TPTP Problem File: COM163^1.p
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%------------------------------------------------------------------------------
% File : COM163^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Binary decision diagram 95
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [OS08] Ortner & Schirmer (2008), BDD Normalisation
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : bindag__95.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.1.0
% Syntax : Number of formulae : 340 ( 55 unt; 57 typ; 0 def)
% Number of atoms : 945 ( 226 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 3514 ( 96 ~; 25 |; 58 &;2805 @)
% ( 0 <=>; 530 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 246 ( 246 >; 0 *; 0 +; 0 <<)
% Number of symbols : 57 ( 54 usr; 6 con; 0-4 aty)
% Number of variables : 987 ( 32 ^; 872 !; 38 ?; 987 :)
% ( 45 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:45:00.755
%------------------------------------------------------------------------------
%----Could-be-implicit typings (5)
thf(ty_t_BinDag__Mirabelle__rybootvolr_Odag,type,
binDag_Mirabelle_dag: $tType ).
thf(ty_t_Simpl__Heap_Oref,type,
simpl_ref: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
%----Explicit typings (52)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Fields_Ofield,type,
field:
!>[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_Ono__bot,type,
no_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[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_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Ozero__less__one,type,
zero_less_one:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Fields_Odivision__ring,type,
division_ring:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Osemidom__divide,type,
semidom_divide:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord1659791738miring:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit1656338222tinuum:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit1037483654norder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_BinDag__Mirabelle__rybootvolr_ODAG,type,
binDag_Mirabelle_DAG: binDag_Mirabelle_dag > $o ).
thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_ONode,type,
binDag476092410e_Node: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > binDag_Mirabelle_dag ).
thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_OTip,type,
binDag_Mirabelle_Tip: binDag_Mirabelle_dag ).
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 ) ).
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 ) ).
thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_Osize__dag,type,
binDag1924123185ze_dag: binDag_Mirabelle_dag > nat ).
thf(sy_c_BinDag__Mirabelle__rybootvolr_Oset__of,type,
binDag1380252983set_of: binDag_Mirabelle_dag > ( set @ simpl_ref ) ).
thf(sy_c_BinDag__Mirabelle__rybootvolr_Osubdag,type,
binDag786255756subdag: binDag_Mirabelle_dag > binDag_Mirabelle_dag > $o ).
thf(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_Groups_Osgn__class_Osgn,type,
sgn_sgn:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Orderings_Oord__class_OLeast,type,
ord_Least:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oorder__class_Oantimono,type,
order_antimono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
order_strict_mono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oordering,type,
ordering:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Rings_Odivide__class_Odivide,type,
divide_divide:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
zero_neq_one_of_bool:
!>[A: $tType] : ( $o > A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_x,type,
x: binDag_Mirabelle_dag ).
thf(sy_v_y,type,
y: binDag_Mirabelle_dag ).
thf(sy_v_z,type,
z: binDag_Mirabelle_dag ).
%----Relevant facts (256)
thf(fact_0_True,axiom,
x = y ).
% True
thf(fact_1_y__z,axiom,
ord_less_eq @ binDag_Mirabelle_dag @ y @ z ).
% y_z
thf(fact_2_x__y,axiom,
ord_less_eq @ binDag_Mirabelle_dag @ x @ y ).
% x_y
thf(fact_3_le__dag__refl,axiom,
! [X: binDag_Mirabelle_dag] : ( ord_less_eq @ binDag_Mirabelle_dag @ X @ X ) ).
% le_dag_refl
thf(fact_4_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_5_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_6_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_7_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B] :
( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_8_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% le_fun_def
thf(fact_9_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X2: B,Y: B] :
( ( ord_less_eq @ B @ X2 @ Y )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_10_order__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
=> ( ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_11_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X2: B,Y: B] :
( ( ord_less_eq @ B @ X2 @ Y )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_12_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > B,C: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_13_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y2: A,Z: A] : Y2 = Z )
= ( ^ [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% eq_iff
thf(fact_14_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ( ord_less_eq @ A @ Y4 @ X )
=> ( X = Y4 ) ) ) ) ).
% antisym
thf(fact_15_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
| ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% linear
thf(fact_16_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 ) ) ) ) ).
% dual_order.antisym
thf(fact_17_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 ) ) ) ) ).
% dual_order.trans
thf(fact_18_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: A,B3: A] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_19_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_20_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ( ord_less_eq @ A @ Y4 @ Z2 )
=> ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% order_trans
thf(fact_21_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 ) ) ) ) ).
% order_class.order.antisym
thf(fact_22_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 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_23_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 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_24_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y4: A,X: A] :
( ( ord_less_eq @ A @ Y4 @ X )
=> ( ( ord_less_eq @ A @ X @ Y4 )
= ( X = Y4 ) ) ) ) ).
% antisym_conv
thf(fact_25_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A,Z2: A] :
( ( ( ord_less_eq @ A @ X @ Y4 )
=> ~ ( ord_less_eq @ A @ Y4 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y4 @ X )
=> ~ ( ord_less_eq @ A @ X @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y4 ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y4 )
=> ~ ( ord_less_eq @ A @ Y4 @ X ) )
=> ( ( ( ord_less_eq @ A @ Y4 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y4 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_26_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 ) ) ) ) ).
% order.trans
thf(fact_27_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ~ ( ord_less_eq @ A @ X @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% le_cases
thf(fact_28_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( X = Y4 )
=> ( ord_less_eq @ A @ X @ Y4 ) ) ) ).
% eq_refl
thf(fact_29_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ( ( order_antimono @ A @ B )
= ( ^ [F2: A > B] :
! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ Y3 ) @ ( F2 @ X3 ) ) ) ) ) ) ).
% antimono_def
thf(fact_30_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B] :
( ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X2 ) ) )
=> ( order_antimono @ A @ B @ F ) ) ) ).
% antimonoI
thf(fact_31_antimonoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B,X: A,Y4: A] :
( ( order_antimono @ A @ B @ F )
=> ( ( ord_less_eq @ A @ X @ Y4 )
=> ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X ) ) ) ) ) ).
% antimonoE
thf(fact_32_antimonoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B,X: A,Y4: A] :
( ( order_antimono @ A @ B @ F )
=> ( ( ord_less_eq @ A @ X @ Y4 )
=> ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X ) ) ) ) ) ).
% antimonoD
thf(fact_33_strict__mono__less__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B,X: A,Y4: A] :
( ( order_strict_mono @ A @ B @ F )
=> ( ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y4 ) )
= ( ord_less_eq @ A @ X @ Y4 ) ) ) ) ).
% strict_mono_less_eq
thf(fact_34_le__dag__def,axiom,
( ( ord_less_eq @ binDag_Mirabelle_dag )
= ( ^ [S: binDag_Mirabelle_dag,T: binDag_Mirabelle_dag] :
( ( S = T )
| ( ord_less @ binDag_Mirabelle_dag @ S @ T ) ) ) ) ).
% le_dag_def
thf(fact_35_LeastI2__wellorder__ex,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,Q: A > $o] :
( ? [X1: A] : ( P @ X1 )
=> ( ! [A3: A] :
( ( P @ A3 )
=> ( ! [B4: A] :
( ( P @ B4 )
=> ( ord_less_eq @ A @ A3 @ B4 ) )
=> ( Q @ A3 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_wellorder_ex
thf(fact_36_LeastI2__wellorder,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,A2: A,Q: A > $o] :
( ( P @ A2 )
=> ( ! [A3: A] :
( ( P @ A3 )
=> ( ! [B4: A] :
( ( P @ B4 )
=> ( ord_less_eq @ A @ A3 @ B4 ) )
=> ( Q @ A3 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_wellorder
thf(fact_37_Least__equality,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [P: A > $o,X: A] :
( ( P @ X )
=> ( ! [Y: A] :
( ( P @ Y )
=> ( ord_less_eq @ A @ X @ Y ) )
=> ( ( ord_Least @ A @ P )
= X ) ) ) ) ).
% Least_equality
thf(fact_38_LeastI2__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ! [Y: A] :
( ( P @ Y )
=> ( ord_less_eq @ A @ X @ Y ) )
=> ( ! [X2: A] :
( ( P @ X2 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ A @ X2 @ Y5 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ) ).
% LeastI2_order
thf(fact_39_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C )
=> ( ! [X2: B,Y: B] :
( ( ord_less @ B @ X2 @ Y )
=> ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_40_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > B,C: B] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_41_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( ord_less @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C )
=> ( ! [X2: B,Y: B] :
( ( ord_less @ B @ X2 @ Y )
=> ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_42_order__less__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > C2,C: C2] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
=> ( ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_less_subst2
thf(fact_43_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [Y: A] : ( ord_less @ A @ Y @ X ) ) ).
% lt_ex
thf(fact_44_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [X12: A] : ( ord_less @ A @ X @ X12 ) ) ).
% gt_ex
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X2: A] :
( ( F @ X2 )
= ( G @ X2 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_neqE,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( X != Y4 )
=> ( ~ ( ord_less @ A @ X @ Y4 )
=> ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% neqE
thf(fact_50_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( X != Y4 )
= ( ( ord_less @ A @ X @ Y4 )
| ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% neq_iff
thf(fact_51_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% order.asym
thf(fact_52_dense,axiom,
! [A: $tType] :
( ( dense_order @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ? [Z3: A] :
( ( ord_less @ A @ X @ Z3 )
& ( ord_less @ A @ Z3 @ Y4 ) ) ) ) ).
% dense
thf(fact_53_LeastI2,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,A2: A,Q: A > $o] :
( ( P @ A2 )
=> ( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2
thf(fact_54_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ( X != Y4 ) ) ) ).
% less_imp_neq
thf(fact_55_strict__monoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B,X: A,Y4: A] :
( ( order_strict_mono @ A @ B @ F )
=> ( ( ord_less @ A @ X @ Y4 )
=> ( ord_less @ B @ ( F @ X ) @ ( F @ Y4 ) ) ) ) ) ).
% strict_monoD
thf(fact_56_strict__monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B] :
( ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( order_strict_mono @ A @ B @ F ) ) ) ).
% strict_monoI
thf(fact_57_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ~ ( ord_less @ A @ Y4 @ X ) ) ) ).
% less_asym
thf(fact_58_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% less_asym'
thf(fact_59_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A,Z2: A] :
( ( ord_less @ A @ X @ Y4 )
=> ( ( ord_less @ A @ Y4 @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% less_trans
thf(fact_60_LeastI__ex,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o] :
( ? [X1: A] : ( P @ X1 )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% LeastI_ex
thf(fact_61_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ord_less @ A @ X @ Y4 )
| ( X = Y4 )
| ( ord_less @ A @ Y4 @ X ) ) ) ).
% less_linear
thf(fact_62_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% less_irrefl
thf(fact_63_LeastI2__ex,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,Q: A > $o] :
( ? [X1: A] : ( P @ X1 )
=> ( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_ex
thf(fact_64_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 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_65_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 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_66_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% dual_order.asym
thf(fact_67_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ( X != Y4 ) ) ) ).
% less_imp_not_eq
thf(fact_68_strict__mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ( ( order_strict_mono @ A @ B )
= ( ^ [F2: A > B] :
! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ) ).
% strict_mono_def
thf(fact_69_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ~ ( ord_less @ A @ Y4 @ X ) ) ) ).
% less_not_sym
thf(fact_70_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,A2: A] :
( ! [X2: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X2 )
=> ( P @ Y5 ) )
=> ( P @ X2 ) )
=> ( P @ A2 ) ) ) ).
% less_induct
thf(fact_71_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y4: A,X: A] :
( ~ ( ord_less @ A @ Y4 @ X )
=> ( ( ~ ( ord_less @ A @ X @ Y4 ) )
= ( X = Y4 ) ) ) ) ).
% antisym_conv3
thf(fact_72_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ( Y4 != X ) ) ) ).
% less_imp_not_eq2
thf(fact_73_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A,P: $o] :
( ( ord_less @ A @ X @ Y4 )
=> ( ( ord_less @ A @ Y4 @ X )
=> P ) ) ) ).
% less_imp_triv
thf(fact_74_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ~ ( ord_less @ A @ X @ Y4 )
=> ( ( X != Y4 )
=> ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% linorder_cases
thf(fact_75_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_76_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 ) ) ) ) ).
% order.strict_trans
thf(fact_77_strict__mono__less,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B,X: A,Y4: A] :
( ( order_strict_mono @ A @ B @ F )
=> ( ( ord_less @ B @ ( F @ X ) @ ( F @ Y4 ) )
= ( ord_less @ A @ X @ Y4 ) ) ) ) ).
% strict_mono_less
thf(fact_78_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ~ ( ord_less @ A @ Y4 @ X ) ) ) ).
% less_imp_not_less
thf(fact_79_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 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_80_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ~ ( ord_less @ A @ X @ Y4 ) )
= ( ( ord_less @ A @ Y4 @ X )
| ( X = Y4 ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_81_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( A2 != B2 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_82_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( A2 != B2 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_83_strict__mono__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B,X: A,Y4: A] :
( ( order_strict_mono @ A @ B @ F )
=> ( ( ( F @ X )
= ( F @ Y4 ) )
= ( X = Y4 ) ) ) ) ).
% strict_mono_eq
thf(fact_84_leD,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y4: A,X: A] :
( ( ord_less_eq @ A @ Y4 @ X )
=> ~ ( ord_less @ A @ X @ Y4 ) ) ) ).
% leD
thf(fact_85_leI,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ~ ( ord_less @ A @ X @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% leI
thf(fact_86_le__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ) ).
% le_less
thf(fact_87_less__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ) ).
% less_le
thf(fact_88_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C )
=> ( ! [X2: B,Y: B] :
( ( ord_less @ B @ X2 @ Y )
=> ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_89_order__le__less__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
=> ( ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_90_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( ord_less @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X2: B,Y: B] :
( ( ord_less_eq @ B @ X2 @ Y )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_91_order__less__le__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > C2,C: C2] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
=> ( ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_92_not__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y4 ) )
= ( ord_less @ A @ Y4 @ X ) ) ) ).
% not_le
thf(fact_93_not__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ~ ( ord_less @ A @ X @ Y4 ) )
= ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% not_less
thf(fact_94_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 ) ) ) ) ).
% le_neq_trans
thf(fact_95_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ( ord_less_eq @ A @ X @ Y4 ) ) ) ).
% less_imp_le
thf(fact_96_antisym__conv1,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ~ ( ord_less @ A @ X @ Y4 )
=> ( ( ord_less_eq @ A @ X @ Y4 )
= ( X = Y4 ) ) ) ) ).
% antisym_conv1
thf(fact_97_antisym__conv2,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ( ~ ( ord_less @ A @ X @ Y4 ) )
= ( X = Y4 ) ) ) ) ).
% antisym_conv2
thf(fact_98_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ( ord_less @ A @ Y4 @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% le_less_trans
thf(fact_99_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A,Z2: A] :
( ( ord_less @ A @ X @ Y4 )
=> ( ( ord_less_eq @ A @ Y4 @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% less_le_trans
thf(fact_100_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z2: A,Y4: A] :
( ! [X2: A] :
( ( ord_less @ A @ Z2 @ X2 )
=> ( ord_less_eq @ A @ Y4 @ X2 ) )
=> ( ord_less_eq @ A @ Y4 @ Z2 ) ) ) ).
% dense_ge
thf(fact_101_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Y4: A,Z2: A] :
( ! [X2: A] :
( ( ord_less @ A @ X2 @ Y4 )
=> ( ord_less_eq @ A @ X2 @ Z2 ) )
=> ( ord_less_eq @ A @ Y4 @ Z2 ) ) ) ).
% dense_le
thf(fact_102_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
| ( ord_less @ A @ Y4 @ X ) ) ) ).
% le_less_linear
thf(fact_103_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ( ord_less @ A @ X @ Y4 )
| ( X = Y4 ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_104_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
& ~ ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% less_le_not_le
thf(fact_105_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y4: A,X: A] :
( ~ ( ord_less_eq @ A @ Y4 @ X )
=> ( ord_less @ A @ X @ Y4 ) ) ) ).
% not_le_imp_less
thf(fact_106_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 ) ) ) ) ).
% order.strict_trans1
thf(fact_107_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 ) ) ) ) ).
% order.strict_trans2
thf(fact_108_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B5: A] :
( ( ord_less @ A @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_109_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_110_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 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_111_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 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_112_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z2: A,X: A,Y4: A] :
( ( ord_less @ A @ Z2 @ X )
=> ( ! [W: A] :
( ( ord_less @ A @ Z2 @ W )
=> ( ( ord_less @ A @ W @ X )
=> ( ord_less_eq @ A @ Y4 @ W ) ) )
=> ( ord_less_eq @ A @ Y4 @ Z2 ) ) ) ) ).
% dense_ge_bounded
thf(fact_113_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A,Z2: A] :
( ( ord_less @ A @ X @ Y4 )
=> ( ! [W: A] :
( ( ord_less @ A @ X @ W )
=> ( ( ord_less @ A @ W @ Y4 )
=> ( ord_less_eq @ A @ W @ Z2 ) ) )
=> ( ord_less_eq @ A @ Y4 @ Z2 ) ) ) ) ).
% dense_le_bounded
thf(fact_114_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 ) ) ) ).
% order.strict_implies_order
thf(fact_115_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( ( ord_less @ A @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_116_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A5: A] :
( ( ord_less_eq @ A @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_117_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 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_118_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 ) ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_119_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z3: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z3 )
=> ~ ( ord_less_eq @ A @ T2 @ X4 ) ) ) ).
% minf(8)
thf(fact_120_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z3: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z3 )
=> ( ord_less_eq @ A @ X4 @ T2 ) ) ) ).
% minf(6)
thf(fact_121_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z3: A] :
! [X4: A] :
( ( ord_less @ A @ Z3 @ X4 )
=> ( ord_less_eq @ A @ T2 @ X4 ) ) ) ).
% pinf(8)
thf(fact_122_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z3: A] :
! [X4: A] :
( ( ord_less @ A @ Z3 @ X4 )
=> ~ ( ord_less_eq @ A @ X4 @ T2 ) ) ) ).
% pinf(6)
thf(fact_123_complete__interval,axiom,
! [A: $tType] :
( ( condit1037483654norder @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,P: A > $o] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C3: A] :
( ( ord_less_eq @ A @ A2 @ C3 )
& ( ord_less_eq @ A @ C3 @ B2 )
& ! [X4: A] :
( ( ( ord_less_eq @ A @ A2 @ X4 )
& ( ord_less @ A @ X4 @ C3 ) )
=> ( P @ X4 ) )
& ! [D: A] :
( ! [X2: A] :
( ( ( ord_less_eq @ A @ A2 @ X2 )
& ( ord_less @ A @ X2 @ D ) )
=> ( P @ X2 ) )
=> ( ord_less_eq @ A @ D @ C3 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_124_less__dag__def,axiom,
( ( ord_less @ binDag_Mirabelle_dag )
= ( ^ [S: binDag_Mirabelle_dag,T: binDag_Mirabelle_dag] : ( binDag786255756subdag @ T @ S ) ) ) ).
% less_dag_def
thf(fact_125_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
=> ! [A2: A] :
? [B3: A] :
( ( ord_less @ A @ A2 @ B3 )
| ( ord_less @ A @ B3 @ A2 ) ) ) ).
% ex_gt_or_lt
thf(fact_126_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( X != Y4 )
=> ( ~ ( ord_less @ A @ X @ Y4 )
=> ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_127_subdag__neq,axiom,
! [T2: binDag_Mirabelle_dag,S2: binDag_Mirabelle_dag] :
( ( binDag786255756subdag @ T2 @ S2 )
=> ( T2 != S2 ) ) ).
% subdag_neq
thf(fact_128_subdag__trans,axiom,
! [T2: binDag_Mirabelle_dag,S2: binDag_Mirabelle_dag,R: binDag_Mirabelle_dag] :
( ( binDag786255756subdag @ T2 @ S2 )
=> ( ( binDag786255756subdag @ S2 @ R )
=> ( binDag786255756subdag @ T2 @ R ) ) ) ).
% subdag_trans
thf(fact_129_subdag__not__sym,axiom,
! [S2: binDag_Mirabelle_dag,T2: binDag_Mirabelle_dag] :
( ( binDag786255756subdag @ S2 @ T2 )
=> ~ ( binDag786255756subdag @ T2 @ S2 ) ) ).
% subdag_not_sym
thf(fact_130_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
& ~ ( ord_less_eq @ ( A > B ) @ G2 @ F2 ) ) ) ) ) ).
% less_fun_def
thf(fact_131_minf_I11_J,axiom,
! [C2: $tType,D2: $tType] :
( ( ord @ C2 @ ( type2 @ C2 ) )
=> ! [F3: D2] :
? [Z3: C2] :
! [X4: C2] :
( ( ord_less @ C2 @ X4 @ Z3 )
=> ( F3 = F3 ) ) ) ).
% minf(11)
thf(fact_132_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z3: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z3 )
=> ~ ( ord_less @ A @ T2 @ X4 ) ) ) ).
% minf(7)
thf(fact_133_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z3: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z3 )
=> ( ord_less @ A @ X4 @ T2 ) ) ) ).
% minf(5)
thf(fact_134_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z3: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z3 )
=> ( X4 != T2 ) ) ) ).
% minf(4)
thf(fact_135_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z3: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z3 )
=> ( X4 != T2 ) ) ) ).
% minf(3)
thf(fact_136_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z3 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_137_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z3 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_138_pinf_I11_J,axiom,
! [C2: $tType,D2: $tType] :
( ( ord @ C2 @ ( type2 @ C2 ) )
=> ! [F3: D2] :
? [Z3: C2] :
! [X4: C2] :
( ( ord_less @ C2 @ Z3 @ X4 )
=> ( F3 = F3 ) ) ) ).
% pinf(11)
thf(fact_139_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z3: A] :
! [X4: A] :
( ( ord_less @ A @ Z3 @ X4 )
=> ( ord_less @ A @ T2 @ X4 ) ) ) ).
% pinf(7)
thf(fact_140_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z3: A] :
! [X4: A] :
( ( ord_less @ A @ Z3 @ X4 )
=> ~ ( ord_less @ A @ X4 @ T2 ) ) ) ).
% pinf(5)
thf(fact_141_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z3: A] :
! [X4: A] :
( ( ord_less @ A @ Z3 @ X4 )
=> ( X4 != T2 ) ) ) ).
% pinf(4)
thf(fact_142_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z3: A] :
! [X4: A] :
( ( ord_less @ A @ Z3 @ X4 )
=> ( X4 != T2 ) ) ) ).
% pinf(3)
thf(fact_143_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: A] :
! [X4: A] :
( ( ord_less @ A @ Z3 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_144_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: A] :
! [X4: A] :
( ( ord_less @ A @ Z3 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_145_dependent__wellorder__choice,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: ( A > B ) > A > B > $o] :
( ! [R2: B,F4: A > B,G3: A > B,X2: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X2 )
=> ( ( F4 @ Y5 )
= ( G3 @ Y5 ) ) )
=> ( ( P @ F4 @ X2 @ R2 )
= ( P @ G3 @ X2 @ R2 ) ) )
=> ( ! [X2: A,F4: A > B] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X2 )
=> ( P @ F4 @ Y5 @ ( F4 @ Y5 ) ) )
=> ? [X1: B] : ( P @ F4 @ X2 @ X1 ) )
=> ? [F4: A > B] :
! [X4: A] : ( P @ F4 @ X4 @ ( F4 @ X4 ) ) ) ) ) ).
% dependent_wellorder_choice
thf(fact_146_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [X4: A] :
? [Y: A] : ( ord_less @ A @ Y @ X4 ) ) ).
% linordered_field_no_lb
thf(fact_147_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [X4: A] :
? [X12: A] : ( ord_less @ A @ X4 @ X12 ) ) ).
% linordered_field_no_ub
thf(fact_148_subdag_Osimps_I1_J,axiom,
! [T2: binDag_Mirabelle_dag] :
~ ( binDag786255756subdag @ binDag_Mirabelle_Tip @ T2 ) ).
% subdag.simps(1)
thf(fact_149_DAG_Osimps_I1_J,axiom,
binDag_Mirabelle_DAG @ binDag_Mirabelle_Tip ).
% DAG.simps(1)
thf(fact_150_dag_Osimps_I4_J,axiom,
! [A: $tType,F1: A,F22: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A] :
( ( binDag1297733282se_dag @ A @ F1 @ F22 @ binDag_Mirabelle_Tip )
= F1 ) ).
% dag.simps(4)
thf(fact_151_dag_Osimps_I6_J,axiom,
! [A: $tType,F1: A,F22: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A > A > A] :
( ( binDag1442713106ec_dag @ A @ F1 @ F22 @ binDag_Mirabelle_Tip )
= F1 ) ).
% dag.simps(6)
thf(fact_152_subdag__size,axiom,
! [T2: binDag_Mirabelle_dag,S2: binDag_Mirabelle_dag] :
( ( binDag786255756subdag @ T2 @ S2 )
=> ( ord_less @ nat @ ( size_size @ binDag_Mirabelle_dag @ S2 ) @ ( size_size @ binDag_Mirabelle_dag @ T2 ) ) ) ).
% subdag_size
thf(fact_153_dag_Osimps_I7_J,axiom,
! [A: $tType,F1: A,F22: 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 @ F22 @ ( binDag476092410e_Node @ X21 @ X22 @ X23 ) )
= ( F22 @ X21 @ X22 @ X23 @ ( binDag1442713106ec_dag @ A @ F1 @ F22 @ X21 ) @ ( binDag1442713106ec_dag @ A @ F1 @ F22 @ X23 ) ) ) ).
% dag.simps(7)
thf(fact_154_dag_Osimps_I5_J,axiom,
! [A: $tType,F1: A,F22: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A,X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag] :
( ( binDag1297733282se_dag @ A @ F1 @ F22 @ ( binDag476092410e_Node @ X21 @ X22 @ X23 ) )
= ( F22 @ X21 @ X22 @ X23 ) ) ).
% dag.simps(5)
thf(fact_155_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 ) ) ) ).
% dag.inject
thf(fact_156_subdag__NodeD,axiom,
! [T2: binDag_Mirabelle_dag,Lt: binDag_Mirabelle_dag,A2: simpl_ref,Rt: binDag_Mirabelle_dag] :
( ( binDag786255756subdag @ T2 @ ( binDag476092410e_Node @ Lt @ A2 @ Rt ) )
=> ( ( binDag786255756subdag @ T2 @ Lt )
& ( binDag786255756subdag @ T2 @ Rt ) ) ) ).
% subdag_NodeD
thf(fact_157_subdag_Osimps_I2_J,axiom,
! [L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag,T2: binDag_Mirabelle_dag] :
( ( binDag786255756subdag @ ( binDag476092410e_Node @ L @ A2 @ R ) @ T2 )
= ( ( T2 = L )
| ( T2 = R )
| ( binDag786255756subdag @ L @ T2 )
| ( binDag786255756subdag @ R @ T2 ) ) ) ).
% subdag.simps(2)
thf(fact_158_dag_Oexhaust,axiom,
! [Y4: binDag_Mirabelle_dag] :
( ( Y4 != binDag_Mirabelle_Tip )
=> ~ ! [X212: binDag_Mirabelle_dag,X222: simpl_ref,X232: binDag_Mirabelle_dag] :
( Y4
!= ( binDag476092410e_Node @ X212 @ X222 @ X232 ) ) ) ).
% dag.exhaust
thf(fact_159_dag_Oinduct,axiom,
! [P: binDag_Mirabelle_dag > $o,Dag: binDag_Mirabelle_dag] :
( ( P @ binDag_Mirabelle_Tip )
=> ( ! [X12: binDag_Mirabelle_dag,X24: simpl_ref,X32: binDag_Mirabelle_dag] :
( ( P @ X12 )
=> ( ( P @ X32 )
=> ( P @ ( binDag476092410e_Node @ X12 @ X24 @ X32 ) ) ) )
=> ( P @ Dag ) ) ) ).
% dag.induct
thf(fact_160_dag_Odistinct_I1_J,axiom,
! [X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag] :
( binDag_Mirabelle_Tip
!= ( binDag476092410e_Node @ X21 @ X22 @ X23 ) ) ).
% dag.distinct(1)
thf(fact_161_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 ) ) ) ).
% DAG.simps(2)
thf(fact_162_dag_Osize_I3_J,axiom,
( ( size_size @ binDag_Mirabelle_dag @ binDag_Mirabelle_Tip )
= ( zero_zero @ nat ) ) ).
% dag.size(3)
thf(fact_163_order_Oordering__axioms,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ordering @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% order.ordering_axioms
thf(fact_164_ordering_Onot__eq__order__implies__strict,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( A2 != B2 )
=> ( ( Less_eq @ A2 @ B2 )
=> ( Less @ A2 @ B2 ) ) ) ) ).
% ordering.not_eq_order_implies_strict
thf(fact_165_ordering_Ostrict__implies__not__eq,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less @ A2 @ B2 )
=> ( A2 != B2 ) ) ) ).
% ordering.strict_implies_not_eq
thf(fact_166_ordering_Ostrict__implies__order,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less @ A2 @ B2 )
=> ( Less_eq @ A2 @ B2 ) ) ) ).
% ordering.strict_implies_order
thf(fact_167_ordering_Ostrict__iff__order,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less @ A2 @ B2 )
= ( ( Less_eq @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% ordering.strict_iff_order
thf(fact_168_ordering_Oorder__iff__strict,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less_eq @ A2 @ B2 )
= ( ( Less @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% ordering.order_iff_strict
thf(fact_169_ordering_Ostrict__trans2,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A,C: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less @ A2 @ B2 )
=> ( ( Less_eq @ B2 @ C )
=> ( Less @ A2 @ C ) ) ) ) ).
% ordering.strict_trans2
thf(fact_170_ordering_Ostrict__trans1,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A,C: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less_eq @ A2 @ B2 )
=> ( ( Less @ B2 @ C )
=> ( Less @ A2 @ C ) ) ) ) ).
% ordering.strict_trans1
thf(fact_171_ordering_Ostrict__trans,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A,C: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less @ A2 @ B2 )
=> ( ( Less @ B2 @ C )
=> ( Less @ A2 @ C ) ) ) ) ).
% ordering.strict_trans
thf(fact_172_ordering_Oantisym,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less_eq @ A2 @ B2 )
=> ( ( Less_eq @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% ordering.antisym
thf(fact_173_ordering_Oirrefl,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ~ ( Less @ A2 @ A2 ) ) ).
% ordering.irrefl
thf(fact_174_ordering_Otrans,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A,C: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less_eq @ A2 @ B2 )
=> ( ( Less_eq @ B2 @ C )
=> ( Less_eq @ A2 @ C ) ) ) ) ).
% ordering.trans
thf(fact_175_ordering_Ointro,axiom,
! [A: $tType,Less: A > A > $o,Less_eq: A > A > $o] :
( ! [A3: A,B3: A] :
( ( Less @ A3 @ B3 )
= ( ( Less_eq @ A3 @ B3 )
& ( A3 != B3 ) ) )
=> ( ! [A3: A] : ( Less_eq @ A3 @ A3 )
=> ( ! [A3: A,B3: A] :
( ( Less_eq @ A3 @ B3 )
=> ( ( Less_eq @ B3 @ A3 )
=> ( A3 = B3 ) ) )
=> ( ! [A3: A,B3: A,C3: A] :
( ( Less_eq @ A3 @ B3 )
=> ( ( Less_eq @ B3 @ C3 )
=> ( Less_eq @ A3 @ C3 ) ) )
=> ( ordering @ A @ Less_eq @ Less ) ) ) ) ) ).
% ordering.intro
thf(fact_176_ordering_Orefl,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( Less_eq @ A2 @ A2 ) ) ).
% ordering.refl
thf(fact_177_ordering_Oasym,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less @ A2 @ B2 )
=> ~ ( Less @ B2 @ A2 ) ) ) ).
% ordering.asym
thf(fact_178_ordering__def,axiom,
! [A: $tType] :
( ( ordering @ A )
= ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o] :
( ! [A5: A,B5: A] :
( ( Less2 @ A5 @ B5 )
= ( ( Less_eq2 @ A5 @ B5 )
& ( A5 != B5 ) ) )
& ! [A5: A] : ( Less_eq2 @ A5 @ A5 )
& ! [A5: A,B5: A] :
( ( Less_eq2 @ A5 @ B5 )
=> ( ( Less_eq2 @ B5 @ A5 )
=> ( A5 = B5 ) ) )
& ! [A5: A,B5: A,C4: A] :
( ( Less_eq2 @ A5 @ B5 )
=> ( ( Less_eq2 @ B5 @ C4 )
=> ( Less_eq2 @ A5 @ C4 ) ) ) ) ) ) ).
% ordering_def
thf(fact_179_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_180_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_181_dag_Osize__gen_I1_J,axiom,
( ( binDag1924123185ze_dag @ binDag_Mirabelle_Tip )
= ( zero_zero @ nat ) ) ).
% dag.size_gen(1)
thf(fact_182_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_183_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N )
= ( N
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_184_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [M: A,N: A] :
( ( ord_less @ A @ M @ N )
=> ( N
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_185_not__less__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_186_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( N
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% gr_zeroI
thf(fact_187_less__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% less_numeral_extra(3)
thf(fact_188_le__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% le_numeral_extra(3)
thf(fact_189_sgn__greater,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( sgn_sgn @ A @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% sgn_greater
thf(fact_190_sgn__sgn,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( sgn_sgn @ A @ ( sgn_sgn @ A @ A2 ) )
= ( sgn_sgn @ A @ A2 ) ) ) ).
% sgn_sgn
thf(fact_191_sgn__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( sgn_sgn @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% sgn_less
thf(fact_192_sgn__0__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ( sgn_sgn @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% sgn_0_0
thf(fact_193_sgn__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( sgn_sgn @ A @ A2 )
= ( one_one @ A ) ) ) ) ).
% sgn_pos
thf(fact_194_of__bool__eq_I1_J,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ( ( zero_neq_one_of_bool @ A @ $false )
= ( zero_zero @ A ) ) ) ).
% of_bool_eq(1)
thf(fact_195_of__bool__eq_I2_J,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ( ( zero_neq_one_of_bool @ A @ $true )
= ( one_one @ A ) ) ) ).
% of_bool_eq(2)
thf(fact_196_less__numeral__extra_I2_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% less_numeral_extra(2)
thf(fact_197_less__numeral__extra_I1_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(1)
thf(fact_198_le__numeral__extra_I1_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% le_numeral_extra(1)
thf(fact_199_le__numeral__extra_I2_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% le_numeral_extra(2)
thf(fact_200_less__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(4)
thf(fact_201_le__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% le_numeral_extra(4)
thf(fact_202_of__bool__eq__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ! [P3: $o,Q3: $o] :
( ( ( zero_neq_one_of_bool @ A @ P3 )
= ( zero_neq_one_of_bool @ A @ Q3 ) )
= ( P3 = Q3 ) ) ) ).
% of_bool_eq_iff
thf(fact_203_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A @ ( type2 @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_204_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A @ ( type2 @ A ) )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_205_zero__le__one,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A @ ( type2 @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_le_one
thf(fact_206_not__one__le__zero,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A @ ( type2 @ A ) )
=> ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_le_zero
thf(fact_207_of__bool__def,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ( ( zero_neq_one_of_bool @ A )
= ( ^ [P4: $o] : ( if @ A @ P4 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_bool_def
thf(fact_208_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_209_split__of__bool,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P3: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P3 ) )
= ( ( P3
=> ( P @ ( one_one @ A ) ) )
& ( ~ P3
=> ( P @ ( zero_zero @ A ) ) ) ) ) ) ).
% split_of_bool
thf(fact_210_split__of__bool__asm,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P3: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P3 ) )
= ( ~ ( ( P3
& ~ ( P @ ( one_one @ A ) ) )
| ( ~ P3
& ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_211_sgn__1__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ( sgn_sgn @ A @ A2 )
= ( one_one @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% sgn_1_pos
thf(fact_212_sgn__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( sgn_sgn @ A @ A2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% sgn_neg
thf(fact_213_divide__le__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% divide_le_eq_1_neg
thf(fact_214_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% neg_le_iff_le
thf(fact_215_neg__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% neg_less_iff_less
thf(fact_216_division__ring__divide__zero,axiom,
! [A: $tType] :
( ( division_ring @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% division_ring_divide_zero
thf(fact_217_divide__cancel__right,axiom,
! [A: $tType] :
( ( field @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ C )
= ( divide_divide @ A @ B2 @ C ) )
= ( ( C
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% divide_cancel_right
thf(fact_218_divide__cancel__left,axiom,
! [A: $tType] :
( ( field @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,B2: A] :
( ( ( divide_divide @ A @ C @ A2 )
= ( divide_divide @ A @ C @ B2 ) )
= ( ( C
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% divide_cancel_left
thf(fact_219_divide__eq__0__iff,axiom,
! [A: $tType] :
( ( field @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_eq_0_iff
thf(fact_220_divide__zero__left,axiom,
! [A: $tType] :
( ( semidom_divide @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% divide_zero_left
thf(fact_221_divide__zero,axiom,
! [A: $tType] :
( ( semidom_divide @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% divide_zero
thf(fact_222_divide__1,axiom,
! [A: $tType] :
( ( semidom_divide @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% divide_1
thf(fact_223_neg__0__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_le_iff_le
thf(fact_224_neg__le__0__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_le_0_iff_le
thf(fact_225_less__eq__neg__nonpos,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% less_eq_neg_nonpos
thf(fact_226_neg__less__eq__nonneg,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_less_eq_nonneg
thf(fact_227_neg__less__0__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_less_0_iff_less
thf(fact_228_neg__0__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_less_iff_less
thf(fact_229_neg__less__pos,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_less_pos
thf(fact_230_less__neg__neg,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% less_neg_neg
thf(fact_231_zero__eq__1__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% zero_eq_1_divide_iff
thf(fact_232_one__divide__eq__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ( divide_divide @ A @ ( one_one @ A ) @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% one_divide_eq_0_iff
thf(fact_233_eq__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ B2 @ A2 ) )
= ( ( A2
!= ( zero_zero @ A ) )
& ( A2 = B2 ) ) ) ) ).
% eq_divide_eq_1
thf(fact_234_divide__eq__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ( divide_divide @ A @ B2 @ A2 )
= ( one_one @ A ) )
= ( ( A2
!= ( zero_zero @ A ) )
& ( A2 = B2 ) ) ) ) ).
% divide_eq_eq_1
thf(fact_235_divide__self__if,axiom,
! [A: $tType] :
( ( division_ring @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ( A2
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) )
& ( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ A2 )
= ( one_one @ A ) ) ) ) ) ).
% divide_self_if
thf(fact_236_divide__self,axiom,
! [A: $tType] :
( ( division_ring @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ A2 )
= ( one_one @ A ) ) ) ) ).
% divide_self
thf(fact_237_one__eq__divide__iff,axiom,
! [A: $tType] :
( ( field @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ A2 @ B2 ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A2 = B2 ) ) ) ) ).
% one_eq_divide_iff
thf(fact_238_divide__eq__1__iff,axiom,
! [A: $tType] :
( ( field @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ B2 )
= ( one_one @ A ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A2 = B2 ) ) ) ) ).
% divide_eq_1_iff
thf(fact_239_div__self,axiom,
! [A: $tType] :
( ( semidom_divide @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ A2 )
= ( one_one @ A ) ) ) ) ).
% div_self
thf(fact_240_divide__minus1,axiom,
! [A: $tType] :
( ( field @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( divide_divide @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ X ) ) ) ).
% divide_minus1
thf(fact_241_divide__le__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% divide_le_0_1_iff
thf(fact_242_zero__le__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_le_divide_1_iff
thf(fact_243_divide__less__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% divide_less_0_1_iff
thf(fact_244_divide__less__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% divide_less_eq_1_neg
thf(fact_245_divide__less__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% divide_less_eq_1_pos
thf(fact_246_less__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% less_divide_eq_1_neg
thf(fact_247_less__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% less_divide_eq_1_pos
thf(fact_248_zero__less__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_less_divide_1_iff
thf(fact_249_le__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% le_divide_eq_1_pos
thf(fact_250_le__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% le_divide_eq_1_neg
thf(fact_251_divide__le__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% divide_le_eq_1_pos
thf(fact_252_less__minus__one__simps_I4_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% less_minus_one_simps(4)
thf(fact_253_less__minus__one__simps_I2_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ).
% less_minus_one_simps(2)
thf(fact_254_le__minus__one__simps_I2_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ).
% le_minus_one_simps(2)
thf(fact_255_le__minus__one__simps_I4_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% le_minus_one_simps(4)
%----Type constructors (23)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A6: $tType,A7: $tType] :
( ( preorder @ A7 @ ( type2 @ A7 ) )
=> ( preorder @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A6: $tType,A7: $tType] :
( ( order @ A7 @ ( type2 @ A7 ) )
=> ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A6: $tType,A7: $tType] :
( ( ord @ A7 @ ( type2 @ A7 ) )
=> ( ord @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit1037483654norder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
linord1659791738miring @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide,axiom,
semidom_divide @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one,axiom,
zero_less_one @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one,axiom,
zero_neq_one @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
preorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
no_top @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
! [A6: $tType] : ( preorder @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
! [A6: $tType] : ( ord @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_8,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
ord @ $o @ ( type2 @ $o ) ).
%----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,X: A,Y4: A] :
( ( if @ A @ $false @ X @ Y4 )
= Y4 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y4: A] :
( ( if @ A @ $true @ X @ Y4 )
= X ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq @ binDag_Mirabelle_dag @ x @ z ).
%------------------------------------------------------------------------------