TPTP Problem File: COM152^1.p
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%------------------------------------------------------------------------------
% File : COM152^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Abstract completeness 27
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [BPT14] Blanchette et al. (2014), Abstract Completeness
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : abstract_completeness__27.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.1.0
% Syntax : Number of formulae : 333 ( 110 unt; 62 typ; 0 def)
% Number of atoms : 945 ( 261 equ; 4 cnn)
% Maximal formula atoms : 15 ( 3 avg)
% Number of connectives : 5222 ( 52 ~; 5 |; 44 &;4825 @)
% ( 0 <=>; 296 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 10 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 1173 (1173 >; 0 *; 0 +; 0 <<)
% Number of symbols : 65 ( 61 usr; 5 con; 0-8 aty)
% Number of variables : 1504 ( 198 ^;1207 !; 11 ?;1504 :)
% ( 88 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:53:12.254
%------------------------------------------------------------------------------
%----Could-be-implicit typings (6)
thf(ty_t_Abstract__Completeness__Mirabelle__wdxnrclvrt_Otree,type,
abstra2103299360e_tree: $tType > $tType ).
thf(ty_t_Sum__Type_Osum,type,
sum_sum: $tType > $tType > $tType ).
thf(ty_t_Stream_Ostream,type,
stream: $tType > $tType ).
thf(ty_t_FSet_Ofset,type,
fset: $tType > $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (56)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_Oipath,type,
abstra313004635_ipath:
!>[A: $tType] : ( ( abstra2103299360e_tree @ A ) > ( stream @ A ) > $o ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_Otfinite,type,
abstra668420080finite:
!>[A: $tType] : ( ( abstra2103299360e_tree @ A ) > $o ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_Otree_ONode,type,
abstra388494275e_Node:
!>[A: $tType] : ( A > ( fset @ ( abstra2103299360e_tree @ A ) ) > ( abstra2103299360e_tree @ A ) ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_Otree_Ocase__tree,type,
abstra457966479e_tree:
!>[A: $tType,B: $tType] : ( ( A > ( fset @ ( abstra2103299360e_tree @ A ) ) > B ) > ( abstra2103299360e_tree @ A ) > B ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_Otree_Ocont,type,
abstra1749095923e_cont:
!>[A: $tType] : ( ( abstra2103299360e_tree @ A ) > ( fset @ ( abstra2103299360e_tree @ A ) ) ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_Otree_Ocorec__tree,type,
abstra1151671297c_tree:
!>[C: $tType,A: $tType] : ( ( C > A ) > ( C > ( fset @ ( sum_sum @ ( abstra2103299360e_tree @ A ) @ C ) ) ) > C > ( abstra2103299360e_tree @ A ) ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_Otree_Opred__tree,type,
abstra1615255520d_tree:
!>[A: $tType] : ( ( A > $o ) > ( abstra2103299360e_tree @ A ) > $o ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_Otree_Orel__tree,type,
abstra2101783510l_tree:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( abstra2103299360e_tree @ A ) > ( abstra2103299360e_tree @ B ) > $o ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_Otree_Oroot,type,
abstra573067619e_root:
!>[A: $tType] : ( ( abstra2103299360e_tree @ A ) > A ) ).
thf(sy_c_BNF__Def_Oeq__onp,type,
bNF_eq_onp:
!>[A: $tType] : ( ( A > $o ) > A > A > $o ) ).
thf(sy_c_BNF__Def_Orel__fun,type,
bNF_rel_fun:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( A > B ) > ( C > D ) > $o ) ).
thf(sy_c_BNF__Def_Orel__sum,type,
bNF_rel_sum:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( sum_sum @ A @ B ) > ( sum_sum @ C @ D ) > $o ) ).
thf(sy_c_Basic__BNFs_Opred__sum,type,
basic_pred_sum:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( sum_sum @ A @ B ) > $o ) ).
thf(sy_c_Complete__Partial__Order_Omonotone,type,
comple1396247847notone:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > B > $o ) > ( A > B ) > $o ) ).
thf(sy_c_FSet_OfBall,type,
fBall:
!>[A: $tType] : ( ( fset @ A ) > ( A > $o ) > $o ) ).
thf(sy_c_FSet_OfBex,type,
fBex:
!>[A: $tType] : ( ( fset @ A ) > ( A > $o ) > $o ) ).
thf(sy_c_FSet_Ofbind,type,
fbind:
!>[A: $tType,B: $tType] : ( ( fset @ A ) > ( A > ( fset @ B ) ) > ( fset @ B ) ) ).
thf(sy_c_FSet_Offilter,type,
ffilter:
!>[A: $tType] : ( ( A > $o ) > ( fset @ A ) > ( fset @ A ) ) ).
thf(sy_c_FSet_Ofimage,type,
fimage:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( fset @ A ) > ( fset @ B ) ) ).
thf(sy_c_FSet_Ofinsert,type,
finsert:
!>[A: $tType] : ( A > ( fset @ A ) > ( fset @ A ) ) ).
thf(sy_c_FSet_Ofmember,type,
fmember:
!>[A: $tType] : ( A > ( fset @ A ) > $o ) ).
thf(sy_c_FSet_Ofset_OFSet_Opred__fset,type,
pred_fset:
!>[A: $tType] : ( ( A > $o ) > ( fset @ A ) > $o ) ).
thf(sy_c_FSet_Orel__fset,type,
rel_fset:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( fset @ A ) > ( fset @ B ) > $o ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Fun_Ofun__upd,type,
fun_upd:
!>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
thf(sy_c_Fun_Omap__fun,type,
map_fun:
!>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( C > A ) > ( B > D ) > ( A > B ) > C > D ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Linear__Temporal__Logic__on__Streams_Onxt,type,
linear1494993505on_nxt:
!>[A: $tType,B: $tType] : ( ( ( stream @ A ) > B ) > ( stream @ A ) > B ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Partial__Function_Ofun__ord,type,
partial_fun_ord:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( C > A ) > ( C > B ) > $o ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Quotient_OBex1__rel,type,
bex1_rel:
!>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > $o ) ).
thf(sy_c_Quotient_OQuotient3,type,
quotient3:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > $o ) ).
thf(sy_c_Stream_Osdrop__while,type,
sdrop_while:
!>[A: $tType] : ( ( A > $o ) > ( stream @ A ) > ( stream @ A ) ) ).
thf(sy_c_Stream_Osfilter,type,
sfilter:
!>[A: $tType] : ( ( A > $o ) > ( stream @ A ) > ( stream @ A ) ) ).
thf(sy_c_Stream_Osinterleave,type,
sinterleave:
!>[A: $tType] : ( ( stream @ A ) > ( stream @ A ) > ( stream @ A ) ) ).
thf(sy_c_Stream_Osmap2,type,
smap2:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( stream @ A ) > ( stream @ B ) > ( stream @ C ) ) ).
thf(sy_c_Stream_Osmember,type,
smember:
!>[A: $tType] : ( A > ( stream @ A ) > $o ) ).
thf(sy_c_Stream_Ostream_OSCons,type,
sCons:
!>[A: $tType] : ( A > ( stream @ A ) > ( stream @ A ) ) ).
thf(sy_c_Stream_Ostream_Ocase__stream,type,
case_stream:
!>[A: $tType,B: $tType] : ( ( A > ( stream @ A ) > B ) > ( stream @ A ) > B ) ).
thf(sy_c_Stream_Ostream_Ocorec__stream,type,
corec_stream:
!>[C: $tType,A: $tType] : ( ( C > A ) > ( C > $o ) > ( C > ( stream @ A ) ) > ( C > C ) > C > ( stream @ A ) ) ).
thf(sy_c_Stream_Ostream_Opred__stream,type,
pred_stream:
!>[A: $tType] : ( ( A > $o ) > ( stream @ A ) > $o ) ).
thf(sy_c_Stream_Ostream_Oshd,type,
shd:
!>[A: $tType] : ( ( stream @ A ) > A ) ).
thf(sy_c_Stream_Ostream_Osmap,type,
smap:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( stream @ A ) > ( stream @ Aa ) ) ).
thf(sy_c_Stream_Ostream_Ostl,type,
stl:
!>[A: $tType] : ( ( stream @ A ) > ( stream @ A ) ) ).
thf(sy_c_Stream_Ostream_Ostream__all2,type,
stream_all2:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( stream @ A ) > ( stream @ B ) > $o ) ).
thf(sy_c_Sum__Type_Osum_Ocase__sum,type,
sum_case_sum:
!>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( B > C ) > ( sum_sum @ A @ B ) > C ) ).
thf(sy_c_Transfer_Obi__total,type,
bi_total:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Transfer_Obi__unique,type,
bi_unique:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_v_steps,type,
steps: stream @ a ).
thf(sy_v_t,type,
t: abstra2103299360e_tree @ a ).
%----Relevant facts (256)
thf(fact_0_tfinite_Ocases,axiom,
! [A: $tType,A2: abstra2103299360e_tree @ A] :
( ( abstra668420080finite @ A @ A2 )
=> ! [T: abstra2103299360e_tree @ A] :
( ( fmember @ ( abstra2103299360e_tree @ A ) @ T @ ( abstra1749095923e_cont @ A @ A2 ) )
=> ( abstra668420080finite @ A @ T ) ) ) ).
% tfinite.cases
thf(fact_1_tfinite_Osimps,axiom,
! [A: $tType] :
( ( abstra668420080finite @ A )
= ( ^ [A3: abstra2103299360e_tree @ A] :
? [T2: abstra2103299360e_tree @ A] :
( ( A3 = T2 )
& ! [X: abstra2103299360e_tree @ A] :
( ( fmember @ ( abstra2103299360e_tree @ A ) @ X @ ( abstra1749095923e_cont @ A @ T2 ) )
=> ( abstra668420080finite @ A @ X ) ) ) ) ) ).
% tfinite.simps
thf(fact_2_tfinite_Oinducts,axiom,
! [A: $tType,X2: abstra2103299360e_tree @ A,P: ( abstra2103299360e_tree @ A ) > $o] :
( ( abstra668420080finite @ A @ X2 )
=> ( ! [T3: abstra2103299360e_tree @ A] :
( ! [T: abstra2103299360e_tree @ A] :
( ( fmember @ ( abstra2103299360e_tree @ A ) @ T @ ( abstra1749095923e_cont @ A @ T3 ) )
=> ( abstra668420080finite @ A @ T ) )
=> ( ! [T: abstra2103299360e_tree @ A] :
( ( fmember @ ( abstra2103299360e_tree @ A ) @ T @ ( abstra1749095923e_cont @ A @ T3 ) )
=> ( P @ T ) )
=> ( P @ T3 ) ) )
=> ( P @ X2 ) ) ) ).
% tfinite.inducts
thf(fact_3_tfinite,axiom,
! [A: $tType,T4: abstra2103299360e_tree @ A] :
( ! [T5: abstra2103299360e_tree @ A] :
( ( fmember @ ( abstra2103299360e_tree @ A ) @ T5 @ ( abstra1749095923e_cont @ A @ T4 ) )
=> ( abstra668420080finite @ A @ T5 ) )
=> ( abstra668420080finite @ A @ T4 ) ) ).
% tfinite
thf(fact_4_tree_Ocorec__disc,axiom,
! [A: $tType,C: $tType] :
( ( abstra1151671297c_tree @ C @ A )
= ( abstra1151671297c_tree @ C @ A ) ) ).
% tree.corec_disc
thf(fact_5_tree_Oinject,axiom,
! [A: $tType,X1: A,X22: fset @ ( abstra2103299360e_tree @ A ),Y1: A,Y2: fset @ ( abstra2103299360e_tree @ A )] :
( ( ( abstra388494275e_Node @ A @ X1 @ X22 )
= ( abstra388494275e_Node @ A @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% tree.inject
thf(fact_6_tree_Osel_I2_J,axiom,
! [A: $tType,X1: A,X22: fset @ ( abstra2103299360e_tree @ A )] :
( ( abstra1749095923e_cont @ A @ ( abstra388494275e_Node @ A @ X1 @ X22 ) )
= X22 ) ).
% tree.sel(2)
thf(fact_7_tree_Oexhaust,axiom,
! [A: $tType,Y: abstra2103299360e_tree @ A] :
~ ! [X12: A,X23: fset @ ( abstra2103299360e_tree @ A )] :
( Y
!= ( abstra388494275e_Node @ A @ X12 @ X23 ) ) ).
% tree.exhaust
thf(fact_8_tree_Ocollapse,axiom,
! [A: $tType,Tree: abstra2103299360e_tree @ A] :
( ( abstra388494275e_Node @ A @ ( abstra573067619e_root @ A @ Tree ) @ ( abstra1749095923e_cont @ A @ Tree ) )
= Tree ) ).
% tree.collapse
thf(fact_9_fset__eqI,axiom,
! [A: $tType,A4: fset @ A,B2: fset @ A] :
( ! [X3: A] :
( ( fmember @ A @ X3 @ A4 )
= ( fmember @ A @ X3 @ B2 ) )
=> ( A4 = B2 ) ) ).
% fset_eqI
thf(fact_10_fequalityCE,axiom,
! [A: $tType,A4: fset @ A,B2: fset @ A,C2: A] :
( ( A4 = B2 )
=> ( ( ( fmember @ A @ C2 @ A4 )
=> ~ ( fmember @ A @ C2 @ B2 ) )
=> ~ ( ~ ( fmember @ A @ C2 @ A4 )
=> ( fmember @ A @ C2 @ B2 ) ) ) ) ).
% fequalityCE
thf(fact_11_fset__choice,axiom,
! [B: $tType,A: $tType,A4: fset @ A,P: A > B > $o] :
( ! [X3: A] :
( ( fmember @ A @ X3 @ A4 )
=> ? [X13: B] : ( P @ X3 @ X13 ) )
=> ? [F: A > B] :
! [X4: A] :
( ( fmember @ A @ X4 @ A4 )
=> ( P @ X4 @ ( F @ X4 ) ) ) ) ).
% fset_choice
thf(fact_12_eq__fmem__trans,axiom,
! [A: $tType,A2: A,B3: A,A4: fset @ A] :
( ( A2 = B3 )
=> ( ( fmember @ A @ B3 @ A4 )
=> ( fmember @ A @ A2 @ A4 ) ) ) ).
% eq_fmem_trans
thf(fact_13_eqfset__imp__iff,axiom,
! [A: $tType,A4: fset @ A,B2: fset @ A,X2: A] :
( ( A4 = B2 )
=> ( ( fmember @ A @ X2 @ A4 )
= ( fmember @ A @ X2 @ B2 ) ) ) ).
% eqfset_imp_iff
thf(fact_14_if__split__fmem1,axiom,
! [A: $tType,Q: $o,X2: A,Y: A,B3: fset @ A] :
( ( fmember @ A @ ( if @ A @ Q @ X2 @ Y ) @ B3 )
= ( ( Q
=> ( fmember @ A @ X2 @ B3 ) )
& ( ~ Q
=> ( fmember @ A @ Y @ B3 ) ) ) ) ).
% if_split_fmem1
thf(fact_15_if__split__fmem2,axiom,
! [A: $tType,A2: A,Q: $o,X2: fset @ A,Y: fset @ A] :
( ( fmember @ A @ A2 @ ( if @ ( fset @ A ) @ Q @ X2 @ Y ) )
= ( ( Q
=> ( fmember @ A @ A2 @ X2 ) )
& ( ~ Q
=> ( fmember @ A @ A2 @ Y ) ) ) ) ).
% if_split_fmem2
thf(fact_16_eqfelem__imp__iff,axiom,
! [A: $tType,X2: A,Y: A,A4: fset @ A] :
( ( X2 = Y )
=> ( ( fmember @ A @ X2 @ A4 )
= ( fmember @ A @ Y @ A4 ) ) ) ).
% eqfelem_imp_iff
thf(fact_17_tree_Oexhaust__sel,axiom,
! [A: $tType,Tree: abstra2103299360e_tree @ A] :
( Tree
= ( abstra388494275e_Node @ A @ ( abstra573067619e_root @ A @ Tree ) @ ( abstra1749095923e_cont @ A @ Tree ) ) ) ).
% tree.exhaust_sel
thf(fact_18_tree_Ocase,axiom,
! [B: $tType,A: $tType,F2: A > ( fset @ ( abstra2103299360e_tree @ A ) ) > B,X1: A,X22: fset @ ( abstra2103299360e_tree @ A )] :
( ( abstra457966479e_tree @ A @ B @ F2 @ ( abstra388494275e_Node @ A @ X1 @ X22 ) )
= ( F2 @ X1 @ X22 ) ) ).
% tree.case
thf(fact_19_tree_Ocase__eq__if,axiom,
! [B: $tType,A: $tType] :
( ( abstra457966479e_tree @ A @ B )
= ( ^ [F3: A > ( fset @ ( abstra2103299360e_tree @ A ) ) > B,Tree2: abstra2103299360e_tree @ A] : ( F3 @ ( abstra573067619e_root @ A @ Tree2 ) @ ( abstra1749095923e_cont @ A @ Tree2 ) ) ) ) ).
% tree.case_eq_if
thf(fact_20_tree_Oexpand,axiom,
! [A: $tType,Tree: abstra2103299360e_tree @ A,Tree3: abstra2103299360e_tree @ A] :
( ( ( ( abstra573067619e_root @ A @ Tree )
= ( abstra573067619e_root @ A @ Tree3 ) )
& ( ( abstra1749095923e_cont @ A @ Tree )
= ( abstra1749095923e_cont @ A @ Tree3 ) ) )
=> ( Tree = Tree3 ) ) ).
% tree.expand
thf(fact_21_tree_Osel_I1_J,axiom,
! [A: $tType,X1: A,X22: fset @ ( abstra2103299360e_tree @ A )] :
( ( abstra573067619e_root @ A @ ( abstra388494275e_Node @ A @ X1 @ X22 ) )
= X1 ) ).
% tree.sel(1)
thf(fact_22_tree_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F2: A > ( fset @ ( abstra2103299360e_tree @ A ) ) > B,Tree: abstra2103299360e_tree @ A] :
( ( P @ ( abstra457966479e_tree @ A @ B @ F2 @ Tree ) )
= ( ~ ( ( Tree
= ( abstra388494275e_Node @ A @ ( abstra573067619e_root @ A @ Tree ) @ ( abstra1749095923e_cont @ A @ Tree ) ) )
& ~ ( P @ ( F2 @ ( abstra573067619e_root @ A @ Tree ) @ ( abstra1749095923e_cont @ A @ Tree ) ) ) ) ) ) ).
% tree.split_sel_asm
thf(fact_23_tree_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F2: A > ( fset @ ( abstra2103299360e_tree @ A ) ) > B,Tree: abstra2103299360e_tree @ A] :
( ( P @ ( abstra457966479e_tree @ A @ B @ F2 @ Tree ) )
= ( ( Tree
= ( abstra388494275e_Node @ A @ ( abstra573067619e_root @ A @ Tree ) @ ( abstra1749095923e_cont @ A @ Tree ) ) )
=> ( P @ ( F2 @ ( abstra573067619e_root @ A @ Tree ) @ ( abstra1749095923e_cont @ A @ Tree ) ) ) ) ) ).
% tree.split_sel
thf(fact_24_tree_Ocorec__sel_I1_J,axiom,
! [A: $tType,C: $tType,G1: C > A,G2: C > ( fset @ ( sum_sum @ ( abstra2103299360e_tree @ A ) @ C ) ),A2: C] :
( ( abstra573067619e_root @ A @ ( abstra1151671297c_tree @ C @ A @ G1 @ G2 @ A2 ) )
= ( G1 @ A2 ) ) ).
% tree.corec_sel(1)
thf(fact_25_ffmember__filter,axiom,
! [A: $tType,X2: A,P: A > $o,A4: fset @ A] :
( ( fmember @ A @ X2 @ ( ffilter @ A @ P @ A4 ) )
= ( ( fmember @ A @ X2 @ A4 )
& ( P @ X2 ) ) ) ).
% ffmember_filter
thf(fact_26_ipath_Ocoinduct,axiom,
! [A: $tType,X5: ( abstra2103299360e_tree @ A ) > ( stream @ A ) > $o,X2: abstra2103299360e_tree @ A,Xa: stream @ A] :
( ( X5 @ X2 @ Xa )
=> ( ! [X3: abstra2103299360e_tree @ A,Xa2: stream @ A] :
( ( X5 @ X3 @ Xa2 )
=> ? [T6: abstra2103299360e_tree @ A,Steps: stream @ A,T: abstra2103299360e_tree @ A] :
( ( X3 = T6 )
& ( Xa2 = Steps )
& ( ( abstra573067619e_root @ A @ T6 )
= ( shd @ A @ Steps ) )
& ( fmember @ ( abstra2103299360e_tree @ A ) @ T @ ( abstra1749095923e_cont @ A @ T6 ) )
& ( ( X5 @ T @ ( stl @ A @ Steps ) )
| ( abstra313004635_ipath @ A @ T @ ( stl @ A @ Steps ) ) ) ) )
=> ( abstra313004635_ipath @ A @ X2 @ Xa ) ) ) ).
% ipath.coinduct
thf(fact_27_ipath_Ointros,axiom,
! [A: $tType,T4: abstra2103299360e_tree @ A,Steps2: stream @ A,T7: abstra2103299360e_tree @ A] :
( ( ( abstra573067619e_root @ A @ T4 )
= ( shd @ A @ Steps2 ) )
=> ( ( fmember @ ( abstra2103299360e_tree @ A ) @ T7 @ ( abstra1749095923e_cont @ A @ T4 ) )
=> ( ( abstra313004635_ipath @ A @ T7 @ ( stl @ A @ Steps2 ) )
=> ( abstra313004635_ipath @ A @ T4 @ Steps2 ) ) ) ) ).
% ipath.intros
thf(fact_28_ipath_Osimps,axiom,
! [A: $tType] :
( ( abstra313004635_ipath @ A )
= ( ^ [A1: abstra2103299360e_tree @ A,A22: stream @ A] :
? [T2: abstra2103299360e_tree @ A,Steps3: stream @ A,T8: abstra2103299360e_tree @ A] :
( ( A1 = T2 )
& ( A22 = Steps3 )
& ( ( abstra573067619e_root @ A @ T2 )
= ( shd @ A @ Steps3 ) )
& ( fmember @ ( abstra2103299360e_tree @ A ) @ T8 @ ( abstra1749095923e_cont @ A @ T2 ) )
& ( abstra313004635_ipath @ A @ T8 @ ( stl @ A @ Steps3 ) ) ) ) ) ).
% ipath.simps
thf(fact_29_ipath_Ocases,axiom,
! [A: $tType,A12: abstra2103299360e_tree @ A,A23: stream @ A] :
( ( abstra313004635_ipath @ A @ A12 @ A23 )
=> ~ ( ( ( abstra573067619e_root @ A @ A12 )
= ( shd @ A @ A23 ) )
=> ! [T5: abstra2103299360e_tree @ A] :
( ( fmember @ ( abstra2103299360e_tree @ A ) @ T5 @ ( abstra1749095923e_cont @ A @ A12 ) )
=> ~ ( abstra313004635_ipath @ A @ T5 @ ( stl @ A @ A23 ) ) ) ) ) ).
% ipath.cases
thf(fact_30_tree_Ocoinduct,axiom,
! [A: $tType,R: ( abstra2103299360e_tree @ A ) > ( abstra2103299360e_tree @ A ) > $o,Tree: abstra2103299360e_tree @ A,Tree3: abstra2103299360e_tree @ A] :
( ( R @ Tree @ Tree3 )
=> ( ! [Tree4: abstra2103299360e_tree @ A,Tree5: abstra2103299360e_tree @ A] :
( ( R @ Tree4 @ Tree5 )
=> ( ( ( abstra573067619e_root @ A @ Tree4 )
= ( abstra573067619e_root @ A @ Tree5 ) )
& ( rel_fset @ ( abstra2103299360e_tree @ A ) @ ( abstra2103299360e_tree @ A ) @ R @ ( abstra1749095923e_cont @ A @ Tree4 ) @ ( abstra1749095923e_cont @ A @ Tree5 ) ) ) )
=> ( Tree = Tree3 ) ) ) ).
% tree.coinduct
thf(fact_31_fBallI,axiom,
! [A: $tType,A4: fset @ A,P: A > $o] :
( ! [X3: A] :
( ( fmember @ A @ X3 @ A4 )
=> ( P @ X3 ) )
=> ( fBall @ A @ A4 @ P ) ) ).
% fBallI
thf(fact_32_fBex__triv__one__point2,axiom,
! [A: $tType,A4: fset @ A,A2: A] :
( ( fBex @ A @ A4
@ ( ^ [Y3: A,Z: A] : Y3 = Z
@ A2 ) )
= ( fmember @ A @ A2 @ A4 ) ) ).
% fBex_triv_one_point2
thf(fact_33_fBexI,axiom,
! [A: $tType,P: A > $o,X2: A,A4: fset @ A] :
( ( P @ X2 )
=> ( ( fmember @ A @ X2 @ A4 )
=> ( fBex @ A @ A4 @ P ) ) ) ).
% fBexI
thf(fact_34_eq__ffilter,axiom,
! [A: $tType,P: A > $o,A4: fset @ A,Q: A > $o] :
( ( ( ffilter @ A @ P @ A4 )
= ( ffilter @ A @ Q @ A4 ) )
= ( ! [X: A] :
( ( fmember @ A @ X @ A4 )
=> ( ( P @ X )
= ( Q @ X ) ) ) ) ) ).
% eq_ffilter
thf(fact_35_fset_Orel__refl,axiom,
! [B: $tType,Ra: B > B > $o,X2: fset @ B] :
( ! [X3: B] : ( Ra @ X3 @ X3 )
=> ( rel_fset @ B @ B @ Ra @ X2 @ X2 ) ) ).
% fset.rel_refl
thf(fact_36_fset_Orel__eq,axiom,
! [A: $tType] :
( ( rel_fset @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [Y3: fset @ A,Z: fset @ A] : Y3 = Z ) ) ).
% fset.rel_eq
thf(fact_37_fBexE,axiom,
! [A: $tType,A4: fset @ A,P: A > $o] :
( ( fBex @ A @ A4 @ P )
=> ~ ! [X3: A] :
( ( fmember @ A @ X3 @ A4 )
=> ~ ( P @ X3 ) ) ) ).
% fBexE
thf(fact_38_fBex__cong,axiom,
! [A: $tType,A4: fset @ A,B2: fset @ A,P: A > $o,Q: A > $o] :
( ( A4 = B2 )
=> ( ! [X3: A] :
( ( fmember @ A @ X3 @ B2 )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( fBex @ A @ A4 @ P )
= ( fBex @ A @ B2 @ Q ) ) ) ) ).
% fBex_cong
thf(fact_39_rev__fBexI,axiom,
! [A: $tType,X2: A,A4: fset @ A,P: A > $o] :
( ( fmember @ A @ X2 @ A4 )
=> ( ( P @ X2 )
=> ( fBex @ A @ A4 @ P ) ) ) ).
% rev_fBexI
thf(fact_40_fBallE,axiom,
! [A: $tType,A4: fset @ A,P: A > $o,X2: A] :
( ( fBall @ A @ A4 @ P )
=> ( ~ ( P @ X2 )
=> ~ ( fmember @ A @ X2 @ A4 ) ) ) ).
% fBallE
thf(fact_41_fbspec,axiom,
! [A: $tType,A4: fset @ A,P: A > $o,X2: A] :
( ( fBall @ A @ A4 @ P )
=> ( ( fmember @ A @ X2 @ A4 )
=> ( P @ X2 ) ) ) ).
% fbspec
thf(fact_42_fBall__cong,axiom,
! [A: $tType,A4: fset @ A,B2: fset @ A,P: A > $o,Q: A > $o] :
( ( A4 = B2 )
=> ( ! [X3: A] :
( ( fmember @ A @ X3 @ B2 )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( fBall @ A @ A4 @ P )
= ( fBall @ A @ B2 @ Q ) ) ) ) ).
% fBall_cong
thf(fact_43_stream_Oexpand,axiom,
! [A: $tType,Stream: stream @ A,Stream2: stream @ A] :
( ( ( ( shd @ A @ Stream )
= ( shd @ A @ Stream2 ) )
& ( ( stl @ A @ Stream )
= ( stl @ A @ Stream2 ) ) )
=> ( Stream = Stream2 ) ) ).
% stream.expand
thf(fact_44_stream_Ocoinduct,axiom,
! [A: $tType,R: ( stream @ A ) > ( stream @ A ) > $o,Stream: stream @ A,Stream2: stream @ A] :
( ( R @ Stream @ Stream2 )
=> ( ! [Stream3: stream @ A,Stream4: stream @ A] :
( ( R @ Stream3 @ Stream4 )
=> ( ( ( shd @ A @ Stream3 )
= ( shd @ A @ Stream4 ) )
& ( R @ ( stl @ A @ Stream3 ) @ ( stl @ A @ Stream4 ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct
thf(fact_45_ext,axiom,
! [B: $tType,A: $tType,F2: A > B,G: A > B] :
( ! [X3: A] :
( ( F2 @ X3 )
= ( G @ X3 ) )
=> ( F2 = G ) ) ).
% ext
thf(fact_46_stream_Ocoinduct__strong,axiom,
! [A: $tType,R: ( stream @ A ) > ( stream @ A ) > $o,Stream: stream @ A,Stream2: stream @ A] :
( ( R @ Stream @ Stream2 )
=> ( ! [Stream3: stream @ A,Stream4: stream @ A] :
( ( R @ Stream3 @ Stream4 )
=> ( ( ( shd @ A @ Stream3 )
= ( shd @ A @ Stream4 ) )
& ( ( R @ ( stl @ A @ Stream3 ) @ ( stl @ A @ Stream4 ) )
| ( ( stl @ A @ Stream3 )
= ( stl @ A @ Stream4 ) ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct_strong
thf(fact_47_tree_Orel__coinduct,axiom,
! [A: $tType,B: $tType,P: ( abstra2103299360e_tree @ A ) > ( abstra2103299360e_tree @ B ) > $o,X2: abstra2103299360e_tree @ A,Y: abstra2103299360e_tree @ B,R: A > B > $o] :
( ( P @ X2 @ Y )
=> ( ! [Tree4: abstra2103299360e_tree @ A,Tree5: abstra2103299360e_tree @ B] :
( ( P @ Tree4 @ Tree5 )
=> ( ( R @ ( abstra573067619e_root @ A @ Tree4 ) @ ( abstra573067619e_root @ B @ Tree5 ) )
& ( rel_fset @ ( abstra2103299360e_tree @ A ) @ ( abstra2103299360e_tree @ B ) @ P @ ( abstra1749095923e_cont @ A @ Tree4 ) @ ( abstra1749095923e_cont @ B @ Tree5 ) ) ) )
=> ( abstra2101783510l_tree @ A @ B @ R @ X2 @ Y ) ) ) ).
% tree.rel_coinduct
thf(fact_48_tree_Orel__sel,axiom,
! [B: $tType,A: $tType] :
( ( abstra2101783510l_tree @ A @ B )
= ( ^ [R2: A > B > $o,A3: abstra2103299360e_tree @ A,B4: abstra2103299360e_tree @ B] :
( ( R2 @ ( abstra573067619e_root @ A @ A3 ) @ ( abstra573067619e_root @ B @ B4 ) )
& ( rel_fset @ ( abstra2103299360e_tree @ A ) @ ( abstra2103299360e_tree @ B ) @ ( abstra2101783510l_tree @ A @ B @ R2 ) @ ( abstra1749095923e_cont @ A @ A3 ) @ ( abstra1749095923e_cont @ B @ B4 ) ) ) ) ) ).
% tree.rel_sel
thf(fact_49_tree_Orel__inject,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X1: A,X22: fset @ ( abstra2103299360e_tree @ A ),Y1: B,Y2: fset @ ( abstra2103299360e_tree @ B )] :
( ( abstra2101783510l_tree @ A @ B @ R @ ( abstra388494275e_Node @ A @ X1 @ X22 ) @ ( abstra388494275e_Node @ B @ Y1 @ Y2 ) )
= ( ( R @ X1 @ Y1 )
& ( rel_fset @ ( abstra2103299360e_tree @ A ) @ ( abstra2103299360e_tree @ B ) @ ( abstra2101783510l_tree @ A @ B @ R ) @ X22 @ Y2 ) ) ) ).
% tree.rel_inject
thf(fact_50_tree_Orel__cases,axiom,
! [A: $tType,B: $tType,R: A > B > $o,A2: abstra2103299360e_tree @ A,B3: abstra2103299360e_tree @ B] :
( ( abstra2101783510l_tree @ A @ B @ R @ A2 @ B3 )
=> ~ ! [X1a: A,X2a: fset @ ( abstra2103299360e_tree @ A )] :
( ( A2
= ( abstra388494275e_Node @ A @ X1a @ X2a ) )
=> ! [Y1a: B,Y2a: fset @ ( abstra2103299360e_tree @ B )] :
( ( B3
= ( abstra388494275e_Node @ B @ Y1a @ Y2a ) )
=> ( ( R @ X1a @ Y1a )
=> ~ ( rel_fset @ ( abstra2103299360e_tree @ A ) @ ( abstra2103299360e_tree @ B ) @ ( abstra2101783510l_tree @ A @ B @ R ) @ X2a @ Y2a ) ) ) ) ) ).
% tree.rel_cases
thf(fact_51_tree_Orel__intros,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X1: A,Y1: B,X22: fset @ ( abstra2103299360e_tree @ A ),Y2: fset @ ( abstra2103299360e_tree @ B )] :
( ( R @ X1 @ Y1 )
=> ( ( rel_fset @ ( abstra2103299360e_tree @ A ) @ ( abstra2103299360e_tree @ B ) @ ( abstra2101783510l_tree @ A @ B @ R ) @ X22 @ Y2 )
=> ( abstra2101783510l_tree @ A @ B @ R @ ( abstra388494275e_Node @ A @ X1 @ X22 ) @ ( abstra388494275e_Node @ B @ Y1 @ Y2 ) ) ) ) ).
% tree.rel_intros
thf(fact_52_subset__ffilter,axiom,
! [A: $tType,P: A > $o,A4: fset @ A,Q: A > $o] :
( ( ord_less_eq @ ( fset @ A ) @ ( ffilter @ A @ P @ A4 ) @ ( ffilter @ A @ Q @ A4 ) )
= ( ! [X: A] :
( ( fmember @ A @ X @ A4 )
=> ( ( P @ X )
=> ( Q @ X ) ) ) ) ) ).
% subset_ffilter
thf(fact_53_fsubset__antisym,axiom,
! [A: $tType,A4: fset @ A,B2: fset @ A] :
( ( ord_less_eq @ ( fset @ A ) @ A4 @ B2 )
=> ( ( ord_less_eq @ ( fset @ A ) @ B2 @ A4 )
=> ( A4 = B2 ) ) ) ).
% fsubset_antisym
thf(fact_54_fsubsetI,axiom,
! [A: $tType,A4: fset @ A,B2: fset @ A] :
( ! [X3: A] :
( ( fmember @ A @ X3 @ A4 )
=> ( fmember @ A @ X3 @ B2 ) )
=> ( ord_less_eq @ ( fset @ A ) @ A4 @ B2 ) ) ).
% fsubsetI
thf(fact_55_tree_Orel__refl,axiom,
! [B: $tType,Ra: B > B > $o,X2: abstra2103299360e_tree @ B] :
( ! [X3: B] : ( Ra @ X3 @ X3 )
=> ( abstra2101783510l_tree @ B @ B @ Ra @ X2 @ X2 ) ) ).
% tree.rel_refl
thf(fact_56_tree_Orel__eq,axiom,
! [A: $tType] :
( ( abstra2101783510l_tree @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [Y3: abstra2103299360e_tree @ A,Z: abstra2103299360e_tree @ A] : Y3 = Z ) ) ).
% tree.rel_eq
thf(fact_57_fequalityE,axiom,
! [A: $tType,A4: fset @ A,B2: fset @ A] :
( ( A4 = B2 )
=> ~ ( ( ord_less_eq @ ( fset @ A ) @ A4 @ B2 )
=> ~ ( ord_less_eq @ ( fset @ A ) @ B2 @ A4 ) ) ) ).
% fequalityE
thf(fact_58_fequalityD1,axiom,
! [A: $tType,A4: fset @ A,B2: fset @ A] :
( ( A4 = B2 )
=> ( ord_less_eq @ ( fset @ A ) @ A4 @ B2 ) ) ).
% fequalityD1
thf(fact_59_fequalityD2,axiom,
! [A: $tType,A4: fset @ A,B2: fset @ A] :
( ( A4 = B2 )
=> ( ord_less_eq @ ( fset @ A ) @ B2 @ A4 ) ) ).
% fequalityD2
thf(fact_60_fsubset__refl,axiom,
! [A: $tType,A4: fset @ A] : ( ord_less_eq @ ( fset @ A ) @ A4 @ A4 ) ).
% fsubset_refl
thf(fact_61_fsubset__trans,axiom,
! [A: $tType,A4: fset @ A,B2: fset @ A,C3: fset @ A] :
( ( ord_less_eq @ ( fset @ A ) @ A4 @ B2 )
=> ( ( ord_less_eq @ ( fset @ A ) @ B2 @ C3 )
=> ( ord_less_eq @ ( fset @ A ) @ A4 @ C3 ) ) ) ).
% fsubset_trans
thf(fact_62_fset__eq__fsubset,axiom,
! [A: $tType] :
( ( ^ [Y3: fset @ A,Z: fset @ A] : Y3 = Z )
= ( ^ [A5: fset @ A,B5: fset @ A] :
( ( ord_less_eq @ ( fset @ A ) @ A5 @ B5 )
& ( ord_less_eq @ ( fset @ A ) @ B5 @ A5 ) ) ) ) ).
% fset_eq_fsubset
thf(fact_63_fset__rev__mp,axiom,
! [A: $tType,X2: A,A4: fset @ A,B2: fset @ A] :
( ( fmember @ A @ X2 @ A4 )
=> ( ( ord_less_eq @ ( fset @ A ) @ A4 @ B2 )
=> ( fmember @ A @ X2 @ B2 ) ) ) ).
% fset_rev_mp
thf(fact_64_fsubsetD,axiom,
! [A: $tType,A4: fset @ A,B2: fset @ A,C2: A] :
( ( ord_less_eq @ ( fset @ A ) @ A4 @ B2 )
=> ( ( fmember @ A @ C2 @ A4 )
=> ( fmember @ A @ C2 @ B2 ) ) ) ).
% fsubsetD
thf(fact_65_fin__mono,axiom,
! [A: $tType,A4: fset @ A,B2: fset @ A,X2: A] :
( ( ord_less_eq @ ( fset @ A ) @ A4 @ B2 )
=> ( ( fmember @ A @ X2 @ A4 )
=> ( fmember @ A @ X2 @ B2 ) ) ) ).
% fin_mono
thf(fact_66_fset__mp,axiom,
! [A: $tType,A4: fset @ A,B2: fset @ A,X2: A] :
( ( ord_less_eq @ ( fset @ A ) @ A4 @ B2 )
=> ( ( fmember @ A @ X2 @ A4 )
=> ( fmember @ A @ X2 @ B2 ) ) ) ).
% fset_mp
thf(fact_67_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% order_refl
thf(fact_68_stream_Ocase__eq__if,axiom,
! [B: $tType,A: $tType] :
( ( case_stream @ A @ B )
= ( ^ [F3: A > ( stream @ A ) > B,Stream5: stream @ A] : ( F3 @ ( shd @ A @ Stream5 ) @ ( stl @ A @ Stream5 ) ) ) ) ).
% stream.case_eq_if
thf(fact_69_sdrop__while_Osimps,axiom,
! [A: $tType] :
( ( sdrop_while @ A )
= ( ^ [P2: A > $o,S: stream @ A] : ( if @ ( stream @ A ) @ ( P2 @ ( shd @ A @ S ) ) @ ( sdrop_while @ A @ P2 @ ( stl @ A @ S ) ) @ S ) ) ) ).
% sdrop_while.simps
thf(fact_70_tree_Osel__transfer_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o] : ( bNF_rel_fun @ ( abstra2103299360e_tree @ A ) @ ( abstra2103299360e_tree @ B ) @ ( fset @ ( abstra2103299360e_tree @ A ) ) @ ( fset @ ( abstra2103299360e_tree @ B ) ) @ ( abstra2101783510l_tree @ A @ B @ R ) @ ( rel_fset @ ( abstra2103299360e_tree @ A ) @ ( abstra2103299360e_tree @ B ) @ ( abstra2101783510l_tree @ A @ B @ R ) ) @ ( abstra1749095923e_cont @ A ) @ ( abstra1749095923e_cont @ B ) ) ).
% tree.sel_transfer(2)
thf(fact_71_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_72_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_73_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A2: A,B3: A] :
( ! [A6: A,B6: A] :
( ( ord_less_eq @ A @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: A,B6: A] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A2 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_74_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_75_fset_Orel__mono,axiom,
! [B: $tType,A: $tType,R: A > B > $o,Ra: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R @ Ra )
=> ( ord_less_eq @ ( ( fset @ A ) > ( fset @ B ) > $o ) @ ( rel_fset @ A @ B @ R ) @ ( rel_fset @ A @ B @ Ra ) ) ) ).
% fset.rel_mono
thf(fact_76_tree_Orel__mono,axiom,
! [B: $tType,A: $tType,R: A > B > $o,Ra: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R @ Ra )
=> ( ord_less_eq @ ( ( abstra2103299360e_tree @ A ) > ( abstra2103299360e_tree @ B ) > $o ) @ ( abstra2101783510l_tree @ A @ B @ R ) @ ( abstra2101783510l_tree @ A @ B @ Ra ) ) ) ).
% tree.rel_mono
thf(fact_77_tree_Osel__transfer_I1_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o] : ( bNF_rel_fun @ ( abstra2103299360e_tree @ A ) @ ( abstra2103299360e_tree @ B ) @ A @ B @ ( abstra2101783510l_tree @ A @ B @ R ) @ R @ ( abstra573067619e_root @ A ) @ ( abstra573067619e_root @ B ) ) ).
% tree.sel_transfer(1)
thf(fact_78_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F2: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G )
=> ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funD
thf(fact_79_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F2: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G )
=> ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funE
thf(fact_80_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F2: A > B,G: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F2 @ G ) ) ) ).
% le_funI
thf(fact_81_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F3: A > B,G3: A > B] :
! [X: A] : ( ord_less_eq @ B @ ( F3 @ X ) @ ( G3 @ X ) ) ) ) ) ).
% le_fun_def
thf(fact_82_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F2: B > A,B3: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F2 @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X3: B,Y4: B] :
( ( ord_less_eq @ B @ X3 @ Y4 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_83_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B3: A,F2: A > C,C2: C] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ C @ ( F2 @ B3 ) @ C2 )
=> ( ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ C @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_84_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F2: B > A,B3: B,C2: B] :
( ( A2
= ( F2 @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X3: B,Y4: B] :
( ( ord_less_eq @ B @ X3 @ Y4 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_85_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B3: A,F2: A > B,C2: B] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ( F2 @ B3 )
= C2 )
=> ( ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_86_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [X: A,Y5: A] :
( ( ord_less_eq @ A @ X @ Y5 )
& ( ord_less_eq @ A @ Y5 @ X ) ) ) ) ) ).
% eq_iff
thf(fact_87_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ X2 )
=> ( X2 = Y ) ) ) ) ).
% antisym
thf(fact_88_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
| ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% linear
thf(fact_89_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A] :
( ( X2 = Y )
=> ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% eq_refl
thf(fact_90_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A] :
( ~ ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% le_cases
thf(fact_91_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% order.trans
thf(fact_92_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A,Z2: A] :
( ( ( ord_less_eq @ A @ X2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X2 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_93_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y )
= ( X2 = Y ) ) ) ) ).
% antisym_conv
thf(fact_94_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( A2 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_95_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_96_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_97_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less_eq @ A @ X2 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_98_rel__funI,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A4: A > B > $o,B2: C > D > $o,F2: A > C,G: B > D] :
( ! [X3: A,Y4: B] :
( ( A4 @ X3 @ Y4 )
=> ( B2 @ ( F2 @ X3 ) @ ( G @ Y4 ) ) )
=> ( bNF_rel_fun @ A @ B @ C @ D @ A4 @ B2 @ F2 @ G ) ) ).
% rel_funI
thf(fact_99_stream_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F2: A > ( stream @ A ) > B,Stream: stream @ A] :
( ( P @ ( case_stream @ A @ B @ F2 @ Stream ) )
= ( ( Stream
= ( sCons @ A @ ( shd @ A @ Stream ) @ ( stl @ A @ Stream ) ) )
=> ( P @ ( F2 @ ( shd @ A @ Stream ) @ ( stl @ A @ Stream ) ) ) ) ) ).
% stream.split_sel
thf(fact_100_stream_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F2: A > ( stream @ A ) > B,Stream: stream @ A] :
( ( P @ ( case_stream @ A @ B @ F2 @ Stream ) )
= ( ~ ( ( Stream
= ( sCons @ A @ ( shd @ A @ Stream ) @ ( stl @ A @ Stream ) ) )
& ~ ( P @ ( F2 @ ( shd @ A @ Stream ) @ ( stl @ A @ Stream ) ) ) ) ) ) ).
% stream.split_sel_asm
thf(fact_101_tree_Ocorec__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,S2: C > D > $o,R: A > B > $o] : ( bNF_rel_fun @ ( C > A ) @ ( D > B ) @ ( ( C > ( fset @ ( sum_sum @ ( abstra2103299360e_tree @ A ) @ C ) ) ) > C > ( abstra2103299360e_tree @ A ) ) @ ( ( D > ( fset @ ( sum_sum @ ( abstra2103299360e_tree @ B ) @ D ) ) ) > D > ( abstra2103299360e_tree @ B ) ) @ ( bNF_rel_fun @ C @ D @ A @ B @ S2 @ R ) @ ( bNF_rel_fun @ ( C > ( fset @ ( sum_sum @ ( abstra2103299360e_tree @ A ) @ C ) ) ) @ ( D > ( fset @ ( sum_sum @ ( abstra2103299360e_tree @ B ) @ D ) ) ) @ ( C > ( abstra2103299360e_tree @ A ) ) @ ( D > ( abstra2103299360e_tree @ B ) ) @ ( bNF_rel_fun @ C @ D @ ( fset @ ( sum_sum @ ( abstra2103299360e_tree @ A ) @ C ) ) @ ( fset @ ( sum_sum @ ( abstra2103299360e_tree @ B ) @ D ) ) @ S2 @ ( rel_fset @ ( sum_sum @ ( abstra2103299360e_tree @ A ) @ C ) @ ( sum_sum @ ( abstra2103299360e_tree @ B ) @ D ) @ ( bNF_rel_sum @ ( abstra2103299360e_tree @ A ) @ ( abstra2103299360e_tree @ B ) @ C @ D @ ( abstra2101783510l_tree @ A @ B @ R ) @ S2 ) ) ) @ ( bNF_rel_fun @ C @ D @ ( abstra2103299360e_tree @ A ) @ ( abstra2103299360e_tree @ B ) @ S2 @ ( abstra2101783510l_tree @ A @ B @ R ) ) ) @ ( abstra1151671297c_tree @ C @ A ) @ ( abstra1151671297c_tree @ D @ B ) ) ).
% tree.corec_transfer
thf(fact_102_fun_Orel__mono,axiom,
! [D: $tType,B: $tType,A: $tType,R: A > B > $o,Ra: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R @ Ra )
=> ( ord_less_eq @ ( ( D > A ) > ( D > B ) > $o )
@ ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y3: D,Z: D] : Y3 = Z
@ R )
@ ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y3: D,Z: D] : Y3 = Z
@ Ra ) ) ) ).
% fun.rel_mono
thf(fact_103_fun__mono,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,C3: A > B > $o,A4: A > B > $o,B2: C > D > $o,D2: C > D > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ C3 @ A4 )
=> ( ( ord_less_eq @ ( C > D > $o ) @ B2 @ D2 )
=> ( ord_less_eq @ ( ( A > C ) > ( B > D ) > $o ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A4 @ B2 ) @ ( bNF_rel_fun @ A @ B @ C @ D @ C3 @ D2 ) ) ) ) ).
% fun_mono
thf(fact_104_stream_Oinject,axiom,
! [A: $tType,X1: A,X22: stream @ A,Y1: A,Y2: stream @ A] :
( ( ( sCons @ A @ X1 @ X22 )
= ( sCons @ A @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% stream.inject
thf(fact_105_predicate2I,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
( ! [X3: A,Y4: B] :
( ( P @ X3 @ Y4 )
=> ( Q @ X3 @ Y4 ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% predicate2I
thf(fact_106_stream_Ocollapse,axiom,
! [A: $tType,Stream: stream @ A] :
( ( sCons @ A @ ( shd @ A @ Stream ) @ ( stl @ A @ Stream ) )
= Stream ) ).
% stream.collapse
thf(fact_107_fset_Orel__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: A > C > $o,Sc: B > D > $o] :
( bNF_rel_fun @ ( A > B > $o ) @ ( C > D > $o ) @ ( ( fset @ A ) > ( fset @ B ) > $o ) @ ( ( fset @ C ) > ( fset @ D ) > $o )
@ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( D > $o ) @ Sa
@ ( bNF_rel_fun @ B @ D @ $o @ $o @ Sc
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ ( fset @ A ) @ ( fset @ C ) @ ( ( fset @ B ) > $o ) @ ( ( fset @ D ) > $o ) @ ( rel_fset @ A @ C @ Sa )
@ ( bNF_rel_fun @ ( fset @ B ) @ ( fset @ D ) @ $o @ $o @ ( rel_fset @ B @ D @ Sc )
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( rel_fset @ A @ B )
@ ( rel_fset @ C @ D ) ) ).
% fset.rel_transfer
thf(fact_108_rel__fset__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A4: A > C > $o,B2: B > D > $o] :
( bNF_rel_fun @ ( A > B > $o ) @ ( C > D > $o ) @ ( ( fset @ A ) > ( fset @ B ) > $o ) @ ( ( fset @ C ) > ( fset @ D ) > $o )
@ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( D > $o ) @ A4
@ ( bNF_rel_fun @ B @ D @ $o @ $o @ B2
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ ( fset @ A ) @ ( fset @ C ) @ ( ( fset @ B ) > $o ) @ ( ( fset @ D ) > $o ) @ ( rel_fset @ A @ C @ A4 )
@ ( bNF_rel_fun @ ( fset @ B ) @ ( fset @ D ) @ $o @ $o @ ( rel_fset @ B @ D @ B2 )
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( rel_fset @ A @ B )
@ ( rel_fset @ C @ D ) ) ).
% rel_fset_transfer
thf(fact_109_fBall__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( bNF_rel_fun @ ( fset @ A ) @ ( fset @ B ) @ ( ( A > $o ) > $o ) @ ( ( B > $o ) > $o ) @ ( rel_fset @ A @ B @ A4 )
@ ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A4
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( fBall @ A )
@ ( fBall @ B ) ) ).
% fBall_transfer
thf(fact_110_fBex__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( bNF_rel_fun @ ( fset @ A ) @ ( fset @ B ) @ ( ( A > $o ) > $o ) @ ( ( B > $o ) > $o ) @ ( rel_fset @ A @ B @ A4 )
@ ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A4
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( fBex @ A )
@ ( fBex @ B ) ) ).
% fBex_transfer
thf(fact_111_tree_Orel__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: A > C > $o,Sc: B > D > $o] :
( bNF_rel_fun @ ( A > B > $o ) @ ( C > D > $o ) @ ( ( abstra2103299360e_tree @ A ) > ( abstra2103299360e_tree @ B ) > $o ) @ ( ( abstra2103299360e_tree @ C ) > ( abstra2103299360e_tree @ D ) > $o )
@ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( D > $o ) @ Sa
@ ( bNF_rel_fun @ B @ D @ $o @ $o @ Sc
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ ( abstra2103299360e_tree @ A ) @ ( abstra2103299360e_tree @ C ) @ ( ( abstra2103299360e_tree @ B ) > $o ) @ ( ( abstra2103299360e_tree @ D ) > $o ) @ ( abstra2101783510l_tree @ A @ C @ Sa )
@ ( bNF_rel_fun @ ( abstra2103299360e_tree @ B ) @ ( abstra2103299360e_tree @ D ) @ $o @ $o @ ( abstra2101783510l_tree @ B @ D @ Sc )
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( abstra2101783510l_tree @ A @ B )
@ ( abstra2101783510l_tree @ C @ D ) ) ).
% tree.rel_transfer
thf(fact_112_sum_Orel__mono,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R1: A > C > $o,R1a: A > C > $o,R22: B > D > $o,R2a: B > D > $o] :
( ( ord_less_eq @ ( A > C > $o ) @ R1 @ R1a )
=> ( ( ord_less_eq @ ( B > D > $o ) @ R22 @ R2a )
=> ( ord_less_eq @ ( ( sum_sum @ A @ B ) > ( sum_sum @ C @ D ) > $o ) @ ( bNF_rel_sum @ A @ C @ B @ D @ R1 @ R22 ) @ ( bNF_rel_sum @ A @ C @ B @ D @ R1a @ R2a ) ) ) ) ).
% sum.rel_mono
thf(fact_113_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X2: A,Y: B,Q: A > B > $o] :
( ( P @ X2 @ Y )
=> ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( Q @ X2 @ Y ) ) ) ).
% rev_predicate2D
thf(fact_114_predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X2: A,Y: B] :
( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( ( P @ X2 @ Y )
=> ( Q @ X2 @ Y ) ) ) ).
% predicate2D
thf(fact_115_refl__ge__eq,axiom,
! [A: $tType,R: A > A > $o] :
( ! [X3: A] : ( R @ X3 @ X3 )
=> ( ord_less_eq @ ( A > A > $o )
@ ^ [Y3: A,Z: A] : Y3 = Z
@ R ) ) ).
% refl_ge_eq
thf(fact_116_ge__eq__refl,axiom,
! [A: $tType,R: A > A > $o,X2: A] :
( ( ord_less_eq @ ( A > A > $o )
@ ^ [Y3: A,Z: A] : Y3 = Z
@ R )
=> ( R @ X2 @ X2 ) ) ).
% ge_eq_refl
thf(fact_117_sum_Orel__transfer,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,H: $tType,F4: $tType,G4: $tType,E: $tType,S1a: A > E > $o,S1c: C > G4 > $o,S2a: B > F4 > $o,S2c: D > H > $o] :
( bNF_rel_fun @ ( A > C > $o ) @ ( E > G4 > $o ) @ ( ( B > D > $o ) > ( sum_sum @ A @ B ) > ( sum_sum @ C @ D ) > $o ) @ ( ( F4 > H > $o ) > ( sum_sum @ E @ F4 ) > ( sum_sum @ G4 @ H ) > $o )
@ ( bNF_rel_fun @ A @ E @ ( C > $o ) @ ( G4 > $o ) @ S1a
@ ( bNF_rel_fun @ C @ G4 @ $o @ $o @ S1c
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ ( B > D > $o ) @ ( F4 > H > $o ) @ ( ( sum_sum @ A @ B ) > ( sum_sum @ C @ D ) > $o ) @ ( ( sum_sum @ E @ F4 ) > ( sum_sum @ G4 @ H ) > $o )
@ ( bNF_rel_fun @ B @ F4 @ ( D > $o ) @ ( H > $o ) @ S2a
@ ( bNF_rel_fun @ D @ H @ $o @ $o @ S2c
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ ( sum_sum @ A @ B ) @ ( sum_sum @ E @ F4 ) @ ( ( sum_sum @ C @ D ) > $o ) @ ( ( sum_sum @ G4 @ H ) > $o ) @ ( bNF_rel_sum @ A @ E @ B @ F4 @ S1a @ S2a )
@ ( bNF_rel_fun @ ( sum_sum @ C @ D ) @ ( sum_sum @ G4 @ H ) @ $o @ $o @ ( bNF_rel_sum @ C @ G4 @ D @ H @ S1c @ S2c )
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) ) )
@ ( bNF_rel_sum @ A @ C @ B @ D )
@ ( bNF_rel_sum @ E @ G4 @ F4 @ H ) ) ).
% sum.rel_transfer
thf(fact_118_stream_Oexhaust,axiom,
! [A: $tType,Y: stream @ A] :
~ ! [X12: A,X23: stream @ A] :
( Y
!= ( sCons @ A @ X12 @ X23 ) ) ).
% stream.exhaust
thf(fact_119_stream_Osel_I2_J,axiom,
! [A: $tType,X1: A,X22: stream @ A] :
( ( stl @ A @ ( sCons @ A @ X1 @ X22 ) )
= X22 ) ).
% stream.sel(2)
thf(fact_120_stream_Osel_I1_J,axiom,
! [A: $tType,X1: A,X22: stream @ A] :
( ( shd @ A @ ( sCons @ A @ X1 @ X22 ) )
= X1 ) ).
% stream.sel(1)
thf(fact_121_tree_Ocase__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R: A > B > $o,S2: C > D > $o] : ( bNF_rel_fun @ ( A > ( fset @ ( abstra2103299360e_tree @ A ) ) > C ) @ ( B > ( fset @ ( abstra2103299360e_tree @ B ) ) > D ) @ ( ( abstra2103299360e_tree @ A ) > C ) @ ( ( abstra2103299360e_tree @ B ) > D ) @ ( bNF_rel_fun @ A @ B @ ( ( fset @ ( abstra2103299360e_tree @ A ) ) > C ) @ ( ( fset @ ( abstra2103299360e_tree @ B ) ) > D ) @ R @ ( bNF_rel_fun @ ( fset @ ( abstra2103299360e_tree @ A ) ) @ ( fset @ ( abstra2103299360e_tree @ B ) ) @ C @ D @ ( rel_fset @ ( abstra2103299360e_tree @ A ) @ ( abstra2103299360e_tree @ B ) @ ( abstra2101783510l_tree @ A @ B @ R ) ) @ S2 ) ) @ ( bNF_rel_fun @ ( abstra2103299360e_tree @ A ) @ ( abstra2103299360e_tree @ B ) @ C @ D @ ( abstra2101783510l_tree @ A @ B @ R ) @ S2 ) @ ( abstra457966479e_tree @ A @ C ) @ ( abstra457966479e_tree @ B @ D ) ) ).
% tree.case_transfer
thf(fact_122_stream_Ocase,axiom,
! [B: $tType,A: $tType,F2: A > ( stream @ A ) > B,X1: A,X22: stream @ A] :
( ( case_stream @ A @ B @ F2 @ ( sCons @ A @ X1 @ X22 ) )
= ( F2 @ X1 @ X22 ) ) ).
% stream.case
thf(fact_123_sdrop__while__SCons,axiom,
! [A: $tType,P: A > $o,A2: A,S3: stream @ A] :
( ( ( P @ A2 )
=> ( ( sdrop_while @ A @ P @ ( sCons @ A @ A2 @ S3 ) )
= ( sdrop_while @ A @ P @ S3 ) ) )
& ( ~ ( P @ A2 )
=> ( ( sdrop_while @ A @ P @ ( sCons @ A @ A2 @ S3 ) )
= ( sCons @ A @ A2 @ S3 ) ) ) ) ).
% sdrop_while_SCons
thf(fact_124_tree_Octr__transfer,axiom,
! [A: $tType,B: $tType,R: A > B > $o] : ( bNF_rel_fun @ A @ B @ ( ( fset @ ( abstra2103299360e_tree @ A ) ) > ( abstra2103299360e_tree @ A ) ) @ ( ( fset @ ( abstra2103299360e_tree @ B ) ) > ( abstra2103299360e_tree @ B ) ) @ R @ ( bNF_rel_fun @ ( fset @ ( abstra2103299360e_tree @ A ) ) @ ( fset @ ( abstra2103299360e_tree @ B ) ) @ ( abstra2103299360e_tree @ A ) @ ( abstra2103299360e_tree @ B ) @ ( rel_fset @ ( abstra2103299360e_tree @ A ) @ ( abstra2103299360e_tree @ B ) @ ( abstra2101783510l_tree @ A @ B @ R ) ) @ ( abstra2101783510l_tree @ A @ B @ R ) ) @ ( abstra388494275e_Node @ A ) @ ( abstra388494275e_Node @ B ) ) ).
% tree.ctr_transfer
thf(fact_125_stream_Oexhaust__sel,axiom,
! [A: $tType,Stream: stream @ A] :
( Stream
= ( sCons @ A @ ( shd @ A @ Stream ) @ ( stl @ A @ Stream ) ) ) ).
% stream.exhaust_sel
thf(fact_126_fun_Orel__transfer,axiom,
! [B: $tType,A: $tType,C: $tType,E: $tType,D: $tType,Sa: A > C > $o,Sc: B > E > $o] :
( bNF_rel_fun @ ( A > B > $o ) @ ( C > E > $o ) @ ( ( D > A ) > ( D > B ) > $o ) @ ( ( D > C ) > ( D > E ) > $o )
@ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( E > $o ) @ Sa
@ ( bNF_rel_fun @ B @ E @ $o @ $o @ Sc
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ ( D > A ) @ ( D > C ) @ ( ( D > B ) > $o ) @ ( ( D > E ) > $o )
@ ( bNF_rel_fun @ D @ D @ A @ C
@ ^ [Y3: D,Z: D] : Y3 = Z
@ Sa )
@ ( bNF_rel_fun @ ( D > B ) @ ( D > E ) @ $o @ $o
@ ( bNF_rel_fun @ D @ D @ B @ E
@ ^ [Y3: D,Z: D] : Y3 = Z
@ Sc )
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y3: D,Z: D] : Y3 = Z )
@ ( bNF_rel_fun @ D @ D @ C @ E
@ ^ [Y3: D,Z: D] : Y3 = Z ) ) ).
% fun.rel_transfer
thf(fact_127_fun_Orel__refl,axiom,
! [B: $tType,D: $tType,Ra: B > B > $o,X2: D > B] :
( ! [X3: B] : ( Ra @ X3 @ X3 )
=> ( bNF_rel_fun @ D @ D @ B @ B
@ ^ [Y3: D,Z: D] : Y3 = Z
@ Ra
@ X2
@ X2 ) ) ).
% fun.rel_refl
thf(fact_128_fun_Orel__eq,axiom,
! [A: $tType,D: $tType] :
( ( bNF_rel_fun @ D @ D @ A @ A
@ ^ [Y3: D,Z: D] : Y3 = Z
@ ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [Y3: D > A,Z: D > A] : Y3 = Z ) ) ).
% fun.rel_eq
thf(fact_129_rel__fun__mono_H,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,Y6: A > B > $o,X5: A > B > $o,A4: C > D > $o,B2: C > D > $o,F2: A > C,G: B > D] :
( ! [X3: A,Y4: B] :
( ( Y6 @ X3 @ Y4 )
=> ( X5 @ X3 @ Y4 ) )
=> ( ! [X3: C,Y4: D] :
( ( A4 @ X3 @ Y4 )
=> ( B2 @ X3 @ Y4 ) )
=> ( ( bNF_rel_fun @ A @ B @ C @ D @ X5 @ A4 @ F2 @ G )
=> ( bNF_rel_fun @ A @ B @ C @ D @ Y6 @ B2 @ F2 @ G ) ) ) ) ).
% rel_fun_mono'
thf(fact_130_rel__fun__mono,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,X5: A > B > $o,A4: C > D > $o,F2: A > C,G: B > D,Y6: A > B > $o,B2: C > D > $o] :
( ( bNF_rel_fun @ A @ B @ C @ D @ X5 @ A4 @ F2 @ G )
=> ( ! [X3: A,Y4: B] :
( ( Y6 @ X3 @ Y4 )
=> ( X5 @ X3 @ Y4 ) )
=> ( ! [X3: C,Y4: D] :
( ( A4 @ X3 @ Y4 )
=> ( B2 @ X3 @ Y4 ) )
=> ( bNF_rel_fun @ A @ B @ C @ D @ Y6 @ B2 @ F2 @ G ) ) ) ) ).
% rel_fun_mono
thf(fact_131_apply__rsp_H,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R1: A > B > $o,R22: C > D > $o,F2: A > C,G: B > D,X2: A,Y: B] :
( ( bNF_rel_fun @ A @ B @ C @ D @ R1 @ R22 @ F2 @ G )
=> ( ( R1 @ X2 @ Y )
=> ( R22 @ ( F2 @ X2 ) @ ( G @ Y ) ) ) ) ).
% apply_rsp'
thf(fact_132_rel__funD,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A4: A > B > $o,B2: C > D > $o,F2: A > C,G: B > D,X2: A,Y: B] :
( ( bNF_rel_fun @ A @ B @ C @ D @ A4 @ B2 @ F2 @ G )
=> ( ( A4 @ X2 @ Y )
=> ( B2 @ ( F2 @ X2 ) @ ( G @ Y ) ) ) ) ).
% rel_funD
thf(fact_133_Let__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A4: A > B > $o,B2: C > D > $o] :
( bNF_rel_fun @ A @ B @ ( ( A > C ) > C ) @ ( ( B > D ) > D ) @ A4 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ C @ D @ ( bNF_rel_fun @ A @ B @ C @ D @ A4 @ B2 ) @ B2 )
@ ^ [S: A,F3: A > C] : ( F3 @ S )
@ ^ [S: B,F3: B > D] : ( F3 @ S ) ) ).
% Let_transfer
thf(fact_134_let__rsp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R1: A > B > $o,R22: C > D > $o] :
( bNF_rel_fun @ A @ B @ ( ( A > C ) > C ) @ ( ( B > D ) > D ) @ R1 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ C @ D @ ( bNF_rel_fun @ A @ B @ C @ D @ R1 @ R22 ) @ R22 )
@ ^ [S: A,F3: A > C] : ( F3 @ S )
@ ^ [S: B,F3: B > D] : ( F3 @ S ) ) ).
% let_rsp
thf(fact_135_smember__code,axiom,
! [A: $tType,X2: A,Y: A,S3: stream @ A] :
( ( smember @ A @ X2 @ ( sCons @ A @ Y @ S3 ) )
= ( ( X2 != Y )
=> ( smember @ A @ X2 @ S3 ) ) ) ).
% smember_code
thf(fact_136_rel__fun__def__butlast,axiom,
! [B: $tType,D: $tType,C: $tType,E: $tType,F4: $tType,A: $tType,R: A > B > $o,S2: C > E > $o,T9: D > F4 > $o,F2: A > C > D,G: B > E > F4] :
( ( bNF_rel_fun @ A @ B @ ( C > D ) @ ( E > F4 ) @ R @ ( bNF_rel_fun @ C @ E @ D @ F4 @ S2 @ T9 ) @ F2 @ G )
= ( ! [X: A,Y5: B] :
( ( R @ X @ Y5 )
=> ( bNF_rel_fun @ C @ E @ D @ F4 @ S2 @ T9 @ ( F2 @ X ) @ ( G @ Y5 ) ) ) ) ) ).
% rel_fun_def_butlast
thf(fact_137_If__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( bNF_rel_fun @ $o @ $o @ ( A > A > A ) @ ( B > B > B )
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A4 @ ( bNF_rel_fun @ A @ B @ A @ B @ A4 @ A4 ) )
@ ( if @ A )
@ ( if @ B ) ) ).
% If_transfer
thf(fact_138_rel__funE,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A4: A > B > $o,B2: C > D > $o,F2: A > C,G: B > D,X2: A,Y: B] :
( ( bNF_rel_fun @ A @ B @ C @ D @ A4 @ B2 @ F2 @ G )
=> ( ( A4 @ X2 @ Y )
=> ( B2 @ ( F2 @ X2 ) @ ( G @ Y ) ) ) ) ).
% rel_funE
thf(fact_139_rel__funD2,axiom,
! [B: $tType,C: $tType,A: $tType,A4: A > A > $o,B2: B > C > $o,F2: A > B,G: A > C,X2: A] :
( ( bNF_rel_fun @ A @ A @ B @ C @ A4 @ B2 @ F2 @ G )
=> ( ( A4 @ X2 @ X2 )
=> ( B2 @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ).
% rel_funD2
thf(fact_140_predicate2D__conj,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,R: $o,X2: A,Y: B] :
( ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
& R )
=> ( R
& ( ( P @ X2 @ Y )
=> ( Q @ X2 @ Y ) ) ) ) ).
% predicate2D_conj
thf(fact_141_case__sum__transfer,axiom,
! [A: $tType,B: $tType,E: $tType,F4: $tType,D: $tType,C: $tType,R: A > C > $o,T9: B > D > $o,S2: E > F4 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > B ) > ( sum_sum @ A @ E ) > B ) @ ( ( F4 > D ) > ( sum_sum @ C @ F4 ) > D ) @ ( bNF_rel_fun @ A @ C @ B @ D @ R @ T9 ) @ ( bNF_rel_fun @ ( E > B ) @ ( F4 > D ) @ ( ( sum_sum @ A @ E ) > B ) @ ( ( sum_sum @ C @ F4 ) > D ) @ ( bNF_rel_fun @ E @ F4 @ B @ D @ S2 @ T9 ) @ ( bNF_rel_fun @ ( sum_sum @ A @ E ) @ ( sum_sum @ C @ F4 ) @ B @ D @ ( bNF_rel_sum @ A @ C @ E @ F4 @ R @ S2 ) @ T9 ) ) @ ( sum_case_sum @ A @ B @ E ) @ ( sum_case_sum @ C @ D @ F4 ) ) ).
% case_sum_transfer
thf(fact_142_sinterleave_Ocode,axiom,
! [A: $tType] :
( ( sinterleave @ A )
= ( ^ [S1: stream @ A,S22: stream @ A] : ( sCons @ A @ ( shd @ A @ S1 ) @ ( sinterleave @ A @ S22 @ ( stl @ A @ S1 ) ) ) ) ) ).
% sinterleave.code
thf(fact_143_tree_Opred__transfer,axiom,
! [A: $tType,B: $tType,R: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( abstra2103299360e_tree @ A ) > $o ) @ ( ( abstra2103299360e_tree @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ R
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( bNF_rel_fun @ ( abstra2103299360e_tree @ A ) @ ( abstra2103299360e_tree @ B ) @ $o @ $o @ ( abstra2101783510l_tree @ A @ B @ R )
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( abstra1615255520d_tree @ A )
@ ( abstra1615255520d_tree @ B ) ) ).
% tree.pred_transfer
thf(fact_144_fset_Opred__transfer,axiom,
! [A: $tType,B: $tType,R: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( fset @ A ) > $o ) @ ( ( fset @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ R
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( bNF_rel_fun @ ( fset @ A ) @ ( fset @ B ) @ $o @ $o @ ( rel_fset @ A @ B @ R )
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( pred_fset @ A )
@ ( pred_fset @ B ) ) ).
% fset.pred_transfer
thf(fact_145_tree_Opred__inject,axiom,
! [A: $tType,P: A > $o,A2: A,Aa2: fset @ ( abstra2103299360e_tree @ A )] :
( ( abstra1615255520d_tree @ A @ P @ ( abstra388494275e_Node @ A @ A2 @ Aa2 ) )
= ( ( P @ A2 )
& ( pred_fset @ ( abstra2103299360e_tree @ A ) @ ( abstra1615255520d_tree @ A @ P ) @ Aa2 ) ) ) ).
% tree.pred_inject
thf(fact_146_sinterleave_Osimps_I2_J,axiom,
! [A: $tType,S12: stream @ A,S23: stream @ A] :
( ( stl @ A @ ( sinterleave @ A @ S12 @ S23 ) )
= ( sinterleave @ A @ S23 @ ( stl @ A @ S12 ) ) ) ).
% sinterleave.simps(2)
thf(fact_147_sinterleave_Osimps_I1_J,axiom,
! [A: $tType,S12: stream @ A,S23: stream @ A] :
( ( shd @ A @ ( sinterleave @ A @ S12 @ S23 ) )
= ( shd @ A @ S12 ) ) ).
% sinterleave.simps(1)
thf(fact_148_sinterleave__code,axiom,
! [A: $tType,X2: A,S12: stream @ A,S23: stream @ A] :
( ( sinterleave @ A @ ( sCons @ A @ X2 @ S12 ) @ S23 )
= ( sCons @ A @ X2 @ ( sinterleave @ A @ S23 @ S12 ) ) ) ).
% sinterleave_code
thf(fact_149_sum_Opred__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > C > $o,R22: B > D > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( C > $o ) @ ( ( B > $o ) > ( sum_sum @ A @ B ) > $o ) @ ( ( D > $o ) > ( sum_sum @ C @ D ) > $o )
@ ( bNF_rel_fun @ A @ C @ $o @ $o @ R1
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( bNF_rel_fun @ ( B > $o ) @ ( D > $o ) @ ( ( sum_sum @ A @ B ) > $o ) @ ( ( sum_sum @ C @ D ) > $o )
@ ( bNF_rel_fun @ B @ D @ $o @ $o @ R22
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( bNF_rel_fun @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) @ $o @ $o @ ( bNF_rel_sum @ A @ C @ B @ D @ R1 @ R22 )
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( basic_pred_sum @ A @ B )
@ ( basic_pred_sum @ C @ D ) ) ).
% sum.pred_transfer
thf(fact_150_smap2_Ocode,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( smap2 @ A @ B @ C )
= ( ^ [F3: A > B > C,S1: stream @ A,S22: stream @ B] : ( sCons @ C @ ( F3 @ ( shd @ A @ S1 ) @ ( shd @ B @ S22 ) ) @ ( smap2 @ A @ B @ C @ F3 @ ( stl @ A @ S1 ) @ ( stl @ B @ S22 ) ) ) ) ) ).
% smap2.code
thf(fact_151_fmember__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( ( bi_unique @ A @ B @ A4 )
=> ( bNF_rel_fun @ A @ B @ ( ( fset @ A ) > $o ) @ ( ( fset @ B ) > $o ) @ A4
@ ( bNF_rel_fun @ ( fset @ A ) @ ( fset @ B ) @ $o @ $o @ ( rel_fset @ A @ B @ A4 )
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( fmember @ A )
@ ( fmember @ B ) ) ) ).
% fmember_transfer
thf(fact_152_fset_Obi__unique__rel,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( bi_unique @ A @ B @ R )
=> ( bi_unique @ ( fset @ A ) @ ( fset @ B ) @ ( rel_fset @ A @ B @ R ) ) ) ).
% fset.bi_unique_rel
thf(fact_153_tree_Obi__unique__rel,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( bi_unique @ A @ B @ R )
=> ( bi_unique @ ( abstra2103299360e_tree @ A ) @ ( abstra2103299360e_tree @ B ) @ ( abstra2101783510l_tree @ A @ B @ R ) ) ) ).
% tree.bi_unique_rel
thf(fact_154_smap2_Osimps_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,F2: A > B > C,S12: stream @ A,S23: stream @ B] :
( ( stl @ C @ ( smap2 @ A @ B @ C @ F2 @ S12 @ S23 ) )
= ( smap2 @ A @ B @ C @ F2 @ ( stl @ A @ S12 ) @ ( stl @ B @ S23 ) ) ) ).
% smap2.simps(2)
thf(fact_155_smap2_Osimps_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,F2: A > B > C,S12: stream @ A,S23: stream @ B] :
( ( shd @ C @ ( smap2 @ A @ B @ C @ F2 @ S12 @ S23 ) )
= ( F2 @ ( shd @ A @ S12 ) @ ( shd @ B @ S23 ) ) ) ).
% smap2.simps(1)
thf(fact_156_smap2__unfold,axiom,
! [B: $tType,A: $tType,C: $tType,F2: B > C > A,A2: B,S12: stream @ B,B3: C,S23: stream @ C] :
( ( smap2 @ B @ C @ A @ F2 @ ( sCons @ B @ A2 @ S12 ) @ ( sCons @ C @ B3 @ S23 ) )
= ( sCons @ A @ ( F2 @ A2 @ B3 ) @ ( smap2 @ B @ C @ A @ F2 @ S12 @ S23 ) ) ) ).
% smap2_unfold
thf(fact_157_eq__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( ( bi_unique @ A @ B @ A4 )
=> ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A4
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A4
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ^ [Y3: A,Z: A] : Y3 = Z
@ ^ [Y3: B,Z: B] : Y3 = Z ) ) ).
% eq_transfer
thf(fact_158_bi__unique__alt__def2,axiom,
! [B: $tType,A: $tType] :
( ( bi_unique @ A @ B )
= ( ^ [R2: A > B > $o] :
( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ R2
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ R2
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ^ [Y3: A,Z: A] : Y3 = Z
@ ^ [Y3: B,Z: B] : Y3 = Z ) ) ) ).
% bi_unique_alt_def2
thf(fact_159_ffilter__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( ( bi_unique @ A @ B @ A4 )
=> ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( fset @ A ) > ( fset @ A ) ) @ ( ( fset @ B ) > ( fset @ B ) )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A4
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( bNF_rel_fun @ ( fset @ A ) @ ( fset @ B ) @ ( fset @ A ) @ ( fset @ B ) @ ( rel_fset @ A @ B @ A4 ) @ ( rel_fset @ A @ B @ A4 ) )
@ ( ffilter @ A )
@ ( ffilter @ B ) ) ) ).
% ffilter_transfer
thf(fact_160_fun__upd__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A4: A > B > $o,B2: C > D > $o] :
( ( bi_unique @ A @ B @ A4 )
=> ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ ( A > C > A > C ) @ ( B > D > B > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A4 @ B2 ) @ ( bNF_rel_fun @ A @ B @ ( C > A > C ) @ ( D > B > D ) @ A4 @ ( bNF_rel_fun @ C @ D @ ( A > C ) @ ( B > D ) @ B2 @ ( bNF_rel_fun @ A @ B @ C @ D @ A4 @ B2 ) ) ) @ ( fun_upd @ A @ C ) @ ( fun_upd @ B @ D ) ) ) ).
% fun_upd_transfer
thf(fact_161_eq__onp__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( ( bi_unique @ A @ B @ A4 )
=> ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( A > A > $o ) @ ( B > B > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A4
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A4
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A4
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_eq_onp @ A )
@ ( bNF_eq_onp @ B ) ) ) ).
% eq_onp_transfer
thf(fact_162_FSet_Obind__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A4: A > B > $o,B2: C > D > $o] : ( bNF_rel_fun @ ( fset @ A ) @ ( fset @ B ) @ ( ( A > ( fset @ C ) ) > ( fset @ C ) ) @ ( ( B > ( fset @ D ) ) > ( fset @ D ) ) @ ( rel_fset @ A @ B @ A4 ) @ ( bNF_rel_fun @ ( A > ( fset @ C ) ) @ ( B > ( fset @ D ) ) @ ( fset @ C ) @ ( fset @ D ) @ ( bNF_rel_fun @ A @ B @ ( fset @ C ) @ ( fset @ D ) @ A4 @ ( rel_fset @ C @ D @ B2 ) ) @ ( rel_fset @ C @ D @ B2 ) ) @ ( fbind @ A @ C ) @ ( fbind @ B @ D ) ) ).
% FSet.bind_transfer
thf(fact_163_fset_Orel__eq__onp,axiom,
! [A: $tType,P: A > $o] :
( ( rel_fset @ A @ A @ ( bNF_eq_onp @ A @ P ) )
= ( bNF_eq_onp @ ( fset @ A ) @ ( pred_fset @ A @ P ) ) ) ).
% fset.rel_eq_onp
thf(fact_164_eq__onp__le__eq,axiom,
! [A: $tType,P: A > $o] :
( ord_less_eq @ ( A > A > $o ) @ ( bNF_eq_onp @ A @ P )
@ ^ [Y3: A,Z: A] : Y3 = Z ) ).
% eq_onp_le_eq
thf(fact_165_tree_Orel__eq__onp,axiom,
! [A: $tType,P: A > $o] :
( ( abstra2101783510l_tree @ A @ A @ ( bNF_eq_onp @ A @ P ) )
= ( bNF_eq_onp @ ( abstra2103299360e_tree @ A ) @ ( abstra1615255520d_tree @ A @ P ) ) ) ).
% tree.rel_eq_onp
thf(fact_166_fset_Opred__rel,axiom,
! [A: $tType] :
( ( pred_fset @ A )
= ( ^ [P2: A > $o,X: fset @ A] : ( rel_fset @ A @ A @ ( bNF_eq_onp @ A @ P2 ) @ X @ X ) ) ) ).
% fset.pred_rel
thf(fact_167_tree_Opred__rel,axiom,
! [A: $tType] :
( ( abstra1615255520d_tree @ A )
= ( ^ [P2: A > $o,X: abstra2103299360e_tree @ A] : ( abstra2101783510l_tree @ A @ A @ ( bNF_eq_onp @ A @ P2 ) @ X @ X ) ) ) ).
% tree.pred_rel
thf(fact_168_fimage__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A4: A > C > $o,B2: B > D > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( fset @ A ) > ( fset @ B ) ) @ ( ( fset @ C ) > ( fset @ D ) ) @ ( bNF_rel_fun @ A @ C @ B @ D @ A4 @ B2 ) @ ( bNF_rel_fun @ ( fset @ A ) @ ( fset @ C ) @ ( fset @ B ) @ ( fset @ D ) @ ( rel_fset @ A @ C @ A4 ) @ ( rel_fset @ B @ D @ B2 ) ) @ ( fimage @ A @ B ) @ ( fimage @ C @ D ) ) ).
% fimage_transfer
thf(fact_169_fset_Omap__transfer,axiom,
! [A: $tType,B: $tType,F4: $tType,E: $tType,Rb: A > E > $o,Sd: B > F4 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( E > F4 ) @ ( ( fset @ A ) > ( fset @ B ) ) @ ( ( fset @ E ) > ( fset @ F4 ) ) @ ( bNF_rel_fun @ A @ E @ B @ F4 @ Rb @ Sd ) @ ( bNF_rel_fun @ ( fset @ A ) @ ( fset @ E ) @ ( fset @ B ) @ ( fset @ F4 ) @ ( rel_fset @ A @ E @ Rb ) @ ( rel_fset @ B @ F4 @ Sd ) ) @ ( fimage @ A @ B ) @ ( fimage @ E @ F4 ) ) ).
% fset.map_transfer
thf(fact_170_if__rsp,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A] :
( ( quotient3 @ A @ B @ R @ Abs @ Rep )
=> ( bNF_rel_fun @ $o @ $o @ ( A > A > A ) @ ( A > A > A )
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ ( bNF_rel_fun @ A @ A @ ( A > A ) @ ( A > A ) @ R @ ( bNF_rel_fun @ A @ A @ A @ A @ R @ R ) )
@ ( if @ A )
@ ( if @ A ) ) ) ).
% if_rsp
thf(fact_171_fimage__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F2: B > A,X2: B,A4: fset @ B] :
( ( B3
= ( F2 @ X2 ) )
=> ( ( fmember @ B @ X2 @ A4 )
=> ( fmember @ A @ B3 @ ( fimage @ B @ A @ F2 @ A4 ) ) ) ) ).
% fimage_eqI
thf(fact_172_apply__rspQ3,axiom,
! [B: $tType,C: $tType,A: $tType,R1: A > A > $o,Abs1: A > B,Rep1: B > A,R22: C > C > $o,F2: A > C,G: A > C,X2: A,Y: A] :
( ( quotient3 @ A @ B @ R1 @ Abs1 @ Rep1 )
=> ( ( bNF_rel_fun @ A @ A @ C @ C @ R1 @ R22 @ F2 @ G )
=> ( ( R1 @ X2 @ Y )
=> ( R22 @ ( F2 @ X2 ) @ ( G @ Y ) ) ) ) ) ).
% apply_rspQ3
thf(fact_173_apply__rspQ3_H_H,axiom,
! [C: $tType,A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,S2: C > C > $o,F2: A > C,X2: B] :
( ( quotient3 @ A @ B @ R @ Abs @ Rep )
=> ( ( bNF_rel_fun @ A @ A @ C @ C @ R @ S2 @ F2 @ F2 )
=> ( S2 @ ( F2 @ ( Rep @ X2 ) ) @ ( F2 @ ( Rep @ X2 ) ) ) ) ) ).
% apply_rspQ3''
thf(fact_174_rev__fimage__eqI,axiom,
! [B: $tType,A: $tType,X2: A,A4: fset @ A,B3: B,F2: A > B] :
( ( fmember @ A @ X2 @ A4 )
=> ( ( B3
= ( F2 @ X2 ) )
=> ( fmember @ B @ B3 @ ( fimage @ A @ B @ F2 @ A4 ) ) ) ) ).
% rev_fimage_eqI
thf(fact_175_fimage__cong,axiom,
! [B: $tType,A: $tType,M: fset @ A,N: fset @ A,F2: A > B,G: A > B] :
( ( M = N )
=> ( ! [X3: A] :
( ( fmember @ A @ X3 @ N )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( fimage @ A @ B @ F2 @ M )
= ( fimage @ A @ B @ G @ N ) ) ) ) ).
% fimage_cong
thf(fact_176_fimageI,axiom,
! [B: $tType,A: $tType,X2: A,A4: fset @ A,F2: A > B] :
( ( fmember @ A @ X2 @ A4 )
=> ( fmember @ B @ ( F2 @ X2 ) @ ( fimage @ A @ B @ F2 @ A4 ) ) ) ).
% fimageI
thf(fact_177_fimageE,axiom,
! [A: $tType,B: $tType,B3: A,F2: B > A,A4: fset @ B] :
( ( fmember @ A @ B3 @ ( fimage @ B @ A @ F2 @ A4 ) )
=> ~ ! [X3: B] :
( ( B3
= ( F2 @ X3 ) )
=> ~ ( fmember @ B @ X3 @ A4 ) ) ) ).
% fimageE
thf(fact_178_subset__fimage__iff,axiom,
! [A: $tType,B: $tType,B2: fset @ A,F2: B > A,A4: fset @ B] :
( ( ord_less_eq @ ( fset @ A ) @ B2 @ ( fimage @ B @ A @ F2 @ A4 ) )
= ( ? [AA: fset @ B] :
( ( ord_less_eq @ ( fset @ B ) @ AA @ A4 )
& ( B2
= ( fimage @ B @ A @ F2 @ AA ) ) ) ) ) ).
% subset_fimage_iff
thf(fact_179_fimage__mono,axiom,
! [B: $tType,A: $tType,A4: fset @ A,B2: fset @ A,F2: A > B] :
( ( ord_less_eq @ ( fset @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ ( fset @ B ) @ ( fimage @ A @ B @ F2 @ A4 ) @ ( fimage @ A @ B @ F2 @ B2 ) ) ) ).
% fimage_mono
thf(fact_180_fimage__fsubsetI,axiom,
! [A: $tType,B: $tType,A4: fset @ A,F2: A > B,B2: fset @ B] :
( ! [X3: A] :
( ( fmember @ A @ X3 @ A4 )
=> ( fmember @ B @ ( F2 @ X3 ) @ B2 ) )
=> ( ord_less_eq @ ( fset @ B ) @ ( fimage @ A @ B @ F2 @ A4 ) @ B2 ) ) ).
% fimage_fsubsetI
thf(fact_181_quot__rel__rsp,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A] :
( ( quotient3 @ A @ B @ R @ Abs @ Rep )
=> ( bNF_rel_fun @ A @ A @ ( A > $o ) @ ( A > $o ) @ R
@ ( bNF_rel_fun @ A @ A @ $o @ $o @ R
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ R
@ R ) ) ).
% quot_rel_rsp
thf(fact_182_bex1__rel__rsp,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Absf: A > B,Repf: B > A] :
( ( quotient3 @ A @ B @ R @ Absf @ Repf )
=> ( bNF_rel_fun @ ( A > $o ) @ ( A > $o ) @ $o @ $o
@ ( bNF_rel_fun @ A @ A @ $o @ $o @ R
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ ( bex1_rel @ A @ R )
@ ( bex1_rel @ A @ R ) ) ) ).
% bex1_rel_rsp
thf(fact_183_map__fun__parametric,axiom,
! [A: $tType,B: $tType,E: $tType,F4: $tType,H: $tType,G4: $tType,D: $tType,C: $tType,A4: A > C > $o,B2: B > D > $o,C3: E > G4 > $o,D2: F4 > H > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > F4 ) > ( B > E ) > A > F4 ) @ ( ( G4 > H ) > ( D > G4 ) > C > H ) @ ( bNF_rel_fun @ A @ C @ B @ D @ A4 @ B2 ) @ ( bNF_rel_fun @ ( E > F4 ) @ ( G4 > H ) @ ( ( B > E ) > A > F4 ) @ ( ( D > G4 ) > C > H ) @ ( bNF_rel_fun @ E @ G4 @ F4 @ H @ C3 @ D2 ) @ ( bNF_rel_fun @ ( B > E ) @ ( D > G4 ) @ ( A > F4 ) @ ( C > H ) @ ( bNF_rel_fun @ B @ D @ E @ G4 @ B2 @ C3 ) @ ( bNF_rel_fun @ A @ C @ F4 @ H @ A4 @ D2 ) ) ) @ ( map_fun @ A @ B @ E @ F4 ) @ ( map_fun @ C @ D @ G4 @ H ) ) ).
% map_fun_parametric
thf(fact_184_finsert__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] : ( bNF_rel_fun @ A @ B @ ( ( fset @ A ) > ( fset @ A ) ) @ ( ( fset @ B ) > ( fset @ B ) ) @ A4 @ ( bNF_rel_fun @ ( fset @ A ) @ ( fset @ B ) @ ( fset @ A ) @ ( fset @ B ) @ ( rel_fset @ A @ B @ A4 ) @ ( rel_fset @ A @ B @ A4 ) ) @ ( finsert @ A ) @ ( finsert @ B ) ) ).
% finsert_transfer
thf(fact_185_finsert__absorb2,axiom,
! [A: $tType,X2: A,A4: fset @ A] :
( ( finsert @ A @ X2 @ ( finsert @ A @ X2 @ A4 ) )
= ( finsert @ A @ X2 @ A4 ) ) ).
% finsert_absorb2
thf(fact_186_finsertCI,axiom,
! [A: $tType,A2: A,B2: fset @ A,B3: A] :
( ( ~ ( fmember @ A @ A2 @ B2 )
=> ( A2 = B3 ) )
=> ( fmember @ A @ A2 @ ( finsert @ A @ B3 @ B2 ) ) ) ).
% finsertCI
thf(fact_187_finsert__iff,axiom,
! [A: $tType,A2: A,B3: A,A4: fset @ A] :
( ( fmember @ A @ A2 @ ( finsert @ A @ B3 @ A4 ) )
= ( ( A2 = B3 )
| ( fmember @ A @ A2 @ A4 ) ) ) ).
% finsert_iff
thf(fact_188_fimage__finsert,axiom,
! [A: $tType,B: $tType,F2: B > A,A2: B,B2: fset @ B] :
( ( fimage @ B @ A @ F2 @ ( finsert @ B @ A2 @ B2 ) )
= ( finsert @ A @ ( F2 @ A2 ) @ ( fimage @ B @ A @ F2 @ B2 ) ) ) ).
% fimage_finsert
thf(fact_189_finsert__fsubset,axiom,
! [A: $tType,X2: A,A4: fset @ A,B2: fset @ A] :
( ( ord_less_eq @ ( fset @ A ) @ ( finsert @ A @ X2 @ A4 ) @ B2 )
= ( ( fmember @ A @ X2 @ B2 )
& ( ord_less_eq @ ( fset @ A ) @ A4 @ B2 ) ) ) ).
% finsert_fsubset
thf(fact_190_finsert__fimage,axiom,
! [B: $tType,A: $tType,X2: A,A4: fset @ A,F2: A > B] :
( ( fmember @ A @ X2 @ A4 )
=> ( ( finsert @ B @ ( F2 @ X2 ) @ ( fimage @ A @ B @ F2 @ A4 ) )
= ( fimage @ A @ B @ F2 @ A4 ) ) ) ).
% finsert_fimage
thf(fact_191_finsert__mono,axiom,
! [A: $tType,C3: fset @ A,D2: fset @ A,A2: A] :
( ( ord_less_eq @ ( fset @ A ) @ C3 @ D2 )
=> ( ord_less_eq @ ( fset @ A ) @ ( finsert @ A @ A2 @ C3 ) @ ( finsert @ A @ A2 @ D2 ) ) ) ).
% finsert_mono
thf(fact_192_fsubset__finsertI,axiom,
! [A: $tType,B2: fset @ A,A2: A] : ( ord_less_eq @ ( fset @ A ) @ B2 @ ( finsert @ A @ A2 @ B2 ) ) ).
% fsubset_finsertI
thf(fact_193_fsubset__finsertI2,axiom,
! [A: $tType,A4: fset @ A,B2: fset @ A,B3: A] :
( ( ord_less_eq @ ( fset @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ ( fset @ A ) @ A4 @ ( finsert @ A @ B3 @ B2 ) ) ) ).
% fsubset_finsertI2
thf(fact_194_finsert__commute,axiom,
! [A: $tType,X2: A,Y: A,A4: fset @ A] :
( ( finsert @ A @ X2 @ ( finsert @ A @ Y @ A4 ) )
= ( finsert @ A @ Y @ ( finsert @ A @ X2 @ A4 ) ) ) ).
% finsert_commute
thf(fact_195_finsertE,axiom,
! [A: $tType,A2: A,B3: A,A4: fset @ A] :
( ( fmember @ A @ A2 @ ( finsert @ A @ B3 @ A4 ) )
=> ( ( A2 != B3 )
=> ( fmember @ A @ A2 @ A4 ) ) ) ).
% finsertE
thf(fact_196_finsertI1,axiom,
! [A: $tType,A2: A,B2: fset @ A] : ( fmember @ A @ A2 @ ( finsert @ A @ A2 @ B2 ) ) ).
% finsertI1
thf(fact_197_finsertI2,axiom,
! [A: $tType,A2: A,B2: fset @ A,B3: A] :
( ( fmember @ A @ A2 @ B2 )
=> ( fmember @ A @ A2 @ ( finsert @ A @ B3 @ B2 ) ) ) ).
% finsertI2
thf(fact_198_set__finsert,axiom,
! [A: $tType,X2: A,A4: fset @ A] :
( ( fmember @ A @ X2 @ A4 )
=> ~ ! [B7: fset @ A] :
( ( A4
= ( finsert @ A @ X2 @ B7 ) )
=> ( fmember @ A @ X2 @ B7 ) ) ) ).
% set_finsert
thf(fact_199_finsert__ident,axiom,
! [A: $tType,X2: A,A4: fset @ A,B2: fset @ A] :
( ~ ( fmember @ A @ X2 @ A4 )
=> ( ~ ( fmember @ A @ X2 @ B2 )
=> ( ( ( finsert @ A @ X2 @ A4 )
= ( finsert @ A @ X2 @ B2 ) )
= ( A4 = B2 ) ) ) ) ).
% finsert_ident
thf(fact_200_finsert__absorb,axiom,
! [A: $tType,A2: A,A4: fset @ A] :
( ( fmember @ A @ A2 @ A4 )
=> ( ( finsert @ A @ A2 @ A4 )
= A4 ) ) ).
% finsert_absorb
thf(fact_201_mk__disjoint__finsert,axiom,
! [A: $tType,A2: A,A4: fset @ A] :
( ( fmember @ A @ A2 @ A4 )
=> ? [B7: fset @ A] :
( ( A4
= ( finsert @ A @ A2 @ B7 ) )
& ~ ( fmember @ A @ A2 @ B7 ) ) ) ).
% mk_disjoint_finsert
thf(fact_202_fsubset__finsert,axiom,
! [A: $tType,X2: A,A4: fset @ A,B2: fset @ A] :
( ~ ( fmember @ A @ X2 @ A4 )
=> ( ( ord_less_eq @ ( fset @ A ) @ A4 @ ( finsert @ A @ X2 @ B2 ) )
= ( ord_less_eq @ ( fset @ A ) @ A4 @ B2 ) ) ) ).
% fsubset_finsert
thf(fact_203_fun__quotient3,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,R22: C > C > $o,Abs2: C > D,Rep2: D > C] :
( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
=> ( ( quotient3 @ C @ D @ R22 @ Abs2 @ Rep2 )
=> ( quotient3 @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ A @ A @ C @ C @ R1 @ R22 ) @ ( map_fun @ B @ A @ C @ D @ Rep12 @ Abs2 ) @ ( map_fun @ A @ B @ D @ C @ Abs12 @ Rep2 ) ) ) ) ).
% fun_quotient3
thf(fact_204_sfilter__P,axiom,
! [A: $tType,P: A > $o,S3: stream @ A] :
( ( P @ ( shd @ A @ S3 ) )
=> ( ( sfilter @ A @ P @ S3 )
= ( sCons @ A @ ( shd @ A @ S3 ) @ ( sfilter @ A @ P @ ( stl @ A @ S3 ) ) ) ) ) ).
% sfilter_P
thf(fact_205_smap__ctr,axiom,
! [B: $tType,A: $tType,F2: B > A,S3: stream @ B,X2: A,S4: stream @ A] :
( ( ( smap @ B @ A @ F2 @ S3 )
= ( sCons @ A @ X2 @ S4 ) )
= ( ( ( F2 @ ( shd @ B @ S3 ) )
= X2 )
& ( ( smap @ B @ A @ F2 @ ( stl @ B @ S3 ) )
= S4 ) ) ) ).
% smap_ctr
thf(fact_206_stream_Omap__sel_I2_J,axiom,
! [B: $tType,A: $tType,F2: A > B,A2: stream @ A] :
( ( stl @ B @ ( smap @ A @ B @ F2 @ A2 ) )
= ( smap @ A @ B @ F2 @ ( stl @ A @ A2 ) ) ) ).
% stream.map_sel(2)
thf(fact_207_stream_Omap__sel_I1_J,axiom,
! [B: $tType,A: $tType,F2: A > B,A2: stream @ A] :
( ( shd @ B @ ( smap @ A @ B @ F2 @ A2 ) )
= ( F2 @ ( shd @ A @ A2 ) ) ) ).
% stream.map_sel(1)
thf(fact_208_sfilter__not__P,axiom,
! [A: $tType,P: A > $o,S3: stream @ A] :
( ~ ( P @ ( shd @ A @ S3 ) )
=> ( ( sfilter @ A @ P @ S3 )
= ( sfilter @ A @ P @ ( stl @ A @ S3 ) ) ) ) ).
% sfilter_not_P
thf(fact_209_stream_Omap,axiom,
! [B: $tType,A: $tType,F2: A > B,X1: A,X22: stream @ A] :
( ( smap @ A @ B @ F2 @ ( sCons @ A @ X1 @ X22 ) )
= ( sCons @ B @ ( F2 @ X1 ) @ ( smap @ A @ B @ F2 @ X22 ) ) ) ).
% stream.map
thf(fact_210_sfilter__Stream,axiom,
! [A: $tType,P: A > $o,X2: A,S3: stream @ A] :
( ( ( P @ X2 )
=> ( ( sfilter @ A @ P @ ( sCons @ A @ X2 @ S3 ) )
= ( sCons @ A @ X2 @ ( sfilter @ A @ P @ S3 ) ) ) )
& ( ~ ( P @ X2 )
=> ( ( sfilter @ A @ P @ ( sCons @ A @ X2 @ S3 ) )
= ( sfilter @ A @ P @ S3 ) ) ) ) ).
% sfilter_Stream
thf(fact_211_sfilter_Ocode,axiom,
! [A: $tType] :
( ( sfilter @ A )
= ( ^ [P2: A > $o,S: stream @ A] : ( sCons @ A @ ( shd @ A @ ( sdrop_while @ A @ ( comp @ $o @ $o @ A @ (~) @ P2 ) @ S ) ) @ ( sfilter @ A @ P2 @ ( stl @ A @ ( sdrop_while @ A @ ( comp @ $o @ $o @ A @ (~) @ P2 ) @ S ) ) ) ) ) ) ).
% sfilter.code
thf(fact_212_nxt_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( linear1494993505on_nxt @ A @ B )
= ( ^ [Phi: ( stream @ A ) > B,Xs: stream @ A] : ( Phi @ ( stl @ A @ Xs ) ) ) ) ).
% nxt.simps
thf(fact_213_fset_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: B > C,F2: A > B,V: fset @ A] :
( ( fimage @ B @ C @ G @ ( fimage @ A @ B @ F2 @ V ) )
= ( fimage @ A @ C @ ( comp @ B @ C @ A @ G @ F2 ) @ V ) ) ).
% fset.map_comp
thf(fact_214_stream_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: B > C,F2: A > B,V: stream @ A] :
( ( smap @ B @ C @ G @ ( smap @ A @ B @ F2 @ V ) )
= ( smap @ A @ C @ ( comp @ B @ C @ A @ G @ F2 ) @ V ) ) ).
% stream.map_comp
thf(fact_215_fset_Opred__map,axiom,
! [B: $tType,A: $tType,Q: B > $o,F2: A > B,X2: fset @ A] :
( ( pred_fset @ B @ Q @ ( fimage @ A @ B @ F2 @ X2 ) )
= ( pred_fset @ A @ ( comp @ B @ $o @ A @ Q @ F2 ) @ X2 ) ) ).
% fset.pred_map
thf(fact_216_fun_Omap__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,G4: $tType,F4: $tType,Rb: A > F4 > $o,Sd: B > G4 > $o] :
( bNF_rel_fun @ ( A > B ) @ ( F4 > G4 ) @ ( ( D > A ) > D > B ) @ ( ( D > F4 ) > D > G4 ) @ ( bNF_rel_fun @ A @ F4 @ B @ G4 @ Rb @ Sd )
@ ( bNF_rel_fun @ ( D > A ) @ ( D > F4 ) @ ( D > B ) @ ( D > G4 )
@ ( bNF_rel_fun @ D @ D @ A @ F4
@ ^ [Y3: D,Z: D] : Y3 = Z
@ Rb )
@ ( bNF_rel_fun @ D @ D @ B @ G4
@ ^ [Y3: D,Z: D] : Y3 = Z
@ Sd ) )
@ ( comp @ A @ B @ D )
@ ( comp @ F4 @ G4 @ D ) ) ).
% fun.map_transfer
thf(fact_217_o__rsp_I2_J,axiom,
! [E: $tType,F4: $tType,H: $tType,G4: $tType,R1: E > F4 > $o] :
( bNF_rel_fun @ ( G4 > H ) @ ( G4 > H ) @ ( ( E > G4 ) > E > H ) @ ( ( F4 > G4 ) > F4 > H )
@ ^ [Y3: G4 > H,Z: G4 > H] : Y3 = Z
@ ( bNF_rel_fun @ ( E > G4 ) @ ( F4 > G4 ) @ ( E > H ) @ ( F4 > H )
@ ( bNF_rel_fun @ E @ F4 @ G4 @ G4 @ R1
@ ^ [Y3: G4,Z: G4] : Y3 = Z )
@ ( bNF_rel_fun @ E @ F4 @ H @ H @ R1
@ ^ [Y3: H,Z: H] : Y3 = Z ) )
@ ( comp @ G4 @ H @ E )
@ ( comp @ G4 @ H @ F4 ) ) ).
% o_rsp(2)
thf(fact_218_o__rsp_I1_J,axiom,
! [A: $tType,B: $tType,E: $tType,F4: $tType,D: $tType,C: $tType,R22: A > C > $o,R3: B > D > $o,R1: E > F4 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > A ) > E > B ) @ ( ( F4 > C ) > F4 > D ) @ ( bNF_rel_fun @ A @ C @ B @ D @ R22 @ R3 ) @ ( bNF_rel_fun @ ( E > A ) @ ( F4 > C ) @ ( E > B ) @ ( F4 > D ) @ ( bNF_rel_fun @ E @ F4 @ A @ C @ R1 @ R22 ) @ ( bNF_rel_fun @ E @ F4 @ B @ D @ R1 @ R3 ) ) @ ( comp @ A @ B @ E ) @ ( comp @ C @ D @ F4 ) ) ).
% o_rsp(1)
thf(fact_219_comp__transfer,axiom,
! [A: $tType,B: $tType,E: $tType,F4: $tType,D: $tType,C: $tType,B2: A > C > $o,C3: B > D > $o,A4: E > F4 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > A ) > E > B ) @ ( ( F4 > C ) > F4 > D ) @ ( bNF_rel_fun @ A @ C @ B @ D @ B2 @ C3 ) @ ( bNF_rel_fun @ ( E > A ) @ ( F4 > C ) @ ( E > B ) @ ( F4 > D ) @ ( bNF_rel_fun @ E @ F4 @ A @ C @ A4 @ B2 ) @ ( bNF_rel_fun @ E @ F4 @ B @ D @ A4 @ C3 ) ) @ ( comp @ A @ B @ E ) @ ( comp @ C @ D @ F4 ) ) ).
% comp_transfer
thf(fact_220_sfilter_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,S3: stream @ A] :
( ( stl @ A @ ( sfilter @ A @ P @ S3 ) )
= ( sfilter @ A @ P @ ( stl @ A @ ( sdrop_while @ A @ ( comp @ $o @ $o @ A @ (~) @ P ) @ S3 ) ) ) ) ).
% sfilter.simps(2)
thf(fact_221_sfilter_Osimps_I1_J,axiom,
! [A: $tType,P: A > $o,S3: stream @ A] :
( ( shd @ A @ ( sfilter @ A @ P @ S3 ) )
= ( shd @ A @ ( sdrop_while @ A @ ( comp @ $o @ $o @ A @ (~) @ P ) @ S3 ) ) ) ).
% sfilter.simps(1)
thf(fact_222_nxt_Oelims,axiom,
! [B: $tType,A: $tType,X2: ( stream @ A ) > B,Xa: stream @ A,Y: B] :
( ( ( linear1494993505on_nxt @ A @ B @ X2 @ Xa )
= Y )
=> ( Y
= ( X2 @ ( stl @ A @ Xa ) ) ) ) ).
% nxt.elims
thf(fact_223_stream_Omap__o__corec,axiom,
! [A: $tType,B: $tType,C: $tType,F2: A > B,G: C > A,Ga: C > $o,Gb: C > ( stream @ A ),Gc: C > C] :
( ( comp @ ( stream @ A ) @ ( stream @ B ) @ C @ ( smap @ A @ B @ F2 ) @ ( corec_stream @ C @ A @ G @ Ga @ Gb @ Gc ) )
= ( corec_stream @ C @ B @ ( comp @ A @ B @ C @ F2 @ G ) @ Ga @ ( comp @ ( stream @ A ) @ ( stream @ B ) @ C @ ( smap @ A @ B @ F2 ) @ Gb ) @ Gc ) ) ).
% stream.map_o_corec
thf(fact_224_stream_Opred__map,axiom,
! [B: $tType,A: $tType,Q: B > $o,F2: A > B,X2: stream @ A] :
( ( pred_stream @ B @ Q @ ( smap @ A @ B @ F2 @ X2 ) )
= ( pred_stream @ A @ ( comp @ B @ $o @ A @ Q @ F2 ) @ X2 ) ) ).
% stream.pred_map
thf(fact_225_stream_Opred__inject,axiom,
! [A: $tType,P: A > $o,A2: A,Aa2: stream @ A] :
( ( pred_stream @ A @ P @ ( sCons @ A @ A2 @ Aa2 ) )
= ( ( P @ A2 )
& ( pred_stream @ A @ P @ Aa2 ) ) ) ).
% stream.pred_inject
thf(fact_226_stream_Ocorec__sel_I2_J,axiom,
! [A: $tType,C: $tType,Q2: C > $o,A2: C,G1: C > A,G21: C > ( stream @ A ),G22: C > C] :
( ( ( Q2 @ A2 )
=> ( ( stl @ A @ ( corec_stream @ C @ A @ G1 @ Q2 @ G21 @ G22 @ A2 ) )
= ( G21 @ A2 ) ) )
& ( ~ ( Q2 @ A2 )
=> ( ( stl @ A @ ( corec_stream @ C @ A @ G1 @ Q2 @ G21 @ G22 @ A2 ) )
= ( corec_stream @ C @ A @ G1 @ Q2 @ G21 @ G22 @ ( G22 @ A2 ) ) ) ) ) ).
% stream.corec_sel(2)
thf(fact_227_stream_Ocorec__sel_I1_J,axiom,
! [A: $tType,C: $tType,G1: C > A,Q2: C > $o,G21: C > ( stream @ A ),G22: C > C,A2: C] :
( ( shd @ A @ ( corec_stream @ C @ A @ G1 @ Q2 @ G21 @ G22 @ A2 ) )
= ( G1 @ A2 ) ) ).
% stream.corec_sel(1)
thf(fact_228_stream_Ocorec__code,axiom,
! [A: $tType,C: $tType] :
( ( corec_stream @ C @ A )
= ( ^ [G12: C > A,Q22: C > $o,G212: C > ( stream @ A ),G222: C > C,A3: C] : ( sCons @ A @ ( G12 @ A3 ) @ ( if @ ( stream @ A ) @ ( Q22 @ A3 ) @ ( G212 @ A3 ) @ ( corec_stream @ C @ A @ G12 @ Q22 @ G212 @ G222 @ ( G222 @ A3 ) ) ) ) ) ) ).
% stream.corec_code
thf(fact_229_stream_Ocorec__disc,axiom,
! [A: $tType,C: $tType] :
( ( corec_stream @ C @ A )
= ( corec_stream @ C @ A ) ) ).
% stream.corec_disc
thf(fact_230_fun__ord__parametric,axiom,
! [C: $tType,D: $tType,A: $tType,B: $tType,F4: $tType,E: $tType,C3: A > B > $o,A4: C > E > $o,B2: D > F4 > $o] :
( ( bi_total @ A @ B @ C3 )
=> ( bNF_rel_fun @ ( C > D > $o ) @ ( E > F4 > $o ) @ ( ( A > C ) > ( A > D ) > $o ) @ ( ( B > E ) > ( B > F4 ) > $o )
@ ( bNF_rel_fun @ C @ E @ ( D > $o ) @ ( F4 > $o ) @ A4
@ ( bNF_rel_fun @ D @ F4 @ $o @ $o @ B2
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ ( A > C ) @ ( B > E ) @ ( ( A > D ) > $o ) @ ( ( B > F4 ) > $o ) @ ( bNF_rel_fun @ A @ B @ C @ E @ C3 @ A4 )
@ ( bNF_rel_fun @ ( A > D ) @ ( B > F4 ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ D @ F4 @ C3 @ B2 )
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( partial_fun_ord @ C @ D @ A )
@ ( partial_fun_ord @ E @ F4 @ B ) ) ) ).
% fun_ord_parametric
thf(fact_231_stream_Ocorec__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,S2: C > D > $o,R: A > B > $o] :
( bNF_rel_fun @ ( C > A ) @ ( D > B ) @ ( ( C > $o ) > ( C > ( stream @ A ) ) > ( C > C ) > C > ( stream @ A ) ) @ ( ( D > $o ) > ( D > ( stream @ B ) ) > ( D > D ) > D > ( stream @ B ) ) @ ( bNF_rel_fun @ C @ D @ A @ B @ S2 @ R )
@ ( bNF_rel_fun @ ( C > $o ) @ ( D > $o ) @ ( ( C > ( stream @ A ) ) > ( C > C ) > C > ( stream @ A ) ) @ ( ( D > ( stream @ B ) ) > ( D > D ) > D > ( stream @ B ) )
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ S2
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( bNF_rel_fun @ ( C > ( stream @ A ) ) @ ( D > ( stream @ B ) ) @ ( ( C > C ) > C > ( stream @ A ) ) @ ( ( D > D ) > D > ( stream @ B ) ) @ ( bNF_rel_fun @ C @ D @ ( stream @ A ) @ ( stream @ B ) @ S2 @ ( stream_all2 @ A @ B @ R ) ) @ ( bNF_rel_fun @ ( C > C ) @ ( D > D ) @ ( C > ( stream @ A ) ) @ ( D > ( stream @ B ) ) @ ( bNF_rel_fun @ C @ D @ C @ D @ S2 @ S2 ) @ ( bNF_rel_fun @ C @ D @ ( stream @ A ) @ ( stream @ B ) @ S2 @ ( stream_all2 @ A @ B @ R ) ) ) ) )
@ ( corec_stream @ C @ A )
@ ( corec_stream @ D @ B ) ) ).
% stream.corec_transfer
thf(fact_232_stream_Orel__inject,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X1: A,X22: stream @ A,Y1: B,Y2: stream @ B] :
( ( stream_all2 @ A @ B @ R @ ( sCons @ A @ X1 @ X22 ) @ ( sCons @ B @ Y1 @ Y2 ) )
= ( ( R @ X1 @ Y1 )
& ( stream_all2 @ A @ B @ R @ X22 @ Y2 ) ) ) ).
% stream.rel_inject
thf(fact_233_stream_Opred__transfer,axiom,
! [A: $tType,B: $tType,R: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( stream @ A ) > $o ) @ ( ( stream @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ R
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( bNF_rel_fun @ ( stream @ A ) @ ( stream @ B ) @ $o @ $o @ ( stream_all2 @ A @ B @ R )
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( pred_stream @ A )
@ ( pred_stream @ B ) ) ).
% stream.pred_transfer
thf(fact_234_stream_Orel__eq__onp,axiom,
! [A: $tType,P: A > $o] :
( ( stream_all2 @ A @ A @ ( bNF_eq_onp @ A @ P ) )
= ( bNF_eq_onp @ ( stream @ A ) @ ( pred_stream @ A @ P ) ) ) ).
% stream.rel_eq_onp
thf(fact_235_stream_Opred__rel,axiom,
! [A: $tType] :
( ( pred_stream @ A )
= ( ^ [P2: A > $o,X: stream @ A] : ( stream_all2 @ A @ A @ ( bNF_eq_onp @ A @ P2 ) @ X @ X ) ) ) ).
% stream.pred_rel
thf(fact_236_stream_Omap__transfer,axiom,
! [A: $tType,B: $tType,F4: $tType,E: $tType,Rb: A > E > $o,Sd: B > F4 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( E > F4 ) @ ( ( stream @ A ) > ( stream @ B ) ) @ ( ( stream @ E ) > ( stream @ F4 ) ) @ ( bNF_rel_fun @ A @ E @ B @ F4 @ Rb @ Sd ) @ ( bNF_rel_fun @ ( stream @ A ) @ ( stream @ E ) @ ( stream @ B ) @ ( stream @ F4 ) @ ( stream_all2 @ A @ E @ Rb ) @ ( stream_all2 @ B @ F4 @ Sd ) ) @ ( smap @ A @ B ) @ ( smap @ E @ F4 ) ) ).
% stream.map_transfer
thf(fact_237_stream_Orel__mono,axiom,
! [B: $tType,A: $tType,R: A > B > $o,Ra: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R @ Ra )
=> ( ord_less_eq @ ( ( stream @ A ) > ( stream @ B ) > $o ) @ ( stream_all2 @ A @ B @ R ) @ ( stream_all2 @ A @ B @ Ra ) ) ) ).
% stream.rel_mono
thf(fact_238_stream_Obi__total__rel,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( bi_total @ A @ B @ R )
=> ( bi_total @ ( stream @ A ) @ ( stream @ B ) @ ( stream_all2 @ A @ B @ R ) ) ) ).
% stream.bi_total_rel
thf(fact_239_stream_Orel__eq,axiom,
! [A: $tType] :
( ( stream_all2 @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [Y3: stream @ A,Z: stream @ A] : Y3 = Z ) ) ).
% stream.rel_eq
thf(fact_240_stream_Orel__refl,axiom,
! [B: $tType,Ra: B > B > $o,X2: stream @ B] :
( ! [X3: B] : ( Ra @ X3 @ X3 )
=> ( stream_all2 @ B @ B @ Ra @ X2 @ X2 ) ) ).
% stream.rel_refl
thf(fact_241_tree_Obi__total__rel,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( bi_total @ A @ B @ R )
=> ( bi_total @ ( abstra2103299360e_tree @ A ) @ ( abstra2103299360e_tree @ B ) @ ( abstra2101783510l_tree @ A @ B @ R ) ) ) ).
% tree.bi_total_rel
thf(fact_242_fset_Obi__total__rel,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( bi_total @ A @ B @ R )
=> ( bi_total @ ( fset @ A ) @ ( fset @ B ) @ ( rel_fset @ A @ B @ R ) ) ) ).
% fset.bi_total_rel
thf(fact_243_stream_Orel__intros,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X1: A,Y1: B,X22: stream @ A,Y2: stream @ B] :
( ( R @ X1 @ Y1 )
=> ( ( stream_all2 @ A @ B @ R @ X22 @ Y2 )
=> ( stream_all2 @ A @ B @ R @ ( sCons @ A @ X1 @ X22 ) @ ( sCons @ B @ Y1 @ Y2 ) ) ) ) ).
% stream.rel_intros
thf(fact_244_stream_Orel__cases,axiom,
! [A: $tType,B: $tType,R: A > B > $o,A2: stream @ A,B3: stream @ B] :
( ( stream_all2 @ A @ B @ R @ A2 @ B3 )
=> ~ ! [X1a: A,X2a: stream @ A] :
( ( A2
= ( sCons @ A @ X1a @ X2a ) )
=> ! [Y1a: B,Y2a: stream @ B] :
( ( B3
= ( sCons @ B @ Y1a @ Y2a ) )
=> ( ( R @ X1a @ Y1a )
=> ~ ( stream_all2 @ A @ B @ R @ X2a @ Y2a ) ) ) ) ) ).
% stream.rel_cases
thf(fact_245_stream_Obi__unique__rel,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( bi_unique @ A @ B @ R )
=> ( bi_unique @ ( stream @ A ) @ ( stream @ B ) @ ( stream_all2 @ A @ B @ R ) ) ) ).
% stream.bi_unique_rel
thf(fact_246_stream_Orel__sel,axiom,
! [B: $tType,A: $tType] :
( ( stream_all2 @ A @ B )
= ( ^ [R2: A > B > $o,A3: stream @ A,B4: stream @ B] :
( ( R2 @ ( shd @ A @ A3 ) @ ( shd @ B @ B4 ) )
& ( stream_all2 @ A @ B @ R2 @ ( stl @ A @ A3 ) @ ( stl @ B @ B4 ) ) ) ) ) ).
% stream.rel_sel
thf(fact_247_stream_Orel__coinduct,axiom,
! [A: $tType,B: $tType,P: ( stream @ A ) > ( stream @ B ) > $o,X2: stream @ A,Y: stream @ B,R: A > B > $o] :
( ( P @ X2 @ Y )
=> ( ! [Stream3: stream @ A,Stream4: stream @ B] :
( ( P @ Stream3 @ Stream4 )
=> ( ( R @ ( shd @ A @ Stream3 ) @ ( shd @ B @ Stream4 ) )
& ( P @ ( stl @ A @ Stream3 ) @ ( stl @ B @ Stream4 ) ) ) )
=> ( stream_all2 @ A @ B @ R @ X2 @ Y ) ) ) ).
% stream.rel_coinduct
thf(fact_248_stream_Orel__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: A > C > $o,Sc: B > D > $o] :
( bNF_rel_fun @ ( A > B > $o ) @ ( C > D > $o ) @ ( ( stream @ A ) > ( stream @ B ) > $o ) @ ( ( stream @ C ) > ( stream @ D ) > $o )
@ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( D > $o ) @ Sa
@ ( bNF_rel_fun @ B @ D @ $o @ $o @ Sc
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ ( stream @ A ) @ ( stream @ C ) @ ( ( stream @ B ) > $o ) @ ( ( stream @ D ) > $o ) @ ( stream_all2 @ A @ C @ Sa )
@ ( bNF_rel_fun @ ( stream @ B ) @ ( stream @ D ) @ $o @ $o @ ( stream_all2 @ B @ D @ Sc )
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( stream_all2 @ A @ B )
@ ( stream_all2 @ C @ D ) ) ).
% stream.rel_transfer
thf(fact_249_bi__unique__fun,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A4: A > B > $o,B2: C > D > $o] :
( ( bi_total @ A @ B @ A4 )
=> ( ( bi_unique @ C @ D @ B2 )
=> ( bi_unique @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A4 @ B2 ) ) ) ) ).
% bi_unique_fun
thf(fact_250_stream_Ocase__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R: A > B > $o,S2: C > D > $o] : ( bNF_rel_fun @ ( A > ( stream @ A ) > C ) @ ( B > ( stream @ B ) > D ) @ ( ( stream @ A ) > C ) @ ( ( stream @ B ) > D ) @ ( bNF_rel_fun @ A @ B @ ( ( stream @ A ) > C ) @ ( ( stream @ B ) > D ) @ R @ ( bNF_rel_fun @ ( stream @ A ) @ ( stream @ B ) @ C @ D @ ( stream_all2 @ A @ B @ R ) @ S2 ) ) @ ( bNF_rel_fun @ ( stream @ A ) @ ( stream @ B ) @ C @ D @ ( stream_all2 @ A @ B @ R ) @ S2 ) @ ( case_stream @ A @ C ) @ ( case_stream @ B @ D ) ) ).
% stream.case_transfer
thf(fact_251_bi__total__fun,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A4: A > B > $o,B2: C > D > $o] :
( ( bi_unique @ A @ B @ A4 )
=> ( ( bi_total @ C @ D @ B2 )
=> ( bi_total @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A4 @ B2 ) ) ) ) ).
% bi_total_fun
thf(fact_252_stream_Octr__transfer,axiom,
! [A: $tType,B: $tType,R: A > B > $o] : ( bNF_rel_fun @ A @ B @ ( ( stream @ A ) > ( stream @ A ) ) @ ( ( stream @ B ) > ( stream @ B ) ) @ R @ ( bNF_rel_fun @ ( stream @ A ) @ ( stream @ B ) @ ( stream @ A ) @ ( stream @ B ) @ ( stream_all2 @ A @ B @ R ) @ ( stream_all2 @ A @ B @ R ) ) @ ( sCons @ A ) @ ( sCons @ B ) ) ).
% stream.ctr_transfer
thf(fact_253_stream_Osel__transfer_I1_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o] : ( bNF_rel_fun @ ( stream @ A ) @ ( stream @ B ) @ A @ B @ ( stream_all2 @ A @ B @ R ) @ R @ ( shd @ A ) @ ( shd @ B ) ) ).
% stream.sel_transfer(1)
thf(fact_254_stream_Osel__transfer_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o] : ( bNF_rel_fun @ ( stream @ A ) @ ( stream @ B ) @ ( stream @ A ) @ ( stream @ B ) @ ( stream_all2 @ A @ B @ R ) @ ( stream_all2 @ A @ B @ R ) @ ( stl @ A ) @ ( stl @ B ) ) ).
% stream.sel_transfer(2)
thf(fact_255_monotone__parametric,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A4: A > B > $o,B2: C > D > $o] :
( ( bi_total @ A @ B @ A4 )
=> ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ ( ( C > C > $o ) > ( A > C ) > $o ) @ ( ( D > D > $o ) > ( B > D ) > $o )
@ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A4
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A4
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ ( C > C > $o ) @ ( D > D > $o ) @ ( ( A > C ) > $o ) @ ( ( B > D ) > $o )
@ ( bNF_rel_fun @ C @ D @ ( C > $o ) @ ( D > $o ) @ B2
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ B2
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ C @ D @ A4 @ B2 )
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( comple1396247847notone @ A @ C )
@ ( comple1396247847notone @ B @ D ) ) ) ).
% monotone_parametric
%----Type constructors (10)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A8: $tType] :
( ( preorder @ A8 @ ( type2 @ A8 ) )
=> ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A7: $tType,A8: $tType] :
( ( order @ A8 @ ( type2 @ A8 ) )
=> ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 @ ( type2 @ A8 ) )
=> ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_1,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_2,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_3,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_FSet_Ofset___Orderings_Opreorder_4,axiom,
! [A7: $tType] : ( preorder @ ( fset @ A7 ) @ ( type2 @ ( fset @ A7 ) ) ) ).
thf(tcon_FSet_Ofset___Orderings_Oorder_5,axiom,
! [A7: $tType] : ( order @ ( fset @ A7 ) @ ( type2 @ ( fset @ A7 ) ) ) ).
thf(tcon_FSet_Ofset___Orderings_Oord_6,axiom,
! [A7: $tType] : ( ord @ ( fset @ A7 ) @ ( type2 @ ( fset @ A7 ) ) ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X2: A,Y: A] :
( ( if @ A @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X2: A,Y: A] :
( ( if @ A @ $true @ X2 @ Y )
= X2 ) ).
%----Conjectures (2)
thf(conj_0,hypothesis,
abstra668420080finite @ a @ t ).
thf(conj_1,conjecture,
~ ( abstra313004635_ipath @ a @ t @ steps ) ).
%------------------------------------------------------------------------------