%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : COM170^1 : 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 : n002.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:41:50 AM UTC 2026
% Result : Theorem 2.98s 1.87s
% Output : Refutation 2.98s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : COM170^1 : TPTP v9.2.1. Released v7.0.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n002.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 04:15:48 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.29/0.48 ---- Original DTF file ---
% 0.29/0.48 thf(spec,logic,$$dhol).
% 0.29/0.48 %------------------------------------------------------------------------------
% 0.29/0.48 % File : COM170^1 : TPTP v9.2.1. Released v7.0.0.
% 0.29/0.48 % Domain : Computing Theory
% 0.29/0.48 % Problem : Binary decision diagram 356
% 0.29/0.48 % Version : [Bla16] axioms : Especial.
% 0.29/0.48 % English :
% 0.29/0.48
% 0.29/0.48 % Refs : [OS08] Ortner & Schirmer (2008), BDD Normalisation
% 0.29/0.48 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.29/0.48 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.29/0.48 % Source : [Bla16]
% 0.29/0.48 % Names : bindag__356.p [Bla16]
% 0.29/0.48
% 0.29/0.48 % Status : Theorem
% 0.29/0.48 % Rating : 0.00 v8.1.0, 0.25 v7.5.0, 0.00 v7.2.0, 0.25 v7.1.0
% 0.29/0.48 % Syntax : Number of formulae : 333 ( 90 unt; 53 typ; 0 def)
% 0.29/0.48 % Number of atoms : 876 ( 226 equ; 0 cnn)
% 0.29/0.48 % Maximal formula atoms : 13 ( 3 avg)
% 0.29/0.48 % Number of connectives : 3297 ( 105 ~; 26 |; 47 &;2695 @)
% 0.29/0.48 % ( 0 <=>; 424 =>; 0 <=; 0 <~>)
% 0.29/0.48 % Maximal formula depth : 21 ( 8 avg)
% 0.29/0.48 % Number of types : 3 ( 2 usr)
% 0.29/0.48 % Number of type conns : 237 ( 237 >; 0 *; 0 +; 0 <<)
% 0.29/0.48 % Number of symbols : 54 ( 51 usr; 9 con; 0-6 aty)
% 0.29/0.48 % Number of variables : 1056 ( 64 ^; 918 !; 38 ?;1056 :)
% 0.29/0.48 % ( 36 !>; 0 ?*; 0 @-; 0 @+)
% 0.29/0.48 % SPC : TH1_THM_EQU_NAR
% 0.29/0.48
% 0.29/0.48 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.29/0.48 % 2016-07-13 14:46:21.240
% 0.29/0.48 %------------------------------------------------------------------------------
% 0.29/0.48 %----Could-be-implicit typings (5)
% 0.29/0.48 thf(ty_t_BinDag__Mirabelle__rybootvolr_Odag,type,
% 0.29/0.48 binDag_Mirabelle_dag: $tType ).
% 0.29/0.48
% 0.29/0.48 thf(ty_t_Simpl__Heap_Oref,type,
% 0.29/0.48 simpl_ref: $tType ).
% 0.29/0.48
% 0.29/0.48 thf(ty_t_List_Olist,type,
% 0.29/0.48 list: $tType > $tType ).
% 0.29/0.48
% 0.29/0.48 thf(ty_t_Set_Oset,type,
% 0.29/0.48 set: $tType > $tType ).
% 0.29/0.48
% 0.29/0.48 thf(ty_t_itself,type,
% 0.29/0.48 itself: $tType > $tType ).
% 0.29/0.48
% 0.29/0.48 %----Explicit typings (48)
% 0.29/0.48 thf(sy_cl_HOL_Otype,type,
% 0.29/0.48 type:
% 0.29/0.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_cl_Orderings_Obot,type,
% 0.29/0.48 bot:
% 0.29/0.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_cl_Orderings_Oord,type,
% 0.29/0.48 ord:
% 0.29/0.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_cl_Orderings_Oorder,type,
% 0.29/0.48 order:
% 0.29/0.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_cl_Orderings_Ono__bot,type,
% 0.29/0.48 no_bot:
% 0.29/0.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_cl_Orderings_Ono__top,type,
% 0.29/0.48 no_top:
% 0.29/0.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_cl_Orderings_Olinorder,type,
% 0.29/0.48 linorder:
% 0.29/0.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_cl_Orderings_Opreorder,type,
% 0.29/0.48 preorder:
% 0.29/0.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_cl_Orderings_Oorder__bot,type,
% 0.29/0.48 order_bot:
% 0.29/0.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_cl_Orderings_Owellorder,type,
% 0.29/0.48 wellorder:
% 0.29/0.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_cl_Orderings_Odense__order,type,
% 0.29/0.48 dense_order:
% 0.29/0.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.29/0.48 dense_linorder:
% 0.29/0.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 0.29/0.48 condit1656338222tinuum:
% 0.29/0.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 0.29/0.48 condit1037483654norder:
% 0.29/0.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_c_BinDag__Mirabelle__rybootvolr_ODAG,type,
% 0.29/0.48 binDag_Mirabelle_DAG: binDag_Mirabelle_dag > $o ).
% 0.29/0.48
% 0.29/0.48 thf(sy_c_BinDag__Mirabelle__rybootvolr_ODag,type,
% 0.29/0.48 binDag_Mirabelle_Dag: simpl_ref > ( simpl_ref > simpl_ref ) > ( simpl_ref > simpl_ref ) > binDag_Mirabelle_dag > $o ).
% 0.29/0.48
% 0.29/0.48 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_ONode,type,
% 0.29/0.48 binDag476092410e_Node: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > binDag_Mirabelle_dag ).
% 0.29/0.48
% 0.29/0.48 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_OTip,type,
% 0.29/0.48 binDag_Mirabelle_Tip: binDag_Mirabelle_dag ).
% 0.29/0.48
% 0.29/0.48 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_Ocase__dag,type,
% 0.29/0.48 binDag1297733282se_dag:
% 0.29/0.48 !>[A: $tType] : ( A > ( binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A ) > binDag_Mirabelle_dag > A ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_Orec__dag,type,
% 0.29/0.48 binDag1442713106ec_dag:
% 0.29/0.48 !>[A: $tType] : ( A > ( binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A > A > A ) > binDag_Mirabelle_dag > A ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_c_BinDag__Mirabelle__rybootvolr_Oset__of,type,
% 0.29/0.48 binDag1380252983set_of: binDag_Mirabelle_dag > ( set @ simpl_ref ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_c_BinDag__Mirabelle__rybootvolr_Osubdag,type,
% 0.29/0.48 binDag786255756subdag: binDag_Mirabelle_dag > binDag_Mirabelle_dag > $o ).
% 0.29/0.48
% 0.29/0.48 thf(sy_c_Finite__Set_Ofinite,type,
% 0.29/0.48 finite_finite:
% 0.29/0.48 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_c_Fun_Ofun__upd,type,
% 0.29/0.48 fun_upd:
% 0.29/0.48 !>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_c_Fun_Ooverride__on,type,
% 0.29/0.48 override_on:
% 0.29/0.48 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > B ) > ( set @ A ) > A > B ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_c_Fun_Oswap,type,
% 0.29/0.48 swap:
% 0.29/0.48 !>[A: $tType,B: $tType] : ( A > A > ( A > B ) > A > B ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_c_If,type,
% 0.29/0.48 if:
% 0.29/0.48 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_c_List_Ocoset,type,
% 0.29/0.48 coset:
% 0.29/0.48 !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
% 0.29/0.48 linord467138063of_set:
% 0.29/0.48 !>[A: $tType] : ( ( set @ A ) > ( list @ A ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_c_List_Olist_ONil,type,
% 0.29/0.48 nil:
% 0.29/0.48 !>[A: $tType] : ( list @ A ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_c_List_Olist_Oset,type,
% 0.29/0.48 set2:
% 0.29/0.48 !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_c_List_Onull,type,
% 0.29/0.48 null:
% 0.29/0.48 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.29/0.48 bot_bot:
% 0.29/0.48 !>[A: $tType] : A ).
% 0.29/0.48
% 0.29/0.48 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.29/0.48 ord_less:
% 0.29/0.48 !>[A: $tType] : ( A > A > $o ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.29/0.48 ord_less_eq:
% 0.29/0.48 !>[A: $tType] : ( A > A > $o ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_c_Pure_Otype,type,
% 0.29/0.48 type2:
% 0.29/0.48 !>[A: $tType] : ( itself @ A ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_c_Set_OCollect,type,
% 0.29/0.48 collect:
% 0.29/0.48 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_c_Set_Ois__empty,type,
% 0.29/0.48 is_empty:
% 0.29/0.48 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_c_Simpl__Heap_ONull,type,
% 0.29/0.48 simpl_Null: simpl_ref ).
% 0.29/0.48
% 0.29/0.48 thf(sy_c_Simpl__Heap_Onew,type,
% 0.29/0.48 simpl_new: ( set @ simpl_ref ) > simpl_ref ).
% 0.29/0.48
% 0.29/0.48 thf(sy_c_member,type,
% 0.29/0.48 member:
% 0.29/0.48 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.29/0.48
% 0.29/0.48 thf(sy_v_l,type,
% 0.29/0.48 l: simpl_ref > simpl_ref ).
% 0.29/0.48
% 0.29/0.48 thf(sy_v_lt____,type,
% 0.29/0.48 lt: binDag_Mirabelle_dag ).
% 0.29/0.48
% 0.29/0.48 thf(sy_v_p,type,
% 0.29/0.48 p: simpl_ref ).
% 0.29/0.48
% 0.29/0.48 thf(sy_v_r,type,
% 0.29/0.48 r: simpl_ref > simpl_ref ).
% 0.29/0.48
% 0.29/0.48 thf(sy_v_rt____,type,
% 0.29/0.48 rt: binDag_Mirabelle_dag ).
% 0.29/0.48
% 0.29/0.48 thf(sy_v_t,type,
% 0.29/0.48 t: binDag_Mirabelle_dag ).
% 0.29/0.48
% 0.29/0.48 thf(sy_v_thesis____,type,
% 0.29/0.48 thesis: $o ).
% 0.29/0.48
% 0.29/0.48 %----Relevant facts (256)
% 0.29/0.48 thf(fact_0_assms,axiom,
% 0.29/0.48 binDag_Mirabelle_Dag @ ( r @ p ) @ l @ r @ t ).
% 0.29/0.48
% 0.29/0.48 % assms
% 0.29/0.48 thf(fact_1_calculation,axiom,
% 0.29/0.48 ! [Lt: binDag_Mirabelle_dag,Rt: binDag_Mirabelle_dag] :
% 0.29/0.48 ( t
% 0.29/0.48 != ( binDag476092410e_Node @ Lt @ p @ Rt ) ) ).
% 0.29/0.48
% 0.29/0.48 % calculation
% 0.29/0.48 thf(fact_2_subdag,axiom,
% 0.29/0.48 binDag786255756subdag @ t @ ( binDag476092410e_Node @ lt @ p @ rt ) ).
% 0.29/0.48
% 0.29/0.48 % subdag
% 0.29/0.48 thf(fact_3_dag_Oinject,axiom,
% 0.29/0.48 ! [X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag,Y21: binDag_Mirabelle_dag,Y22: simpl_ref,Y23: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( ( binDag476092410e_Node @ X21 @ X22 @ X23 )
% 0.29/0.48 = ( binDag476092410e_Node @ Y21 @ Y22 @ Y23 ) )
% 0.29/0.48 = ( ( X21 = Y21 )
% 0.29/0.48 & ( X22 = Y22 )
% 0.29/0.48 & ( X23 = Y23 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % dag.inject
% 0.29/0.48 thf(fact_4_Dag__unique,axiom,
% 0.29/0.48 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T1: binDag_Mirabelle_dag,T2: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( binDag_Mirabelle_Dag @ P @ L @ R @ T1 )
% 0.29/0.48 => ( ( binDag_Mirabelle_Dag @ P @ L @ R @ T2 )
% 0.29/0.48 => ( T1 = T2 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Dag_unique
% 0.29/0.48 thf(fact_5_Dag__unique1,axiom,
% 0.29/0.48 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( binDag_Mirabelle_Dag @ P @ L @ R @ T )
% 0.29/0.48 => ? [X: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( binDag_Mirabelle_Dag @ P @ L @ R @ X )
% 0.29/0.48 & ! [Y: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( binDag_Mirabelle_Dag @ P @ L @ R @ Y )
% 0.29/0.48 => ( Y = X ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Dag_unique1
% 0.29/0.48 thf(fact_6_subdag_Osimps_I2_J,axiom,
% 0.29/0.48 ! [L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag,T: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( binDag786255756subdag @ ( binDag476092410e_Node @ L @ A2 @ R ) @ T )
% 0.29/0.48 = ( ( T = L )
% 0.29/0.48 | ( T = R )
% 0.29/0.48 | ( binDag786255756subdag @ L @ T )
% 0.29/0.48 | ( binDag786255756subdag @ R @ T ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % subdag.simps(2)
% 0.29/0.48 thf(fact_7_subdag__NodeD,axiom,
% 0.29/0.48 ! [T: binDag_Mirabelle_dag,Lt2: binDag_Mirabelle_dag,A2: simpl_ref,Rt2: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( binDag786255756subdag @ T @ ( binDag476092410e_Node @ Lt2 @ A2 @ Rt2 ) )
% 0.29/0.48 => ( ( binDag786255756subdag @ T @ Lt2 )
% 0.29/0.48 & ( binDag786255756subdag @ T @ Rt2 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % subdag_NodeD
% 0.29/0.48 thf(fact_8_Dag__subdag,axiom,
% 0.29/0.48 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag,S: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( binDag_Mirabelle_Dag @ P @ L @ R @ T )
% 0.29/0.48 => ( ( binDag786255756subdag @ T @ S )
% 0.29/0.48 => ? [Q: simpl_ref] : ( binDag_Mirabelle_Dag @ Q @ L @ R @ S ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Dag_subdag
% 0.29/0.48 thf(fact_9_Dag__Ref,axiom,
% 0.29/0.48 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( P != simpl_Null )
% 0.29/0.48 => ( ( binDag_Mirabelle_Dag @ P @ L @ R @ T )
% 0.29/0.48 = ( ? [Lt3: binDag_Mirabelle_dag,Rt3: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( T
% 0.29/0.48 = ( binDag476092410e_Node @ Lt3 @ P @ Rt3 ) )
% 0.29/0.48 & ( binDag_Mirabelle_Dag @ ( L @ P ) @ L @ R @ Lt3 )
% 0.29/0.48 & ( binDag_Mirabelle_Dag @ ( R @ P ) @ L @ R @ Rt3 ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Dag_Ref
% 0.29/0.48 thf(fact_10_Dag_Osimps_I2_J,axiom,
% 0.29/0.48 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,Lt2: binDag_Mirabelle_dag,A2: simpl_ref,Rt2: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( binDag_Mirabelle_Dag @ P @ L @ R @ ( binDag476092410e_Node @ Lt2 @ A2 @ Rt2 ) )
% 0.29/0.48 = ( ( P = A2 )
% 0.29/0.48 & ( P != simpl_Null )
% 0.29/0.48 & ( binDag_Mirabelle_Dag @ ( L @ P ) @ L @ R @ Lt2 )
% 0.29/0.48 & ( binDag_Mirabelle_Dag @ ( R @ P ) @ L @ R @ Rt2 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Dag.simps(2)
% 0.29/0.48 thf(fact_11_dag_Osimps_I7_J,axiom,
% 0.29/0.48 ! [A: $tType,F1: A,F2: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A > A > A,X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( binDag1442713106ec_dag @ A @ F1 @ F2 @ ( binDag476092410e_Node @ X21 @ X22 @ X23 ) )
% 0.29/0.48 = ( F2 @ X21 @ X22 @ X23 @ ( binDag1442713106ec_dag @ A @ F1 @ F2 @ X21 ) @ ( binDag1442713106ec_dag @ A @ F1 @ F2 @ X23 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % dag.simps(7)
% 0.29/0.48 thf(fact_12_dag_Osimps_I5_J,axiom,
% 0.29/0.48 ! [A: $tType,F1: A,F2: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A,X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( binDag1297733282se_dag @ A @ F1 @ F2 @ ( binDag476092410e_Node @ X21 @ X22 @ X23 ) )
% 0.29/0.48 = ( F2 @ X21 @ X22 @ X23 ) ) ).
% 0.29/0.48
% 0.29/0.48 % dag.simps(5)
% 0.29/0.48 thf(fact_13_Dag__root__not__in__subdag__l,axiom,
% 0.29/0.48 ! [L: simpl_ref > simpl_ref,P: simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( binDag_Mirabelle_Dag @ ( L @ P ) @ L @ R @ T )
% 0.29/0.48 => ~ ( member @ simpl_ref @ P @ ( binDag1380252983set_of @ T ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Dag_root_not_in_subdag_l
% 0.29/0.48 thf(fact_14_dag_Odistinct_I1_J,axiom,
% 0.29/0.48 ! [X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag] :
% 0.29/0.48 ( binDag_Mirabelle_Tip
% 0.29/0.48 != ( binDag476092410e_Node @ X21 @ X22 @ X23 ) ) ).
% 0.29/0.48
% 0.29/0.48 % dag.distinct(1)
% 0.29/0.48 thf(fact_15_dag_Oinduct,axiom,
% 0.29/0.48 ! [P2: binDag_Mirabelle_dag > $o,Dag: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( P2 @ binDag_Mirabelle_Tip )
% 0.29/0.48 => ( ! [X1: binDag_Mirabelle_dag,X2: simpl_ref,X3: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( P2 @ X1 )
% 0.29/0.48 => ( ( P2 @ X3 )
% 0.29/0.48 => ( P2 @ ( binDag476092410e_Node @ X1 @ X2 @ X3 ) ) ) )
% 0.29/0.48 => ( P2 @ Dag ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % dag.induct
% 0.29/0.48 thf(fact_16_dag_Oexhaust,axiom,
% 0.29/0.48 ! [Y2: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( Y2 != binDag_Mirabelle_Tip )
% 0.29/0.48 => ~ ! [X212: binDag_Mirabelle_dag,X222: simpl_ref,X232: binDag_Mirabelle_dag] :
% 0.29/0.48 ( Y2
% 0.29/0.48 != ( binDag476092410e_Node @ X212 @ X222 @ X232 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % dag.exhaust
% 0.29/0.48 thf(fact_17_subdag__neq,axiom,
% 0.29/0.48 ! [T: binDag_Mirabelle_dag,S: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( binDag786255756subdag @ T @ S )
% 0.29/0.48 => ( T != S ) ) ).
% 0.29/0.48
% 0.29/0.48 % subdag_neq
% 0.29/0.48 thf(fact_18_Null__notin__Dag,axiom,
% 0.29/0.48 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( binDag_Mirabelle_Dag @ P @ L @ R @ T )
% 0.29/0.48 => ~ ( member @ simpl_ref @ simpl_Null @ ( binDag1380252983set_of @ T ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Null_notin_Dag
% 0.29/0.48 thf(fact_19_Dag__Null,axiom,
% 0.29/0.48 ! [L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( binDag_Mirabelle_Dag @ simpl_Null @ L @ R @ T )
% 0.29/0.48 = ( T = binDag_Mirabelle_Tip ) ) ).
% 0.29/0.48
% 0.29/0.48 % Dag_Null
% 0.29/0.48 thf(fact_20_dag_Osimps_I6_J,axiom,
% 0.29/0.48 ! [A: $tType,F1: A,F2: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A > A > A] :
% 0.29/0.48 ( ( binDag1442713106ec_dag @ A @ F1 @ F2 @ binDag_Mirabelle_Tip )
% 0.29/0.48 = F1 ) ).
% 0.29/0.48
% 0.29/0.48 % dag.simps(6)
% 0.29/0.48 thf(fact_21_dag_Osimps_I4_J,axiom,
% 0.29/0.48 ! [A: $tType,F1: A,F2: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A] :
% 0.29/0.48 ( ( binDag1297733282se_dag @ A @ F1 @ F2 @ binDag_Mirabelle_Tip )
% 0.29/0.48 = F1 ) ).
% 0.29/0.48
% 0.29/0.48 % dag.simps(4)
% 0.29/0.48 thf(fact_22_subdag_Osimps_I1_J,axiom,
% 0.29/0.48 ! [T: binDag_Mirabelle_dag] :
% 0.29/0.48 ~ ( binDag786255756subdag @ binDag_Mirabelle_Tip @ T ) ).
% 0.29/0.48
% 0.29/0.48 % subdag.simps(1)
% 0.29/0.48 thf(fact_23_Dag_Osimps_I1_J,axiom,
% 0.29/0.48 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref] :
% 0.29/0.48 ( ( binDag_Mirabelle_Dag @ P @ L @ R @ binDag_Mirabelle_Tip )
% 0.29/0.48 = ( P = simpl_Null ) ) ).
% 0.29/0.48
% 0.29/0.48 % Dag.simps(1)
% 0.29/0.48 thf(fact_24_in__set__of__decomp,axiom,
% 0.29/0.48 ! [P: simpl_ref,T: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( member @ simpl_ref @ P @ ( binDag1380252983set_of @ T ) )
% 0.29/0.48 = ( ? [L2: binDag_Mirabelle_dag,R2: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( T
% 0.29/0.48 = ( binDag476092410e_Node @ L2 @ P @ R2 ) )
% 0.29/0.48 | ( binDag786255756subdag @ T @ ( binDag476092410e_Node @ L2 @ P @ R2 ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % in_set_of_decomp
% 0.29/0.48 thf(fact_25_subdag__not__sym,axiom,
% 0.29/0.48 ! [S: binDag_Mirabelle_dag,T: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( binDag786255756subdag @ S @ T )
% 0.29/0.48 => ~ ( binDag786255756subdag @ T @ S ) ) ).
% 0.29/0.48
% 0.29/0.48 % subdag_not_sym
% 0.29/0.48 thf(fact_26_subdag__trans,axiom,
% 0.29/0.48 ! [T: binDag_Mirabelle_dag,S: binDag_Mirabelle_dag,R: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( binDag786255756subdag @ T @ S )
% 0.29/0.48 => ( ( binDag786255756subdag @ S @ R )
% 0.29/0.48 => ( binDag786255756subdag @ T @ R ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % subdag_trans
% 0.29/0.48 thf(fact_27_DAG_Osimps_I2_J,axiom,
% 0.29/0.48 ! [L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( binDag_Mirabelle_DAG @ ( binDag476092410e_Node @ L @ A2 @ R ) )
% 0.29/0.48 = ( ~ ( member @ simpl_ref @ A2 @ ( binDag1380252983set_of @ L ) )
% 0.29/0.48 & ~ ( member @ simpl_ref @ A2 @ ( binDag1380252983set_of @ R ) )
% 0.29/0.48 & ( binDag_Mirabelle_DAG @ L )
% 0.29/0.48 & ( binDag_Mirabelle_DAG @ R ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % DAG.simps(2)
% 0.29/0.48 thf(fact_28_Dag__upd__same__r,axiom,
% 0.29/0.48 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( binDag_Mirabelle_Dag @ P @ L @ ( fun_upd @ simpl_ref @ simpl_ref @ R @ P @ P ) @ T )
% 0.29/0.48 = ( ( P = simpl_Null )
% 0.29/0.48 & ( T = binDag_Mirabelle_Tip ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Dag_upd_same_r
% 0.29/0.48 thf(fact_29_Dag__upd__same__l,axiom,
% 0.29/0.48 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( binDag_Mirabelle_Dag @ P @ ( fun_upd @ simpl_ref @ simpl_ref @ L @ P @ P ) @ R @ T )
% 0.29/0.48 = ( ( P = simpl_Null )
% 0.29/0.48 & ( T = binDag_Mirabelle_Tip ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Dag_upd_same_l
% 0.29/0.48 thf(fact_30_notin__Dag__update__r,axiom,
% 0.29/0.48 ! [Q2: simpl_ref,T: binDag_Mirabelle_dag,P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,Y2: simpl_ref] :
% 0.29/0.48 ( ~ ( member @ simpl_ref @ Q2 @ ( binDag1380252983set_of @ T ) )
% 0.29/0.48 => ( ( binDag_Mirabelle_Dag @ P @ L @ ( fun_upd @ simpl_ref @ simpl_ref @ R @ Q2 @ Y2 ) @ T )
% 0.29/0.48 = ( binDag_Mirabelle_Dag @ P @ L @ R @ T ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % notin_Dag_update_r
% 0.29/0.48 thf(fact_31_notin__Dag__update__l,axiom,
% 0.29/0.48 ! [Q2: simpl_ref,T: binDag_Mirabelle_dag,P: simpl_ref,L: simpl_ref > simpl_ref,Y2: simpl_ref,R: simpl_ref > simpl_ref] :
% 0.29/0.48 ( ~ ( member @ simpl_ref @ Q2 @ ( binDag1380252983set_of @ T ) )
% 0.29/0.48 => ( ( binDag_Mirabelle_Dag @ P @ ( fun_upd @ simpl_ref @ simpl_ref @ L @ Q2 @ Y2 ) @ R @ T )
% 0.29/0.48 = ( binDag_Mirabelle_Dag @ P @ L @ R @ T ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % notin_Dag_update_l
% 0.29/0.48 thf(fact_32_Dag__update__rI,axiom,
% 0.29/0.48 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag,Q2: simpl_ref,Y2: simpl_ref] :
% 0.29/0.48 ( ( binDag_Mirabelle_Dag @ P @ L @ R @ T )
% 0.29/0.48 => ( ~ ( member @ simpl_ref @ Q2 @ ( binDag1380252983set_of @ T ) )
% 0.29/0.48 => ( binDag_Mirabelle_Dag @ P @ L @ ( fun_upd @ simpl_ref @ simpl_ref @ R @ Q2 @ Y2 ) @ T ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Dag_update_rI
% 0.29/0.48 thf(fact_33_Dag__update__lI,axiom,
% 0.29/0.48 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag,Q2: simpl_ref,Y2: simpl_ref] :
% 0.29/0.48 ( ( binDag_Mirabelle_Dag @ P @ L @ R @ T )
% 0.29/0.48 => ( ~ ( member @ simpl_ref @ Q2 @ ( binDag1380252983set_of @ T ) )
% 0.29/0.48 => ( binDag_Mirabelle_Dag @ P @ ( fun_upd @ simpl_ref @ simpl_ref @ L @ Q2 @ Y2 ) @ R @ T ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Dag_update_lI
% 0.29/0.48 thf(fact_34_DAG_Osimps_I1_J,axiom,
% 0.29/0.48 binDag_Mirabelle_DAG @ binDag_Mirabelle_Tip ).
% 0.29/0.48
% 0.29/0.48 % DAG.simps(1)
% 0.29/0.48 thf(fact_35_Dag__upd__same__r__lemma,axiom,
% 0.29/0.48 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( P != simpl_Null )
% 0.29/0.48 => ~ ( binDag_Mirabelle_Dag @ P @ L @ ( fun_upd @ simpl_ref @ simpl_ref @ R @ P @ P ) @ T ) ) ).
% 0.29/0.48
% 0.29/0.48 % Dag_upd_same_r_lemma
% 0.29/0.48 thf(fact_36_Dag__upd__same__l__lemma,axiom,
% 0.29/0.48 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( P != simpl_Null )
% 0.29/0.48 => ~ ( binDag_Mirabelle_Dag @ P @ ( fun_upd @ simpl_ref @ simpl_ref @ L @ P @ P ) @ R @ T ) ) ).
% 0.29/0.48
% 0.29/0.48 % Dag_upd_same_l_lemma
% 0.29/0.48 thf(fact_37_fun__upd__apply,axiom,
% 0.29/0.48 ! [A: $tType,B: $tType] :
% 0.29/0.48 ( ( fun_upd @ B @ A )
% 0.29/0.48 = ( ^ [F: B > A,X4: B,Y3: A,Z: B] : ( if @ A @ ( Z = X4 ) @ Y3 @ ( F @ Z ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % fun_upd_apply
% 0.29/0.48 thf(fact_38_fun__upd__triv,axiom,
% 0.29/0.48 ! [B: $tType,A: $tType,F3: A > B,X5: A] :
% 0.29/0.48 ( ( fun_upd @ A @ B @ F3 @ X5 @ ( F3 @ X5 ) )
% 0.29/0.48 = F3 ) ).
% 0.29/0.48
% 0.29/0.48 % fun_upd_triv
% 0.29/0.48 thf(fact_39_fun__upd__upd,axiom,
% 0.29/0.48 ! [A: $tType,B: $tType,F3: A > B,X5: A,Y2: B,Z2: B] :
% 0.29/0.48 ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ F3 @ X5 @ Y2 ) @ X5 @ Z2 )
% 0.29/0.48 = ( fun_upd @ A @ B @ F3 @ X5 @ Z2 ) ) ).
% 0.29/0.48
% 0.29/0.48 % fun_upd_upd
% 0.29/0.48 thf(fact_40_fun__upd__idem__iff,axiom,
% 0.29/0.48 ! [A: $tType,B: $tType,F3: A > B,X5: A,Y2: B] :
% 0.29/0.48 ( ( ( fun_upd @ A @ B @ F3 @ X5 @ Y2 )
% 0.29/0.48 = F3 )
% 0.29/0.48 = ( ( F3 @ X5 )
% 0.29/0.48 = Y2 ) ) ).
% 0.29/0.48
% 0.29/0.48 % fun_upd_idem_iff
% 0.29/0.48 thf(fact_41_fun__upd__twist,axiom,
% 0.29/0.48 ! [A: $tType,B: $tType,A2: A,C: A,M: A > B,B2: B,D: B] :
% 0.29/0.48 ( ( A2 != C )
% 0.29/0.48 => ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M @ A2 @ B2 ) @ C @ D )
% 0.29/0.48 = ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M @ C @ D ) @ A2 @ B2 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % fun_upd_twist
% 0.29/0.48 thf(fact_42_fun__upd__other,axiom,
% 0.29/0.48 ! [B: $tType,A: $tType,Z2: A,X5: A,F3: A > B,Y2: B] :
% 0.29/0.48 ( ( Z2 != X5 )
% 0.29/0.48 => ( ( fun_upd @ A @ B @ F3 @ X5 @ Y2 @ Z2 )
% 0.29/0.48 = ( F3 @ Z2 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % fun_upd_other
% 0.29/0.48 thf(fact_43_fun__upd__same,axiom,
% 0.29/0.48 ! [B: $tType,A: $tType,F3: B > A,X5: B,Y2: A] :
% 0.29/0.48 ( ( fun_upd @ B @ A @ F3 @ X5 @ Y2 @ X5 )
% 0.29/0.48 = Y2 ) ).
% 0.29/0.48
% 0.29/0.48 % fun_upd_same
% 0.29/0.48 thf(fact_44_fun__upd__idem,axiom,
% 0.29/0.48 ! [A: $tType,B: $tType,F3: B > A,X5: B,Y2: A] :
% 0.29/0.48 ( ( ( F3 @ X5 )
% 0.29/0.48 = Y2 )
% 0.29/0.48 => ( ( fun_upd @ B @ A @ F3 @ X5 @ Y2 )
% 0.29/0.48 = F3 ) ) ).
% 0.29/0.48
% 0.29/0.48 % fun_upd_idem
% 0.29/0.48 thf(fact_45_mem__Collect__eq,axiom,
% 0.29/0.48 ! [A: $tType,A2: A,P2: A > $o] :
% 0.29/0.48 ( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
% 0.29/0.48 = ( P2 @ A2 ) ) ).
% 0.29/0.48
% 0.29/0.48 % mem_Collect_eq
% 0.29/0.48 thf(fact_46_Collect__mem__eq,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A] :
% 0.29/0.48 ( ( collect @ A
% 0.29/0.48 @ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
% 0.29/0.48 = A3 ) ).
% 0.29/0.48
% 0.29/0.48 % Collect_mem_eq
% 0.29/0.48 thf(fact_47_Collect__cong,axiom,
% 0.29/0.48 ! [A: $tType,P2: A > $o,Q3: A > $o] :
% 0.29/0.48 ( ! [X: A] :
% 0.29/0.48 ( ( P2 @ X )
% 0.29/0.48 = ( Q3 @ X ) )
% 0.29/0.48 => ( ( collect @ A @ P2 )
% 0.29/0.48 = ( collect @ A @ Q3 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Collect_cong
% 0.29/0.48 thf(fact_48_ext,axiom,
% 0.29/0.48 ! [B: $tType,A: $tType,F3: A > B,G: A > B] :
% 0.29/0.48 ( ! [X: A] :
% 0.29/0.48 ( ( F3 @ X )
% 0.29/0.48 = ( G @ X ) )
% 0.29/0.48 => ( F3 = G ) ) ).
% 0.29/0.48
% 0.29/0.48 % ext
% 0.29/0.48 thf(fact_49_fun__upd__eqD,axiom,
% 0.29/0.48 ! [A: $tType,B: $tType,F3: A > B,X5: A,Y2: B,G: A > B,Z2: B] :
% 0.29/0.48 ( ( ( fun_upd @ A @ B @ F3 @ X5 @ Y2 )
% 0.29/0.48 = ( fun_upd @ A @ B @ G @ X5 @ Z2 ) )
% 0.29/0.48 => ( Y2 = Z2 ) ) ).
% 0.29/0.48
% 0.29/0.48 % fun_upd_eqD
% 0.29/0.48 thf(fact_50_fun__upd__def,axiom,
% 0.29/0.48 ! [B: $tType,A: $tType] :
% 0.29/0.48 ( ( fun_upd @ A @ B )
% 0.29/0.48 = ( ^ [F: A > B,A4: A,B3: B,X4: A] : ( if @ B @ ( X4 = A4 ) @ B3 @ ( F @ X4 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % fun_upd_def
% 0.29/0.48 thf(fact_51_set__of__Tip,axiom,
% 0.29/0.48 ( ( binDag1380252983set_of @ binDag_Mirabelle_Tip )
% 0.29/0.48 = ( bot_bot @ ( set @ simpl_ref ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % set_of_Tip
% 0.29/0.48 thf(fact_52_DAG__less,axiom,
% 0.29/0.48 ! [Y2: binDag_Mirabelle_dag,X5: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( binDag_Mirabelle_DAG @ Y2 )
% 0.29/0.48 => ( ( ord_less @ binDag_Mirabelle_dag @ X5 @ Y2 )
% 0.29/0.48 => ( binDag_Mirabelle_DAG @ X5 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % DAG_less
% 0.29/0.48 thf(fact_53_Dag__update__l__new,axiom,
% 0.29/0.48 ! [T: binDag_Mirabelle_dag,Alloc: list @ simpl_ref,P: simpl_ref,L: simpl_ref > simpl_ref,X5: simpl_ref,R: simpl_ref > simpl_ref] :
% 0.29/0.48 ( ( ord_less_eq @ ( set @ simpl_ref ) @ ( binDag1380252983set_of @ T ) @ ( set2 @ simpl_ref @ Alloc ) )
% 0.29/0.48 => ( ( binDag_Mirabelle_Dag @ P @ ( fun_upd @ simpl_ref @ simpl_ref @ L @ ( simpl_new @ ( set2 @ simpl_ref @ Alloc ) ) @ X5 ) @ R @ T )
% 0.29/0.48 = ( binDag_Mirabelle_Dag @ P @ L @ R @ T ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Dag_update_l_new
% 0.29/0.48 thf(fact_54_Dag__update__r__new,axiom,
% 0.29/0.48 ! [T: binDag_Mirabelle_dag,Alloc: list @ simpl_ref,P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,X5: simpl_ref] :
% 0.29/0.48 ( ( ord_less_eq @ ( set @ simpl_ref ) @ ( binDag1380252983set_of @ T ) @ ( set2 @ simpl_ref @ Alloc ) )
% 0.29/0.48 => ( ( binDag_Mirabelle_Dag @ P @ L @ ( fun_upd @ simpl_ref @ simpl_ref @ R @ ( simpl_new @ ( set2 @ simpl_ref @ Alloc ) ) @ X5 ) @ T )
% 0.29/0.48 = ( binDag_Mirabelle_Dag @ P @ L @ R @ T ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Dag_update_r_new
% 0.29/0.48 thf(fact_55_swap__def,axiom,
% 0.29/0.48 ! [B: $tType,A: $tType] :
% 0.29/0.48 ( ( swap @ A @ B )
% 0.29/0.48 = ( ^ [A4: A,B3: A,F: A > B] : ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ F @ A4 @ ( F @ B3 ) ) @ B3 @ ( F @ A4 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % swap_def
% 0.29/0.48 thf(fact_56_less__dag__Tip,axiom,
% 0.29/0.48 ! [X5: binDag_Mirabelle_dag] :
% 0.29/0.48 ~ ( ord_less @ binDag_Mirabelle_dag @ X5 @ binDag_Mirabelle_Tip ) ).
% 0.29/0.48
% 0.29/0.48 % less_dag_Tip
% 0.29/0.48 thf(fact_57_less__dag__def,axiom,
% 0.29/0.48 ( ( ord_less @ binDag_Mirabelle_dag )
% 0.29/0.48 = ( ^ [S2: binDag_Mirabelle_dag,T3: binDag_Mirabelle_dag] : ( binDag786255756subdag @ T3 @ S2 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % less_dag_def
% 0.29/0.48 thf(fact_58_less__dag__Node_H,axiom,
% 0.29/0.48 ! [X5: binDag_Mirabelle_dag,L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( ord_less @ binDag_Mirabelle_dag @ X5 @ ( binDag476092410e_Node @ L @ A2 @ R ) )
% 0.29/0.48 = ( ( X5 = L )
% 0.29/0.48 | ( X5 = R )
% 0.29/0.48 | ( ord_less @ binDag_Mirabelle_dag @ X5 @ L )
% 0.29/0.48 | ( ord_less @ binDag_Mirabelle_dag @ X5 @ R ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % less_dag_Node'
% 0.29/0.48 thf(fact_59_swap__apply_I3_J,axiom,
% 0.29/0.48 ! [A: $tType,B: $tType,C: B,A2: B,B2: B,F3: B > A] :
% 0.29/0.48 ( ( C != A2 )
% 0.29/0.48 => ( ( C != B2 )
% 0.29/0.48 => ( ( swap @ B @ A @ A2 @ B2 @ F3 @ C )
% 0.29/0.48 = ( F3 @ C ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % swap_apply(3)
% 0.29/0.48 thf(fact_60_swap__apply_I2_J,axiom,
% 0.29/0.48 ! [A: $tType,B: $tType,A2: B,B2: B,F3: B > A] :
% 0.29/0.48 ( ( swap @ B @ A @ A2 @ B2 @ F3 @ B2 )
% 0.29/0.48 = ( F3 @ A2 ) ) ).
% 0.29/0.48
% 0.29/0.48 % swap_apply(2)
% 0.29/0.48 thf(fact_61_swap__apply_I1_J,axiom,
% 0.29/0.48 ! [A: $tType,B: $tType,A2: B,B2: B,F3: B > A] :
% 0.29/0.48 ( ( swap @ B @ A @ A2 @ B2 @ F3 @ A2 )
% 0.29/0.48 = ( F3 @ B2 ) ) ).
% 0.29/0.48
% 0.29/0.48 % swap_apply(1)
% 0.29/0.48 thf(fact_62_swap__self,axiom,
% 0.29/0.48 ! [B: $tType,A: $tType,A2: A,F3: A > B] :
% 0.29/0.48 ( ( swap @ A @ B @ A2 @ A2 @ F3 )
% 0.29/0.48 = F3 ) ).
% 0.29/0.48
% 0.29/0.48 % swap_self
% 0.29/0.48 thf(fact_63_swap__nilpotent,axiom,
% 0.29/0.48 ! [B: $tType,A: $tType,A2: A,B2: A,F3: A > B] :
% 0.29/0.48 ( ( swap @ A @ B @ A2 @ B2 @ ( swap @ A @ B @ A2 @ B2 @ F3 ) )
% 0.29/0.48 = F3 ) ).
% 0.29/0.48
% 0.29/0.48 % swap_nilpotent
% 0.29/0.48 thf(fact_64_swap__triple,axiom,
% 0.29/0.48 ! [B: $tType,A: $tType,A2: A,C: A,B2: A,F3: A > B] :
% 0.29/0.48 ( ( A2 != C )
% 0.29/0.48 => ( ( B2 != C )
% 0.29/0.48 => ( ( swap @ A @ B @ A2 @ B2 @ ( swap @ A @ B @ B2 @ C @ ( swap @ A @ B @ A2 @ B2 @ F3 ) ) )
% 0.29/0.48 = ( swap @ A @ B @ A2 @ C @ F3 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % swap_triple
% 0.29/0.48 thf(fact_65_swap__commute,axiom,
% 0.29/0.48 ! [B: $tType,A: $tType] :
% 0.29/0.48 ( ( swap @ A @ B )
% 0.29/0.48 = ( ^ [A4: A,B3: A] : ( swap @ A @ B @ B3 @ A4 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % swap_commute
% 0.29/0.48 thf(fact_66_less__dag__set__of,axiom,
% 0.29/0.48 ! [X5: binDag_Mirabelle_dag,Y2: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( ord_less @ binDag_Mirabelle_dag @ X5 @ Y2 )
% 0.29/0.48 => ( ord_less_eq @ ( set @ simpl_ref ) @ ( binDag1380252983set_of @ X5 ) @ ( binDag1380252983set_of @ Y2 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % less_dag_set_of
% 0.29/0.48 thf(fact_67_less__Node__dag,axiom,
% 0.29/0.48 ! [L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag,X5: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( ord_less @ binDag_Mirabelle_dag @ ( binDag476092410e_Node @ L @ A2 @ R ) @ X5 )
% 0.29/0.48 => ( ( ord_less @ binDag_Mirabelle_dag @ L @ X5 )
% 0.29/0.48 & ( ord_less @ binDag_Mirabelle_dag @ R @ X5 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % less_Node_dag
% 0.29/0.48 thf(fact_68_empty__subsetI,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% 0.29/0.48
% 0.29/0.48 % empty_subsetI
% 0.29/0.48 thf(fact_69_subset__empty,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A] :
% 0.29/0.48 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 0.29/0.48 = ( A3
% 0.29/0.48 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % subset_empty
% 0.29/0.48 thf(fact_70_subset__antisym,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A,B4: set @ A] :
% 0.29/0.48 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 0.29/0.48 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ A3 )
% 0.29/0.48 => ( A3 = B4 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % subset_antisym
% 0.29/0.48 thf(fact_71_subsetI,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A,B4: set @ A] :
% 0.29/0.48 ( ! [X: A] :
% 0.29/0.48 ( ( member @ A @ X @ A3 )
% 0.29/0.48 => ( member @ A @ X @ B4 ) )
% 0.29/0.48 => ( ord_less_eq @ ( set @ A ) @ A3 @ B4 ) ) ).
% 0.29/0.48
% 0.29/0.48 % subsetI
% 0.29/0.48 thf(fact_72_empty__Collect__eq,axiom,
% 0.29/0.48 ! [A: $tType,P2: A > $o] :
% 0.29/0.48 ( ( ( bot_bot @ ( set @ A ) )
% 0.29/0.48 = ( collect @ A @ P2 ) )
% 0.29/0.48 = ( ! [X4: A] :
% 0.29/0.48 ~ ( P2 @ X4 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % empty_Collect_eq
% 0.29/0.48 thf(fact_73_Collect__empty__eq,axiom,
% 0.29/0.48 ! [A: $tType,P2: A > $o] :
% 0.29/0.48 ( ( ( collect @ A @ P2 )
% 0.29/0.48 = ( bot_bot @ ( set @ A ) ) )
% 0.29/0.48 = ( ! [X4: A] :
% 0.29/0.48 ~ ( P2 @ X4 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Collect_empty_eq
% 0.29/0.48 thf(fact_74_all__not__in__conv,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A] :
% 0.29/0.48 ( ( ! [X4: A] :
% 0.29/0.48 ~ ( member @ A @ X4 @ A3 ) )
% 0.29/0.48 = ( A3
% 0.29/0.48 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % all_not_in_conv
% 0.29/0.48 thf(fact_75_empty__iff,axiom,
% 0.29/0.48 ! [A: $tType,C: A] :
% 0.29/0.48 ~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % empty_iff
% 0.29/0.48 thf(fact_76_emptyE,axiom,
% 0.29/0.48 ! [A: $tType,A2: A] :
% 0.29/0.48 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % emptyE
% 0.29/0.48 thf(fact_77_equals0D,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A,A2: A] :
% 0.29/0.48 ( ( A3
% 0.29/0.48 = ( bot_bot @ ( set @ A ) ) )
% 0.29/0.48 => ~ ( member @ A @ A2 @ A3 ) ) ).
% 0.29/0.48
% 0.29/0.48 % equals0D
% 0.29/0.48 thf(fact_78_equals0I,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A] :
% 0.29/0.48 ( ! [Y4: A] :
% 0.29/0.48 ~ ( member @ A @ Y4 @ A3 )
% 0.29/0.48 => ( A3
% 0.29/0.48 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % equals0I
% 0.29/0.48 thf(fact_79_ex__in__conv,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A] :
% 0.29/0.48 ( ( ? [X4: A] : ( member @ A @ X4 @ A3 ) )
% 0.29/0.48 = ( A3
% 0.29/0.48 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % ex_in_conv
% 0.29/0.48 thf(fact_80_set__mp,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A,B4: set @ A,X5: A] :
% 0.29/0.48 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 0.29/0.48 => ( ( member @ A @ X5 @ A3 )
% 0.29/0.48 => ( member @ A @ X5 @ B4 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % set_mp
% 0.29/0.48 thf(fact_81_in__mono,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A,B4: set @ A,X5: A] :
% 0.29/0.48 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 0.29/0.48 => ( ( member @ A @ X5 @ A3 )
% 0.29/0.48 => ( member @ A @ X5 @ B4 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % in_mono
% 0.29/0.48 thf(fact_82_subsetD,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A,B4: set @ A,C: A] :
% 0.29/0.48 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 0.29/0.48 => ( ( member @ A @ C @ A3 )
% 0.29/0.48 => ( member @ A @ C @ B4 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % subsetD
% 0.29/0.48 thf(fact_83_subsetCE,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A,B4: set @ A,C: A] :
% 0.29/0.48 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 0.29/0.48 => ( ( member @ A @ C @ A3 )
% 0.29/0.48 => ( member @ A @ C @ B4 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % subsetCE
% 0.29/0.48 thf(fact_84_equalityE,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A,B4: set @ A] :
% 0.29/0.48 ( ( A3 = B4 )
% 0.29/0.48 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 0.29/0.48 => ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A3 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % equalityE
% 0.29/0.48 thf(fact_85_subset__eq,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( ord_less_eq @ ( set @ A ) )
% 0.29/0.48 = ( ^ [A5: set @ A,B5: set @ A] :
% 0.29/0.48 ! [X4: A] :
% 0.29/0.48 ( ( member @ A @ X4 @ A5 )
% 0.29/0.48 => ( member @ A @ X4 @ B5 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % subset_eq
% 0.29/0.48 thf(fact_86_equalityD1,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A,B4: set @ A] :
% 0.29/0.48 ( ( A3 = B4 )
% 0.29/0.48 => ( ord_less_eq @ ( set @ A ) @ A3 @ B4 ) ) ).
% 0.29/0.48
% 0.29/0.48 % equalityD1
% 0.29/0.48 thf(fact_87_equalityD2,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A,B4: set @ A] :
% 0.29/0.48 ( ( A3 = B4 )
% 0.29/0.48 => ( ord_less_eq @ ( set @ A ) @ B4 @ A3 ) ) ).
% 0.29/0.48
% 0.29/0.48 % equalityD2
% 0.29/0.48 thf(fact_88_set__rev__mp,axiom,
% 0.29/0.48 ! [A: $tType,X5: A,A3: set @ A,B4: set @ A] :
% 0.29/0.48 ( ( member @ A @ X5 @ A3 )
% 0.29/0.48 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 0.29/0.48 => ( member @ A @ X5 @ B4 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % set_rev_mp
% 0.29/0.48 thf(fact_89_subset__iff,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( ord_less_eq @ ( set @ A ) )
% 0.29/0.48 = ( ^ [A5: set @ A,B5: set @ A] :
% 0.29/0.48 ! [T3: A] :
% 0.29/0.48 ( ( member @ A @ T3 @ A5 )
% 0.29/0.48 => ( member @ A @ T3 @ B5 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % subset_iff
% 0.29/0.48 thf(fact_90_rev__subsetD,axiom,
% 0.29/0.48 ! [A: $tType,C: A,A3: set @ A,B4: set @ A] :
% 0.29/0.48 ( ( member @ A @ C @ A3 )
% 0.29/0.48 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 0.29/0.48 => ( member @ A @ C @ B4 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % rev_subsetD
% 0.29/0.48 thf(fact_91_subset__refl,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% 0.29/0.48
% 0.29/0.48 % subset_refl
% 0.29/0.48 thf(fact_92_Collect__mono,axiom,
% 0.29/0.48 ! [A: $tType,P2: A > $o,Q3: A > $o] :
% 0.29/0.48 ( ! [X: A] :
% 0.29/0.48 ( ( P2 @ X )
% 0.29/0.48 => ( Q3 @ X ) )
% 0.29/0.48 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q3 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Collect_mono
% 0.29/0.48 thf(fact_93_subset__trans,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A,B4: set @ A,C2: set @ A] :
% 0.29/0.48 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 0.29/0.48 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ C2 )
% 0.29/0.48 => ( ord_less_eq @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % subset_trans
% 0.29/0.48 thf(fact_94_set__eq__subset,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( ^ [Y5: set @ A,Z3: set @ A] : ( Y5 = Z3 ) )
% 0.29/0.48 = ( ^ [A5: set @ A,B5: set @ A] :
% 0.29/0.48 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.29/0.48 & ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % set_eq_subset
% 0.29/0.48 thf(fact_95_contra__subsetD,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A,B4: set @ A,C: A] :
% 0.29/0.48 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 0.29/0.48 => ( ~ ( member @ A @ C @ B4 )
% 0.29/0.48 => ~ ( member @ A @ C @ A3 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % contra_subsetD
% 0.29/0.48 thf(fact_96_Collect__mono__iff,axiom,
% 0.29/0.48 ! [A: $tType,P2: A > $o,Q3: A > $o] :
% 0.29/0.48 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q3 ) )
% 0.29/0.48 = ( ! [X4: A] :
% 0.29/0.48 ( ( P2 @ X4 )
% 0.29/0.48 => ( Q3 @ X4 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Collect_mono_iff
% 0.29/0.48 thf(fact_97_bot__apply,axiom,
% 0.29/0.48 ! [C3: $tType,D2: $tType] :
% 0.29/0.48 ( ( bot @ C3 @ ( type2 @ C3 ) )
% 0.29/0.48 => ( ( bot_bot @ ( D2 > C3 ) )
% 0.29/0.48 = ( ^ [X4: D2] : ( bot_bot @ C3 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % bot_apply
% 0.29/0.48 thf(fact_98_order__refl,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.48 => ! [X5: A] : ( ord_less_eq @ A @ X5 @ X5 ) ) ).
% 0.29/0.48
% 0.29/0.48 % order_refl
% 0.29/0.48 thf(fact_99_subset__code_I1_J,axiom,
% 0.29/0.48 ! [A: $tType,Xs: list @ A,B4: set @ A] :
% 0.29/0.48 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B4 )
% 0.29/0.48 = ( ! [X4: A] :
% 0.29/0.48 ( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
% 0.29/0.48 => ( member @ A @ X4 @ B4 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % subset_code(1)
% 0.29/0.48 thf(fact_100_subset__emptyI,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A] :
% 0.29/0.48 ( ! [X: A] :
% 0.29/0.48 ~ ( member @ A @ X @ A3 )
% 0.29/0.48 => ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % subset_emptyI
% 0.29/0.48 thf(fact_101_bot_Oextremum__strict,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( order_bot @ A @ ( type2 @ A ) )
% 0.29/0.48 => ! [A2: A] :
% 0.29/0.48 ~ ( ord_less @ A @ A2 @ ( bot_bot @ A ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % bot.extremum_strict
% 0.29/0.48 thf(fact_102_bot_Onot__eq__extremum,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( order_bot @ A @ ( type2 @ A ) )
% 0.29/0.48 => ! [A2: A] :
% 0.29/0.48 ( ( A2
% 0.29/0.48 != ( bot_bot @ A ) )
% 0.29/0.48 = ( ord_less @ A @ ( bot_bot @ A ) @ A2 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % bot.not_eq_extremum
% 0.29/0.48 thf(fact_103_psubsetI,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A,B4: set @ A] :
% 0.29/0.48 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 0.29/0.48 => ( ( A3 != B4 )
% 0.29/0.48 => ( ord_less @ ( set @ A ) @ A3 @ B4 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % psubsetI
% 0.29/0.48 thf(fact_104_bot__set__def,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( bot_bot @ ( set @ A ) )
% 0.29/0.48 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % bot_set_def
% 0.29/0.48 thf(fact_105_not__psubset__empty,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A] :
% 0.29/0.48 ~ ( ord_less @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % not_psubset_empty
% 0.29/0.48 thf(fact_106_psubsetE,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A,B4: set @ A] :
% 0.29/0.48 ( ( ord_less @ ( set @ A ) @ A3 @ B4 )
% 0.29/0.48 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 0.29/0.48 => ( ord_less_eq @ ( set @ A ) @ B4 @ A3 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % psubsetE
% 0.29/0.48 thf(fact_107_psubset__eq,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( ord_less @ ( set @ A ) )
% 0.29/0.48 = ( ^ [A5: set @ A,B5: set @ A] :
% 0.29/0.48 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.29/0.48 & ( A5 != B5 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % psubset_eq
% 0.29/0.48 thf(fact_108_psubset__imp__subset,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A,B4: set @ A] :
% 0.29/0.48 ( ( ord_less @ ( set @ A ) @ A3 @ B4 )
% 0.29/0.48 => ( ord_less_eq @ ( set @ A ) @ A3 @ B4 ) ) ).
% 0.29/0.48
% 0.29/0.48 % psubset_imp_subset
% 0.29/0.48 thf(fact_109_psubset__subset__trans,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A,B4: set @ A,C2: set @ A] :
% 0.29/0.48 ( ( ord_less @ ( set @ A ) @ A3 @ B4 )
% 0.29/0.48 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ C2 )
% 0.29/0.48 => ( ord_less @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % psubset_subset_trans
% 0.29/0.48 thf(fact_110_subset__not__subset__eq,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( ord_less @ ( set @ A ) )
% 0.29/0.48 = ( ^ [A5: set @ A,B5: set @ A] :
% 0.29/0.48 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.29/0.48 & ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % subset_not_subset_eq
% 0.29/0.48 thf(fact_111_subset__psubset__trans,axiom,
% 0.29/0.48 ! [A: $tType,A3: set @ A,B4: set @ A,C2: set @ A] :
% 0.29/0.48 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 0.29/0.48 => ( ( ord_less @ ( set @ A ) @ B4 @ C2 )
% 0.29/0.48 => ( ord_less @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % subset_psubset_trans
% 0.29/0.48 thf(fact_112_subset__iff__psubset__eq,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( ord_less_eq @ ( set @ A ) )
% 0.29/0.48 = ( ^ [A5: set @ A,B5: set @ A] :
% 0.29/0.48 ( ( ord_less @ ( set @ A ) @ A5 @ B5 )
% 0.29/0.48 | ( A5 = B5 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % subset_iff_psubset_eq
% 0.29/0.48 thf(fact_113_dag__less__le,axiom,
% 0.29/0.48 ( ( ord_less @ binDag_Mirabelle_dag )
% 0.29/0.48 = ( ^ [X4: binDag_Mirabelle_dag,Y3: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( ord_less_eq @ binDag_Mirabelle_dag @ X4 @ Y3 )
% 0.29/0.48 & ( X4 != Y3 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % dag_less_le
% 0.29/0.48 thf(fact_114_le__dag__def,axiom,
% 0.29/0.48 ( ( ord_less_eq @ binDag_Mirabelle_dag )
% 0.29/0.48 = ( ^ [S2: binDag_Mirabelle_dag,T3: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( S2 = T3 )
% 0.29/0.48 | ( ord_less @ binDag_Mirabelle_dag @ S2 @ T3 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % le_dag_def
% 0.29/0.48 thf(fact_115_less__dag__Node,axiom,
% 0.29/0.48 ! [X5: binDag_Mirabelle_dag,L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( ord_less @ binDag_Mirabelle_dag @ X5 @ ( binDag476092410e_Node @ L @ A2 @ R ) )
% 0.29/0.48 = ( ( ord_less_eq @ binDag_Mirabelle_dag @ X5 @ L )
% 0.29/0.48 | ( ord_less_eq @ binDag_Mirabelle_dag @ X5 @ R ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % less_dag_Node
% 0.29/0.48 thf(fact_116_le__dag__set__of,axiom,
% 0.29/0.48 ! [X5: binDag_Mirabelle_dag,Y2: binDag_Mirabelle_dag] :
% 0.29/0.48 ( ( ord_less_eq @ binDag_Mirabelle_dag @ X5 @ Y2 )
% 0.29/0.48 => ( ord_less_eq @ ( set @ simpl_ref ) @ ( binDag1380252983set_of @ X5 ) @ ( binDag1380252983set_of @ Y2 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % le_dag_set_of
% 0.29/0.48 thf(fact_117_dual__order_Oantisym,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.48 => ! [B2: A,A2: A] :
% 0.29/0.48 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.29/0.48 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.48 => ( A2 = B2 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % dual_order.antisym
% 0.29/0.48 thf(fact_118_dual__order_Otrans,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.48 => ! [B2: A,A2: A,C: A] :
% 0.29/0.48 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.29/0.48 => ( ( ord_less_eq @ A @ C @ B2 )
% 0.29/0.48 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % dual_order.trans
% 0.29/0.48 thf(fact_119_linorder__wlog,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.48 => ! [P2: A > A > $o,A2: A,B2: A] :
% 0.29/0.48 ( ! [A6: A,B6: A] :
% 0.29/0.48 ( ( ord_less_eq @ A @ A6 @ B6 )
% 0.29/0.48 => ( P2 @ A6 @ B6 ) )
% 0.29/0.48 => ( ! [A6: A,B6: A] :
% 0.29/0.48 ( ( P2 @ B6 @ A6 )
% 0.29/0.48 => ( P2 @ A6 @ B6 ) )
% 0.29/0.48 => ( P2 @ A2 @ B2 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % linorder_wlog
% 0.29/0.48 thf(fact_120_dual__order_Orefl,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.48 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.29/0.48
% 0.29/0.48 % dual_order.refl
% 0.29/0.48 thf(fact_121_order__trans,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.48 => ! [X5: A,Y2: A,Z2: A] :
% 0.29/0.48 ( ( ord_less_eq @ A @ X5 @ Y2 )
% 0.29/0.48 => ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 0.29/0.48 => ( ord_less_eq @ A @ X5 @ Z2 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % order_trans
% 0.29/0.48 thf(fact_122_order__class_Oorder_Oantisym,axiom,
% 0.29/0.48 ! [A: $tType] :
% 0.29/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.48 => ! [A2: A,B2: A] :
% 0.29/0.48 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.48 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.29/0.48 => ( A2 = B2 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % order_class.order.antisym
% 0.29/0.48 thf(fact_123_ord__le__eq__trans,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( ord @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [A2: A,B2: A,C: A] :
% 0.29/0.49 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.49 => ( ( B2 = C )
% 0.29/0.49 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % ord_le_eq_trans
% 0.29/0.49 thf(fact_124_ord__eq__le__trans,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( ord @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [A2: A,B2: A,C: A] :
% 0.29/0.49 ( ( A2 = B2 )
% 0.29/0.49 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.29/0.49 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % ord_eq_le_trans
% 0.29/0.49 thf(fact_125_antisym__conv,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [Y2: A,X5: A] :
% 0.29/0.49 ( ( ord_less_eq @ A @ Y2 @ X5 )
% 0.29/0.49 => ( ( ord_less_eq @ A @ X5 @ Y2 )
% 0.29/0.49 = ( X5 = Y2 ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % antisym_conv
% 0.29/0.49 thf(fact_126_le__cases3,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A,Y2: A,Z2: A] :
% 0.29/0.49 ( ( ( ord_less_eq @ A @ X5 @ Y2 )
% 0.29/0.49 => ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
% 0.29/0.49 => ( ( ( ord_less_eq @ A @ Y2 @ X5 )
% 0.29/0.49 => ~ ( ord_less_eq @ A @ X5 @ Z2 ) )
% 0.29/0.49 => ( ( ( ord_less_eq @ A @ X5 @ Z2 )
% 0.29/0.49 => ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
% 0.29/0.49 => ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
% 0.29/0.49 => ~ ( ord_less_eq @ A @ Y2 @ X5 ) )
% 0.29/0.49 => ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 0.29/0.49 => ~ ( ord_less_eq @ A @ Z2 @ X5 ) )
% 0.29/0.49 => ~ ( ( ord_less_eq @ A @ Z2 @ X5 )
% 0.29/0.49 => ~ ( ord_less_eq @ A @ X5 @ Y2 ) ) ) ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % le_cases3
% 0.29/0.49 thf(fact_127_order_Otrans,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [A2: A,B2: A,C: A] :
% 0.29/0.49 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.49 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.29/0.49 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % order.trans
% 0.29/0.49 thf(fact_128_le__cases,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A,Y2: A] :
% 0.29/0.49 ( ~ ( ord_less_eq @ A @ X5 @ Y2 )
% 0.29/0.49 => ( ord_less_eq @ A @ Y2 @ X5 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % le_cases
% 0.29/0.49 thf(fact_129_eq__refl,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A,Y2: A] :
% 0.29/0.49 ( ( X5 = Y2 )
% 0.29/0.49 => ( ord_less_eq @ A @ X5 @ Y2 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % eq_refl
% 0.29/0.49 thf(fact_130_linear,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A,Y2: A] :
% 0.29/0.49 ( ( ord_less_eq @ A @ X5 @ Y2 )
% 0.29/0.49 | ( ord_less_eq @ A @ Y2 @ X5 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % linear
% 0.29/0.49 thf(fact_131_antisym,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A,Y2: A] :
% 0.29/0.49 ( ( ord_less_eq @ A @ X5 @ Y2 )
% 0.29/0.49 => ( ( ord_less_eq @ A @ Y2 @ X5 )
% 0.29/0.49 => ( X5 = Y2 ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % antisym
% 0.29/0.49 thf(fact_132_eq__iff,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.49 => ( ( ^ [Y5: A,Z3: A] : ( Y5 = Z3 ) )
% 0.29/0.49 = ( ^ [X4: A,Y3: A] :
% 0.29/0.49 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.29/0.49 & ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % eq_iff
% 0.29/0.49 thf(fact_133_ord__le__eq__subst,axiom,
% 0.29/0.49 ! [A: $tType,B: $tType] :
% 0.29/0.49 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.49 & ( ord @ A @ ( type2 @ A ) ) )
% 0.29/0.49 => ! [A2: A,B2: A,F3: A > B,C: B] :
% 0.29/0.49 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.49 => ( ( ( F3 @ B2 )
% 0.29/0.49 = C )
% 0.29/0.49 => ( ! [X: A,Y4: A] :
% 0.29/0.49 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.29/0.49 => ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
% 0.29/0.49 => ( ord_less_eq @ B @ ( F3 @ A2 ) @ C ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % ord_le_eq_subst
% 0.29/0.49 thf(fact_134_ord__eq__le__subst,axiom,
% 0.29/0.49 ! [A: $tType,B: $tType] :
% 0.29/0.49 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.49 & ( ord @ A @ ( type2 @ A ) ) )
% 0.29/0.49 => ! [A2: A,F3: B > A,B2: B,C: B] :
% 0.29/0.49 ( ( A2
% 0.29/0.49 = ( F3 @ B2 ) )
% 0.29/0.49 => ( ( ord_less_eq @ B @ B2 @ C )
% 0.29/0.49 => ( ! [X: B,Y4: B] :
% 0.29/0.49 ( ( ord_less_eq @ B @ X @ Y4 )
% 0.29/0.49 => ( ord_less_eq @ A @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
% 0.29/0.49 => ( ord_less_eq @ A @ A2 @ ( F3 @ C ) ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % ord_eq_le_subst
% 0.29/0.49 thf(fact_135_order__subst2,axiom,
% 0.29/0.49 ! [A: $tType,C3: $tType] :
% 0.29/0.49 ( ( ( order @ C3 @ ( type2 @ C3 ) )
% 0.29/0.49 & ( order @ A @ ( type2 @ A ) ) )
% 0.29/0.49 => ! [A2: A,B2: A,F3: A > C3,C: C3] :
% 0.29/0.49 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.49 => ( ( ord_less_eq @ C3 @ ( F3 @ B2 ) @ C )
% 0.29/0.49 => ( ! [X: A,Y4: A] :
% 0.29/0.49 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.29/0.49 => ( ord_less_eq @ C3 @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
% 0.29/0.49 => ( ord_less_eq @ C3 @ ( F3 @ A2 ) @ C ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % order_subst2
% 0.29/0.49 thf(fact_136_order__subst1,axiom,
% 0.29/0.49 ! [A: $tType,B: $tType] :
% 0.29/0.49 ( ( ( order @ B @ ( type2 @ B ) )
% 0.29/0.49 & ( order @ A @ ( type2 @ A ) ) )
% 0.29/0.49 => ! [A2: A,F3: B > A,B2: B,C: B] :
% 0.29/0.49 ( ( ord_less_eq @ A @ A2 @ ( F3 @ B2 ) )
% 0.29/0.49 => ( ( ord_less_eq @ B @ B2 @ C )
% 0.29/0.49 => ( ! [X: B,Y4: B] :
% 0.29/0.49 ( ( ord_less_eq @ B @ X @ Y4 )
% 0.29/0.49 => ( ord_less_eq @ A @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
% 0.29/0.49 => ( ord_less_eq @ A @ A2 @ ( F3 @ C ) ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % order_subst1
% 0.29/0.49 thf(fact_137_le__fun__def,axiom,
% 0.29/0.49 ! [B: $tType,A: $tType] :
% 0.29/0.49 ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.49 => ( ( ord_less_eq @ ( A > B ) )
% 0.29/0.49 = ( ^ [F: A > B,G2: A > B] :
% 0.29/0.49 ! [X4: A] : ( ord_less_eq @ B @ ( F @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % le_fun_def
% 0.29/0.49 thf(fact_138_le__funI,axiom,
% 0.29/0.49 ! [B: $tType,A: $tType] :
% 0.29/0.49 ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.49 => ! [F3: A > B,G: A > B] :
% 0.29/0.49 ( ! [X: A] : ( ord_less_eq @ B @ ( F3 @ X ) @ ( G @ X ) )
% 0.29/0.49 => ( ord_less_eq @ ( A > B ) @ F3 @ G ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % le_funI
% 0.29/0.49 thf(fact_139_le__funE,axiom,
% 0.29/0.49 ! [B: $tType,A: $tType] :
% 0.29/0.49 ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.49 => ! [F3: A > B,G: A > B,X5: A] :
% 0.29/0.49 ( ( ord_less_eq @ ( A > B ) @ F3 @ G )
% 0.29/0.49 => ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( G @ X5 ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % le_funE
% 0.29/0.49 thf(fact_140_le__funD,axiom,
% 0.29/0.49 ! [B: $tType,A: $tType] :
% 0.29/0.49 ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.49 => ! [F3: A > B,G: A > B,X5: A] :
% 0.29/0.49 ( ( ord_less_eq @ ( A > B ) @ F3 @ G )
% 0.29/0.49 => ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( G @ X5 ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % le_funD
% 0.29/0.49 thf(fact_141_dual__order_Ostrict__implies__not__eq,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [B2: A,A2: A] :
% 0.29/0.49 ( ( ord_less @ A @ B2 @ A2 )
% 0.29/0.49 => ( A2 != B2 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % dual_order.strict_implies_not_eq
% 0.29/0.49 thf(fact_142_order_Ostrict__implies__not__eq,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [A2: A,B2: A] :
% 0.29/0.49 ( ( ord_less @ A @ A2 @ B2 )
% 0.29/0.49 => ( A2 != B2 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % order.strict_implies_not_eq
% 0.29/0.49 thf(fact_143_not__less__iff__gr__or__eq,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A,Y2: A] :
% 0.29/0.49 ( ( ~ ( ord_less @ A @ X5 @ Y2 ) )
% 0.29/0.49 = ( ( ord_less @ A @ Y2 @ X5 )
% 0.29/0.49 | ( X5 = Y2 ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % not_less_iff_gr_or_eq
% 0.29/0.49 thf(fact_144_dual__order_Ostrict__trans,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [B2: A,A2: A,C: A] :
% 0.29/0.49 ( ( ord_less @ A @ B2 @ A2 )
% 0.29/0.49 => ( ( ord_less @ A @ C @ B2 )
% 0.29/0.49 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % dual_order.strict_trans
% 0.29/0.49 thf(fact_145_less__imp__not__less,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A,Y2: A] :
% 0.29/0.49 ( ( ord_less @ A @ X5 @ Y2 )
% 0.29/0.49 => ~ ( ord_less @ A @ Y2 @ X5 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % less_imp_not_less
% 0.29/0.49 thf(fact_146_order_Ostrict__trans,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [A2: A,B2: A,C: A] :
% 0.29/0.49 ( ( ord_less @ A @ A2 @ B2 )
% 0.29/0.49 => ( ( ord_less @ A @ B2 @ C )
% 0.29/0.49 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % order.strict_trans
% 0.29/0.49 thf(fact_147_dual__order_Oirrefl,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [A2: A] :
% 0.29/0.49 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 0.29/0.49
% 0.29/0.49 % dual_order.irrefl
% 0.29/0.49 thf(fact_148_linorder__cases,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A,Y2: A] :
% 0.29/0.49 ( ~ ( ord_less @ A @ X5 @ Y2 )
% 0.29/0.49 => ( ( X5 != Y2 )
% 0.29/0.49 => ( ord_less @ A @ Y2 @ X5 ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % linorder_cases
% 0.29/0.49 thf(fact_149_less__imp__triv,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A,Y2: A,P2: $o] :
% 0.29/0.49 ( ( ord_less @ A @ X5 @ Y2 )
% 0.29/0.49 => ( ( ord_less @ A @ Y2 @ X5 )
% 0.29/0.49 => P2 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % less_imp_triv
% 0.29/0.49 thf(fact_150_less__imp__not__eq2,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A,Y2: A] :
% 0.29/0.49 ( ( ord_less @ A @ X5 @ Y2 )
% 0.29/0.49 => ( Y2 != X5 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % less_imp_not_eq2
% 0.29/0.49 thf(fact_151_antisym__conv3,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [Y2: A,X5: A] :
% 0.29/0.49 ( ~ ( ord_less @ A @ Y2 @ X5 )
% 0.29/0.49 => ( ( ~ ( ord_less @ A @ X5 @ Y2 ) )
% 0.29/0.49 = ( X5 = Y2 ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % antisym_conv3
% 0.29/0.49 thf(fact_152_less__induct,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [P2: A > $o,A2: A] :
% 0.29/0.49 ( ! [X: A] :
% 0.29/0.49 ( ! [Y: A] :
% 0.29/0.49 ( ( ord_less @ A @ Y @ X )
% 0.29/0.49 => ( P2 @ Y ) )
% 0.29/0.49 => ( P2 @ X ) )
% 0.29/0.49 => ( P2 @ A2 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % less_induct
% 0.29/0.49 thf(fact_153_less__not__sym,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A,Y2: A] :
% 0.29/0.49 ( ( ord_less @ A @ X5 @ Y2 )
% 0.29/0.49 => ~ ( ord_less @ A @ Y2 @ X5 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % less_not_sym
% 0.29/0.49 thf(fact_154_less__imp__not__eq,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A,Y2: A] :
% 0.29/0.49 ( ( ord_less @ A @ X5 @ Y2 )
% 0.29/0.49 => ( X5 != Y2 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % less_imp_not_eq
% 0.29/0.49 thf(fact_155_dual__order_Oasym,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [B2: A,A2: A] :
% 0.29/0.49 ( ( ord_less @ A @ B2 @ A2 )
% 0.29/0.49 => ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % dual_order.asym
% 0.29/0.49 thf(fact_156_ord__less__eq__trans,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( ord @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [A2: A,B2: A,C: A] :
% 0.29/0.49 ( ( ord_less @ A @ A2 @ B2 )
% 0.29/0.49 => ( ( B2 = C )
% 0.29/0.49 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % ord_less_eq_trans
% 0.29/0.49 thf(fact_157_ord__eq__less__trans,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( ord @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [A2: A,B2: A,C: A] :
% 0.29/0.49 ( ( A2 = B2 )
% 0.29/0.49 => ( ( ord_less @ A @ B2 @ C )
% 0.29/0.49 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % ord_eq_less_trans
% 0.29/0.49 thf(fact_158_less__irrefl,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A] :
% 0.29/0.49 ~ ( ord_less @ A @ X5 @ X5 ) ) ).
% 0.29/0.49
% 0.29/0.49 % less_irrefl
% 0.29/0.49 thf(fact_159_less__linear,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A,Y2: A] :
% 0.29/0.49 ( ( ord_less @ A @ X5 @ Y2 )
% 0.29/0.49 | ( X5 = Y2 )
% 0.29/0.49 | ( ord_less @ A @ Y2 @ X5 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % less_linear
% 0.29/0.49 thf(fact_160_less__trans,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A,Y2: A,Z2: A] :
% 0.29/0.49 ( ( ord_less @ A @ X5 @ Y2 )
% 0.29/0.49 => ( ( ord_less @ A @ Y2 @ Z2 )
% 0.29/0.49 => ( ord_less @ A @ X5 @ Z2 ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % less_trans
% 0.29/0.49 thf(fact_161_less__asym_H,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [A2: A,B2: A] :
% 0.29/0.49 ( ( ord_less @ A @ A2 @ B2 )
% 0.29/0.49 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % less_asym'
% 0.29/0.49 thf(fact_162_less__asym,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A,Y2: A] :
% 0.29/0.49 ( ( ord_less @ A @ X5 @ Y2 )
% 0.29/0.49 => ~ ( ord_less @ A @ Y2 @ X5 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % less_asym
% 0.29/0.49 thf(fact_163_less__imp__neq,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A,Y2: A] :
% 0.29/0.49 ( ( ord_less @ A @ X5 @ Y2 )
% 0.29/0.49 => ( X5 != Y2 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % less_imp_neq
% 0.29/0.49 thf(fact_164_dense,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( dense_order @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A,Y2: A] :
% 0.29/0.49 ( ( ord_less @ A @ X5 @ Y2 )
% 0.29/0.49 => ? [Z4: A] :
% 0.29/0.49 ( ( ord_less @ A @ X5 @ Z4 )
% 0.29/0.49 & ( ord_less @ A @ Z4 @ Y2 ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % dense
% 0.29/0.49 thf(fact_165_order_Oasym,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [A2: A,B2: A] :
% 0.29/0.49 ( ( ord_less @ A @ A2 @ B2 )
% 0.29/0.49 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % order.asym
% 0.29/0.49 thf(fact_166_neq__iff,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A,Y2: A] :
% 0.29/0.49 ( ( X5 != Y2 )
% 0.29/0.49 = ( ( ord_less @ A @ X5 @ Y2 )
% 0.29/0.49 | ( ord_less @ A @ Y2 @ X5 ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % neq_iff
% 0.29/0.49 thf(fact_167_neqE,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A,Y2: A] :
% 0.29/0.49 ( ( X5 != Y2 )
% 0.29/0.49 => ( ~ ( ord_less @ A @ X5 @ Y2 )
% 0.29/0.49 => ( ord_less @ A @ Y2 @ X5 ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % neqE
% 0.29/0.49 thf(fact_168_gt__ex,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( no_top @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A] :
% 0.29/0.49 ? [X1: A] : ( ord_less @ A @ X5 @ X1 ) ) ).
% 0.29/0.49
% 0.29/0.49 % gt_ex
% 0.29/0.49 thf(fact_169_lt__ex,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( no_bot @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A] :
% 0.29/0.49 ? [Y4: A] : ( ord_less @ A @ Y4 @ X5 ) ) ).
% 0.29/0.49
% 0.29/0.49 % lt_ex
% 0.29/0.49 thf(fact_170_order__less__subst2,axiom,
% 0.29/0.49 ! [A: $tType,C3: $tType] :
% 0.29/0.49 ( ( ( order @ C3 @ ( type2 @ C3 ) )
% 0.29/0.49 & ( order @ A @ ( type2 @ A ) ) )
% 0.29/0.49 => ! [A2: A,B2: A,F3: A > C3,C: C3] :
% 0.29/0.49 ( ( ord_less @ A @ A2 @ B2 )
% 0.29/0.49 => ( ( ord_less @ C3 @ ( F3 @ B2 ) @ C )
% 0.29/0.49 => ( ! [X: A,Y4: A] :
% 0.29/0.49 ( ( ord_less @ A @ X @ Y4 )
% 0.29/0.49 => ( ord_less @ C3 @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
% 0.29/0.49 => ( ord_less @ C3 @ ( F3 @ A2 ) @ C ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % order_less_subst2
% 0.29/0.49 thf(fact_171_order__less__subst1,axiom,
% 0.29/0.49 ! [A: $tType,B: $tType] :
% 0.29/0.49 ( ( ( order @ B @ ( type2 @ B ) )
% 0.29/0.49 & ( order @ A @ ( type2 @ A ) ) )
% 0.29/0.49 => ! [A2: A,F3: B > A,B2: B,C: B] :
% 0.29/0.49 ( ( ord_less @ A @ A2 @ ( F3 @ B2 ) )
% 0.29/0.49 => ( ( ord_less @ B @ B2 @ C )
% 0.29/0.49 => ( ! [X: B,Y4: B] :
% 0.29/0.49 ( ( ord_less @ B @ X @ Y4 )
% 0.29/0.49 => ( ord_less @ A @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
% 0.29/0.49 => ( ord_less @ A @ A2 @ ( F3 @ C ) ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % order_less_subst1
% 0.29/0.49 thf(fact_172_ord__less__eq__subst,axiom,
% 0.29/0.49 ! [A: $tType,B: $tType] :
% 0.29/0.49 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.49 & ( ord @ A @ ( type2 @ A ) ) )
% 0.29/0.49 => ! [A2: A,B2: A,F3: A > B,C: B] :
% 0.29/0.49 ( ( ord_less @ A @ A2 @ B2 )
% 0.29/0.49 => ( ( ( F3 @ B2 )
% 0.29/0.49 = C )
% 0.29/0.49 => ( ! [X: A,Y4: A] :
% 0.29/0.49 ( ( ord_less @ A @ X @ Y4 )
% 0.29/0.49 => ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
% 0.29/0.49 => ( ord_less @ B @ ( F3 @ A2 ) @ C ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % ord_less_eq_subst
% 0.29/0.49 thf(fact_173_ord__eq__less__subst,axiom,
% 0.29/0.49 ! [A: $tType,B: $tType] :
% 0.29/0.49 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.49 & ( ord @ A @ ( type2 @ A ) ) )
% 0.29/0.49 => ! [A2: A,F3: B > A,B2: B,C: B] :
% 0.29/0.49 ( ( A2
% 0.29/0.49 = ( F3 @ B2 ) )
% 0.29/0.49 => ( ( ord_less @ B @ B2 @ C )
% 0.29/0.49 => ( ! [X: B,Y4: B] :
% 0.29/0.49 ( ( ord_less @ B @ X @ Y4 )
% 0.29/0.49 => ( ord_less @ A @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
% 0.29/0.49 => ( ord_less @ A @ A2 @ ( F3 @ C ) ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % ord_eq_less_subst
% 0.29/0.49 thf(fact_174_bot__fun__def,axiom,
% 0.29/0.49 ! [B: $tType,A: $tType] :
% 0.29/0.49 ( ( bot @ B @ ( type2 @ B ) )
% 0.29/0.49 => ( ( bot_bot @ ( A > B ) )
% 0.29/0.49 = ( ^ [X4: A] : ( bot_bot @ B ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % bot_fun_def
% 0.29/0.49 thf(fact_175_less__DAG__set__of,axiom,
% 0.29/0.49 ! [X5: binDag_Mirabelle_dag,Y2: binDag_Mirabelle_dag] :
% 0.29/0.49 ( ( ord_less @ binDag_Mirabelle_dag @ X5 @ Y2 )
% 0.29/0.49 => ( ( binDag_Mirabelle_DAG @ Y2 )
% 0.29/0.49 => ( ord_less @ ( set @ simpl_ref ) @ ( binDag1380252983set_of @ X5 ) @ ( binDag1380252983set_of @ Y2 ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % less_DAG_set_of
% 0.29/0.49 thf(fact_176_order_Onot__eq__order__implies__strict,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [A2: A,B2: A] :
% 0.29/0.49 ( ( A2 != B2 )
% 0.29/0.49 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.49 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % order.not_eq_order_implies_strict
% 0.29/0.49 thf(fact_177_dual__order_Ostrict__implies__order,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [B2: A,A2: A] :
% 0.29/0.49 ( ( ord_less @ A @ B2 @ A2 )
% 0.29/0.49 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % dual_order.strict_implies_order
% 0.29/0.49 thf(fact_178_dual__order_Ostrict__iff__order,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.49 => ( ( ord_less @ A )
% 0.29/0.49 = ( ^ [B3: A,A4: A] :
% 0.29/0.49 ( ( ord_less_eq @ A @ B3 @ A4 )
% 0.29/0.49 & ( A4 != B3 ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % dual_order.strict_iff_order
% 0.29/0.49 thf(fact_179_dual__order_Oorder__iff__strict,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.49 => ( ( ord_less_eq @ A )
% 0.29/0.49 = ( ^ [B3: A,A4: A] :
% 0.29/0.49 ( ( ord_less @ A @ B3 @ A4 )
% 0.29/0.49 | ( A4 = B3 ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % dual_order.order_iff_strict
% 0.29/0.49 thf(fact_180_order_Ostrict__implies__order,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [A2: A,B2: A] :
% 0.29/0.49 ( ( ord_less @ A @ A2 @ B2 )
% 0.29/0.49 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % order.strict_implies_order
% 0.29/0.49 thf(fact_181_dense__le__bounded,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A,Y2: A,Z2: A] :
% 0.29/0.49 ( ( ord_less @ A @ X5 @ Y2 )
% 0.29/0.49 => ( ! [W: A] :
% 0.29/0.49 ( ( ord_less @ A @ X5 @ W )
% 0.29/0.49 => ( ( ord_less @ A @ W @ Y2 )
% 0.29/0.49 => ( ord_less_eq @ A @ W @ Z2 ) ) )
% 0.29/0.49 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % dense_le_bounded
% 0.29/0.49 thf(fact_182_dense__ge__bounded,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [Z2: A,X5: A,Y2: A] :
% 0.29/0.49 ( ( ord_less @ A @ Z2 @ X5 )
% 0.29/0.49 => ( ! [W: A] :
% 0.29/0.49 ( ( ord_less @ A @ Z2 @ W )
% 0.29/0.49 => ( ( ord_less @ A @ W @ X5 )
% 0.29/0.49 => ( ord_less_eq @ A @ Y2 @ W ) ) )
% 0.29/0.49 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % dense_ge_bounded
% 0.29/0.49 thf(fact_183_dual__order_Ostrict__trans2,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [B2: A,A2: A,C: A] :
% 0.29/0.49 ( ( ord_less @ A @ B2 @ A2 )
% 0.29/0.49 => ( ( ord_less_eq @ A @ C @ B2 )
% 0.29/0.49 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % dual_order.strict_trans2
% 0.29/0.49 thf(fact_184_dual__order_Ostrict__trans1,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [B2: A,A2: A,C: A] :
% 0.29/0.49 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.29/0.49 => ( ( ord_less @ A @ C @ B2 )
% 0.29/0.49 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % dual_order.strict_trans1
% 0.29/0.49 thf(fact_185_order_Ostrict__iff__order,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.49 => ( ( ord_less @ A )
% 0.29/0.49 = ( ^ [A4: A,B3: A] :
% 0.29/0.49 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.29/0.49 & ( A4 != B3 ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % order.strict_iff_order
% 0.29/0.49 thf(fact_186_order_Oorder__iff__strict,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.49 => ( ( ord_less_eq @ A )
% 0.29/0.49 = ( ^ [A4: A,B3: A] :
% 0.29/0.49 ( ( ord_less @ A @ A4 @ B3 )
% 0.29/0.49 | ( A4 = B3 ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % order.order_iff_strict
% 0.29/0.49 thf(fact_187_order_Ostrict__trans2,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [A2: A,B2: A,C: A] :
% 0.29/0.49 ( ( ord_less @ A @ A2 @ B2 )
% 0.29/0.49 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.29/0.49 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % order.strict_trans2
% 0.29/0.49 thf(fact_188_order_Ostrict__trans1,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [A2: A,B2: A,C: A] :
% 0.29/0.49 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.49 => ( ( ord_less @ A @ B2 @ C )
% 0.29/0.49 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % order.strict_trans1
% 0.29/0.49 thf(fact_189_not__le__imp__less,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [Y2: A,X5: A] :
% 0.29/0.49 ( ~ ( ord_less_eq @ A @ Y2 @ X5 )
% 0.29/0.49 => ( ord_less @ A @ X5 @ Y2 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % not_le_imp_less
% 0.29/0.49 thf(fact_190_less__le__not__le,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ( ( ord_less @ A )
% 0.29/0.49 = ( ^ [X4: A,Y3: A] :
% 0.29/0.49 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.29/0.49 & ~ ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % less_le_not_le
% 0.29/0.49 thf(fact_191_le__imp__less__or__eq,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A,Y2: A] :
% 0.29/0.49 ( ( ord_less_eq @ A @ X5 @ Y2 )
% 0.29/0.49 => ( ( ord_less @ A @ X5 @ Y2 )
% 0.29/0.49 | ( X5 = Y2 ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % le_imp_less_or_eq
% 0.29/0.49 thf(fact_192_le__less__linear,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A,Y2: A] :
% 0.29/0.49 ( ( ord_less_eq @ A @ X5 @ Y2 )
% 0.29/0.49 | ( ord_less @ A @ Y2 @ X5 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % le_less_linear
% 0.29/0.49 thf(fact_193_dense__le,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [Y2: A,Z2: A] :
% 0.29/0.49 ( ! [X: A] :
% 0.29/0.49 ( ( ord_less @ A @ X @ Y2 )
% 0.29/0.49 => ( ord_less_eq @ A @ X @ Z2 ) )
% 0.29/0.49 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % dense_le
% 0.29/0.49 thf(fact_194_dense__ge,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [Z2: A,Y2: A] :
% 0.29/0.49 ( ! [X: A] :
% 0.29/0.49 ( ( ord_less @ A @ Z2 @ X )
% 0.29/0.49 => ( ord_less_eq @ A @ Y2 @ X ) )
% 0.29/0.49 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % dense_ge
% 0.29/0.49 thf(fact_195_less__le__trans,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A,Y2: A,Z2: A] :
% 0.29/0.49 ( ( ord_less @ A @ X5 @ Y2 )
% 0.29/0.49 => ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 0.29/0.49 => ( ord_less @ A @ X5 @ Z2 ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % less_le_trans
% 0.29/0.49 thf(fact_196_le__less__trans,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A,Y2: A,Z2: A] :
% 0.29/0.49 ( ( ord_less_eq @ A @ X5 @ Y2 )
% 0.29/0.49 => ( ( ord_less @ A @ Y2 @ Z2 )
% 0.29/0.49 => ( ord_less @ A @ X5 @ Z2 ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % le_less_trans
% 0.29/0.49 thf(fact_197_antisym__conv2,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A,Y2: A] :
% 0.29/0.49 ( ( ord_less_eq @ A @ X5 @ Y2 )
% 0.29/0.49 => ( ( ~ ( ord_less @ A @ X5 @ Y2 ) )
% 0.29/0.49 = ( X5 = Y2 ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % antisym_conv2
% 0.29/0.49 thf(fact_198_antisym__conv1,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A,Y2: A] :
% 0.29/0.49 ( ~ ( ord_less @ A @ X5 @ Y2 )
% 0.29/0.49 => ( ( ord_less_eq @ A @ X5 @ Y2 )
% 0.29/0.49 = ( X5 = Y2 ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % antisym_conv1
% 0.29/0.49 thf(fact_199_less__imp__le,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A,Y2: A] :
% 0.29/0.49 ( ( ord_less @ A @ X5 @ Y2 )
% 0.29/0.49 => ( ord_less_eq @ A @ X5 @ Y2 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % less_imp_le
% 0.29/0.49 thf(fact_200_le__neq__trans,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [A2: A,B2: A] :
% 0.29/0.49 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.49 => ( ( A2 != B2 )
% 0.29/0.49 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % le_neq_trans
% 0.29/0.49 thf(fact_201_not__less,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A,Y2: A] :
% 0.29/0.49 ( ( ~ ( ord_less @ A @ X5 @ Y2 ) )
% 0.29/0.49 = ( ord_less_eq @ A @ Y2 @ X5 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % not_less
% 0.29/0.49 thf(fact_202_not__le,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A,Y2: A] :
% 0.29/0.49 ( ( ~ ( ord_less_eq @ A @ X5 @ Y2 ) )
% 0.29/0.49 = ( ord_less @ A @ Y2 @ X5 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % not_le
% 0.29/0.49 thf(fact_203_order__less__le__subst2,axiom,
% 0.29/0.49 ! [A: $tType,C3: $tType] :
% 0.29/0.49 ( ( ( order @ C3 @ ( type2 @ C3 ) )
% 0.29/0.49 & ( order @ A @ ( type2 @ A ) ) )
% 0.29/0.49 => ! [A2: A,B2: A,F3: A > C3,C: C3] :
% 0.29/0.49 ( ( ord_less @ A @ A2 @ B2 )
% 0.29/0.49 => ( ( ord_less_eq @ C3 @ ( F3 @ B2 ) @ C )
% 0.29/0.49 => ( ! [X: A,Y4: A] :
% 0.29/0.49 ( ( ord_less @ A @ X @ Y4 )
% 0.29/0.49 => ( ord_less @ C3 @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
% 0.29/0.49 => ( ord_less @ C3 @ ( F3 @ A2 ) @ C ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % order_less_le_subst2
% 0.29/0.49 thf(fact_204_order__less__le__subst1,axiom,
% 0.29/0.49 ! [A: $tType,B: $tType] :
% 0.29/0.49 ( ( ( order @ B @ ( type2 @ B ) )
% 0.29/0.49 & ( order @ A @ ( type2 @ A ) ) )
% 0.29/0.49 => ! [A2: A,F3: B > A,B2: B,C: B] :
% 0.29/0.49 ( ( ord_less @ A @ A2 @ ( F3 @ B2 ) )
% 0.29/0.49 => ( ( ord_less_eq @ B @ B2 @ C )
% 0.29/0.49 => ( ! [X: B,Y4: B] :
% 0.29/0.49 ( ( ord_less_eq @ B @ X @ Y4 )
% 0.29/0.49 => ( ord_less_eq @ A @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
% 0.29/0.49 => ( ord_less @ A @ A2 @ ( F3 @ C ) ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % order_less_le_subst1
% 0.29/0.49 thf(fact_205_order__le__less__subst2,axiom,
% 0.29/0.49 ! [A: $tType,C3: $tType] :
% 0.29/0.49 ( ( ( order @ C3 @ ( type2 @ C3 ) )
% 0.29/0.49 & ( order @ A @ ( type2 @ A ) ) )
% 0.29/0.49 => ! [A2: A,B2: A,F3: A > C3,C: C3] :
% 0.29/0.49 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.49 => ( ( ord_less @ C3 @ ( F3 @ B2 ) @ C )
% 0.29/0.49 => ( ! [X: A,Y4: A] :
% 0.29/0.49 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.29/0.49 => ( ord_less_eq @ C3 @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
% 0.29/0.49 => ( ord_less @ C3 @ ( F3 @ A2 ) @ C ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % order_le_less_subst2
% 0.29/0.49 thf(fact_206_order__le__less__subst1,axiom,
% 0.29/0.49 ! [A: $tType,B: $tType] :
% 0.29/0.49 ( ( ( order @ B @ ( type2 @ B ) )
% 0.29/0.49 & ( order @ A @ ( type2 @ A ) ) )
% 0.29/0.49 => ! [A2: A,F3: B > A,B2: B,C: B] :
% 0.29/0.49 ( ( ord_less_eq @ A @ A2 @ ( F3 @ B2 ) )
% 0.29/0.49 => ( ( ord_less @ B @ B2 @ C )
% 0.29/0.49 => ( ! [X: B,Y4: B] :
% 0.29/0.49 ( ( ord_less @ B @ X @ Y4 )
% 0.29/0.49 => ( ord_less @ A @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
% 0.29/0.49 => ( ord_less @ A @ A2 @ ( F3 @ C ) ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % order_le_less_subst1
% 0.29/0.49 thf(fact_207_less__le,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.49 => ( ( ord_less @ A )
% 0.29/0.49 = ( ^ [X4: A,Y3: A] :
% 0.29/0.49 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.29/0.49 & ( X4 != Y3 ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % less_le
% 0.29/0.49 thf(fact_208_le__less,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.49 => ( ( ord_less_eq @ A )
% 0.29/0.49 = ( ^ [X4: A,Y3: A] :
% 0.29/0.49 ( ( ord_less @ A @ X4 @ Y3 )
% 0.29/0.49 | ( X4 = Y3 ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % le_less
% 0.29/0.49 thf(fact_209_leI,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [X5: A,Y2: A] :
% 0.29/0.49 ( ~ ( ord_less @ A @ X5 @ Y2 )
% 0.29/0.49 => ( ord_less_eq @ A @ Y2 @ X5 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % leI
% 0.29/0.49 thf(fact_210_leD,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [Y2: A,X5: A] :
% 0.29/0.49 ( ( ord_less_eq @ A @ Y2 @ X5 )
% 0.29/0.49 => ~ ( ord_less @ A @ X5 @ Y2 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % leD
% 0.29/0.49 thf(fact_211_bot_Oextremum__uniqueI,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( order_bot @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [A2: A] :
% 0.29/0.49 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 0.29/0.49 => ( A2
% 0.29/0.49 = ( bot_bot @ A ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % bot.extremum_uniqueI
% 0.29/0.49 thf(fact_212_bot_Oextremum__unique,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( order_bot @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [A2: A] :
% 0.29/0.49 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 0.29/0.49 = ( A2
% 0.29/0.49 = ( bot_bot @ A ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % bot.extremum_unique
% 0.29/0.49 thf(fact_213_bot_Oextremum,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( order_bot @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% 0.29/0.49
% 0.29/0.49 % bot.extremum
% 0.29/0.49 thf(fact_214_complete__interval,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( condit1037483654norder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [A2: A,B2: A,P2: A > $o] :
% 0.29/0.49 ( ( ord_less @ A @ A2 @ B2 )
% 0.29/0.49 => ( ( P2 @ A2 )
% 0.29/0.49 => ( ~ ( P2 @ B2 )
% 0.29/0.49 => ? [C4: A] :
% 0.29/0.49 ( ( ord_less_eq @ A @ A2 @ C4 )
% 0.29/0.49 & ( ord_less_eq @ A @ C4 @ B2 )
% 0.29/0.49 & ! [X6: A] :
% 0.29/0.49 ( ( ( ord_less_eq @ A @ A2 @ X6 )
% 0.29/0.49 & ( ord_less @ A @ X6 @ C4 ) )
% 0.29/0.49 => ( P2 @ X6 ) )
% 0.29/0.49 & ! [D3: A] :
% 0.29/0.49 ( ! [X: A] :
% 0.29/0.49 ( ( ( ord_less_eq @ A @ A2 @ X )
% 0.29/0.49 & ( ord_less @ A @ X @ D3 ) )
% 0.29/0.49 => ( P2 @ X ) )
% 0.29/0.49 => ( ord_less_eq @ A @ D3 @ C4 ) ) ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % complete_interval
% 0.29/0.49 thf(fact_215_pinf_I6_J,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [T: A] :
% 0.29/0.49 ? [Z4: A] :
% 0.29/0.49 ! [X6: A] :
% 0.29/0.49 ( ( ord_less @ A @ Z4 @ X6 )
% 0.29/0.49 => ~ ( ord_less_eq @ A @ X6 @ T ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % pinf(6)
% 0.29/0.49 thf(fact_216_pinf_I8_J,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [T: A] :
% 0.29/0.49 ? [Z4: A] :
% 0.29/0.49 ! [X6: A] :
% 0.29/0.49 ( ( ord_less @ A @ Z4 @ X6 )
% 0.29/0.49 => ( ord_less_eq @ A @ T @ X6 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % pinf(8)
% 0.29/0.49 thf(fact_217_minf_I6_J,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [T: A] :
% 0.29/0.49 ? [Z4: A] :
% 0.29/0.49 ! [X6: A] :
% 0.29/0.49 ( ( ord_less @ A @ X6 @ Z4 )
% 0.29/0.49 => ( ord_less_eq @ A @ X6 @ T ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % minf(6)
% 0.29/0.49 thf(fact_218_minf_I8_J,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [T: A] :
% 0.29/0.49 ? [Z4: A] :
% 0.29/0.49 ! [X6: A] :
% 0.29/0.49 ( ( ord_less @ A @ X6 @ Z4 )
% 0.29/0.49 => ~ ( ord_less_eq @ A @ T @ X6 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % minf(8)
% 0.29/0.49 thf(fact_219_le__dag__antisym,axiom,
% 0.29/0.49 ! [X5: binDag_Mirabelle_dag,Y2: binDag_Mirabelle_dag] :
% 0.29/0.49 ( ( ord_less_eq @ binDag_Mirabelle_dag @ X5 @ Y2 )
% 0.29/0.49 => ( ( ord_less_eq @ binDag_Mirabelle_dag @ Y2 @ X5 )
% 0.29/0.49 => ( X5 = Y2 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % le_dag_antisym
% 0.29/0.49 thf(fact_220_le__dag__trans,axiom,
% 0.29/0.49 ! [X5: binDag_Mirabelle_dag,Y2: binDag_Mirabelle_dag,Z2: binDag_Mirabelle_dag] :
% 0.29/0.49 ( ( ord_less_eq @ binDag_Mirabelle_dag @ X5 @ Y2 )
% 0.29/0.49 => ( ( ord_less_eq @ binDag_Mirabelle_dag @ Y2 @ Z2 )
% 0.29/0.49 => ( ord_less_eq @ binDag_Mirabelle_dag @ X5 @ Z2 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % le_dag_trans
% 0.29/0.49 thf(fact_221_le__dag__refl,axiom,
% 0.29/0.49 ! [X5: binDag_Mirabelle_dag] : ( ord_less_eq @ binDag_Mirabelle_dag @ X5 @ X5 ) ).
% 0.29/0.49
% 0.29/0.49 % le_dag_refl
% 0.29/0.49 thf(fact_222_less__fun__def,axiom,
% 0.29/0.49 ! [B: $tType,A: $tType] :
% 0.29/0.49 ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.49 => ( ( ord_less @ ( A > B ) )
% 0.29/0.49 = ( ^ [F: A > B,G2: A > B] :
% 0.29/0.49 ( ( ord_less_eq @ ( A > B ) @ F @ G2 )
% 0.29/0.49 & ~ ( ord_less_eq @ ( A > B ) @ G2 @ F ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % less_fun_def
% 0.29/0.49 thf(fact_223_psubset__trans,axiom,
% 0.29/0.49 ! [A: $tType,A3: set @ A,B4: set @ A,C2: set @ A] :
% 0.29/0.49 ( ( ord_less @ ( set @ A ) @ A3 @ B4 )
% 0.29/0.49 => ( ( ord_less @ ( set @ A ) @ B4 @ C2 )
% 0.29/0.49 => ( ord_less @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % psubset_trans
% 0.29/0.49 thf(fact_224_psubsetD,axiom,
% 0.29/0.49 ! [A: $tType,A3: set @ A,B4: set @ A,C: A] :
% 0.29/0.49 ( ( ord_less @ ( set @ A ) @ A3 @ B4 )
% 0.29/0.49 => ( ( member @ A @ C @ A3 )
% 0.29/0.49 => ( member @ A @ C @ B4 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % psubsetD
% 0.29/0.49 thf(fact_225_minf_I11_J,axiom,
% 0.29/0.49 ! [C3: $tType,D2: $tType] :
% 0.29/0.49 ( ( ord @ C3 @ ( type2 @ C3 ) )
% 0.29/0.49 => ! [F4: D2] :
% 0.29/0.49 ? [Z4: C3] :
% 0.29/0.49 ! [X6: C3] :
% 0.29/0.49 ( ( ord_less @ C3 @ X6 @ Z4 )
% 0.29/0.49 => ( F4 = F4 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % minf(11)
% 0.29/0.49 thf(fact_226_minf_I7_J,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [T: A] :
% 0.29/0.49 ? [Z4: A] :
% 0.29/0.49 ! [X6: A] :
% 0.29/0.49 ( ( ord_less @ A @ X6 @ Z4 )
% 0.29/0.49 => ~ ( ord_less @ A @ T @ X6 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % minf(7)
% 0.29/0.49 thf(fact_227_minf_I5_J,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [T: A] :
% 0.29/0.49 ? [Z4: A] :
% 0.29/0.49 ! [X6: A] :
% 0.29/0.49 ( ( ord_less @ A @ X6 @ Z4 )
% 0.29/0.49 => ( ord_less @ A @ X6 @ T ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % minf(5)
% 0.29/0.49 thf(fact_228_minf_I4_J,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [T: A] :
% 0.29/0.49 ? [Z4: A] :
% 0.29/0.49 ! [X6: A] :
% 0.29/0.49 ( ( ord_less @ A @ X6 @ Z4 )
% 0.29/0.49 => ( X6 != T ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % minf(4)
% 0.29/0.49 thf(fact_229_minf_I3_J,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [T: A] :
% 0.29/0.49 ? [Z4: A] :
% 0.29/0.49 ! [X6: A] :
% 0.29/0.49 ( ( ord_less @ A @ X6 @ Z4 )
% 0.29/0.49 => ( X6 != T ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % minf(3)
% 0.29/0.49 thf(fact_230_minf_I2_J,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [P2: A > $o,P3: A > $o,Q3: A > $o,Q4: A > $o] :
% 0.29/0.49 ( ? [Z5: A] :
% 0.29/0.49 ! [X: A] :
% 0.29/0.49 ( ( ord_less @ A @ X @ Z5 )
% 0.29/0.49 => ( ( P2 @ X )
% 0.29/0.49 = ( P3 @ X ) ) )
% 0.29/0.49 => ( ? [Z5: A] :
% 0.29/0.49 ! [X: A] :
% 0.29/0.49 ( ( ord_less @ A @ X @ Z5 )
% 0.29/0.49 => ( ( Q3 @ X )
% 0.29/0.49 = ( Q4 @ X ) ) )
% 0.29/0.49 => ? [Z4: A] :
% 0.29/0.49 ! [X6: A] :
% 0.29/0.49 ( ( ord_less @ A @ X6 @ Z4 )
% 0.29/0.49 => ( ( ( P2 @ X6 )
% 0.29/0.49 | ( Q3 @ X6 ) )
% 0.29/0.49 = ( ( P3 @ X6 )
% 0.29/0.49 | ( Q4 @ X6 ) ) ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % minf(2)
% 0.29/0.49 thf(fact_231_minf_I1_J,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [P2: A > $o,P3: A > $o,Q3: A > $o,Q4: A > $o] :
% 0.29/0.49 ( ? [Z5: A] :
% 0.29/0.49 ! [X: A] :
% 0.29/0.49 ( ( ord_less @ A @ X @ Z5 )
% 0.29/0.49 => ( ( P2 @ X )
% 0.29/0.49 = ( P3 @ X ) ) )
% 0.29/0.49 => ( ? [Z5: A] :
% 0.29/0.49 ! [X: A] :
% 0.29/0.49 ( ( ord_less @ A @ X @ Z5 )
% 0.29/0.49 => ( ( Q3 @ X )
% 0.29/0.49 = ( Q4 @ X ) ) )
% 0.29/0.49 => ? [Z4: A] :
% 0.29/0.49 ! [X6: A] :
% 0.29/0.49 ( ( ord_less @ A @ X6 @ Z4 )
% 0.29/0.49 => ( ( ( P2 @ X6 )
% 0.29/0.49 & ( Q3 @ X6 ) )
% 0.29/0.49 = ( ( P3 @ X6 )
% 0.29/0.49 & ( Q4 @ X6 ) ) ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % minf(1)
% 0.29/0.49 thf(fact_232_pinf_I11_J,axiom,
% 0.29/0.49 ! [C3: $tType,D2: $tType] :
% 0.29/0.49 ( ( ord @ C3 @ ( type2 @ C3 ) )
% 0.29/0.49 => ! [F4: D2] :
% 0.29/0.49 ? [Z4: C3] :
% 0.29/0.49 ! [X6: C3] :
% 0.29/0.49 ( ( ord_less @ C3 @ Z4 @ X6 )
% 0.29/0.49 => ( F4 = F4 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % pinf(11)
% 0.29/0.49 thf(fact_233_pinf_I7_J,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [T: A] :
% 0.29/0.49 ? [Z4: A] :
% 0.29/0.49 ! [X6: A] :
% 0.29/0.49 ( ( ord_less @ A @ Z4 @ X6 )
% 0.29/0.49 => ( ord_less @ A @ T @ X6 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % pinf(7)
% 0.29/0.49 thf(fact_234_pinf_I5_J,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [T: A] :
% 0.29/0.49 ? [Z4: A] :
% 0.29/0.49 ! [X6: A] :
% 0.29/0.49 ( ( ord_less @ A @ Z4 @ X6 )
% 0.29/0.49 => ~ ( ord_less @ A @ X6 @ T ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % pinf(5)
% 0.29/0.49 thf(fact_235_pinf_I4_J,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [T: A] :
% 0.29/0.49 ? [Z4: A] :
% 0.29/0.49 ! [X6: A] :
% 0.29/0.49 ( ( ord_less @ A @ Z4 @ X6 )
% 0.29/0.49 => ( X6 != T ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % pinf(4)
% 0.29/0.49 thf(fact_236_pinf_I3_J,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [T: A] :
% 0.29/0.49 ? [Z4: A] :
% 0.29/0.49 ! [X6: A] :
% 0.29/0.49 ( ( ord_less @ A @ Z4 @ X6 )
% 0.29/0.49 => ( X6 != T ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % pinf(3)
% 0.29/0.49 thf(fact_237_pinf_I2_J,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [P2: A > $o,P3: A > $o,Q3: A > $o,Q4: A > $o] :
% 0.29/0.49 ( ? [Z5: A] :
% 0.29/0.49 ! [X: A] :
% 0.29/0.49 ( ( ord_less @ A @ Z5 @ X )
% 0.29/0.49 => ( ( P2 @ X )
% 0.29/0.49 = ( P3 @ X ) ) )
% 0.29/0.49 => ( ? [Z5: A] :
% 0.29/0.49 ! [X: A] :
% 0.29/0.49 ( ( ord_less @ A @ Z5 @ X )
% 0.29/0.49 => ( ( Q3 @ X )
% 0.29/0.49 = ( Q4 @ X ) ) )
% 0.29/0.49 => ? [Z4: A] :
% 0.29/0.49 ! [X6: A] :
% 0.29/0.49 ( ( ord_less @ A @ Z4 @ X6 )
% 0.29/0.49 => ( ( ( P2 @ X6 )
% 0.29/0.49 | ( Q3 @ X6 ) )
% 0.29/0.49 = ( ( P3 @ X6 )
% 0.29/0.49 | ( Q4 @ X6 ) ) ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % pinf(2)
% 0.29/0.49 thf(fact_238_pinf_I1_J,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [P2: A > $o,P3: A > $o,Q3: A > $o,Q4: A > $o] :
% 0.29/0.49 ( ? [Z5: A] :
% 0.29/0.49 ! [X: A] :
% 0.29/0.49 ( ( ord_less @ A @ Z5 @ X )
% 0.29/0.49 => ( ( P2 @ X )
% 0.29/0.49 = ( P3 @ X ) ) )
% 0.29/0.49 => ( ? [Z5: A] :
% 0.29/0.49 ! [X: A] :
% 0.29/0.49 ( ( ord_less @ A @ Z5 @ X )
% 0.29/0.49 => ( ( Q3 @ X )
% 0.29/0.49 = ( Q4 @ X ) ) )
% 0.29/0.49 => ? [Z4: A] :
% 0.29/0.49 ! [X6: A] :
% 0.29/0.49 ( ( ord_less @ A @ Z4 @ X6 )
% 0.29/0.49 => ( ( ( P2 @ X6 )
% 0.29/0.49 & ( Q3 @ X6 ) )
% 0.29/0.49 = ( ( P3 @ X6 )
% 0.29/0.49 & ( Q4 @ X6 ) ) ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % pinf(1)
% 0.29/0.49 thf(fact_239_ex__gt__or__lt,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [A2: A] :
% 0.29/0.49 ? [B6: A] :
% 0.29/0.49 ( ( ord_less @ A @ A2 @ B6 )
% 0.29/0.49 | ( ord_less @ A @ B6 @ A2 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % ex_gt_or_lt
% 0.29/0.49 thf(fact_240_bot__empty__eq,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( bot_bot @ ( A > $o ) )
% 0.29/0.49 = ( ^ [X4: A] : ( member @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % bot_empty_eq
% 0.29/0.49 thf(fact_241_Collect__empty__eq__bot,axiom,
% 0.29/0.49 ! [A: $tType,P2: A > $o] :
% 0.29/0.49 ( ( ( collect @ A @ P2 )
% 0.29/0.49 = ( bot_bot @ ( set @ A ) ) )
% 0.29/0.49 = ( P2
% 0.29/0.49 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % Collect_empty_eq_bot
% 0.29/0.49 thf(fact_242_override__on__emptyset,axiom,
% 0.29/0.49 ! [B: $tType,A: $tType,F3: A > B,G: A > B] :
% 0.29/0.49 ( ( override_on @ A @ B @ F3 @ G @ ( bot_bot @ ( set @ A ) ) )
% 0.29/0.49 = F3 ) ).
% 0.29/0.49
% 0.29/0.49 % override_on_emptyset
% 0.29/0.49 thf(fact_243_Set_Ois__empty__def,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( is_empty @ A )
% 0.29/0.49 = ( ^ [A5: set @ A] :
% 0.29/0.49 ( A5
% 0.29/0.49 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % Set.is_empty_def
% 0.29/0.49 thf(fact_244_override__on__apply__notin,axiom,
% 0.29/0.49 ! [B: $tType,A: $tType,A2: A,A3: set @ A,F3: A > B,G: A > B] :
% 0.29/0.49 ( ~ ( member @ A @ A2 @ A3 )
% 0.29/0.49 => ( ( override_on @ A @ B @ F3 @ G @ A3 @ A2 )
% 0.29/0.49 = ( F3 @ A2 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % override_on_apply_notin
% 0.29/0.49 thf(fact_245_override__on__apply__in,axiom,
% 0.29/0.49 ! [B: $tType,A: $tType,A2: A,A3: set @ A,F3: A > B,G: A > B] :
% 0.29/0.49 ( ( member @ A @ A2 @ A3 )
% 0.29/0.49 => ( ( override_on @ A @ B @ F3 @ G @ A3 @ A2 )
% 0.29/0.49 = ( G @ A2 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % override_on_apply_in
% 0.29/0.49 thf(fact_246_override__on__def,axiom,
% 0.29/0.49 ! [B: $tType,A: $tType] :
% 0.29/0.49 ( ( override_on @ A @ B )
% 0.29/0.49 = ( ^ [F: A > B,G2: A > B,A5: set @ A,A4: A] : ( if @ B @ ( member @ A @ A4 @ A5 ) @ ( G2 @ A4 ) @ ( F @ A4 ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % override_on_def
% 0.29/0.49 thf(fact_247_is__empty__set,axiom,
% 0.29/0.49 ! [A: $tType,Xs: list @ A] :
% 0.29/0.49 ( ( is_empty @ A @ ( set2 @ A @ Xs ) )
% 0.29/0.49 = ( null @ A @ Xs ) ) ).
% 0.29/0.49
% 0.29/0.49 % is_empty_set
% 0.29/0.49 thf(fact_248_subset__code_I2_J,axiom,
% 0.29/0.49 ! [B: $tType,A3: set @ B,Ys: list @ B] :
% 0.29/0.49 ( ( ord_less_eq @ ( set @ B ) @ A3 @ ( coset @ B @ Ys ) )
% 0.29/0.49 = ( ! [X4: B] :
% 0.29/0.49 ( ( member @ B @ X4 @ ( set2 @ B @ Ys ) )
% 0.29/0.49 => ~ ( member @ B @ X4 @ A3 ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % subset_code(2)
% 0.29/0.49 thf(fact_249_subset__code_I3_J,axiom,
% 0.29/0.49 ! [C3: $tType] :
% 0.29/0.49 ~ ( ord_less_eq @ ( set @ C3 ) @ ( coset @ C3 @ ( nil @ C3 ) ) @ ( set2 @ C3 @ ( nil @ C3 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % subset_code(3)
% 0.29/0.49 thf(fact_250_set__empty2,axiom,
% 0.29/0.49 ! [A: $tType,Xs: list @ A] :
% 0.29/0.49 ( ( ( bot_bot @ ( set @ A ) )
% 0.29/0.49 = ( set2 @ A @ Xs ) )
% 0.29/0.49 = ( Xs
% 0.29/0.49 = ( nil @ A ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % set_empty2
% 0.29/0.49 thf(fact_251_set__empty,axiom,
% 0.29/0.49 ! [A: $tType,Xs: list @ A] :
% 0.29/0.49 ( ( ( set2 @ A @ Xs )
% 0.29/0.49 = ( bot_bot @ ( set @ A ) ) )
% 0.29/0.49 = ( Xs
% 0.29/0.49 = ( nil @ A ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % set_empty
% 0.29/0.49 thf(fact_252_empty__set,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( bot_bot @ ( set @ A ) )
% 0.29/0.49 = ( set2 @ A @ ( nil @ A ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % empty_set
% 0.29/0.49 thf(fact_253_sorted__list__of__set__empty,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ( ( linord467138063of_set @ A @ ( bot_bot @ ( set @ A ) ) )
% 0.29/0.49 = ( nil @ A ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % sorted_list_of_set_empty
% 0.29/0.49 thf(fact_254_sorted__list__of__set__eq__Nil__iff,axiom,
% 0.29/0.49 ! [A: $tType] :
% 0.29/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.49 => ! [A3: set @ A] :
% 0.29/0.49 ( ( finite_finite @ A @ A3 )
% 0.29/0.49 => ( ( ( linord467138063of_set @ A @ A3 )
% 0.29/0.49 = ( nil @ A ) )
% 0.29/0.49 = ( A3
% 0.29/0.49 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 % sorted_list_of_set_eq_Nil_iff
% 0.29/0.49 thf(fact_255_List_Ofinite__set,axiom,
% 0.29/0.49 ! [A: $tType,Xs: list @ A] : ( finite_finite @ A @ ( set2 @ A @ Xs ) ) ).
% 0.29/0.49
% 0.29/0.49 % List.finite_set
% 0.29/0.49
% 0.29/0.49 %----Type constructors (19)
% 0.29/0.49 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 0.29/0.49 ! [A7: $tType,A8: $tType] :
% 0.29/0.49 ( ( order_bot @ A8 @ ( type2 @ A8 ) )
% 0.29/0.49 => ( order_bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.29/0.49 ! [A7: $tType,A8: $tType] :
% 0.29/0.49 ( ( preorder @ A8 @ ( type2 @ A8 ) )
% 0.29/0.49 => ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.29/0.49 ! [A7: $tType,A8: $tType] :
% 0.29/0.49 ( ( order @ A8 @ ( type2 @ A8 ) )
% 0.29/0.49 => ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 thf(tcon_fun___Orderings_Oord,axiom,
% 0.29/0.49 ! [A7: $tType,A8: $tType] :
% 0.29/0.49 ( ( ord @ A8 @ ( type2 @ A8 ) )
% 0.29/0.49 => ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 thf(tcon_fun___Orderings_Obot,axiom,
% 0.29/0.49 ! [A7: $tType,A8: $tType] :
% 0.29/0.49 ( ( bot @ A8 @ ( type2 @ A8 ) )
% 0.29/0.49 => ( bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.29/0.49
% 0.29/0.49 thf(tcon_Set_Oset___Orderings_Oorder__bot_1,axiom,
% 0.29/0.49 ! [A7: $tType] : ( order_bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 thf(tcon_Set_Oset___Orderings_Opreorder_2,axiom,
% 0.29/0.49 ! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 thf(tcon_Set_Oset___Orderings_Oorder_3,axiom,
% 0.29/0.49 ! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 thf(tcon_Set_Oset___Orderings_Oord_4,axiom,
% 0.29/0.49 ! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 thf(tcon_Set_Oset___Orderings_Obot_5,axiom,
% 0.29/0.49 ! [A7: $tType] : ( bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.29/0.49
% 0.29/0.49 thf(tcon_HOL_Obool___Orderings_Oorder__bot_6,axiom,
% 0.29/0.49 order_bot @ $o @ ( type2 @ $o ) ).
% 0.29/0.49
% 0.29/0.49 thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
% 0.29/0.49 preorder @ $o @ ( type2 @ $o ) ).
% 0.29/0.49
% 0.29/0.49 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.29/0.49 linorder @ $o @ ( type2 @ $o ) ).
% 0.29/0.49
% 0.29/0.49 thf(tcon_HOL_Obool___Orderings_Oorder_8,axiom,
% 0.29/0.49 order @ $o @ ( type2 @ $o ) ).
% 0.29/0.49
% 0.29/0.49 thf(tcon_HOL_Obool___Orderings_Oord_9,axiom,
% 0.29/0.49 ord @ $o @ ( type2 @ $o ) ).
% 0.29/0.49
% 0.29/0.49 thf(tcon_HOL_Obool___Orderings_Obot_10,axiom,
% 0.29/0.49 bot @ $o @ ( type2 @ $o ) ).
% 0.29/0.49
% 0.29/0.49 thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Opreorder_11,axiom,
% 0.29/0.49 preorder @ binDag_Mirabelle_dag @ ( type2 @ binDag_Mirabelle_dag ) ).
% 0.29/0.49
% 0.29/0.49 thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Oorder_12,axiom,
% 0.29/0.49 order @ binDag_Mirabelle_dag @ ( type2 @ binDag_Mirabelle_dag ) ).
% 0.29/0.49
% 0.29/0.49 thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Oord_13,axiom,
% 0.29/0.49 ord @ binDag_Mirabelle_dag @ ( type2 @ binDag_Mirabelle_dag ) ).
% 0.29/0.49
% 0.29/0.49 %----Helper facts (3)
% 0.29/0.49 thf(help_If_3_1_T,axiom,
% 0.29/0.49 ! [P2: $o] :
% 0.29/0.49 ( ( P2 = $true )
% 0.29/0.49 | ( P2 = $false ) ) ).
% 0.29/0.49
% 0.29/0.49 thf(help_If_2_1_T,axiom,
% 0.29/0.49 ! [A: $tType,X5: A,Y2: A] :
% 0.29/0.49 ( ( if @ A @ $false @ X5 @ Y2 )
% 0.29/0.49 = Y2 ) ).
% 0.29/0.49
% 0.29/0.49 thf(help_If_1_1_T,axiom,
% 0.29/0.49 ! [A: $tType,X5: A,Y2: A] :
% 0.29/0.49 ( ( if @ A @ $true @ X5 @ Y2 )
% 0.29/0.49 = X5 ) ).
% 0.29/0.49
% 0.29/0.49 %----Conjectures (2)
% 0.29/0.49 thf(conj_0,hypothesis,
% 0.29/0.49 ! [Q5: simpl_ref] :
% 0.29/0.49 ( ( binDag_Mirabelle_Dag @ Q5 @ l @ r @ ( binDag476092410e_Node @ lt @ p @ rt ) )
% 0.29/0.49 => thesis ) ).
% 0.29/0.49
% 0.29/0.49 thf(conj_1,conjecture,
% 0.29/0.49 thesis ).
% 0.29/0.49
% 0.29/0.49 %------------------------------------------------------------------------------
% 0.29/0.49 ------------------------
% 2.85/1.76 ---- Embedded in THF ---
% 2.85/1.76 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.85/1.76 %%% Generated on Tue Feb 24 04:15:49 EST 2026
% 2.85/1.76 %%% using '$$dhol' embedding, version 1.3.0.
% 2.85/1.76 %%% Logic specification used:
% 2.85/1.76 %%% thf(spec, logic, $$dhol).
% 2.85/1.76
% 2.85/1.76 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.CLaN6L8RqG/DTF2DTF22996.p
% 2.85/1.76 thf(ty_t_BinDag__Mirabelle__rybootvolr_Odag, type, binDag_Mirabelle_dag: $tType).
% 2.85/1.76 thf(ty_t_Simpl__Heap_Oref, type, simpl_ref: $tType).
% 2.85/1.76 thf(ty_t_List_Olist, type, list: ($tType > $tType)).
% 2.85/1.76 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.85/1.76 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.85/1.76 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.76 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.76 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.76 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.76 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.76 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.76 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.76 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.76 thf(sy_cl_Orderings_Oorder__bot, type, order_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.76 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.76 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.76 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.76 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum, type, condit1656338222tinuum: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.76 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder, type, condit1037483654norder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.76 thf(sy_c_BinDag__Mirabelle__rybootvolr_ODAG, type, binDag_Mirabelle_DAG: (binDag_Mirabelle_dag > $o)).
% 2.85/1.76 thf(sy_c_BinDag__Mirabelle__rybootvolr_ODag, type, binDag_Mirabelle_Dag: (simpl_ref > ((simpl_ref > simpl_ref) > ((simpl_ref > simpl_ref) > (binDag_Mirabelle_dag > $o))))).
% 2.85/1.76 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_ONode, type, binDag476092410e_Node: (binDag_Mirabelle_dag > (simpl_ref > (binDag_Mirabelle_dag > binDag_Mirabelle_dag)))).
% 2.85/1.76 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_OTip, type, binDag_Mirabelle_Tip: binDag_Mirabelle_dag).
% 2.85/1.76 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_Ocase__dag, type, binDag1297733282se_dag: (!> [A:$tType]: ((A > ((binDag_Mirabelle_dag > (simpl_ref > (binDag_Mirabelle_dag > A))) > (binDag_Mirabelle_dag > A)))))).
% 2.85/1.76 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_Orec__dag, type, binDag1442713106ec_dag: (!> [A:$tType]: ((A > ((binDag_Mirabelle_dag > (simpl_ref > (binDag_Mirabelle_dag > (A > (A > A))))) > (binDag_Mirabelle_dag > A)))))).
% 2.85/1.76 thf(sy_c_BinDag__Mirabelle__rybootvolr_Oset__of, type, binDag1380252983set_of: (binDag_Mirabelle_dag > (set @ simpl_ref))).
% 2.85/1.76 thf(sy_c_BinDag__Mirabelle__rybootvolr_Osubdag, type, binDag786255756subdag: (binDag_Mirabelle_dag > (binDag_Mirabelle_dag > $o))).
% 2.85/1.76 thf(sy_c_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.85/1.76 thf(sy_c_Fun_Ofun__upd, type, fun_upd: (!> [A:$tType,B:$tType]: (((A > B) > (A > (B > (A > B))))))).
% 2.85/1.76 thf(sy_c_Fun_Ooverride__on, type, override_on: (!> [A:$tType,B:$tType]: (((A > B) > ((A > B) > ((set @ A) > (A > B))))))).
% 2.85/1.76 thf(sy_c_Fun_Oswap, type, swap: (!> [A:$tType,B:$tType]: ((A > (A > ((A > B) > (A > B))))))).
% 2.85/1.76 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.85/1.76 thf(sy_c_List_Ocoset, type, coset: (!> [A:$tType]: (((list @ A) > (set @ A))))).
% 2.85/1.76 thf(sy_c_List_Olinorder__class_Osorted__list__of__set, type, linord467138063of_set: (!> [A:$tType]: (((set @ A) > (list @ A))))).
% 2.85/1.76 thf(sy_c_List_Olist_ONil, type, nil: (!> [A:$tType]: ((list @ A)))).
% 2.85/1.76 thf(sy_c_List_Olist_Oset, type, set2: (!> [A:$tType]: (((list @ A) > (set @ A))))).
% 2.85/1.76 thf(sy_c_List_Onull, type, null: (!> [A:$tType]: (((list @ A) > $o)))).
% 2.85/1.76 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 2.85/1.76 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.85/1.76 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.85/1.76 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 2.85/1.76 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.85/1.76 thf(sy_c_Set_Ois__empty, type, is_empty: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.85/1.76 thf(sy_c_Simpl__Heap_ONull, type, simpl_Null: simpl_ref).
% 2.85/1.76 thf(sy_c_Simpl__Heap_Onew, type, simpl_new: ((set @ simpl_ref) > simpl_ref)).
% 2.85/1.76 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.85/1.76 thf(sy_v_l, type, l: (simpl_ref > simpl_ref)).
% 2.85/1.76 thf(sy_v_lt____, type, lt: binDag_Mirabelle_dag).
% 2.85/1.76 thf(sy_v_p, type, p: simpl_ref).
% 2.85/1.76 thf(sy_v_r, type, r: (simpl_ref > simpl_ref)).
% 2.85/1.76 thf(sy_v_rt____, type, rt: binDag_Mirabelle_dag).
% 2.85/1.76 thf(sy_v_t, type, t: binDag_Mirabelle_dag).
% 2.85/1.76 thf(sy_v_thesis____, type, thesis: $o).
% 2.85/1.76 thf(fact_0_assms, axiom, ((((binDag_Mirabelle_Dag @ (r @ p)) @ l) @ r) @ t)).
% 2.85/1.76 thf(fact_1_calculation, axiom, (! [Lt:binDag_Mirabelle_dag,Rt:binDag_Mirabelle_dag]: ((t != (((binDag476092410e_Node @ Lt) @ p) @ Rt))))).
% 2.85/1.76 thf(fact_2_subdag, axiom, ((binDag786255756subdag @ t) @ (((binDag476092410e_Node @ lt) @ p) @ rt))).
% 2.85/1.76 thf(fact_3_dag_Oinject, axiom, (! [X21:binDag_Mirabelle_dag,X22:simpl_ref,X23:binDag_Mirabelle_dag,Y21:binDag_Mirabelle_dag,Y22:simpl_ref,Y23:binDag_Mirabelle_dag]: ((((((binDag476092410e_Node @ X21) @ X22) @ X23) = (((binDag476092410e_Node @ Y21) @ Y22) @ Y23)) = ((X21 = Y21) & ((X22 = Y22) & (X23 = Y23))))))).
% 2.85/1.76 thf(fact_4_Dag__unique, axiom, (! [P:simpl_ref,L:(simpl_ref > simpl_ref),R:(simpl_ref > simpl_ref),T1:binDag_Mirabelle_dag,T2:binDag_Mirabelle_dag]: ((((((binDag_Mirabelle_Dag @ P) @ L) @ R) @ T1) => (((((binDag_Mirabelle_Dag @ P) @ L) @ R) @ T2) => (T1 = T2)))))).
% 2.85/1.76 thf(fact_5_Dag__unique1, axiom, (! [P:simpl_ref,L:(simpl_ref > simpl_ref),R:(simpl_ref > simpl_ref),T:binDag_Mirabelle_dag]: ((((((binDag_Mirabelle_Dag @ P) @ L) @ R) @ T) => (? [X:binDag_Mirabelle_dag]: ((((((binDag_Mirabelle_Dag @ P) @ L) @ R) @ X) & (! [Y:binDag_Mirabelle_dag]: ((((((binDag_Mirabelle_Dag @ P) @ L) @ R) @ Y) => (Y = X))))))))))).
% 2.85/1.76 thf(fact_6_subdag_Osimps_I2_J, axiom, (! [L:binDag_Mirabelle_dag,A2:simpl_ref,R:binDag_Mirabelle_dag,T:binDag_Mirabelle_dag]: ((((binDag786255756subdag @ (((binDag476092410e_Node @ L) @ A2) @ R)) @ T) = ((T = L) | ((T = R) | (((binDag786255756subdag @ L) @ T) | ((binDag786255756subdag @ R) @ T)))))))).
% 2.85/1.76 thf(fact_7_subdag__NodeD, axiom, (! [T:binDag_Mirabelle_dag,Lt2:binDag_Mirabelle_dag,A2:simpl_ref,Rt2:binDag_Mirabelle_dag]: ((((binDag786255756subdag @ T) @ (((binDag476092410e_Node @ Lt2) @ A2) @ Rt2)) => (((binDag786255756subdag @ T) @ Lt2) & ((binDag786255756subdag @ T) @ Rt2)))))).
% 2.85/1.76 thf(fact_8_Dag__subdag, axiom, (! [P:simpl_ref,L:(simpl_ref > simpl_ref),R:(simpl_ref > simpl_ref),T:binDag_Mirabelle_dag,S:binDag_Mirabelle_dag]: ((((((binDag_Mirabelle_Dag @ P) @ L) @ R) @ T) => (((binDag786255756subdag @ T) @ S) => (? [Q:simpl_ref]: (((((binDag_Mirabelle_Dag @ Q) @ L) @ R) @ S)))))))).
% 2.85/1.76 thf(fact_9_Dag__Ref, axiom, (! [P:simpl_ref,L:(simpl_ref > simpl_ref),R:(simpl_ref > simpl_ref),T:binDag_Mirabelle_dag]: (((P != simpl_Null) => (((((binDag_Mirabelle_Dag @ P) @ L) @ R) @ T) = (? [Lt3:binDag_Mirabelle_dag,Rt3:binDag_Mirabelle_dag]: (((T = (((binDag476092410e_Node @ Lt3) @ P) @ Rt3)) & (((((binDag_Mirabelle_Dag @ (L @ P)) @ L) @ R) @ Lt3) & ((((binDag_Mirabelle_Dag @ (R @ P)) @ L) @ R) @ Rt3)))))))))).
% 2.85/1.76 thf(fact_10_Dag_Osimps_I2_J, axiom, (! [P:simpl_ref,L:(simpl_ref > simpl_ref),R:(simpl_ref > simpl_ref),Lt2:binDag_Mirabelle_dag,A2:simpl_ref,Rt2:binDag_Mirabelle_dag]: ((((((binDag_Mirabelle_Dag @ P) @ L) @ R) @ (((binDag476092410e_Node @ Lt2) @ A2) @ Rt2)) = ((P = A2) & ((P != simpl_Null) & (((((binDag_Mirabelle_Dag @ (L @ P)) @ L) @ R) @ Lt2) & ((((binDag_Mirabelle_Dag @ (R @ P)) @ L) @ R) @ Rt2)))))))).
% 2.85/1.76 thf(fact_11_dag_Osimps_I7_J, axiom, (! [A:$tType,F1:A,F2:(binDag_Mirabelle_dag > (simpl_ref > (binDag_Mirabelle_dag > (A > (A > A))))),X21:binDag_Mirabelle_dag,X22:simpl_ref,X23:binDag_Mirabelle_dag]: ((((((binDag1442713106ec_dag @ A) @ F1) @ F2) @ (((binDag476092410e_Node @ X21) @ X22) @ X23)) = (((((F2 @ X21) @ X22) @ X23) @ ((((binDag1442713106ec_dag @ A) @ F1) @ F2) @ X21)) @ ((((binDag1442713106ec_dag @ A) @ F1) @ F2) @ X23)))))).
% 2.85/1.76 thf(fact_12_dag_Osimps_I5_J, axiom, (! [A:$tType,F1:A,F2:(binDag_Mirabelle_dag > (simpl_ref > (binDag_Mirabelle_dag > A))),X21:binDag_Mirabelle_dag,X22:simpl_ref,X23:binDag_Mirabelle_dag]: ((((((binDag1297733282se_dag @ A) @ F1) @ F2) @ (((binDag476092410e_Node @ X21) @ X22) @ X23)) = (((F2 @ X21) @ X22) @ X23))))).
% 2.85/1.76 thf(fact_13_Dag__root__not__in__subdag__l, axiom, (! [L:(simpl_ref > simpl_ref),P:simpl_ref,R:(simpl_ref > simpl_ref),T:binDag_Mirabelle_dag]: ((((((binDag_Mirabelle_Dag @ (L @ P)) @ L) @ R) @ T) => (~ ((((member @ simpl_ref) @ P) @ (binDag1380252983set_of @ T)))))))).
% 2.85/1.76 thf(fact_14_dag_Odistinct_I1_J, axiom, (! [X21:binDag_Mirabelle_dag,X22:simpl_ref,X23:binDag_Mirabelle_dag]: ((binDag_Mirabelle_Tip != (((binDag476092410e_Node @ X21) @ X22) @ X23))))).
% 2.85/1.76 thf(fact_15_dag_Oinduct, axiom, (! [P2:(binDag_Mirabelle_dag > $o),Dag:binDag_Mirabelle_dag]: (((P2 @ binDag_Mirabelle_Tip) => ((! [X1:binDag_Mirabelle_dag,X2:simpl_ref,X3:binDag_Mirabelle_dag]: (((P2 @ X1) => ((P2 @ X3) => (P2 @ (((binDag476092410e_Node @ X1) @ X2) @ X3)))))) => (P2 @ Dag)))))).
% 2.85/1.76 thf(fact_16_dag_Oexhaust, axiom, (! [Y2:binDag_Mirabelle_dag]: (((Y2 != binDag_Mirabelle_Tip) => (~ ((! [X212:binDag_Mirabelle_dag,X222:simpl_ref,X232:binDag_Mirabelle_dag]: ((Y2 != (((binDag476092410e_Node @ X212) @ X222) @ X232)))))))))).
% 2.85/1.76 thf(fact_17_subdag__neq, axiom, (! [T:binDag_Mirabelle_dag,S:binDag_Mirabelle_dag]: ((((binDag786255756subdag @ T) @ S) => (T != S))))).
% 2.85/1.76 thf(fact_18_Null__notin__Dag, axiom, (! [P:simpl_ref,L:(simpl_ref > simpl_ref),R:(simpl_ref > simpl_ref),T:binDag_Mirabelle_dag]: ((((((binDag_Mirabelle_Dag @ P) @ L) @ R) @ T) => (~ ((((member @ simpl_ref) @ simpl_Null) @ (binDag1380252983set_of @ T)))))))).
% 2.85/1.76 thf(fact_19_Dag__Null, axiom, (! [L:(simpl_ref > simpl_ref),R:(simpl_ref > simpl_ref),T:binDag_Mirabelle_dag]: ((((((binDag_Mirabelle_Dag @ simpl_Null) @ L) @ R) @ T) = (T = binDag_Mirabelle_Tip))))).
% 2.85/1.76 thf(fact_20_dag_Osimps_I6_J, axiom, (! [A:$tType,F1:A,F2:(binDag_Mirabelle_dag > (simpl_ref > (binDag_Mirabelle_dag > (A > (A > A)))))]: ((((((binDag1442713106ec_dag @ A) @ F1) @ F2) @ binDag_Mirabelle_Tip) = F1)))).
% 2.85/1.76 thf(fact_21_dag_Osimps_I4_J, axiom, (! [A:$tType,F1:A,F2:(binDag_Mirabelle_dag > (simpl_ref > (binDag_Mirabelle_dag > A)))]: ((((((binDag1297733282se_dag @ A) @ F1) @ F2) @ binDag_Mirabelle_Tip) = F1)))).
% 2.85/1.76 thf(fact_22_subdag_Osimps_I1_J, axiom, (! [T:binDag_Mirabelle_dag]: ((~ (((binDag786255756subdag @ binDag_Mirabelle_Tip) @ T)))))).
% 2.85/1.76 thf(fact_23_Dag_Osimps_I1_J, axiom, (! [P:simpl_ref,L:(simpl_ref > simpl_ref),R:(simpl_ref > simpl_ref)]: ((((((binDag_Mirabelle_Dag @ P) @ L) @ R) @ binDag_Mirabelle_Tip) = (P = simpl_Null))))).
% 2.85/1.76 thf(fact_24_in__set__of__decomp, axiom, (! [P:simpl_ref,T:binDag_Mirabelle_dag]: (((((member @ simpl_ref) @ P) @ (binDag1380252983set_of @ T)) = (? [L2:binDag_Mirabelle_dag,R2:binDag_Mirabelle_dag]: (((T = (((binDag476092410e_Node @ L2) @ P) @ R2)) | ((binDag786255756subdag @ T) @ (((binDag476092410e_Node @ L2) @ P) @ R2))))))))).
% 2.85/1.76 thf(fact_25_subdag__not__sym, axiom, (! [S:binDag_Mirabelle_dag,T:binDag_Mirabelle_dag]: ((((binDag786255756subdag @ S) @ T) => (~ (((binDag786255756subdag @ T) @ S))))))).
% 2.85/1.76 thf(fact_26_subdag__trans, axiom, (! [T:binDag_Mirabelle_dag,S:binDag_Mirabelle_dag,R:binDag_Mirabelle_dag]: ((((binDag786255756subdag @ T) @ S) => (((binDag786255756subdag @ S) @ R) => ((binDag786255756subdag @ T) @ R)))))).
% 2.85/1.76 thf(fact_27_DAG_Osimps_I2_J, axiom, (! [L:binDag_Mirabelle_dag,A2:simpl_ref,R:binDag_Mirabelle_dag]: (((binDag_Mirabelle_DAG @ (((binDag476092410e_Node @ L) @ A2) @ R)) = ((~ ((((member @ simpl_ref) @ A2) @ (binDag1380252983set_of @ L)))) & ((~ ((((member @ simpl_ref) @ A2) @ (binDag1380252983set_of @ R)))) & ((binDag_Mirabelle_DAG @ L) & (binDag_Mirabelle_DAG @ R)))))))).
% 2.85/1.76 thf(fact_28_Dag__upd__same__r, axiom, (! [P:simpl_ref,L:(simpl_ref > simpl_ref),R:(simpl_ref > simpl_ref),T:binDag_Mirabelle_dag]: ((((((binDag_Mirabelle_Dag @ P) @ L) @ (((((fun_upd @ simpl_ref) @ simpl_ref) @ R) @ P) @ P)) @ T) = ((P = simpl_Null) & (T = binDag_Mirabelle_Tip)))))).
% 2.85/1.76 thf(fact_29_Dag__upd__same__l, axiom, (! [P:simpl_ref,L:(simpl_ref > simpl_ref),R:(simpl_ref > simpl_ref),T:binDag_Mirabelle_dag]: ((((((binDag_Mirabelle_Dag @ P) @ (((((fun_upd @ simpl_ref) @ simpl_ref) @ L) @ P) @ P)) @ R) @ T) = ((P = simpl_Null) & (T = binDag_Mirabelle_Tip)))))).
% 2.85/1.76 thf(fact_30_notin__Dag__update__r, axiom, (! [Q2:simpl_ref,T:binDag_Mirabelle_dag,P:simpl_ref,L:(simpl_ref > simpl_ref),R:(simpl_ref > simpl_ref),Y2:simpl_ref]: (((~ ((((member @ simpl_ref) @ Q2) @ (binDag1380252983set_of @ T)))) => (((((binDag_Mirabelle_Dag @ P) @ L) @ (((((fun_upd @ simpl_ref) @ simpl_ref) @ R) @ Q2) @ Y2)) @ T) = ((((binDag_Mirabelle_Dag @ P) @ L) @ R) @ T)))))).
% 2.85/1.76 thf(fact_31_notin__Dag__update__l, axiom, (! [Q2:simpl_ref,T:binDag_Mirabelle_dag,P:simpl_ref,L:(simpl_ref > simpl_ref),Y2:simpl_ref,R:(simpl_ref > simpl_ref)]: (((~ ((((member @ simpl_ref) @ Q2) @ (binDag1380252983set_of @ T)))) => (((((binDag_Mirabelle_Dag @ P) @ (((((fun_upd @ simpl_ref) @ simpl_ref) @ L) @ Q2) @ Y2)) @ R) @ T) = ((((binDag_Mirabelle_Dag @ P) @ L) @ R) @ T)))))).
% 2.85/1.76 thf(fact_32_Dag__update__rI, axiom, (! [P:simpl_ref,L:(simpl_ref > simpl_ref),R:(simpl_ref > simpl_ref),T:binDag_Mirabelle_dag,Q2:simpl_ref,Y2:simpl_ref]: ((((((binDag_Mirabelle_Dag @ P) @ L) @ R) @ T) => ((~ ((((member @ simpl_ref) @ Q2) @ (binDag1380252983set_of @ T)))) => ((((binDag_Mirabelle_Dag @ P) @ L) @ (((((fun_upd @ simpl_ref) @ simpl_ref) @ R) @ Q2) @ Y2)) @ T)))))).
% 2.85/1.76 thf(fact_33_Dag__update__lI, axiom, (! [P:simpl_ref,L:(simpl_ref > simpl_ref),R:(simpl_ref > simpl_ref),T:binDag_Mirabelle_dag,Q2:simpl_ref,Y2:simpl_ref]: ((((((binDag_Mirabelle_Dag @ P) @ L) @ R) @ T) => ((~ ((((member @ simpl_ref) @ Q2) @ (binDag1380252983set_of @ T)))) => ((((binDag_Mirabelle_Dag @ P) @ (((((fun_upd @ simpl_ref) @ simpl_ref) @ L) @ Q2) @ Y2)) @ R) @ T)))))).
% 2.85/1.76 thf(fact_34_DAG_Osimps_I1_J, axiom, (binDag_Mirabelle_DAG @ binDag_Mirabelle_Tip)).
% 2.85/1.76 thf(fact_35_Dag__upd__same__r__lemma, axiom, (! [P:simpl_ref,L:(simpl_ref > simpl_ref),R:(simpl_ref > simpl_ref),T:binDag_Mirabelle_dag]: (((P != simpl_Null) => (~ (((((binDag_Mirabelle_Dag @ P) @ L) @ (((((fun_upd @ simpl_ref) @ simpl_ref) @ R) @ P) @ P)) @ T))))))).
% 2.85/1.76 thf(fact_36_Dag__upd__same__l__lemma, axiom, (! [P:simpl_ref,L:(simpl_ref > simpl_ref),R:(simpl_ref > simpl_ref),T:binDag_Mirabelle_dag]: (((P != simpl_Null) => (~ (((((binDag_Mirabelle_Dag @ P) @ (((((fun_upd @ simpl_ref) @ simpl_ref) @ L) @ P) @ P)) @ R) @ T))))))).
% 2.85/1.76 thf(fact_37_fun__upd__apply, axiom, (! [A:$tType,B:$tType]: ((((fun_upd @ B) @ A) = (^ [F:(B > A),X4:B,Y3:A,Z:B]: (((((if @ A) @ (Z = X4)) @ Y3) @ (F @ Z)))))))).
% 2.85/1.76 thf(fact_38_fun__upd__triv, axiom, (! [B:$tType,A:$tType,F3:(A > B),X5:A]: (((((((fun_upd @ A) @ B) @ F3) @ X5) @ (F3 @ X5)) = F3)))).
% 2.85/1.76 thf(fact_39_fun__upd__upd, axiom, (! [A:$tType,B:$tType,F3:(A > B),X5:A,Y2:B,Z2:B]: (((((((fun_upd @ A) @ B) @ (((((fun_upd @ A) @ B) @ F3) @ X5) @ Y2)) @ X5) @ Z2) = (((((fun_upd @ A) @ B) @ F3) @ X5) @ Z2))))).
% 2.85/1.76 thf(fact_40_fun__upd__idem__iff, axiom, (! [A:$tType,B:$tType,F3:(A > B),X5:A,Y2:B]: ((((((((fun_upd @ A) @ B) @ F3) @ X5) @ Y2) = F3) = ((F3 @ X5) = Y2))))).
% 2.85/1.76 thf(fact_41_fun__upd__twist, axiom, (! [A:$tType,B:$tType,A2:A,C:A,M:(A > B),B2:B,D:B]: (((A2 != C) => ((((((fun_upd @ A) @ B) @ (((((fun_upd @ A) @ B) @ M) @ A2) @ B2)) @ C) @ D) = (((((fun_upd @ A) @ B) @ (((((fun_upd @ A) @ B) @ M) @ C) @ D)) @ A2) @ B2)))))).
% 2.85/1.76 thf(fact_42_fun__upd__other, axiom, (! [B:$tType,A:$tType,Z2:A,X5:A,F3:(A > B),Y2:B]: (((Z2 != X5) => (((((((fun_upd @ A) @ B) @ F3) @ X5) @ Y2) @ Z2) = (F3 @ Z2)))))).
% 2.85/1.76 thf(fact_43_fun__upd__same, axiom, (! [B:$tType,A:$tType,F3:(B > A),X5:B,Y2:A]: ((((((((fun_upd @ B) @ A) @ F3) @ X5) @ Y2) @ X5) = Y2)))).
% 2.85/1.76 thf(fact_44_fun__upd__idem, axiom, (! [A:$tType,B:$tType,F3:(B > A),X5:B,Y2:A]: ((((F3 @ X5) = Y2) => ((((((fun_upd @ B) @ A) @ F3) @ X5) @ Y2) = F3))))).
% 2.85/1.76 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P2:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P2)) = (P2 @ A2))))).
% 2.85/1.76 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X4:A]: ((((member @ A) @ X4) @ A3)))) = A3)))).
% 2.85/1.76 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P2:(A > $o),Q3:(A > $o)]: (((! [X:A]: (((P2 @ X) = (Q3 @ X)))) => (((collect @ A) @ P2) = ((collect @ A) @ Q3)))))).
% 2.85/1.76 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F3:(A > B),G:(A > B)]: (((! [X:A]: (((F3 @ X) = (G @ X)))) => (F3 = G))))).
% 2.85/1.76 thf(fact_49_fun__upd__eqD, axiom, (! [A:$tType,B:$tType,F3:(A > B),X5:A,Y2:B,G:(A > B),Z2:B]: ((((((((fun_upd @ A) @ B) @ F3) @ X5) @ Y2) = (((((fun_upd @ A) @ B) @ G) @ X5) @ Z2)) => (Y2 = Z2))))).
% 2.85/1.76 thf(fact_50_fun__upd__def, axiom, (! [B:$tType,A:$tType]: ((((fun_upd @ A) @ B) = (^ [F:(A > B),A4:A,B3:B,X4:A]: (((((if @ B) @ (X4 = A4)) @ B3) @ (F @ X4)))))))).
% 2.85/1.76 thf(fact_51_set__of__Tip, axiom, ((binDag1380252983set_of @ binDag_Mirabelle_Tip) = (bot_bot @ (set @ simpl_ref)))).
% 2.85/1.76 thf(fact_52_DAG__less, axiom, (! [Y2:binDag_Mirabelle_dag,X5:binDag_Mirabelle_dag]: (((binDag_Mirabelle_DAG @ Y2) => ((((ord_less @ binDag_Mirabelle_dag) @ X5) @ Y2) => (binDag_Mirabelle_DAG @ X5)))))).
% 2.85/1.76 thf(fact_53_Dag__update__l__new, axiom, (! [T:binDag_Mirabelle_dag,Alloc:(list @ simpl_ref),P:simpl_ref,L:(simpl_ref > simpl_ref),X5:simpl_ref,R:(simpl_ref > simpl_ref)]: (((((ord_less_eq @ (set @ simpl_ref)) @ (binDag1380252983set_of @ T)) @ ((set2 @ simpl_ref) @ Alloc)) => (((((binDag_Mirabelle_Dag @ P) @ (((((fun_upd @ simpl_ref) @ simpl_ref) @ L) @ (simpl_new @ ((set2 @ simpl_ref) @ Alloc))) @ X5)) @ R) @ T) = ((((binDag_Mirabelle_Dag @ P) @ L) @ R) @ T)))))).
% 2.85/1.76 thf(fact_54_Dag__update__r__new, axiom, (! [T:binDag_Mirabelle_dag,Alloc:(list @ simpl_ref),P:simpl_ref,L:(simpl_ref > simpl_ref),R:(simpl_ref > simpl_ref),X5:simpl_ref]: (((((ord_less_eq @ (set @ simpl_ref)) @ (binDag1380252983set_of @ T)) @ ((set2 @ simpl_ref) @ Alloc)) => (((((binDag_Mirabelle_Dag @ P) @ L) @ (((((fun_upd @ simpl_ref) @ simpl_ref) @ R) @ (simpl_new @ ((set2 @ simpl_ref) @ Alloc))) @ X5)) @ T) = ((((binDag_Mirabelle_Dag @ P) @ L) @ R) @ T)))))).
% 2.85/1.76 thf(fact_55_swap__def, axiom, (! [B:$tType,A:$tType]: ((((swap @ A) @ B) = (^ [A4:A,B3:A,F:(A > B)]: ((((((fun_upd @ A) @ B) @ (((((fun_upd @ A) @ B) @ F) @ A4) @ (F @ B3))) @ B3) @ (F @ A4)))))))).
% 2.85/1.76 thf(fact_56_less__dag__Tip, axiom, (! [X5:binDag_Mirabelle_dag]: ((~ ((((ord_less @ binDag_Mirabelle_dag) @ X5) @ binDag_Mirabelle_Tip)))))).
% 2.85/1.76 thf(fact_57_less__dag__def, axiom, ((ord_less @ binDag_Mirabelle_dag) = (^ [S2:binDag_Mirabelle_dag,T3:binDag_Mirabelle_dag]: (((binDag786255756subdag @ T3) @ S2))))).
% 2.85/1.76 thf(fact_58_less__dag__Node_H, axiom, (! [X5:binDag_Mirabelle_dag,L:binDag_Mirabelle_dag,A2:simpl_ref,R:binDag_Mirabelle_dag]: (((((ord_less @ binDag_Mirabelle_dag) @ X5) @ (((binDag476092410e_Node @ L) @ A2) @ R)) = ((X5 = L) | ((X5 = R) | ((((ord_less @ binDag_Mirabelle_dag) @ X5) @ L) | (((ord_less @ binDag_Mirabelle_dag) @ X5) @ R)))))))).
% 2.85/1.76 thf(fact_59_swap__apply_I3_J, axiom, (! [A:$tType,B:$tType,C:B,A2:B,B2:B,F3:(B > A)]: (((C != A2) => ((C != B2) => (((((((swap @ B) @ A) @ A2) @ B2) @ F3) @ C) = (F3 @ C))))))).
% 2.85/1.76 thf(fact_60_swap__apply_I2_J, axiom, (! [A:$tType,B:$tType,A2:B,B2:B,F3:(B > A)]: ((((((((swap @ B) @ A) @ A2) @ B2) @ F3) @ B2) = (F3 @ A2))))).
% 2.85/1.76 thf(fact_61_swap__apply_I1_J, axiom, (! [A:$tType,B:$tType,A2:B,B2:B,F3:(B > A)]: ((((((((swap @ B) @ A) @ A2) @ B2) @ F3) @ A2) = (F3 @ B2))))).
% 2.85/1.76 thf(fact_62_swap__self, axiom, (! [B:$tType,A:$tType,A2:A,F3:(A > B)]: (((((((swap @ A) @ B) @ A2) @ A2) @ F3) = F3)))).
% 2.85/1.76 thf(fact_63_swap__nilpotent, axiom, (! [B:$tType,A:$tType,A2:A,B2:A,F3:(A > B)]: (((((((swap @ A) @ B) @ A2) @ B2) @ (((((swap @ A) @ B) @ A2) @ B2) @ F3)) = F3)))).
% 2.85/1.76 thf(fact_64_swap__triple, axiom, (! [B:$tType,A:$tType,A2:A,C:A,B2:A,F3:(A > B)]: (((A2 != C) => ((B2 != C) => ((((((swap @ A) @ B) @ A2) @ B2) @ (((((swap @ A) @ B) @ B2) @ C) @ (((((swap @ A) @ B) @ A2) @ B2) @ F3))) = (((((swap @ A) @ B) @ A2) @ C) @ F3))))))).
% 2.85/1.76 thf(fact_65_swap__commute, axiom, (! [B:$tType,A:$tType]: ((((swap @ A) @ B) = (^ [A4:A,B3:A]: (((((swap @ A) @ B) @ B3) @ A4))))))).
% 2.85/1.76 thf(fact_66_less__dag__set__of, axiom, (! [X5:binDag_Mirabelle_dag,Y2:binDag_Mirabelle_dag]: (((((ord_less @ binDag_Mirabelle_dag) @ X5) @ Y2) => (((ord_less_eq @ (set @ simpl_ref)) @ (binDag1380252983set_of @ X5)) @ (binDag1380252983set_of @ Y2)))))).
% 2.85/1.76 thf(fact_67_less__Node__dag, axiom, (! [L:binDag_Mirabelle_dag,A2:simpl_ref,R:binDag_Mirabelle_dag,X5:binDag_Mirabelle_dag]: (((((ord_less @ binDag_Mirabelle_dag) @ (((binDag476092410e_Node @ L) @ A2) @ R)) @ X5) => ((((ord_less @ binDag_Mirabelle_dag) @ L) @ X5) & (((ord_less @ binDag_Mirabelle_dag) @ R) @ X5)))))).
% 2.85/1.76 thf(fact_68_empty__subsetI, axiom, (! [A:$tType,A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (bot_bot @ (set @ A))) @ A3)))).
% 2.85/1.76 thf(fact_69_subset__empty, axiom, (! [A:$tType,A3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ (bot_bot @ (set @ A))) = (A3 = (bot_bot @ (set @ A))))))).
% 2.85/1.76 thf(fact_70_subset__antisym, axiom, (! [A:$tType,A3:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B4) => ((((ord_less_eq @ (set @ A)) @ B4) @ A3) => (A3 = B4)))))).
% 2.85/1.76 thf(fact_71_subsetI, axiom, (! [A:$tType,A3:(set @ A),B4:(set @ A)]: (((! [X:A]: (((((member @ A) @ X) @ A3) => (((member @ A) @ X) @ B4)))) => (((ord_less_eq @ (set @ A)) @ A3) @ B4))))).
% 2.85/1.76 thf(fact_72_empty__Collect__eq, axiom, (! [A:$tType,P2:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P2)) = (! [X4:A]: ((~ ((P2 @ X4))))))))).
% 2.85/1.76 thf(fact_73_Collect__empty__eq, axiom, (! [A:$tType,P2:(A > $o)]: (((((collect @ A) @ P2) = (bot_bot @ (set @ A))) = (! [X4:A]: ((~ ((P2 @ X4))))))))).
% 2.85/1.76 thf(fact_74_all__not__in__conv, axiom, (! [A:$tType,A3:(set @ A)]: (((! [X4:A]: ((~ ((((member @ A) @ X4) @ A3))))) = (A3 = (bot_bot @ (set @ A))))))).
% 2.85/1.76 thf(fact_75_empty__iff, axiom, (! [A:$tType,C:A]: ((~ ((((member @ A) @ C) @ (bot_bot @ (set @ A)))))))).
% 2.85/1.76 thf(fact_76_emptyE, axiom, (! [A:$tType,A2:A]: ((~ ((((member @ A) @ A2) @ (bot_bot @ (set @ A)))))))).
% 2.85/1.76 thf(fact_77_equals0D, axiom, (! [A:$tType,A3:(set @ A),A2:A]: (((A3 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A2) @ A3))))))).
% 2.85/1.76 thf(fact_78_equals0I, axiom, (! [A:$tType,A3:(set @ A)]: (((! [Y4:A]: ((~ ((((member @ A) @ Y4) @ A3))))) => (A3 = (bot_bot @ (set @ A))))))).
% 2.85/1.76 thf(fact_79_ex__in__conv, axiom, (! [A:$tType,A3:(set @ A)]: (((? [X4:A]: ((((member @ A) @ X4) @ A3))) = (A3 != (bot_bot @ (set @ A))))))).
% 2.85/1.76 thf(fact_80_set__mp, axiom, (! [A:$tType,A3:(set @ A),B4:(set @ A),X5:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B4) => ((((member @ A) @ X5) @ A3) => (((member @ A) @ X5) @ B4)))))).
% 2.85/1.76 thf(fact_81_in__mono, axiom, (! [A:$tType,A3:(set @ A),B4:(set @ A),X5:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B4) => ((((member @ A) @ X5) @ A3) => (((member @ A) @ X5) @ B4)))))).
% 2.85/1.76 thf(fact_82_subsetD, axiom, (! [A:$tType,A3:(set @ A),B4:(set @ A),C:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B4) => ((((member @ A) @ C) @ A3) => (((member @ A) @ C) @ B4)))))).
% 2.85/1.76 thf(fact_83_subsetCE, axiom, (! [A:$tType,A3:(set @ A),B4:(set @ A),C:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B4) => ((((member @ A) @ C) @ A3) => (((member @ A) @ C) @ B4)))))).
% 2.85/1.76 thf(fact_84_equalityE, axiom, (! [A:$tType,A3:(set @ A),B4:(set @ A)]: (((A3 = B4) => (~ (((((ord_less_eq @ (set @ A)) @ A3) @ B4) => (~ ((((ord_less_eq @ (set @ A)) @ B4) @ A3)))))))))).
% 2.85/1.76 thf(fact_85_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A5:(set @ A),B5:(set @ A)]: ((! [X4:A]: (((((member @ A) @ X4) @ A5) => (((member @ A) @ X4) @ B5)))))))))).
% 2.85/1.76 thf(fact_86_equalityD1, axiom, (! [A:$tType,A3:(set @ A),B4:(set @ A)]: (((A3 = B4) => (((ord_less_eq @ (set @ A)) @ A3) @ B4))))).
% 2.85/1.76 thf(fact_87_equalityD2, axiom, (! [A:$tType,A3:(set @ A),B4:(set @ A)]: (((A3 = B4) => (((ord_less_eq @ (set @ A)) @ B4) @ A3))))).
% 2.85/1.76 thf(fact_88_set__rev__mp, axiom, (! [A:$tType,X5:A,A3:(set @ A),B4:(set @ A)]: (((((member @ A) @ X5) @ A3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B4) => (((member @ A) @ X5) @ B4)))))).
% 2.85/1.76 thf(fact_89_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A5:(set @ A),B5:(set @ A)]: ((! [T3:A]: (((((member @ A) @ T3) @ A5) => (((member @ A) @ T3) @ B5)))))))))).
% 2.85/1.76 thf(fact_90_rev__subsetD, axiom, (! [A:$tType,C:A,A3:(set @ A),B4:(set @ A)]: (((((member @ A) @ C) @ A3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B4) => (((member @ A) @ C) @ B4)))))).
% 2.85/1.76 thf(fact_91_subset__refl, axiom, (! [A:$tType,A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A3) @ A3)))).
% 2.85/1.76 thf(fact_92_Collect__mono, axiom, (! [A:$tType,P2:(A > $o),Q3:(A > $o)]: (((! [X:A]: (((P2 @ X) => (Q3 @ X)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P2)) @ ((collect @ A) @ Q3)))))).
% 2.85/1.76 thf(fact_93_subset__trans, axiom, (! [A:$tType,A3:(set @ A),B4:(set @ A),C2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B4) => ((((ord_less_eq @ (set @ A)) @ B4) @ C2) => (((ord_less_eq @ (set @ A)) @ A3) @ C2)))))).
% 2.85/1.76 thf(fact_94_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y5:(set @ A),Z3:(set @ A)]: ((Y5 = Z3))) = (^ [A5:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) & (((ord_less_eq @ (set @ A)) @ B5) @ A5)))))))).
% 2.85/1.76 thf(fact_95_contra__subsetD, axiom, (! [A:$tType,A3:(set @ A),B4:(set @ A),C:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B4) => ((~ ((((member @ A) @ C) @ B4))) => (~ ((((member @ A) @ C) @ A3)))))))).
% 2.85/1.76 thf(fact_96_Collect__mono__iff, axiom, (! [A:$tType,P2:(A > $o),Q3:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P2)) @ ((collect @ A) @ Q3)) = (! [X4:A]: (((P2 @ X4) => (Q3 @ X4)))))))).
% 2.85/1.76 thf(fact_97_bot__apply, axiom, (! [C3:$tType,D2:$tType]: ((((bot @ C3) @ (type2 @ C3)) => ((bot_bot @ (D2 > C3)) = (^ [X4:D2]: ((bot_bot @ C3)))))))).
% 2.85/1.76 thf(fact_98_order__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X5:A]: ((((ord_less_eq @ A) @ X5) @ X5))))))).
% 2.85/1.76 thf(fact_99_subset__code_I1_J, axiom, (! [A:$tType,Xs:(list @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs)) @ B4) = (! [X4:A]: (((((member @ A) @ X4) @ ((set2 @ A) @ Xs)) => (((member @ A) @ X4) @ B4)))))))).
% 2.85/1.76 thf(fact_100_subset__emptyI, axiom, (! [A:$tType,A3:(set @ A)]: (((! [X:A]: ((~ ((((member @ A) @ X) @ A3))))) => (((ord_less_eq @ (set @ A)) @ A3) @ (bot_bot @ (set @ A))))))).
% 2.85/1.76 thf(fact_101_bot_Oextremum__strict, axiom, (! [A:$tType]: ((((order_bot @ A) @ (type2 @ A)) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ (bot_bot @ A)))))))))).
% 2.85/1.76 thf(fact_102_bot_Onot__eq__extremum, axiom, (! [A:$tType]: ((((order_bot @ A) @ (type2 @ A)) => (! [A2:A]: (((A2 != (bot_bot @ A)) = (((ord_less @ A) @ (bot_bot @ A)) @ A2)))))))).
% 2.85/1.76 thf(fact_103_psubsetI, axiom, (! [A:$tType,A3:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B4) => ((A3 != B4) => (((ord_less @ (set @ A)) @ A3) @ B4)))))).
% 2.85/1.76 thf(fact_104_bot__set__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (bot_bot @ (A > $o))))))).
% 2.85/1.76 thf(fact_105_not__psubset__empty, axiom, (! [A:$tType,A3:(set @ A)]: ((~ ((((ord_less @ (set @ A)) @ A3) @ (bot_bot @ (set @ A)))))))).
% 2.85/1.76 thf(fact_106_psubsetE, axiom, (! [A:$tType,A3:(set @ A),B4:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B4) => (~ (((((ord_less_eq @ (set @ A)) @ A3) @ B4) => (((ord_less_eq @ (set @ A)) @ B4) @ A3)))))))).
% 2.85/1.76 thf(fact_107_psubset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A5:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) & (A5 != B5)))))))).
% 2.85/1.76 thf(fact_108_psubset__imp__subset, axiom, (! [A:$tType,A3:(set @ A),B4:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B4) => (((ord_less_eq @ (set @ A)) @ A3) @ B4))))).
% 2.85/1.76 thf(fact_109_psubset__subset__trans, axiom, (! [A:$tType,A3:(set @ A),B4:(set @ A),C2:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B4) => ((((ord_less_eq @ (set @ A)) @ B4) @ C2) => (((ord_less @ (set @ A)) @ A3) @ C2)))))).
% 2.85/1.76 thf(fact_110_subset__not__subset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A5:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) & (~ ((((ord_less_eq @ (set @ A)) @ B5) @ A5)))))))))).
% 2.85/1.76 thf(fact_111_subset__psubset__trans, axiom, (! [A:$tType,A3:(set @ A),B4:(set @ A),C2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B4) => ((((ord_less @ (set @ A)) @ B4) @ C2) => (((ord_less @ (set @ A)) @ A3) @ C2)))))).
% 2.85/1.76 thf(fact_112_subset__iff__psubset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A5:(set @ A),B5:(set @ A)]: (((((ord_less @ (set @ A)) @ A5) @ B5) | (A5 = B5)))))))).
% 2.85/1.76 thf(fact_113_dag__less__le, axiom, ((ord_less @ binDag_Mirabelle_dag) = (^ [X4:binDag_Mirabelle_dag,Y3:binDag_Mirabelle_dag]: (((((ord_less_eq @ binDag_Mirabelle_dag) @ X4) @ Y3) & (X4 != Y3)))))).
% 2.85/1.76 thf(fact_114_le__dag__def, axiom, ((ord_less_eq @ binDag_Mirabelle_dag) = (^ [S2:binDag_Mirabelle_dag,T3:binDag_Mirabelle_dag]: (((S2 = T3) | (((ord_less @ binDag_Mirabelle_dag) @ S2) @ T3)))))).
% 2.85/1.76 thf(fact_115_less__dag__Node, axiom, (! [X5:binDag_Mirabelle_dag,L:binDag_Mirabelle_dag,A2:simpl_ref,R:binDag_Mirabelle_dag]: (((((ord_less @ binDag_Mirabelle_dag) @ X5) @ (((binDag476092410e_Node @ L) @ A2) @ R)) = ((((ord_less_eq @ binDag_Mirabelle_dag) @ X5) @ L) | (((ord_less_eq @ binDag_Mirabelle_dag) @ X5) @ R)))))).
% 2.85/1.76 thf(fact_116_le__dag__set__of, axiom, (! [X5:binDag_Mirabelle_dag,Y2:binDag_Mirabelle_dag]: (((((ord_less_eq @ binDag_Mirabelle_dag) @ X5) @ Y2) => (((ord_less_eq @ (set @ simpl_ref)) @ (binDag1380252983set_of @ X5)) @ (binDag1380252983set_of @ Y2)))))).
% 2.85/1.76 thf(fact_117_dual__order_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ A2) @ B2) => (A2 = B2))))))))).
% 2.85/1.76 thf(fact_118_dual__order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C) @ B2) => (((ord_less_eq @ A) @ C) @ A2))))))))).
% 2.85/1.76 thf(fact_119_linorder__wlog, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P2:(A > (A > $o)),A2:A,B2:A]: (((! [A6:A,B6:A]: (((((ord_less_eq @ A) @ A6) @ B6) => ((P2 @ A6) @ B6)))) => ((! [A6:A,B6:A]: ((((P2 @ B6) @ A6) => ((P2 @ A6) @ B6)))) => ((P2 @ A2) @ B2))))))))).
% 2.85/1.76 thf(fact_120_dual__order_Orefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 2.85/1.76 thf(fact_121_order__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X5:A,Y2:A,Z2:A]: (((((ord_less_eq @ A) @ X5) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ Z2) => (((ord_less_eq @ A) @ X5) @ Z2))))))))).
% 2.85/1.76 thf(fact_122_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ A2) => (A2 = B2))))))))).
% 2.85/1.76 thf(fact_123_ord__le__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((B2 = C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.85/1.76 thf(fact_124_ord__eq__le__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((A2 = B2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.85/1.76 thf(fact_125_antisym__conv, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [Y2:A,X5:A]: (((((ord_less_eq @ A) @ Y2) @ X5) => ((((ord_less_eq @ A) @ X5) @ Y2) = (X5 = Y2))))))))).
% 2.85/1.76 thf(fact_126_le__cases3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X5:A,Y2:A,Z2:A]: ((((((ord_less_eq @ A) @ X5) @ Y2) => (~ ((((ord_less_eq @ A) @ Y2) @ Z2)))) => (((((ord_less_eq @ A) @ Y2) @ X5) => (~ ((((ord_less_eq @ A) @ X5) @ Z2)))) => (((((ord_less_eq @ A) @ X5) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y2)))) => (((((ord_less_eq @ A) @ Z2) @ Y2) => (~ ((((ord_less_eq @ A) @ Y2) @ X5)))) => (((((ord_less_eq @ A) @ Y2) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X5)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X5) => (~ ((((ord_less_eq @ A) @ X5) @ Y2))))))))))))))))).
% 2.85/1.76 thf(fact_127_order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.85/1.76 thf(fact_128_le__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X5:A,Y2:A]: (((~ ((((ord_less_eq @ A) @ X5) @ Y2))) => (((ord_less_eq @ A) @ Y2) @ X5)))))))).
% 2.85/1.76 thf(fact_129_eq__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X5:A,Y2:A]: (((X5 = Y2) => (((ord_less_eq @ A) @ X5) @ Y2)))))))).
% 2.85/1.77 thf(fact_130_linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X5:A,Y2:A]: (((((ord_less_eq @ A) @ X5) @ Y2) | (((ord_less_eq @ A) @ Y2) @ X5)))))))).
% 2.85/1.77 thf(fact_131_antisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X5:A,Y2:A]: (((((ord_less_eq @ A) @ X5) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ X5) => (X5 = Y2))))))))).
% 2.85/1.77 thf(fact_132_eq__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((^ [Y5:A,Z3:A]: ((Y5 = Z3))) = (^ [X4:A,Y3:A]: (((((ord_less_eq @ A) @ X4) @ Y3) & (((ord_less_eq @ A) @ Y3) @ X4))))))))).
% 2.85/1.77 thf(fact_133_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F3:(A > B),C:B]: (((((ord_less_eq @ A) @ A2) @ B2) => (((F3 @ B2) = C) => ((! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) => (((ord_less_eq @ B) @ (F3 @ X)) @ (F3 @ Y4))))) => (((ord_less_eq @ B) @ (F3 @ A2)) @ C)))))))))).
% 2.85/1.77 thf(fact_134_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,F3:(B > A),B2:B,C:B]: (((A2 = (F3 @ B2)) => ((((ord_less_eq @ B) @ B2) @ C) => ((! [X:B,Y4:B]: (((((ord_less_eq @ B) @ X) @ Y4) => (((ord_less_eq @ A) @ (F3 @ X)) @ (F3 @ Y4))))) => (((ord_less_eq @ A) @ A2) @ (F3 @ C))))))))))).
% 2.85/1.77 thf(fact_135_order__subst2, axiom, (! [A:$tType,C3:$tType]: (((((order @ C3) @ (type2 @ C3)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F3:(A > C3),C:C3]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ C3) @ (F3 @ B2)) @ C) => ((! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) => (((ord_less_eq @ C3) @ (F3 @ X)) @ (F3 @ Y4))))) => (((ord_less_eq @ C3) @ (F3 @ A2)) @ C)))))))))).
% 2.85/1.77 thf(fact_136_order__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F3:(B > A),B2:B,C:B]: (((((ord_less_eq @ A) @ A2) @ (F3 @ B2)) => ((((ord_less_eq @ B) @ B2) @ C) => ((! [X:B,Y4:B]: (((((ord_less_eq @ B) @ X) @ Y4) => (((ord_less_eq @ A) @ (F3 @ X)) @ (F3 @ Y4))))) => (((ord_less_eq @ A) @ A2) @ (F3 @ C))))))))))).
% 2.85/1.77 thf(fact_137_le__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less_eq @ (A > B)) = (^ [F:(A > B),G2:(A > B)]: ((! [X4:A]: ((((ord_less_eq @ B) @ (F @ X4)) @ (G2 @ X4))))))))))).
% 2.85/1.77 thf(fact_138_le__funI, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F3:(A > B),G:(A > B)]: (((! [X:A]: ((((ord_less_eq @ B) @ (F3 @ X)) @ (G @ X)))) => (((ord_less_eq @ (A > B)) @ F3) @ G)))))))).
% 2.85/1.77 thf(fact_139_le__funE, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F3:(A > B),G:(A > B),X5:A]: (((((ord_less_eq @ (A > B)) @ F3) @ G) => (((ord_less_eq @ B) @ (F3 @ X5)) @ (G @ X5))))))))).
% 2.85/1.77 thf(fact_140_le__funD, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F3:(A > B),G:(A > B),X5:A]: (((((ord_less_eq @ (A > B)) @ F3) @ G) => (((ord_less_eq @ B) @ (F3 @ X5)) @ (G @ X5))))))))).
% 2.85/1.77 thf(fact_141_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => (A2 != B2)))))))).
% 2.85/1.77 thf(fact_142_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (A2 != B2)))))))).
% 2.85/1.77 thf(fact_143_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X5:A,Y2:A]: (((~ ((((ord_less @ A) @ X5) @ Y2))) = ((((ord_less @ A) @ Y2) @ X5) | (X5 = Y2))))))))).
% 2.85/1.77 thf(fact_144_dual__order_Ostrict__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((ord_less @ A) @ B2) @ A2) => ((((ord_less @ A) @ C) @ B2) => (((ord_less @ A) @ C) @ A2))))))))).
% 2.85/1.77 thf(fact_145_less__imp__not__less, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X5:A,Y2:A]: (((((ord_less @ A) @ X5) @ Y2) => (~ ((((ord_less @ A) @ Y2) @ X5)))))))))).
% 2.85/1.77 thf(fact_146_order_Ostrict__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ B2) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.85/1.77 thf(fact_147_dual__order_Oirrefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ A2))))))))).
% 2.85/1.77 thf(fact_148_linorder__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X5:A,Y2:A]: (((~ ((((ord_less @ A) @ X5) @ Y2))) => ((X5 != Y2) => (((ord_less @ A) @ Y2) @ X5))))))))).
% 2.85/1.77 thf(fact_149_less__imp__triv, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X5:A,Y2:A,P2:$o]: (((((ord_less @ A) @ X5) @ Y2) => ((((ord_less @ A) @ Y2) @ X5) => P2)))))))).
% 2.85/1.77 thf(fact_150_less__imp__not__eq2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X5:A,Y2:A]: (((((ord_less @ A) @ X5) @ Y2) => (Y2 != X5)))))))).
% 2.85/1.77 thf(fact_151_antisym__conv3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y2:A,X5:A]: (((~ ((((ord_less @ A) @ Y2) @ X5))) => ((~ ((((ord_less @ A) @ X5) @ Y2))) = (X5 = Y2))))))))).
% 2.85/1.77 thf(fact_152_less__induct, axiom, (! [A:$tType]: ((((wellorder @ A) @ (type2 @ A)) => (! [P2:(A > $o),A2:A]: (((! [X:A]: (((! [Y:A]: (((((ord_less @ A) @ Y) @ X) => (P2 @ Y)))) => (P2 @ X)))) => (P2 @ A2)))))))).
% 2.85/1.77 thf(fact_153_less__not__sym, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X5:A,Y2:A]: (((((ord_less @ A) @ X5) @ Y2) => (~ ((((ord_less @ A) @ Y2) @ X5)))))))))).
% 2.85/1.77 thf(fact_154_less__imp__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X5:A,Y2:A]: (((((ord_less @ A) @ X5) @ Y2) => (X5 != Y2)))))))).
% 2.85/1.77 thf(fact_155_dual__order_Oasym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => (~ ((((ord_less @ A) @ A2) @ B2)))))))))).
% 2.85/1.77 thf(fact_156_ord__less__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ B2) => ((B2 = C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.85/1.77 thf(fact_157_ord__eq__less__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((A2 = B2) => ((((ord_less @ A) @ B2) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.85/1.77 thf(fact_158_less__irrefl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X5:A]: ((~ ((((ord_less @ A) @ X5) @ X5))))))))).
% 2.85/1.77 thf(fact_159_less__linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X5:A,Y2:A]: (((((ord_less @ A) @ X5) @ Y2) | ((X5 = Y2) | (((ord_less @ A) @ Y2) @ X5))))))))).
% 2.85/1.77 thf(fact_160_less__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X5:A,Y2:A,Z2:A]: (((((ord_less @ A) @ X5) @ Y2) => ((((ord_less @ A) @ Y2) @ Z2) => (((ord_less @ A) @ X5) @ Z2))))))))).
% 2.85/1.77 thf(fact_161_less__asym_H, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (~ ((((ord_less @ A) @ B2) @ A2)))))))))).
% 2.85/1.77 thf(fact_162_less__asym, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X5:A,Y2:A]: (((((ord_less @ A) @ X5) @ Y2) => (~ ((((ord_less @ A) @ Y2) @ X5)))))))))).
% 2.85/1.77 thf(fact_163_less__imp__neq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X5:A,Y2:A]: (((((ord_less @ A) @ X5) @ Y2) => (X5 != Y2)))))))).
% 2.85/1.77 thf(fact_164_dense, axiom, (! [A:$tType]: ((((dense_order @ A) @ (type2 @ A)) => (! [X5:A,Y2:A]: (((((ord_less @ A) @ X5) @ Y2) => (? [Z4:A]: (((((ord_less @ A) @ X5) @ Z4) & (((ord_less @ A) @ Z4) @ Y2))))))))))).
% 2.85/1.77 thf(fact_165_order_Oasym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (~ ((((ord_less @ A) @ B2) @ A2)))))))))).
% 2.85/1.77 thf(fact_166_neq__iff, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X5:A,Y2:A]: (((X5 != Y2) = ((((ord_less @ A) @ X5) @ Y2) | (((ord_less @ A) @ Y2) @ X5))))))))).
% 2.85/1.77 thf(fact_167_neqE, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X5:A,Y2:A]: (((X5 != Y2) => ((~ ((((ord_less @ A) @ X5) @ Y2))) => (((ord_less @ A) @ Y2) @ X5))))))))).
% 2.85/1.77 thf(fact_168_gt__ex, axiom, (! [A:$tType]: ((((no_top @ A) @ (type2 @ A)) => (! [X5:A]: ((? [X1:A]: ((((ord_less @ A) @ X5) @ X1))))))))).
% 2.85/1.77 thf(fact_169_lt__ex, axiom, (! [A:$tType]: ((((no_bot @ A) @ (type2 @ A)) => (! [X5:A]: ((? [Y4:A]: ((((ord_less @ A) @ Y4) @ X5))))))))).
% 2.85/1.77 thf(fact_170_order__less__subst2, axiom, (! [A:$tType,C3:$tType]: (((((order @ C3) @ (type2 @ C3)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F3:(A > C3),C:C3]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ C3) @ (F3 @ B2)) @ C) => ((! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (((ord_less @ C3) @ (F3 @ X)) @ (F3 @ Y4))))) => (((ord_less @ C3) @ (F3 @ A2)) @ C)))))))))).
% 2.85/1.77 thf(fact_171_order__less__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F3:(B > A),B2:B,C:B]: (((((ord_less @ A) @ A2) @ (F3 @ B2)) => ((((ord_less @ B) @ B2) @ C) => ((! [X:B,Y4:B]: (((((ord_less @ B) @ X) @ Y4) => (((ord_less @ A) @ (F3 @ X)) @ (F3 @ Y4))))) => (((ord_less @ A) @ A2) @ (F3 @ C))))))))))).
% 2.85/1.77 thf(fact_172_ord__less__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F3:(A > B),C:B]: (((((ord_less @ A) @ A2) @ B2) => (((F3 @ B2) = C) => ((! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (((ord_less @ B) @ (F3 @ X)) @ (F3 @ Y4))))) => (((ord_less @ B) @ (F3 @ A2)) @ C)))))))))).
% 2.85/1.77 thf(fact_173_ord__eq__less__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,F3:(B > A),B2:B,C:B]: (((A2 = (F3 @ B2)) => ((((ord_less @ B) @ B2) @ C) => ((! [X:B,Y4:B]: (((((ord_less @ B) @ X) @ Y4) => (((ord_less @ A) @ (F3 @ X)) @ (F3 @ Y4))))) => (((ord_less @ A) @ A2) @ (F3 @ C))))))))))).
% 2.85/1.77 thf(fact_174_bot__fun__def, axiom, (! [B:$tType,A:$tType]: ((((bot @ B) @ (type2 @ B)) => ((bot_bot @ (A > B)) = (^ [X4:A]: ((bot_bot @ B)))))))).
% 2.85/1.77 thf(fact_175_less__DAG__set__of, axiom, (! [X5:binDag_Mirabelle_dag,Y2:binDag_Mirabelle_dag]: (((((ord_less @ binDag_Mirabelle_dag) @ X5) @ Y2) => ((binDag_Mirabelle_DAG @ Y2) => (((ord_less @ (set @ simpl_ref)) @ (binDag1380252983set_of @ X5)) @ (binDag1380252983set_of @ Y2))))))).
% 2.85/1.77 thf(fact_176_order_Onot__eq__order__implies__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((A2 != B2) => ((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less @ A) @ A2) @ B2))))))))).
% 2.85/1.77 thf(fact_177_dual__order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => (((ord_less_eq @ A) @ B2) @ A2)))))))).
% 2.85/1.77 thf(fact_178_dual__order_Ostrict__iff__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [B3:A,A4:A]: (((((ord_less_eq @ A) @ B3) @ A4) & (A4 != B3))))))))).
% 2.85/1.77 thf(fact_179_dual__order_Oorder__iff__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [B3:A,A4:A]: (((((ord_less @ A) @ B3) @ A4) | (A4 = B3))))))))).
% 2.85/1.77 thf(fact_180_order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.85/1.77 thf(fact_181_dense__le__bounded, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [X5:A,Y2:A,Z2:A]: (((((ord_less @ A) @ X5) @ Y2) => ((! [W:A]: (((((ord_less @ A) @ X5) @ W) => ((((ord_less @ A) @ W) @ Y2) => (((ord_less_eq @ A) @ W) @ Z2))))) => (((ord_less_eq @ A) @ Y2) @ Z2))))))))).
% 2.85/1.77 thf(fact_182_dense__ge__bounded, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Z2:A,X5:A,Y2:A]: (((((ord_less @ A) @ Z2) @ X5) => ((! [W:A]: (((((ord_less @ A) @ Z2) @ W) => ((((ord_less @ A) @ W) @ X5) => (((ord_less_eq @ A) @ Y2) @ W))))) => (((ord_less_eq @ A) @ Y2) @ Z2))))))))).
% 2.85/1.77 thf(fact_183_dual__order_Ostrict__trans2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((ord_less @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C) @ B2) => (((ord_less @ A) @ C) @ A2))))))))).
% 2.85/1.77 thf(fact_184_dual__order_Ostrict__trans1, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less @ A) @ C) @ B2) => (((ord_less @ A) @ C) @ A2))))))))).
% 2.85/1.77 thf(fact_185_order_Ostrict__iff__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [A4:A,B3:A]: (((((ord_less_eq @ A) @ A4) @ B3) & (A4 != B3))))))))).
% 2.85/1.77 thf(fact_186_order_Oorder__iff__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [A4:A,B3:A]: (((((ord_less @ A) @ A4) @ B3) | (A4 = B3))))))))).
% 2.85/1.77 thf(fact_187_order_Ostrict__trans2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.85/1.77 thf(fact_188_order_Ostrict__trans1, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less @ A) @ B2) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.85/1.77 thf(fact_189_not__le__imp__less, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y2:A,X5:A]: (((~ ((((ord_less_eq @ A) @ Y2) @ X5))) => (((ord_less @ A) @ X5) @ Y2)))))))).
% 2.85/1.77 thf(fact_190_less__le__not__le, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [X4:A,Y3:A]: (((((ord_less_eq @ A) @ X4) @ Y3) & (~ ((((ord_less_eq @ A) @ Y3) @ X4))))))))))).
% 2.85/1.77 thf(fact_191_le__imp__less__or__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X5:A,Y2:A]: (((((ord_less_eq @ A) @ X5) @ Y2) => ((((ord_less @ A) @ X5) @ Y2) | (X5 = Y2))))))))).
% 2.85/1.77 thf(fact_192_le__less__linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X5:A,Y2:A]: (((((ord_less_eq @ A) @ X5) @ Y2) | (((ord_less @ A) @ Y2) @ X5)))))))).
% 2.85/1.77 thf(fact_193_dense__le, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Y2:A,Z2:A]: (((! [X:A]: (((((ord_less @ A) @ X) @ Y2) => (((ord_less_eq @ A) @ X) @ Z2)))) => (((ord_less_eq @ A) @ Y2) @ Z2)))))))).
% 2.85/1.77 thf(fact_194_dense__ge, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Z2:A,Y2:A]: (((! [X:A]: (((((ord_less @ A) @ Z2) @ X) => (((ord_less_eq @ A) @ Y2) @ X)))) => (((ord_less_eq @ A) @ Y2) @ Z2)))))))).
% 2.85/1.77 thf(fact_195_less__le__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X5:A,Y2:A,Z2:A]: (((((ord_less @ A) @ X5) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ Z2) => (((ord_less @ A) @ X5) @ Z2))))))))).
% 2.85/1.77 thf(fact_196_le__less__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X5:A,Y2:A,Z2:A]: (((((ord_less_eq @ A) @ X5) @ Y2) => ((((ord_less @ A) @ Y2) @ Z2) => (((ord_less @ A) @ X5) @ Z2))))))))).
% 2.85/1.77 thf(fact_197_antisym__conv2, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X5:A,Y2:A]: (((((ord_less_eq @ A) @ X5) @ Y2) => ((~ ((((ord_less @ A) @ X5) @ Y2))) = (X5 = Y2))))))))).
% 2.85/1.77 thf(fact_198_antisym__conv1, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X5:A,Y2:A]: (((~ ((((ord_less @ A) @ X5) @ Y2))) => ((((ord_less_eq @ A) @ X5) @ Y2) = (X5 = Y2))))))))).
% 2.85/1.77 thf(fact_199_less__imp__le, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X5:A,Y2:A]: (((((ord_less @ A) @ X5) @ Y2) => (((ord_less_eq @ A) @ X5) @ Y2)))))))).
% 2.85/1.77 thf(fact_200_le__neq__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((A2 != B2) => (((ord_less @ A) @ A2) @ B2))))))))).
% 2.85/1.77 thf(fact_201_not__less, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X5:A,Y2:A]: (((~ ((((ord_less @ A) @ X5) @ Y2))) = (((ord_less_eq @ A) @ Y2) @ X5)))))))).
% 2.85/1.77 thf(fact_202_not__le, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X5:A,Y2:A]: (((~ ((((ord_less_eq @ A) @ X5) @ Y2))) = (((ord_less @ A) @ Y2) @ X5)))))))).
% 2.85/1.77 thf(fact_203_order__less__le__subst2, axiom, (! [A:$tType,C3:$tType]: (((((order @ C3) @ (type2 @ C3)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F3:(A > C3),C:C3]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less_eq @ C3) @ (F3 @ B2)) @ C) => ((! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (((ord_less @ C3) @ (F3 @ X)) @ (F3 @ Y4))))) => (((ord_less @ C3) @ (F3 @ A2)) @ C)))))))))).
% 2.85/1.77 thf(fact_204_order__less__le__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F3:(B > A),B2:B,C:B]: (((((ord_less @ A) @ A2) @ (F3 @ B2)) => ((((ord_less_eq @ B) @ B2) @ C) => ((! [X:B,Y4:B]: (((((ord_less_eq @ B) @ X) @ Y4) => (((ord_less_eq @ A) @ (F3 @ X)) @ (F3 @ Y4))))) => (((ord_less @ A) @ A2) @ (F3 @ C))))))))))).
% 2.85/1.77 thf(fact_205_order__le__less__subst2, axiom, (! [A:$tType,C3:$tType]: (((((order @ C3) @ (type2 @ C3)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F3:(A > C3),C:C3]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less @ C3) @ (F3 @ B2)) @ C) => ((! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) => (((ord_less_eq @ C3) @ (F3 @ X)) @ (F3 @ Y4))))) => (((ord_less @ C3) @ (F3 @ A2)) @ C)))))))))).
% 2.85/1.77 thf(fact_206_order__le__less__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F3:(B > A),B2:B,C:B]: (((((ord_less_eq @ A) @ A2) @ (F3 @ B2)) => ((((ord_less @ B) @ B2) @ C) => ((! [X:B,Y4:B]: (((((ord_less @ B) @ X) @ Y4) => (((ord_less @ A) @ (F3 @ X)) @ (F3 @ Y4))))) => (((ord_less @ A) @ A2) @ (F3 @ C))))))))))).
% 2.85/1.77 thf(fact_207_less__le, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [X4:A,Y3:A]: (((((ord_less_eq @ A) @ X4) @ Y3) & (X4 != Y3))))))))).
% 2.85/1.77 thf(fact_208_le__less, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [X4:A,Y3:A]: (((((ord_less @ A) @ X4) @ Y3) | (X4 = Y3))))))))).
% 2.85/1.77 thf(fact_209_leI, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X5:A,Y2:A]: (((~ ((((ord_less @ A) @ X5) @ Y2))) => (((ord_less_eq @ A) @ Y2) @ X5)))))))).
% 2.85/1.77 thf(fact_210_leD, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y2:A,X5:A]: (((((ord_less_eq @ A) @ Y2) @ X5) => (~ ((((ord_less @ A) @ X5) @ Y2)))))))))).
% 2.85/1.77 thf(fact_211_bot_Oextremum__uniqueI, axiom, (! [A:$tType]: ((((order_bot @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ (bot_bot @ A)) => (A2 = (bot_bot @ A))))))))).
% 2.85/1.77 thf(fact_212_bot_Oextremum__unique, axiom, (! [A:$tType]: ((((order_bot @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ (bot_bot @ A)) = (A2 = (bot_bot @ A))))))))).
% 2.85/1.77 thf(fact_213_bot_Oextremum, axiom, (! [A:$tType]: ((((order_bot @ A) @ (type2 @ A)) => (! [A2:A]: ((((ord_less_eq @ A) @ (bot_bot @ A)) @ A2))))))).
% 2.85/1.77 thf(fact_214_complete__interval, axiom, (! [A:$tType]: ((((condit1037483654norder @ A) @ (type2 @ A)) => (! [A2:A,B2:A,P2:(A > $o)]: (((((ord_less @ A) @ A2) @ B2) => ((P2 @ A2) => ((~ ((P2 @ B2))) => (? [C4:A]: (((((ord_less_eq @ A) @ A2) @ C4) & ((((ord_less_eq @ A) @ C4) @ B2) & ((! [X6:A]: ((((((ord_less_eq @ A) @ A2) @ X6) & (((ord_less @ A) @ X6) @ C4)) => (P2 @ X6)))) & (! [D3:A]: (((! [X:A]: ((((((ord_less_eq @ A) @ A2) @ X) & (((ord_less @ A) @ X) @ D3)) => (P2 @ X)))) => (((ord_less_eq @ A) @ D3) @ C4)))))))))))))))))).
% 2.85/1.77 thf(fact_215_pinf_I6_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z4:A]: ((! [X6:A]: (((((ord_less @ A) @ Z4) @ X6) => (~ ((((ord_less_eq @ A) @ X6) @ T)))))))))))))).
% 2.85/1.77 thf(fact_216_pinf_I8_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z4:A]: ((! [X6:A]: (((((ord_less @ A) @ Z4) @ X6) => (((ord_less_eq @ A) @ T) @ X6)))))))))))).
% 2.85/1.77 thf(fact_217_minf_I6_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z4:A]: ((! [X6:A]: (((((ord_less @ A) @ X6) @ Z4) => (((ord_less_eq @ A) @ X6) @ T)))))))))))).
% 2.85/1.77 thf(fact_218_minf_I8_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z4:A]: ((! [X6:A]: (((((ord_less @ A) @ X6) @ Z4) => (~ ((((ord_less_eq @ A) @ T) @ X6)))))))))))))).
% 2.85/1.77 thf(fact_219_le__dag__antisym, axiom, (! [X5:binDag_Mirabelle_dag,Y2:binDag_Mirabelle_dag]: (((((ord_less_eq @ binDag_Mirabelle_dag) @ X5) @ Y2) => ((((ord_less_eq @ binDag_Mirabelle_dag) @ Y2) @ X5) => (X5 = Y2)))))).
% 2.85/1.77 thf(fact_220_le__dag__trans, axiom, (! [X5:binDag_Mirabelle_dag,Y2:binDag_Mirabelle_dag,Z2:binDag_Mirabelle_dag]: (((((ord_less_eq @ binDag_Mirabelle_dag) @ X5) @ Y2) => ((((ord_less_eq @ binDag_Mirabelle_dag) @ Y2) @ Z2) => (((ord_less_eq @ binDag_Mirabelle_dag) @ X5) @ Z2)))))).
% 2.85/1.77 thf(fact_221_le__dag__refl, axiom, (! [X5:binDag_Mirabelle_dag]: ((((ord_less_eq @ binDag_Mirabelle_dag) @ X5) @ X5)))).
% 2.85/1.77 thf(fact_222_less__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less @ (A > B)) = (^ [F:(A > B),G2:(A > B)]: (((((ord_less_eq @ (A > B)) @ F) @ G2) & (~ ((((ord_less_eq @ (A > B)) @ G2) @ F))))))))))).
% 2.85/1.77 thf(fact_223_psubset__trans, axiom, (! [A:$tType,A3:(set @ A),B4:(set @ A),C2:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B4) => ((((ord_less @ (set @ A)) @ B4) @ C2) => (((ord_less @ (set @ A)) @ A3) @ C2)))))).
% 2.85/1.77 thf(fact_224_psubsetD, axiom, (! [A:$tType,A3:(set @ A),B4:(set @ A),C:A]: (((((ord_less @ (set @ A)) @ A3) @ B4) => ((((member @ A) @ C) @ A3) => (((member @ A) @ C) @ B4)))))).
% 2.85/1.77 thf(fact_225_minf_I11_J, axiom, (! [C3:$tType,D2:$tType]: ((((ord @ C3) @ (type2 @ C3)) => (! [F4:D2]: ((? [Z4:C3]: ((! [X6:C3]: (((((ord_less @ C3) @ X6) @ Z4) => (F4 = F4)))))))))))).
% 2.85/1.77 thf(fact_226_minf_I7_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z4:A]: ((! [X6:A]: (((((ord_less @ A) @ X6) @ Z4) => (~ ((((ord_less @ A) @ T) @ X6)))))))))))))).
% 2.85/1.77 thf(fact_227_minf_I5_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z4:A]: ((! [X6:A]: (((((ord_less @ A) @ X6) @ Z4) => (((ord_less @ A) @ X6) @ T)))))))))))).
% 2.85/1.77 thf(fact_228_minf_I4_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z4:A]: ((! [X6:A]: (((((ord_less @ A) @ X6) @ Z4) => (X6 != T)))))))))))).
% 2.85/1.77 thf(fact_229_minf_I3_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z4:A]: ((! [X6:A]: (((((ord_less @ A) @ X6) @ Z4) => (X6 != T)))))))))))).
% 2.85/1.77 thf(fact_230_minf_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P2:(A > $o),P3:(A > $o),Q3:(A > $o),Q4:(A > $o)]: (((? [Z5:A]: ((! [X:A]: (((((ord_less @ A) @ X) @ Z5) => ((P2 @ X) = (P3 @ X))))))) => ((? [Z5:A]: ((! [X:A]: (((((ord_less @ A) @ X) @ Z5) => ((Q3 @ X) = (Q4 @ X))))))) => (? [Z4:A]: ((! [X6:A]: (((((ord_less @ A) @ X6) @ Z4) => (((P2 @ X6) | (Q3 @ X6)) = ((P3 @ X6) | (Q4 @ X6)))))))))))))))).
% 2.85/1.77 thf(fact_231_minf_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P2:(A > $o),P3:(A > $o),Q3:(A > $o),Q4:(A > $o)]: (((? [Z5:A]: ((! [X:A]: (((((ord_less @ A) @ X) @ Z5) => ((P2 @ X) = (P3 @ X))))))) => ((? [Z5:A]: ((! [X:A]: (((((ord_less @ A) @ X) @ Z5) => ((Q3 @ X) = (Q4 @ X))))))) => (? [Z4:A]: ((! [X6:A]: (((((ord_less @ A) @ X6) @ Z4) => (((P2 @ X6) & (Q3 @ X6)) = ((P3 @ X6) & (Q4 @ X6)))))))))))))))).
% 2.85/1.77 thf(fact_232_pinf_I11_J, axiom, (! [C3:$tType,D2:$tType]: ((((ord @ C3) @ (type2 @ C3)) => (! [F4:D2]: ((? [Z4:C3]: ((! [X6:C3]: (((((ord_less @ C3) @ Z4) @ X6) => (F4 = F4)))))))))))).
% 2.85/1.77 thf(fact_233_pinf_I7_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z4:A]: ((! [X6:A]: (((((ord_less @ A) @ Z4) @ X6) => (((ord_less @ A) @ T) @ X6)))))))))))).
% 2.85/1.77 thf(fact_234_pinf_I5_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z4:A]: ((! [X6:A]: (((((ord_less @ A) @ Z4) @ X6) => (~ ((((ord_less @ A) @ X6) @ T)))))))))))))).
% 2.85/1.77 thf(fact_235_pinf_I4_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z4:A]: ((! [X6:A]: (((((ord_less @ A) @ Z4) @ X6) => (X6 != T)))))))))))).
% 2.85/1.77 thf(fact_236_pinf_I3_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z4:A]: ((! [X6:A]: (((((ord_less @ A) @ Z4) @ X6) => (X6 != T)))))))))))).
% 2.85/1.77 thf(fact_237_pinf_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P2:(A > $o),P3:(A > $o),Q3:(A > $o),Q4:(A > $o)]: (((? [Z5:A]: ((! [X:A]: (((((ord_less @ A) @ Z5) @ X) => ((P2 @ X) = (P3 @ X))))))) => ((? [Z5:A]: ((! [X:A]: (((((ord_less @ A) @ Z5) @ X) => ((Q3 @ X) = (Q4 @ X))))))) => (? [Z4:A]: ((! [X6:A]: (((((ord_less @ A) @ Z4) @ X6) => (((P2 @ X6) | (Q3 @ X6)) = ((P3 @ X6) | (Q4 @ X6)))))))))))))))).
% 2.85/1.77 thf(fact_238_pinf_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P2:(A > $o),P3:(A > $o),Q3:(A > $o),Q4:(A > $o)]: (((? [Z5:A]: ((! [X:A]: (((((ord_less @ A) @ Z5) @ X) => ((P2 @ X) = (P3 @ X))))))) => ((? [Z5:A]: ((! [X:A]: (((((ord_less @ A) @ Z5) @ X) => ((Q3 @ X) = (Q4 @ X))))))) => (? [Z4:A]: ((! [X6:A]: (((((ord_less @ A) @ Z4) @ X6) => (((P2 @ X6) & (Q3 @ X6)) = ((P3 @ X6) & (Q4 @ X6)))))))))))))))).
% 2.85/1.77 thf(fact_239_ex__gt__or__lt, axiom, (! [A:$tType]: ((((condit1656338222tinuum @ A) @ (type2 @ A)) => (! [A2:A]: ((? [B6:A]: (((((ord_less @ A) @ A2) @ B6) | (((ord_less @ A) @ B6) @ A2)))))))))).
% 2.85/1.77 thf(fact_240_bot__empty__eq, axiom, (! [A:$tType]: (((bot_bot @ (A > $o)) = (^ [X4:A]: ((((member @ A) @ X4) @ (bot_bot @ (set @ A))))))))).
% 2.85/1.77 thf(fact_241_Collect__empty__eq__bot, axiom, (! [A:$tType,P2:(A > $o)]: (((((collect @ A) @ P2) = (bot_bot @ (set @ A))) = (P2 = (bot_bot @ (A > $o))))))).
% 2.85/1.77 thf(fact_242_override__on__emptyset, axiom, (! [B:$tType,A:$tType,F3:(A > B),G:(A > B)]: (((((((override_on @ A) @ B) @ F3) @ G) @ (bot_bot @ (set @ A))) = F3)))).
% 2.85/1.77 thf(fact_243_Set_Ois__empty__def, axiom, (! [A:$tType]: (((is_empty @ A) = (^ [A5:(set @ A)]: ((A5 = (bot_bot @ (set @ A))))))))).
% 2.85/1.77 thf(fact_244_override__on__apply__notin, axiom, (! [B:$tType,A:$tType,A2:A,A3:(set @ A),F3:(A > B),G:(A > B)]: (((~ ((((member @ A) @ A2) @ A3))) => (((((((override_on @ A) @ B) @ F3) @ G) @ A3) @ A2) = (F3 @ A2)))))).
% 2.85/1.77 thf(fact_245_override__on__apply__in, axiom, (! [B:$tType,A:$tType,A2:A,A3:(set @ A),F3:(A > B),G:(A > B)]: (((((member @ A) @ A2) @ A3) => (((((((override_on @ A) @ B) @ F3) @ G) @ A3) @ A2) = (G @ A2)))))).
% 2.85/1.77 thf(fact_246_override__on__def, axiom, (! [B:$tType,A:$tType]: ((((override_on @ A) @ B) = (^ [F:(A > B),G2:(A > B),A5:(set @ A),A4:A]: (((((if @ B) @ (((member @ A) @ A4) @ A5)) @ (G2 @ A4)) @ (F @ A4)))))))).
% 2.85/1.77 thf(fact_247_is__empty__set, axiom, (! [A:$tType,Xs:(list @ A)]: ((((is_empty @ A) @ ((set2 @ A) @ Xs)) = ((null @ A) @ Xs))))).
% 2.85/1.77 thf(fact_248_subset__code_I2_J, axiom, (! [B:$tType,A3:(set @ B),Ys:(list @ B)]: (((((ord_less_eq @ (set @ B)) @ A3) @ ((coset @ B) @ Ys)) = (! [X4:B]: (((((member @ B) @ X4) @ ((set2 @ B) @ Ys)) => (~ ((((member @ B) @ X4) @ A3)))))))))).
% 2.85/1.77 thf(fact_249_subset__code_I3_J, axiom, (! [C3:$tType]: ((~ ((((ord_less_eq @ (set @ C3)) @ ((coset @ C3) @ (nil @ C3))) @ ((set2 @ C3) @ (nil @ C3)))))))).
% 2.85/1.77 thf(fact_250_set__empty2, axiom, (! [A:$tType,Xs:(list @ A)]: ((((bot_bot @ (set @ A)) = ((set2 @ A) @ Xs)) = (Xs = (nil @ A)))))).
% 2.85/1.77 thf(fact_251_set__empty, axiom, (! [A:$tType,Xs:(list @ A)]: (((((set2 @ A) @ Xs) = (bot_bot @ (set @ A))) = (Xs = (nil @ A)))))).
% 2.85/1.77 thf(fact_252_empty__set, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((set2 @ A) @ (nil @ A)))))).
% 2.85/1.77 thf(fact_253_sorted__list__of__set__empty, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (((linord467138063of_set @ A) @ (bot_bot @ (set @ A))) = (nil @ A)))))).
% 2.85/1.77 thf(fact_254_sorted__list__of__set__eq__Nil__iff, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [A3:(set @ A)]: ((((finite_finite @ A) @ A3) => ((((linord467138063of_set @ A) @ A3) = (nil @ A)) = (A3 = (bot_bot @ (set @ A))))))))))).
% 2.85/1.77 thf(fact_255_List_Ofinite__set, axiom, (! [A:$tType,Xs:(list @ A)]: (((finite_finite @ A) @ ((set2 @ A) @ Xs))))).
% 2.85/1.77 thf(tcon_fun___Orderings_Oorder__bot, axiom, (! [A7:$tType,A8:$tType]: ((((order_bot @ A8) @ (type2 @ A8)) => ((order_bot @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.85/1.77 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A7:$tType,A8:$tType]: ((((preorder @ A8) @ (type2 @ A8)) => ((preorder @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.85/1.77 thf(tcon_fun___Orderings_Oorder, axiom, (! [A7:$tType,A8:$tType]: ((((order @ A8) @ (type2 @ A8)) => ((order @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.85/1.77 thf(tcon_fun___Orderings_Oord, axiom, (! [A7:$tType,A8:$tType]: ((((ord @ A8) @ (type2 @ A8)) => ((ord @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.85/1.77 thf(tcon_fun___Orderings_Obot, axiom, (! [A7:$tType,A8:$tType]: ((((bot @ A8) @ (type2 @ A8)) => ((bot @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.85/1.77 thf(tcon_Set_Oset___Orderings_Oorder__bot_1, axiom, (! [A7:$tType]: (((order_bot @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.85/1.77 thf(tcon_Set_Oset___Orderings_Opreorder_2, axiom, (! [A7:$tType]: (((preorder @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.85/1.77 thf(tcon_Set_Oset___Orderings_Oorder_3, axiom, (! [A7:$tType]: (((order @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.85/1.77 thf(tcon_Set_Oset___Orderings_Oord_4, axiom, (! [A7:$tType]: (((ord @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.85/1.77 thf(tcon_Set_Oset___Orderings_Obot_5, axiom, (! [A7:$tType]: (((bot @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.85/1.77 thf(tcon_HOL_Obool___Orderings_Oorder__bot_6, axiom, ((order_bot @ $o) @ (type2 @ $o))).
% 2.85/1.77 thf(tcon_HOL_Obool___Orderings_Opreorder_7, axiom, ((preorder @ $o) @ (type2 @ $o))).
% 2.85/1.77 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, ((linorder @ $o) @ (type2 @ $o))).
% 2.85/1.77 thf(tcon_HOL_Obool___Orderings_Oorder_8, axiom, ((order @ $o) @ (type2 @ $o))).
% 2.85/1.77 thf(tcon_HOL_Obool___Orderings_Oord_9, axiom, ((ord @ $o) @ (type2 @ $o))).
% 2.85/1.77 thf(tcon_HOL_Obool___Orderings_Obot_10, axiom, ((bot @ $o) @ (type2 @ $o))).
% 2.85/1.77 thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Opreorder_11, axiom, ((preorder @ binDag_Mirabelle_dag) @ (type2 @ binDag_Mirabelle_dag))).
% 2.85/1.77 thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Oorder_12, axiom, ((order @ binDag_Mirabelle_dag) @ (type2 @ binDag_Mirabelle_dag))).
% 2.85/1.77 thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Oord_13, axiom, ((ord @ binDag_Mirabelle_dag) @ (type2 @ binDag_Mirabelle_dag))).
% 2.85/1.77 thf(help_If_3_1_T, axiom, (! [P2:$o]: (((P2 = $true) | (P2 = $false))))).
% 2.85/1.77 thf(help_If_2_1_T, axiom, (! [A:$tType,X5:A,Y2:A]: ((((((if @ A) @ $false) @ X5) @ Y2) = Y2)))).
% 2.85/1.77 thf(help_If_1_1_T, axiom, (! [A:$tType,X5:A,Y2:A]: ((((((if @ A) @ $true) @ X5) @ Y2) = X5)))).
% 2.85/1.77 thf(conj_0, hypothesis, (! [Q5:simpl_ref]: ((((((binDag_Mirabelle_Dag @ Q5) @ l) @ r) @ (((binDag476092410e_Node @ lt) @ p) @ rt)) => thesis)))).
% 2.85/1.77 thf(conj_1, conjecture, thesis).
% 2.85/1.77 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.CLaN6L8RqG/DTF2DTF22996.p
% 2.85/1.77 ------------------------
% 2.98/1.81 ---- Cleaned THF ---
% 2.98/1.81 thf(ty_t_BinDag__Mirabelle__rybootvolr_Odag,type,
% 2.98/1.81 binDag_Mirabelle_dag: $tType ).
% 2.98/1.81
% 2.98/1.81 thf(ty_t_Simpl__Heap_Oref,type,
% 2.98/1.81 simpl_ref: $tType ).
% 2.98/1.81
% 2.98/1.81 thf(ty_t_List_Olist,type,
% 2.98/1.81 list: $tType > $tType ).
% 2.98/1.81
% 2.98/1.81 thf(ty_t_Set_Oset,type,
% 2.98/1.81 set: $tType > $tType ).
% 2.98/1.81
% 2.98/1.81 thf(ty_t_itself,type,
% 2.98/1.81 itself: $tType > $tType ).
% 2.98/1.81
% 2.98/1.81 thf(sy_cl_HOL_Otype,type,
% 2.98/1.81 type:
% 2.98/1.81 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_cl_Orderings_Obot,type,
% 2.98/1.81 bot:
% 2.98/1.81 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_cl_Orderings_Oord,type,
% 2.98/1.81 ord:
% 2.98/1.81 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_cl_Orderings_Oorder,type,
% 2.98/1.81 order:
% 2.98/1.81 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_cl_Orderings_Ono__bot,type,
% 2.98/1.81 no_bot:
% 2.98/1.81 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_cl_Orderings_Ono__top,type,
% 2.98/1.81 no_top:
% 2.98/1.81 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_cl_Orderings_Olinorder,type,
% 2.98/1.81 linorder:
% 2.98/1.81 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_cl_Orderings_Opreorder,type,
% 2.98/1.81 preorder:
% 2.98/1.81 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_cl_Orderings_Oorder__bot,type,
% 2.98/1.81 order_bot:
% 2.98/1.81 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_cl_Orderings_Owellorder,type,
% 2.98/1.81 wellorder:
% 2.98/1.81 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_cl_Orderings_Odense__order,type,
% 2.98/1.81 dense_order:
% 2.98/1.81 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_cl_Orderings_Odense__linorder,type,
% 2.98/1.81 dense_linorder:
% 2.98/1.81 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 2.98/1.81 condit1656338222tinuum:
% 2.98/1.81 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 2.98/1.81 condit1037483654norder:
% 2.98/1.81 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_c_BinDag__Mirabelle__rybootvolr_ODAG,type,
% 2.98/1.81 binDag_Mirabelle_DAG: binDag_Mirabelle_dag > $o ).
% 2.98/1.81
% 2.98/1.81 thf(sy_c_BinDag__Mirabelle__rybootvolr_ODag,type,
% 2.98/1.81 binDag_Mirabelle_Dag: simpl_ref > ( simpl_ref > simpl_ref ) > ( simpl_ref > simpl_ref ) > binDag_Mirabelle_dag > $o ).
% 2.98/1.81
% 2.98/1.81 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_ONode,type,
% 2.98/1.81 binDag476092410e_Node: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > binDag_Mirabelle_dag ).
% 2.98/1.81
% 2.98/1.81 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_OTip,type,
% 2.98/1.81 binDag_Mirabelle_Tip: binDag_Mirabelle_dag ).
% 2.98/1.81
% 2.98/1.81 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_Ocase__dag,type,
% 2.98/1.81 binDag1297733282se_dag:
% 2.98/1.81 !>[A: $tType] : ( A > ( binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A ) > binDag_Mirabelle_dag > A ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_Orec__dag,type,
% 2.98/1.81 binDag1442713106ec_dag:
% 2.98/1.81 !>[A: $tType] : ( A > ( binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A > A > A ) > binDag_Mirabelle_dag > A ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_c_BinDag__Mirabelle__rybootvolr_Oset__of,type,
% 2.98/1.81 binDag1380252983set_of: binDag_Mirabelle_dag > ( set @ simpl_ref ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_c_BinDag__Mirabelle__rybootvolr_Osubdag,type,
% 2.98/1.81 binDag786255756subdag: binDag_Mirabelle_dag > binDag_Mirabelle_dag > $o ).
% 2.98/1.81
% 2.98/1.81 thf(sy_c_Finite__Set_Ofinite,type,
% 2.98/1.81 finite_finite:
% 2.98/1.81 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_c_Fun_Ofun__upd,type,
% 2.98/1.81 fun_upd:
% 2.98/1.81 !>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_c_Fun_Ooverride__on,type,
% 2.98/1.81 override_on:
% 2.98/1.81 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > B ) > ( set @ A ) > A > B ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_c_Fun_Oswap,type,
% 2.98/1.81 swap:
% 2.98/1.81 !>[A: $tType,B: $tType] : ( A > A > ( A > B ) > A > B ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_c_If,type,
% 2.98/1.81 if:
% 2.98/1.81 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_c_List_Ocoset,type,
% 2.98/1.81 coset:
% 2.98/1.81 !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
% 2.98/1.81 linord467138063of_set:
% 2.98/1.81 !>[A: $tType] : ( ( set @ A ) > ( list @ A ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_c_List_Olist_ONil,type,
% 2.98/1.81 nil:
% 2.98/1.81 !>[A: $tType] : ( list @ A ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_c_List_Olist_Oset,type,
% 2.98/1.81 set2:
% 2.98/1.81 !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_c_List_Onull,type,
% 2.98/1.81 null:
% 2.98/1.81 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_c_Orderings_Obot__class_Obot,type,
% 2.98/1.81 bot_bot:
% 2.98/1.81 !>[A: $tType] : A ).
% 2.98/1.81
% 2.98/1.81 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.98/1.81 ord_less:
% 2.98/1.81 !>[A: $tType] : ( A > A > $o ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.98/1.81 ord_less_eq:
% 2.98/1.81 !>[A: $tType] : ( A > A > $o ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_c_Pure_Otype,type,
% 2.98/1.81 type2:
% 2.98/1.81 !>[A: $tType] : ( itself @ A ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_c_Set_OCollect,type,
% 2.98/1.81 collect:
% 2.98/1.81 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_c_Set_Ois__empty,type,
% 2.98/1.81 is_empty:
% 2.98/1.81 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_c_Simpl__Heap_ONull,type,
% 2.98/1.81 simpl_Null: simpl_ref ).
% 2.98/1.81
% 2.98/1.81 thf(sy_c_Simpl__Heap_Onew,type,
% 2.98/1.81 simpl_new: ( set @ simpl_ref ) > simpl_ref ).
% 2.98/1.81
% 2.98/1.81 thf(sy_c_member,type,
% 2.98/1.81 member:
% 2.98/1.81 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.98/1.81
% 2.98/1.81 thf(sy_v_l,type,
% 2.98/1.81 l: simpl_ref > simpl_ref ).
% 2.98/1.81
% 2.98/1.81 thf(sy_v_lt____,type,
% 2.98/1.81 lt: binDag_Mirabelle_dag ).
% 2.98/1.81
% 2.98/1.81 thf(sy_v_p,type,
% 2.98/1.81 p: simpl_ref ).
% 2.98/1.81
% 2.98/1.81 thf(sy_v_r,type,
% 2.98/1.81 r: simpl_ref > simpl_ref ).
% 2.98/1.81
% 2.98/1.81 thf(sy_v_rt____,type,
% 2.98/1.81 rt: binDag_Mirabelle_dag ).
% 2.98/1.81
% 2.98/1.81 thf(sy_v_t,type,
% 2.98/1.81 t: binDag_Mirabelle_dag ).
% 2.98/1.81
% 2.98/1.81 thf(sy_v_thesis____,type,
% 2.98/1.81 thesis: $o ).
% 2.98/1.81
% 2.98/1.81 thf(fact_0_assms,axiom,
% 2.98/1.81 binDag_Mirabelle_Dag @ ( r @ p ) @ l @ r @ t ).
% 2.98/1.81
% 2.98/1.81 thf(fact_1_calculation,axiom,
% 2.98/1.81 ! [Lt: binDag_Mirabelle_dag,Rt: binDag_Mirabelle_dag] :
% 2.98/1.81 ( t
% 2.98/1.81 != ( binDag476092410e_Node @ Lt @ p @ Rt ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_2_subdag,axiom,
% 2.98/1.81 binDag786255756subdag @ t @ ( binDag476092410e_Node @ lt @ p @ rt ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_3_dag_Oinject,axiom,
% 2.98/1.81 ! [X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag,Y21: binDag_Mirabelle_dag,Y22: simpl_ref,Y23: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( ( binDag476092410e_Node @ X21 @ X22 @ X23 )
% 2.98/1.81 = ( binDag476092410e_Node @ Y21 @ Y22 @ Y23 ) )
% 2.98/1.81 = ( ( X21 = Y21 )
% 2.98/1.81 & ( X22 = Y22 )
% 2.98/1.81 & ( X23 = Y23 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_4_Dag__unique,axiom,
% 2.98/1.81 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T1: binDag_Mirabelle_dag,T2: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( binDag_Mirabelle_Dag @ P @ L @ R @ T1 )
% 2.98/1.81 => ( ( binDag_Mirabelle_Dag @ P @ L @ R @ T2 )
% 2.98/1.81 => ( T1 = T2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_5_Dag__unique1,axiom,
% 2.98/1.81 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( binDag_Mirabelle_Dag @ P @ L @ R @ T )
% 2.98/1.81 => ? [X: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( binDag_Mirabelle_Dag @ P @ L @ R @ X )
% 2.98/1.81 & ! [Y: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( binDag_Mirabelle_Dag @ P @ L @ R @ Y )
% 2.98/1.81 => ( Y = X ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_6_subdag_Osimps_I2_J,axiom,
% 2.98/1.81 ! [L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag,T: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( binDag786255756subdag @ ( binDag476092410e_Node @ L @ A2 @ R ) @ T )
% 2.98/1.81 = ( ( T = L )
% 2.98/1.81 | ( T = R )
% 2.98/1.81 | ( binDag786255756subdag @ L @ T )
% 2.98/1.81 | ( binDag786255756subdag @ R @ T ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_7_subdag__NodeD,axiom,
% 2.98/1.81 ! [T: binDag_Mirabelle_dag,Lt2: binDag_Mirabelle_dag,A2: simpl_ref,Rt2: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( binDag786255756subdag @ T @ ( binDag476092410e_Node @ Lt2 @ A2 @ Rt2 ) )
% 2.98/1.81 => ( ( binDag786255756subdag @ T @ Lt2 )
% 2.98/1.81 & ( binDag786255756subdag @ T @ Rt2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_8_Dag__subdag,axiom,
% 2.98/1.81 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag,S: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( binDag_Mirabelle_Dag @ P @ L @ R @ T )
% 2.98/1.81 => ( ( binDag786255756subdag @ T @ S )
% 2.98/1.81 => ? [Q: simpl_ref] : ( binDag_Mirabelle_Dag @ Q @ L @ R @ S ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_9_Dag__Ref,axiom,
% 2.98/1.81 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( P != simpl_Null )
% 2.98/1.81 => ( ( binDag_Mirabelle_Dag @ P @ L @ R @ T )
% 2.98/1.81 = ( ? [Lt3: binDag_Mirabelle_dag,Rt3: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( T
% 2.98/1.81 = ( binDag476092410e_Node @ Lt3 @ P @ Rt3 ) )
% 2.98/1.81 & ( binDag_Mirabelle_Dag @ ( L @ P ) @ L @ R @ Lt3 )
% 2.98/1.81 & ( binDag_Mirabelle_Dag @ ( R @ P ) @ L @ R @ Rt3 ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_10_Dag_Osimps_I2_J,axiom,
% 2.98/1.81 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,Lt2: binDag_Mirabelle_dag,A2: simpl_ref,Rt2: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( binDag_Mirabelle_Dag @ P @ L @ R @ ( binDag476092410e_Node @ Lt2 @ A2 @ Rt2 ) )
% 2.98/1.81 = ( ( P = A2 )
% 2.98/1.81 & ( P != simpl_Null )
% 2.98/1.81 & ( binDag_Mirabelle_Dag @ ( L @ P ) @ L @ R @ Lt2 )
% 2.98/1.81 & ( binDag_Mirabelle_Dag @ ( R @ P ) @ L @ R @ Rt2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_11_dag_Osimps_I7_J,axiom,
% 2.98/1.81 ! [A: $tType,F1: A,F2: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A > A > A,X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( binDag1442713106ec_dag @ A @ F1 @ F2 @ ( binDag476092410e_Node @ X21 @ X22 @ X23 ) )
% 2.98/1.81 = ( F2 @ X21 @ X22 @ X23 @ ( binDag1442713106ec_dag @ A @ F1 @ F2 @ X21 ) @ ( binDag1442713106ec_dag @ A @ F1 @ F2 @ X23 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_12_dag_Osimps_I5_J,axiom,
% 2.98/1.81 ! [A: $tType,F1: A,F2: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A,X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( binDag1297733282se_dag @ A @ F1 @ F2 @ ( binDag476092410e_Node @ X21 @ X22 @ X23 ) )
% 2.98/1.81 = ( F2 @ X21 @ X22 @ X23 ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_13_Dag__root__not__in__subdag__l,axiom,
% 2.98/1.81 ! [L: simpl_ref > simpl_ref,P: simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( binDag_Mirabelle_Dag @ ( L @ P ) @ L @ R @ T )
% 2.98/1.81 => ~ ( member @ simpl_ref @ P @ ( binDag1380252983set_of @ T ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_14_dag_Odistinct_I1_J,axiom,
% 2.98/1.81 ! [X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag] :
% 2.98/1.81 ( binDag_Mirabelle_Tip
% 2.98/1.81 != ( binDag476092410e_Node @ X21 @ X22 @ X23 ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_15_dag_Oinduct,axiom,
% 2.98/1.81 ! [P2: binDag_Mirabelle_dag > $o,Dag: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( P2 @ binDag_Mirabelle_Tip )
% 2.98/1.81 => ( ! [X1: binDag_Mirabelle_dag,X2: simpl_ref,X3: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( P2 @ X1 )
% 2.98/1.81 => ( ( P2 @ X3 )
% 2.98/1.81 => ( P2 @ ( binDag476092410e_Node @ X1 @ X2 @ X3 ) ) ) )
% 2.98/1.81 => ( P2 @ Dag ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_16_dag_Oexhaust,axiom,
% 2.98/1.81 ! [Y2: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( Y2 != binDag_Mirabelle_Tip )
% 2.98/1.81 => ~ ! [X212: binDag_Mirabelle_dag,X222: simpl_ref,X232: binDag_Mirabelle_dag] :
% 2.98/1.81 ( Y2
% 2.98/1.81 != ( binDag476092410e_Node @ X212 @ X222 @ X232 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_17_subdag__neq,axiom,
% 2.98/1.81 ! [T: binDag_Mirabelle_dag,S: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( binDag786255756subdag @ T @ S )
% 2.98/1.81 => ( T != S ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_18_Null__notin__Dag,axiom,
% 2.98/1.81 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( binDag_Mirabelle_Dag @ P @ L @ R @ T )
% 2.98/1.81 => ~ ( member @ simpl_ref @ simpl_Null @ ( binDag1380252983set_of @ T ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_19_Dag__Null,axiom,
% 2.98/1.81 ! [L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( binDag_Mirabelle_Dag @ simpl_Null @ L @ R @ T )
% 2.98/1.81 = ( T = binDag_Mirabelle_Tip ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_20_dag_Osimps_I6_J,axiom,
% 2.98/1.81 ! [A: $tType,F1: A,F2: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A > A > A] :
% 2.98/1.81 ( ( binDag1442713106ec_dag @ A @ F1 @ F2 @ binDag_Mirabelle_Tip )
% 2.98/1.81 = F1 ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_21_dag_Osimps_I4_J,axiom,
% 2.98/1.81 ! [A: $tType,F1: A,F2: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A] :
% 2.98/1.81 ( ( binDag1297733282se_dag @ A @ F1 @ F2 @ binDag_Mirabelle_Tip )
% 2.98/1.81 = F1 ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_22_subdag_Osimps_I1_J,axiom,
% 2.98/1.81 ! [T: binDag_Mirabelle_dag] :
% 2.98/1.81 ~ ( binDag786255756subdag @ binDag_Mirabelle_Tip @ T ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_23_Dag_Osimps_I1_J,axiom,
% 2.98/1.81 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref] :
% 2.98/1.81 ( ( binDag_Mirabelle_Dag @ P @ L @ R @ binDag_Mirabelle_Tip )
% 2.98/1.81 = ( P = simpl_Null ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_24_in__set__of__decomp,axiom,
% 2.98/1.81 ! [P: simpl_ref,T: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( member @ simpl_ref @ P @ ( binDag1380252983set_of @ T ) )
% 2.98/1.81 = ( ? [L2: binDag_Mirabelle_dag,R2: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( T
% 2.98/1.81 = ( binDag476092410e_Node @ L2 @ P @ R2 ) )
% 2.98/1.81 | ( binDag786255756subdag @ T @ ( binDag476092410e_Node @ L2 @ P @ R2 ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_25_subdag__not__sym,axiom,
% 2.98/1.81 ! [S: binDag_Mirabelle_dag,T: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( binDag786255756subdag @ S @ T )
% 2.98/1.81 => ~ ( binDag786255756subdag @ T @ S ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_26_subdag__trans,axiom,
% 2.98/1.81 ! [T: binDag_Mirabelle_dag,S: binDag_Mirabelle_dag,R: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( binDag786255756subdag @ T @ S )
% 2.98/1.81 => ( ( binDag786255756subdag @ S @ R )
% 2.98/1.81 => ( binDag786255756subdag @ T @ R ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_27_DAG_Osimps_I2_J,axiom,
% 2.98/1.81 ! [L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( binDag_Mirabelle_DAG @ ( binDag476092410e_Node @ L @ A2 @ R ) )
% 2.98/1.81 = ( ~ ( member @ simpl_ref @ A2 @ ( binDag1380252983set_of @ L ) )
% 2.98/1.81 & ~ ( member @ simpl_ref @ A2 @ ( binDag1380252983set_of @ R ) )
% 2.98/1.81 & ( binDag_Mirabelle_DAG @ L )
% 2.98/1.81 & ( binDag_Mirabelle_DAG @ R ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_28_Dag__upd__same__r,axiom,
% 2.98/1.81 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( binDag_Mirabelle_Dag @ P @ L @ ( fun_upd @ simpl_ref @ simpl_ref @ R @ P @ P ) @ T )
% 2.98/1.81 = ( ( P = simpl_Null )
% 2.98/1.81 & ( T = binDag_Mirabelle_Tip ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_29_Dag__upd__same__l,axiom,
% 2.98/1.81 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( binDag_Mirabelle_Dag @ P @ ( fun_upd @ simpl_ref @ simpl_ref @ L @ P @ P ) @ R @ T )
% 2.98/1.81 = ( ( P = simpl_Null )
% 2.98/1.81 & ( T = binDag_Mirabelle_Tip ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_30_notin__Dag__update__r,axiom,
% 2.98/1.81 ! [Q2: simpl_ref,T: binDag_Mirabelle_dag,P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,Y2: simpl_ref] :
% 2.98/1.81 ( ~ ( member @ simpl_ref @ Q2 @ ( binDag1380252983set_of @ T ) )
% 2.98/1.81 => ( ( binDag_Mirabelle_Dag @ P @ L @ ( fun_upd @ simpl_ref @ simpl_ref @ R @ Q2 @ Y2 ) @ T )
% 2.98/1.81 = ( binDag_Mirabelle_Dag @ P @ L @ R @ T ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_31_notin__Dag__update__l,axiom,
% 2.98/1.81 ! [Q2: simpl_ref,T: binDag_Mirabelle_dag,P: simpl_ref,L: simpl_ref > simpl_ref,Y2: simpl_ref,R: simpl_ref > simpl_ref] :
% 2.98/1.81 ( ~ ( member @ simpl_ref @ Q2 @ ( binDag1380252983set_of @ T ) )
% 2.98/1.81 => ( ( binDag_Mirabelle_Dag @ P @ ( fun_upd @ simpl_ref @ simpl_ref @ L @ Q2 @ Y2 ) @ R @ T )
% 2.98/1.81 = ( binDag_Mirabelle_Dag @ P @ L @ R @ T ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_32_Dag__update__rI,axiom,
% 2.98/1.81 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag,Q2: simpl_ref,Y2: simpl_ref] :
% 2.98/1.81 ( ( binDag_Mirabelle_Dag @ P @ L @ R @ T )
% 2.98/1.81 => ( ~ ( member @ simpl_ref @ Q2 @ ( binDag1380252983set_of @ T ) )
% 2.98/1.81 => ( binDag_Mirabelle_Dag @ P @ L @ ( fun_upd @ simpl_ref @ simpl_ref @ R @ Q2 @ Y2 ) @ T ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_33_Dag__update__lI,axiom,
% 2.98/1.81 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag,Q2: simpl_ref,Y2: simpl_ref] :
% 2.98/1.81 ( ( binDag_Mirabelle_Dag @ P @ L @ R @ T )
% 2.98/1.81 => ( ~ ( member @ simpl_ref @ Q2 @ ( binDag1380252983set_of @ T ) )
% 2.98/1.81 => ( binDag_Mirabelle_Dag @ P @ ( fun_upd @ simpl_ref @ simpl_ref @ L @ Q2 @ Y2 ) @ R @ T ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_34_DAG_Osimps_I1_J,axiom,
% 2.98/1.81 binDag_Mirabelle_DAG @ binDag_Mirabelle_Tip ).
% 2.98/1.81
% 2.98/1.81 thf(fact_35_Dag__upd__same__r__lemma,axiom,
% 2.98/1.81 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( P != simpl_Null )
% 2.98/1.81 => ~ ( binDag_Mirabelle_Dag @ P @ L @ ( fun_upd @ simpl_ref @ simpl_ref @ R @ P @ P ) @ T ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_36_Dag__upd__same__l__lemma,axiom,
% 2.98/1.81 ! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( P != simpl_Null )
% 2.98/1.81 => ~ ( binDag_Mirabelle_Dag @ P @ ( fun_upd @ simpl_ref @ simpl_ref @ L @ P @ P ) @ R @ T ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_37_fun__upd__apply,axiom,
% 2.98/1.81 ! [A: $tType,B: $tType] :
% 2.98/1.81 ( ( fun_upd @ B @ A )
% 2.98/1.81 = ( ^ [F: B > A,X4: B,Y3: A,Z: B] : ( if @ A @ ( Z = X4 ) @ Y3 @ ( F @ Z ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_38_fun__upd__triv,axiom,
% 2.98/1.81 ! [B: $tType,A: $tType,F3: A > B,X5: A] :
% 2.98/1.81 ( ( fun_upd @ A @ B @ F3 @ X5 @ ( F3 @ X5 ) )
% 2.98/1.81 = F3 ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_39_fun__upd__upd,axiom,
% 2.98/1.81 ! [A: $tType,B: $tType,F3: A > B,X5: A,Y2: B,Z2: B] :
% 2.98/1.81 ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ F3 @ X5 @ Y2 ) @ X5 @ Z2 )
% 2.98/1.81 = ( fun_upd @ A @ B @ F3 @ X5 @ Z2 ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_40_fun__upd__idem__iff,axiom,
% 2.98/1.81 ! [A: $tType,B: $tType,F3: A > B,X5: A,Y2: B] :
% 2.98/1.81 ( ( ( fun_upd @ A @ B @ F3 @ X5 @ Y2 )
% 2.98/1.81 = F3 )
% 2.98/1.81 = ( ( F3 @ X5 )
% 2.98/1.81 = Y2 ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_41_fun__upd__twist,axiom,
% 2.98/1.81 ! [A: $tType,B: $tType,A2: A,C: A,M: A > B,B2: B,D: B] :
% 2.98/1.81 ( ( A2 != C )
% 2.98/1.81 => ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M @ A2 @ B2 ) @ C @ D )
% 2.98/1.81 = ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M @ C @ D ) @ A2 @ B2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_42_fun__upd__other,axiom,
% 2.98/1.81 ! [B: $tType,A: $tType,Z2: A,X5: A,F3: A > B,Y2: B] :
% 2.98/1.81 ( ( Z2 != X5 )
% 2.98/1.81 => ( ( fun_upd @ A @ B @ F3 @ X5 @ Y2 @ Z2 )
% 2.98/1.81 = ( F3 @ Z2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_43_fun__upd__same,axiom,
% 2.98/1.81 ! [B: $tType,A: $tType,F3: B > A,X5: B,Y2: A] :
% 2.98/1.81 ( ( fun_upd @ B @ A @ F3 @ X5 @ Y2 @ X5 )
% 2.98/1.81 = Y2 ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_44_fun__upd__idem,axiom,
% 2.98/1.81 ! [A: $tType,B: $tType,F3: B > A,X5: B,Y2: A] :
% 2.98/1.81 ( ( ( F3 @ X5 )
% 2.98/1.81 = Y2 )
% 2.98/1.81 => ( ( fun_upd @ B @ A @ F3 @ X5 @ Y2 )
% 2.98/1.81 = F3 ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_45_mem__Collect__eq,axiom,
% 2.98/1.81 ! [A: $tType,A2: A,P2: A > $o] :
% 2.98/1.81 ( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
% 2.98/1.81 = ( P2 @ A2 ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_46_Collect__mem__eq,axiom,
% 2.98/1.81 ! [A: $tType,A3: set @ A] :
% 2.98/1.81 ( ( collect @ A
% 2.98/1.81 @ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
% 2.98/1.81 = A3 ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_47_Collect__cong,axiom,
% 2.98/1.81 ! [A: $tType,P2: A > $o,Q3: A > $o] :
% 2.98/1.81 ( ! [X: A] :
% 2.98/1.81 ( ( P2 @ X )
% 2.98/1.81 = ( Q3 @ X ) )
% 2.98/1.81 => ( ( collect @ A @ P2 )
% 2.98/1.81 = ( collect @ A @ Q3 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_48_ext,axiom,
% 2.98/1.81 ! [B: $tType,A: $tType,F3: A > B,G: A > B] :
% 2.98/1.81 ( ! [X: A] :
% 2.98/1.81 ( ( F3 @ X )
% 2.98/1.81 = ( G @ X ) )
% 2.98/1.81 => ( F3 = G ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_49_fun__upd__eqD,axiom,
% 2.98/1.81 ! [A: $tType,B: $tType,F3: A > B,X5: A,Y2: B,G: A > B,Z2: B] :
% 2.98/1.81 ( ( ( fun_upd @ A @ B @ F3 @ X5 @ Y2 )
% 2.98/1.81 = ( fun_upd @ A @ B @ G @ X5 @ Z2 ) )
% 2.98/1.81 => ( Y2 = Z2 ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_50_fun__upd__def,axiom,
% 2.98/1.81 ! [B: $tType,A: $tType] :
% 2.98/1.81 ( ( fun_upd @ A @ B )
% 2.98/1.81 = ( ^ [F: A > B,A4: A,B3: B,X4: A] : ( if @ B @ ( X4 = A4 ) @ B3 @ ( F @ X4 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_51_set__of__Tip,axiom,
% 2.98/1.81 ( ( binDag1380252983set_of @ binDag_Mirabelle_Tip )
% 2.98/1.81 = ( bot_bot @ ( set @ simpl_ref ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_52_DAG__less,axiom,
% 2.98/1.81 ! [Y2: binDag_Mirabelle_dag,X5: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( binDag_Mirabelle_DAG @ Y2 )
% 2.98/1.81 => ( ( ord_less @ binDag_Mirabelle_dag @ X5 @ Y2 )
% 2.98/1.81 => ( binDag_Mirabelle_DAG @ X5 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_53_Dag__update__l__new,axiom,
% 2.98/1.81 ! [T: binDag_Mirabelle_dag,Alloc: list @ simpl_ref,P: simpl_ref,L: simpl_ref > simpl_ref,X5: simpl_ref,R: simpl_ref > simpl_ref] :
% 2.98/1.81 ( ( ord_less_eq @ ( set @ simpl_ref ) @ ( binDag1380252983set_of @ T ) @ ( set2 @ simpl_ref @ Alloc ) )
% 2.98/1.81 => ( ( binDag_Mirabelle_Dag @ P @ ( fun_upd @ simpl_ref @ simpl_ref @ L @ ( simpl_new @ ( set2 @ simpl_ref @ Alloc ) ) @ X5 ) @ R @ T )
% 2.98/1.81 = ( binDag_Mirabelle_Dag @ P @ L @ R @ T ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_54_Dag__update__r__new,axiom,
% 2.98/1.81 ! [T: binDag_Mirabelle_dag,Alloc: list @ simpl_ref,P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,X5: simpl_ref] :
% 2.98/1.81 ( ( ord_less_eq @ ( set @ simpl_ref ) @ ( binDag1380252983set_of @ T ) @ ( set2 @ simpl_ref @ Alloc ) )
% 2.98/1.81 => ( ( binDag_Mirabelle_Dag @ P @ L @ ( fun_upd @ simpl_ref @ simpl_ref @ R @ ( simpl_new @ ( set2 @ simpl_ref @ Alloc ) ) @ X5 ) @ T )
% 2.98/1.81 = ( binDag_Mirabelle_Dag @ P @ L @ R @ T ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_55_swap__def,axiom,
% 2.98/1.81 ! [B: $tType,A: $tType] :
% 2.98/1.81 ( ( swap @ A @ B )
% 2.98/1.81 = ( ^ [A4: A,B3: A,F: A > B] : ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ F @ A4 @ ( F @ B3 ) ) @ B3 @ ( F @ A4 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_56_less__dag__Tip,axiom,
% 2.98/1.81 ! [X5: binDag_Mirabelle_dag] :
% 2.98/1.81 ~ ( ord_less @ binDag_Mirabelle_dag @ X5 @ binDag_Mirabelle_Tip ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_57_less__dag__def,axiom,
% 2.98/1.81 ( ( ord_less @ binDag_Mirabelle_dag )
% 2.98/1.81 = ( ^ [S2: binDag_Mirabelle_dag,T3: binDag_Mirabelle_dag] : ( binDag786255756subdag @ T3 @ S2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_58_less__dag__Node_H,axiom,
% 2.98/1.81 ! [X5: binDag_Mirabelle_dag,L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( ord_less @ binDag_Mirabelle_dag @ X5 @ ( binDag476092410e_Node @ L @ A2 @ R ) )
% 2.98/1.81 = ( ( X5 = L )
% 2.98/1.81 | ( X5 = R )
% 2.98/1.81 | ( ord_less @ binDag_Mirabelle_dag @ X5 @ L )
% 2.98/1.81 | ( ord_less @ binDag_Mirabelle_dag @ X5 @ R ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_59_swap__apply_I3_J,axiom,
% 2.98/1.81 ! [A: $tType,B: $tType,C: B,A2: B,B2: B,F3: B > A] :
% 2.98/1.81 ( ( C != A2 )
% 2.98/1.81 => ( ( C != B2 )
% 2.98/1.81 => ( ( swap @ B @ A @ A2 @ B2 @ F3 @ C )
% 2.98/1.81 = ( F3 @ C ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_60_swap__apply_I2_J,axiom,
% 2.98/1.81 ! [A: $tType,B: $tType,A2: B,B2: B,F3: B > A] :
% 2.98/1.81 ( ( swap @ B @ A @ A2 @ B2 @ F3 @ B2 )
% 2.98/1.81 = ( F3 @ A2 ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_61_swap__apply_I1_J,axiom,
% 2.98/1.81 ! [A: $tType,B: $tType,A2: B,B2: B,F3: B > A] :
% 2.98/1.81 ( ( swap @ B @ A @ A2 @ B2 @ F3 @ A2 )
% 2.98/1.81 = ( F3 @ B2 ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_62_swap__self,axiom,
% 2.98/1.81 ! [B: $tType,A: $tType,A2: A,F3: A > B] :
% 2.98/1.81 ( ( swap @ A @ B @ A2 @ A2 @ F3 )
% 2.98/1.81 = F3 ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_63_swap__nilpotent,axiom,
% 2.98/1.81 ! [B: $tType,A: $tType,A2: A,B2: A,F3: A > B] :
% 2.98/1.81 ( ( swap @ A @ B @ A2 @ B2 @ ( swap @ A @ B @ A2 @ B2 @ F3 ) )
% 2.98/1.81 = F3 ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_64_swap__triple,axiom,
% 2.98/1.81 ! [B: $tType,A: $tType,A2: A,C: A,B2: A,F3: A > B] :
% 2.98/1.81 ( ( A2 != C )
% 2.98/1.81 => ( ( B2 != C )
% 2.98/1.81 => ( ( swap @ A @ B @ A2 @ B2 @ ( swap @ A @ B @ B2 @ C @ ( swap @ A @ B @ A2 @ B2 @ F3 ) ) )
% 2.98/1.81 = ( swap @ A @ B @ A2 @ C @ F3 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_65_swap__commute,axiom,
% 2.98/1.81 ! [B: $tType,A: $tType] :
% 2.98/1.81 ( ( swap @ A @ B )
% 2.98/1.81 = ( ^ [A4: A,B3: A] : ( swap @ A @ B @ B3 @ A4 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_66_less__dag__set__of,axiom,
% 2.98/1.81 ! [X5: binDag_Mirabelle_dag,Y2: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( ord_less @ binDag_Mirabelle_dag @ X5 @ Y2 )
% 2.98/1.81 => ( ord_less_eq @ ( set @ simpl_ref ) @ ( binDag1380252983set_of @ X5 ) @ ( binDag1380252983set_of @ Y2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_67_less__Node__dag,axiom,
% 2.98/1.81 ! [L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag,X5: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( ord_less @ binDag_Mirabelle_dag @ ( binDag476092410e_Node @ L @ A2 @ R ) @ X5 )
% 2.98/1.81 => ( ( ord_less @ binDag_Mirabelle_dag @ L @ X5 )
% 2.98/1.81 & ( ord_less @ binDag_Mirabelle_dag @ R @ X5 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_68_empty__subsetI,axiom,
% 2.98/1.81 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_69_subset__empty,axiom,
% 2.98/1.81 ! [A: $tType,A3: set @ A] :
% 2.98/1.81 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 2.98/1.81 = ( A3
% 2.98/1.81 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_70_subset__antisym,axiom,
% 2.98/1.81 ! [A: $tType,A3: set @ A,B4: set @ A] :
% 2.98/1.81 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 2.98/1.81 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ A3 )
% 2.98/1.81 => ( A3 = B4 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_71_subsetI,axiom,
% 2.98/1.81 ! [A: $tType,A3: set @ A,B4: set @ A] :
% 2.98/1.81 ( ! [X: A] :
% 2.98/1.81 ( ( member @ A @ X @ A3 )
% 2.98/1.81 => ( member @ A @ X @ B4 ) )
% 2.98/1.81 => ( ord_less_eq @ ( set @ A ) @ A3 @ B4 ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_72_empty__Collect__eq,axiom,
% 2.98/1.81 ! [A: $tType,P2: A > $o] :
% 2.98/1.81 ( ( ( bot_bot @ ( set @ A ) )
% 2.98/1.81 = ( collect @ A @ P2 ) )
% 2.98/1.81 = ( ! [X4: A] :
% 2.98/1.81 ~ ( P2 @ X4 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_73_Collect__empty__eq,axiom,
% 2.98/1.81 ! [A: $tType,P2: A > $o] :
% 2.98/1.81 ( ( ( collect @ A @ P2 )
% 2.98/1.81 = ( bot_bot @ ( set @ A ) ) )
% 2.98/1.81 = ( ! [X4: A] :
% 2.98/1.81 ~ ( P2 @ X4 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_74_all__not__in__conv,axiom,
% 2.98/1.81 ! [A: $tType,A3: set @ A] :
% 2.98/1.81 ( ( ! [X4: A] :
% 2.98/1.81 ~ ( member @ A @ X4 @ A3 ) )
% 2.98/1.81 = ( A3
% 2.98/1.81 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_75_empty__iff,axiom,
% 2.98/1.81 ! [A: $tType,C: A] :
% 2.98/1.81 ~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_76_emptyE,axiom,
% 2.98/1.81 ! [A: $tType,A2: A] :
% 2.98/1.81 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_77_equals0D,axiom,
% 2.98/1.81 ! [A: $tType,A3: set @ A,A2: A] :
% 2.98/1.81 ( ( A3
% 2.98/1.81 = ( bot_bot @ ( set @ A ) ) )
% 2.98/1.81 => ~ ( member @ A @ A2 @ A3 ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_78_equals0I,axiom,
% 2.98/1.81 ! [A: $tType,A3: set @ A] :
% 2.98/1.81 ( ! [Y4: A] :
% 2.98/1.81 ~ ( member @ A @ Y4 @ A3 )
% 2.98/1.81 => ( A3
% 2.98/1.81 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_79_ex__in__conv,axiom,
% 2.98/1.81 ! [A: $tType,A3: set @ A] :
% 2.98/1.81 ( ( ? [X4: A] : ( member @ A @ X4 @ A3 ) )
% 2.98/1.81 = ( A3
% 2.98/1.81 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_80_set__mp,axiom,
% 2.98/1.81 ! [A: $tType,A3: set @ A,B4: set @ A,X5: A] :
% 2.98/1.81 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 2.98/1.81 => ( ( member @ A @ X5 @ A3 )
% 2.98/1.81 => ( member @ A @ X5 @ B4 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_81_in__mono,axiom,
% 2.98/1.81 ! [A: $tType,A3: set @ A,B4: set @ A,X5: A] :
% 2.98/1.81 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 2.98/1.81 => ( ( member @ A @ X5 @ A3 )
% 2.98/1.81 => ( member @ A @ X5 @ B4 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_82_subsetD,axiom,
% 2.98/1.81 ! [A: $tType,A3: set @ A,B4: set @ A,C: A] :
% 2.98/1.81 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 2.98/1.81 => ( ( member @ A @ C @ A3 )
% 2.98/1.81 => ( member @ A @ C @ B4 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_83_subsetCE,axiom,
% 2.98/1.81 ! [A: $tType,A3: set @ A,B4: set @ A,C: A] :
% 2.98/1.81 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 2.98/1.81 => ( ( member @ A @ C @ A3 )
% 2.98/1.81 => ( member @ A @ C @ B4 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_84_equalityE,axiom,
% 2.98/1.81 ! [A: $tType,A3: set @ A,B4: set @ A] :
% 2.98/1.81 ( ( A3 = B4 )
% 2.98/1.81 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 2.98/1.81 => ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A3 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_85_subset__eq,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( ord_less_eq @ ( set @ A ) )
% 2.98/1.81 = ( ^ [A5: set @ A,B5: set @ A] :
% 2.98/1.81 ! [X4: A] :
% 2.98/1.81 ( ( member @ A @ X4 @ A5 )
% 2.98/1.81 => ( member @ A @ X4 @ B5 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_86_equalityD1,axiom,
% 2.98/1.81 ! [A: $tType,A3: set @ A,B4: set @ A] :
% 2.98/1.81 ( ( A3 = B4 )
% 2.98/1.81 => ( ord_less_eq @ ( set @ A ) @ A3 @ B4 ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_87_equalityD2,axiom,
% 2.98/1.81 ! [A: $tType,A3: set @ A,B4: set @ A] :
% 2.98/1.81 ( ( A3 = B4 )
% 2.98/1.81 => ( ord_less_eq @ ( set @ A ) @ B4 @ A3 ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_88_set__rev__mp,axiom,
% 2.98/1.81 ! [A: $tType,X5: A,A3: set @ A,B4: set @ A] :
% 2.98/1.81 ( ( member @ A @ X5 @ A3 )
% 2.98/1.81 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 2.98/1.81 => ( member @ A @ X5 @ B4 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_89_subset__iff,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( ord_less_eq @ ( set @ A ) )
% 2.98/1.81 = ( ^ [A5: set @ A,B5: set @ A] :
% 2.98/1.81 ! [T3: A] :
% 2.98/1.81 ( ( member @ A @ T3 @ A5 )
% 2.98/1.81 => ( member @ A @ T3 @ B5 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_90_rev__subsetD,axiom,
% 2.98/1.81 ! [A: $tType,C: A,A3: set @ A,B4: set @ A] :
% 2.98/1.81 ( ( member @ A @ C @ A3 )
% 2.98/1.81 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 2.98/1.81 => ( member @ A @ C @ B4 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_91_subset__refl,axiom,
% 2.98/1.81 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_92_Collect__mono,axiom,
% 2.98/1.81 ! [A: $tType,P2: A > $o,Q3: A > $o] :
% 2.98/1.81 ( ! [X: A] :
% 2.98/1.81 ( ( P2 @ X )
% 2.98/1.81 => ( Q3 @ X ) )
% 2.98/1.81 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q3 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_93_subset__trans,axiom,
% 2.98/1.81 ! [A: $tType,A3: set @ A,B4: set @ A,C2: set @ A] :
% 2.98/1.81 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 2.98/1.81 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ C2 )
% 2.98/1.81 => ( ord_less_eq @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_94_set__eq__subset,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( ^ [Y5: set @ A,Z3: set @ A] : ( Y5 = Z3 ) )
% 2.98/1.81 = ( ^ [A5: set @ A,B5: set @ A] :
% 2.98/1.81 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.98/1.81 & ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_95_contra__subsetD,axiom,
% 2.98/1.81 ! [A: $tType,A3: set @ A,B4: set @ A,C: A] :
% 2.98/1.81 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 2.98/1.81 => ( ~ ( member @ A @ C @ B4 )
% 2.98/1.81 => ~ ( member @ A @ C @ A3 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_96_Collect__mono__iff,axiom,
% 2.98/1.81 ! [A: $tType,P2: A > $o,Q3: A > $o] :
% 2.98/1.81 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q3 ) )
% 2.98/1.81 = ( ! [X4: A] :
% 2.98/1.81 ( ( P2 @ X4 )
% 2.98/1.81 => ( Q3 @ X4 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_97_bot__apply,axiom,
% 2.98/1.81 ! [C3: $tType,D2: $tType] :
% 2.98/1.81 ( ( bot @ C3 @ ( type2 @ C3 ) )
% 2.98/1.81 => ( ( bot_bot @ ( D2 > C3 ) )
% 2.98/1.81 = ( ^ [X4: D2] : ( bot_bot @ C3 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_98_order__refl,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( preorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A] : ( ord_less_eq @ A @ X5 @ X5 ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_99_subset__code_I1_J,axiom,
% 2.98/1.81 ! [A: $tType,Xs: list @ A,B4: set @ A] :
% 2.98/1.81 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B4 )
% 2.98/1.81 = ( ! [X4: A] :
% 2.98/1.81 ( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
% 2.98/1.81 => ( member @ A @ X4 @ B4 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_100_subset__emptyI,axiom,
% 2.98/1.81 ! [A: $tType,A3: set @ A] :
% 2.98/1.81 ( ! [X: A] :
% 2.98/1.81 ~ ( member @ A @ X @ A3 )
% 2.98/1.81 => ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_101_bot_Oextremum__strict,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order_bot @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [A2: A] :
% 2.98/1.81 ~ ( ord_less @ A @ A2 @ ( bot_bot @ A ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_102_bot_Onot__eq__extremum,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order_bot @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [A2: A] :
% 2.98/1.81 ( ( A2
% 2.98/1.81 != ( bot_bot @ A ) )
% 2.98/1.81 = ( ord_less @ A @ ( bot_bot @ A ) @ A2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_103_psubsetI,axiom,
% 2.98/1.81 ! [A: $tType,A3: set @ A,B4: set @ A] :
% 2.98/1.81 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 2.98/1.81 => ( ( A3 != B4 )
% 2.98/1.81 => ( ord_less @ ( set @ A ) @ A3 @ B4 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_104_bot__set__def,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( bot_bot @ ( set @ A ) )
% 2.98/1.81 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_105_not__psubset__empty,axiom,
% 2.98/1.81 ! [A: $tType,A3: set @ A] :
% 2.98/1.81 ~ ( ord_less @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_106_psubsetE,axiom,
% 2.98/1.81 ! [A: $tType,A3: set @ A,B4: set @ A] :
% 2.98/1.81 ( ( ord_less @ ( set @ A ) @ A3 @ B4 )
% 2.98/1.81 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 2.98/1.81 => ( ord_less_eq @ ( set @ A ) @ B4 @ A3 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_107_psubset__eq,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( ord_less @ ( set @ A ) )
% 2.98/1.81 = ( ^ [A5: set @ A,B5: set @ A] :
% 2.98/1.81 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.98/1.81 & ( A5 != B5 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_108_psubset__imp__subset,axiom,
% 2.98/1.81 ! [A: $tType,A3: set @ A,B4: set @ A] :
% 2.98/1.81 ( ( ord_less @ ( set @ A ) @ A3 @ B4 )
% 2.98/1.81 => ( ord_less_eq @ ( set @ A ) @ A3 @ B4 ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_109_psubset__subset__trans,axiom,
% 2.98/1.81 ! [A: $tType,A3: set @ A,B4: set @ A,C2: set @ A] :
% 2.98/1.81 ( ( ord_less @ ( set @ A ) @ A3 @ B4 )
% 2.98/1.81 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ C2 )
% 2.98/1.81 => ( ord_less @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_110_subset__not__subset__eq,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( ord_less @ ( set @ A ) )
% 2.98/1.81 = ( ^ [A5: set @ A,B5: set @ A] :
% 2.98/1.81 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.98/1.81 & ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_111_subset__psubset__trans,axiom,
% 2.98/1.81 ! [A: $tType,A3: set @ A,B4: set @ A,C2: set @ A] :
% 2.98/1.81 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
% 2.98/1.81 => ( ( ord_less @ ( set @ A ) @ B4 @ C2 )
% 2.98/1.81 => ( ord_less @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_112_subset__iff__psubset__eq,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( ord_less_eq @ ( set @ A ) )
% 2.98/1.81 = ( ^ [A5: set @ A,B5: set @ A] :
% 2.98/1.81 ( ( ord_less @ ( set @ A ) @ A5 @ B5 )
% 2.98/1.81 | ( A5 = B5 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_113_dag__less__le,axiom,
% 2.98/1.81 ( ( ord_less @ binDag_Mirabelle_dag )
% 2.98/1.81 = ( ^ [X4: binDag_Mirabelle_dag,Y3: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( ord_less_eq @ binDag_Mirabelle_dag @ X4 @ Y3 )
% 2.98/1.81 & ( X4 != Y3 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_114_le__dag__def,axiom,
% 2.98/1.81 ( ( ord_less_eq @ binDag_Mirabelle_dag )
% 2.98/1.81 = ( ^ [S2: binDag_Mirabelle_dag,T3: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( S2 = T3 )
% 2.98/1.81 | ( ord_less @ binDag_Mirabelle_dag @ S2 @ T3 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_115_less__dag__Node,axiom,
% 2.98/1.81 ! [X5: binDag_Mirabelle_dag,L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( ord_less @ binDag_Mirabelle_dag @ X5 @ ( binDag476092410e_Node @ L @ A2 @ R ) )
% 2.98/1.81 = ( ( ord_less_eq @ binDag_Mirabelle_dag @ X5 @ L )
% 2.98/1.81 | ( ord_less_eq @ binDag_Mirabelle_dag @ X5 @ R ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_116_le__dag__set__of,axiom,
% 2.98/1.81 ! [X5: binDag_Mirabelle_dag,Y2: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( ord_less_eq @ binDag_Mirabelle_dag @ X5 @ Y2 )
% 2.98/1.81 => ( ord_less_eq @ ( set @ simpl_ref ) @ ( binDag1380252983set_of @ X5 ) @ ( binDag1380252983set_of @ Y2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_117_dual__order_Oantisym,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [B2: A,A2: A] :
% 2.98/1.81 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.98/1.81 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.98/1.81 => ( A2 = B2 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_118_dual__order_Otrans,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [B2: A,A2: A,C: A] :
% 2.98/1.81 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.98/1.81 => ( ( ord_less_eq @ A @ C @ B2 )
% 2.98/1.81 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_119_linorder__wlog,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [P2: A > A > $o,A2: A,B2: A] :
% 2.98/1.81 ( ! [A6: A,B6: A] :
% 2.98/1.81 ( ( ord_less_eq @ A @ A6 @ B6 )
% 2.98/1.81 => ( P2 @ A6 @ B6 ) )
% 2.98/1.81 => ( ! [A6: A,B6: A] :
% 2.98/1.81 ( ( P2 @ B6 @ A6 )
% 2.98/1.81 => ( P2 @ A6 @ B6 ) )
% 2.98/1.81 => ( P2 @ A2 @ B2 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_120_dual__order_Orefl,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_121_order__trans,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( preorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A,Y2: A,Z2: A] :
% 2.98/1.81 ( ( ord_less_eq @ A @ X5 @ Y2 )
% 2.98/1.81 => ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 2.98/1.81 => ( ord_less_eq @ A @ X5 @ Z2 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_122_order__class_Oorder_Oantisym,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [A2: A,B2: A] :
% 2.98/1.81 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.98/1.81 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.98/1.81 => ( A2 = B2 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_123_ord__le__eq__trans,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( ord @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [A2: A,B2: A,C: A] :
% 2.98/1.81 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.98/1.81 => ( ( B2 = C )
% 2.98/1.81 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_124_ord__eq__le__trans,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( ord @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [A2: A,B2: A,C: A] :
% 2.98/1.81 ( ( A2 = B2 )
% 2.98/1.81 => ( ( ord_less_eq @ A @ B2 @ C )
% 2.98/1.81 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_125_antisym__conv,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [Y2: A,X5: A] :
% 2.98/1.81 ( ( ord_less_eq @ A @ Y2 @ X5 )
% 2.98/1.81 => ( ( ord_less_eq @ A @ X5 @ Y2 )
% 2.98/1.81 = ( X5 = Y2 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_126_le__cases3,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A,Y2: A,Z2: A] :
% 2.98/1.81 ( ( ( ord_less_eq @ A @ X5 @ Y2 )
% 2.98/1.81 => ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
% 2.98/1.81 => ( ( ( ord_less_eq @ A @ Y2 @ X5 )
% 2.98/1.81 => ~ ( ord_less_eq @ A @ X5 @ Z2 ) )
% 2.98/1.81 => ( ( ( ord_less_eq @ A @ X5 @ Z2 )
% 2.98/1.81 => ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
% 2.98/1.81 => ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
% 2.98/1.81 => ~ ( ord_less_eq @ A @ Y2 @ X5 ) )
% 2.98/1.81 => ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 2.98/1.81 => ~ ( ord_less_eq @ A @ Z2 @ X5 ) )
% 2.98/1.81 => ~ ( ( ord_less_eq @ A @ Z2 @ X5 )
% 2.98/1.81 => ~ ( ord_less_eq @ A @ X5 @ Y2 ) ) ) ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_127_order_Otrans,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [A2: A,B2: A,C: A] :
% 2.98/1.81 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.98/1.81 => ( ( ord_less_eq @ A @ B2 @ C )
% 2.98/1.81 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_128_le__cases,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A,Y2: A] :
% 2.98/1.81 ( ~ ( ord_less_eq @ A @ X5 @ Y2 )
% 2.98/1.81 => ( ord_less_eq @ A @ Y2 @ X5 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_129_eq__refl,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( preorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A,Y2: A] :
% 2.98/1.81 ( ( X5 = Y2 )
% 2.98/1.81 => ( ord_less_eq @ A @ X5 @ Y2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_130_linear,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A,Y2: A] :
% 2.98/1.81 ( ( ord_less_eq @ A @ X5 @ Y2 )
% 2.98/1.81 | ( ord_less_eq @ A @ Y2 @ X5 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_131_antisym,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A,Y2: A] :
% 2.98/1.81 ( ( ord_less_eq @ A @ X5 @ Y2 )
% 2.98/1.81 => ( ( ord_less_eq @ A @ Y2 @ X5 )
% 2.98/1.81 => ( X5 = Y2 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_132_eq__iff,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ( ( ^ [Y5: A,Z3: A] : ( Y5 = Z3 ) )
% 2.98/1.81 = ( ^ [X4: A,Y3: A] :
% 2.98/1.81 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 2.98/1.81 & ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_133_ord__le__eq__subst,axiom,
% 2.98/1.81 ! [A: $tType,B: $tType] :
% 2.98/1.81 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.98/1.81 & ( ord @ A @ ( type2 @ A ) ) )
% 2.98/1.81 => ! [A2: A,B2: A,F3: A > B,C: B] :
% 2.98/1.81 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.98/1.81 => ( ( ( F3 @ B2 )
% 2.98/1.81 = C )
% 2.98/1.81 => ( ! [X: A,Y4: A] :
% 2.98/1.81 ( ( ord_less_eq @ A @ X @ Y4 )
% 2.98/1.81 => ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
% 2.98/1.81 => ( ord_less_eq @ B @ ( F3 @ A2 ) @ C ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_134_ord__eq__le__subst,axiom,
% 2.98/1.81 ! [A: $tType,B: $tType] :
% 2.98/1.81 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.98/1.81 & ( ord @ A @ ( type2 @ A ) ) )
% 2.98/1.81 => ! [A2: A,F3: B > A,B2: B,C: B] :
% 2.98/1.81 ( ( A2
% 2.98/1.81 = ( F3 @ B2 ) )
% 2.98/1.81 => ( ( ord_less_eq @ B @ B2 @ C )
% 2.98/1.81 => ( ! [X: B,Y4: B] :
% 2.98/1.81 ( ( ord_less_eq @ B @ X @ Y4 )
% 2.98/1.81 => ( ord_less_eq @ A @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
% 2.98/1.81 => ( ord_less_eq @ A @ A2 @ ( F3 @ C ) ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_135_order__subst2,axiom,
% 2.98/1.81 ! [A: $tType,C3: $tType] :
% 2.98/1.81 ( ( ( order @ C3 @ ( type2 @ C3 ) )
% 2.98/1.81 & ( order @ A @ ( type2 @ A ) ) )
% 2.98/1.81 => ! [A2: A,B2: A,F3: A > C3,C: C3] :
% 2.98/1.81 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.98/1.81 => ( ( ord_less_eq @ C3 @ ( F3 @ B2 ) @ C )
% 2.98/1.81 => ( ! [X: A,Y4: A] :
% 2.98/1.81 ( ( ord_less_eq @ A @ X @ Y4 )
% 2.98/1.81 => ( ord_less_eq @ C3 @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
% 2.98/1.81 => ( ord_less_eq @ C3 @ ( F3 @ A2 ) @ C ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_136_order__subst1,axiom,
% 2.98/1.81 ! [A: $tType,B: $tType] :
% 2.98/1.81 ( ( ( order @ B @ ( type2 @ B ) )
% 2.98/1.81 & ( order @ A @ ( type2 @ A ) ) )
% 2.98/1.81 => ! [A2: A,F3: B > A,B2: B,C: B] :
% 2.98/1.81 ( ( ord_less_eq @ A @ A2 @ ( F3 @ B2 ) )
% 2.98/1.81 => ( ( ord_less_eq @ B @ B2 @ C )
% 2.98/1.81 => ( ! [X: B,Y4: B] :
% 2.98/1.81 ( ( ord_less_eq @ B @ X @ Y4 )
% 2.98/1.81 => ( ord_less_eq @ A @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
% 2.98/1.81 => ( ord_less_eq @ A @ A2 @ ( F3 @ C ) ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_137_le__fun__def,axiom,
% 2.98/1.81 ! [B: $tType,A: $tType] :
% 2.98/1.81 ( ( ord @ B @ ( type2 @ B ) )
% 2.98/1.81 => ( ( ord_less_eq @ ( A > B ) )
% 2.98/1.81 = ( ^ [F: A > B,G2: A > B] :
% 2.98/1.81 ! [X4: A] : ( ord_less_eq @ B @ ( F @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_138_le__funI,axiom,
% 2.98/1.81 ! [B: $tType,A: $tType] :
% 2.98/1.81 ( ( ord @ B @ ( type2 @ B ) )
% 2.98/1.81 => ! [F3: A > B,G: A > B] :
% 2.98/1.81 ( ! [X: A] : ( ord_less_eq @ B @ ( F3 @ X ) @ ( G @ X ) )
% 2.98/1.81 => ( ord_less_eq @ ( A > B ) @ F3 @ G ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_139_le__funE,axiom,
% 2.98/1.81 ! [B: $tType,A: $tType] :
% 2.98/1.81 ( ( ord @ B @ ( type2 @ B ) )
% 2.98/1.81 => ! [F3: A > B,G: A > B,X5: A] :
% 2.98/1.81 ( ( ord_less_eq @ ( A > B ) @ F3 @ G )
% 2.98/1.81 => ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( G @ X5 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_140_le__funD,axiom,
% 2.98/1.81 ! [B: $tType,A: $tType] :
% 2.98/1.81 ( ( ord @ B @ ( type2 @ B ) )
% 2.98/1.81 => ! [F3: A > B,G: A > B,X5: A] :
% 2.98/1.81 ( ( ord_less_eq @ ( A > B ) @ F3 @ G )
% 2.98/1.81 => ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( G @ X5 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_141_dual__order_Ostrict__implies__not__eq,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [B2: A,A2: A] :
% 2.98/1.81 ( ( ord_less @ A @ B2 @ A2 )
% 2.98/1.81 => ( A2 != B2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_142_order_Ostrict__implies__not__eq,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [A2: A,B2: A] :
% 2.98/1.81 ( ( ord_less @ A @ A2 @ B2 )
% 2.98/1.81 => ( A2 != B2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_143_not__less__iff__gr__or__eq,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A,Y2: A] :
% 2.98/1.81 ( ( ~ ( ord_less @ A @ X5 @ Y2 ) )
% 2.98/1.81 = ( ( ord_less @ A @ Y2 @ X5 )
% 2.98/1.81 | ( X5 = Y2 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_144_dual__order_Ostrict__trans,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [B2: A,A2: A,C: A] :
% 2.98/1.81 ( ( ord_less @ A @ B2 @ A2 )
% 2.98/1.81 => ( ( ord_less @ A @ C @ B2 )
% 2.98/1.81 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_145_less__imp__not__less,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( preorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A,Y2: A] :
% 2.98/1.81 ( ( ord_less @ A @ X5 @ Y2 )
% 2.98/1.81 => ~ ( ord_less @ A @ Y2 @ X5 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_146_order_Ostrict__trans,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [A2: A,B2: A,C: A] :
% 2.98/1.81 ( ( ord_less @ A @ A2 @ B2 )
% 2.98/1.81 => ( ( ord_less @ A @ B2 @ C )
% 2.98/1.81 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_147_dual__order_Oirrefl,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [A2: A] :
% 2.98/1.81 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_148_linorder__cases,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A,Y2: A] :
% 2.98/1.81 ( ~ ( ord_less @ A @ X5 @ Y2 )
% 2.98/1.81 => ( ( X5 != Y2 )
% 2.98/1.81 => ( ord_less @ A @ Y2 @ X5 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_149_less__imp__triv,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( preorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A,Y2: A,P2: $o] :
% 2.98/1.81 ( ( ord_less @ A @ X5 @ Y2 )
% 2.98/1.81 => ( ( ord_less @ A @ Y2 @ X5 )
% 2.98/1.81 => P2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_150_less__imp__not__eq2,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A,Y2: A] :
% 2.98/1.81 ( ( ord_less @ A @ X5 @ Y2 )
% 2.98/1.81 => ( Y2 != X5 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_151_antisym__conv3,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [Y2: A,X5: A] :
% 2.98/1.81 ( ~ ( ord_less @ A @ Y2 @ X5 )
% 2.98/1.81 => ( ( ~ ( ord_less @ A @ X5 @ Y2 ) )
% 2.98/1.81 = ( X5 = Y2 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_152_less__induct,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( wellorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [P2: A > $o,A2: A] :
% 2.98/1.81 ( ! [X: A] :
% 2.98/1.81 ( ! [Y: A] :
% 2.98/1.81 ( ( ord_less @ A @ Y @ X )
% 2.98/1.81 => ( P2 @ Y ) )
% 2.98/1.81 => ( P2 @ X ) )
% 2.98/1.81 => ( P2 @ A2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_153_less__not__sym,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( preorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A,Y2: A] :
% 2.98/1.81 ( ( ord_less @ A @ X5 @ Y2 )
% 2.98/1.81 => ~ ( ord_less @ A @ Y2 @ X5 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_154_less__imp__not__eq,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A,Y2: A] :
% 2.98/1.81 ( ( ord_less @ A @ X5 @ Y2 )
% 2.98/1.81 => ( X5 != Y2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_155_dual__order_Oasym,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [B2: A,A2: A] :
% 2.98/1.81 ( ( ord_less @ A @ B2 @ A2 )
% 2.98/1.81 => ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_156_ord__less__eq__trans,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( ord @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [A2: A,B2: A,C: A] :
% 2.98/1.81 ( ( ord_less @ A @ A2 @ B2 )
% 2.98/1.81 => ( ( B2 = C )
% 2.98/1.81 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_157_ord__eq__less__trans,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( ord @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [A2: A,B2: A,C: A] :
% 2.98/1.81 ( ( A2 = B2 )
% 2.98/1.81 => ( ( ord_less @ A @ B2 @ C )
% 2.98/1.81 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_158_less__irrefl,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( preorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A] :
% 2.98/1.81 ~ ( ord_less @ A @ X5 @ X5 ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_159_less__linear,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A,Y2: A] :
% 2.98/1.81 ( ( ord_less @ A @ X5 @ Y2 )
% 2.98/1.81 | ( X5 = Y2 )
% 2.98/1.81 | ( ord_less @ A @ Y2 @ X5 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_160_less__trans,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( preorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A,Y2: A,Z2: A] :
% 2.98/1.81 ( ( ord_less @ A @ X5 @ Y2 )
% 2.98/1.81 => ( ( ord_less @ A @ Y2 @ Z2 )
% 2.98/1.81 => ( ord_less @ A @ X5 @ Z2 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_161_less__asym_H,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( preorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [A2: A,B2: A] :
% 2.98/1.81 ( ( ord_less @ A @ A2 @ B2 )
% 2.98/1.81 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_162_less__asym,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( preorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A,Y2: A] :
% 2.98/1.81 ( ( ord_less @ A @ X5 @ Y2 )
% 2.98/1.81 => ~ ( ord_less @ A @ Y2 @ X5 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_163_less__imp__neq,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A,Y2: A] :
% 2.98/1.81 ( ( ord_less @ A @ X5 @ Y2 )
% 2.98/1.81 => ( X5 != Y2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_164_dense,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( dense_order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A,Y2: A] :
% 2.98/1.81 ( ( ord_less @ A @ X5 @ Y2 )
% 2.98/1.81 => ? [Z4: A] :
% 2.98/1.81 ( ( ord_less @ A @ X5 @ Z4 )
% 2.98/1.81 & ( ord_less @ A @ Z4 @ Y2 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_165_order_Oasym,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [A2: A,B2: A] :
% 2.98/1.81 ( ( ord_less @ A @ A2 @ B2 )
% 2.98/1.81 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_166_neq__iff,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A,Y2: A] :
% 2.98/1.81 ( ( X5 != Y2 )
% 2.98/1.81 = ( ( ord_less @ A @ X5 @ Y2 )
% 2.98/1.81 | ( ord_less @ A @ Y2 @ X5 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_167_neqE,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A,Y2: A] :
% 2.98/1.81 ( ( X5 != Y2 )
% 2.98/1.81 => ( ~ ( ord_less @ A @ X5 @ Y2 )
% 2.98/1.81 => ( ord_less @ A @ Y2 @ X5 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_168_gt__ex,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( no_top @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A] :
% 2.98/1.81 ? [X1: A] : ( ord_less @ A @ X5 @ X1 ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_169_lt__ex,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( no_bot @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A] :
% 2.98/1.81 ? [Y4: A] : ( ord_less @ A @ Y4 @ X5 ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_170_order__less__subst2,axiom,
% 2.98/1.81 ! [A: $tType,C3: $tType] :
% 2.98/1.81 ( ( ( order @ C3 @ ( type2 @ C3 ) )
% 2.98/1.81 & ( order @ A @ ( type2 @ A ) ) )
% 2.98/1.81 => ! [A2: A,B2: A,F3: A > C3,C: C3] :
% 2.98/1.81 ( ( ord_less @ A @ A2 @ B2 )
% 2.98/1.81 => ( ( ord_less @ C3 @ ( F3 @ B2 ) @ C )
% 2.98/1.81 => ( ! [X: A,Y4: A] :
% 2.98/1.81 ( ( ord_less @ A @ X @ Y4 )
% 2.98/1.81 => ( ord_less @ C3 @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
% 2.98/1.81 => ( ord_less @ C3 @ ( F3 @ A2 ) @ C ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_171_order__less__subst1,axiom,
% 2.98/1.81 ! [A: $tType,B: $tType] :
% 2.98/1.81 ( ( ( order @ B @ ( type2 @ B ) )
% 2.98/1.81 & ( order @ A @ ( type2 @ A ) ) )
% 2.98/1.81 => ! [A2: A,F3: B > A,B2: B,C: B] :
% 2.98/1.81 ( ( ord_less @ A @ A2 @ ( F3 @ B2 ) )
% 2.98/1.81 => ( ( ord_less @ B @ B2 @ C )
% 2.98/1.81 => ( ! [X: B,Y4: B] :
% 2.98/1.81 ( ( ord_less @ B @ X @ Y4 )
% 2.98/1.81 => ( ord_less @ A @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
% 2.98/1.81 => ( ord_less @ A @ A2 @ ( F3 @ C ) ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_172_ord__less__eq__subst,axiom,
% 2.98/1.81 ! [A: $tType,B: $tType] :
% 2.98/1.81 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.98/1.81 & ( ord @ A @ ( type2 @ A ) ) )
% 2.98/1.81 => ! [A2: A,B2: A,F3: A > B,C: B] :
% 2.98/1.81 ( ( ord_less @ A @ A2 @ B2 )
% 2.98/1.81 => ( ( ( F3 @ B2 )
% 2.98/1.81 = C )
% 2.98/1.81 => ( ! [X: A,Y4: A] :
% 2.98/1.81 ( ( ord_less @ A @ X @ Y4 )
% 2.98/1.81 => ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
% 2.98/1.81 => ( ord_less @ B @ ( F3 @ A2 ) @ C ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_173_ord__eq__less__subst,axiom,
% 2.98/1.81 ! [A: $tType,B: $tType] :
% 2.98/1.81 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.98/1.81 & ( ord @ A @ ( type2 @ A ) ) )
% 2.98/1.81 => ! [A2: A,F3: B > A,B2: B,C: B] :
% 2.98/1.81 ( ( A2
% 2.98/1.81 = ( F3 @ B2 ) )
% 2.98/1.81 => ( ( ord_less @ B @ B2 @ C )
% 2.98/1.81 => ( ! [X: B,Y4: B] :
% 2.98/1.81 ( ( ord_less @ B @ X @ Y4 )
% 2.98/1.81 => ( ord_less @ A @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
% 2.98/1.81 => ( ord_less @ A @ A2 @ ( F3 @ C ) ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_174_bot__fun__def,axiom,
% 2.98/1.81 ! [B: $tType,A: $tType] :
% 2.98/1.81 ( ( bot @ B @ ( type2 @ B ) )
% 2.98/1.81 => ( ( bot_bot @ ( A > B ) )
% 2.98/1.81 = ( ^ [X4: A] : ( bot_bot @ B ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_175_less__DAG__set__of,axiom,
% 2.98/1.81 ! [X5: binDag_Mirabelle_dag,Y2: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( ord_less @ binDag_Mirabelle_dag @ X5 @ Y2 )
% 2.98/1.81 => ( ( binDag_Mirabelle_DAG @ Y2 )
% 2.98/1.81 => ( ord_less @ ( set @ simpl_ref ) @ ( binDag1380252983set_of @ X5 ) @ ( binDag1380252983set_of @ Y2 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_176_order_Onot__eq__order__implies__strict,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [A2: A,B2: A] :
% 2.98/1.81 ( ( A2 != B2 )
% 2.98/1.81 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.98/1.81 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_177_dual__order_Ostrict__implies__order,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [B2: A,A2: A] :
% 2.98/1.81 ( ( ord_less @ A @ B2 @ A2 )
% 2.98/1.81 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_178_dual__order_Ostrict__iff__order,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ( ( ord_less @ A )
% 2.98/1.81 = ( ^ [B3: A,A4: A] :
% 2.98/1.81 ( ( ord_less_eq @ A @ B3 @ A4 )
% 2.98/1.81 & ( A4 != B3 ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_179_dual__order_Oorder__iff__strict,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ( ( ord_less_eq @ A )
% 2.98/1.81 = ( ^ [B3: A,A4: A] :
% 2.98/1.81 ( ( ord_less @ A @ B3 @ A4 )
% 2.98/1.81 | ( A4 = B3 ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_180_order_Ostrict__implies__order,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [A2: A,B2: A] :
% 2.98/1.81 ( ( ord_less @ A @ A2 @ B2 )
% 2.98/1.81 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_181_dense__le__bounded,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A,Y2: A,Z2: A] :
% 2.98/1.81 ( ( ord_less @ A @ X5 @ Y2 )
% 2.98/1.81 => ( ! [W: A] :
% 2.98/1.81 ( ( ord_less @ A @ X5 @ W )
% 2.98/1.81 => ( ( ord_less @ A @ W @ Y2 )
% 2.98/1.81 => ( ord_less_eq @ A @ W @ Z2 ) ) )
% 2.98/1.81 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_182_dense__ge__bounded,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [Z2: A,X5: A,Y2: A] :
% 2.98/1.81 ( ( ord_less @ A @ Z2 @ X5 )
% 2.98/1.81 => ( ! [W: A] :
% 2.98/1.81 ( ( ord_less @ A @ Z2 @ W )
% 2.98/1.81 => ( ( ord_less @ A @ W @ X5 )
% 2.98/1.81 => ( ord_less_eq @ A @ Y2 @ W ) ) )
% 2.98/1.81 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_183_dual__order_Ostrict__trans2,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [B2: A,A2: A,C: A] :
% 2.98/1.81 ( ( ord_less @ A @ B2 @ A2 )
% 2.98/1.81 => ( ( ord_less_eq @ A @ C @ B2 )
% 2.98/1.81 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_184_dual__order_Ostrict__trans1,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [B2: A,A2: A,C: A] :
% 2.98/1.81 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.98/1.81 => ( ( ord_less @ A @ C @ B2 )
% 2.98/1.81 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_185_order_Ostrict__iff__order,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ( ( ord_less @ A )
% 2.98/1.81 = ( ^ [A4: A,B3: A] :
% 2.98/1.81 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.98/1.81 & ( A4 != B3 ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_186_order_Oorder__iff__strict,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ( ( ord_less_eq @ A )
% 2.98/1.81 = ( ^ [A4: A,B3: A] :
% 2.98/1.81 ( ( ord_less @ A @ A4 @ B3 )
% 2.98/1.81 | ( A4 = B3 ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_187_order_Ostrict__trans2,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [A2: A,B2: A,C: A] :
% 2.98/1.81 ( ( ord_less @ A @ A2 @ B2 )
% 2.98/1.81 => ( ( ord_less_eq @ A @ B2 @ C )
% 2.98/1.81 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_188_order_Ostrict__trans1,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [A2: A,B2: A,C: A] :
% 2.98/1.81 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.98/1.81 => ( ( ord_less @ A @ B2 @ C )
% 2.98/1.81 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_189_not__le__imp__less,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [Y2: A,X5: A] :
% 2.98/1.81 ( ~ ( ord_less_eq @ A @ Y2 @ X5 )
% 2.98/1.81 => ( ord_less @ A @ X5 @ Y2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_190_less__le__not__le,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( preorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ( ( ord_less @ A )
% 2.98/1.81 = ( ^ [X4: A,Y3: A] :
% 2.98/1.81 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 2.98/1.81 & ~ ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_191_le__imp__less__or__eq,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A,Y2: A] :
% 2.98/1.81 ( ( ord_less_eq @ A @ X5 @ Y2 )
% 2.98/1.81 => ( ( ord_less @ A @ X5 @ Y2 )
% 2.98/1.81 | ( X5 = Y2 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_192_le__less__linear,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A,Y2: A] :
% 2.98/1.81 ( ( ord_less_eq @ A @ X5 @ Y2 )
% 2.98/1.81 | ( ord_less @ A @ Y2 @ X5 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_193_dense__le,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [Y2: A,Z2: A] :
% 2.98/1.81 ( ! [X: A] :
% 2.98/1.81 ( ( ord_less @ A @ X @ Y2 )
% 2.98/1.81 => ( ord_less_eq @ A @ X @ Z2 ) )
% 2.98/1.81 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_194_dense__ge,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [Z2: A,Y2: A] :
% 2.98/1.81 ( ! [X: A] :
% 2.98/1.81 ( ( ord_less @ A @ Z2 @ X )
% 2.98/1.81 => ( ord_less_eq @ A @ Y2 @ X ) )
% 2.98/1.81 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_195_less__le__trans,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( preorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A,Y2: A,Z2: A] :
% 2.98/1.81 ( ( ord_less @ A @ X5 @ Y2 )
% 2.98/1.81 => ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 2.98/1.81 => ( ord_less @ A @ X5 @ Z2 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_196_le__less__trans,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( preorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A,Y2: A,Z2: A] :
% 2.98/1.81 ( ( ord_less_eq @ A @ X5 @ Y2 )
% 2.98/1.81 => ( ( ord_less @ A @ Y2 @ Z2 )
% 2.98/1.81 => ( ord_less @ A @ X5 @ Z2 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_197_antisym__conv2,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A,Y2: A] :
% 2.98/1.81 ( ( ord_less_eq @ A @ X5 @ Y2 )
% 2.98/1.81 => ( ( ~ ( ord_less @ A @ X5 @ Y2 ) )
% 2.98/1.81 = ( X5 = Y2 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_198_antisym__conv1,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A,Y2: A] :
% 2.98/1.81 ( ~ ( ord_less @ A @ X5 @ Y2 )
% 2.98/1.81 => ( ( ord_less_eq @ A @ X5 @ Y2 )
% 2.98/1.81 = ( X5 = Y2 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_199_less__imp__le,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( preorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A,Y2: A] :
% 2.98/1.81 ( ( ord_less @ A @ X5 @ Y2 )
% 2.98/1.81 => ( ord_less_eq @ A @ X5 @ Y2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_200_le__neq__trans,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [A2: A,B2: A] :
% 2.98/1.81 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.98/1.81 => ( ( A2 != B2 )
% 2.98/1.81 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_201_not__less,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A,Y2: A] :
% 2.98/1.81 ( ( ~ ( ord_less @ A @ X5 @ Y2 ) )
% 2.98/1.81 = ( ord_less_eq @ A @ Y2 @ X5 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_202_not__le,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A,Y2: A] :
% 2.98/1.81 ( ( ~ ( ord_less_eq @ A @ X5 @ Y2 ) )
% 2.98/1.81 = ( ord_less @ A @ Y2 @ X5 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_203_order__less__le__subst2,axiom,
% 2.98/1.81 ! [A: $tType,C3: $tType] :
% 2.98/1.81 ( ( ( order @ C3 @ ( type2 @ C3 ) )
% 2.98/1.81 & ( order @ A @ ( type2 @ A ) ) )
% 2.98/1.81 => ! [A2: A,B2: A,F3: A > C3,C: C3] :
% 2.98/1.81 ( ( ord_less @ A @ A2 @ B2 )
% 2.98/1.81 => ( ( ord_less_eq @ C3 @ ( F3 @ B2 ) @ C )
% 2.98/1.81 => ( ! [X: A,Y4: A] :
% 2.98/1.81 ( ( ord_less @ A @ X @ Y4 )
% 2.98/1.81 => ( ord_less @ C3 @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
% 2.98/1.81 => ( ord_less @ C3 @ ( F3 @ A2 ) @ C ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_204_order__less__le__subst1,axiom,
% 2.98/1.81 ! [A: $tType,B: $tType] :
% 2.98/1.81 ( ( ( order @ B @ ( type2 @ B ) )
% 2.98/1.81 & ( order @ A @ ( type2 @ A ) ) )
% 2.98/1.81 => ! [A2: A,F3: B > A,B2: B,C: B] :
% 2.98/1.81 ( ( ord_less @ A @ A2 @ ( F3 @ B2 ) )
% 2.98/1.81 => ( ( ord_less_eq @ B @ B2 @ C )
% 2.98/1.81 => ( ! [X: B,Y4: B] :
% 2.98/1.81 ( ( ord_less_eq @ B @ X @ Y4 )
% 2.98/1.81 => ( ord_less_eq @ A @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
% 2.98/1.81 => ( ord_less @ A @ A2 @ ( F3 @ C ) ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_205_order__le__less__subst2,axiom,
% 2.98/1.81 ! [A: $tType,C3: $tType] :
% 2.98/1.81 ( ( ( order @ C3 @ ( type2 @ C3 ) )
% 2.98/1.81 & ( order @ A @ ( type2 @ A ) ) )
% 2.98/1.81 => ! [A2: A,B2: A,F3: A > C3,C: C3] :
% 2.98/1.81 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.98/1.81 => ( ( ord_less @ C3 @ ( F3 @ B2 ) @ C )
% 2.98/1.81 => ( ! [X: A,Y4: A] :
% 2.98/1.81 ( ( ord_less_eq @ A @ X @ Y4 )
% 2.98/1.81 => ( ord_less_eq @ C3 @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
% 2.98/1.81 => ( ord_less @ C3 @ ( F3 @ A2 ) @ C ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_206_order__le__less__subst1,axiom,
% 2.98/1.81 ! [A: $tType,B: $tType] :
% 2.98/1.81 ( ( ( order @ B @ ( type2 @ B ) )
% 2.98/1.81 & ( order @ A @ ( type2 @ A ) ) )
% 2.98/1.81 => ! [A2: A,F3: B > A,B2: B,C: B] :
% 2.98/1.81 ( ( ord_less_eq @ A @ A2 @ ( F3 @ B2 ) )
% 2.98/1.81 => ( ( ord_less @ B @ B2 @ C )
% 2.98/1.81 => ( ! [X: B,Y4: B] :
% 2.98/1.81 ( ( ord_less @ B @ X @ Y4 )
% 2.98/1.81 => ( ord_less @ A @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
% 2.98/1.81 => ( ord_less @ A @ A2 @ ( F3 @ C ) ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_207_less__le,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ( ( ord_less @ A )
% 2.98/1.81 = ( ^ [X4: A,Y3: A] :
% 2.98/1.81 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 2.98/1.81 & ( X4 != Y3 ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_208_le__less,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order @ A @ ( type2 @ A ) )
% 2.98/1.81 => ( ( ord_less_eq @ A )
% 2.98/1.81 = ( ^ [X4: A,Y3: A] :
% 2.98/1.81 ( ( ord_less @ A @ X4 @ Y3 )
% 2.98/1.81 | ( X4 = Y3 ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_209_leI,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [X5: A,Y2: A] :
% 2.98/1.81 ( ~ ( ord_less @ A @ X5 @ Y2 )
% 2.98/1.81 => ( ord_less_eq @ A @ Y2 @ X5 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_210_leD,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [Y2: A,X5: A] :
% 2.98/1.81 ( ( ord_less_eq @ A @ Y2 @ X5 )
% 2.98/1.81 => ~ ( ord_less @ A @ X5 @ Y2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_211_bot_Oextremum__uniqueI,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order_bot @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [A2: A] :
% 2.98/1.81 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 2.98/1.81 => ( A2
% 2.98/1.81 = ( bot_bot @ A ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_212_bot_Oextremum__unique,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order_bot @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [A2: A] :
% 2.98/1.81 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 2.98/1.81 = ( A2
% 2.98/1.81 = ( bot_bot @ A ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_213_bot_Oextremum,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( order_bot @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_214_complete__interval,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( condit1037483654norder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [A2: A,B2: A,P2: A > $o] :
% 2.98/1.81 ( ( ord_less @ A @ A2 @ B2 )
% 2.98/1.81 => ( ( P2 @ A2 )
% 2.98/1.81 => ( ~ ( P2 @ B2 )
% 2.98/1.81 => ? [C4: A] :
% 2.98/1.81 ( ( ord_less_eq @ A @ A2 @ C4 )
% 2.98/1.81 & ( ord_less_eq @ A @ C4 @ B2 )
% 2.98/1.81 & ! [X6: A] :
% 2.98/1.81 ( ( ( ord_less_eq @ A @ A2 @ X6 )
% 2.98/1.81 & ( ord_less @ A @ X6 @ C4 ) )
% 2.98/1.81 => ( P2 @ X6 ) )
% 2.98/1.81 & ! [D3: A] :
% 2.98/1.81 ( ! [X: A] :
% 2.98/1.81 ( ( ( ord_less_eq @ A @ A2 @ X )
% 2.98/1.81 & ( ord_less @ A @ X @ D3 ) )
% 2.98/1.81 => ( P2 @ X ) )
% 2.98/1.81 => ( ord_less_eq @ A @ D3 @ C4 ) ) ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_215_pinf_I6_J,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [T: A] :
% 2.98/1.81 ? [Z4: A] :
% 2.98/1.81 ! [X6: A] :
% 2.98/1.81 ( ( ord_less @ A @ Z4 @ X6 )
% 2.98/1.81 => ~ ( ord_less_eq @ A @ X6 @ T ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_216_pinf_I8_J,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [T: A] :
% 2.98/1.81 ? [Z4: A] :
% 2.98/1.81 ! [X6: A] :
% 2.98/1.81 ( ( ord_less @ A @ Z4 @ X6 )
% 2.98/1.81 => ( ord_less_eq @ A @ T @ X6 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_217_minf_I6_J,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [T: A] :
% 2.98/1.81 ? [Z4: A] :
% 2.98/1.81 ! [X6: A] :
% 2.98/1.81 ( ( ord_less @ A @ X6 @ Z4 )
% 2.98/1.81 => ( ord_less_eq @ A @ X6 @ T ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_218_minf_I8_J,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [T: A] :
% 2.98/1.81 ? [Z4: A] :
% 2.98/1.81 ! [X6: A] :
% 2.98/1.81 ( ( ord_less @ A @ X6 @ Z4 )
% 2.98/1.81 => ~ ( ord_less_eq @ A @ T @ X6 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_219_le__dag__antisym,axiom,
% 2.98/1.81 ! [X5: binDag_Mirabelle_dag,Y2: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( ord_less_eq @ binDag_Mirabelle_dag @ X5 @ Y2 )
% 2.98/1.81 => ( ( ord_less_eq @ binDag_Mirabelle_dag @ Y2 @ X5 )
% 2.98/1.81 => ( X5 = Y2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_220_le__dag__trans,axiom,
% 2.98/1.81 ! [X5: binDag_Mirabelle_dag,Y2: binDag_Mirabelle_dag,Z2: binDag_Mirabelle_dag] :
% 2.98/1.81 ( ( ord_less_eq @ binDag_Mirabelle_dag @ X5 @ Y2 )
% 2.98/1.81 => ( ( ord_less_eq @ binDag_Mirabelle_dag @ Y2 @ Z2 )
% 2.98/1.81 => ( ord_less_eq @ binDag_Mirabelle_dag @ X5 @ Z2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_221_le__dag__refl,axiom,
% 2.98/1.81 ! [X5: binDag_Mirabelle_dag] : ( ord_less_eq @ binDag_Mirabelle_dag @ X5 @ X5 ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_222_less__fun__def,axiom,
% 2.98/1.81 ! [B: $tType,A: $tType] :
% 2.98/1.81 ( ( ord @ B @ ( type2 @ B ) )
% 2.98/1.81 => ( ( ord_less @ ( A > B ) )
% 2.98/1.81 = ( ^ [F: A > B,G2: A > B] :
% 2.98/1.81 ( ( ord_less_eq @ ( A > B ) @ F @ G2 )
% 2.98/1.81 & ~ ( ord_less_eq @ ( A > B ) @ G2 @ F ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_223_psubset__trans,axiom,
% 2.98/1.81 ! [A: $tType,A3: set @ A,B4: set @ A,C2: set @ A] :
% 2.98/1.81 ( ( ord_less @ ( set @ A ) @ A3 @ B4 )
% 2.98/1.81 => ( ( ord_less @ ( set @ A ) @ B4 @ C2 )
% 2.98/1.81 => ( ord_less @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_224_psubsetD,axiom,
% 2.98/1.81 ! [A: $tType,A3: set @ A,B4: set @ A,C: A] :
% 2.98/1.81 ( ( ord_less @ ( set @ A ) @ A3 @ B4 )
% 2.98/1.81 => ( ( member @ A @ C @ A3 )
% 2.98/1.81 => ( member @ A @ C @ B4 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_225_minf_I11_J,axiom,
% 2.98/1.81 ! [C3: $tType,D2: $tType] :
% 2.98/1.81 ( ( ord @ C3 @ ( type2 @ C3 ) )
% 2.98/1.81 => ! [F4: D2] :
% 2.98/1.81 ? [Z4: C3] :
% 2.98/1.81 ! [X6: C3] :
% 2.98/1.81 ( ( ord_less @ C3 @ X6 @ Z4 )
% 2.98/1.81 => ( F4 = F4 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_226_minf_I7_J,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [T: A] :
% 2.98/1.81 ? [Z4: A] :
% 2.98/1.81 ! [X6: A] :
% 2.98/1.81 ( ( ord_less @ A @ X6 @ Z4 )
% 2.98/1.81 => ~ ( ord_less @ A @ T @ X6 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_227_minf_I5_J,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [T: A] :
% 2.98/1.81 ? [Z4: A] :
% 2.98/1.81 ! [X6: A] :
% 2.98/1.81 ( ( ord_less @ A @ X6 @ Z4 )
% 2.98/1.81 => ( ord_less @ A @ X6 @ T ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_228_minf_I4_J,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [T: A] :
% 2.98/1.81 ? [Z4: A] :
% 2.98/1.81 ! [X6: A] :
% 2.98/1.81 ( ( ord_less @ A @ X6 @ Z4 )
% 2.98/1.81 => ( X6 != T ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_229_minf_I3_J,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [T: A] :
% 2.98/1.81 ? [Z4: A] :
% 2.98/1.81 ! [X6: A] :
% 2.98/1.81 ( ( ord_less @ A @ X6 @ Z4 )
% 2.98/1.81 => ( X6 != T ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_230_minf_I2_J,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [P2: A > $o,P3: A > $o,Q3: A > $o,Q4: A > $o] :
% 2.98/1.81 ( ? [Z5: A] :
% 2.98/1.81 ! [X: A] :
% 2.98/1.81 ( ( ord_less @ A @ X @ Z5 )
% 2.98/1.81 => ( ( P2 @ X )
% 2.98/1.81 = ( P3 @ X ) ) )
% 2.98/1.81 => ( ? [Z5: A] :
% 2.98/1.81 ! [X: A] :
% 2.98/1.81 ( ( ord_less @ A @ X @ Z5 )
% 2.98/1.81 => ( ( Q3 @ X )
% 2.98/1.81 = ( Q4 @ X ) ) )
% 2.98/1.81 => ? [Z4: A] :
% 2.98/1.81 ! [X6: A] :
% 2.98/1.81 ( ( ord_less @ A @ X6 @ Z4 )
% 2.98/1.81 => ( ( ( P2 @ X6 )
% 2.98/1.81 | ( Q3 @ X6 ) )
% 2.98/1.81 = ( ( P3 @ X6 )
% 2.98/1.81 | ( Q4 @ X6 ) ) ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_231_minf_I1_J,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [P2: A > $o,P3: A > $o,Q3: A > $o,Q4: A > $o] :
% 2.98/1.81 ( ? [Z5: A] :
% 2.98/1.81 ! [X: A] :
% 2.98/1.81 ( ( ord_less @ A @ X @ Z5 )
% 2.98/1.81 => ( ( P2 @ X )
% 2.98/1.81 = ( P3 @ X ) ) )
% 2.98/1.81 => ( ? [Z5: A] :
% 2.98/1.81 ! [X: A] :
% 2.98/1.81 ( ( ord_less @ A @ X @ Z5 )
% 2.98/1.81 => ( ( Q3 @ X )
% 2.98/1.81 = ( Q4 @ X ) ) )
% 2.98/1.81 => ? [Z4: A] :
% 2.98/1.81 ! [X6: A] :
% 2.98/1.81 ( ( ord_less @ A @ X6 @ Z4 )
% 2.98/1.81 => ( ( ( P2 @ X6 )
% 2.98/1.81 & ( Q3 @ X6 ) )
% 2.98/1.81 = ( ( P3 @ X6 )
% 2.98/1.81 & ( Q4 @ X6 ) ) ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_232_pinf_I11_J,axiom,
% 2.98/1.81 ! [C3: $tType,D2: $tType] :
% 2.98/1.81 ( ( ord @ C3 @ ( type2 @ C3 ) )
% 2.98/1.81 => ! [F4: D2] :
% 2.98/1.81 ? [Z4: C3] :
% 2.98/1.81 ! [X6: C3] :
% 2.98/1.81 ( ( ord_less @ C3 @ Z4 @ X6 )
% 2.98/1.81 => ( F4 = F4 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_233_pinf_I7_J,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [T: A] :
% 2.98/1.81 ? [Z4: A] :
% 2.98/1.81 ! [X6: A] :
% 2.98/1.81 ( ( ord_less @ A @ Z4 @ X6 )
% 2.98/1.81 => ( ord_less @ A @ T @ X6 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_234_pinf_I5_J,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [T: A] :
% 2.98/1.81 ? [Z4: A] :
% 2.98/1.81 ! [X6: A] :
% 2.98/1.81 ( ( ord_less @ A @ Z4 @ X6 )
% 2.98/1.81 => ~ ( ord_less @ A @ X6 @ T ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_235_pinf_I4_J,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [T: A] :
% 2.98/1.81 ? [Z4: A] :
% 2.98/1.81 ! [X6: A] :
% 2.98/1.81 ( ( ord_less @ A @ Z4 @ X6 )
% 2.98/1.81 => ( X6 != T ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_236_pinf_I3_J,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [T: A] :
% 2.98/1.81 ? [Z4: A] :
% 2.98/1.81 ! [X6: A] :
% 2.98/1.81 ( ( ord_less @ A @ Z4 @ X6 )
% 2.98/1.81 => ( X6 != T ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_237_pinf_I2_J,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [P2: A > $o,P3: A > $o,Q3: A > $o,Q4: A > $o] :
% 2.98/1.81 ( ? [Z5: A] :
% 2.98/1.81 ! [X: A] :
% 2.98/1.81 ( ( ord_less @ A @ Z5 @ X )
% 2.98/1.81 => ( ( P2 @ X )
% 2.98/1.81 = ( P3 @ X ) ) )
% 2.98/1.81 => ( ? [Z5: A] :
% 2.98/1.81 ! [X: A] :
% 2.98/1.81 ( ( ord_less @ A @ Z5 @ X )
% 2.98/1.81 => ( ( Q3 @ X )
% 2.98/1.81 = ( Q4 @ X ) ) )
% 2.98/1.81 => ? [Z4: A] :
% 2.98/1.81 ! [X6: A] :
% 2.98/1.81 ( ( ord_less @ A @ Z4 @ X6 )
% 2.98/1.81 => ( ( ( P2 @ X6 )
% 2.98/1.81 | ( Q3 @ X6 ) )
% 2.98/1.81 = ( ( P3 @ X6 )
% 2.98/1.81 | ( Q4 @ X6 ) ) ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_238_pinf_I1_J,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [P2: A > $o,P3: A > $o,Q3: A > $o,Q4: A > $o] :
% 2.98/1.81 ( ? [Z5: A] :
% 2.98/1.81 ! [X: A] :
% 2.98/1.81 ( ( ord_less @ A @ Z5 @ X )
% 2.98/1.81 => ( ( P2 @ X )
% 2.98/1.81 = ( P3 @ X ) ) )
% 2.98/1.81 => ( ? [Z5: A] :
% 2.98/1.81 ! [X: A] :
% 2.98/1.81 ( ( ord_less @ A @ Z5 @ X )
% 2.98/1.81 => ( ( Q3 @ X )
% 2.98/1.81 = ( Q4 @ X ) ) )
% 2.98/1.81 => ? [Z4: A] :
% 2.98/1.81 ! [X6: A] :
% 2.98/1.81 ( ( ord_less @ A @ Z4 @ X6 )
% 2.98/1.81 => ( ( ( P2 @ X6 )
% 2.98/1.81 & ( Q3 @ X6 ) )
% 2.98/1.81 = ( ( P3 @ X6 )
% 2.98/1.81 & ( Q4 @ X6 ) ) ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_239_ex__gt__or__lt,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [A2: A] :
% 2.98/1.81 ? [B6: A] :
% 2.98/1.81 ( ( ord_less @ A @ A2 @ B6 )
% 2.98/1.81 | ( ord_less @ A @ B6 @ A2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_240_bot__empty__eq,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( bot_bot @ ( A > $o ) )
% 2.98/1.81 = ( ^ [X4: A] : ( member @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_241_Collect__empty__eq__bot,axiom,
% 2.98/1.81 ! [A: $tType,P2: A > $o] :
% 2.98/1.81 ( ( ( collect @ A @ P2 )
% 2.98/1.81 = ( bot_bot @ ( set @ A ) ) )
% 2.98/1.81 = ( P2
% 2.98/1.81 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_242_override__on__emptyset,axiom,
% 2.98/1.81 ! [B: $tType,A: $tType,F3: A > B,G: A > B] :
% 2.98/1.81 ( ( override_on @ A @ B @ F3 @ G @ ( bot_bot @ ( set @ A ) ) )
% 2.98/1.81 = F3 ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_243_Set_Ois__empty__def,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( is_empty @ A )
% 2.98/1.81 = ( ^ [A5: set @ A] :
% 2.98/1.81 ( A5
% 2.98/1.81 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_244_override__on__apply__notin,axiom,
% 2.98/1.81 ! [B: $tType,A: $tType,A2: A,A3: set @ A,F3: A > B,G: A > B] :
% 2.98/1.81 ( ~ ( member @ A @ A2 @ A3 )
% 2.98/1.81 => ( ( override_on @ A @ B @ F3 @ G @ A3 @ A2 )
% 2.98/1.81 = ( F3 @ A2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_245_override__on__apply__in,axiom,
% 2.98/1.81 ! [B: $tType,A: $tType,A2: A,A3: set @ A,F3: A > B,G: A > B] :
% 2.98/1.81 ( ( member @ A @ A2 @ A3 )
% 2.98/1.81 => ( ( override_on @ A @ B @ F3 @ G @ A3 @ A2 )
% 2.98/1.81 = ( G @ A2 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_246_override__on__def,axiom,
% 2.98/1.81 ! [B: $tType,A: $tType] :
% 2.98/1.81 ( ( override_on @ A @ B )
% 2.98/1.81 = ( ^ [F: A > B,G2: A > B,A5: set @ A,A4: A] : ( if @ B @ ( member @ A @ A4 @ A5 ) @ ( G2 @ A4 ) @ ( F @ A4 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_247_is__empty__set,axiom,
% 2.98/1.81 ! [A: $tType,Xs: list @ A] :
% 2.98/1.81 ( ( is_empty @ A @ ( set2 @ A @ Xs ) )
% 2.98/1.81 = ( null @ A @ Xs ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_248_subset__code_I2_J,axiom,
% 2.98/1.81 ! [B: $tType,A3: set @ B,Ys: list @ B] :
% 2.98/1.81 ( ( ord_less_eq @ ( set @ B ) @ A3 @ ( coset @ B @ Ys ) )
% 2.98/1.81 = ( ! [X4: B] :
% 2.98/1.81 ( ( member @ B @ X4 @ ( set2 @ B @ Ys ) )
% 2.98/1.81 => ~ ( member @ B @ X4 @ A3 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_249_subset__code_I3_J,axiom,
% 2.98/1.81 ! [C3: $tType] :
% 2.98/1.81 ~ ( ord_less_eq @ ( set @ C3 ) @ ( coset @ C3 @ ( nil @ C3 ) ) @ ( set2 @ C3 @ ( nil @ C3 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_250_set__empty2,axiom,
% 2.98/1.81 ! [A: $tType,Xs: list @ A] :
% 2.98/1.81 ( ( ( bot_bot @ ( set @ A ) )
% 2.98/1.81 = ( set2 @ A @ Xs ) )
% 2.98/1.81 = ( Xs
% 2.98/1.81 = ( nil @ A ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_251_set__empty,axiom,
% 2.98/1.81 ! [A: $tType,Xs: list @ A] :
% 2.98/1.81 ( ( ( set2 @ A @ Xs )
% 2.98/1.81 = ( bot_bot @ ( set @ A ) ) )
% 2.98/1.81 = ( Xs
% 2.98/1.81 = ( nil @ A ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_252_empty__set,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( bot_bot @ ( set @ A ) )
% 2.98/1.81 = ( set2 @ A @ ( nil @ A ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_253_sorted__list__of__set__empty,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ( ( linord467138063of_set @ A @ ( bot_bot @ ( set @ A ) ) )
% 2.98/1.81 = ( nil @ A ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_254_sorted__list__of__set__eq__Nil__iff,axiom,
% 2.98/1.81 ! [A: $tType] :
% 2.98/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 2.98/1.81 => ! [A3: set @ A] :
% 2.98/1.81 ( ( finite_finite @ A @ A3 )
% 2.98/1.81 => ( ( ( linord467138063of_set @ A @ A3 )
% 2.98/1.81 = ( nil @ A ) )
% 2.98/1.81 = ( A3
% 2.98/1.81 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(fact_255_List_Ofinite__set,axiom,
% 2.98/1.81 ! [A: $tType,Xs: list @ A] : ( finite_finite @ A @ ( set2 @ A @ Xs ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 2.98/1.81 ! [A7: $tType,A8: $tType] :
% 2.98/1.81 ( ( order_bot @ A8 @ ( type2 @ A8 ) )
% 2.98/1.81 => ( order_bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.98/1.81 ! [A7: $tType,A8: $tType] :
% 2.98/1.81 ( ( preorder @ A8 @ ( type2 @ A8 ) )
% 2.98/1.81 => ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.98/1.81 ! [A7: $tType,A8: $tType] :
% 2.98/1.81 ( ( order @ A8 @ ( type2 @ A8 ) )
% 2.98/1.81 => ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(tcon_fun___Orderings_Oord,axiom,
% 2.98/1.81 ! [A7: $tType,A8: $tType] :
% 2.98/1.81 ( ( ord @ A8 @ ( type2 @ A8 ) )
% 2.98/1.81 => ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(tcon_fun___Orderings_Obot,axiom,
% 2.98/1.81 ! [A7: $tType,A8: $tType] :
% 2.98/1.81 ( ( bot @ A8 @ ( type2 @ A8 ) )
% 2.98/1.81 => ( bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(tcon_Set_Oset___Orderings_Oorder__bot_1,axiom,
% 2.98/1.81 ! [A7: $tType] : ( order_bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(tcon_Set_Oset___Orderings_Opreorder_2,axiom,
% 2.98/1.81 ! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(tcon_Set_Oset___Orderings_Oorder_3,axiom,
% 2.98/1.81 ! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(tcon_Set_Oset___Orderings_Oord_4,axiom,
% 2.98/1.81 ! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(tcon_Set_Oset___Orderings_Obot_5,axiom,
% 2.98/1.81 ! [A7: $tType] : ( bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(tcon_HOL_Obool___Orderings_Oorder__bot_6,axiom,
% 2.98/1.81 order_bot @ $o @ ( type2 @ $o ) ).
% 2.98/1.81
% 2.98/1.81 thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
% 2.98/1.81 preorder @ $o @ ( type2 @ $o ) ).
% 2.98/1.81
% 2.98/1.81 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 2.98/1.81 linorder @ $o @ ( type2 @ $o ) ).
% 2.98/1.81
% 2.98/1.81 thf(tcon_HOL_Obool___Orderings_Oorder_8,axiom,
% 2.98/1.81 order @ $o @ ( type2 @ $o ) ).
% 2.98/1.81
% 2.98/1.81 thf(tcon_HOL_Obool___Orderings_Oord_9,axiom,
% 2.98/1.81 ord @ $o @ ( type2 @ $o ) ).
% 2.98/1.81
% 2.98/1.81 thf(tcon_HOL_Obool___Orderings_Obot_10,axiom,
% 2.98/1.81 bot @ $o @ ( type2 @ $o ) ).
% 2.98/1.81
% 2.98/1.81 thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Opreorder_11,axiom,
% 2.98/1.81 preorder @ binDag_Mirabelle_dag @ ( type2 @ binDag_Mirabelle_dag ) ).
% 2.98/1.81
% 2.98/1.81 thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Oorder_12,axiom,
% 2.98/1.81 order @ binDag_Mirabelle_dag @ ( type2 @ binDag_Mirabelle_dag ) ).
% 2.98/1.81
% 2.98/1.81 thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Oord_13,axiom,
% 2.98/1.81 ord @ binDag_Mirabelle_dag @ ( type2 @ binDag_Mirabelle_dag ) ).
% 2.98/1.81
% 2.98/1.81 thf(help_If_3_1_T,axiom,
% 2.98/1.81 ! [P2: $o] :
% 2.98/1.81 ( ( P2 = $true )
% 2.98/1.81 | ( P2 = $false ) ) ).
% 2.98/1.81
% 2.98/1.81 thf(help_If_2_1_T,axiom,
% 2.98/1.81 ! [A: $tType,X5: A,Y2: A] :
% 2.98/1.81 ( ( if @ A @ $false @ X5 @ Y2 )
% 2.98/1.81 = Y2 ) ).
% 2.98/1.81
% 2.98/1.81 thf(help_If_1_1_T,axiom,
% 2.98/1.81 ! [A: $tType,X5: A,Y2: A] :
% 2.98/1.81 ( ( if @ A @ $true @ X5 @ Y2 )
% 2.98/1.81 = X5 ) ).
% 2.98/1.81
% 2.98/1.81 thf(conj_0,hypothesis,
% 2.98/1.81 ! [Q5: simpl_ref] :
% 2.98/1.81 ( ( binDag_Mirabelle_Dag @ Q5 @ l @ r @ ( binDag476092410e_Node @ lt @ p @ rt ) )
% 2.98/1.81 => thesis ) ).
% 2.98/1.81
% 2.98/1.81 thf(conj_1,conjecture,
% 2.98/1.81 thesis ).
% 2.98/1.81 ------------------------
% 2.98/1.85 % (23159)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_22996 for (2999ds/275Mi)
% 2.98/1.85 % (23155)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_22996 for (2999ds/4Mi)
% 2.98/1.85 % (23157)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_22996 for (2999ds/2Mi)
% 2.98/1.85 % (23154)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_22996 for (2999ds/183Mi)
% 2.98/1.85 % (23156)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_22996 for (2999ds/27Mi)
% 2.98/1.85 % (23160)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_22996 for (2999ds/18Mi)
% 2.98/1.85 % (23158)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_22996 for (2999ds/2Mi)
% 2.98/1.85 % (23157)Instruction limit reached!
% 2.98/1.85 % (23157)------------------------------
% 2.98/1.85 % (23157)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.85 % (23157)Termination reason: Unknown
% 2.98/1.85 % (23157)Termination phase: shuffling
% 2.98/1.85 % (23158)Instruction limit reached!
% 2.98/1.85 % (23158)------------------------------
% 2.98/1.85 % (23158)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.85
% 2.98/1.85 % (23157)Memory used [KB]: 1279
% 2.98/1.85 % (23157)Time elapsed: 0.003 s
% 2.98/1.85 % (23157)Instructions burned: 2 (million)
% 2.98/1.85 % (23157)------------------------------
% 2.98/1.85 % (23157)------------------------------
% 2.98/1.85 % (23158)Termination reason: Unknown
% 2.98/1.85 % (23158)Termination phase: shuffling
% 2.98/1.85
% 2.98/1.85 % (23158)Memory used [KB]: 1279
% 2.98/1.85 % (23158)Time elapsed: 0.003 s
% 2.98/1.85 % (23158)Instructions burned: 2 (million)
% 2.98/1.85 % (23158)------------------------------
% 2.98/1.85 % (23158)------------------------------
% 2.98/1.85 % (23155)Instruction limit reached!
% 2.98/1.85 % (23155)------------------------------
% 2.98/1.85 % (23155)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.85 % (23155)Termination reason: Unknown
% 2.98/1.85 % (23155)Termination phase: shuffling
% 2.98/1.85
% 2.98/1.85 % (23155)Memory used [KB]: 1407
% 2.98/1.85 % (23155)Time elapsed: 0.004 s
% 2.98/1.85 % (23155)Instructions burned: 4 (million)
% 2.98/1.85 % (23155)------------------------------
% 2.98/1.85 % (23155)------------------------------
% 2.98/1.85 % (23160)Instruction limit reached!
% 2.98/1.85 % (23160)------------------------------
% 2.98/1.85 % (23160)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.85 % (23160)Termination reason: Unknown
% 2.98/1.85 % (23160)Termination phase: shuffling
% 2.98/1.85
% 2.98/1.85 % (23160)Memory used [KB]: 1663
% 2.98/1.85 % (23160)Time elapsed: 0.011 s
% 2.98/1.85 % (23160)Instructions burned: 18 (million)
% 2.98/1.85 % (23160)------------------------------
% 2.98/1.85 % (23160)------------------------------
% 2.98/1.86 % (23156)Instruction limit reached!
% 2.98/1.86 % (23156)------------------------------
% 2.98/1.86 % (23156)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.86 % (23156)Termination reason: Unknown
% 2.98/1.86 % (23156)Termination phase: Property scanning
% 2.98/1.86
% 2.98/1.86 % (23156)Memory used [KB]: 1663
% 2.98/1.86 % (23156)Time elapsed: 0.016 s
% 2.98/1.86 % (23156)Instructions burned: 28 (million)
% 2.98/1.86 % (23156)------------------------------
% 2.98/1.86 % (23156)------------------------------
% 2.98/1.86 % (23161)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_22996 for (2999ds/3Mi)
% 2.98/1.86 % (23162)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_22996 for (2999ds/37Mi)
% 2.98/1.86 % (23163)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_22996 for (2999ds/15Mi)
% 2.98/1.86 % (23161)Instruction limit reached!
% 2.98/1.86 % (23161)------------------------------
% 2.98/1.86 % (23161)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.86 % (23161)Termination reason: Unknown
% 2.98/1.86 % (23161)Termination phase: shuffling
% 2.98/1.86
% 2.98/1.86 % (23161)Memory used [KB]: 1407
% 2.98/1.86 % (23161)Time elapsed: 0.003 s
% 2.98/1.86 % (23161)Instructions burned: 3 (million)
% 2.98/1.86 % (23161)------------------------------
% 2.98/1.86 % (23161)------------------------------
% 2.98/1.87 % (23159)First to succeed.
% 2.98/1.87 % (23159)Refutation found. Thanks to Tanya!
% 2.98/1.87 % SZS status Theorem for DTF2THF_22996
% 2.98/1.87 % SZS output start Proof for DTF2THF_22996
% 2.98/1.87 thf(type_def_5, type, itself: ($tType) > $tType).
% 2.98/1.87 thf(type_def_7, type, binDag_Mirabelle_dag: $tType).
% 2.98/1.87 thf(type_def_8, type, simpl_ref: $tType).
% 2.98/1.87 thf(type_def_9, type, set: ($tType) > $tType).
% 2.98/1.87 thf(type_def_10, type, list: ($tType) > $tType).
% 2.98/1.87 thf(func_def_0, type, binDag_Mirabelle_dag: $tType).
% 2.98/1.87 thf(func_def_1, type, simpl_ref: $tType).
% 2.98/1.87 thf(func_def_2, type, list: ($tType) > $tType).
% 2.98/1.87 thf(func_def_3, type, set: ($tType) > $tType).
% 2.98/1.87 thf(func_def_4, type, itself: ($tType) > $tType).
% 2.98/1.87 thf(func_def_5, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.98/1.87 thf(func_def_6, type, bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.98/1.87 thf(func_def_7, type, ord: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.98/1.87 thf(func_def_8, type, order: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.98/1.87 thf(func_def_9, type, no_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.98/1.87 thf(func_def_10, type, no_top: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.98/1.87 thf(func_def_11, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.98/1.87 thf(func_def_12, type, preorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.98/1.87 thf(func_def_13, type, order_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.98/1.87 thf(func_def_14, type, wellorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.98/1.87 thf(func_def_15, type, dense_order: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.98/1.87 thf(func_def_16, type, dense_linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.98/1.87 thf(func_def_17, type, condit1656338222tinuum: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.98/1.87 thf(func_def_18, type, condit1037483654norder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.98/1.87 thf(func_def_19, type, binDag_Mirabelle_DAG: binDag_Mirabelle_dag > $o).
% 2.98/1.87 thf(func_def_20, type, binDag_Mirabelle_Dag: simpl_ref > (simpl_ref > simpl_ref) > (simpl_ref > simpl_ref) > binDag_Mirabelle_dag > $o).
% 2.98/1.87 thf(func_def_21, type, binDag476092410e_Node: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > binDag_Mirabelle_dag).
% 2.98/1.87 thf(func_def_22, type, binDag_Mirabelle_Tip: binDag_Mirabelle_dag).
% 2.98/1.87 thf(func_def_23, type, binDag1297733282se_dag: !>[X0: $tType]:(X0 > (binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > X0) > binDag_Mirabelle_dag > X0)).
% 2.98/1.87 thf(func_def_24, type, binDag1442713106ec_dag: !>[X0: $tType]:(X0 > (binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > X0 > X0 > X0) > binDag_Mirabelle_dag > X0)).
% 2.98/1.87 thf(func_def_25, type, binDag1380252983set_of: binDag_Mirabelle_dag > set @ simpl_ref).
% 2.98/1.87 thf(func_def_26, type, binDag786255756subdag: binDag_Mirabelle_dag > binDag_Mirabelle_dag > $o).
% 2.98/1.87 thf(func_def_27, type, finite_finite: !>[X0: $tType]:(set @ X0 > $o)).
% 2.98/1.87 thf(func_def_28, type, fun_upd: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0 > X1 > X0 > X1)).
% 2.98/1.87 thf(func_def_29, type, override_on: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X0 > X1) > set @ X0 > X0 > X1)).
% 2.98/1.87 thf(func_def_30, type, swap: !>[X0: $tType, X1: $tType]:(X0 > X0 > (X0 > X1) > X0 > X1)).
% 2.98/1.87 thf(func_def_31, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 2.98/1.87 thf(func_def_32, type, coset: !>[X0: $tType]:(list @ X0 > set @ X0)).
% 2.98/1.87 thf(func_def_33, type, linord467138063of_set: !>[X0: $tType]:(set @ X0 > list @ X0)).
% 2.98/1.87 thf(func_def_34, type, nil: !>[X0: $tType]:(list @ X0)).
% 2.98/1.87 thf(func_def_35, type, set2: !>[X0: $tType]:(list @ X0 > set @ X0)).
% 2.98/1.87 thf(func_def_36, type, null: !>[X0: $tType]:(list @ X0 > $o)).
% 2.98/1.87 thf(func_def_37, type, bot_bot: !>[X0: $tType]:(X0)).
% 2.98/1.87 thf(func_def_38, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.98/1.87 thf(func_def_39, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.98/1.87 thf(func_def_40, type, type2: !>[X0: $tType]:(itself @ X0)).
% 2.98/1.87 thf(func_def_41, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.98/1.87 thf(func_def_42, type, is_empty: !>[X0: $tType]:(set @ X0 > $o)).
% 2.98/1.87 thf(func_def_43, type, simpl_Null: simpl_ref).
% 2.98/1.87 thf(func_def_44, type, simpl_new: set @ simpl_ref > simpl_ref).
% 2.98/1.87 thf(func_def_45, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.98/1.87 thf(func_def_46, type, l: simpl_ref > simpl_ref).
% 2.98/1.87 thf(func_def_47, type, lt: binDag_Mirabelle_dag).
% 2.98/1.87 thf(func_def_48, type, p: simpl_ref).
% 2.98/1.87 thf(func_def_49, type, r: simpl_ref > simpl_ref).
% 2.98/1.87 thf(func_def_50, type, rt: binDag_Mirabelle_dag).
% 2.98/1.87 thf(func_def_51, type, t: binDag_Mirabelle_dag).
% 2.98/1.87 thf(func_def_67, type, sK0: binDag_Mirabelle_dag > (simpl_ref > simpl_ref) > (simpl_ref > simpl_ref) > simpl_ref).
% 2.98/1.87 thf(func_def_68, type, sK1: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.98/1.87 thf(func_def_69, type, sK2: (simpl_ref > simpl_ref) > (simpl_ref > simpl_ref) > simpl_ref > binDag_Mirabelle_dag).
% 2.98/1.87 thf(func_def_70, type, sK3: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag).
% 2.98/1.87 thf(func_def_71, type, sK4: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag).
% 2.98/1.87 thf(func_def_73, type, ph6: !>[X0: $tType]:(X0)).
% 2.98/1.87 thf(f891,plain,(
% 2.98/1.87 $false),
% 2.98/1.87 inference(subsumption_resolution,[status(thm)],[f888,f855])).
% 2.98/1.87 thf(f855,plain,(
% 2.98/1.87 ((binDag786255756subdag @ t @ (binDag476092410e_Node @ lt @ p @ rt)) = $true)),
% 2.98/1.87 inference(cnf_transformation,[status(thm)],[f450])).
% 2.98/1.87 thf(f450,plain,(
% 2.98/1.87 ((binDag786255756subdag @ t @ (binDag476092410e_Node @ lt @ p @ rt)) = $true)),
% 2.98/1.87 inference(fool_elimination,[status(thm)],[f449])).
% 2.98/1.87 thf(f449,plain,(
% 2.98/1.87 (binDag786255756subdag @ t @ (binDag476092410e_Node @ lt @ p @ rt))),
% 2.98/1.87 inference(rectify,[status(thm)],[f3])).
% 2.98/1.87 thf(f3,axiom,(
% 2.98/1.87 (binDag786255756subdag @ t @ (binDag476092410e_Node @ lt @ p @ rt))),
% 2.98/1.87 file('/export/starexec/sandbox/tmp/tmp.CLaN6L8RqG/DTF2THF_22996.p',fact_2_subdag)).
% 2.98/1.87 thf(f888,plain,(
% 2.98/1.87 ((binDag786255756subdag @ t @ (binDag476092410e_Node @ lt @ p @ rt)) != $true)),
% 2.98/1.87 inference(trivial_inequality_removal,[status(thm)],[f885])).
% 2.98/1.87 thf(f885,plain,(
% 2.98/1.87 ((binDag786255756subdag @ t @ (binDag476092410e_Node @ lt @ p @ rt)) != $true) | ($true != $true)),
% 2.98/1.87 inference(superposition,[status(thm)],[f884,f857])).
% 2.98/1.87 thf(f857,plain,(
% 2.98/1.87 ((binDag_Mirabelle_Dag @ (r @ p) @ l @ r @ t) = $true)),
% 2.98/1.87 inference(cnf_transformation,[status(thm)],[f404])).
% 2.98/1.87 thf(f404,plain,(
% 2.98/1.87 ((binDag_Mirabelle_Dag @ (r @ p) @ l @ r @ t) = $true)),
% 2.98/1.87 inference(fool_elimination,[status(thm)],[f403])).
% 2.98/1.87 thf(f403,plain,(
% 2.98/1.87 (binDag_Mirabelle_Dag @ (r @ p) @ l @ r @ t)),
% 2.98/1.87 inference(rectify,[status(thm)],[f1])).
% 2.98/1.87 thf(f1,axiom,(
% 2.98/1.87 (binDag_Mirabelle_Dag @ (r @ p) @ l @ r @ t)),
% 2.98/1.87 file('/export/starexec/sandbox/tmp/tmp.CLaN6L8RqG/DTF2THF_22996.p',fact_0_assms)).
% 2.98/1.87 thf(f884,plain,(
% 2.98/1.87 ( ! [X0 : binDag_Mirabelle_dag,X1 : simpl_ref] : (((binDag_Mirabelle_Dag @ X1 @ l @ r @ X0) != $true) | ((binDag786255756subdag @ X0 @ (binDag476092410e_Node @ lt @ p @ rt)) != $true)) )),
% 2.98/1.87 inference(trivial_inequality_removal,[status(thm)],[f883])).
% 2.98/1.87 thf(f883,plain,(
% 2.98/1.87 ( ! [X0 : binDag_Mirabelle_dag,X1 : simpl_ref] : (($true != $true) | ((binDag_Mirabelle_Dag @ X1 @ l @ r @ X0) != $true) | ((binDag786255756subdag @ X0 @ (binDag476092410e_Node @ lt @ p @ rt)) != $true)) )),
% 2.98/1.87 inference(superposition,[status(thm)],[f882,f831])).
% 2.98/1.87 thf(f831,plain,(
% 2.98/1.87 ( ! [X2 : simpl_ref > simpl_ref,X3 : binDag_Mirabelle_dag,X0 : simpl_ref,X1 : simpl_ref > simpl_ref,X4 : binDag_Mirabelle_dag] : (((binDag_Mirabelle_Dag @ (sK0 @ X4 @ X2 @ X1) @ X1 @ X2 @ X4) = $true) | ($true != (binDag786255756subdag @ X3 @ X4)) | ($true != (binDag_Mirabelle_Dag @ X0 @ X1 @ X2 @ X3))) )),
% 2.98/1.87 inference(cnf_transformation,[status(thm)],[f812])).
% 2.98/1.87 thf(f812,plain,(
% 2.98/1.87 ! [X0 : simpl_ref,X1 : simpl_ref > simpl_ref,X2 : simpl_ref > simpl_ref,X3 : binDag_Mirabelle_dag,X4 : binDag_Mirabelle_dag] : (((binDag_Mirabelle_Dag @ (sK0 @ X4 @ X2 @ X1) @ X1 @ X2 @ X4) = $true) | ($true != (binDag786255756subdag @ X3 @ X4)) | ($true != (binDag_Mirabelle_Dag @ X0 @ X1 @ X2 @ X3)))),
% 2.98/1.87 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0])],[f804,f811])).
% 2.98/1.87 thf(f811,plain,(
% 2.98/1.87 ! [X1 : simpl_ref > simpl_ref,X2 : simpl_ref > simpl_ref,X4 : binDag_Mirabelle_dag] : (? [X5 : simpl_ref] : ((binDag_Mirabelle_Dag @ X5 @ X1 @ X2 @ X4) = $true) => ((binDag_Mirabelle_Dag @ (sK0 @ X4 @ X2 @ X1) @ X1 @ X2 @ X4) = $true))),
% 2.98/1.87 introduced(definition,[],[choice_axiom])).
% 2.98/1.87 thf(f804,plain,(
% 2.98/1.87 ! [X0 : simpl_ref,X1 : simpl_ref > simpl_ref,X2 : simpl_ref > simpl_ref,X3 : binDag_Mirabelle_dag,X4 : binDag_Mirabelle_dag] : (? [X5 : simpl_ref] : ((binDag_Mirabelle_Dag @ X5 @ X1 @ X2 @ X4) = $true) | ($true != (binDag786255756subdag @ X3 @ X4)) | ($true != (binDag_Mirabelle_Dag @ X0 @ X1 @ X2 @ X3)))),
% 2.98/1.87 inference(flattening,[status(thm)],[f803])).
% 2.98/1.87 thf(f803,plain,(
% 2.98/1.87 ! [X0 : simpl_ref,X1 : simpl_ref > simpl_ref,X2 : simpl_ref > simpl_ref,X3 : binDag_Mirabelle_dag,X4 : binDag_Mirabelle_dag] : ((? [X5 : simpl_ref] : ((binDag_Mirabelle_Dag @ X5 @ X1 @ X2 @ X4) = $true) | ($true != (binDag786255756subdag @ X3 @ X4))) | ($true != (binDag_Mirabelle_Dag @ X0 @ X1 @ X2 @ X3)))),
% 2.98/1.87 inference(ennf_transformation,[status(thm)],[f434])).
% 2.98/1.87 thf(f434,plain,(
% 2.98/1.87 ! [X0 : simpl_ref,X1 : simpl_ref > simpl_ref,X2 : simpl_ref > simpl_ref,X3 : binDag_Mirabelle_dag,X4 : binDag_Mirabelle_dag] : (($true = (binDag_Mirabelle_Dag @ X0 @ X1 @ X2 @ X3)) => (($true = (binDag786255756subdag @ X3 @ X4)) => ? [X5 : simpl_ref] : ((binDag_Mirabelle_Dag @ X5 @ X1 @ X2 @ X4) = $true)))),
% 2.98/1.87 inference(fool_elimination,[status(thm)],[f433])).
% 2.98/1.87 thf(f433,plain,(
% 2.98/1.87 ! [X0 : simpl_ref,X1 : simpl_ref > simpl_ref,X2 : simpl_ref > simpl_ref,X3 : binDag_Mirabelle_dag,X4 : binDag_Mirabelle_dag] : ((binDag_Mirabelle_Dag @ X0 @ X1 @ X2 @ X3) => ((binDag786255756subdag @ X3 @ X4) => ? [X5 : simpl_ref] : (binDag_Mirabelle_Dag @ X5 @ X1 @ X2 @ X4)))),
% 2.98/1.87 inference(rectify,[status(thm)],[f9])).
% 2.98/1.87 thf(f9,axiom,(
% 2.98/1.87 ! [X10 : simpl_ref,X11 : simpl_ref > simpl_ref,X12 : simpl_ref > simpl_ref,X15 : binDag_Mirabelle_dag,X21 : binDag_Mirabelle_dag] : ((binDag_Mirabelle_Dag @ X10 @ X11 @ X12 @ X15) => ((binDag786255756subdag @ X15 @ X21) => ? [X22 : simpl_ref] : (binDag_Mirabelle_Dag @ X22 @ X11 @ X12 @ X21)))),
% 2.98/1.87 file('/export/starexec/sandbox/tmp/tmp.CLaN6L8RqG/DTF2THF_22996.p',fact_8_Dag__subdag)).
% 2.98/1.87 thf(f882,plain,(
% 2.98/1.87 ( ! [X0 : simpl_ref] : (($true != (binDag_Mirabelle_Dag @ X0 @ l @ r @ (binDag476092410e_Node @ lt @ p @ rt)))) )),
% 2.98/1.87 inference(subsumption_resolution,[status(thm)],[f846,f858])).
% 2.98/1.87 thf(f858,plain,(
% 2.98/1.87 (thesis != $true)),
% 2.98/1.87 inference(cnf_transformation,[status(thm)],[f787])).
% 2.98/1.87 thf(f787,plain,(
% 2.98/1.87 (thesis != $true)),
% 2.98/1.87 inference(flattening,[status(thm)],[f746])).
% 2.98/1.87 thf(f746,plain,(
% 2.98/1.87 ~(thesis = $true)),
% 2.98/1.87 inference(fool_elimination,[status(thm)],[f745])).
% 2.98/1.87 thf(f745,plain,(
% 2.98/1.87 ~thesis),
% 2.98/1.87 inference(rectify,[status(thm)],[f281])).
% 2.98/1.87 thf(f281,negated_conjecture,(
% 2.98/1.87 ~thesis),
% 2.98/1.87 inference(negated_conjecture,[status(cth)],[f280])).
% 2.98/1.87 thf(f280,conjecture,(
% 2.98/1.87 thesis),
% 2.98/1.87 file('/export/starexec/sandbox/tmp/tmp.CLaN6L8RqG/DTF2THF_22996.p',conj_1)).
% 2.98/1.87 thf(f846,plain,(
% 2.98/1.87 ( ! [X0 : simpl_ref] : ((thesis = $true) | ($true != (binDag_Mirabelle_Dag @ X0 @ l @ r @ (binDag476092410e_Node @ lt @ p @ rt)))) )),
% 2.98/1.87 inference(cnf_transformation,[status(thm)],[f800])).
% 2.98/1.87 thf(f800,plain,(
% 2.98/1.87 ! [X0 : simpl_ref] : ((thesis = $true) | ($true != (binDag_Mirabelle_Dag @ X0 @ l @ r @ (binDag476092410e_Node @ lt @ p @ rt))))),
% 2.98/1.87 inference(ennf_transformation,[status(thm)],[f686])).
% 2.98/1.87 thf(f686,plain,(
% 2.98/1.87 ! [X0 : simpl_ref] : (($true = (binDag_Mirabelle_Dag @ X0 @ l @ r @ (binDag476092410e_Node @ lt @ p @ rt))) => (thesis = $true))),
% 2.98/1.87 inference(fool_elimination,[status(thm)],[f685])).
% 2.98/1.87 thf(f685,plain,(
% 2.98/1.87 ! [X0 : simpl_ref] : ((binDag_Mirabelle_Dag @ X0 @ l @ r @ (binDag476092410e_Node @ lt @ p @ rt)) => thesis)),
% 2.98/1.87 inference(rectify,[status(thm)],[f279])).
% 2.98/1.87 thf(f279,axiom,(
% 2.98/1.87 ! [X83 : simpl_ref] : ((binDag_Mirabelle_Dag @ X83 @ l @ r @ (binDag476092410e_Node @ lt @ p @ rt)) => thesis)),
% 2.98/1.87 file('/export/starexec/sandbox/tmp/tmp.CLaN6L8RqG/DTF2THF_22996.p',conj_0)).
% 2.98/1.87 % SZS output end Proof for DTF2THF_22996
% 2.98/1.87 % (23159)------------------------------
% 2.98/1.87 % (23159)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.87 % (23159)Termination reason: Refutation
% 2.98/1.87
% 2.98/1.87 % (23159)Memory used [KB]: 6396
% 2.98/1.87 % (23159)Time elapsed: 0.025 s
% 2.98/1.87 % (23159)Instructions burned: 44 (million)
% 2.98/1.87 % (23159)------------------------------
% 2.98/1.87 % (23159)------------------------------
% 2.98/1.87 % (23153)Success in time 0.053 s
%------------------------------------------------------------------------------