @ B8) @ C4)))))).
% 2.54/1.49 thf(fact_108_ball__Un, axiom, (! [A:$tType,A7:(set @ A),B8:(set @ A),P:(A > $o)]: (((! [X5:A]: (((((member @ A) @ X5) @ (((sup_sup @ (set @ A)) @ A7) @ B8)) => (P @ X5)))) = ((! [X5:A]: (((((member @ A) @ X5) @ A7) => (P @ X5)))) & (! [X5:A]: (((((member @ A) @ X5) @ B8) => (P @ X5))))))))).
% 2.54/1.49 thf(fact_109_bex__Un, axiom, (! [A:$tType,A7:(set @ A),B8:(set @ A),P:(A > $o)]: (((? [X5:A]: (((((member @ A) @ X5) @ (((sup_sup @ (set @ A)) @ A7) @ B8)) & (P @ X5)))) = ((? [X5:A]: (((((member @ A) @ X5) @ A7) & (P @ X5)))) | (? [X5:A]: (((((member @ A) @ X5) @ B8) & (P @ X5))))))))).
% 2.54/1.49 thf(fact_110_UnI2, axiom, (! [A:$tType,C3:A,B8:(set @ A),A7:(set @ A)]: (((((member @ A) @ C3) @ B8) => (((member @ A) @ C3) @ (((sup_sup @ (set @ A)) @ A7) @ B8)))))).
% 2.54/1.49 thf(fact_111_UnI1, axiom, (! [A:$tType,C3:A,A7:(set @ A),B8:(set @ A)]: (((((member @ A) @ C3) @ A7) => (((member @ A) @ C3) @ (((sup_sup @ (set @ A)) @ A7) @ B8)))))).
% 2.54/1.49 thf(fact_112_UnE, axiom, (! [A:$tType,C3:A,A7:(set @ A),B8:(set @ A)]: (((((member @ A) @ C3) @ (((sup_sup @ (set @ A)) @ A7) @ B8)) => ((~ ((((member @ A) @ C3) @ A7))) => (((member @ A) @ C3) @ B8)))))).
% 2.54/1.49 thf(fact_113_vimage__Collect, axiom, (! [B:$tType,A:$tType,P:(B > $o),F3:(A > B),Q:(A > $o)]: (((! [X3:A]: (((P @ (F3 @ X3)) = (Q @ X3)))) => (((((vimage @ A) @ B) @ F3) @ ((collect @ B) @ P)) = ((collect @ A) @ Q)))))).
% 2.54/1.49 thf(fact_114_vimageI2, axiom, (! [B:$tType,A:$tType,F3:(B > A),A2:B,A7:(set @ A)]: (((((member @ A) @ (F3 @ A2)) @ A7) => (((member @ B) @ A2) @ ((((vimage @ B) @ A) @ F3) @ A7)))))).
% 2.54/1.49 thf(fact_115_vimageE, axiom, (! [A:$tType,B:$tType,A2:A,F3:(A > B),B8:(set @ B)]: (((((member @ A) @ A2) @ ((((vimage @ A) @ B) @ F3) @ B8)) => (((member @ B) @ (F3 @ A2)) @ B8))))).
% 2.54/1.49 thf(fact_116_vimageD, axiom, (! [A:$tType,B:$tType,A2:A,F3:(A > B),A7:(set @ B)]: (((((member @ A) @ A2) @ ((((vimage @ A) @ B) @ F3) @ A7)) => (((member @ B) @ (F3 @ A2)) @ A7))))).
% 2.54/1.49 thf(fact_117_split__cong, axiom, (! [C:$tType,B:$tType,A:$tType,Q2:((product_prod @ A) @ B),F3:(A > (B > C)),G3:(A > (B > C)),P2:((product_prod @ A) @ B)]: (((! [X3:A,Y3:B]: (((((((product_Pair @ A) @ B) @ X3) @ Y3) = Q2) => (((F3 @ X3) @ Y3) = ((G3 @ X3) @ Y3))))) => ((P2 = Q2) => ((((((product_case_prod @ A) @ B) @ C) @ F3) @ P2) = (((((product_case_prod @ A) @ B) @ C) @ G3) @ Q2))))))).
% 2.54/1.49 thf(fact_118_case__sum__expand__Inr__pointfree, axiom, (! [B:$tType,C:$tType,A:$tType,F3:(((sum_sum @ A) @ C) > B),G3:(A > B)]: ((((((((comp @ ((sum_sum @ A) @ C)) @ B) @ A) @ F3) @ ((sum_Inl @ A) @ C)) = G3) => ((((((sum_case_sum @ A) @ B) @ C) @ G3) @ (((((comp @ ((sum_sum @ A) @ C)) @ B) @ C) @ F3) @ ((sum_Inr @ C) @ A))) = F3))))).
% 2.54/1.49 thf(fact_119_case__sum__expand__Inr_H, axiom, (! [B:$tType,C:$tType,A:$tType,F3:(((sum_sum @ A) @ C) > B),G3:(A > B),H:(C > B)]: ((((((((comp @ ((sum_sum @ A) @ C)) @ B) @ A) @ F3) @ ((sum_Inl @ A) @ C)) = G3) => ((H = (((((comp @ ((sum_sum @ A) @ C)) @ B) @ C) @ F3) @ ((sum_Inr @ C) @ A))) = ((((((sum_case_sum @ A) @ B) @ C) @ G3) @ H) = F3)))))).
% 2.54/1.49 thf(fact_120_case__sum__expand__Inr, axiom, (! [B:$tType,C:$tType,A:$tType,F3:(((sum_sum @ A) @ C) > B),G3:(A > B),X:((sum_sum @ A) @ C)]: ((((((((comp @ ((sum_sum @ A) @ C)) @ B) @ A) @ F3) @ ((sum_Inl @ A) @ C)) = G3) => ((F3 @ X) = ((((((sum_case_sum @ A) @ B) @ C) @ G3) @ (((((comp @ ((sum_sum @ A) @ C)) @ B) @ C) @ F3) @ ((sum_Inr @ C) @ A))) @ X)))))).
% 2.54/1.49 thf(fact_121_scomp__apply, axiom, (! [A:$tType,D:$tType,C:$tType,B:$tType]: ((((((product_scomp @ B) @ C) @ D) @ A) = (^ [F4:(B > ((product_prod @ C) @ D)),G5:(C > (D > A)),X5:B]: ((((((product_case_prod @ C) @ D) @ A) @ G5) @ (F4 @ X5)))))))).
% 2.54/1.49 thf(fact_122_pair__in__swap__image, axiom, (! [A:$tType,B:$tType,Y:A,X:B,A7:(set @ ((product_prod @ B) @ A))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Y) @ X)) @ ((((image @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((product_swap @ B) @ A)) @ A7)) = (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X) @ Y)) @ A7))))).
% 2.54/1.49 thf(fact_123_image__eqI, axiom, (! [A:$tType,B:$tType,B2:A,F3:(B > A),X:B,A7:(set @ B)]: (((B2 = (F3 @ X)) => ((((member @ B) @ X) @ A7) => (((member @ A) @ B2) @ ((((image @ B) @ A) @ F3) @ A7))))))).
% 2.54/1.49 thf(fact_124_o__case__sum, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,H:(D > C),F3:(A > D),G3:(B > D)]: (((((((comp @ D) @ C) @ ((sum_sum @ A) @ B)) @ H) @ (((((sum_case_sum @ A) @ D) @ B) @ F3) @ G3)) = (((((sum_case_sum @ A) @ C) @ B) @ (((((comp @ D) @ C) @ A) @ H) @ F3)) @ (((((comp @ D) @ C) @ B) @ H) @ G3)))))).
% 2.54/1.49 thf(fact_125_rev__image__eqI, axiom, (! [B:$tType,A:$tType,X:A,A7:(set @ A),B2:B,F3:(A > B)]: (((((member @ A) @ X) @ A7) => ((B2 = (F3 @ X)) => (((member @ B) @ B2) @ ((((image @ A) @ B) @ F3) @ A7))))))).
% 2.54/1.49 thf(fact_126_ball__imageD, axiom, (! [A:$tType,B:$tType,F3:(B > A),A7:(set @ B),P:(A > $o)]: (((! [X3:A]: (((((member @ A) @ X3) @ ((((image @ B) @ A) @ F3) @ A7)) => (P @ X3)))) => (! [X6:B]: (((((member @ B) @ X6) @ A7) => (P @ (F3 @ X6))))))))).
% 2.54/1.49 thf(fact_127_image__cong, axiom, (! [B:$tType,A:$tType,M:(set @ A),N:(set @ A),F3:(A > B),G3:(A > B)]: (((M = N) => ((! [X3:A]: (((((member @ A) @ X3) @ N) => ((F3 @ X3) = (G3 @ X3))))) => (((((image @ A) @ B) @ F3) @ M) = ((((image @ A) @ B) @ G3) @ N))))))).
% 2.54/1.49 thf(fact_128_bex__imageD, axiom, (! [A:$tType,B:$tType,F3:(B > A),A7:(set @ B),P:(A > $o)]: (((? [X6:A]: (((((member @ A) @ X6) @ ((((image @ B) @ A) @ F3) @ A7)) & (P @ X6)))) => (? [X3:B]: (((((member @ B) @ X3) @ A7) & (P @ (F3 @ X3))))))))).
% 2.54/1.49 thf(fact_129_image__iff, axiom, (! [A:$tType,B:$tType,Z:A,F3:(B > A),A7:(set @ B)]: (((((member @ A) @ Z) @ ((((image @ B) @ A) @ F3) @ A7)) = (? [X5:B]: (((((member @ B) @ X5) @ A7) & (Z = (F3 @ X5))))))))).
% 2.54/1.49 thf(fact_130_imageI, axiom, (! [B:$tType,A:$tType,X:A,A7:(set @ A),F3:(A > B)]: (((((member @ A) @ X) @ A7) => (((member @ B) @ (F3 @ X)) @ ((((image @ A) @ B) @ F3) @ A7)))))).
% 2.54/1.49 thf(fact_131_rewriteR__comp__comp2, axiom, (! [C:$tType,B:$tType,E:$tType,D:$tType,A:$tType,G3:(C > B),H:(A > C),R1:(D > B),R22:(A > D),F3:(B > E),L:(D > E)]: ((((((((comp @ C) @ B) @ A) @ G3) @ H) = (((((comp @ D) @ B) @ A) @ R1) @ R22)) => (((((((comp @ B) @ E) @ D) @ F3) @ R1) = L) => ((((((comp @ C) @ E) @ A) @ (((((comp @ B) @ E) @ C) @ F3) @ G3)) @ H) = (((((comp @ D) @ E) @ A) @ L) @ R22))))))).
% 2.54/1.49 thf(fact_132_rewriteL__comp__comp2, axiom, (! [A:$tType,C:$tType,B:$tType,D:$tType,E:$tType,F3:(C > B),G3:(A > C),L1:(D > B),L2:(A > D),H:(E > A),R:(E > D)]: ((((((((comp @ C) @ B) @ A) @ F3) @ G3) = (((((comp @ D) @ B) @ A) @ L1) @ L2)) => (((((((comp @ A) @ D) @ E) @ L2) @ H) = R) => ((((((comp @ C) @ B) @ E) @ F3) @ (((((comp @ A) @ C) @ E) @ G3) @ H)) = (((((comp @ D) @ B) @ E) @ L1) @ R))))))).
% 2.54/1.49 thf(fact_133_rewriteR__comp__comp, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,G3:(C > B),H:(A > C),R:(A > B),F3:(B > D)]: ((((((((comp @ C) @ B) @ A) @ G3) @ H) = R) => ((((((comp @ C) @ D) @ A) @ (((((comp @ B) @ D) @ C) @ F3) @ G3)) @ H) = (((((comp @ B) @ D) @ A) @ F3) @ R)))))).
% 2.54/1.49 thf(fact_134_rewriteL__comp__comp, axiom, (! [C:$tType,B:$tType,A:$tType,D:$tType,F3:(C > B),G3:(A > C),L:(A > B),H:(D > A)]: ((((((((comp @ C) @ B) @ A) @ F3) @ G3) = L) => ((((((comp @ C) @ B) @ D) @ F3) @ (((((comp @ A) @ C) @ D) @ G3) @ H)) = (((((comp @ A) @ B) @ D) @ L) @ H)))))).
% 2.54/1.49 thf(fact_135_comp__cong, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,E:$tType,F3:(B > A),G3:(C > B),X:C,F5:(D > A),G4:(E > D),X7:E]: ((((F3 @ (G3 @ X)) = (F5 @ (G4 @ X7))) => (((((((comp @ B) @ A) @ C) @ F3) @ G3) @ X) = ((((((comp @ D) @ A) @ E) @ F5) @ G4) @ X7)))))).
% 2.54/1.49 thf(fact_136_image__Un, axiom, (! [A:$tType,B:$tType,F3:(B > A),A7:(set @ B),B8:(set @ B)]: ((((((image @ B) @ A) @ F3) @ (((sup_sup @ (set @ B)) @ A7) @ B8)) = (((sup_sup @ (set @ A)) @ ((((image @ B) @ A) @ F3) @ A7)) @ ((((image @ B) @ A) @ F3) @ B8)))))).
% 2.54/1.49 thf(fact_137_scomp__Pair, axiom, (! [C:$tType,B:$tType,A:$tType,X:(A > ((product_prod @ B) @ C))]: ((((((((product_scomp @ A) @ B) @ C) @ ((product_prod @ B) @ C)) @ X) @ ((product_Pair @ B) @ C)) = X)))).
% 2.54/1.49 thf(fact_138_Pair__scomp, axiom, (! [A:$tType,B:$tType,C:$tType,X:C,F3:(C > (A > B))]: ((((((((product_scomp @ A) @ C) @ A) @ B) @ (((product_Pair @ C) @ A) @ X)) @ F3) = (F3 @ X))))).
% 2.54/1.49 thf(fact_139_scomp__def, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType]: ((((((product_scomp @ A) @ B) @ C) @ D) = (^ [F4:(A > ((product_prod @ B) @ C)),G5:(B > (C > D)),X5:A]: ((((((product_case_prod @ B) @ C) @ D) @ G5) @ (F4 @ X5)))))))).
% 2.54/1.49 thf(fact_140_case__sum__o__inj_I1_J, axiom, (! [C:$tType,B:$tType,A:$tType,F3:(A > B),G3:(C > B)]: (((((((comp @ ((sum_sum @ A) @ C)) @ B) @ A) @ (((((sum_case_sum @ A) @ B) @ C) @ F3) @ G3)) @ ((sum_Inl @ A) @ C)) = F3)))).
% 2.54/1.49 thf(fact_141_case__sum__o__inj_I2_J, axiom, (! [A:$tType,B:$tType,C:$tType,F3:(A > B),G3:(C > B)]: (((((((comp @ ((sum_sum @ A) @ C)) @ B) @ C) @ (((((sum_case_sum @ A) @ B) @ C) @ F3) @ G3)) @ ((sum_Inr @ C) @ A)) = G3)))).
% 2.54/1.49 thf(fact_142_comp__apply, axiom, (! [C:$tType,A:$tType,B:$tType]: (((((comp @ B) @ A) @ C) = (^ [F4:(B > A),G5:(C > B),X5:C]: ((F4 @ (G5 @ X5)))))))).
% 2.54/1.49 thf(fact_143_set_Ocomp, axiom, (! [A:$tType,B:$tType,C:$tType,F3:(C > B),G3:(B > A)]: (((((((comp @ (set @ B)) @ (set @ C)) @ (set @ A)) @ (((vimage @ C) @ B) @ F3)) @ (((vimage @ B) @ A) @ G3)) = (((vimage @ C) @ A) @ (((((comp @ B) @ A) @ C) @ G3) @ F3)))))).
% 2.54/1.49 thf(fact_144_comp__apply__eq, axiom, (! [B:$tType,D:$tType,A:$tType,C:$tType,F3:(B > A),G3:(C > B),X:C,H:(D > A),K:(C > D)]: ((((F3 @ (G3 @ X)) = (H @ (K @ X))) => (((((((comp @ B) @ A) @ C) @ F3) @ G3) @ X) = ((((((comp @ D) @ A) @ C) @ H) @ K) @ X)))))).
% 2.54/1.49 thf(fact_145_comp__eq__dest__lhs, axiom, (! [C:$tType,B:$tType,A:$tType,A2:(C > B),B2:(A > C),C3:(A > B),V:A]: ((((((((comp @ C) @ B) @ A) @ A2) @ B2) = C3) => ((A2 @ (B2 @ V)) = (C3 @ V)))))).
% 2.54/1.49 thf(fact_146_comp__eq__elim, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,A2:(C > B),B2:(A > C),C3:(D > B),D3:(A > D)]: ((((((((comp @ C) @ B) @ A) @ A2) @ B2) = (((((comp @ D) @ B) @ A) @ C3) @ D3)) => (! [V2:A]: (((A2 @ (B2 @ V2)) = (C3 @ (D3 @ V2))))))))).
% 2.54/1.49 thf(fact_147_comp__eq__dest, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,A2:(C > B),B2:(A > C),C3:(D > B),D3:(A > D),V:A]: ((((((((comp @ C) @ B) @ A) @ A2) @ B2) = (((((comp @ D) @ B) @ A) @ C3) @ D3)) => ((A2 @ (B2 @ V)) = (C3 @ (D3 @ V))))))).
% 2.54/1.49 thf(fact_148_comp__assoc, axiom, (! [B:$tType,D:$tType,C:$tType,A:$tType,F3:(D > B),G3:(C > D),H:(A > C)]: (((((((comp @ C) @ B) @ A) @ (((((comp @ D) @ B) @ C) @ F3) @ G3)) @ H) = (((((comp @ D) @ B) @ A) @ F3) @ (((((comp @ C) @ D) @ A) @ G3) @ H)))))).
% 2.54/1.49 thf(fact_149_comp__def, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((comp @ B) @ C) @ A) = (^ [F4:(B > C),G5:(A > B),X5:A]: ((F4 @ (G5 @ X5)))))))).
% 2.54/1.49 thf(fact_150_image__eq__imp__comp, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,F3:(B > A),A7:(set @ B),G3:(C > A),B8:(set @ C),H:(A > D)]: (((((((image @ B) @ A) @ F3) @ A7) = ((((image @ C) @ A) @ G3) @ B8)) => (((((image @ B) @ D) @ (((((comp @ A) @ D) @ B) @ H) @ F3)) @ A7) = ((((image @ C) @ D) @ (((((comp @ A) @ D) @ C) @ H) @ G3)) @ B8)))))).
% 2.54/1.49 thf(fact_151_image__comp, axiom, (! [B:$tType,A:$tType,C:$tType,F3:(B > A),G3:(C > B),R:(set @ C)]: ((((((image @ B) @ A) @ F3) @ ((((image @ C) @ B) @ G3) @ R)) = ((((image @ C) @ A) @ (((((comp @ B) @ A) @ C) @ F3) @ G3)) @ R))))).
% 2.54/1.49 thf(fact_152_set_Ocompositionality, axiom, (! [C:$tType,B:$tType,A:$tType,F3:(C > B),G3:(B > A),Set:(set @ A)]: ((((((vimage @ C) @ B) @ F3) @ ((((vimage @ B) @ A) @ G3) @ Set)) = ((((vimage @ C) @ A) @ (((((comp @ B) @ A) @ C) @ G3) @ F3)) @ Set))))).
% 2.54/1.49 thf(fact_153_vimage__comp, axiom, (! [A:$tType,B:$tType,C:$tType,F3:(A > B),G3:(B > C),X:(set @ C)]: ((((((vimage @ A) @ B) @ F3) @ ((((vimage @ B) @ C) @ G3) @ X)) = ((((vimage @ A) @ C) @ (((((comp @ B) @ C) @ A) @ G3) @ F3)) @ X))))).
% 2.54/1.49 thf(fact_154_Sup_OSUP__image, axiom, (! [B:$tType,A:$tType,C:$tType,Sup:((set @ A) > A),G3:(B > A),F3:(C > B),A7:(set @ C)]: (((Sup @ ((((image @ B) @ A) @ G3) @ ((((image @ C) @ B) @ F3) @ A7))) = (Sup @ ((((image @ C) @ A) @ (((((comp @ B) @ A) @ C) @ G3) @ F3)) @ A7)))))).
% 2.54/1.49 thf(fact_155_Inf_OINF__image, axiom, (! [B:$tType,A:$tType,C:$tType,Inf:((set @ A) > A),G3:(B > A),F3:(C > B),A7:(set @ C)]: (((Inf @ ((((image @ B) @ A) @ G3) @ ((((image @ C) @ B) @ F3) @ A7))) = (Inf @ ((((image @ C) @ A) @ (((((comp @ B) @ A) @ C) @ G3) @ F3)) @ A7)))))).
% 2.54/1.49 thf(fact_156_Plus__def, axiom, (! [B:$tType,A:$tType]: ((((sum_Plus @ A) @ B) = (^ [A9:(set @ A),B9:(set @ B)]: ((((sup_sup @ (set @ ((sum_sum @ A) @ B))) @ ((((image @ A) @ ((sum_sum @ A) @ B)) @ ((sum_Inl @ A) @ B)) @ A9)) @ ((((image @ B) @ ((sum_sum @ A) @ B)) @ ((sum_Inr @ B) @ A)) @ B9)))))))).
% 2.54/1.49 thf(fact_157_surj__swap, axiom, (! [B:$tType,A:$tType]: ((((((image @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((product_swap @ B) @ A)) @ (top_top @ (set @ ((product_prod @ B) @ A)))) = (top_top @ (set @ ((product_prod @ A) @ B))))))).
% 2.54/1.49 thf(fact_158_UNIV__I, axiom, (! [A:$tType,X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A)))))).
% 2.54/1.49 thf(fact_159_sup__top__left, axiom, (! [A:$tType]: ((((bounded_lattice_top @ A) @ (type2 @ A)) => (! [X:A]: (((((sup_sup @ A) @ (top_top @ A)) @ X) = (top_top @ A)))))))).
% 2.54/1.49 thf(fact_160_sup__top__right, axiom, (! [A:$tType]: ((((bounded_lattice_top @ A) @ (type2 @ A)) => (! [X:A]: (((((sup_sup @ A) @ X) @ (top_top @ A)) = (top_top @ A)))))))).
% 2.54/1.49 thf(fact_161_vimage__UNIV, axiom, (! [B:$tType,A:$tType,F3:(A > B)]: ((((((vimage @ A) @ B) @ F3) @ (top_top @ (set @ B))) = (top_top @ (set @ A)))))).
% 2.54/1.49 thf(fact_162_UNIV__Plus__UNIV, axiom, (! [B:$tType,A:$tType]: ((((((sum_Plus @ A) @ B) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) = (top_top @ (set @ ((sum_sum @ A) @ B))))))).
% 2.54/1.49 thf(fact_163_InrI, axiom, (! [B:$tType,A:$tType,B2:A,B8:(set @ A),A7:(set @ B)]: (((((member @ A) @ B2) @ B8) => (((member @ ((sum_sum @ B) @ A)) @ (((sum_Inr @ A) @ B) @ B2)) @ ((((sum_Plus @ B) @ A) @ A7) @ B8)))))).
% 2.54/1.49 thf(fact_164_InlI, axiom, (! [A:$tType,B:$tType,A2:A,A7:(set @ A),B8:(set @ B)]: (((((member @ A) @ A2) @ A7) => (((member @ ((sum_sum @ A) @ B)) @ (((sum_Inl @ A) @ B) @ A2)) @ ((((sum_Plus @ A) @ B) @ A7) @ B8)))))).
% 2.54/1.49 thf(fact_165_range__eqI, axiom, (! [A:$tType,B:$tType,B2:A,F3:(B > A),X:B]: (((B2 = (F3 @ X)) => (((member @ A) @ B2) @ ((((image @ B) @ A) @ F3) @ (top_top @ (set @ B)))))))).
% 2.54/1.49 thf(fact_166_rangeI, axiom, (! [A:$tType,B:$tType,F3:(B > A),X:B]: ((((member @ A) @ (F3 @ X)) @ ((((image @ B) @ A) @ F3) @ (top_top @ (set @ B))))))).
% 2.54/1.49 thf(fact_167_surjD, axiom, (! [A:$tType,B:$tType,F3:(B > A),Y:A]: (((((((image @ B) @ A) @ F3) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (? [X3:B]: ((Y = (F3 @ X3)))))))).
% 2.54/1.49 thf(fact_168_surjE, axiom, (! [A:$tType,B:$tType,F3:(B > A),Y:A]: (((((((image @ B) @ A) @ F3) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (~ ((! [X3:B]: ((Y != (F3 @ X3)))))))))).
% 2.54/1.49 thf(fact_169_surjI, axiom, (! [B:$tType,A:$tType,G3:(B > A),F3:(A > B)]: (((! [X3:A]: (((G3 @ (F3 @ X3)) = X3))) => (((((image @ B) @ A) @ G3) @ (top_top @ (set @ B))) = (top_top @ (set @ A))))))).
% 2.54/1.49 thf(fact_170_surj__def, axiom, (! [B:$tType,A:$tType,F3:(B > A)]: (((((((image @ B) @ A) @ F3) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) = (! [Y4:A]: ((? [X5:B]: ((Y4 = (F3 @ X5)))))))))).
% 2.54/1.49 thf(fact_171_UNIV__witness, axiom, (! [A:$tType]: ((? [X3:A]: ((((member @ A) @ X3) @ (top_top @ (set @ A)))))))).
% 2.54/1.49 thf(fact_172_UNIV__eq__I, axiom, (! [A:$tType,A7:(set @ A)]: (((! [X3:A]: ((((member @ A) @ X3) @ A7))) => ((top_top @ (set @ A)) = A7))))).
% 2.54/1.49 thf(fact_173_Un__UNIV__left, axiom, (! [A:$tType,B8:(set @ A)]: (((((sup_sup @ (set @ A)) @ (top_top @ (set @ A))) @ B8) = (top_top @ (set @ A)))))).
% 2.54/1.49 thf(fact_174_Un__UNIV__right, axiom, (! [A:$tType,A7:(set @ A)]: (((((sup_sup @ (set @ A)) @ A7) @ (top_top @ (set @ A))) = (top_top @ (set @ A)))))).
% 2.54/1.49 thf(fact_175_UNIV__sum, axiom, (! [A:$tType,B:$tType]: (((top_top @ (set @ ((sum_sum @ A) @ B))) = (((sup_sup @ (set @ ((sum_sum @ A) @ B))) @ ((((image @ A) @ ((sum_sum @ A) @ B)) @ ((sum_Inl @ A) @ B)) @ (top_top @ (set @ A)))) @ ((((image @ B) @ ((sum_sum @ A) @ B)) @ ((sum_Inr @ B) @ A)) @ (top_top @ (set @ B)))))))).
% 2.54/1.49 thf(fact_176_surj__fun__eq, axiom, (! [B:$tType,C:$tType,A:$tType,F3:(B > A),X8:(set @ B),G1:(A > C),G22:(A > C)]: (((((((image @ B) @ A) @ F3) @ X8) = (top_top @ (set @ A))) => ((! [X3:B]: (((((member @ B) @ X3) @ X8) => (((((((comp @ A) @ C) @ B) @ G1) @ F3) @ X3) = ((((((comp @ A) @ C) @ B) @ G22) @ F3) @ X3))))) => (G1 = G22)))))).
% 2.54/1.49 thf(fact_177_comp__surj, axiom, (! [B:$tType,A:$tType,C:$tType,F3:(B > A),G3:(A > C)]: (((((((image @ B) @ A) @ F3) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => ((((((image @ A) @ C) @ G3) @ (top_top @ (set @ A))) = (top_top @ (set @ C))) => (((((image @ B) @ C) @ (((((comp @ A) @ C) @ B) @ G3) @ F3)) @ (top_top @ (set @ B))) = (top_top @ (set @ C)))))))).
% 2.54/1.49 thf(fact_178_surj__image__vimage__eq, axiom, (! [B:$tType,A:$tType,F3:(B > A),A7:(set @ A)]: (((((((image @ B) @ A) @ F3) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (((((image @ B) @ A) @ F3) @ ((((vimage @ B) @ A) @ F3) @ A7)) = A7))))).
% 2.54/1.49 thf(fact_179_PlusE, axiom, (! [A:$tType,B:$tType,U:((sum_sum @ A) @ B),A7:(set @ A),B8:(set @ B)]: (((((member @ ((sum_sum @ A) @ B)) @ U) @ ((((sum_Plus @ A) @ B) @ A7) @ B8)) => ((! [X3:A]: (((((member @ A) @ X3) @ A7) => (U != (((sum_Inl @ A) @ B) @ X3))))) => (~ ((! [Y3:B]: (((((member @ B) @ Y3) @ B8) => (U != (((sum_Inr @ B) @ A) @ Y3)))))))))))).
% 2.54/1.49 thf(fact_180_Inf_OINF__cong, axiom, (! [A:$tType,B:$tType,A7:(set @ B),B8:(set @ B),C4:(B > A),D4:(B > A),Inf:((set @ A) > A)]: (((A7 = B8) => ((! [X3:B]: (((((member @ B) @ X3) @ B8) => ((C4 @ X3) = (D4 @ X3))))) => ((Inf @ ((((image @ B) @ A) @ C4) @ A7)) = (Inf @ ((((image @ B) @ A) @ D4) @ B8)))))))).
% 2.54/1.49 thf(fact_181_Sup_OSUP__cong, axiom, (! [A:$tType,B:$tType,A7:(set @ B),B8:(set @ B),C4:(B > A),D4:(B > A),Sup:((set @ A) > A)]: (((A7 = B8) => ((! [X3:B]: (((((member @ B) @ X3) @ B8) => ((C4 @ X3) = (D4 @ X3))))) => ((Sup @ ((((image @ B) @ A) @ C4) @ A7)) = (Sup @ ((((image @ B) @ A) @ D4) @ B8)))))))).
% 2.54/1.49 thf(fact_182_fun_Oset__map, axiom, (! [B:$tType,A:$tType,D:$tType,F3:(A > B),V:(D > A)]: ((((((image @ D) @ B) @ (((((comp @ A) @ B) @ D) @ F3) @ V)) @ (top_top @ (set @ D))) = ((((image @ A) @ B) @ F3) @ ((((image @ D) @ A) @ V) @ (top_top @ (set @ D)))))))).
% 2.54/1.49 thf(fact_183_fun_Omap__cong, axiom, (! [B:$tType,A:$tType,D:$tType,X:(D > A),Ya:(D > A),F3:(A > B),G3:(A > B)]: (((X = Ya) => ((! [Z2:A]: (((((member @ A) @ Z2) @ ((((image @ D) @ A) @ Ya) @ (top_top @ (set @ D)))) => ((F3 @ Z2) = (G3 @ Z2))))) => ((((((comp @ A) @ B) @ D) @ F3) @ X) = (((((comp @ A) @ B) @ D) @ G3) @ Ya))))))).
% 2.54/1.49 thf(fact_184_fun_Omap__cong0, axiom, (! [B:$tType,A:$tType,D:$tType,X:(D > A),F3:(A > B),G3:(A > B)]: (((! [Z2:A]: (((((member @ A) @ Z2) @ ((((image @ D) @ A) @ X) @ (top_top @ (set @ D)))) => ((F3 @ Z2) = (G3 @ Z2))))) => ((((((comp @ A) @ B) @ D) @ F3) @ X) = (((((comp @ A) @ B) @ D) @ G3) @ X)))))).
% 2.54/1.49 thf(fact_185_top__set__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (top_top @ (A > $o))))))).
% 2.54/1.49 thf(fact_186_fun_Omap__comp, axiom, (! [B:$tType,C:$tType,A:$tType,D:$tType,G3:(B > C),F3:(A > B),V:(D > A)]: (((((((comp @ B) @ C) @ D) @ G3) @ (((((comp @ A) @ B) @ D) @ F3) @ V)) = (((((comp @ A) @ C) @ D) @ (((((comp @ B) @ C) @ A) @ G3) @ F3)) @ V))))).
% 2.54/1.49 thf(fact_187_fun_Oinj__map__strong, axiom, (! [B:$tType,A:$tType,D:$tType,X:(D > A),Xa:(D > A),F3:(A > B),Fa:(A > B)]: (((! [Z2:A,Za:A]: (((((member @ A) @ Z2) @ ((((image @ D) @ A) @ X) @ (top_top @ (set @ D)))) => ((((member @ A) @ Za) @ ((((image @ D) @ A) @ Xa) @ (top_top @ (set @ D)))) => (((F3 @ Z2) = (Fa @ Za)) => (Z2 = Za)))))) => (((((((comp @ A) @ B) @ D) @ F3) @ X) = (((((comp @ A) @ B) @ D) @ Fa) @ Xa)) => (X = Xa)))))).
% 2.54/1.49 thf(fact_188_map__prod__surj, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,F3:(A > B),G3:(C > D)]: (((((((image @ A) @ B) @ F3) @ (top_top @ (set @ A))) = (top_top @ (set @ B))) => ((((((image @ C) @ D) @ G3) @ (top_top @ (set @ C))) = (top_top @ (set @ D))) => (((((image @ ((product_prod @ A) @ C)) @ ((product_prod @ B) @ D)) @ ((((((product_map_prod @ A) @ B) @ C) @ D) @ F3) @ G3)) @ (top_top @ (set @ ((product_prod @ A) @ C)))) = (top_top @ (set @ ((product_prod @ B) @ D))))))))).
% 2.54/1.49 thf(fact_189_type__copy__set__map0, axiom, (! [A:$tType,B:$tType,D:$tType,E:$tType,C:$tType,F:$tType,Rep:(A > B),Abs:(B > A),S3:(B > (set @ D)),M:(C > B),F3:(E > D),S4:(C > (set @ E)),G3:(F > C)]: (((((((type_definition @ A) @ B) @ Rep) @ Abs) @ (top_top @ (set @ B))) => (((((((comp @ B) @ (set @ D)) @ C) @ S3) @ M) = (((((comp @ (set @ E)) @ (set @ D)) @ C) @ (((image @ E) @ D) @ F3)) @ S4)) => ((((((comp @ A) @ (set @ D)) @ F) @ (((((comp @ B) @ (set @ D)) @ A) @ S3) @ Rep)) @ (((((comp @ C) @ A) @ F) @ (((((comp @ B) @ A) @ C) @ Abs) @ M)) @ G3)) = (((((comp @ (set @ E)) @ (set @ D)) @ F) @ (((image @ E) @ D) @ F3)) @ (((((comp @ C) @ (set @ E)) @ F) @ S4) @ G3)))))))).
% 2.54/1.49 thf(fact_190_image__vimage__eq, axiom, (! [A:$tType,B:$tType,F3:(B > A),A7:(set @ A)]: ((((((image @ B) @ A) @ F3) @ ((((vimage @ B) @ A) @ F3) @ A7)) = (((inf_inf @ (set @ A)) @ A7) @ ((((image @ B) @ A) @ F3) @ (top_top @ (set @ B)))))))).
% 2.54/1.49 thf(fact_191_inf__right__idem, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ X) @ Y)) @ Y) = (((inf_inf @ A) @ X) @ Y)))))))).
% 2.54/1.49 thf(fact_192_inf_Oright__idem, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ B2) = (((inf_inf @ A) @ A2) @ B2)))))))).
% 2.54/1.49 thf(fact_193_inf__left__idem, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((inf_inf @ A) @ X) @ (((inf_inf @ A) @ X) @ Y)) = (((inf_inf @ A) @ X) @ Y)))))))).
% 2.54/1.49 thf(fact_194_inf_Oleft__idem, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((inf_inf @ A) @ A2) @ (((inf_inf @ A) @ A2) @ B2)) = (((inf_inf @ A) @ A2) @ B2)))))))).
% 2.54/1.49 thf(fact_195_inf__idem, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [X:A]: (((((inf_inf @ A) @ X) @ X) = X))))))).
% 2.54/1.49 thf(fact_196_inf_Oidem, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [A2:A]: (((((inf_inf @ A) @ A2) @ A2) = A2))))))).
% 2.54/1.49 thf(fact_197_inf__apply, axiom, (! [B:$tType,A:$tType]: ((((semilattice_inf @ B) @ (type2 @ B)) => ((inf_inf @ (A > B)) = (^ [F4:(A > B),G5:(A > B),X5:A]: ((((inf_inf @ B) @ (F4 @ X5)) @ (G5 @ X5))))))))).
% 2.54/1.49 thf(fact_198_Int__iff, axiom, (! [A:$tType,C3:A,A7:(set @ A),B8:(set @ A)]: (((((member @ A) @ C3) @ (((inf_inf @ (set @ A)) @ A7) @ B8)) = ((((member @ A) @ C3) @ A7) & (((member @ A) @ C3) @ B8)))))).
% 2.54/1.49 thf(fact_199_IntI, axiom, (! [A:$tType,C3:A,A7:(set @ A),B8:(set @ A)]: (((((member @ A) @ C3) @ A7) => ((((member @ A) @ C3) @ B8) => (((member @ A) @ C3) @ (((inf_inf @ (set @ A)) @ A7) @ B8))))))).
% 2.54/1.49 thf(fact_200_sup__inf__absorb, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((sup_sup @ A) @ X) @ (((inf_inf @ A) @ X) @ Y)) = X))))))).
% 2.54/1.49 thf(fact_201_inf__sup__absorb, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((inf_inf @ A) @ X) @ (((sup_sup @ A) @ X) @ Y)) = X))))))).
% 2.54/1.49 thf(fact_202_inf__top_Oright__neutral, axiom, (! [A:$tType]: ((((bounde1561333602nf_top @ A) @ (type2 @ A)) => (! [A2:A]: (((((inf_inf @ A) @ A2) @ (top_top @ A)) = A2))))))).
% 2.54/1.49 thf(fact_203_inf__top_Oleft__neutral, axiom, (! [A:$tType]: ((((bounde1561333602nf_top @ A) @ (type2 @ A)) => (! [A2:A]: (((((inf_inf @ A) @ (top_top @ A)) @ A2) = A2))))))).
% 2.54/1.49 thf(fact_204_inf__eq__top__iff, axiom, (! [A:$tType]: ((((bounded_lattice_top @ A) @ (type2 @ A)) => (! [X:A,Y:A]: ((((((inf_inf @ A) @ X) @ Y) = (top_top @ A)) = ((X = (top_top @ A)) & (Y = (top_top @ A)))))))))).
% 2.54/1.49 thf(fact_205_Int__UNIV, axiom, (! [A:$tType,A7:(set @ A),B8:(set @ A)]: ((((((inf_inf @ (set @ A)) @ A7) @ B8) = (top_top @ (set @ A))) = ((A7 = (top_top @ (set @ A))) & (B8 = (top_top @ (set @ A)))))))).
% 2.54/1.49 thf(fact_206_vimage__Int, axiom, (! [A:$tType,B:$tType,F3:(A > B),A7:(set @ B),B8:(set @ B)]: ((((((vimage @ A) @ B) @ F3) @ (((inf_inf @ (set @ B)) @ A7) @ B8)) = (((inf_inf @ (set @ A)) @ ((((vimage @ A) @ B) @ F3) @ A7)) @ ((((vimage @ A) @ B) @ F3) @ B8)))))).
% 2.54/1.49 thf(fact_207_map__prod__simp, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,F3:(C > A),G3:(D > B),A2:C,B2:D]: (((((((((product_map_prod @ C) @ A) @ D) @ B) @ F3) @ G3) @ ((((product_Pair @ C) @ D) @ A2) @ B2)) = ((((product_Pair @ A) @ B) @ (F3 @ A2)) @ (G3 @ B2)))))).
% 2.54/1.49 thf(fact_208_map__prod__imageI, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,A2:A,B2:B,R2:(set @ ((product_prod @ A) @ B)),F3:(A > C),G3:(B > D)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ R2) => (((member @ ((product_prod @ C) @ D)) @ ((((product_Pair @ C) @ D) @ (F3 @ A2)) @ (G3 @ B2))) @ ((((image @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)) @ ((((((product_map_prod @ A) @ C) @ B) @ D) @ F3) @ G3)) @ R2)))))).
% 2.54/1.49 thf(fact_209_type__copy__wit, axiom, (! [A:$tType,C:$tType,B:$tType,Rep:(A > B),Abs:(B > A),X:C,S3:(B > (set @ C)),Y:B]: (((((((type_definition @ A) @ B) @ Rep) @ Abs) @ (top_top @ (set @ B))) => ((((member @ C) @ X) @ ((((((comp @ B) @ (set @ C)) @ A) @ S3) @ Rep) @ (Abs @ Y))) => (((member @ C) @ X) @ (S3 @ Y))))))).
% 2.54/1.49 thf(fact_210_type__copy__map__comp0, axiom, (! [F:$tType,D:$tType,B:$tType,A:$tType,C:$tType,E:$tType,Rep:(A > B),Abs:(B > A),M:(C > D),M1:(B > D),M2:(C > B),F3:(D > F),G3:(E > C)]: (((((((type_definition @ A) @ B) @ Rep) @ Abs) @ (top_top @ (set @ B))) => ((M = (((((comp @ B) @ D) @ C) @ M1) @ M2)) => ((((((comp @ C) @ F) @ E) @ (((((comp @ D) @ F) @ C) @ F3) @ M)) @ G3) = (((((comp @ A) @ F) @ E) @ (((((comp @ B) @ F) @ A) @ (((((comp @ D) @ F) @ B) @ F3) @ M1)) @ Rep)) @ (((((comp @ C) @ A) @ E) @ (((((comp @ B) @ A) @ C) @ Abs) @ M2)) @ G3)))))))).
% 2.54/1.49 thf(fact_211_inf__top__right, axiom, (! [A:$tType]: ((((bounded_lattice_top @ A) @ (type2 @ A)) => (! [X:A]: (((((inf_inf @ A) @ X) @ (top_top @ A)) = X))))))).
% 2.54/1.49 thf(fact_212_inf__top__left, axiom, (! [A:$tType]: ((((bounded_lattice_top @ A) @ (type2 @ A)) => (! [X:A]: (((((inf_inf @ A) @ (top_top @ A)) @ X) = X))))))).
% 2.54/1.49 thf(fact_213_Int__UNIV__right, axiom, (! [A:$tType,A7:(set @ A)]: (((((inf_inf @ (set @ A)) @ A7) @ (top_top @ (set @ A))) = A7)))).
% 2.54/1.49 thf(fact_214_Int__UNIV__left, axiom, (! [A:$tType,B8:(set @ A)]: (((((inf_inf @ (set @ A)) @ (top_top @ (set @ A))) @ B8) = B8)))).
% 2.54/1.49 thf(fact_215_type__copy__obj__one__point__absE, axiom, (! [A:$tType,B:$tType,Rep:(A > B),Abs:(B > A),S:A]: (((((((type_definition @ A) @ B) @ Rep) @ Abs) @ (top_top @ (set @ B))) => (~ ((! [X3:B]: ((S != (Abs @ X3)))))))))).
% 2.54/1.49 thf(fact_216_top__conj_I1_J, axiom, (! [A:$tType,X:A,P:$o]: (((((top_top @ (A > $o)) @ X) & P) = P)))).
% 2.54/1.49 thf(fact_217_top__conj_I2_J, axiom, (! [A:$tType,P:$o,X:A]: (((P & ((top_top @ (A > $o)) @ X)) = P)))).
% 2.54/1.49 thf(fact_218_prod_Omap__comp, axiom, (! [D:$tType,F:$tType,E:$tType,C:$tType,B:$tType,A:$tType,G1:(C > E),G22:(D > F),F1:(A > C),F22:(B > D),V:((product_prod @ A) @ B)]: (((((((((product_map_prod @ C) @ E) @ D) @ F) @ G1) @ G22) @ (((((((product_map_prod @ A) @ C) @ B) @ D) @ F1) @ F22) @ V)) = (((((((product_map_prod @ A) @ E) @ B) @ F) @ (((((comp @ C) @ E) @ A) @ G1) @ F1)) @ (((((comp @ D) @ F) @ B) @ G22) @ F22)) @ V))))).
% 2.54/1.49 thf(fact_219_map__prod_Ocompositionality, axiom, (! [D:$tType,F:$tType,E:$tType,C:$tType,B:$tType,A:$tType,F3:(C > E),G3:(D > F),H:(A > C),I:(B > D),Prod:((product_prod @ A) @ B)]: (((((((((product_map_prod @ C) @ E) @ D) @ F) @ F3) @ G3) @ (((((((product_map_prod @ A) @ C) @ B) @ D) @ H) @ I) @ Prod)) = (((((((product_map_prod @ A) @ E) @ B) @ F) @ (((((comp @ C) @ E) @ A) @ F3) @ H)) @ (((((comp @ D) @ F) @ B) @ G3) @ I)) @ Prod))))).
% 2.54/1.49 thf(fact_220_map__prod__compose, axiom, (! [D:$tType,C:$tType,A:$tType,E:$tType,F:$tType,B:$tType,F1:(E > C),F22:(A > E),G1:(F > D),G22:(B > F)]: ((((((((product_map_prod @ A) @ C) @ B) @ D) @ (((((comp @ E) @ C) @ A) @ F1) @ F22)) @ (((((comp @ F) @ D) @ B) @ G1) @ G22)) = (((((comp @ ((product_prod @ E) @ F)) @ ((product_prod @ C) @ D)) @ ((product_prod @ A) @ B)) @ ((((((product_map_prod @ E) @ C) @ F) @ D) @ F1) @ G1)) @ ((((((product_map_prod @ A) @ E) @ B) @ F) @ F22) @ G22)))))).
% 2.54/1.49 thf(fact_221_map__prod_Ocomp, axiom, (! [A:$tType,C:$tType,E:$tType,F:$tType,D:$tType,B:$tType,F3:(C > E),G3:(D > F),H:(A > C),I:(B > D)]: (((((((comp @ ((product_prod @ C) @ D)) @ ((product_prod @ E) @ F)) @ ((product_prod @ A) @ B)) @ ((((((product_map_prod @ C) @ E) @ D) @ F) @ F3) @ G3)) @ ((((((product_map_prod @ A) @ C) @ B) @ D) @ H) @ I)) = ((((((product_map_prod @ A) @ E) @ B) @ F) @ (((((comp @ C) @ E) @ A) @ F3) @ H)) @ (((((comp @ D) @ F) @ B) @ G3) @ I)))))).
% 2.54/1.49 thf(fact_222_inf__left__commute, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((inf_inf @ A) @ X) @ (((inf_inf @ A) @ Y) @ Z)) = (((inf_inf @ A) @ Y) @ (((inf_inf @ A) @ X) @ Z))))))))).
% 2.54/1.49 thf(fact_223_inf_Oleft__commute, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C3:A]: (((((inf_inf @ A) @ B2) @ (((inf_inf @ A) @ A2) @ C3)) = (((inf_inf @ A) @ A2) @ (((inf_inf @ A) @ B2) @ C3))))))))).
% 2.54/1.49 thf(fact_224_inf__commute, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => ((inf_inf @ A) = (^ [X5:A,Y4:A]: ((((inf_inf @ A) @ Y4) @ X5)))))))).
% 2.54/1.49 thf(fact_225_inf_Ocommute, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => ((inf_inf @ A) = (^ [A8:A,B7:A]: ((((inf_inf @ A) @ B7) @ A8)))))))).
% 2.54/1.49 thf(fact_226_inf__assoc, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ X) @ Y)) @ Z) = (((inf_inf @ A) @ X) @ (((inf_inf @ A) @ Y) @ Z))))))))).
% 2.54/1.49 thf(fact_227_inf_Oassoc, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C3:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ C3) = (((inf_inf @ A) @ A2) @ (((inf_inf @ A) @ B2) @ C3))))))))).
% 2.54/1.49 thf(fact_228_Int__left__commute, axiom, (! [A:$tType,A7:(set @ A),B8:(set @ A),C4:(set @ A)]: (((((inf_inf @ (set @ A)) @ A7) @ (((inf_inf @ (set @ A)) @ B8) @ C4)) = (((inf_inf @ (set @ A)) @ B8) @ (((inf_inf @ (set @ A)) @ A7) @ C4)))))).
% 2.54/1.49 thf(fact_229_inf__fun__def, axiom, (! [B:$tType,A:$tType]: ((((semilattice_inf @ B) @ (type2 @ B)) => ((inf_inf @ (A > B)) = (^ [F4:(A > B),G5:(A > B),X5:A]: ((((inf_inf @ B) @ (F4 @ X5)) @ (G5 @ X5))))))))).
% 2.54/1.49 thf(fact_230_Int__left__absorb, axiom, (! [A:$tType,A7:(set @ A),B8:(set @ A)]: (((((inf_inf @ (set @ A)) @ A7) @ (((inf_inf @ (set @ A)) @ A7) @ B8)) = (((inf_inf @ (set @ A)) @ A7) @ B8))))).
% 2.54/1.49 thf(fact_231_Int__commute, axiom, (! [A:$tType]: (((inf_inf @ (set @ A)) = (^ [A9:(set @ A),B9:(set @ A)]: ((((inf_inf @ (set @ A)) @ B9) @ A9))))))).
% 2.54/1.49 thf(fact_232_Int__absorb, axiom, (! [A:$tType,A7:(set @ A)]: (((((inf_inf @ (set @ A)) @ A7) @ A7) = A7)))).
% 2.54/1.49 thf(fact_233_Int__assoc, axiom, (! [A:$tType,A7:(set @ A),B8:(set @ A),C4:(set @ A)]: (((((inf_inf @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A7) @ B8)) @ C4) = (((inf_inf @ (set @ A)) @ A7) @ (((inf_inf @ (set @ A)) @ B8) @ C4)))))).
% 2.54/1.49 thf(fact_234_IntD2, axiom, (! [A:$tType,C3:A,A7:(set @ A),B8:(set @ A)]: (((((member @ A) @ C3) @ (((inf_inf @ (set @ A)) @ A7) @ B8)) => (((member @ A) @ C3) @ B8))))).
% 2.54/1.49 thf(fact_235_IntD1, axiom, (! [A:$tType,C3:A,A7:(set @ A),B8:(set @ A)]: (((((member @ A) @ C3) @ (((inf_inf @ (set @ A)) @ A7) @ B8)) => (((member @ A) @ C3) @ A7))))).
% 2.54/1.49 thf(fact_236_IntE, axiom, (! [A:$tType,C3:A,A7:(set @ A),B8:(set @ A)]: (((((member @ A) @ C3) @ (((inf_inf @ (set @ A)) @ A7) @ B8)) => (~ (((((member @ A) @ C3) @ A7) => (~ ((((member @ A) @ C3) @ B8)))))))))).
% 2.54/1.49 thf(fact_237_inf__sup__aci_I1_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => ((inf_inf @ A) = (^ [X5:A,Y4:A]: ((((inf_inf @ A) @ Y4) @ X5)))))))).
% 2.54/1.49 thf(fact_238_inf__sup__aci_I2_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ X) @ Y)) @ Z) = (((inf_inf @ A) @ X) @ (((inf_inf @ A) @ Y) @ Z))))))))).
% 2.54/1.49 thf(fact_239_inf__sup__aci_I3_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((inf_inf @ A) @ X) @ (((inf_inf @ A) @ Y) @ Z)) = (((inf_inf @ A) @ Y) @ (((inf_inf @ A) @ X) @ Z))))))))).
% 2.54/1.49 thf(fact_240_inf__sup__aci_I4_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((inf_inf @ A) @ X) @ (((inf_inf @ A) @ X) @ Y)) = (((inf_inf @ A) @ X) @ Y)))))))).
% 2.54/1.49 thf(fact_241_vimage__inter__cong, axiom, (! [B:$tType,A:$tType,S3:(set @ A),F3:(A > B),G3:(A > B),Y:(set @ B)]: (((! [W:A]: (((((member @ A) @ W) @ S3) => ((F3 @ W) = (G3 @ W))))) => ((((inf_inf @ (set @ A)) @ ((((vimage @ A) @ B) @ F3) @ Y)) @ S3) = (((inf_inf @ (set @ A)) @ ((((vimage @ A) @ B) @ G3) @ Y)) @ S3)))))).
% 2.54/1.49 thf(fact_242_sup__inf__distrib2, axiom, (! [A:$tType]: ((((distrib_lattice @ A) @ (type2 @ A)) => (! [Y:A,Z:A,X:A]: (((((sup_sup @ A) @ (((inf_inf @ A) @ Y) @ Z)) @ X) = (((inf_inf @ A) @ (((sup_sup @ A) @ Y) @ X)) @ (((sup_sup @ A) @ Z) @ X))))))))).
% 2.54/1.49 thf(fact_243_sup__inf__distrib1, axiom, (! [A:$tType]: ((((distrib_lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((sup_sup @ A) @ X) @ (((inf_inf @ A) @ Y) @ Z)) = (((inf_inf @ A) @ (((sup_sup @ A) @ X) @ Y)) @ (((sup_sup @ A) @ X) @ Z))))))))).
% 2.54/1.49 thf(fact_244_inf__sup__distrib2, axiom, (! [A:$tType]: ((((distrib_lattice @ A) @ (type2 @ A)) => (! [Y:A,Z:A,X:A]: (((((inf_inf @ A) @ (((sup_sup @ A) @ Y) @ Z)) @ X) = (((sup_sup @ A) @ (((inf_inf @ A) @ Y) @ X)) @ (((inf_inf @ A) @ Z) @ X))))))))).
% 2.54/1.49 thf(fact_245_inf__sup__distrib1, axiom, (! [A:$tType]: ((((distrib_lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((inf_inf @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z)) = (((sup_sup @ A) @ (((inf_inf @ A) @ X) @ Y)) @ (((inf_inf @ A) @ X) @ Z))))))))).
% 2.54/1.49 thf(fact_246_distrib__imp2, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((! [X3:A,Y3:A,Z2:A]: (((((sup_sup @ A) @ X3) @ (((inf_inf @ A) @ Y3) @ Z2)) = (((inf_inf @ A) @ (((sup_sup @ A) @ X3) @ Y3)) @ (((sup_sup @ A) @ X3) @ Z2))))) => ((((inf_inf @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z)) = (((sup_sup @ A) @ (((inf_inf @ A) @ X) @ Y)) @ (((inf_inf @ A) @ X) @ Z)))))))))).
% 2.54/1.49 thf(fact_247_distrib__imp1, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((! [X3:A,Y3:A,Z2:A]: (((((inf_inf @ A) @ X3) @ (((sup_sup @ A) @ Y3) @ Z2)) = (((sup_sup @ A) @ (((inf_inf @ A) @ X3) @ Y3)) @ (((inf_inf @ A) @ X3) @ Z2))))) => ((((sup_sup @ A) @ X) @ (((inf_inf @ A) @ Y) @ Z)) = (((inf_inf @ A) @ (((sup_sup @ A) @ X) @ Y)) @ (((sup_sup @ A) @ X) @ Z)))))))))).
% 2.54/1.49 thf(fact_248_Un__Int__distrib2, axiom, (! [A:$tType,B8:(set @ A),C4:(set @ A),A7:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((inf_inf @ (set @ A)) @ B8) @ C4)) @ A7) = (((inf_inf @ (set @ A)) @ (((sup_sup @ (set @ A)) @ B8) @ A7)) @ (((sup_sup @ (set @ A)) @ C4) @ A7)))))).
% 2.54/1.49 thf(fact_249_Int__Un__distrib2, axiom, (! [A:$tType,B8:(set @ A),C4:(set @ A),A7:(set @ A)]: (((((inf_inf @ (set @ A)) @ (((sup_sup @ (set @ A)) @ B8) @ C4)) @ A7) = (((sup_sup @ (set @ A)) @ (((inf_inf @ (set @ A)) @ B8) @ A7)) @ (((inf_inf @ (set @ A)) @ C4) @ A7)))))).
% 2.54/1.49 thf(fact_250_Un__Int__distrib, axiom, (! [A:$tType,A7:(set @ A),B8:(set @ A),C4:(set @ A)]: (((((sup_sup @ (set @ A)) @ A7) @ (((inf_inf @ (set @ A)) @ B8) @ C4)) = (((inf_inf @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A7) @ B8)) @ (((sup_sup @ (set @ A)) @ A7) @ C4)))))).
% 2.54/1.49 thf(fact_251_Int__Un__distrib, axiom, (! [A:$tType,A7:(set @ A),B8:(set @ A),C4:(set @ A)]: (((((inf_inf @ (set @ A)) @ A7) @ (((sup_sup @ (set @ A)) @ B8) @ C4)) = (((sup_sup @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A7) @ B8)) @ (((inf_inf @ (set @ A)) @ A7) @ C4)))))).
% 2.54/1.49 thf(fact_252_Un__Int__crazy, axiom, (! [A:$tType,A7:(set @ A),B8:(set @ A),C4:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((sup_sup @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A7) @ B8)) @ (((inf_inf @ (set @ A)) @ B8) @ C4))) @ (((inf_inf @ (set @ A)) @ C4) @ A7)) = (((inf_inf @ (set @ A)) @ (((inf_inf @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A7) @ B8)) @ (((sup_sup @ (set @ A)) @ B8) @ C4))) @ (((sup_sup @ (set @ A)) @ C4) @ A7)))))).
% 2.54/1.49 thf(fact_253_type__copy__map__comp0__undo, axiom, (! [E:$tType,A:$tType,C:$tType,B:$tType,D:$tType,F:$tType,Rep:(A > B),Abs:(B > A),Rep2:(C > D),Abs2:(D > C),Rep3:(E > F),Abs3:(F > E),M:(F > D),M1:(B > D),M2:(F > B)]: (((((((type_definition @ A) @ B) @ Rep) @ Abs) @ (top_top @ (set @ B))) => ((((((type_definition @ C) @ D) @ Rep2) @ Abs2) @ (top_top @ (set @ D))) => ((((((type_definition @ E) @ F) @ Rep3) @ Abs3) @ (top_top @ (set @ F))) => (((((((comp @ F) @ C) @ E) @ (((((comp @ D) @ C) @ F) @ Abs2) @ M)) @ Rep3) = (((((comp @ A) @ C) @ E) @ (((((comp @ B) @ C) @ A) @ (((((comp @ D) @ C) @ B) @ Abs2) @ M1)) @ Rep)) @ (((((comp @ F) @ A) @ E) @ (((((comp @ B) @ A) @ F) @ Abs) @ M2)) @ Rep3))) => ((((((comp @ B) @ D) @ F) @ M1) @ M2) = M)))))))).
% 2.54/1.49 thf(fact_254_prod__fun__imageE, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,C3:((product_prod @ A) @ B),F3:(C > A),G3:(D > B),R2:(set @ ((product_prod @ C) @ D))]: (((((member @ ((product_prod @ A) @ B)) @ C3) @ ((((image @ ((product_prod @ C) @ D)) @ ((product_prod @ A) @ B)) @ ((((((product_map_prod @ C) @ A) @ D) @ B) @ F3) @ G3)) @ R2)) => (~ ((! [X3:C,Y3:D]: (((C3 = ((((product_Pair @ A) @ B) @ (F3 @ X3)) @ (G3 @ Y3))) => (~ ((((member @ ((product_prod @ C) @ D)) @ ((((product_Pair @ C) @ D) @ X3) @ Y3)) @ R2)))))))))))).
% 2.54/1.49 thf(fact_255_type__copy__map__cong0, axiom, (! [B:$tType,D:$tType,E:$tType,A:$tType,C:$tType,M:(B > A),G3:(C > B),X:C,N:(D > A),H:(C > D),F3:(A > E)]: ((((M @ (G3 @ X)) = (N @ (H @ X))) => (((((((comp @ B) @ E) @ C) @ (((((comp @ A) @ E) @ B) @ F3) @ M)) @ G3) @ X) = ((((((comp @ D) @ E) @ C) @ (((((comp @ A) @ E) @ D) @ F3) @ N)) @ H) @ X)))))).
% 2.54/1.49 thf(tcon_HOL_Obool___Lattices_Obounded__lattice, axiom, ((bounded_lattice @ $o) @ (type2 @ $o))).
% 2.54/1.49 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1, axiom, (! [A5:$tType]: (((bounded_lattice @ (set @ A5)) @ (type2 @ (set @ A5)))))).
% 2.54/1.49 thf(tcon_fun___Lattices_Obounded__lattice_2, axiom, (! [A5:$tType,A10:$tType]: ((((bounded_lattice @ A10) @ (type2 @ A10)) => ((bounded_lattice @ (A5 > A10)) @ (type2 @ (A5 > A10))))))).
% 2.54/1.49 thf(tcon_fun___Lattices_Obounded__semilattice__inf__top, axiom, (! [A5:$tType,A10:$tType]: ((((bounded_lattice @ A10) @ (type2 @ A10)) => ((bounde1561333602nf_top @ (A5 > A10)) @ (type2 @ (A5 > A10))))))).
% 2.54/1.49 thf(tcon_fun___Lattices_Obounded__lattice__top, axiom, (! [A5:$tType,A10:$tType]: ((((bounded_lattice @ A10) @ (type2 @ A10)) => ((bounded_lattice_top @ (A5 > A10)) @ (type2 @ (A5 > A10))))))).
% 2.54/1.49 thf(tcon_fun___Lattices_Osemilattice__sup, axiom, (! [A5:$tType,A10:$tType]: ((((semilattice_sup @ A10) @ (type2 @ A10)) => ((semilattice_sup @ (A5 > A10)) @ (type2 @ (A5 > A10))))))).
% 2.54/1.49 thf(tcon_fun___Lattices_Osemilattice__inf, axiom, (! [A5:$tType,A10:$tType]: ((((semilattice_inf @ A10) @ (type2 @ A10)) => ((semilattice_inf @ (A5 > A10)) @ (type2 @ (A5 > A10))))))).
% 2.54/1.49 thf(tcon_fun___Lattices_Odistrib__lattice, axiom, (! [A5:$tType,A10:$tType]: ((((distrib_lattice @ A10) @ (type2 @ A10)) => ((distrib_lattice @ (A5 > A10)) @ (type2 @ (A5 > A10))))))).
% 2.54/1.49 thf(tcon_fun___Lattices_Olattice, axiom, (! [A5:$tType,A10:$tType]: ((((lattice @ A10) @ (type2 @ A10)) => ((lattice @ (A5 > A10)) @ (type2 @ (A5 > A10))))))).
% 2.54/1.49 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_3, axiom, (! [A5:$tType]: (((bounde1561333602nf_top @ (set @ A5)) @ (type2 @ (set @ A5)))))).
% 2.54/1.49 thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_4, axiom, (! [A5:$tType]: (((bounded_lattice_top @ (set @ A5)) @ (type2 @ (set @ A5)))))).
% 2.54/1.49 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_5, axiom, (! [A5:$tType]: (((semilattice_sup @ (set @ A5)) @ (type2 @ (set @ A5)))))).
% 2.54/1.49 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_6, axiom, (! [A5:$tType]: (((semilattice_inf @ (set @ A5)) @ (type2 @ (set @ A5)))))).
% 2.54/1.49 thf(tcon_Set_Oset___Lattices_Odistrib__lattice_7, axiom, (! [A5:$tType]: (((distrib_lattice @ (set @ A5)) @ (type2 @ (set @ A5)))))).
% 2.54/1.49 thf(tcon_Set_Oset___Lattices_Olattice_8, axiom, (! [A5:$tType]: (((lattice @ (set @ A5)) @ (type2 @ (set @ A5)))))).
% 2.54/1.49 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_9, axiom, ((bounde1561333602nf_top @ $o) @ (type2 @ $o))).
% 2.54/1.49 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_10, axiom, ((bounded_lattice_top @ $o) @ (type2 @ $o))).
% 2.54/1.49 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_11, axiom, ((semilattice_sup @ $o) @ (type2 @ $o))).
% 2.54/1.49 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_12, axiom, ((semilattice_inf @ $o) @ (type2 @ $o))).
% 2.54/1.49 thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_13, axiom, ((distrib_lattice @ $o) @ (type2 @ $o))).
% 2.54/1.49 thf(tcon_HOL_Obool___Lattices_Olattice_14, axiom, ((lattice @ $o) @ (type2 @ $o))).
% 2.54/1.49 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 2.54/1.49 thf(help_If_2_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $false) @ X) @ Y) = Y)))).
% 2.54/1.49 thf(help_If_1_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $true) @ X) @ Y) = X)))).
% 2.54/1.49 thf(conj_0, hypothesis, $true).
% 2.54/1.49 thf(conj_1, hypothesis, $true).
% 2.54/1.49 thf(conj_2, hypothesis, $true).
% 2.54/1.49 thf(conj_3, hypothesis, $true).
% 2.54/1.49 thf(conj_4, conjecture, (? [Tr14:dtree,Tr24:dtree,Tr33:dtree]: (((trA = (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ Tr14) @ Tr24))) @ Tr33))) & ((parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ tr1) @ (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ tr2) @ tr3)))) = (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ Tr14) @ (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ Tr24) @ Tr33))))))))).
% 2.54/1.49 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.sn5SmONUSK/DTF2DTF10309.p
% 2.54/1.49 ------------------------
% 2.83/1.52 ---- Cleaned THF ---
% 2.83/1.52 thf(ty_t_Product__Type_Oprod,type,
% 2.83/1.52 product_prod: $tType > $tType > $tType ).
% 2.83/1.52
% 2.83/1.52 thf(ty_t_Sum__Type_Osum,type,
% 2.83/1.52 sum_sum: $tType > $tType > $tType ).
% 2.83/1.52
% 2.83/1.52 thf(ty_t_DTree_Odtree,type,
% 2.83/1.52 dtree: $tType ).
% 2.83/1.52
% 2.83/1.52 thf(ty_t_Set_Oset,type,
% 2.83/1.52 set: $tType > $tType ).
% 2.83/1.52
% 2.83/1.52 thf(ty_t_DTree_OT,type,
% 2.83/1.52 t: $tType ).
% 2.83/1.52
% 2.83/1.52 thf(ty_t_itself,type,
% 2.83/1.52 itself: $tType > $tType ).
% 2.83/1.52
% 2.83/1.52 thf(sy_cl_Lattices_Obounded__lattice,type,
% 2.83/1.52 bounded_lattice:
% 2.83/1.52 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_cl_HOL_Otype,type,
% 2.83/1.52 type:
% 2.83/1.52 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_cl_Lattices_Olattice,type,
% 2.83/1.52 lattice:
% 2.83/1.52 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_cl_Lattices_Odistrib__lattice,type,
% 2.83/1.52 distrib_lattice:
% 2.83/1.52 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_cl_Lattices_Osemilattice__inf,type,
% 2.83/1.52 semilattice_inf:
% 2.83/1.52 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 2.83/1.52 semilattice_sup:
% 2.83/1.52 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_cl_Lattices_Obounded__lattice__top,type,
% 2.83/1.52 bounded_lattice_top:
% 2.83/1.52 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_cl_Lattices_Obounded__semilattice__inf__top,type,
% 2.83/1.52 bounde1561333602nf_top:
% 2.83/1.52 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
% 2.83/1.52 bNF_Greatest_image2:
% 2.83/1.52 !>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_c_DTree_Ocont,type,
% 2.83/1.52 cont: dtree > ( set @ ( sum_sum @ t @ dtree ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_c_Fun_Ocomp,type,
% 2.83/1.52 comp:
% 2.83/1.52 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_c_If,type,
% 2.83/1.52 if:
% 2.83/1.52 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 2.83/1.52 inf_inf:
% 2.83/1.52 !>[A: $tType] : ( A > A > A ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_c_Lattices_Osup__class_Osup,type,
% 2.83/1.52 sup_sup:
% 2.83/1.52 !>[A: $tType] : ( A > A > A ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_c_Orderings_Otop__class_Otop,type,
% 2.83/1.52 top_top:
% 2.83/1.52 !>[A: $tType] : A ).
% 2.83/1.52
% 2.83/1.52 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar,type,
% 2.83/1.52 parall1899940088le_par: ( product_prod @ dtree @ dtree ) > dtree ).
% 2.83/1.52
% 2.83/1.52 thf(sy_c_Product__Type_OPair,type,
% 2.83/1.52 product_Pair:
% 2.83/1.52 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_c_Product__Type_Ocurry,type,
% 2.83/1.52 product_curry:
% 2.83/1.52 !>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 2.83/1.52 produc2004651681e_prod:
% 2.83/1.52 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_c_Product__Type_Omap__prod,type,
% 2.83/1.52 product_map_prod:
% 2.83/1.52 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 2.83/1.52 product_rec_prod:
% 2.83/1.52 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
% 2.83/1.52 product_case_prod:
% 2.83/1.52 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 2.83/1.52 product_swap:
% 2.83/1.52 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_c_Product__Type_Oscomp,type,
% 2.83/1.52 product_scomp:
% 2.83/1.52 !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( B > C > D ) > A > D ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_c_Pure_Otype,type,
% 2.83/1.52 type2:
% 2.83/1.52 !>[A: $tType] : ( itself @ A ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_c_Set_OCollect,type,
% 2.83/1.52 collect:
% 2.83/1.52 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_c_Set_Oimage,type,
% 2.83/1.52 image:
% 2.83/1.52 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_c_Set_Ovimage,type,
% 2.83/1.52 vimage:
% 2.83/1.52 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ B ) > ( set @ A ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_c_Sum__Type_OInl,type,
% 2.83/1.52 sum_Inl:
% 2.83/1.52 !>[A: $tType,B: $tType] : ( A > ( sum_sum @ A @ B ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_c_Sum__Type_OInr,type,
% 2.83/1.52 sum_Inr:
% 2.83/1.52 !>[B: $tType,A: $tType] : ( B > ( sum_sum @ A @ B ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_c_Sum__Type_OPlus,type,
% 2.83/1.52 sum_Plus:
% 2.83/1.52 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( sum_sum @ A @ B ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_c_Sum__Type_OSuml,type,
% 2.83/1.52 sum_Suml:
% 2.83/1.52 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( sum_sum @ A @ B ) > C ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_c_Sum__Type_OSumr,type,
% 2.83/1.52 sum_Sumr:
% 2.83/1.52 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( sum_sum @ A @ B ) > C ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_c_Sum__Type_Oold_Osum_Orec__sum,type,
% 2.83/1.52 sum_rec_sum:
% 2.83/1.52 !>[A: $tType,T: $tType,B: $tType] : ( ( A > T ) > ( B > T ) > ( sum_sum @ A @ B ) > T ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_c_Sum__Type_Osum_Ocase__sum,type,
% 2.83/1.52 sum_case_sum:
% 2.83/1.52 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( B > C ) > ( sum_sum @ A @ B ) > C ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_c_Typedef_Otype__definition,type,
% 2.83/1.52 type_definition:
% 2.83/1.52 !>[B: $tType,A: $tType] : ( ( B > A ) > ( A > B ) > ( set @ A ) > $o ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_c_member,type,
% 2.83/1.52 member:
% 2.83/1.52 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.83/1.52
% 2.83/1.52 thf(sy_v_tr1_H____,type,
% 2.83/1.52 tr1: dtree ).
% 2.83/1.52
% 2.83/1.52 thf(sy_v_tr1a____,type,
% 2.83/1.52 tr1a: dtree ).
% 2.83/1.52
% 2.83/1.52 thf(sy_v_tr2_H____,type,
% 2.83/1.52 tr2: dtree ).
% 2.83/1.52
% 2.83/1.52 thf(sy_v_tr2a____,type,
% 2.83/1.52 tr2a: dtree ).
% 2.83/1.52
% 2.83/1.52 thf(sy_v_tr3_H____,type,
% 2.83/1.52 tr3: dtree ).
% 2.83/1.52
% 2.83/1.52 thf(sy_v_tr3a____,type,
% 2.83/1.52 tr3a: dtree ).
% 2.83/1.52
% 2.83/1.52 thf(sy_v_trA_H____,type,
% 2.83/1.52 trA: dtree ).
% 2.83/1.52
% 2.83/1.52 thf(sy_v_trA____,type,
% 2.83/1.52 trA2: dtree ).
% 2.83/1.52
% 2.83/1.52 thf(sy_v_trB____,type,
% 2.83/1.52 trB: dtree ).
% 2.83/1.52
% 2.83/1.52 thf(fact_0__092_060open_062_092_060exists_062tr1_Atr2_Atr3_O_AtrA_A_061_A_Itr1_A_092_060parallel_062_Atr2_J_A_092_060parallel_062_Atr3_A_092_060and_062_AtrB_A_061_Atr1_A_092_060parallel_062_Atr2_A_092_060parallel_062_Atr3_092_060close_062,axiom,
% 2.83/1.52 ? [Tr1: dtree,Tr2: dtree,Tr3: dtree] :
% 2.83/1.52 ( ( trA2
% 2.83/1.52 = ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) ) @ Tr3 ) ) )
% 2.83/1.52 & ( trB
% 2.83/1.52 = ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr1 @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr2 @ Tr3 ) ) ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_1__092_060open_062Inr_Atr1_H_A_092_060in_062_Acont_Atr1_092_060close_062,axiom,
% 2.83/1.52 member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ tr1 ) @ ( cont @ tr1a ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_2__092_060open_062Inr_Atr2_H_A_092_060in_062_Acont_Atr2_092_060close_062,axiom,
% 2.83/1.52 member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ tr2 ) @ ( cont @ tr2a ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_3__092_060open_062Inr_Atr3_H_A_092_060in_062_Acont_Atr3_092_060close_062,axiom,
% 2.83/1.52 member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ tr3 ) @ ( cont @ tr3a ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_4__092_060open_062trA_H_A_061_A_Itr1_H_A_092_060parallel_062_Atr2_H_J_A_092_060parallel_062_Atr3_H_092_060close_062,axiom,
% 2.83/1.52 ( trA
% 2.83/1.52 = ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ tr1 @ tr2 ) ) @ tr3 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_5__092_060open_062Inr_AtrA_H_A_092_060in_062_Acont_A_I_Itr1_A_092_060parallel_062_Atr2_J_A_092_060parallel_062_Atr3_J_092_060close_062,axiom,
% 2.83/1.52 member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ trA ) @ ( cont @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ tr1a @ tr2a ) ) @ tr3a ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_6__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062tr1_H_Atr2_H_Atr3_H_O_A_092_060lbrakk_062trA_H_A_061_A_Itr1_H_A_092_060parallel_062_Atr2_H_J_A_092_060parallel_062_Atr3_H_059_AInr_Atr1_H_A_092_060in_062_Acont_Atr1_059_AInr_Atr2_H_A_092_060in_062_Acont_Atr2_059_AInr_Atr3_H_A_092_060in_062_Acont_Atr3_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 2.83/1.52 ~ ! [Tr12: dtree,Tr22: dtree,Tr32: dtree] :
% 2.83/1.52 ( ( trA
% 2.83/1.52 = ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) ) @ Tr32 ) ) )
% 2.83/1.52 => ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ tr1a ) )
% 2.83/1.52 => ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr22 ) @ ( cont @ tr2a ) )
% 2.83/1.52 => ~ ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr32 ) @ ( cont @ tr3a ) ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_7_par__com,axiom,
% 2.83/1.52 ! [Tr13: dtree,Tr23: dtree] :
% 2.83/1.52 ( ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr13 @ Tr23 ) )
% 2.83/1.52 = ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr23 @ Tr13 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_8_par__r_Ocases,axiom,
% 2.83/1.52 ! [X: product_prod @ dtree @ dtree] :
% 2.83/1.52 ~ ! [Tr1: dtree,Tr2: dtree] :
% 2.83/1.52 ( X
% 2.83/1.52 != ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_9_par__r_Oinduct,axiom,
% 2.83/1.52 ! [P: ( product_prod @ dtree @ dtree ) > $o,A0: product_prod @ dtree @ dtree] :
% 2.83/1.52 ( ! [Tr1: dtree,Tr2: dtree] : ( P @ ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) )
% 2.83/1.52 => ( P @ A0 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_10_sum_Oinject_I2_J,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,X2: B,Y2: B] :
% 2.83/1.52 ( ( ( sum_Inr @ B @ A @ X2 )
% 2.83/1.52 = ( sum_Inr @ B @ A @ Y2 ) )
% 2.83/1.52 = ( X2 = Y2 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_11_old_Osum_Oinject_I2_J,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,B2: B,B3: B] :
% 2.83/1.52 ( ( ( sum_Inr @ B @ A @ B2 )
% 2.83/1.52 = ( sum_Inr @ B @ A @ B3 ) )
% 2.83/1.52 = ( B2 = B3 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_12_prod_Oinject,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
% 2.83/1.52 ( ( ( product_Pair @ A @ B @ X1 @ X2 )
% 2.83/1.52 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 2.83/1.52 = ( ( X1 = Y1 )
% 2.83/1.52 & ( X2 = Y2 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_13_old_Oprod_Oinject,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 2.83/1.52 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 2.83/1.52 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 2.83/1.52 = ( ( A2 = A3 )
% 2.83/1.52 & ( B2 = B3 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_14_Inr__inject,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,X: B,Y: B] :
% 2.83/1.52 ( ( ( sum_Inr @ B @ A @ X )
% 2.83/1.52 = ( sum_Inr @ B @ A @ Y ) )
% 2.83/1.52 => ( X = Y ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_15_not__arg__cong__Inr,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType,X: A,Y: A] :
% 2.83/1.52 ( ( X != Y )
% 2.83/1.52 => ( ( sum_Inr @ A @ B @ X )
% 2.83/1.52 != ( sum_Inr @ A @ B @ Y ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_16_Inl__in__cont__par,axiom,
% 2.83/1.52 ! [T2: t,Tr13: dtree,Tr23: dtree] :
% 2.83/1.52 ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr13 @ Tr23 ) ) ) )
% 2.83/1.52 = ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ Tr13 ) )
% 2.83/1.52 | ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ Tr23 ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_17_surj__pair,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 2.83/1.52 ? [X3: A,Y3: B] :
% 2.83/1.52 ( P2
% 2.83/1.52 = ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_18_prod__cases,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 2.83/1.52 ( ! [A4: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 2.83/1.52 => ( P @ P2 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_19_Pair__inject,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 2.83/1.52 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 2.83/1.52 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 2.83/1.52 => ~ ( ( A2 = A3 )
% 2.83/1.52 => ( B2 != B3 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_20_prod__cases3,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.83/1.52 ~ ! [A4: A,B4: B,C2: C] :
% 2.83/1.52 ( Y
% 2.83/1.52 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_21_prod__cases4,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.83/1.52 ~ ! [A4: A,B4: B,C2: C,D2: D] :
% 2.83/1.52 ( Y
% 2.83/1.52 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_22_sum_Oinject_I1_J,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType,X1: A,Y1: A] :
% 2.83/1.52 ( ( ( sum_Inl @ A @ B @ X1 )
% 2.83/1.52 = ( sum_Inl @ A @ B @ Y1 ) )
% 2.83/1.52 = ( X1 = Y1 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_23_old_Osum_Oinject_I1_J,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType,A2: A,A3: A] :
% 2.83/1.52 ( ( ( sum_Inl @ A @ B @ A2 )
% 2.83/1.52 = ( sum_Inl @ A @ B @ A3 ) )
% 2.83/1.52 = ( A2 = A3 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_24_Inl__inject,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType,X: A,Y: A] :
% 2.83/1.52 ( ( ( sum_Inl @ A @ B @ X )
% 2.83/1.52 = ( sum_Inl @ A @ B @ Y ) )
% 2.83/1.52 => ( X = Y ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_25_obj__sumE,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,S: sum_sum @ A @ B] :
% 2.83/1.52 ( ! [X3: A] :
% 2.83/1.52 ( S
% 2.83/1.52 != ( sum_Inl @ A @ B @ X3 ) )
% 2.83/1.52 => ~ ! [X3: B] :
% 2.83/1.52 ( S
% 2.83/1.52 != ( sum_Inr @ B @ A @ X3 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_26_old_Osum_Oinducts,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType,P: ( sum_sum @ A @ B ) > $o,Sum: sum_sum @ A @ B] :
% 2.83/1.52 ( ! [A4: A] : ( P @ ( sum_Inl @ A @ B @ A4 ) )
% 2.83/1.52 => ( ! [B4: B] : ( P @ ( sum_Inr @ B @ A @ B4 ) )
% 2.83/1.52 => ( P @ Sum ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_27_old_Osum_Oexhaust,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,Y: sum_sum @ A @ B] :
% 2.83/1.52 ( ! [A4: A] :
% 2.83/1.52 ( Y
% 2.83/1.52 != ( sum_Inl @ A @ B @ A4 ) )
% 2.83/1.52 => ~ ! [B4: B] :
% 2.83/1.52 ( Y
% 2.83/1.52 != ( sum_Inr @ B @ A @ B4 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_28_split__sum__all,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType] :
% 2.83/1.52 ( ( ^ [P3: ( sum_sum @ A @ B ) > $o] :
% 2.83/1.52 ! [X4: sum_sum @ A @ B] : ( P3 @ X4 ) )
% 2.83/1.52 = ( ^ [P4: ( sum_sum @ A @ B ) > $o] :
% 2.83/1.52 ( ! [X5: A] : ( P4 @ ( sum_Inl @ A @ B @ X5 ) )
% 2.83/1.52 & ! [X5: B] : ( P4 @ ( sum_Inr @ B @ A @ X5 ) ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_29_split__sum__ex,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType] :
% 2.83/1.52 ( ( ^ [P3: ( sum_sum @ A @ B ) > $o] :
% 2.83/1.52 ? [X4: sum_sum @ A @ B] : ( P3 @ X4 ) )
% 2.83/1.52 = ( ^ [P4: ( sum_sum @ A @ B ) > $o] :
% 2.83/1.52 ( ? [X5: A] : ( P4 @ ( sum_Inl @ A @ B @ X5 ) )
% 2.83/1.52 | ? [X5: B] : ( P4 @ ( sum_Inr @ B @ A @ X5 ) ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_30_Inr__not__Inl,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType,B2: B,A2: A] :
% 2.83/1.52 ( ( sum_Inr @ B @ A @ B2 )
% 2.83/1.52 != ( sum_Inl @ A @ B @ A2 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_31_sumE,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,S: sum_sum @ A @ B] :
% 2.83/1.52 ( ! [X3: A] :
% 2.83/1.52 ( S
% 2.83/1.52 != ( sum_Inl @ A @ B @ X3 ) )
% 2.83/1.52 => ~ ! [Y3: B] :
% 2.83/1.52 ( S
% 2.83/1.52 != ( sum_Inr @ B @ A @ Y3 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_32_old_Osum_Odistinct_I1_J,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,A2: A,B3: B] :
% 2.83/1.52 ( ( sum_Inl @ A @ B @ A2 )
% 2.83/1.52 != ( sum_Inr @ B @ A @ B3 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_33_old_Osum_Odistinct_I2_J,axiom,
% 2.83/1.52 ! [B5: $tType,A5: $tType,B6: B5,A6: A5] :
% 2.83/1.52 ( ( sum_Inr @ B5 @ A5 @ B6 )
% 2.83/1.52 != ( sum_Inl @ A5 @ B5 @ A6 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_34_sum_Odistinct_I1_J,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,X1: A,X2: B] :
% 2.83/1.52 ( ( sum_Inl @ A @ B @ X1 )
% 2.83/1.52 != ( sum_Inr @ B @ A @ X2 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_35_old_Oprod_Oinducts,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 2.83/1.52 ( ! [A4: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 2.83/1.52 => ( P @ Prod ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_36_old_Oprod_Oexhaust,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 2.83/1.52 ~ ! [A4: A,B4: B] :
% 2.83/1.52 ( Y
% 2.83/1.52 != ( product_Pair @ A @ B @ A4 @ B4 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_37_prod__induct7,axiom,
% 2.83/1.52 ! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 2.83/1.52 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
% 2.83/1.52 => ( P @ X ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_38_prod__induct6,axiom,
% 2.83/1.52 ! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 2.83/1.52 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
% 2.83/1.52 => ( P @ X ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_39_prod__induct5,axiom,
% 2.83/1.52 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 2.83/1.52 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 2.83/1.52 => ( P @ X ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_40_prod__induct4,axiom,
% 2.83/1.52 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.83/1.52 ( ! [A4: A,B4: B,C2: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
% 2.83/1.52 => ( P @ X ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_41_prod__induct3,axiom,
% 2.83/1.52 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.83/1.52 ( ! [A4: A,B4: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
% 2.83/1.52 => ( P @ X ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_42_prod__cases7,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 2.83/1.52 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] :
% 2.83/1.52 ( Y
% 2.83/1.52 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_43_prod__cases6,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 2.83/1.52 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] :
% 2.83/1.52 ( Y
% 2.83/1.52 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_44_prod__cases5,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 2.83/1.52 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E] :
% 2.83/1.52 ( Y
% 2.83/1.52 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_45_mem__Collect__eq,axiom,
% 2.83/1.52 ! [A: $tType,A2: A,P: A > $o] :
% 2.83/1.52 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.83/1.52 = ( P @ A2 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_46_Collect__mem__eq,axiom,
% 2.83/1.52 ! [A: $tType,A7: set @ A] :
% 2.83/1.52 ( ( collect @ A
% 2.83/1.52 @ ^ [X5: A] : ( member @ A @ X5 @ A7 ) )
% 2.83/1.52 = A7 ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_47_Collect__cong,axiom,
% 2.83/1.52 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.83/1.52 ( ! [X3: A] :
% 2.83/1.52 ( ( P @ X3 )
% 2.83/1.52 = ( Q @ X3 ) )
% 2.83/1.52 => ( ( collect @ A @ P )
% 2.83/1.52 = ( collect @ A @ Q ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_48_ext,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType,F3: A > B,G3: A > B] :
% 2.83/1.52 ( ! [X3: A] :
% 2.83/1.52 ( ( F3 @ X3 )
% 2.83/1.52 = ( G3 @ X3 ) )
% 2.83/1.52 => ( F3 = G3 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_49_Inr__Inl__False,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType,X: B,Y: A] :
% 2.83/1.52 ( ( sum_Inr @ B @ A @ X )
% 2.83/1.52 != ( sum_Inl @ A @ B @ Y ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_50_Inl__Inr__False,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,X: A,Y: B] :
% 2.83/1.52 ( ( sum_Inl @ A @ B @ X )
% 2.83/1.52 != ( sum_Inr @ B @ A @ Y ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_51_old_Oprod_Orec,axiom,
% 2.83/1.52 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 2.83/1.52 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 2.83/1.52 = ( F1 @ A2 @ B2 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_52_old_Osum_Osimps_I7_J,axiom,
% 2.83/1.52 ! [B: $tType,T: $tType,A: $tType,F1: A > T,F22: B > T,A2: A] :
% 2.83/1.52 ( ( sum_rec_sum @ A @ T @ B @ F1 @ F22 @ ( sum_Inl @ A @ B @ A2 ) )
% 2.83/1.52 = ( F1 @ A2 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_53_old_Osum_Osimps_I8_J,axiom,
% 2.83/1.52 ! [A: $tType,T: $tType,B: $tType,F1: A > T,F22: B > T,B2: B] :
% 2.83/1.52 ( ( sum_rec_sum @ A @ T @ B @ F1 @ F22 @ ( sum_Inr @ B @ A @ B2 ) )
% 2.83/1.52 = ( F22 @ B2 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_54_obj__sumE__f,axiom,
% 2.83/1.52 ! [A: $tType,C: $tType,B: $tType,S: B,F3: ( sum_sum @ A @ C ) > B,P: $o] :
% 2.83/1.52 ( ! [X3: A] :
% 2.83/1.52 ( ( S
% 2.83/1.52 = ( F3 @ ( sum_Inl @ A @ C @ X3 ) ) )
% 2.83/1.52 => P )
% 2.83/1.52 => ( ! [X3: C] :
% 2.83/1.52 ( ( S
% 2.83/1.52 = ( F3 @ ( sum_Inr @ C @ A @ X3 ) ) )
% 2.83/1.52 => P )
% 2.83/1.52 => ! [X6: sum_sum @ A @ C] :
% 2.83/1.52 ( ( S
% 2.83/1.52 = ( F3 @ X6 ) )
% 2.83/1.52 => P ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_55_internal__case__prod__conv,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B2: C] :
% 2.83/1.52 ( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 2.83/1.52 = ( C3 @ A2 @ B2 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_56_Suml_Osimps,axiom,
% 2.83/1.52 ! [B: $tType,C: $tType,A: $tType,F3: A > C,X: A] :
% 2.83/1.52 ( ( sum_Suml @ A @ C @ B @ F3 @ ( sum_Inl @ A @ B @ X ) )
% 2.83/1.52 = ( F3 @ X ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_57_Sumr_Osimps,axiom,
% 2.83/1.52 ! [A: $tType,C: $tType,B: $tType,F3: B > C,X: B] :
% 2.83/1.52 ( ( sum_Sumr @ B @ C @ A @ F3 @ ( sum_Inr @ B @ A @ X ) )
% 2.83/1.52 = ( F3 @ X ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_58_image2__eqI,axiom,
% 2.83/1.52 ! [A: $tType,C: $tType,B: $tType,B2: A,F3: B > A,X: B,C3: C,G3: B > C,A7: set @ B] :
% 2.83/1.52 ( ( B2
% 2.83/1.52 = ( F3 @ X ) )
% 2.83/1.52 => ( ( C3
% 2.83/1.52 = ( G3 @ X ) )
% 2.83/1.52 => ( ( member @ B @ X @ A7 )
% 2.83/1.52 => ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B2 @ C3 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A7 @ F3 @ G3 ) ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_59_Sumr__inject,axiom,
% 2.83/1.52 ! [A: $tType,C: $tType,B: $tType,F3: B > C,G3: B > C] :
% 2.83/1.52 ( ( ( sum_Sumr @ B @ C @ A @ F3 )
% 2.83/1.52 = ( sum_Sumr @ B @ C @ A @ G3 ) )
% 2.83/1.52 => ( F3 = G3 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_60_Suml__inject,axiom,
% 2.83/1.52 ! [B: $tType,C: $tType,A: $tType,F3: A > C,G3: A > C] :
% 2.83/1.52 ( ( ( sum_Suml @ A @ C @ B @ F3 )
% 2.83/1.52 = ( sum_Suml @ A @ C @ B @ G3 ) )
% 2.83/1.52 => ( F3 = G3 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_61_ssubst__Pair__rhs,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType,R: A,S: B,R2: set @ ( product_prod @ A @ B ),S2: B] :
% 2.83/1.52 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S ) @ R2 )
% 2.83/1.52 => ( ( S2 = S )
% 2.83/1.52 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S2 ) @ R2 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_62_curry__conv,axiom,
% 2.83/1.52 ! [A: $tType,C: $tType,B: $tType] :
% 2.83/1.52 ( ( product_curry @ B @ C @ A )
% 2.83/1.52 = ( ^ [F4: ( product_prod @ B @ C ) > A,A8: B,B7: C] : ( F4 @ ( product_Pair @ B @ C @ A8 @ B7 ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_63_curryI,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 2.83/1.52 ( ( F3 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 2.83/1.52 => ( product_curry @ A @ B @ $o @ F3 @ A2 @ B2 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_64_case__sum__if,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType,C: $tType,P2: $o,F3: B > A,G3: C > A,X: B,Y: C] :
% 2.83/1.52 ( ( P2
% 2.83/1.52 => ( ( sum_case_sum @ B @ A @ C @ F3 @ G3 @ ( if @ ( sum_sum @ B @ C ) @ P2 @ ( sum_Inl @ B @ C @ X ) @ ( sum_Inr @ C @ B @ Y ) ) )
% 2.83/1.52 = ( F3 @ X ) ) )
% 2.83/1.52 & ( ~ P2
% 2.83/1.52 => ( ( sum_case_sum @ B @ A @ C @ F3 @ G3 @ ( if @ ( sum_sum @ B @ C ) @ P2 @ ( sum_Inl @ B @ C @ X ) @ ( sum_Inr @ C @ B @ Y ) ) )
% 2.83/1.52 = ( G3 @ Y ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_65_swap__simp,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,X: B,Y: A] :
% 2.83/1.52 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 2.83/1.52 = ( product_Pair @ A @ B @ Y @ X ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_66_Inl__cont__par,axiom,
% 2.83/1.52 ! [Tr13: dtree,Tr23: dtree] :
% 2.83/1.52 ( ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( cont @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr13 @ Tr23 ) ) ) )
% 2.83/1.52 = ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( sup_sup @ ( set @ ( sum_sum @ t @ dtree ) ) @ ( cont @ Tr13 ) @ ( cont @ Tr23 ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_67_internal__case__prod__def,axiom,
% 2.83/1.52 ! [C: $tType,B: $tType,A: $tType] :
% 2.83/1.52 ( ( produc2004651681e_prod @ A @ B @ C )
% 2.83/1.52 = ( product_case_prod @ A @ B @ C ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_68_swap__swap,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 2.83/1.52 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
% 2.83/1.52 = P2 ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_69_curry__case__prod,axiom,
% 2.83/1.52 ! [C: $tType,B: $tType,A: $tType,F3: A > B > C] :
% 2.83/1.52 ( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F3 ) )
% 2.83/1.52 = F3 ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_70_case__prod__curry,axiom,
% 2.83/1.52 ! [C: $tType,B: $tType,A: $tType,F3: ( product_prod @ A @ B ) > C] :
% 2.83/1.52 ( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F3 ) )
% 2.83/1.52 = F3 ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_71_case__sum__step_I2_J,axiom,
% 2.83/1.52 ! [E: $tType,A: $tType,C: $tType,B: $tType,F3: E > A,F5: B > A,G4: C > A,P2: sum_sum @ B @ C] :
% 2.83/1.52 ( ( sum_case_sum @ E @ A @ ( sum_sum @ B @ C ) @ F3 @ ( sum_case_sum @ B @ A @ C @ F5 @ G4 ) @ ( sum_Inr @ ( sum_sum @ B @ C ) @ E @ P2 ) )
% 2.83/1.52 = ( sum_case_sum @ B @ A @ C @ F5 @ G4 @ P2 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_72_case__sum__step_I1_J,axiom,
% 2.83/1.52 ! [D: $tType,A: $tType,C: $tType,B: $tType,F5: B > A,G4: C > A,G3: D > A,P2: sum_sum @ B @ C] :
% 2.83/1.52 ( ( sum_case_sum @ ( sum_sum @ B @ C ) @ A @ D @ ( sum_case_sum @ B @ A @ C @ F5 @ G4 ) @ G3 @ ( sum_Inl @ ( sum_sum @ B @ C ) @ D @ P2 ) )
% 2.83/1.52 = ( sum_case_sum @ B @ A @ C @ F5 @ G4 @ P2 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_73_case__sum__inject,axiom,
% 2.83/1.52 ! [A: $tType,C: $tType,B: $tType,F1: A > C,F22: B > C,G1: A > C,G22: B > C] :
% 2.83/1.52 ( ( ( sum_case_sum @ A @ C @ B @ F1 @ F22 )
% 2.83/1.52 = ( sum_case_sum @ A @ C @ B @ G1 @ G22 ) )
% 2.83/1.52 => ~ ( ( F1 = G1 )
% 2.83/1.52 => ( F22 != G22 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_74_mem__case__prodE,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType,C: $tType,Z: A,C3: B > C > ( set @ A ),P2: product_prod @ B @ C] :
% 2.83/1.52 ( ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ P2 ) )
% 2.83/1.52 => ~ ! [X3: B,Y3: C] :
% 2.83/1.52 ( ( P2
% 2.83/1.52 = ( product_Pair @ B @ C @ X3 @ Y3 ) )
% 2.83/1.52 => ~ ( member @ A @ Z @ ( C3 @ X3 @ Y3 ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_75_old_Oprod_Ocase,axiom,
% 2.83/1.52 ! [A: $tType,C: $tType,B: $tType,F3: A > B > C,X1: A,X2: B] :
% 2.83/1.52 ( ( product_case_prod @ A @ B @ C @ F3 @ ( product_Pair @ A @ B @ X1 @ X2 ) )
% 2.83/1.52 = ( F3 @ X1 @ X2 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_76_old_Osum_Osimps_I6_J,axiom,
% 2.83/1.52 ! [A: $tType,C: $tType,B: $tType,F1: A > C,F22: B > C,X2: B] :
% 2.83/1.52 ( ( sum_case_sum @ A @ C @ B @ F1 @ F22 @ ( sum_Inr @ B @ A @ X2 ) )
% 2.83/1.52 = ( F22 @ X2 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_77_old_Osum_Osimps_I5_J,axiom,
% 2.83/1.52 ! [B: $tType,C: $tType,A: $tType,F1: A > C,F22: B > C,X1: A] :
% 2.83/1.52 ( ( sum_case_sum @ A @ C @ B @ F1 @ F22 @ ( sum_Inl @ A @ B @ X1 ) )
% 2.83/1.52 = ( F1 @ X1 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_78_curryE,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 2.83/1.52 ( ( product_curry @ A @ B @ $o @ F3 @ A2 @ B2 )
% 2.83/1.52 => ( F3 @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_79_curryD,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 2.83/1.52 ( ( product_curry @ A @ B @ $o @ F3 @ A2 @ B2 )
% 2.83/1.52 => ( F3 @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_80_vimage__Un,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,F3: A > B,A7: set @ B,B8: set @ B] :
% 2.83/1.52 ( ( vimage @ A @ B @ F3 @ ( sup_sup @ ( set @ B ) @ A7 @ B8 ) )
% 2.83/1.52 = ( sup_sup @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ A7 ) @ ( vimage @ A @ B @ F3 @ B8 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_81_Inl__Inr__image__cong,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType,A7: set @ ( sum_sum @ A @ B ),B8: set @ ( sum_sum @ A @ B )] :
% 2.83/1.52 ( ( ( vimage @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ A7 )
% 2.83/1.52 = ( vimage @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ B8 ) )
% 2.83/1.52 => ( ( ( vimage @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ A7 )
% 2.83/1.52 = ( vimage @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ B8 ) )
% 2.83/1.52 => ( A7 = B8 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_82_vimageI,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType,F3: B > A,A2: B,B2: A,B8: set @ A] :
% 2.83/1.52 ( ( ( F3 @ A2 )
% 2.83/1.52 = B2 )
% 2.83/1.52 => ( ( member @ A @ B2 @ B8 )
% 2.83/1.52 => ( member @ B @ A2 @ ( vimage @ B @ A @ F3 @ B8 ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_83_vimage__eq,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,A2: A,F3: A > B,B8: set @ B] :
% 2.83/1.52 ( ( member @ A @ A2 @ ( vimage @ A @ B @ F3 @ B8 ) )
% 2.83/1.52 = ( member @ B @ ( F3 @ A2 ) @ B8 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_84_UnCI,axiom,
% 2.83/1.52 ! [A: $tType,C3: A,B8: set @ A,A7: set @ A] :
% 2.83/1.52 ( ( ~ ( member @ A @ C3 @ B8 )
% 2.83/1.52 => ( member @ A @ C3 @ A7 ) )
% 2.83/1.52 => ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A7 @ B8 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_85_Un__iff,axiom,
% 2.83/1.52 ! [A: $tType,C3: A,A7: set @ A,B8: set @ A] :
% 2.83/1.52 ( ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A7 @ B8 ) )
% 2.83/1.52 = ( ( member @ A @ C3 @ A7 )
% 2.83/1.52 | ( member @ A @ C3 @ B8 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_86_sup__apply,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType] :
% 2.83/1.52 ( ( semilattice_sup @ B @ ( type2 @ B ) )
% 2.83/1.52 => ( ( sup_sup @ ( A > B ) )
% 2.83/1.52 = ( ^ [F4: A > B,G5: A > B,X5: A] : ( sup_sup @ B @ ( F4 @ X5 ) @ ( G5 @ X5 ) ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_87_sup_Oright__idem,axiom,
% 2.83/1.52 ! [A: $tType] :
% 2.83/1.52 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.83/1.52 => ! [A2: A,B2: A] :
% 2.83/1.52 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ B2 )
% 2.83/1.52 = ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_88_sup__left__idem,axiom,
% 2.83/1.52 ! [A: $tType] :
% 2.83/1.52 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.83/1.52 => ! [X: A,Y: A] :
% 2.83/1.52 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 2.83/1.52 = ( sup_sup @ A @ X @ Y ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_89_sup_Oleft__idem,axiom,
% 2.83/1.52 ! [A: $tType] :
% 2.83/1.52 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.83/1.52 => ! [A2: A,B2: A] :
% 2.83/1.52 ( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) )
% 2.83/1.52 = ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_90_sup__idem,axiom,
% 2.83/1.52 ! [A: $tType] :
% 2.83/1.52 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.83/1.52 => ! [X: A] :
% 2.83/1.52 ( ( sup_sup @ A @ X @ X )
% 2.83/1.52 = X ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_91_sup_Oidem,axiom,
% 2.83/1.52 ! [A: $tType] :
% 2.83/1.52 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.83/1.52 => ! [A2: A] :
% 2.83/1.52 ( ( sup_sup @ A @ A2 @ A2 )
% 2.83/1.52 = A2 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_92_sup__left__commute,axiom,
% 2.83/1.52 ! [A: $tType] :
% 2.83/1.52 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.83/1.52 => ! [X: A,Y: A,Z: A] :
% 2.83/1.52 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
% 2.83/1.52 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_93_sup_Oleft__commute,axiom,
% 2.83/1.52 ! [A: $tType] :
% 2.83/1.52 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.83/1.52 => ! [B2: A,A2: A,C3: A] :
% 2.83/1.52 ( ( sup_sup @ A @ B2 @ ( sup_sup @ A @ A2 @ C3 ) )
% 2.83/1.52 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C3 ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_94_sup__commute,axiom,
% 2.83/1.52 ! [A: $tType] :
% 2.83/1.52 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.83/1.52 => ( ( sup_sup @ A )
% 2.83/1.52 = ( ^ [X5: A,Y4: A] : ( sup_sup @ A @ Y4 @ X5 ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_95_sup_Ocommute,axiom,
% 2.83/1.52 ! [A: $tType] :
% 2.83/1.52 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.83/1.52 => ( ( sup_sup @ A )
% 2.83/1.52 = ( ^ [A8: A,B7: A] : ( sup_sup @ A @ B7 @ A8 ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_96_sup__assoc,axiom,
% 2.83/1.52 ! [A: $tType] :
% 2.83/1.52 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.83/1.52 => ! [X: A,Y: A,Z: A] :
% 2.83/1.52 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
% 2.83/1.52 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_97_sup_Oassoc,axiom,
% 2.83/1.52 ! [A: $tType] :
% 2.83/1.52 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.83/1.52 => ! [A2: A,B2: A,C3: A] :
% 2.83/1.52 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ C3 )
% 2.83/1.52 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C3 ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_98_sup__fun__def,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType] :
% 2.83/1.52 ( ( semilattice_sup @ B @ ( type2 @ B ) )
% 2.83/1.52 => ( ( sup_sup @ ( A > B ) )
% 2.83/1.52 = ( ^ [F4: A > B,G5: A > B,X5: A] : ( sup_sup @ B @ ( F4 @ X5 ) @ ( G5 @ X5 ) ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_99_inf__sup__aci_I5_J,axiom,
% 2.83/1.52 ! [A: $tType] :
% 2.83/1.52 ( ( lattice @ A @ ( type2 @ A ) )
% 2.83/1.52 => ( ( sup_sup @ A )
% 2.83/1.52 = ( ^ [X5: A,Y4: A] : ( sup_sup @ A @ Y4 @ X5 ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_100_inf__sup__aci_I6_J,axiom,
% 2.83/1.52 ! [A: $tType] :
% 2.83/1.52 ( ( lattice @ A @ ( type2 @ A ) )
% 2.83/1.52 => ! [X: A,Y: A,Z: A] :
% 2.83/1.52 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
% 2.83/1.52 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_101_inf__sup__aci_I7_J,axiom,
% 2.83/1.52 ! [A: $tType] :
% 2.83/1.52 ( ( lattice @ A @ ( type2 @ A ) )
% 2.83/1.52 => ! [X: A,Y: A,Z: A] :
% 2.83/1.52 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
% 2.83/1.52 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_102_inf__sup__aci_I8_J,axiom,
% 2.83/1.52 ! [A: $tType] :
% 2.83/1.52 ( ( lattice @ A @ ( type2 @ A ) )
% 2.83/1.52 => ! [X: A,Y: A] :
% 2.83/1.52 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 2.83/1.52 = ( sup_sup @ A @ X @ Y ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_103_Un__left__commute,axiom,
% 2.83/1.52 ! [A: $tType,A7: set @ A,B8: set @ A,C4: set @ A] :
% 2.83/1.52 ( ( sup_sup @ ( set @ A ) @ A7 @ ( sup_sup @ ( set @ A ) @ B8 @ C4 ) )
% 2.83/1.52 = ( sup_sup @ ( set @ A ) @ B8 @ ( sup_sup @ ( set @ A ) @ A7 @ C4 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_104_Un__left__absorb,axiom,
% 2.83/1.52 ! [A: $tType,A7: set @ A,B8: set @ A] :
% 2.83/1.52 ( ( sup_sup @ ( set @ A ) @ A7 @ ( sup_sup @ ( set @ A ) @ A7 @ B8 ) )
% 2.83/1.52 = ( sup_sup @ ( set @ A ) @ A7 @ B8 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_105_Un__commute,axiom,
% 2.83/1.52 ! [A: $tType] :
% 2.83/1.52 ( ( sup_sup @ ( set @ A ) )
% 2.83/1.52 = ( ^ [A9: set @ A,B9: set @ A] : ( sup_sup @ ( set @ A ) @ B9 @ A9 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_106_Un__absorb,axiom,
% 2.83/1.52 ! [A: $tType,A7: set @ A] :
% 2.83/1.52 ( ( sup_sup @ ( set @ A ) @ A7 @ A7 )
% 2.83/1.52 = A7 ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_107_Un__assoc,axiom,
% 2.83/1.52 ! [A: $tType,A7: set @ A,B8: set @ A,C4: set @ A] :
% 2.83/1.52 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A7 @ B8 ) @ C4 )
% 2.83/1.52 = ( sup_sup @ ( set @ A ) @ A7 @ ( sup_sup @ ( set @ A ) @ B8 @ C4 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_108_ball__Un,axiom,
% 2.83/1.52 ! [A: $tType,A7: set @ A,B8: set @ A,P: A > $o] :
% 2.83/1.52 ( ( ! [X5: A] :
% 2.83/1.52 ( ( member @ A @ X5 @ ( sup_sup @ ( set @ A ) @ A7 @ B8 ) )
% 2.83/1.52 => ( P @ X5 ) ) )
% 2.83/1.52 = ( ! [X5: A] :
% 2.83/1.52 ( ( member @ A @ X5 @ A7 )
% 2.83/1.52 => ( P @ X5 ) )
% 2.83/1.52 & ! [X5: A] :
% 2.83/1.52 ( ( member @ A @ X5 @ B8 )
% 2.83/1.52 => ( P @ X5 ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_109_bex__Un,axiom,
% 2.83/1.52 ! [A: $tType,A7: set @ A,B8: set @ A,P: A > $o] :
% 2.83/1.52 ( ( ? [X5: A] :
% 2.83/1.52 ( ( member @ A @ X5 @ ( sup_sup @ ( set @ A ) @ A7 @ B8 ) )
% 2.83/1.52 & ( P @ X5 ) ) )
% 2.83/1.52 = ( ? [X5: A] :
% 2.83/1.52 ( ( member @ A @ X5 @ A7 )
% 2.83/1.52 & ( P @ X5 ) )
% 2.83/1.52 | ? [X5: A] :
% 2.83/1.52 ( ( member @ A @ X5 @ B8 )
% 2.83/1.52 & ( P @ X5 ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_110_UnI2,axiom,
% 2.83/1.52 ! [A: $tType,C3: A,B8: set @ A,A7: set @ A] :
% 2.83/1.52 ( ( member @ A @ C3 @ B8 )
% 2.83/1.52 => ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A7 @ B8 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_111_UnI1,axiom,
% 2.83/1.52 ! [A: $tType,C3: A,A7: set @ A,B8: set @ A] :
% 2.83/1.52 ( ( member @ A @ C3 @ A7 )
% 2.83/1.52 => ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A7 @ B8 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_112_UnE,axiom,
% 2.83/1.52 ! [A: $tType,C3: A,A7: set @ A,B8: set @ A] :
% 2.83/1.52 ( ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A7 @ B8 ) )
% 2.83/1.52 => ( ~ ( member @ A @ C3 @ A7 )
% 2.83/1.52 => ( member @ A @ C3 @ B8 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_113_vimage__Collect,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType,P: B > $o,F3: A > B,Q: A > $o] :
% 2.83/1.52 ( ! [X3: A] :
% 2.83/1.52 ( ( P @ ( F3 @ X3 ) )
% 2.83/1.52 = ( Q @ X3 ) )
% 2.83/1.52 => ( ( vimage @ A @ B @ F3 @ ( collect @ B @ P ) )
% 2.83/1.52 = ( collect @ A @ Q ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_114_vimageI2,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType,F3: B > A,A2: B,A7: set @ A] :
% 2.83/1.52 ( ( member @ A @ ( F3 @ A2 ) @ A7 )
% 2.83/1.52 => ( member @ B @ A2 @ ( vimage @ B @ A @ F3 @ A7 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_115_vimageE,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,A2: A,F3: A > B,B8: set @ B] :
% 2.83/1.52 ( ( member @ A @ A2 @ ( vimage @ A @ B @ F3 @ B8 ) )
% 2.83/1.52 => ( member @ B @ ( F3 @ A2 ) @ B8 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_116_vimageD,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,A2: A,F3: A > B,A7: set @ B] :
% 2.83/1.52 ( ( member @ A @ A2 @ ( vimage @ A @ B @ F3 @ A7 ) )
% 2.83/1.52 => ( member @ B @ ( F3 @ A2 ) @ A7 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_117_split__cong,axiom,
% 2.83/1.52 ! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F3: A > B > C,G3: A > B > C,P2: product_prod @ A @ B] :
% 2.83/1.52 ( ! [X3: A,Y3: B] :
% 2.83/1.52 ( ( ( product_Pair @ A @ B @ X3 @ Y3 )
% 2.83/1.52 = Q2 )
% 2.83/1.52 => ( ( F3 @ X3 @ Y3 )
% 2.83/1.52 = ( G3 @ X3 @ Y3 ) ) )
% 2.83/1.52 => ( ( P2 = Q2 )
% 2.83/1.52 => ( ( product_case_prod @ A @ B @ C @ F3 @ P2 )
% 2.83/1.52 = ( product_case_prod @ A @ B @ C @ G3 @ Q2 ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_118_case__sum__expand__Inr__pointfree,axiom,
% 2.83/1.52 ! [B: $tType,C: $tType,A: $tType,F3: ( sum_sum @ A @ C ) > B,G3: A > B] :
% 2.83/1.52 ( ( ( comp @ ( sum_sum @ A @ C ) @ B @ A @ F3 @ ( sum_Inl @ A @ C ) )
% 2.83/1.52 = G3 )
% 2.83/1.52 => ( ( sum_case_sum @ A @ B @ C @ G3 @ ( comp @ ( sum_sum @ A @ C ) @ B @ C @ F3 @ ( sum_Inr @ C @ A ) ) )
% 2.83/1.52 = F3 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_119_case__sum__expand__Inr_H,axiom,
% 2.83/1.52 ! [B: $tType,C: $tType,A: $tType,F3: ( sum_sum @ A @ C ) > B,G3: A > B,H: C > B] :
% 2.83/1.52 ( ( ( comp @ ( sum_sum @ A @ C ) @ B @ A @ F3 @ ( sum_Inl @ A @ C ) )
% 2.83/1.52 = G3 )
% 2.83/1.52 => ( ( H
% 2.83/1.52 = ( comp @ ( sum_sum @ A @ C ) @ B @ C @ F3 @ ( sum_Inr @ C @ A ) ) )
% 2.83/1.52 = ( ( sum_case_sum @ A @ B @ C @ G3 @ H )
% 2.83/1.52 = F3 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_120_case__sum__expand__Inr,axiom,
% 2.83/1.52 ! [B: $tType,C: $tType,A: $tType,F3: ( sum_sum @ A @ C ) > B,G3: A > B,X: sum_sum @ A @ C] :
% 2.83/1.52 ( ( ( comp @ ( sum_sum @ A @ C ) @ B @ A @ F3 @ ( sum_Inl @ A @ C ) )
% 2.83/1.52 = G3 )
% 2.83/1.52 => ( ( F3 @ X )
% 2.83/1.52 = ( sum_case_sum @ A @ B @ C @ G3 @ ( comp @ ( sum_sum @ A @ C ) @ B @ C @ F3 @ ( sum_Inr @ C @ A ) ) @ X ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_121_scomp__apply,axiom,
% 2.83/1.52 ! [A: $tType,D: $tType,C: $tType,B: $tType] :
% 2.83/1.52 ( ( product_scomp @ B @ C @ D @ A )
% 2.83/1.52 = ( ^ [F4: B > ( product_prod @ C @ D ),G5: C > D > A,X5: B] : ( product_case_prod @ C @ D @ A @ G5 @ ( F4 @ X5 ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_122_pair__in__swap__image,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,Y: A,X: B,A7: set @ ( product_prod @ B @ A )] :
% 2.83/1.52 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A7 ) )
% 2.83/1.52 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A7 ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_123_image__eqI,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,B2: A,F3: B > A,X: B,A7: set @ B] :
% 2.83/1.52 ( ( B2
% 2.83/1.52 = ( F3 @ X ) )
% 2.83/1.52 => ( ( member @ B @ X @ A7 )
% 2.83/1.52 => ( member @ A @ B2 @ ( image @ B @ A @ F3 @ A7 ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_124_o__case__sum,axiom,
% 2.83/1.52 ! [A: $tType,C: $tType,D: $tType,B: $tType,H: D > C,F3: A > D,G3: B > D] :
% 2.83/1.52 ( ( comp @ D @ C @ ( sum_sum @ A @ B ) @ H @ ( sum_case_sum @ A @ D @ B @ F3 @ G3 ) )
% 2.83/1.52 = ( sum_case_sum @ A @ C @ B @ ( comp @ D @ C @ A @ H @ F3 ) @ ( comp @ D @ C @ B @ H @ G3 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_125_rev__image__eqI,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType,X: A,A7: set @ A,B2: B,F3: A > B] :
% 2.83/1.52 ( ( member @ A @ X @ A7 )
% 2.83/1.52 => ( ( B2
% 2.83/1.52 = ( F3 @ X ) )
% 2.83/1.52 => ( member @ B @ B2 @ ( image @ A @ B @ F3 @ A7 ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_126_ball__imageD,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,F3: B > A,A7: set @ B,P: A > $o] :
% 2.83/1.52 ( ! [X3: A] :
% 2.83/1.52 ( ( member @ A @ X3 @ ( image @ B @ A @ F3 @ A7 ) )
% 2.83/1.52 => ( P @ X3 ) )
% 2.83/1.52 => ! [X6: B] :
% 2.83/1.52 ( ( member @ B @ X6 @ A7 )
% 2.83/1.52 => ( P @ ( F3 @ X6 ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_127_image__cong,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType,M: set @ A,N: set @ A,F3: A > B,G3: A > B] :
% 2.83/1.52 ( ( M = N )
% 2.83/1.52 => ( ! [X3: A] :
% 2.83/1.52 ( ( member @ A @ X3 @ N )
% 2.83/1.52 => ( ( F3 @ X3 )
% 2.83/1.52 = ( G3 @ X3 ) ) )
% 2.83/1.52 => ( ( image @ A @ B @ F3 @ M )
% 2.83/1.52 = ( image @ A @ B @ G3 @ N ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_128_bex__imageD,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,F3: B > A,A7: set @ B,P: A > $o] :
% 2.83/1.52 ( ? [X6: A] :
% 2.83/1.52 ( ( member @ A @ X6 @ ( image @ B @ A @ F3 @ A7 ) )
% 2.83/1.52 & ( P @ X6 ) )
% 2.83/1.52 => ? [X3: B] :
% 2.83/1.52 ( ( member @ B @ X3 @ A7 )
% 2.83/1.52 & ( P @ ( F3 @ X3 ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_129_image__iff,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,Z: A,F3: B > A,A7: set @ B] :
% 2.83/1.52 ( ( member @ A @ Z @ ( image @ B @ A @ F3 @ A7 ) )
% 2.83/1.52 = ( ? [X5: B] :
% 2.83/1.52 ( ( member @ B @ X5 @ A7 )
% 2.83/1.52 & ( Z
% 2.83/1.52 = ( F3 @ X5 ) ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_130_imageI,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType,X: A,A7: set @ A,F3: A > B] :
% 2.83/1.52 ( ( member @ A @ X @ A7 )
% 2.83/1.52 => ( member @ B @ ( F3 @ X ) @ ( image @ A @ B @ F3 @ A7 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_131_rewriteR__comp__comp2,axiom,
% 2.83/1.52 ! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G3: C > B,H: A > C,R1: D > B,R22: A > D,F3: B > E,L: D > E] :
% 2.83/1.52 ( ( ( comp @ C @ B @ A @ G3 @ H )
% 2.83/1.52 = ( comp @ D @ B @ A @ R1 @ R22 ) )
% 2.83/1.52 => ( ( ( comp @ B @ E @ D @ F3 @ R1 )
% 2.83/1.52 = L )
% 2.83/1.52 => ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F3 @ G3 ) @ H )
% 2.83/1.52 = ( comp @ D @ E @ A @ L @ R22 ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_132_rewriteL__comp__comp2,axiom,
% 2.83/1.52 ! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F3: C > B,G3: A > C,L1: D > B,L2: A > D,H: E > A,R: E > D] :
% 2.83/1.52 ( ( ( comp @ C @ B @ A @ F3 @ G3 )
% 2.83/1.52 = ( comp @ D @ B @ A @ L1 @ L2 ) )
% 2.83/1.52 => ( ( ( comp @ A @ D @ E @ L2 @ H )
% 2.83/1.52 = R )
% 2.83/1.52 => ( ( comp @ C @ B @ E @ F3 @ ( comp @ A @ C @ E @ G3 @ H ) )
% 2.83/1.52 = ( comp @ D @ B @ E @ L1 @ R ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_133_rewriteR__comp__comp,axiom,
% 2.83/1.52 ! [C: $tType,D: $tType,B: $tType,A: $tType,G3: C > B,H: A > C,R: A > B,F3: B > D] :
% 2.83/1.52 ( ( ( comp @ C @ B @ A @ G3 @ H )
% 2.83/1.52 = R )
% 2.83/1.52 => ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F3 @ G3 ) @ H )
% 2.83/1.52 = ( comp @ B @ D @ A @ F3 @ R ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_134_rewriteL__comp__comp,axiom,
% 2.83/1.52 ! [C: $tType,B: $tType,A: $tType,D: $tType,F3: C > B,G3: A > C,L: A > B,H: D > A] :
% 2.83/1.52 ( ( ( comp @ C @ B @ A @ F3 @ G3 )
% 2.83/1.52 = L )
% 2.83/1.52 => ( ( comp @ C @ B @ D @ F3 @ ( comp @ A @ C @ D @ G3 @ H ) )
% 2.83/1.52 = ( comp @ A @ B @ D @ L @ H ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_135_comp__cong,axiom,
% 2.83/1.52 ! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F3: B > A,G3: C > B,X: C,F5: D > A,G4: E > D,X7: E] :
% 2.83/1.52 ( ( ( F3 @ ( G3 @ X ) )
% 2.83/1.52 = ( F5 @ ( G4 @ X7 ) ) )
% 2.83/1.52 => ( ( comp @ B @ A @ C @ F3 @ G3 @ X )
% 2.83/1.52 = ( comp @ D @ A @ E @ F5 @ G4 @ X7 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_136_image__Un,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,F3: B > A,A7: set @ B,B8: set @ B] :
% 2.83/1.52 ( ( image @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ A7 @ B8 ) )
% 2.83/1.52 = ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ F3 @ A7 ) @ ( image @ B @ A @ F3 @ B8 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_137_scomp__Pair,axiom,
% 2.83/1.52 ! [C: $tType,B: $tType,A: $tType,X: A > ( product_prod @ B @ C )] :
% 2.83/1.52 ( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X @ ( product_Pair @ B @ C ) )
% 2.83/1.52 = X ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_138_Pair__scomp,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,C: $tType,X: C,F3: C > A > B] :
% 2.83/1.52 ( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X ) @ F3 )
% 2.83/1.52 = ( F3 @ X ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_139_scomp__def,axiom,
% 2.83/1.52 ! [D: $tType,C: $tType,B: $tType,A: $tType] :
% 2.83/1.52 ( ( product_scomp @ A @ B @ C @ D )
% 2.83/1.52 = ( ^ [F4: A > ( product_prod @ B @ C ),G5: B > C > D,X5: A] : ( product_case_prod @ B @ C @ D @ G5 @ ( F4 @ X5 ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_140_case__sum__o__inj_I1_J,axiom,
% 2.83/1.52 ! [C: $tType,B: $tType,A: $tType,F3: A > B,G3: C > B] :
% 2.83/1.52 ( ( comp @ ( sum_sum @ A @ C ) @ B @ A @ ( sum_case_sum @ A @ B @ C @ F3 @ G3 ) @ ( sum_Inl @ A @ C ) )
% 2.83/1.52 = F3 ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_141_case__sum__o__inj_I2_J,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,C: $tType,F3: A > B,G3: C > B] :
% 2.83/1.52 ( ( comp @ ( sum_sum @ A @ C ) @ B @ C @ ( sum_case_sum @ A @ B @ C @ F3 @ G3 ) @ ( sum_Inr @ C @ A ) )
% 2.83/1.52 = G3 ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_142_comp__apply,axiom,
% 2.83/1.52 ! [C: $tType,A: $tType,B: $tType] :
% 2.83/1.52 ( ( comp @ B @ A @ C )
% 2.83/1.52 = ( ^ [F4: B > A,G5: C > B,X5: C] : ( F4 @ ( G5 @ X5 ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_143_set_Ocomp,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,C: $tType,F3: C > B,G3: B > A] :
% 2.83/1.52 ( ( comp @ ( set @ B ) @ ( set @ C ) @ ( set @ A ) @ ( vimage @ C @ B @ F3 ) @ ( vimage @ B @ A @ G3 ) )
% 2.83/1.52 = ( vimage @ C @ A @ ( comp @ B @ A @ C @ G3 @ F3 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_144_comp__apply__eq,axiom,
% 2.83/1.52 ! [B: $tType,D: $tType,A: $tType,C: $tType,F3: B > A,G3: C > B,X: C,H: D > A,K: C > D] :
% 2.83/1.52 ( ( ( F3 @ ( G3 @ X ) )
% 2.83/1.52 = ( H @ ( K @ X ) ) )
% 2.83/1.52 => ( ( comp @ B @ A @ C @ F3 @ G3 @ X )
% 2.83/1.52 = ( comp @ D @ A @ C @ H @ K @ X ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_145_comp__eq__dest__lhs,axiom,
% 2.83/1.52 ! [C: $tType,B: $tType,A: $tType,A2: C > B,B2: A > C,C3: A > B,V: A] :
% 2.83/1.52 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 2.83/1.52 = C3 )
% 2.83/1.52 => ( ( A2 @ ( B2 @ V ) )
% 2.83/1.52 = ( C3 @ V ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_146_comp__eq__elim,axiom,
% 2.83/1.52 ! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C3: D > B,D3: A > D] :
% 2.83/1.52 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 2.83/1.52 = ( comp @ D @ B @ A @ C3 @ D3 ) )
% 2.83/1.52 => ! [V2: A] :
% 2.83/1.52 ( ( A2 @ ( B2 @ V2 ) )
% 2.83/1.52 = ( C3 @ ( D3 @ V2 ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_147_comp__eq__dest,axiom,
% 2.83/1.52 ! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C3: D > B,D3: A > D,V: A] :
% 2.83/1.52 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 2.83/1.52 = ( comp @ D @ B @ A @ C3 @ D3 ) )
% 2.83/1.52 => ( ( A2 @ ( B2 @ V ) )
% 2.83/1.52 = ( C3 @ ( D3 @ V ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_148_comp__assoc,axiom,
% 2.83/1.52 ! [B: $tType,D: $tType,C: $tType,A: $tType,F3: D > B,G3: C > D,H: A > C] :
% 2.83/1.52 ( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F3 @ G3 ) @ H )
% 2.83/1.52 = ( comp @ D @ B @ A @ F3 @ ( comp @ C @ D @ A @ G3 @ H ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_149_comp__def,axiom,
% 2.83/1.52 ! [A: $tType,C: $tType,B: $tType] :
% 2.83/1.52 ( ( comp @ B @ C @ A )
% 2.83/1.52 = ( ^ [F4: B > C,G5: A > B,X5: A] : ( F4 @ ( G5 @ X5 ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_150_image__eq__imp__comp,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType,D: $tType,C: $tType,F3: B > A,A7: set @ B,G3: C > A,B8: set @ C,H: A > D] :
% 2.83/1.52 ( ( ( image @ B @ A @ F3 @ A7 )
% 2.83/1.52 = ( image @ C @ A @ G3 @ B8 ) )
% 2.83/1.52 => ( ( image @ B @ D @ ( comp @ A @ D @ B @ H @ F3 ) @ A7 )
% 2.83/1.52 = ( image @ C @ D @ ( comp @ A @ D @ C @ H @ G3 ) @ B8 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_151_image__comp,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType,C: $tType,F3: B > A,G3: C > B,R: set @ C] :
% 2.83/1.52 ( ( image @ B @ A @ F3 @ ( image @ C @ B @ G3 @ R ) )
% 2.83/1.52 = ( image @ C @ A @ ( comp @ B @ A @ C @ F3 @ G3 ) @ R ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_152_set_Ocompositionality,axiom,
% 2.83/1.52 ! [C: $tType,B: $tType,A: $tType,F3: C > B,G3: B > A,Set: set @ A] :
% 2.83/1.52 ( ( vimage @ C @ B @ F3 @ ( vimage @ B @ A @ G3 @ Set ) )
% 2.83/1.52 = ( vimage @ C @ A @ ( comp @ B @ A @ C @ G3 @ F3 ) @ Set ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_153_vimage__comp,axiom,
% 2.83/1.52 ! [A: $tType,B: $tType,C: $tType,F3: A > B,G3: B > C,X: set @ C] :
% 2.83/1.52 ( ( vimage @ A @ B @ F3 @ ( vimage @ B @ C @ G3 @ X ) )
% 2.83/1.52 = ( vimage @ A @ C @ ( comp @ B @ C @ A @ G3 @ F3 ) @ X ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_154_Sup_OSUP__image,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType,C: $tType,Sup: ( set @ A ) > A,G3: B > A,F3: C > B,A7: set @ C] :
% 2.83/1.52 ( ( Sup @ ( image @ B @ A @ G3 @ ( image @ C @ B @ F3 @ A7 ) ) )
% 2.83/1.52 = ( Sup @ ( image @ C @ A @ ( comp @ B @ A @ C @ G3 @ F3 ) @ A7 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_155_Inf_OINF__image,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType,C: $tType,Inf: ( set @ A ) > A,G3: B > A,F3: C > B,A7: set @ C] :
% 2.83/1.52 ( ( Inf @ ( image @ B @ A @ G3 @ ( image @ C @ B @ F3 @ A7 ) ) )
% 2.83/1.52 = ( Inf @ ( image @ C @ A @ ( comp @ B @ A @ C @ G3 @ F3 ) @ A7 ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_156_Plus__def,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType] :
% 2.83/1.52 ( ( sum_Plus @ A @ B )
% 2.83/1.52 = ( ^ [A9: set @ A,B9: set @ B] : ( sup_sup @ ( set @ ( sum_sum @ A @ B ) ) @ ( image @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ A9 ) @ ( image @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ B9 ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_157_surj__swap,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType] :
% 2.83/1.52 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 2.83/1.52 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_158_UNIV__I,axiom,
% 2.83/1.52 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_159_sup__top__left,axiom,
% 2.83/1.52 ! [A: $tType] :
% 2.83/1.52 ( ( bounded_lattice_top @ A @ ( type2 @ A ) )
% 2.83/1.52 => ! [X: A] :
% 2.83/1.52 ( ( sup_sup @ A @ ( top_top @ A ) @ X )
% 2.83/1.52 = ( top_top @ A ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_160_sup__top__right,axiom,
% 2.83/1.52 ! [A: $tType] :
% 2.83/1.52 ( ( bounded_lattice_top @ A @ ( type2 @ A ) )
% 2.83/1.52 => ! [X: A] :
% 2.83/1.52 ( ( sup_sup @ A @ X @ ( top_top @ A ) )
% 2.83/1.52 = ( top_top @ A ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_161_vimage__UNIV,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType,F3: A > B] :
% 2.83/1.52 ( ( vimage @ A @ B @ F3 @ ( top_top @ ( set @ B ) ) )
% 2.83/1.52 = ( top_top @ ( set @ A ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_162_UNIV__Plus__UNIV,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType] :
% 2.83/1.52 ( ( sum_Plus @ A @ B @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 2.83/1.52 = ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) ) ).
% 2.83/1.52
% 2.83/1.52 thf(fact_163_InrI,axiom,
% 2.83/1.52 ! [B: $tType,A: $tType,B2: A,B8: set @ A,A7: set @ B] :
% 2.83/1.52 ( ( member @ A @ B2 @ B8 )
% 2.83/1.52 => ( member @ ( sum_sum @ B @ A ) @ ( sum_Inr @ A @ B @ B2 ) @ ( sum_Plus @ B @ A @ A7 @ B8 ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_164_InlI,axiom,
% 2.83/1.53 ! [A: $tType,B: $tType,A2: A,A7: set @ A,B8: set @ B] :
% 2.83/1.53 ( ( member @ A @ A2 @ A7 )
% 2.83/1.53 => ( member @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B @ A2 ) @ ( sum_Plus @ A @ B @ A7 @ B8 ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_165_range__eqI,axiom,
% 2.83/1.53 ! [A: $tType,B: $tType,B2: A,F3: B > A,X: B] :
% 2.83/1.53 ( ( B2
% 2.83/1.53 = ( F3 @ X ) )
% 2.83/1.53 => ( member @ A @ B2 @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_166_rangeI,axiom,
% 2.83/1.53 ! [A: $tType,B: $tType,F3: B > A,X: B] : ( member @ A @ ( F3 @ X ) @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_167_surjD,axiom,
% 2.83/1.53 ! [A: $tType,B: $tType,F3: B > A,Y: A] :
% 2.83/1.53 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 2.83/1.53 = ( top_top @ ( set @ A ) ) )
% 2.83/1.53 => ? [X3: B] :
% 2.83/1.53 ( Y
% 2.83/1.53 = ( F3 @ X3 ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_168_surjE,axiom,
% 2.83/1.53 ! [A: $tType,B: $tType,F3: B > A,Y: A] :
% 2.83/1.53 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 2.83/1.53 = ( top_top @ ( set @ A ) ) )
% 2.83/1.53 => ~ ! [X3: B] :
% 2.83/1.53 ( Y
% 2.83/1.53 != ( F3 @ X3 ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_169_surjI,axiom,
% 2.83/1.53 ! [B: $tType,A: $tType,G3: B > A,F3: A > B] :
% 2.83/1.53 ( ! [X3: A] :
% 2.83/1.53 ( ( G3 @ ( F3 @ X3 ) )
% 2.83/1.53 = X3 )
% 2.83/1.53 => ( ( image @ B @ A @ G3 @ ( top_top @ ( set @ B ) ) )
% 2.83/1.53 = ( top_top @ ( set @ A ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_170_surj__def,axiom,
% 2.83/1.53 ! [B: $tType,A: $tType,F3: B > A] :
% 2.83/1.53 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 2.83/1.53 = ( top_top @ ( set @ A ) ) )
% 2.83/1.53 = ( ! [Y4: A] :
% 2.83/1.53 ? [X5: B] :
% 2.83/1.53 ( Y4
% 2.83/1.53 = ( F3 @ X5 ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_171_UNIV__witness,axiom,
% 2.83/1.53 ! [A: $tType] :
% 2.83/1.53 ? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_172_UNIV__eq__I,axiom,
% 2.83/1.53 ! [A: $tType,A7: set @ A] :
% 2.83/1.53 ( ! [X3: A] : ( member @ A @ X3 @ A7 )
% 2.83/1.53 => ( ( top_top @ ( set @ A ) )
% 2.83/1.53 = A7 ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_173_Un__UNIV__left,axiom,
% 2.83/1.53 ! [A: $tType,B8: set @ A] :
% 2.83/1.53 ( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B8 )
% 2.83/1.53 = ( top_top @ ( set @ A ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_174_Un__UNIV__right,axiom,
% 2.83/1.53 ! [A: $tType,A7: set @ A] :
% 2.83/1.53 ( ( sup_sup @ ( set @ A ) @ A7 @ ( top_top @ ( set @ A ) ) )
% 2.83/1.53 = ( top_top @ ( set @ A ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_175_UNIV__sum,axiom,
% 2.83/1.53 ! [A: $tType,B: $tType] :
% 2.83/1.53 ( ( top_top @ ( set @ ( sum_sum @ A @ B ) ) )
% 2.83/1.53 = ( sup_sup @ ( set @ ( sum_sum @ A @ B ) ) @ ( image @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ ( top_top @ ( set @ A ) ) ) @ ( image @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_176_surj__fun__eq,axiom,
% 2.83/1.53 ! [B: $tType,C: $tType,A: $tType,F3: B > A,X8: set @ B,G1: A > C,G22: A > C] :
% 2.83/1.53 ( ( ( image @ B @ A @ F3 @ X8 )
% 2.83/1.53 = ( top_top @ ( set @ A ) ) )
% 2.83/1.53 => ( ! [X3: B] :
% 2.83/1.53 ( ( member @ B @ X3 @ X8 )
% 2.83/1.53 => ( ( comp @ A @ C @ B @ G1 @ F3 @ X3 )
% 2.83/1.53 = ( comp @ A @ C @ B @ G22 @ F3 @ X3 ) ) )
% 2.83/1.53 => ( G1 = G22 ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_177_comp__surj,axiom,
% 2.83/1.53 ! [B: $tType,A: $tType,C: $tType,F3: B > A,G3: A > C] :
% 2.83/1.53 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 2.83/1.53 = ( top_top @ ( set @ A ) ) )
% 2.83/1.53 => ( ( ( image @ A @ C @ G3 @ ( top_top @ ( set @ A ) ) )
% 2.83/1.53 = ( top_top @ ( set @ C ) ) )
% 2.83/1.53 => ( ( image @ B @ C @ ( comp @ A @ C @ B @ G3 @ F3 ) @ ( top_top @ ( set @ B ) ) )
% 2.83/1.53 = ( top_top @ ( set @ C ) ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_178_surj__image__vimage__eq,axiom,
% 2.83/1.53 ! [B: $tType,A: $tType,F3: B > A,A7: set @ A] :
% 2.83/1.53 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 2.83/1.53 = ( top_top @ ( set @ A ) ) )
% 2.83/1.53 => ( ( image @ B @ A @ F3 @ ( vimage @ B @ A @ F3 @ A7 ) )
% 2.83/1.53 = A7 ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_179_PlusE,axiom,
% 2.83/1.53 ! [A: $tType,B: $tType,U: sum_sum @ A @ B,A7: set @ A,B8: set @ B] :
% 2.83/1.53 ( ( member @ ( sum_sum @ A @ B ) @ U @ ( sum_Plus @ A @ B @ A7 @ B8 ) )
% 2.83/1.53 => ( ! [X3: A] :
% 2.83/1.53 ( ( member @ A @ X3 @ A7 )
% 2.83/1.53 => ( U
% 2.83/1.53 != ( sum_Inl @ A @ B @ X3 ) ) )
% 2.83/1.53 => ~ ! [Y3: B] :
% 2.83/1.53 ( ( member @ B @ Y3 @ B8 )
% 2.83/1.53 => ( U
% 2.83/1.53 != ( sum_Inr @ B @ A @ Y3 ) ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_180_Inf_OINF__cong,axiom,
% 2.83/1.53 ! [A: $tType,B: $tType,A7: set @ B,B8: set @ B,C4: B > A,D4: B > A,Inf: ( set @ A ) > A] :
% 2.83/1.53 ( ( A7 = B8 )
% 2.83/1.53 => ( ! [X3: B] :
% 2.83/1.53 ( ( member @ B @ X3 @ B8 )
% 2.83/1.53 => ( ( C4 @ X3 )
% 2.83/1.53 = ( D4 @ X3 ) ) )
% 2.83/1.53 => ( ( Inf @ ( image @ B @ A @ C4 @ A7 ) )
% 2.83/1.53 = ( Inf @ ( image @ B @ A @ D4 @ B8 ) ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_181_Sup_OSUP__cong,axiom,
% 2.83/1.53 ! [A: $tType,B: $tType,A7: set @ B,B8: set @ B,C4: B > A,D4: B > A,Sup: ( set @ A ) > A] :
% 2.83/1.53 ( ( A7 = B8 )
% 2.83/1.53 => ( ! [X3: B] :
% 2.83/1.53 ( ( member @ B @ X3 @ B8 )
% 2.83/1.53 => ( ( C4 @ X3 )
% 2.83/1.53 = ( D4 @ X3 ) ) )
% 2.83/1.53 => ( ( Sup @ ( image @ B @ A @ C4 @ A7 ) )
% 2.83/1.53 = ( Sup @ ( image @ B @ A @ D4 @ B8 ) ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_182_fun_Oset__map,axiom,
% 2.83/1.53 ! [B: $tType,A: $tType,D: $tType,F3: A > B,V: D > A] :
% 2.83/1.53 ( ( image @ D @ B @ ( comp @ A @ B @ D @ F3 @ V ) @ ( top_top @ ( set @ D ) ) )
% 2.83/1.53 = ( image @ A @ B @ F3 @ ( image @ D @ A @ V @ ( top_top @ ( set @ D ) ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_183_fun_Omap__cong,axiom,
% 2.83/1.53 ! [B: $tType,A: $tType,D: $tType,X: D > A,Ya: D > A,F3: A > B,G3: A > B] :
% 2.83/1.53 ( ( X = Ya )
% 2.83/1.53 => ( ! [Z2: A] :
% 2.83/1.53 ( ( member @ A @ Z2 @ ( image @ D @ A @ Ya @ ( top_top @ ( set @ D ) ) ) )
% 2.83/1.53 => ( ( F3 @ Z2 )
% 2.83/1.53 = ( G3 @ Z2 ) ) )
% 2.83/1.53 => ( ( comp @ A @ B @ D @ F3 @ X )
% 2.83/1.53 = ( comp @ A @ B @ D @ G3 @ Ya ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_184_fun_Omap__cong0,axiom,
% 2.83/1.53 ! [B: $tType,A: $tType,D: $tType,X: D > A,F3: A > B,G3: A > B] :
% 2.83/1.53 ( ! [Z2: A] :
% 2.83/1.53 ( ( member @ A @ Z2 @ ( image @ D @ A @ X @ ( top_top @ ( set @ D ) ) ) )
% 2.83/1.53 => ( ( F3 @ Z2 )
% 2.83/1.53 = ( G3 @ Z2 ) ) )
% 2.83/1.53 => ( ( comp @ A @ B @ D @ F3 @ X )
% 2.83/1.53 = ( comp @ A @ B @ D @ G3 @ X ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_185_top__set__def,axiom,
% 2.83/1.53 ! [A: $tType] :
% 2.83/1.53 ( ( top_top @ ( set @ A ) )
% 2.83/1.53 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_186_fun_Omap__comp,axiom,
% 2.83/1.53 ! [B: $tType,C: $tType,A: $tType,D: $tType,G3: B > C,F3: A > B,V: D > A] :
% 2.83/1.53 ( ( comp @ B @ C @ D @ G3 @ ( comp @ A @ B @ D @ F3 @ V ) )
% 2.83/1.53 = ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G3 @ F3 ) @ V ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_187_fun_Oinj__map__strong,axiom,
% 2.83/1.53 ! [B: $tType,A: $tType,D: $tType,X: D > A,Xa: D > A,F3: A > B,Fa: A > B] :
% 2.83/1.53 ( ! [Z2: A,Za: A] :
% 2.83/1.53 ( ( member @ A @ Z2 @ ( image @ D @ A @ X @ ( top_top @ ( set @ D ) ) ) )
% 2.83/1.53 => ( ( member @ A @ Za @ ( image @ D @ A @ Xa @ ( top_top @ ( set @ D ) ) ) )
% 2.83/1.53 => ( ( ( F3 @ Z2 )
% 2.83/1.53 = ( Fa @ Za ) )
% 2.83/1.53 => ( Z2 = Za ) ) ) )
% 2.83/1.53 => ( ( ( comp @ A @ B @ D @ F3 @ X )
% 2.83/1.53 = ( comp @ A @ B @ D @ Fa @ Xa ) )
% 2.83/1.53 => ( X = Xa ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_188_map__prod__surj,axiom,
% 2.83/1.53 ! [A: $tType,C: $tType,D: $tType,B: $tType,F3: A > B,G3: C > D] :
% 2.83/1.53 ( ( ( image @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 2.83/1.53 = ( top_top @ ( set @ B ) ) )
% 2.83/1.53 => ( ( ( image @ C @ D @ G3 @ ( top_top @ ( set @ C ) ) )
% 2.83/1.53 = ( top_top @ ( set @ D ) ) )
% 2.83/1.53 => ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F3 @ G3 ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
% 2.83/1.53 = ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_189_type__copy__set__map0,axiom,
% 2.83/1.53 ! [A: $tType,B: $tType,D: $tType,E: $tType,C: $tType,F: $tType,Rep: A > B,Abs: B > A,S3: B > ( set @ D ),M: C > B,F3: E > D,S4: C > ( set @ E ),G3: F > C] :
% 2.83/1.53 ( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
% 2.83/1.53 => ( ( ( comp @ B @ ( set @ D ) @ C @ S3 @ M )
% 2.83/1.53 = ( comp @ ( set @ E ) @ ( set @ D ) @ C @ ( image @ E @ D @ F3 ) @ S4 ) )
% 2.83/1.53 => ( ( comp @ A @ ( set @ D ) @ F @ ( comp @ B @ ( set @ D ) @ A @ S3 @ Rep ) @ ( comp @ C @ A @ F @ ( comp @ B @ A @ C @ Abs @ M ) @ G3 ) )
% 2.83/1.53 = ( comp @ ( set @ E ) @ ( set @ D ) @ F @ ( image @ E @ D @ F3 ) @ ( comp @ C @ ( set @ E ) @ F @ S4 @ G3 ) ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_190_image__vimage__eq,axiom,
% 2.83/1.53 ! [A: $tType,B: $tType,F3: B > A,A7: set @ A] :
% 2.83/1.53 ( ( image @ B @ A @ F3 @ ( vimage @ B @ A @ F3 @ A7 ) )
% 2.83/1.53 = ( inf_inf @ ( set @ A ) @ A7 @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_191_inf__right__idem,axiom,
% 2.83/1.53 ! [A: $tType] :
% 2.83/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.83/1.53 => ! [X: A,Y: A] :
% 2.83/1.53 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Y )
% 2.83/1.53 = ( inf_inf @ A @ X @ Y ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_192_inf_Oright__idem,axiom,
% 2.83/1.53 ! [A: $tType] :
% 2.83/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.83/1.53 => ! [A2: A,B2: A] :
% 2.83/1.53 ( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ B2 )
% 2.83/1.53 = ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_193_inf__left__idem,axiom,
% 2.83/1.53 ! [A: $tType] :
% 2.83/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.83/1.53 => ! [X: A,Y: A] :
% 2.83/1.53 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y ) )
% 2.83/1.53 = ( inf_inf @ A @ X @ Y ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_194_inf_Oleft__idem,axiom,
% 2.83/1.53 ! [A: $tType] :
% 2.83/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.83/1.53 => ! [A2: A,B2: A] :
% 2.83/1.53 ( ( inf_inf @ A @ A2 @ ( inf_inf @ A @ A2 @ B2 ) )
% 2.83/1.53 = ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_195_inf__idem,axiom,
% 2.83/1.53 ! [A: $tType] :
% 2.83/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.83/1.53 => ! [X: A] :
% 2.83/1.53 ( ( inf_inf @ A @ X @ X )
% 2.83/1.53 = X ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_196_inf_Oidem,axiom,
% 2.83/1.53 ! [A: $tType] :
% 2.83/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.83/1.53 => ! [A2: A] :
% 2.83/1.53 ( ( inf_inf @ A @ A2 @ A2 )
% 2.83/1.53 = A2 ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_197_inf__apply,axiom,
% 2.83/1.53 ! [B: $tType,A: $tType] :
% 2.83/1.53 ( ( semilattice_inf @ B @ ( type2 @ B ) )
% 2.83/1.53 => ( ( inf_inf @ ( A > B ) )
% 2.83/1.53 = ( ^ [F4: A > B,G5: A > B,X5: A] : ( inf_inf @ B @ ( F4 @ X5 ) @ ( G5 @ X5 ) ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_198_Int__iff,axiom,
% 2.83/1.53 ! [A: $tType,C3: A,A7: set @ A,B8: set @ A] :
% 2.83/1.53 ( ( member @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A7 @ B8 ) )
% 2.83/1.53 = ( ( member @ A @ C3 @ A7 )
% 2.83/1.53 & ( member @ A @ C3 @ B8 ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_199_IntI,axiom,
% 2.83/1.53 ! [A: $tType,C3: A,A7: set @ A,B8: set @ A] :
% 2.83/1.53 ( ( member @ A @ C3 @ A7 )
% 2.83/1.53 => ( ( member @ A @ C3 @ B8 )
% 2.83/1.53 => ( member @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A7 @ B8 ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_200_sup__inf__absorb,axiom,
% 2.83/1.53 ! [A: $tType] :
% 2.83/1.53 ( ( lattice @ A @ ( type2 @ A ) )
% 2.83/1.53 => ! [X: A,Y: A] :
% 2.83/1.53 ( ( sup_sup @ A @ X @ ( inf_inf @ A @ X @ Y ) )
% 2.83/1.53 = X ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_201_inf__sup__absorb,axiom,
% 2.83/1.53 ! [A: $tType] :
% 2.83/1.53 ( ( lattice @ A @ ( type2 @ A ) )
% 2.83/1.53 => ! [X: A,Y: A] :
% 2.83/1.53 ( ( inf_inf @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 2.83/1.53 = X ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_202_inf__top_Oright__neutral,axiom,
% 2.83/1.53 ! [A: $tType] :
% 2.83/1.53 ( ( bounde1561333602nf_top @ A @ ( type2 @ A ) )
% 2.83/1.53 => ! [A2: A] :
% 2.83/1.53 ( ( inf_inf @ A @ A2 @ ( top_top @ A ) )
% 2.83/1.53 = A2 ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_203_inf__top_Oleft__neutral,axiom,
% 2.83/1.53 ! [A: $tType] :
% 2.83/1.53 ( ( bounde1561333602nf_top @ A @ ( type2 @ A ) )
% 2.83/1.53 => ! [A2: A] :
% 2.83/1.53 ( ( inf_inf @ A @ ( top_top @ A ) @ A2 )
% 2.83/1.53 = A2 ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_204_inf__eq__top__iff,axiom,
% 2.83/1.53 ! [A: $tType] :
% 2.83/1.53 ( ( bounded_lattice_top @ A @ ( type2 @ A ) )
% 2.83/1.53 => ! [X: A,Y: A] :
% 2.83/1.53 ( ( ( inf_inf @ A @ X @ Y )
% 2.83/1.53 = ( top_top @ A ) )
% 2.83/1.53 = ( ( X
% 2.83/1.53 = ( top_top @ A ) )
% 2.83/1.53 & ( Y
% 2.83/1.53 = ( top_top @ A ) ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_205_Int__UNIV,axiom,
% 2.83/1.53 ! [A: $tType,A7: set @ A,B8: set @ A] :
% 2.83/1.53 ( ( ( inf_inf @ ( set @ A ) @ A7 @ B8 )
% 2.83/1.53 = ( top_top @ ( set @ A ) ) )
% 2.83/1.53 = ( ( A7
% 2.83/1.53 = ( top_top @ ( set @ A ) ) )
% 2.83/1.53 & ( B8
% 2.83/1.53 = ( top_top @ ( set @ A ) ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_206_vimage__Int,axiom,
% 2.83/1.53 ! [A: $tType,B: $tType,F3: A > B,A7: set @ B,B8: set @ B] :
% 2.83/1.53 ( ( vimage @ A @ B @ F3 @ ( inf_inf @ ( set @ B ) @ A7 @ B8 ) )
% 2.83/1.53 = ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ A7 ) @ ( vimage @ A @ B @ F3 @ B8 ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_207_map__prod__simp,axiom,
% 2.83/1.53 ! [C: $tType,A: $tType,B: $tType,D: $tType,F3: C > A,G3: D > B,A2: C,B2: D] :
% 2.83/1.53 ( ( product_map_prod @ C @ A @ D @ B @ F3 @ G3 @ ( product_Pair @ C @ D @ A2 @ B2 ) )
% 2.83/1.53 = ( product_Pair @ A @ B @ ( F3 @ A2 ) @ ( G3 @ B2 ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_208_map__prod__imageI,axiom,
% 2.83/1.53 ! [D: $tType,C: $tType,B: $tType,A: $tType,A2: A,B2: B,R2: set @ ( product_prod @ A @ B ),F3: A > C,G3: B > D] :
% 2.83/1.53 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R2 )
% 2.83/1.53 => ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F3 @ A2 ) @ ( G3 @ B2 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F3 @ G3 ) @ R2 ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_209_type__copy__wit,axiom,
% 2.83/1.53 ! [A: $tType,C: $tType,B: $tType,Rep: A > B,Abs: B > A,X: C,S3: B > ( set @ C ),Y: B] :
% 2.83/1.53 ( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
% 2.83/1.53 => ( ( member @ C @ X @ ( comp @ B @ ( set @ C ) @ A @ S3 @ Rep @ ( Abs @ Y ) ) )
% 2.83/1.53 => ( member @ C @ X @ ( S3 @ Y ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_210_type__copy__map__comp0,axiom,
% 2.83/1.53 ! [F: $tType,D: $tType,B: $tType,A: $tType,C: $tType,E: $tType,Rep: A > B,Abs: B > A,M: C > D,M1: B > D,M2: C > B,F3: D > F,G3: E > C] :
% 2.83/1.53 ( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
% 2.83/1.53 => ( ( M
% 2.83/1.53 = ( comp @ B @ D @ C @ M1 @ M2 ) )
% 2.83/1.53 => ( ( comp @ C @ F @ E @ ( comp @ D @ F @ C @ F3 @ M ) @ G3 )
% 2.83/1.53 = ( comp @ A @ F @ E @ ( comp @ B @ F @ A @ ( comp @ D @ F @ B @ F3 @ M1 ) @ Rep ) @ ( comp @ C @ A @ E @ ( comp @ B @ A @ C @ Abs @ M2 ) @ G3 ) ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_211_inf__top__right,axiom,
% 2.83/1.53 ! [A: $tType] :
% 2.83/1.53 ( ( bounded_lattice_top @ A @ ( type2 @ A ) )
% 2.83/1.53 => ! [X: A] :
% 2.83/1.53 ( ( inf_inf @ A @ X @ ( top_top @ A ) )
% 2.83/1.53 = X ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_212_inf__top__left,axiom,
% 2.83/1.53 ! [A: $tType] :
% 2.83/1.53 ( ( bounded_lattice_top @ A @ ( type2 @ A ) )
% 2.83/1.53 => ! [X: A] :
% 2.83/1.53 ( ( inf_inf @ A @ ( top_top @ A ) @ X )
% 2.83/1.53 = X ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_213_Int__UNIV__right,axiom,
% 2.83/1.53 ! [A: $tType,A7: set @ A] :
% 2.83/1.53 ( ( inf_inf @ ( set @ A ) @ A7 @ ( top_top @ ( set @ A ) ) )
% 2.83/1.53 = A7 ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_214_Int__UNIV__left,axiom,
% 2.83/1.53 ! [A: $tType,B8: set @ A] :
% 2.83/1.53 ( ( inf_inf @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B8 )
% 2.83/1.53 = B8 ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_215_type__copy__obj__one__point__absE,axiom,
% 2.83/1.53 ! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,S: A] :
% 2.83/1.53 ( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
% 2.83/1.53 => ~ ! [X3: B] :
% 2.83/1.53 ( S
% 2.83/1.53 != ( Abs @ X3 ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_216_top__conj_I1_J,axiom,
% 2.83/1.53 ! [A: $tType,X: A,P: $o] :
% 2.83/1.53 ( ( ( top_top @ ( A > $o ) @ X )
% 2.83/1.53 & P )
% 2.83/1.53 = P ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_217_top__conj_I2_J,axiom,
% 2.83/1.53 ! [A: $tType,P: $o,X: A] :
% 2.83/1.53 ( ( P
% 2.83/1.53 & ( top_top @ ( A > $o ) @ X ) )
% 2.83/1.53 = P ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_218_prod_Omap__comp,axiom,
% 2.83/1.53 ! [D: $tType,F: $tType,E: $tType,C: $tType,B: $tType,A: $tType,G1: C > E,G22: D > F,F1: A > C,F22: B > D,V: product_prod @ A @ B] :
% 2.83/1.53 ( ( product_map_prod @ C @ E @ D @ F @ G1 @ G22 @ ( product_map_prod @ A @ C @ B @ D @ F1 @ F22 @ V ) )
% 2.83/1.53 = ( product_map_prod @ A @ E @ B @ F @ ( comp @ C @ E @ A @ G1 @ F1 ) @ ( comp @ D @ F @ B @ G22 @ F22 ) @ V ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_219_map__prod_Ocompositionality,axiom,
% 2.83/1.53 ! [D: $tType,F: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F3: C > E,G3: D > F,H: A > C,I: B > D,Prod: product_prod @ A @ B] :
% 2.83/1.53 ( ( product_map_prod @ C @ E @ D @ F @ F3 @ G3 @ ( product_map_prod @ A @ C @ B @ D @ H @ I @ Prod ) )
% 2.83/1.53 = ( product_map_prod @ A @ E @ B @ F @ ( comp @ C @ E @ A @ F3 @ H ) @ ( comp @ D @ F @ B @ G3 @ I ) @ Prod ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_220_map__prod__compose,axiom,
% 2.83/1.53 ! [D: $tType,C: $tType,A: $tType,E: $tType,F: $tType,B: $tType,F1: E > C,F22: A > E,G1: F > D,G22: B > F] :
% 2.83/1.53 ( ( product_map_prod @ A @ C @ B @ D @ ( comp @ E @ C @ A @ F1 @ F22 ) @ ( comp @ F @ D @ B @ G1 @ G22 ) )
% 2.83/1.53 = ( comp @ ( product_prod @ E @ F ) @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ E @ C @ F @ D @ F1 @ G1 ) @ ( product_map_prod @ A @ E @ B @ F @ F22 @ G22 ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_221_map__prod_Ocomp,axiom,
% 2.83/1.53 ! [A: $tType,C: $tType,E: $tType,F: $tType,D: $tType,B: $tType,F3: C > E,G3: D > F,H: A > C,I: B > D] :
% 2.83/1.53 ( ( comp @ ( product_prod @ C @ D ) @ ( product_prod @ E @ F ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ E @ D @ F @ F3 @ G3 ) @ ( product_map_prod @ A @ C @ B @ D @ H @ I ) )
% 2.83/1.53 = ( product_map_prod @ A @ E @ B @ F @ ( comp @ C @ E @ A @ F3 @ H ) @ ( comp @ D @ F @ B @ G3 @ I ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_222_inf__left__commute,axiom,
% 2.83/1.53 ! [A: $tType] :
% 2.83/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.83/1.53 => ! [X: A,Y: A,Z: A] :
% 2.83/1.53 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
% 2.83/1.53 = ( inf_inf @ A @ Y @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_223_inf_Oleft__commute,axiom,
% 2.83/1.53 ! [A: $tType] :
% 2.83/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.83/1.53 => ! [B2: A,A2: A,C3: A] :
% 2.83/1.53 ( ( inf_inf @ A @ B2 @ ( inf_inf @ A @ A2 @ C3 ) )
% 2.83/1.53 = ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B2 @ C3 ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_224_inf__commute,axiom,
% 2.83/1.53 ! [A: $tType] :
% 2.83/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.83/1.53 => ( ( inf_inf @ A )
% 2.83/1.53 = ( ^ [X5: A,Y4: A] : ( inf_inf @ A @ Y4 @ X5 ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_225_inf_Ocommute,axiom,
% 2.83/1.53 ! [A: $tType] :
% 2.83/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.83/1.53 => ( ( inf_inf @ A )
% 2.83/1.53 = ( ^ [A8: A,B7: A] : ( inf_inf @ A @ B7 @ A8 ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_226_inf__assoc,axiom,
% 2.83/1.53 ! [A: $tType] :
% 2.83/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.83/1.53 => ! [X: A,Y: A,Z: A] :
% 2.83/1.53 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Z )
% 2.83/1.53 = ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_227_inf_Oassoc,axiom,
% 2.83/1.53 ! [A: $tType] :
% 2.83/1.53 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 2.83/1.53 => ! [A2: A,B2: A,C3: A] :
% 2.83/1.53 ( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C3 )
% 2.83/1.53 = ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B2 @ C3 ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_228_Int__left__commute,axiom,
% 2.83/1.53 ! [A: $tType,A7: set @ A,B8: set @ A,C4: set @ A] :
% 2.83/1.53 ( ( inf_inf @ ( set @ A ) @ A7 @ ( inf_inf @ ( set @ A ) @ B8 @ C4 ) )
% 2.83/1.53 = ( inf_inf @ ( set @ A ) @ B8 @ ( inf_inf @ ( set @ A ) @ A7 @ C4 ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_229_inf__fun__def,axiom,
% 2.83/1.53 ! [B: $tType,A: $tType] :
% 2.83/1.53 ( ( semilattice_inf @ B @ ( type2 @ B ) )
% 2.83/1.53 => ( ( inf_inf @ ( A > B ) )
% 2.83/1.53 = ( ^ [F4: A > B,G5: A > B,X5: A] : ( inf_inf @ B @ ( F4 @ X5 ) @ ( G5 @ X5 ) ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_230_Int__left__absorb,axiom,
% 2.83/1.53 ! [A: $tType,A7: set @ A,B8: set @ A] :
% 2.83/1.53 ( ( inf_inf @ ( set @ A ) @ A7 @ ( inf_inf @ ( set @ A ) @ A7 @ B8 ) )
% 2.83/1.53 = ( inf_inf @ ( set @ A ) @ A7 @ B8 ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_231_Int__commute,axiom,
% 2.83/1.53 ! [A: $tType] :
% 2.83/1.53 ( ( inf_inf @ ( set @ A ) )
% 2.83/1.53 = ( ^ [A9: set @ A,B9: set @ A] : ( inf_inf @ ( set @ A ) @ B9 @ A9 ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_232_Int__absorb,axiom,
% 2.83/1.53 ! [A: $tType,A7: set @ A] :
% 2.83/1.53 ( ( inf_inf @ ( set @ A ) @ A7 @ A7 )
% 2.83/1.53 = A7 ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_233_Int__assoc,axiom,
% 2.83/1.53 ! [A: $tType,A7: set @ A,B8: set @ A,C4: set @ A] :
% 2.83/1.53 ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A7 @ B8 ) @ C4 )
% 2.83/1.53 = ( inf_inf @ ( set @ A ) @ A7 @ ( inf_inf @ ( set @ A ) @ B8 @ C4 ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_234_IntD2,axiom,
% 2.83/1.53 ! [A: $tType,C3: A,A7: set @ A,B8: set @ A] :
% 2.83/1.53 ( ( member @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A7 @ B8 ) )
% 2.83/1.53 => ( member @ A @ C3 @ B8 ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_235_IntD1,axiom,
% 2.83/1.53 ! [A: $tType,C3: A,A7: set @ A,B8: set @ A] :
% 2.83/1.53 ( ( member @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A7 @ B8 ) )
% 2.83/1.53 => ( member @ A @ C3 @ A7 ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_236_IntE,axiom,
% 2.83/1.53 ! [A: $tType,C3: A,A7: set @ A,B8: set @ A] :
% 2.83/1.53 ( ( member @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A7 @ B8 ) )
% 2.83/1.53 => ~ ( ( member @ A @ C3 @ A7 )
% 2.83/1.53 => ~ ( member @ A @ C3 @ B8 ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_237_inf__sup__aci_I1_J,axiom,
% 2.83/1.53 ! [A: $tType] :
% 2.83/1.53 ( ( lattice @ A @ ( type2 @ A ) )
% 2.83/1.53 => ( ( inf_inf @ A )
% 2.83/1.53 = ( ^ [X5: A,Y4: A] : ( inf_inf @ A @ Y4 @ X5 ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_238_inf__sup__aci_I2_J,axiom,
% 2.83/1.53 ! [A: $tType] :
% 2.83/1.53 ( ( lattice @ A @ ( type2 @ A ) )
% 2.83/1.53 => ! [X: A,Y: A,Z: A] :
% 2.83/1.53 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Z )
% 2.83/1.53 = ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_239_inf__sup__aci_I3_J,axiom,
% 2.83/1.53 ! [A: $tType] :
% 2.83/1.53 ( ( lattice @ A @ ( type2 @ A ) )
% 2.83/1.53 => ! [X: A,Y: A,Z: A] :
% 2.83/1.53 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
% 2.83/1.53 = ( inf_inf @ A @ Y @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_240_inf__sup__aci_I4_J,axiom,
% 2.83/1.53 ! [A: $tType] :
% 2.83/1.53 ( ( lattice @ A @ ( type2 @ A ) )
% 2.83/1.53 => ! [X: A,Y: A] :
% 2.83/1.53 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y ) )
% 2.83/1.53 = ( inf_inf @ A @ X @ Y ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_241_vimage__inter__cong,axiom,
% 2.83/1.53 ! [B: $tType,A: $tType,S3: set @ A,F3: A > B,G3: A > B,Y: set @ B] :
% 2.83/1.53 ( ! [W: A] :
% 2.83/1.53 ( ( member @ A @ W @ S3 )
% 2.83/1.53 => ( ( F3 @ W )
% 2.83/1.53 = ( G3 @ W ) ) )
% 2.83/1.53 => ( ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ Y ) @ S3 )
% 2.83/1.53 = ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ G3 @ Y ) @ S3 ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_242_sup__inf__distrib2,axiom,
% 2.83/1.53 ! [A: $tType] :
% 2.83/1.53 ( ( distrib_lattice @ A @ ( type2 @ A ) )
% 2.83/1.53 => ! [Y: A,Z: A,X: A] :
% 2.83/1.53 ( ( sup_sup @ A @ ( inf_inf @ A @ Y @ Z ) @ X )
% 2.83/1.53 = ( inf_inf @ A @ ( sup_sup @ A @ Y @ X ) @ ( sup_sup @ A @ Z @ X ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_243_sup__inf__distrib1,axiom,
% 2.83/1.53 ! [A: $tType] :
% 2.83/1.53 ( ( distrib_lattice @ A @ ( type2 @ A ) )
% 2.83/1.53 => ! [X: A,Y: A,Z: A] :
% 2.83/1.53 ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
% 2.83/1.53 = ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_244_inf__sup__distrib2,axiom,
% 2.83/1.53 ! [A: $tType] :
% 2.83/1.53 ( ( distrib_lattice @ A @ ( type2 @ A ) )
% 2.83/1.53 => ! [Y: A,Z: A,X: A] :
% 2.83/1.53 ( ( inf_inf @ A @ ( sup_sup @ A @ Y @ Z ) @ X )
% 2.83/1.53 = ( sup_sup @ A @ ( inf_inf @ A @ Y @ X ) @ ( inf_inf @ A @ Z @ X ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_245_inf__sup__distrib1,axiom,
% 2.83/1.53 ! [A: $tType] :
% 2.83/1.53 ( ( distrib_lattice @ A @ ( type2 @ A ) )
% 2.83/1.53 => ! [X: A,Y: A,Z: A] :
% 2.83/1.53 ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
% 2.83/1.53 = ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_246_distrib__imp2,axiom,
% 2.83/1.53 ! [A: $tType] :
% 2.83/1.53 ( ( lattice @ A @ ( type2 @ A ) )
% 2.83/1.53 => ! [X: A,Y: A,Z: A] :
% 2.83/1.53 ( ! [X3: A,Y3: A,Z2: A] :
% 2.83/1.53 ( ( sup_sup @ A @ X3 @ ( inf_inf @ A @ Y3 @ Z2 ) )
% 2.83/1.53 = ( inf_inf @ A @ ( sup_sup @ A @ X3 @ Y3 ) @ ( sup_sup @ A @ X3 @ Z2 ) ) )
% 2.83/1.53 => ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
% 2.83/1.53 = ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z ) ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_247_distrib__imp1,axiom,
% 2.83/1.53 ! [A: $tType] :
% 2.83/1.53 ( ( lattice @ A @ ( type2 @ A ) )
% 2.83/1.53 => ! [X: A,Y: A,Z: A] :
% 2.83/1.53 ( ! [X3: A,Y3: A,Z2: A] :
% 2.83/1.53 ( ( inf_inf @ A @ X3 @ ( sup_sup @ A @ Y3 @ Z2 ) )
% 2.83/1.53 = ( sup_sup @ A @ ( inf_inf @ A @ X3 @ Y3 ) @ ( inf_inf @ A @ X3 @ Z2 ) ) )
% 2.83/1.53 => ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
% 2.83/1.53 = ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z ) ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_248_Un__Int__distrib2,axiom,
% 2.83/1.53 ! [A: $tType,B8: set @ A,C4: set @ A,A7: set @ A] :
% 2.83/1.53 ( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B8 @ C4 ) @ A7 )
% 2.83/1.53 = ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B8 @ A7 ) @ ( sup_sup @ ( set @ A ) @ C4 @ A7 ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_249_Int__Un__distrib2,axiom,
% 2.83/1.53 ! [A: $tType,B8: set @ A,C4: set @ A,A7: set @ A] :
% 2.83/1.53 ( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B8 @ C4 ) @ A7 )
% 2.83/1.53 = ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B8 @ A7 ) @ ( inf_inf @ ( set @ A ) @ C4 @ A7 ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_250_Un__Int__distrib,axiom,
% 2.83/1.53 ! [A: $tType,A7: set @ A,B8: set @ A,C4: set @ A] :
% 2.83/1.53 ( ( sup_sup @ ( set @ A ) @ A7 @ ( inf_inf @ ( set @ A ) @ B8 @ C4 ) )
% 2.83/1.53 = ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A7 @ B8 ) @ ( sup_sup @ ( set @ A ) @ A7 @ C4 ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_251_Int__Un__distrib,axiom,
% 2.83/1.53 ! [A: $tType,A7: set @ A,B8: set @ A,C4: set @ A] :
% 2.83/1.53 ( ( inf_inf @ ( set @ A ) @ A7 @ ( sup_sup @ ( set @ A ) @ B8 @ C4 ) )
% 2.83/1.53 = ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A7 @ B8 ) @ ( inf_inf @ ( set @ A ) @ A7 @ C4 ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_252_Un__Int__crazy,axiom,
% 2.83/1.53 ! [A: $tType,A7: set @ A,B8: set @ A,C4: set @ A] :
% 2.83/1.53 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A7 @ B8 ) @ ( inf_inf @ ( set @ A ) @ B8 @ C4 ) ) @ ( inf_inf @ ( set @ A ) @ C4 @ A7 ) )
% 2.83/1.53 = ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A7 @ B8 ) @ ( sup_sup @ ( set @ A ) @ B8 @ C4 ) ) @ ( sup_sup @ ( set @ A ) @ C4 @ A7 ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_253_type__copy__map__comp0__undo,axiom,
% 2.83/1.53 ! [E: $tType,A: $tType,C: $tType,B: $tType,D: $tType,F: $tType,Rep: A > B,Abs: B > A,Rep2: C > D,Abs2: D > C,Rep3: E > F,Abs3: F > E,M: F > D,M1: B > D,M2: F > B] :
% 2.83/1.53 ( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
% 2.83/1.53 => ( ( type_definition @ C @ D @ Rep2 @ Abs2 @ ( top_top @ ( set @ D ) ) )
% 2.83/1.53 => ( ( type_definition @ E @ F @ Rep3 @ Abs3 @ ( top_top @ ( set @ F ) ) )
% 2.83/1.53 => ( ( ( comp @ F @ C @ E @ ( comp @ D @ C @ F @ Abs2 @ M ) @ Rep3 )
% 2.83/1.53 = ( comp @ A @ C @ E @ ( comp @ B @ C @ A @ ( comp @ D @ C @ B @ Abs2 @ M1 ) @ Rep ) @ ( comp @ F @ A @ E @ ( comp @ B @ A @ F @ Abs @ M2 ) @ Rep3 ) ) )
% 2.83/1.53 => ( ( comp @ B @ D @ F @ M1 @ M2 )
% 2.83/1.53 = M ) ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_254_prod__fun__imageE,axiom,
% 2.83/1.53 ! [B: $tType,A: $tType,D: $tType,C: $tType,C3: product_prod @ A @ B,F3: C > A,G3: D > B,R2: set @ ( product_prod @ C @ D )] :
% 2.83/1.53 ( ( member @ ( product_prod @ A @ B ) @ C3 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F3 @ G3 ) @ R2 ) )
% 2.83/1.53 => ~ ! [X3: C,Y3: D] :
% 2.83/1.53 ( ( C3
% 2.83/1.53 = ( product_Pair @ A @ B @ ( F3 @ X3 ) @ ( G3 @ Y3 ) ) )
% 2.83/1.53 => ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X3 @ Y3 ) @ R2 ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(fact_255_type__copy__map__cong0,axiom,
% 2.83/1.53 ! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M: B > A,G3: C > B,X: C,N: D > A,H: C > D,F3: A > E] :
% 2.83/1.53 ( ( ( M @ ( G3 @ X ) )
% 2.83/1.53 = ( N @ ( H @ X ) ) )
% 2.83/1.53 => ( ( comp @ B @ E @ C @ ( comp @ A @ E @ B @ F3 @ M ) @ G3 @ X )
% 2.83/1.53 = ( comp @ D @ E @ C @ ( comp @ A @ E @ D @ F3 @ N ) @ H @ X ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
% 2.83/1.53 bounded_lattice @ $o @ ( type2 @ $o ) ).
% 2.83/1.53
% 2.83/1.53 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
% 2.83/1.53 ! [A5: $tType] : ( bounded_lattice @ ( set @ A5 ) @ ( type2 @ ( set @ A5 ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
% 2.83/1.53 ! [A5: $tType,A10: $tType] :
% 2.83/1.53 ( ( bounded_lattice @ A10 @ ( type2 @ A10 ) )
% 2.83/1.53 => ( bounded_lattice @ ( A5 > A10 ) @ ( type2 @ ( A5 > A10 ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
% 2.83/1.53 ! [A5: $tType,A10: $tType] :
% 2.83/1.53 ( ( bounded_lattice @ A10 @ ( type2 @ A10 ) )
% 2.83/1.53 => ( bounde1561333602nf_top @ ( A5 > A10 ) @ ( type2 @ ( A5 > A10 ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
% 2.83/1.53 ! [A5: $tType,A10: $tType] :
% 2.83/1.53 ( ( bounded_lattice @ A10 @ ( type2 @ A10 ) )
% 2.83/1.53 => ( bounded_lattice_top @ ( A5 > A10 ) @ ( type2 @ ( A5 > A10 ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 2.83/1.53 ! [A5: $tType,A10: $tType] :
% 2.83/1.53 ( ( semilattice_sup @ A10 @ ( type2 @ A10 ) )
% 2.83/1.53 => ( semilattice_sup @ ( A5 > A10 ) @ ( type2 @ ( A5 > A10 ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
% 2.83/1.53 ! [A5: $tType,A10: $tType] :
% 2.83/1.53 ( ( semilattice_inf @ A10 @ ( type2 @ A10 ) )
% 2.83/1.53 => ( semilattice_inf @ ( A5 > A10 ) @ ( type2 @ ( A5 > A10 ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(tcon_fun___Lattices_Odistrib__lattice,axiom,
% 2.83/1.53 ! [A5: $tType,A10: $tType] :
% 2.83/1.53 ( ( distrib_lattice @ A10 @ ( type2 @ A10 ) )
% 2.83/1.53 => ( distrib_lattice @ ( A5 > A10 ) @ ( type2 @ ( A5 > A10 ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(tcon_fun___Lattices_Olattice,axiom,
% 2.83/1.53 ! [A5: $tType,A10: $tType] :
% 2.83/1.53 ( ( lattice @ A10 @ ( type2 @ A10 ) )
% 2.83/1.53 => ( lattice @ ( A5 > A10 ) @ ( type2 @ ( A5 > A10 ) ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_3,axiom,
% 2.83/1.53 ! [A5: $tType] : ( bounde1561333602nf_top @ ( set @ A5 ) @ ( type2 @ ( set @ A5 ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_4,axiom,
% 2.83/1.53 ! [A5: $tType] : ( bounded_lattice_top @ ( set @ A5 ) @ ( type2 @ ( set @ A5 ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_5,axiom,
% 2.83/1.53 ! [A5: $tType] : ( semilattice_sup @ ( set @ A5 ) @ ( type2 @ ( set @ A5 ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_6,axiom,
% 2.83/1.53 ! [A5: $tType] : ( semilattice_inf @ ( set @ A5 ) @ ( type2 @ ( set @ A5 ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(tcon_Set_Oset___Lattices_Odistrib__lattice_7,axiom,
% 2.83/1.53 ! [A5: $tType] : ( distrib_lattice @ ( set @ A5 ) @ ( type2 @ ( set @ A5 ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(tcon_Set_Oset___Lattices_Olattice_8,axiom,
% 2.83/1.53 ! [A5: $tType] : ( lattice @ ( set @ A5 ) @ ( type2 @ ( set @ A5 ) ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_9,axiom,
% 2.83/1.53 bounde1561333602nf_top @ $o @ ( type2 @ $o ) ).
% 2.83/1.53
% 2.83/1.53 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_10,axiom,
% 2.83/1.53 bounded_lattice_top @ $o @ ( type2 @ $o ) ).
% 2.83/1.53
% 2.83/1.53 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_11,axiom,
% 2.83/1.53 semilattice_sup @ $o @ ( type2 @ $o ) ).
% 2.83/1.53
% 2.83/1.53 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_12,axiom,
% 2.83/1.53 semilattice_inf @ $o @ ( type2 @ $o ) ).
% 2.83/1.53
% 2.83/1.53 thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_13,axiom,
% 2.83/1.53 distrib_lattice @ $o @ ( type2 @ $o ) ).
% 2.83/1.53
% 2.83/1.53 thf(tcon_HOL_Obool___Lattices_Olattice_14,axiom,
% 2.83/1.53 lattice @ $o @ ( type2 @ $o ) ).
% 2.83/1.53
% 2.83/1.53 thf(help_If_3_1_T,axiom,
% 2.83/1.53 ! [P: $o] :
% 2.83/1.53 ( ( P = $true )
% 2.83/1.53 | ( P = $false ) ) ).
% 2.83/1.53
% 2.83/1.53 thf(help_If_2_1_T,axiom,
% 2.83/1.53 ! [A: $tType,X: A,Y: A] :
% 2.83/1.53 ( ( if @ A @ $false @ X @ Y )
% 2.83/1.53 = Y ) ).
% 2.83/1.53
% 2.83/1.53 thf(help_If_1_1_T,axiom,
% 2.83/1.53 ! [A: $tType,X: A,Y: A] :
% 2.83/1.53 ( ( if @ A @ $true @ X @ Y )
% 2.83/1.53 = X ) ).
% 2.83/1.53
% 2.83/1.53 thf(conj_0,hypothesis,
% 2.83/1.53 $true ).
% 2.83/1.53
% 2.83/1.53 thf(conj_1,hypothesis,
% 2.83/1.53 $true ).
% 2.83/1.53
% 2.83/1.53 thf(conj_2,hypothesis,
% 2.83/1.53 $true ).
% 2.83/1.53
% 2.83/1.53 thf(conj_3,hypothesis,
% 2.83/1.53 $true ).
% 2.83/1.53
% 2.83/1.53 thf(conj_4,conjecture,
% 2.83/1.53 ? [Tr14: dtree,Tr24: dtree,Tr33: dtree] :
% 2.83/1.53 ( ( trA
% 2.83/1.53 = ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr14 @ Tr24 ) ) @ Tr33 ) ) )
% 2.83/1.53 & ( ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ tr1 @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ tr2 @ tr3 ) ) ) )
% 2.83/1.53 = ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr14 @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr24 @ Tr33 ) ) ) ) ) ) ).
% 2.83/1.53 ------------------------
% 2.83/1.55 % (10431)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_10309 for (2999ds/18Mi)
% 2.83/1.55 % (10425)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_10309 for (2999ds/183Mi)
% 2.83/1.55 % (10429)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_10309 for (2999ds/2Mi)
% 2.83/1.55 % (10428)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_10309 for (2999ds/2Mi)
% 2.83/1.55 % (10428)Instruction limit reached!
% 2.83/1.55 % (10428)------------------------------
% 2.83/1.55 % (10428)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.55 % (10428)Termination reason: Unknown
% 2.83/1.55 % (10428)Termination phase: shuffling
% 2.83/1.55
% 2.83/1.55 % (10428)Memory used [KB]: 1407
% 2.83/1.55 % (10429)Instruction limit reached!
% 2.83/1.55 % (10429)------------------------------
% 2.83/1.55 % (10429)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.55 % (10429)Termination reason: Unknown
% 2.83/1.55 % (10429)Termination phase: shuffling
% 2.83/1.55
% 2.83/1.55 % (10429)Memory used [KB]: 1407
% 2.83/1.55 % (10429)Time elapsed: 0.002 s
% 2.83/1.55 % (10429)Instructions burned: 3 (million)
% 2.83/1.55 % (10429)------------------------------
% 2.83/1.55 % (10429)------------------------------
% 2.83/1.55 % (10428)Time elapsed: 0.002 s
% 2.83/1.55 % (10428)Instructions burned: 3 (million)
% 2.83/1.55 % (10428)------------------------------
% 2.83/1.55 % (10428)------------------------------
% 2.83/1.55 % (10431)Instruction limit reached!
% 2.83/1.55 % (10431)------------------------------
% 2.83/1.55 % (10431)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.55 % (10431)Termination reason: Unknown
% 2.83/1.55 % (10431)Termination phase: shuffling
% 2.83/1.55
% 2.83/1.55 % (10431)Memory used [KB]: 1791
% 2.83/1.55 % (10431)Time elapsed: 0.007 s
% 2.83/1.55 % (10431)Instructions burned: 18 (million)
% 2.83/1.55 % (10431)------------------------------
% 2.83/1.55 % (10431)------------------------------
% 2.83/1.55 % (10430)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_10309 for (2999ds/275Mi)
% 2.83/1.55 % (10426)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_10309 for (2999ds/4Mi)
% 2.83/1.55 % (10427)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_10309 for (2999ds/27Mi)
% 2.83/1.56 % (10426)Instruction limit reached!
% 2.83/1.56 % (10426)------------------------------
% 2.83/1.56 % (10426)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.56 % (10426)Termination reason: Unknown
% 2.83/1.56 % (10426)Termination phase: shuffling
% 2.83/1.56
% 2.83/1.56 % (10426)Memory used [KB]: 1407
% 2.83/1.56 % (10426)Time elapsed: 0.004 s
% 2.83/1.56 % (10426)Instructions burned: 4 (million)
% 2.83/1.56 % (10426)------------------------------
% 2.83/1.56 % (10426)------------------------------
% 2.83/1.56 % (10432)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_10309 for (2999ds/3Mi)
% 2.83/1.56 % (10433)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_10309 for (2999ds/37Mi)
% 2.83/1.56 % (10432)Instruction limit reached!
% 2.83/1.56 % (10432)------------------------------
% 2.83/1.56 % (10432)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.56 % (10432)Termination reason: Unknown
% 2.83/1.56 % (10432)Termination phase: shuffling
% 2.83/1.56
% 2.83/1.56 % (10432)Memory used [KB]: 1407
% 2.83/1.56 % (10432)Time elapsed: 0.003 s
% 2.83/1.56 % (10432)Instructions burned: 4 (million)
% 2.83/1.56 % (10432)------------------------------
% 2.83/1.56 % (10432)------------------------------
% 2.83/1.56 % (10434)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_10309 for (2999ds/15Mi)
% 2.83/1.57 % (10434)Instruction limit reached!
% 2.83/1.57 % (10434)------------------------------
% 2.83/1.57 % (10434)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.57 % (10434)Termination reason: Unknown
% 2.83/1.57 % (10434)Termination phase: shuffling
% 2.83/1.57
% 2.83/1.57 % (10434)Memory used [KB]: 1791
% 2.83/1.57 % (10434)Time elapsed: 0.006 s
% 2.83/1.57 % (10434)Instructions burned: 15 (million)
% 2.83/1.57 % (10434)------------------------------
% 2.83/1.57 % (10434)------------------------------
% 2.83/1.57 % (10436)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_10309 for (2999ds/1041Mi)
% 2.83/1.57 % (10433)Instruction limit reached!
% 2.83/1.57 % (10433)------------------------------
% 2.83/1.57 % (10433)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.57 % (10433)Termination reason: Unknown
% 2.83/1.57 % (10433)Termination phase: shuffling
% 2.83/1.57
% 2.83/1.57 % (10433)Memory used [KB]: 2174
% 2.83/1.57 % (10433)Time elapsed: 0.013 s
% 2.83/1.57 % (10433)Instructions burned: 37 (million)
% 2.83/1.57 % (10433)------------------------------
% 2.83/1.57 % (10433)------------------------------
% 2.83/1.57 % (10427)Instruction limit reached!
% 2.83/1.57 % (10427)------------------------------
% 2.83/1.57 % (10427)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.57 % (10427)Termination reason: Unknown
% 2.83/1.57 % (10427)Termination phase: Property scanning
% 2.83/1.57
% 2.83/1.57 % (10427)Memory used [KB]: 1791
% 2.83/1.57 % (10427)Time elapsed: 0.019 s
% 2.83/1.57 % (10427)Instructions burned: 28 (million)
% 2.83/1.57 % (10427)------------------------------
% 2.83/1.57 % (10427)------------------------------
% 2.83/1.57 % (10435)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_10309 for (2999ds/3Mi)
% 2.83/1.57 % (10435)Instruction limit reached!
% 2.83/1.57 % (10435)------------------------------
% 2.83/1.57 % (10435)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.57 % (10435)Termination reason: Unknown
% 2.83/1.57 % (10435)Termination phase: shuffling
% 2.83/1.58
% 2.83/1.58 % (10435)Memory used [KB]: 1407
% 2.83/1.58 % (10435)Time elapsed: 0.004 s
% 2.83/1.58 % (10435)Instructions burned: 4 (million)
% 2.83/1.58 % (10435)------------------------------
% 2.83/1.58 % (10435)------------------------------
% 2.83/1.58 % (10437)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_10309 for (2999ds/7Mi)
% 2.83/1.58 % (10437)Instruction limit reached!
% 2.83/1.58 % (10437)------------------------------
% 2.83/1.58 % (10437)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.58 % (10437)Termination reason: Unknown
% 2.83/1.58 % (10437)Termination phase: shuffling
% 2.83/1.58
% 2.83/1.58 % (10437)Memory used [KB]: 1535
% 2.83/1.58 % (10437)Time elapsed: 0.003 s
% 2.83/1.58 % (10437)Instructions burned: 7 (million)
% 2.83/1.58 % (10437)------------------------------
% 2.83/1.58 % (10437)------------------------------
% 2.83/1.58 % (10438)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_10309 for (2999ds/16Mi)
% 2.83/1.59 % (10438)Instruction limit reached!
% 2.83/1.59 % (10438)------------------------------
% 2.83/1.59 % (10438)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.59 % (10438)Termination reason: Unknown
% 2.83/1.59 % (10438)Termination phase: shuffling
% 2.83/1.59
% 2.83/1.59 % (10438)Memory used [KB]: 1791
% 2.83/1.59 % (10438)Time elapsed: 0.008 s
% 2.83/1.59 % (10438)Instructions burned: 17 (million)
% 2.83/1.59 % (10438)------------------------------
% 2.83/1.59 % (10438)------------------------------
% 2.83/1.59 % (10441)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_10309 for (2999ds/7Mi)
% 2.83/1.59 % (10441)Instruction limit reached!
% 2.83/1.59 % (10441)------------------------------
% 2.83/1.59 % (10441)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.59 % (10441)Termination reason: Unknown
% 2.83/1.59 % (10441)Termination phase: shuffling
% 2.83/1.59
% 2.83/1.59 % (10441)Memory used [KB]: 1535
% 2.83/1.59 % (10441)Time elapsed: 0.004 s
% 2.83/1.59 % (10441)Instructions burned: 9 (million)
% 2.83/1.59 % (10441)------------------------------
% 2.83/1.59 % (10441)------------------------------
% 2.83/1.59 % (10439)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_10309 for (2999ds/3Mi)
% 2.83/1.59 % (10440)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_10309 for (2999ds/3Mi)
% 2.83/1.59 % (10439)Instruction limit reached!
% 2.83/1.59 % (10439)------------------------------
% 2.83/1.59 % (10439)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.59 % (10439)Termination reason: Unknown
% 2.83/1.59 % (10439)Termination phase: shuffling
% 2.83/1.59
% 2.83/1.59 % (10439)Memory used [KB]: 1407
% 2.83/1.59 % (10439)Time elapsed: 0.005 s
% 2.83/1.59 % (10439)Instructions burned: 3 (million)
% 2.83/1.59 % (10439)------------------------------
% 2.83/1.59 % (10439)------------------------------
% 2.83/1.59 % (10440)Instruction limit reached!
% 2.83/1.59 % (10440)------------------------------
% 2.83/1.59 % (10440)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.59 % (10440)Termination reason: Unknown
% 2.83/1.59 % (10440)Termination phase: shuffling
% 2.83/1.59
% 2.83/1.59 % (10440)Memory used [KB]: 1407
% 2.83/1.59 % (10440)Time elapsed: 0.004 s
% 2.83/1.59 % (10440)Instructions burned: 3 (million)
% 2.83/1.59 % (10440)------------------------------
% 2.83/1.59 % (10440)------------------------------
% 2.83/1.59 % (10430)First to succeed.
% 2.83/1.60 % (10445)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_10309 for (2999ds/3Mi)
% 2.87/1.60 % (10445)Instruction limit reached!
% 2.87/1.60 % (10445)------------------------------
% 2.87/1.60 % (10445)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.60 % (10430)Refutation found. Thanks to Tanya!
% 2.87/1.60 % SZS status Theorem for DTF2THF_10309
% 2.87/1.60 % SZS output start Proof for DTF2THF_10309
% 2.87/1.60 thf(type_def_5, type, itself: ($tType) > $tType).
% 2.87/1.60 thf(type_def_7, type, set: ($tType) > $tType).
% 2.87/1.60 thf(type_def_8, type, product_prod: ($tType * $tType) > $tType).
% 2.87/1.60 thf(type_def_9, type, dtree: $tType).
% 2.87/1.60 thf(type_def_10, type, t: $tType).
% 2.87/1.60 thf(type_def_11, type, sum_sum: ($tType * $tType) > $tType).
% 2.87/1.60 thf(func_def_0, type, product_prod: ($tType * $tType) > $tType).
% 2.87/1.60 thf(func_def_1, type, sum_sum: ($tType * $tType) > $tType).
% 2.87/1.60 thf(func_def_2, type, dtree: $tType).
% 2.87/1.60 thf(func_def_3, type, set: ($tType) > $tType).
% 2.87/1.60 thf(func_def_4, type, t: $tType).
% 2.87/1.60 thf(func_def_5, type, itself: ($tType) > $tType).
% 2.87/1.60 thf(func_def_6, type, bounded_lattice: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.87/1.60 thf(func_def_7, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.87/1.60 thf(func_def_8, type, lattice: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.87/1.60 thf(func_def_9, type, distrib_lattice: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.87/1.60 thf(func_def_10, type, semilattice_inf: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.87/1.60 thf(func_def_11, type, semilattice_sup: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.87/1.60 thf(func_def_12, type, bounded_lattice_top: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.87/1.60 thf(func_def_13, type, bounde1561333602nf_top: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.87/1.60 thf(func_def_14, type, bNF_Greatest_image2: !>[X0: $tType, X1: $tType, X2: $tType]:(set @ X0 > (X0 > X1) > (X0 > X2) > set @ product_prod @ X1 @ X2)).
% 2.87/1.60 thf(func_def_15, type, cont: dtree > set @ sum_sum @ t @ dtree).
% 2.87/1.60 thf(func_def_16, type, comp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X2 > X0) > X2 > X1)).
% 2.87/1.60 thf(func_def_17, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 2.87/1.60 thf(func_def_18, type, inf_inf: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.87/1.60 thf(func_def_19, type, sup_sup: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.87/1.60 thf(func_def_20, type, top_top: !>[X0: $tType]:(X0)).
% 2.87/1.60 thf(func_def_21, type, parall1899940088le_par: product_prod @ dtree @ dtree > dtree).
% 2.87/1.60 thf(func_def_22, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 2.87/1.60 thf(func_def_23, type, product_curry: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X0 @ X1 > X2) > X0 > X1 > X2)).
% 2.87/1.60 thf(func_def_24, type, produc2004651681e_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.87/1.60 thf(func_def_25, type, product_map_prod: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1) > (X2 > X3) > product_prod @ X0 @ X2 > product_prod @ X1 @ X3)).
% 2.87/1.60 thf(func_def_26, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.87/1.60 thf(func_def_27, type, product_case_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.87/1.60 thf(func_def_28, type, product_swap: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > product_prod @ X1 @ X0)).
% 2.87/1.60 thf(func_def_29, type, product_scomp: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > product_prod @ X1 @ X2) > (X1 > X2 > X3) > X0 > X3)).
% 2.87/1.60 thf(func_def_30, type, type2: !>[X0: $tType]:(itself @ X0)).
% 2.87/1.60 thf(func_def_31, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.87/1.60 thf(func_def_32, type, image: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1)).
% 2.87/1.60 thf(func_def_33, type, vimage: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X1 > set @ X0)).
% 2.87/1.60 thf(func_def_34, type, sum_Inl: !>[X0: $tType, X1: $tType]:(X0 > sum_sum @ X0 @ X1)).
% 2.87/1.60 thf(func_def_35, type, sum_Inr: !>[X0: $tType, X1: $tType]:(X0 > sum_sum @ X1 @ X0)).
% 2.87/1.60 thf(func_def_36, type, sum_Plus: !>[X0: $tType, X1: $tType]:(set @ X0 > set @ X1 > set @ sum_sum @ X0 @ X1)).
% 2.87/1.60 thf(func_def_37, type, sum_Suml: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > sum_sum @ X0 @ X2 > X1)).
% 2.87/1.60 thf(func_def_38, type, sum_Sumr: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > sum_sum @ X2 @ X0 > X1)).
% 2.87/1.60 thf(func_def_39, type, sum_rec_sum: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X2 > X1) > sum_sum @ X0 @ X2 > X1)).
% 2.87/1.60 thf(func_def_40, type, sum_case_sum: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X2 > X1) > sum_sum @ X0 @ X2 > X1)).
% 2.87/1.60 thf(func_def_41, type, type_definition: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X1 > X0) > set @ X1 > $o)).
% 2.87/1.60 thf(func_def_42, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.87/1.60 thf(func_def_43, type, tr1: dtree).
% 2.87/1.60 thf(func_def_44, type, tr1a: dtree).
% 2.87/1.60 thf(func_def_45, type, tr2: dtree).
% 2.87/1.60 thf(func_def_46, type, tr2a: dtree).
% 2.87/1.60 thf(func_def_47, type, tr3: dtree).
% 2.87/1.60 thf(func_def_48, type, tr3a: dtree).
% 2.87/1.60 thf(func_def_49, type, trA: dtree).
% 2.87/1.60 thf(func_def_50, type, trA2: dtree).
% 2.87/1.60 thf(func_def_51, type, trB: dtree).
% 2.87/1.60 thf(func_def_63, type, sK0: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X6)).
% 2.87/1.60 thf(func_def_64, type, sK1: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 2.87/1.60 thf(func_def_65, type, sK2: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 2.87/1.60 thf(func_def_66, type, sK3: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.87/1.60 thf(func_def_67, type, sK4: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.87/1.60 thf(func_def_68, type, sK5: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.87/1.60 thf(func_def_69, type, sK6: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.87/1.60 thf(func_def_70, type, sK7: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 2.87/1.60 thf(func_def_71, type, sK8: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 2.87/1.60 thf(func_def_72, type, sK9: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.87/1.60 thf(func_def_73, type, sK10: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.87/1.60 thf(func_def_74, type, sK11: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.87/1.60 thf(func_def_75, type, sK12: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.87/1.60 thf(func_def_76, type, sK13: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 2.87/1.60 thf(func_def_77, type, sK14: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 2.87/1.60 thf(func_def_78, type, sK15: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 2.87/1.60 thf(func_def_79, type, sK16: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 2.87/1.60 thf(func_def_80, type, sK17: product_prod @ dtree @ dtree > dtree).
% 2.87/1.60 thf(func_def_81, type, sK18: product_prod @ dtree @ dtree > dtree).
% 2.87/1.60 thf(func_def_82, type, sK19: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 2.87/1.60 thf(func_def_83, type, sK20: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.87/1.60 thf(func_def_84, type, sK21: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.87/1.60 thf(func_def_85, type, sK22: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.87/1.60 thf(func_def_86, type, sK23: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.87/1.60 thf(func_def_87, type, sK24: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.87/1.60 thf(func_def_88, type, sK25: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.87/1.60 thf(func_def_89, type, sK26: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.87/1.60 thf(func_def_90, type, sK27: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.87/1.60 thf(func_def_91, type, sK28: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X0)).
% 2.87/1.60 thf(func_def_92, type, sK29: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X1)).
% 2.87/1.60 thf(func_def_93, type, sK30: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X2)).
% 2.87/1.60 thf(func_def_94, type, sK31: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X3)).
% 2.87/1.60 thf(func_def_95, type, sK32: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X4)).
% 2.87/1.60 thf(func_def_96, type, sK33: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 2.87/1.60 thf(func_def_97, type, sK34: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 2.87/1.60 thf(func_def_98, type, sK35: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 2.87/1.60 thf(func_def_99, type, sK36: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 2.87/1.60 thf(func_def_100, type, sK37: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X0)).
% 2.87/1.60 thf(func_def_101, type, sK38: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X1)).
% 2.87/1.60 thf(func_def_102, type, sK39: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X2)).
% 2.87/1.60 thf(func_def_103, type, sK40: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X0)).
% 2.87/1.60 thf(func_def_104, type, sK41: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X1)).
% 2.87/1.60 thf(func_def_105, type, sK42: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X2)).
% 2.87/1.60 thf(func_def_106, type, sK43: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X3)).
% 2.87/1.60 thf(func_def_107, type, sK44: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X4)).
% 2.87/1.60 thf(func_def_108, type, sK45: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X5)).
% 2.87/1.60 thf(func_def_109, type, sK46: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X6)).
% 2.87/1.60 thf(func_def_110, type, sK47: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X0)).
% 2.87/1.60 thf(func_def_111, type, sK48: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X1)).
% 2.87/1.60 thf(func_def_112, type, sK49: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X2)).
% 2.87/1.60 thf(func_def_113, type, sK50: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X3)).
% 2.87/1.60 thf(func_def_114, type, sK51: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X4)).
% 2.87/1.60 thf(func_def_115, type, sK52: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X5)).
% 2.87/1.60 thf(func_def_116, type, sK53: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.87/1.60 thf(func_def_117, type, sK54: dtree).
% 2.87/1.60 thf(func_def_118, type, sK55: dtree).
% 2.87/1.60 thf(func_def_119, type, sK56: dtree).
% 2.87/1.60 thf(func_def_120, type, sK57: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.87/1.60 thf(func_def_121, type, sK58: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.87/1.60 thf(func_def_122, type, sK59: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.87/1.60 thf(func_def_123, type, sK60: (product_prod @ dtree @ dtree > $o) > dtree).
% 2.87/1.60 thf(func_def_124, type, sK61: (product_prod @ dtree @ dtree > $o) > dtree).
% 2.87/1.60 thf(func_def_125, type, sK62: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X0)).
% 2.87/1.60 thf(func_def_126, type, sK63: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X1)).
% 2.87/1.60 thf(func_def_127, type, sK64: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X2)).
% 2.87/1.60 thf(func_def_128, type, sK65: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X3)).
% 2.87/1.60 thf(func_def_130, type, ph67: !>[X0: $tType]:(X0)).
% 2.87/1.60 thf(f816,plain,(
% 2.87/1.60 $false),
% 2.87/1.60 inference(trivial_inequality_removal,[status(thm)],[f815])).
% 2.87/1.60 thf(f815,plain,(
% 2.87/1.60 ((parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr1 @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr2 @ tr3)))) != (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr1 @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr2 @ tr3)))))),
% 2.87/1.60 inference(equality_resolution,[status(thm)],[f769])).
% 2.87/1.60 thf(f769,plain,(
% 2.87/1.60 ( ! [X2 : dtree,X0 : dtree,X1 : dtree] : (((parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr3 @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr1 @ tr2)))) != (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X2 @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X0 @ X1))))) | ((parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr1 @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr2 @ tr3)))) != (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X0 @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X1 @ X2)))))) )),
% 2.87/1.60 inference(superposition,[status(thm)],[f761,f714])).
% 2.87/1.60 thf(f714,plain,(
% 2.87/1.60 ( ! [X0 : dtree,X1 : dtree] : (((parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X0 @ X1)) = (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X1 @ X0)))) )),
% 2.87/1.60 inference(cnf_transformation,[status(thm)],[f618])).
% 2.87/1.60 thf(f618,plain,(
% 2.87/1.60 ! [X0 : dtree,X1 : dtree] : ((parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X0 @ X1)) = (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X1 @ X0)))),
% 2.87/1.60 inference(rectify,[status(thm)],[f8])).
% 2.87/1.60 thf(f8,axiom,(
% 2.87/1.60 ! [X11 : dtree,X12 : dtree] : ((parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X11 @ X12)) = (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X12 @ X11)))),
% 2.87/1.60 file('/export/starexec/sandbox/tmp/tmp.sn5SmONUSK/DTF2THF_10309.p',fact_7_par__com)).
% 2.87/1.60 thf(f761,plain,(
% 2.87/1.60 ( ! [X2 : dtree,X0 : dtree,X1 : dtree] : (((parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr3 @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr1 @ tr2)))) != (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X0 @ X1)) @ X2))) | ((parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr1 @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr2 @ tr3)))) != (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X0 @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X1 @ X2)))))) )),
% 2.87/1.60 inference(forward_demodulation,[status(thm)],[f750,f714])).
% 2.87/1.60 thf(f750,plain,(
% 2.87/1.60 ( ! [X2 : dtree,X0 : dtree,X1 : dtree] : (((parall1899940088le_par @ (product_Pair @ dtree @ dtree @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr1 @ tr2)) @ tr3)) != (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X0 @ X1)) @ X2))) | ((parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr1 @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr2 @ tr3)))) != (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X0 @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X1 @ X2)))))) )),
% 2.87/1.60 inference(definition_unfolding,[status(thm)],[f702,f709])).
% 2.87/1.60 thf(f709,plain,(
% 2.87/1.60 (trA = (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr1 @ tr2)) @ tr3)))),
% 2.87/1.60 inference(cnf_transformation,[status(thm)],[f5])).
% 2.87/1.60 thf(f5,axiom,(
% 2.87/1.60 (trA = (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr1 @ tr2)) @ tr3)))),
% 2.87/1.60 file('/export/starexec/sandbox/tmp/tmp.sn5SmONUSK/DTF2THF_10309.p',fact_4__092_060open_062trA_H_A_061_A_Itr1_H_A_092_060parallel_062_Atr2_H_J_A_092_060parallel_062_Atr3_H_092_060close_062)).
% 2.87/1.60 thf(f702,plain,(
% 2.87/1.60 ( ! [X2 : dtree,X0 : dtree,X1 : dtree] : (((parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr1 @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr2 @ tr3)))) != (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X0 @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X1 @ X2))))) | (trA != (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X0 @ X1)) @ X2)))) )),
% 2.87/1.60 inference(cnf_transformation,[status(thm)],[f640])).
% 2.87/1.60 thf(f640,plain,(
% 2.87/1.60 ! [X0 : dtree,X1 : dtree,X2 : dtree] : (((parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr1 @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr2 @ tr3)))) != (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X0 @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X1 @ X2))))) | (trA != (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X0 @ X1)) @ X2))))),
% 2.87/1.60 inference(ennf_transformation,[status(thm)],[f616])).
% 2.87/1.60 thf(f616,plain,(
% 2.87/1.60 ~? [X0 : dtree,X1 : dtree,X2 : dtree] : (((parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr1 @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr2 @ tr3)))) = (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X0 @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X1 @ X2))))) & (trA = (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X0 @ X1)) @ X2))))),
% 2.87/1.60 inference(rectify,[status(thm)],[f286])).
% 2.87/1.60 thf(f286,negated_conjecture,(
% 2.87/1.60 ~? [X113 : dtree,X114 : dtree,X115 : dtree] : ((trA = (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X113 @ X114)) @ X115))) & ((parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr1 @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr2 @ tr3)))) = (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X113 @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X114 @ X115))))))),
% 2.87/1.60 inference(negated_conjecture,[status(cth)],[f285])).
% 2.87/1.60 thf(f285,conjecture,(
% 2.87/1.60 ? [X113 : dtree,X114 : dtree,X115 : dtree] : ((trA = (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X113 @ X114)) @ X115))) & ((parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr1 @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr2 @ tr3)))) = (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X113 @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X114 @ X115))))))),
% 2.87/1.60 file('/export/starexec/sandbox/tmp/tmp.sn5SmONUSK/DTF2THF_10309.p',conj_4)).
% 2.87/1.60 % SZS output end Proof for DTF2THF_10309
% 2.87/1.60 % (10430)------------------------------
% 2.87/1.60 % (10430)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.60 % (10430)Termination reason: Refutation
% 2.87/1.60
% 2.87/1.60 % (10430)Memory used [KB]: 6652
% 2.87/1.60 % (10430)Time elapsed: 0.045 s
% 2.87/1.60 % (10430)Instructions burned: 97 (million)
% 2.87/1.60 % (10430)------------------------------
% 2.87/1.60 % (10430)------------------------------
% 2.87/1.60 % (10424)Success in time 0.061 s
%------------------------------------------------------------------------------