TPTP Problem File: COM164^1.p
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%------------------------------------------------------------------------------
% File : COM164^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Binary decision diagram 141
% 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__141.p [Bla16]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.50 v7.5.0, 0.00 v7.3.0, 0.33 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 364 ( 111 unt; 58 typ; 0 def)
% Number of atoms : 882 ( 194 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3412 ( 111 ~; 27 |; 58 &;2698 @)
% ( 0 <=>; 518 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 269 ( 269 >; 0 *; 0 +; 0 <<)
% Number of symbols : 56 ( 53 usr; 7 con; 0-4 aty)
% Number of variables : 1011 ( 37 ^; 889 !; 44 ?;1011 :)
% ( 41 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:45:12.934
%------------------------------------------------------------------------------
%----Could-be-implicit typings (10)
thf(ty_t_BinDag__Mirabelle__rybootvolr_Odag,type,
binDag_Mirabelle_dag: $tType ).
thf(ty_t_Code__Numeral_Onatural,type,
code_natural: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Simpl__Heap_Oref,type,
simpl_ref: $tType ).
thf(ty_t_Sum__Type_Osum,type,
sum_sum: $tType > $tType > $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_Int_Oint,type,
int: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
%----Explicit typings (48)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Nat_Osize,type,
size:
!>[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_Rings_Osemiring__1,type,
semiring_1:
!>[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_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[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_Rings_Osemiring__1__no__zero__divisors,type,
semiri134348788visors:
!>[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_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Hilbert__Choice_OGreatestM,type,
hilbert_GreatestM:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > $o ) > A ) ).
thf(sy_c_Hilbert__Choice_OLeastM,type,
hilbert_LeastM:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > $o ) > A ) ).
thf(sy_c_List_Ocount__list,type,
count_list:
!>[A: $tType] : ( ( list @ A ) > A > nat ) ).
thf(sy_c_List_Ogen__length,type,
gen_length:
!>[A: $tType] : ( nat > ( list @ A ) > nat ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
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_Oordering,type,
ordering:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Power_Opower__class_Opower,type,
power_power:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Product__Type_Oold_Obool_Orec__bool,type,
product_rec_bool:
!>[T: $tType] : ( T > T > $o > T ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ 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_a,type,
a: simpl_ref ).
thf(sy_v_l,type,
l: binDag_Mirabelle_dag ).
thf(sy_v_r,type,
r: binDag_Mirabelle_dag ).
thf(sy_v_x,type,
x: binDag_Mirabelle_dag ).
%----Relevant facts (255)
thf(fact_0_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_1_le__dag__def,axiom,
( ( ord_less_eq @ binDag_Mirabelle_dag )
= ( ^ [S: binDag_Mirabelle_dag,T2: binDag_Mirabelle_dag] :
( ( S = T2 )
| ( ord_less @ binDag_Mirabelle_dag @ S @ T2 ) ) ) ) ).
% le_dag_def
thf(fact_2_dag__less__le,axiom,
( ( ord_less @ binDag_Mirabelle_dag )
= ( ^ [X: binDag_Mirabelle_dag,Y: binDag_Mirabelle_dag] :
( ( ord_less_eq @ binDag_Mirabelle_dag @ X @ Y )
& ( X != Y ) ) ) ) ).
% dag_less_le
thf(fact_3_le__dag__refl,axiom,
! [X2: binDag_Mirabelle_dag] : ( ord_less_eq @ binDag_Mirabelle_dag @ X2 @ X2 ) ).
% le_dag_refl
thf(fact_4_le__dag__trans,axiom,
! [X2: binDag_Mirabelle_dag,Y2: binDag_Mirabelle_dag,Z: binDag_Mirabelle_dag] :
( ( ord_less_eq @ binDag_Mirabelle_dag @ X2 @ Y2 )
=> ( ( ord_less_eq @ binDag_Mirabelle_dag @ Y2 @ Z )
=> ( ord_less_eq @ binDag_Mirabelle_dag @ X2 @ Z ) ) ) ).
% le_dag_trans
thf(fact_5_le__dag__antisym,axiom,
! [X2: binDag_Mirabelle_dag,Y2: binDag_Mirabelle_dag] :
( ( ord_less_eq @ binDag_Mirabelle_dag @ X2 @ Y2 )
=> ( ( ord_less_eq @ binDag_Mirabelle_dag @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% le_dag_antisym
thf(fact_6_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% order_refl
thf(fact_7_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T3: A] :
? [Z2: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z2 )
=> ~ ( ord_less_eq @ A @ T3 @ X3 ) ) ) ).
% minf(8)
thf(fact_8_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T3: A] :
? [Z2: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z2 )
=> ( ord_less_eq @ A @ X3 @ T3 ) ) ) ).
% minf(6)
thf(fact_9_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T3: A] :
? [Z2: A] :
! [X3: A] :
( ( ord_less @ A @ Z2 @ X3 )
=> ( ord_less_eq @ A @ T3 @ X3 ) ) ) ).
% pinf(8)
thf(fact_10_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T3: A] :
? [Z2: A] :
! [X3: A] :
( ( ord_less @ A @ Z2 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ T3 ) ) ) ).
% pinf(6)
thf(fact_11_leD,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y2: A,X2: A] :
( ( ord_less_eq @ A @ Y2 @ X2 )
=> ~ ( ord_less @ A @ X2 @ Y2 ) ) ) ).
% leD
thf(fact_12_leI,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ~ ( ord_less @ A @ X2 @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% leI
thf(fact_13_le__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
| ( X = Y ) ) ) ) ) ).
% le_less
thf(fact_14_less__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
& ( X != Y ) ) ) ) ) ).
% less_le
thf(fact_15_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 )
=> ( ! [X4: B,Y3: B] :
( ( ord_less @ B @ X4 @ Y3 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_16_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 )
=> ( ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ C2 @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_17_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 )
=> ( ! [X4: B,Y3: B] :
( ( ord_less_eq @ B @ X4 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_18_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_19_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_20_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_21_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_22_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A,Z: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z )
=> ( ord_less_eq @ A @ X2 @ Z ) ) ) ) ).
% order_trans
thf(fact_23_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_24_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_25_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_26_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y2: A,X2: A] :
( ( ord_less_eq @ A @ Y2 @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ) ).
% antisym_conv
thf(fact_27_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A,Z: A] :
( ( ( ord_less_eq @ A @ X2 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Z ) )
=> ( ( ( ord_less_eq @ A @ X2 @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y2 ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X2 ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_28_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_29_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ~ ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% le_cases
thf(fact_30_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ( X2 = Y2 )
=> ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% eq_refl
thf(fact_31_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
| ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% linear
thf(fact_32_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ) ).
% antisym
thf(fact_33_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y4: A,Z3: A] : Y4 = Z3 )
= ( ^ [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
& ( ord_less_eq @ A @ Y @ X ) ) ) ) ) ).
% eq_iff
thf(fact_34_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 )
=> ( ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_35_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 )
=> ( ! [X4: B,Y3: B] :
( ( ord_less_eq @ B @ X4 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_36_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 )
=> ( ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ C2 @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_37_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 )
=> ( ! [X4: B,Y3: B] :
( ( ord_less_eq @ B @ X4 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_38_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G: A > B] :
! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ).
% le_fun_def
thf(fact_39_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G2: A > B] :
( ! [X4: A] : ( ord_less_eq @ B @ ( F @ X4 ) @ ( G2 @ X4 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G2 ) ) ) ).
% le_funI
thf(fact_40_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G2: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G2 )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% le_funE
thf(fact_41_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G2: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G2 )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% le_funD
thf(fact_42_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_43_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_44_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ( ~ ( ord_less @ A @ X2 @ Y2 ) )
= ( ( ord_less @ A @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ) ).
% not_less_iff_gr_or_eq
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
@ ^ [X: A] : ( member @ A @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G2: A > B] :
( ! [X4: A] :
( ( F @ X4 )
= ( G2 @ X4 ) )
=> ( F = G2 ) ) ).
% ext
thf(fact_49_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_50_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% less_imp_not_less
thf(fact_51_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_52_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_53_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ~ ( ord_less @ A @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less @ A @ Y2 @ X2 ) ) ) ) ).
% linorder_cases
thf(fact_54_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A,P: $o] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ X2 )
=> P ) ) ) ).
% less_imp_triv
thf(fact_55_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ) ).
% less_imp_not_eq2
thf(fact_56_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y2: A,X2: A] :
( ~ ( ord_less @ A @ Y2 @ X2 )
=> ( ( ~ ( ord_less @ A @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ) ).
% antisym_conv3
thf(fact_57_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,A2: A] :
( ! [X4: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X4 )
=> ( P @ Y5 ) )
=> ( P @ X4 ) )
=> ( P @ A2 ) ) ) ).
% less_induct
thf(fact_58_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% less_not_sym
thf(fact_59_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ) ).
% less_imp_not_eq
thf(fact_60_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_61_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_62_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_63_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A] :
~ ( ord_less @ A @ X2 @ X2 ) ) ).
% less_irrefl
thf(fact_64_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% less_linear
thf(fact_65_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A,Z: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ Z )
=> ( ord_less @ A @ X2 @ Z ) ) ) ) ).
% less_trans
thf(fact_66_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_67_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% less_asym
thf(fact_68_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ) ).
% less_imp_neq
thf(fact_69_dense,axiom,
! [A: $tType] :
( ( dense_order @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ? [Z2: A] :
( ( ord_less @ A @ X2 @ Z2 )
& ( ord_less @ A @ Z2 @ Y2 ) ) ) ) ).
% dense
thf(fact_70_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_71_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ( X2 != Y2 )
= ( ( ord_less @ A @ X2 @ Y2 )
| ( ord_less @ A @ Y2 @ X2 ) ) ) ) ).
% neq_iff
thf(fact_72_neqE,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ( X2 != Y2 )
=> ( ~ ( ord_less @ A @ X2 @ Y2 )
=> ( ord_less @ A @ Y2 @ X2 ) ) ) ) ).
% neqE
thf(fact_73_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A @ ( type2 @ A ) )
=> ! [X2: A] :
? [X1: A] : ( ord_less @ A @ X2 @ X1 ) ) ).
% gt_ex
thf(fact_74_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A @ ( type2 @ A ) )
=> ! [X2: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X2 ) ) ).
% lt_ex
thf(fact_75_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 )
=> ( ! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( ord_less @ C2 @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_less_subst2
thf(fact_76_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 )
=> ( ! [X4: B,Y3: B] :
( ( ord_less @ B @ X4 @ Y3 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_77_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 )
=> ( ! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( ord_less @ B @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_78_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 )
=> ( ! [X4: B,Y3: B] :
( ( ord_less @ B @ X4 @ Y3 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_79_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z2: A] :
! [X3: A] :
( ( ord_less @ A @ Z2 @ X3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_80_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z2: A] :
! [X3: A] :
( ( ord_less @ A @ Z2 @ X3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_81_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T3: A] :
? [Z2: A] :
! [X3: A] :
( ( ord_less @ A @ Z2 @ X3 )
=> ( X3 != T3 ) ) ) ).
% pinf(3)
thf(fact_82_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T3: A] :
? [Z2: A] :
! [X3: A] :
( ( ord_less @ A @ Z2 @ X3 )
=> ( X3 != T3 ) ) ) ).
% pinf(4)
thf(fact_83_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T3: A] :
? [Z2: A] :
! [X3: A] :
( ( ord_less @ A @ Z2 @ X3 )
=> ~ ( ord_less @ A @ X3 @ T3 ) ) ) ).
% pinf(5)
thf(fact_84_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T3: A] :
? [Z2: A] :
! [X3: A] :
( ( ord_less @ A @ Z2 @ X3 )
=> ( ord_less @ A @ T3 @ X3 ) ) ) ).
% pinf(7)
thf(fact_85_pinf_I11_J,axiom,
! [C2: $tType,D: $tType] :
( ( ord @ C2 @ ( type2 @ C2 ) )
=> ! [F3: D] :
? [Z2: C2] :
! [X3: C2] :
( ( ord_less @ C2 @ Z2 @ X3 )
=> ( F3 = F3 ) ) ) ).
% pinf(11)
thf(fact_86_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z2: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z2 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_87_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z2: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z2 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_88_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T3: A] :
? [Z2: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z2 )
=> ( X3 != T3 ) ) ) ).
% minf(3)
thf(fact_89_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T3: A] :
? [Z2: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z2 )
=> ( X3 != T3 ) ) ) ).
% minf(4)
thf(fact_90_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T3: A] :
? [Z2: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z2 )
=> ( ord_less @ A @ X3 @ T3 ) ) ) ).
% minf(5)
thf(fact_91_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T3: A] :
? [Z2: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z2 )
=> ~ ( ord_less @ A @ T3 @ X3 ) ) ) ).
% minf(7)
thf(fact_92_minf_I11_J,axiom,
! [C2: $tType,D: $tType] :
( ( ord @ C2 @ ( type2 @ C2 ) )
=> ! [F3: D] :
? [Z2: C2] :
! [X3: C2] :
( ( ord_less @ C2 @ X3 @ Z2 )
=> ( F3 = F3 ) ) ) ).
% minf(11)
thf(fact_93_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_94_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_95_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_less_eq @ A @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_96_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_less @ A @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_97_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_98_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A,Z: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( ! [W: A] :
( ( ord_less @ A @ X2 @ W )
=> ( ( ord_less @ A @ W @ Y2 )
=> ( ord_less_eq @ A @ W @ Z ) ) )
=> ( ord_less_eq @ A @ Y2 @ Z ) ) ) ) ).
% dense_le_bounded
thf(fact_99_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z: A,X2: A,Y2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ( ! [W: A] :
( ( ord_less @ A @ Z @ W )
=> ( ( ord_less @ A @ W @ X2 )
=> ( ord_less_eq @ A @ Y2 @ W ) ) )
=> ( ord_less_eq @ A @ Y2 @ Z ) ) ) ) ).
% dense_ge_bounded
thf(fact_100_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_101_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_102_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_103_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_less @ A @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_104_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_105_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_106_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y2: A,X2: A] :
( ~ ( ord_less_eq @ A @ Y2 @ X2 )
=> ( ord_less @ A @ X2 @ Y2 ) ) ) ).
% not_le_imp_less
thf(fact_107_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
& ~ ( ord_less_eq @ A @ Y @ X ) ) ) ) ) ).
% less_le_not_le
thf(fact_108_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less @ A @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_109_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
| ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% le_less_linear
thf(fact_110_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Y2: A,Z: A] :
( ! [X4: A] :
( ( ord_less @ A @ X4 @ Y2 )
=> ( ord_less_eq @ A @ X4 @ Z ) )
=> ( ord_less_eq @ A @ Y2 @ Z ) ) ) ).
% dense_le
thf(fact_111_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z: A,Y2: A] :
( ! [X4: A] :
( ( ord_less @ A @ Z @ X4 )
=> ( ord_less_eq @ A @ Y2 @ X4 ) )
=> ( ord_less_eq @ A @ Y2 @ Z ) ) ) ).
% dense_ge
thf(fact_112_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A,Z: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z )
=> ( ord_less @ A @ X2 @ Z ) ) ) ) ).
% less_le_trans
thf(fact_113_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A,Z: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ Z )
=> ( ord_less @ A @ X2 @ Z ) ) ) ) ).
% le_less_trans
thf(fact_114_antisym__conv2,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ~ ( ord_less @ A @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ) ).
% antisym_conv2
thf(fact_115_antisym__conv1,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ~ ( ord_less @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ) ).
% antisym_conv1
thf(fact_116_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% less_imp_le
thf(fact_117_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_118_not__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ( ~ ( ord_less @ A @ X2 @ Y2 ) )
= ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% not_less
thf(fact_119_not__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ( ~ ( ord_less_eq @ A @ X2 @ Y2 ) )
= ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% not_le
thf(fact_120_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 )
=> ( ! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( ord_less @ C2 @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_121_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 )
& ! [X3: A] :
( ( ( ord_less_eq @ A @ A2 @ X3 )
& ( ord_less @ A @ X3 @ C3 ) )
=> ( P @ X3 ) )
& ! [D2: A] :
( ! [X4: A] :
( ( ( ord_less_eq @ A @ A2 @ X4 )
& ( ord_less @ A @ X4 @ D2 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq @ A @ D2 @ C3 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_122_less__dag__Tip,axiom,
! [X2: binDag_Mirabelle_dag] :
~ ( ord_less @ binDag_Mirabelle_dag @ X2 @ binDag_Mirabelle_Tip ) ).
% less_dag_Tip
thf(fact_123_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_124_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_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 ) )
=> ! [X2: A,Y2: A] :
( ( X2 != Y2 )
=> ( ~ ( ord_less @ A @ X2 @ Y2 )
=> ( ord_less @ A @ Y2 @ X2 ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_127_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [X3: A] :
? [X1: A] : ( ord_less @ A @ X3 @ X1 ) ) ).
% linordered_field_no_ub
thf(fact_128_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [X3: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ).
% linordered_field_no_lb
thf(fact_129_dependent__wellorder__choice,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: ( A > B ) > A > B > $o] :
( ! [R: B,F4: A > B,G3: A > B,X4: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X4 )
=> ( ( F4 @ Y5 )
= ( G3 @ Y5 ) ) )
=> ( ( P @ F4 @ X4 @ R )
= ( P @ G3 @ X4 @ R ) ) )
=> ( ! [X4: A,F4: A > B] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X4 )
=> ( P @ F4 @ Y5 @ ( F4 @ Y5 ) ) )
=> ? [X12: B] : ( P @ F4 @ X4 @ X12 ) )
=> ? [F4: A > B] :
! [X3: A] : ( P @ F4 @ X3 @ ( F4 @ X3 ) ) ) ) ) ).
% dependent_wellorder_choice
thf(fact_130_less__dag__def,axiom,
( ( ord_less @ binDag_Mirabelle_dag )
= ( ^ [S: binDag_Mirabelle_dag,T2: binDag_Mirabelle_dag] : ( binDag786255756subdag @ T2 @ S ) ) ) ).
% less_dag_def
thf(fact_131_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F2: A > B,G: A > B] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G )
& ~ ( ord_less_eq @ ( A > B ) @ G @ F2 ) ) ) ) ) ).
% less_fun_def
thf(fact_132_subdag__not__sym,axiom,
! [S2: binDag_Mirabelle_dag,T3: binDag_Mirabelle_dag] :
( ( binDag786255756subdag @ S2 @ T3 )
=> ~ ( binDag786255756subdag @ T3 @ S2 ) ) ).
% subdag_not_sym
thf(fact_133_subdag__trans,axiom,
! [T3: binDag_Mirabelle_dag,S2: binDag_Mirabelle_dag,R2: binDag_Mirabelle_dag] :
( ( binDag786255756subdag @ T3 @ S2 )
=> ( ( binDag786255756subdag @ S2 @ R2 )
=> ( binDag786255756subdag @ T3 @ R2 ) ) ) ).
% subdag_trans
thf(fact_134_subdag__neq,axiom,
! [T3: binDag_Mirabelle_dag,S2: binDag_Mirabelle_dag] :
( ( binDag786255756subdag @ T3 @ S2 )
=> ( T3 != S2 ) ) ).
% subdag_neq
thf(fact_135_subdag_Osimps_I1_J,axiom,
! [T3: binDag_Mirabelle_dag] :
~ ( binDag786255756subdag @ binDag_Mirabelle_Tip @ T3 ) ).
% subdag.simps(1)
thf(fact_136_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_137_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_138_subdag_Osimps_I2_J,axiom,
! [L: binDag_Mirabelle_dag,A2: simpl_ref,R2: binDag_Mirabelle_dag,T3: binDag_Mirabelle_dag] :
( ( binDag786255756subdag @ ( binDag476092410e_Node @ L @ A2 @ R2 ) @ T3 )
= ( ( T3 = L )
| ( T3 = R2 )
| ( binDag786255756subdag @ L @ T3 )
| ( binDag786255756subdag @ R2 @ T3 ) ) ) ).
% subdag.simps(2)
thf(fact_139_subdag__NodeD,axiom,
! [T3: binDag_Mirabelle_dag,Lt: binDag_Mirabelle_dag,A2: simpl_ref,Rt: binDag_Mirabelle_dag] :
( ( binDag786255756subdag @ T3 @ ( binDag476092410e_Node @ Lt @ A2 @ Rt ) )
=> ( ( binDag786255756subdag @ T3 @ Lt )
& ( binDag786255756subdag @ T3 @ Rt ) ) ) ).
% subdag_NodeD
thf(fact_140_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_141_dag_Oinduct,axiom,
! [P: binDag_Mirabelle_dag > $o,Dag: binDag_Mirabelle_dag] :
( ( P @ binDag_Mirabelle_Tip )
=> ( ! [X1: binDag_Mirabelle_dag,X24: simpl_ref,X32: binDag_Mirabelle_dag] :
( ( P @ X1 )
=> ( ( P @ X32 )
=> ( P @ ( binDag476092410e_Node @ X1 @ X24 @ X32 ) ) ) )
=> ( P @ Dag ) ) ) ).
% dag.induct
thf(fact_142_dag_Oexhaust,axiom,
! [Y2: binDag_Mirabelle_dag] :
( ( Y2 != binDag_Mirabelle_Tip )
=> ~ ! [X212: binDag_Mirabelle_dag,X222: simpl_ref,X232: binDag_Mirabelle_dag] :
( Y2
!= ( binDag476092410e_Node @ X212 @ X222 @ X232 ) ) ) ).
% dag.exhaust
thf(fact_143_DAG_Osimps_I1_J,axiom,
binDag_Mirabelle_DAG @ binDag_Mirabelle_Tip ).
% DAG.simps(1)
thf(fact_144_subdag__size,axiom,
! [T3: binDag_Mirabelle_dag,S2: binDag_Mirabelle_dag] :
( ( binDag786255756subdag @ T3 @ S2 )
=> ( ord_less @ nat @ ( size_size @ binDag_Mirabelle_dag @ S2 ) @ ( size_size @ binDag_Mirabelle_dag @ T3 ) ) ) ).
% subdag_size
thf(fact_145_order_Oordering__axioms,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ordering @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% order.ordering_axioms
thf(fact_146_antimonoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B,X2: A,Y2: A] :
( ( order_antimono @ A @ B @ F )
=> ( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ B @ ( F @ Y2 ) @ ( F @ X2 ) ) ) ) ) ).
% antimonoD
thf(fact_147_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_148_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_149_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_150_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_151_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_152_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_153_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_154_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_155_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_156_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_157_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_158_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_159_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_160_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_161_ordering__def,axiom,
! [A: $tType] :
( ( ordering @ A )
= ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o] :
( ! [A5: A,B4: A] :
( ( Less2 @ A5 @ B4 )
= ( ( Less_eq2 @ A5 @ B4 )
& ( A5 != B4 ) ) )
& ! [A5: A] : ( Less_eq2 @ A5 @ A5 )
& ! [A5: A,B4: A] :
( ( Less_eq2 @ A5 @ B4 )
=> ( ( Less_eq2 @ B4 @ A5 )
=> ( A5 = B4 ) ) )
& ! [A5: A,B4: A,C4: A] :
( ( Less_eq2 @ A5 @ B4 )
=> ( ( Less_eq2 @ B4 @ C4 )
=> ( Less_eq2 @ A5 @ C4 ) ) ) ) ) ) ).
% ordering_def
thf(fact_162_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ( ( order_antimono @ A @ B )
= ( ^ [F2: A > B] :
! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F2 @ Y ) @ ( F2 @ X ) ) ) ) ) ) ).
% antimono_def
thf(fact_163_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B] :
( ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ B @ ( F @ Y3 ) @ ( F @ X4 ) ) )
=> ( order_antimono @ A @ B @ F ) ) ) ).
% antimonoI
thf(fact_164_antimonoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B,X2: A,Y2: A] :
( ( order_antimono @ A @ B @ F )
=> ( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ B @ ( F @ Y2 ) @ ( F @ X2 ) ) ) ) ) ).
% antimonoE
thf(fact_165_DAG_Osimps_I2_J,axiom,
! [L: binDag_Mirabelle_dag,A2: simpl_ref,R2: binDag_Mirabelle_dag] :
( ( binDag_Mirabelle_DAG @ ( binDag476092410e_Node @ L @ A2 @ R2 ) )
= ( ~ ( member @ simpl_ref @ A2 @ ( binDag1380252983set_of @ L ) )
& ~ ( member @ simpl_ref @ A2 @ ( binDag1380252983set_of @ R2 ) )
& ( binDag_Mirabelle_DAG @ L )
& ( binDag_Mirabelle_DAG @ R2 ) ) ) ).
% DAG.simps(2)
thf(fact_166_size__ne__size__imp__ne,axiom,
! [A: $tType] :
( ( size @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ( ( size_size @ A @ X2 )
!= ( size_size @ A @ Y2 ) )
=> ( X2 != Y2 ) ) ) ).
% size_ne_size_imp_ne
thf(fact_167_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
& ( M != N ) ) ) ) ).
% nat_less_le
thf(fact_168_nat__le__linear,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
| ( ord_less_eq @ nat @ N2 @ M2 ) ) ).
% nat_le_linear
thf(fact_169_le__antisym,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( ord_less_eq @ nat @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ).
% le_antisym
thf(fact_170_eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( M2 = N2 )
=> ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% eq_imp_le
thf(fact_171_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ J @ K )
=> ( ord_less_eq @ nat @ I @ K ) ) ) ).
% le_trans
thf(fact_172_le__refl,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ N2 @ N2 ) ).
% le_refl
thf(fact_173_nat__neq__iff,axiom,
! [M2: nat,N2: nat] :
( ( M2 != N2 )
= ( ( ord_less @ nat @ M2 @ N2 )
| ( ord_less @ nat @ N2 @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_174_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_175_less__not__refl2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ N2 @ M2 )
=> ( M2 != N2 ) ) ).
% less_not_refl2
thf(fact_176_less__not__refl3,axiom,
! [S2: nat,T3: nat] :
( ( ord_less @ nat @ S2 @ T3 )
=> ( S2 != T3 ) ) ).
% less_not_refl3
thf(fact_177_measure__induct,axiom,
! [A: $tType,F: A > nat,P: A > $o,A2: A] :
( ! [X4: A] :
( ! [Y5: A] :
( ( ord_less @ nat @ ( F @ Y5 ) @ ( F @ X4 ) )
=> ( P @ Y5 ) )
=> ( P @ X4 ) )
=> ( P @ A2 ) ) ).
% measure_induct
thf(fact_178_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_179_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_180_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_181_linorder__neqE__nat,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less @ nat @ X2 @ Y2 )
=> ( ord_less @ nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_182_measure__induct__rule,axiom,
! [A: $tType,F: A > nat,P: A > $o,A2: A] :
( ! [X4: A] :
( ! [Y5: A] :
( ( ord_less @ nat @ ( F @ Y5 ) @ ( F @ X4 ) )
=> ( P @ Y5 ) )
=> ( P @ X4 ) )
=> ( P @ A2 ) ) ).
% measure_induct_rule
thf(fact_183_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X2: A] :
( ! [X4: A] :
( ~ ( P @ X4 )
=> ? [Y5: A] :
( ( ord_less @ nat @ ( V @ Y5 ) @ ( V @ X4 ) )
& ~ ( P @ Y5 ) ) )
=> ( P @ X2 ) ) ).
% infinite_descent_measure
thf(fact_184_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_185_le__neq__implies__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( M2 != N2 )
=> ( ord_less @ nat @ M2 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_186_less__or__eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( ( ord_less @ nat @ M2 @ N2 )
| ( M2 = N2 ) )
=> ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_187_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
| ( M = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_188_less__imp__le__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_189_ex__has__greatest__nat,axiom,
! [A: $tType,P: A > $o,K: A,M2: A > nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less @ nat @ ( M2 @ Y3 ) @ B2 ) )
=> ? [X4: A] :
( ( P @ X4 )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( M2 @ Y5 ) @ ( M2 @ X4 ) ) ) ) ) ) ).
% ex_has_greatest_nat
thf(fact_190_ex__has__least__nat,axiom,
! [A: $tType,P: A > $o,K: A,M2: A > nat] :
( ( P @ K )
=> ? [X4: A] :
( ( P @ X4 )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( M2 @ X4 ) @ ( M2 @ Y5 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_191_length__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ! [Xs2: list @ A] :
( ! [Ys: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ Ys ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_192_Ex__list__of__length,axiom,
! [A: $tType,N2: nat] :
? [Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ Xs2 )
= N2 ) ).
% Ex_list_of_length
thf(fact_193_neq__if__length__neq,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
!= ( size_size @ ( list @ A ) @ Ys2 ) )
=> ( Xs != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_194_count__le__length,axiom,
! [A: $tType,Xs: list @ A,X2: A] : ( ord_less_eq @ nat @ ( count_list @ A @ Xs @ X2 ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% count_le_length
thf(fact_195_GreatestM__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,M2: A > nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less @ nat @ ( M2 @ Y3 ) @ B2 ) )
=> ( ( P @ ( hilbert_GreatestM @ A @ nat @ M2 @ P ) )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( M2 @ Y5 ) @ ( M2 @ ( hilbert_GreatestM @ A @ nat @ M2 @ P ) ) ) ) ) ) ) ).
% GreatestM_nat_lemma
thf(fact_196_GreatestM__natI,axiom,
! [A: $tType,P: A > $o,K: A,M2: A > nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less @ nat @ ( M2 @ Y3 ) @ B2 ) )
=> ( P @ ( hilbert_GreatestM @ A @ nat @ M2 @ P ) ) ) ) ).
% GreatestM_natI
thf(fact_197_GreatestMI2,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [P: A > $o,X2: A,M2: A > B,Q: A > $o] :
( ( P @ X2 )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ B @ ( M2 @ Y3 ) @ ( M2 @ X2 ) ) )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ B @ ( M2 @ Y5 ) @ ( M2 @ X4 ) ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( hilbert_GreatestM @ A @ B @ M2 @ P ) ) ) ) ) ) ).
% GreatestMI2
thf(fact_198_GreatestM__nat__le,axiom,
! [A: $tType,P: A > $o,X2: A,M2: A > nat,B2: nat] :
( ( P @ X2 )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less @ nat @ ( M2 @ Y3 ) @ B2 ) )
=> ( ord_less_eq @ nat @ ( M2 @ X2 ) @ ( M2 @ ( hilbert_GreatestM @ A @ nat @ M2 @ P ) ) ) ) ) ).
% GreatestM_nat_le
thf(fact_199_LeastMI2,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [P: A > $o,X2: A,M2: A > B,Q: A > $o] :
( ( P @ X2 )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ B @ ( M2 @ X2 ) @ ( M2 @ Y3 ) ) )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ B @ ( M2 @ X4 ) @ ( M2 @ Y5 ) ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( hilbert_LeastM @ A @ B @ M2 @ P ) ) ) ) ) ) ).
% LeastMI2
thf(fact_200_dag_Osize_I3_J,axiom,
( ( size_size @ binDag_Mirabelle_dag @ binDag_Mirabelle_Tip )
= ( zero_zero @ nat ) ) ).
% dag.size(3)
thf(fact_201_neq0__conv,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% neq0_conv
thf(fact_202_le0,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ).
% le0
thf(fact_203_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ~ ( P @ ( zero_zero @ nat ) )
=> ( ( P @ N2 )
=> ? [K2: nat] :
( ( ord_less_eq @ nat @ K2 @ N2 )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_204_infinite__descent0__measure,axiom,
! [A: $tType,V: A > nat,P: A > $o,X2: A] :
( ! [X4: A] :
( ( ( V @ X4 )
= ( zero_zero @ nat ) )
=> ( P @ X4 ) )
=> ( ! [X4: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X4 ) )
=> ( ~ ( P @ X4 )
=> ? [Y5: A] :
( ( ord_less @ nat @ ( V @ Y5 ) @ ( V @ X4 ) )
& ~ ( P @ Y5 ) ) ) )
=> ( P @ X2 ) ) ) ).
% infinite_descent0_measure
thf(fact_205_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_206_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_207_gr__implies__not0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( N2
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_208_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_209_not__less0,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_210_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_211_gr0I,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% gr0I
thf(fact_212_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq @ nat @ N2 @ ( zero_zero @ nat ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_213_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_214_LeastM__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,M2: A > nat] :
( ( P @ K )
=> ( ( P @ ( hilbert_LeastM @ A @ nat @ M2 @ P ) )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( M2 @ ( hilbert_LeastM @ A @ nat @ M2 @ P ) ) @ ( M2 @ Y5 ) ) ) ) ) ).
% LeastM_nat_lemma
thf(fact_215_LeastM__nat__le,axiom,
! [A: $tType,P: A > $o,X2: A,M2: A > nat] :
( ( P @ X2 )
=> ( ord_less_eq @ nat @ ( M2 @ ( hilbert_LeastM @ A @ nat @ M2 @ P ) ) @ ( M2 @ X2 ) ) ) ).
% LeastM_nat_le
thf(fact_216_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N2: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) )
= ( N2
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_217_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N2: A] :
( ( ord_less_eq @ A @ N2 @ ( zero_zero @ A ) )
= ( N2
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_218_LeastM__natI,axiom,
! [A: $tType,P: A > $o,K: A,M2: A > nat] :
( ( P @ K )
=> ( P @ ( hilbert_LeastM @ A @ nat @ M2 @ P ) ) ) ).
% LeastM_natI
thf(fact_219_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [X2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 ) ) ).
% zero_le
thf(fact_220_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N2: A] :
( ( N2
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) ) ) ).
% gr_zeroI
thf(fact_221_not__less__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N2: A] :
~ ( ord_less @ A @ N2 @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_222_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [M2: A,N2: A] :
( ( ord_less @ A @ M2 @ N2 )
=> ( N2
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_223_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N2 )
= ( N2
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_224_natural_Osize_I3_J,axiom,
( ( size_size @ code_natural @ ( zero_zero @ code_natural ) )
= ( zero_zero @ nat ) ) ).
% natural.size(3)
thf(fact_225_bool_Osize_I4_J,axiom,
( ( size_size @ $o @ $false )
= ( zero_zero @ nat ) ) ).
% bool.size(4)
thf(fact_226_bool_Osize_I3_J,axiom,
( ( size_size @ $o @ $true )
= ( zero_zero @ nat ) ) ).
% bool.size(3)
thf(fact_227_prod_Osize__neq,axiom,
! [A: $tType,B: $tType,X2: product_prod @ A @ B] :
( ( size_size @ ( product_prod @ A @ B ) @ X2 )
!= ( zero_zero @ nat ) ) ).
% prod.size_neq
thf(fact_228_sum_Osize__neq,axiom,
! [A: $tType,B: $tType,X2: sum_sum @ A @ B] :
( ( size_size @ ( sum_sum @ A @ B ) @ X2 )
!= ( zero_zero @ nat ) ) ).
% sum.size_neq
thf(fact_229_size__bool,axiom,
( ( size_size @ $o )
= ( ^ [B4: $o] : ( zero_zero @ nat ) ) ) ).
% size_bool
thf(fact_230_size__bool__overloaded__def,axiom,
( ( size_size @ $o )
= ( product_rec_bool @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ).
% size_bool_overloaded_def
thf(fact_231_dag_Osize__gen_I1_J,axiom,
( ( binDag1924123185ze_dag @ binDag_Mirabelle_Tip )
= ( zero_zero @ nat ) ) ).
% dag.size_gen(1)
thf(fact_232_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_233_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_234_length__code,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) )
= ( gen_length @ A @ ( zero_zero @ nat ) ) ) ).
% length_code
thf(fact_235_of__nat__0__less__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N2 ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% of_nat_0_less_iff
thf(fact_236_of__nat__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A @ ( type2 @ A ) )
=> ! [M2: nat,N2: nat] :
( ( ( semiring_1_of_nat @ A @ M2 )
= ( semiring_1_of_nat @ A @ N2 ) )
= ( M2 = N2 ) ) ) ).
% of_nat_eq_iff
thf(fact_237_of__nat__eq__0__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A @ ( type2 @ A ) )
=> ! [M2: nat] :
( ( ( semiring_1_of_nat @ A @ M2 )
= ( zero_zero @ A ) )
= ( M2
= ( zero_zero @ nat ) ) ) ) ).
% of_nat_eq_0_iff
thf(fact_238_of__nat__0__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A @ ( type2 @ A ) )
=> ! [N2: nat] :
( ( ( zero_zero @ A )
= ( semiring_1_of_nat @ A @ N2 ) )
= ( ( zero_zero @ nat )
= N2 ) ) ) ).
% of_nat_0_eq_iff
thf(fact_239_of__nat__0,axiom,
! [A: $tType] :
( ( semiring_1 @ A @ ( type2 @ A ) )
=> ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) ) ) ).
% of_nat_0
thf(fact_240_of__nat__less__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [M2: nat,N2: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N2 ) )
= ( ord_less @ nat @ M2 @ N2 ) ) ) ).
% of_nat_less_iff
thf(fact_241_of__nat__le__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [M2: nat,N2: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N2 ) )
= ( ord_less_eq @ nat @ M2 @ N2 ) ) ) ).
% of_nat_le_iff
thf(fact_242_of__nat__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [M2: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( zero_zero @ A ) )
= ( M2
= ( zero_zero @ nat ) ) ) ) ).
% of_nat_le_0_iff
thf(fact_243_of__nat__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [M2: nat] :
~ ( ord_less @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( zero_zero @ A ) ) ) ).
% of_nat_less_0_iff
thf(fact_244_of__nat__0__le__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% of_nat_0_le_iff
thf(fact_245_of__nat__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [M2: nat,N2: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N2 ) )
=> ( ord_less @ nat @ M2 @ N2 ) ) ) ).
% of_nat_less_imp_less
thf(fact_246_less__imp__of__nat__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ord_less @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ).
% less_imp_of_nat_less
thf(fact_247_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ? [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
& ( K
= ( semiring_1_of_nat @ int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_248_of__nat__zero__less__power__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [X2: nat,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ X2 ) @ N2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X2 )
| ( N2
= ( zero_zero @ nat ) ) ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_249_nat__zero__less__power__iff,axiom,
! [X2: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X2 @ N2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X2 )
| ( N2
= ( zero_zero @ nat ) ) ) ) ).
% nat_zero_less_power_iff
thf(fact_250_power__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri134348788visors @ A @ ( type2 @ A ) )
=> ! [A2: A,N2: nat] :
( ( ( power_power @ A @ A2 @ N2 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% power_eq_0_iff
thf(fact_251_power__eq__iff__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [N2: nat,A2: A,B2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ( power_power @ A @ A2 @ N2 )
= ( power_power @ A @ B2 @ N2 ) )
= ( A2 = B2 ) ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_252_power__eq__imp__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [A2: A,N2: nat,B2: A] :
( ( ( power_power @ A @ A2 @ N2 )
= ( power_power @ A @ B2 @ N2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( A2 = B2 ) ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_253_zero__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [A2: A,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ).
% zero_less_power
thf(fact_254_zero__power,axiom,
! [A: $tType] :
( ( semiring_1 @ A @ ( type2 @ A ) )
=> ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N2 )
= ( zero_zero @ A ) ) ) ) ).
% zero_power
%----Type constructors (50)
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_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit1037483654norder @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
semiri134348788visors @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0 @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Orderings_Opreorder_1,axiom,
preorder @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
linorder @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1 @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
no_top @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
no_bot @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Orderings_Oorder_2,axiom,
order @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Orderings_Oord_3,axiom,
ord @ int @ ( type2 @ int ) ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_4,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_Osemiring__1__no__zero__divisors_5,axiom,
semiri134348788visors @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_6,axiom,
linordered_semidom @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0_7,axiom,
semiring_char_0 @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_8,axiom,
preorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder_9,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Osemiring__1_10,axiom,
semiring_1 @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Ono__top_11,axiom,
no_top @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_12,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_13,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_14,axiom,
! [A6: $tType] : ( preorder @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_15,axiom,
! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_16,axiom,
! [A6: $tType] : ( ord @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_17,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_18,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_19,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_20,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Nat_Osize_21,axiom,
size @ $o @ ( type2 @ $o ) ).
thf(tcon_List_Olist___Nat_Osize_22,axiom,
! [A6: $tType] : ( size @ ( list @ A6 ) @ ( type2 @ ( list @ A6 ) ) ) ).
thf(tcon_Sum__Type_Osum___Nat_Osize_23,axiom,
! [A6: $tType,A7: $tType] : ( size @ ( sum_sum @ A6 @ A7 ) @ ( type2 @ ( sum_sum @ A6 @ A7 ) ) ) ).
thf(tcon_Product__Type_Oprod___Nat_Osize_24,axiom,
! [A6: $tType,A7: $tType] : ( size @ ( product_prod @ A6 @ A7 ) @ ( type2 @ ( product_prod @ A6 @ A7 ) ) ) ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__no__zero__divisors_25,axiom,
semiri134348788visors @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Orderings_Opreorder_26,axiom,
preorder @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Orderings_Olinorder_27,axiom,
linorder @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_28,axiom,
semiring_1 @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Orderings_Oorder_29,axiom,
order @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Orderings_Oord_30,axiom,
ord @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Nat_Osize_31,axiom,
size @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Opreorder_32,axiom,
preorder @ binDag_Mirabelle_dag @ ( type2 @ binDag_Mirabelle_dag ) ).
thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Oorder_33,axiom,
order @ binDag_Mirabelle_dag @ ( type2 @ binDag_Mirabelle_dag ) ).
thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Oord_34,axiom,
ord @ binDag_Mirabelle_dag @ ( type2 @ binDag_Mirabelle_dag ) ).
thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Nat_Osize_35,axiom,
size @ binDag_Mirabelle_dag @ ( type2 @ binDag_Mirabelle_dag ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( ord_less @ binDag_Mirabelle_dag @ x @ ( binDag476092410e_Node @ l @ a @ r ) )
= ( ( ord_less_eq @ binDag_Mirabelle_dag @ x @ l )
| ( ord_less_eq @ binDag_Mirabelle_dag @ x @ r ) ) ) ).
%------------------------------------------------------------------------------