TPTP Problem File: ITP164^2.p
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%------------------------------------------------------------------------------
% File : ITP164^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Refine_Basic problem prob_743__3591834_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Refine_Basic/prob_743__3591834_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 340 ( 158 unt; 39 typ; 0 def)
% Number of atoms : 783 ( 261 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3660 ( 66 ~; 6 |; 40 &;3252 @)
% ( 0 <=>; 296 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 7 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 136 ( 136 >; 0 *; 0 +; 0 <<)
% Number of symbols : 40 ( 37 usr; 2 con; 0-4 aty)
% Number of variables : 1124 ( 61 ^;1011 !; 12 ?;1124 :)
% ( 40 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:19:33.107
%------------------------------------------------------------------------------
% Could-be-implicit typings (5)
thf(ty_t_Refine__Basic__Mirabelle__tqojlsrkwy_Onres,type,
refine1665802226e_nres: $tType > $tType ).
thf(ty_t_Product__Type_Ounit,type,
product_unit: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (34)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple187826305attice:
!>[A: $tType] : $o ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oorder__class_OGreatest,type,
order_Greatest:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Partial__Function_Oflat__ord,type,
partial_flat_ord:
!>[A: $tType] : ( A > A > A > $o ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_ORETURN,type,
refine1687780735RETURN:
!>[A: $tType] : ( A > ( refine1665802226e_nres @ A ) ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oabs__fun,type,
refine81118332bs_fun:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > ( refine1665802226e_nres @ B ) > ( refine1665802226e_nres @ A ) ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Obind,type,
refine463715084e_bind:
!>[B: $tType,A: $tType] : ( ( refine1665802226e_nres @ B ) > ( B > ( refine1665802226e_nres @ A ) ) > ( refine1665802226e_nres @ A ) ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oconc__fun,type,
refine1073749519nc_fun:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( refine1665802226e_nres @ B ) > ( refine1665802226e_nres @ A ) ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oinres,type,
refine1315500908_inres:
!>[A: $tType] : ( ( refine1665802226e_nres @ A ) > A > $o ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onf__inres,type,
refine406925620_inres:
!>[A: $tType] : ( ( refine1665802226e_nres @ A ) > A > $o ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onofail,type,
refine1102455758nofail:
!>[A: $tType] : ( ( refine1665802226e_nres @ A ) > $o ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres_ORES,type,
refine605929679le_RES:
!>[A: $tType] : ( ( set @ A ) > ( refine1665802226e_nres @ A ) ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Othe__RES,type,
refine1672542526he_RES:
!>[A: $tType] : ( ( refine1665802226e_nres @ A ) > ( set @ A ) ) ).
thf(sy_c_Refine__Misc_Ogalois__connection,type,
refine1150083786ection:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( B > A ) > $o ) ).
thf(sy_c_Relation_ODomain,type,
domain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) ) ).
thf(sy_c_Relation_Osingle__valued,type,
single_valued:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_f,type,
f: product_unit > ( refine1665802226e_nres @ a ) ).
% Relevant facts (255)
thf(fact_0_bind__SUCCEED,axiom,
! [B: $tType,A: $tType,F: B > ( refine1665802226e_nres @ A )] :
( ( refine463715084e_bind @ B @ A @ ( bot_bot @ ( refine1665802226e_nres @ B ) ) @ F )
= ( bot_bot @ ( refine1665802226e_nres @ A ) ) ) ).
% bind_SUCCEED
thf(fact_1_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_2_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_3_nres__monad1,axiom,
! [A: $tType,B: $tType,X2: B,F: B > ( refine1665802226e_nres @ A )] :
( ( refine463715084e_bind @ B @ A @ ( refine1687780735RETURN @ B @ X2 ) @ F )
= ( F @ X2 ) ) ).
% nres_monad1
thf(fact_4_nres__monad2,axiom,
! [A: $tType,M: refine1665802226e_nres @ A] :
( ( refine463715084e_bind @ A @ A @ M @ ( refine1687780735RETURN @ A ) )
= M ) ).
% nres_monad2
thf(fact_5_abs__fun__strict,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ A )] :
( ( refine81118332bs_fun @ B @ A @ R @ ( bot_bot @ ( refine1665802226e_nres @ B ) ) )
= ( bot_bot @ ( refine1665802226e_nres @ A ) ) ) ).
% abs_fun_strict
thf(fact_6_conc__fun__strict,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
( ( refine1073749519nc_fun @ A @ B @ R @ ( bot_bot @ ( refine1665802226e_nres @ B ) ) )
= ( bot_bot @ ( refine1665802226e_nres @ A ) ) ) ).
% conc_fun_strict
thf(fact_7_nres__inequalities_I5_J,axiom,
! [C: $tType,X2: C] :
( ( bot_bot @ ( refine1665802226e_nres @ C ) )
!= ( refine1687780735RETURN @ C @ X2 ) ) ).
% nres_inequalities(5)
thf(fact_8_inres__simps_I4_J,axiom,
! [D: $tType] :
( ( refine1315500908_inres @ D @ ( bot_bot @ ( refine1665802226e_nres @ D ) ) )
= ( ^ [Uu: D] : $false ) ) ).
% inres_simps(4)
thf(fact_9_nres__order__simps_I2_J,axiom,
! [B: $tType,M: refine1665802226e_nres @ B] :
( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ M @ ( bot_bot @ ( refine1665802226e_nres @ B ) ) )
= ( M
= ( bot_bot @ ( refine1665802226e_nres @ B ) ) ) ) ).
% nres_order_simps(2)
thf(fact_10_nofail__simps_I4_J,axiom,
! [D: $tType] : ( refine1102455758nofail @ D @ ( bot_bot @ ( refine1665802226e_nres @ D ) ) ) ).
% nofail_simps(4)
thf(fact_11_bind__FAIL,axiom,
! [B: $tType,A: $tType,F: B > ( refine1665802226e_nres @ A )] :
( ( refine463715084e_bind @ B @ A @ ( top_top @ ( refine1665802226e_nres @ B ) ) @ F )
= ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% bind_FAIL
thf(fact_12_nres__order__simps_I1_J,axiom,
! [A: $tType,M: refine1665802226e_nres @ A] : ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( bot_bot @ ( refine1665802226e_nres @ A ) ) @ M ) ).
% nres_order_simps(1)
thf(fact_13_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_14_nres__more__simps_I6_J,axiom,
! [A: $tType,X2: A,Y: A] :
( ( ( refine1687780735RETURN @ A @ X2 )
= ( refine1687780735RETURN @ A @ Y ) )
= ( X2 = Y ) ) ).
% nres_more_simps(6)
thf(fact_15_nres__order__simps_I4_J,axiom,
! [D: $tType,M: refine1665802226e_nres @ D] :
( ( ord_less_eq @ ( refine1665802226e_nres @ D ) @ ( top_top @ ( refine1665802226e_nres @ D ) ) @ M )
= ( M
= ( top_top @ ( refine1665802226e_nres @ D ) ) ) ) ).
% nres_order_simps(4)
thf(fact_16_nofail__simps_I1_J,axiom,
! [A: $tType] :
~ ( refine1102455758nofail @ A @ ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% nofail_simps(1)
thf(fact_17_inres__simps_I1_J,axiom,
! [A: $tType] :
( ( refine1315500908_inres @ A @ ( top_top @ ( refine1665802226e_nres @ A ) ) )
= ( ^ [Uu: A] : $true ) ) ).
% inres_simps(1)
thf(fact_18_conc__fun__fail__iff_I2_J,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S: refine1665802226e_nres @ B] :
( ( ( top_top @ ( refine1665802226e_nres @ A ) )
= ( refine1073749519nc_fun @ A @ B @ R @ S ) )
= ( S
= ( top_top @ ( refine1665802226e_nres @ B ) ) ) ) ).
% conc_fun_fail_iff(2)
thf(fact_19_conc__fun__fail__iff_I1_J,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S: refine1665802226e_nres @ B] :
( ( ( refine1073749519nc_fun @ A @ B @ R @ S )
= ( top_top @ ( refine1665802226e_nres @ A ) ) )
= ( S
= ( top_top @ ( refine1665802226e_nres @ B ) ) ) ) ).
% conc_fun_fail_iff(1)
thf(fact_20_conc__fun__FAIL,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
( ( refine1073749519nc_fun @ A @ B @ R @ ( top_top @ ( refine1665802226e_nres @ B ) ) )
= ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% conc_fun_FAIL
thf(fact_21_nres__order__simps_I20_J,axiom,
! [W: $tType,X2: W,Y: W] :
( ( ord_less_eq @ ( refine1665802226e_nres @ W ) @ ( refine1687780735RETURN @ W @ X2 ) @ ( refine1687780735RETURN @ W @ Y ) )
= ( X2 = Y ) ) ).
% nres_order_simps(20)
thf(fact_22_abs__fun__simps_I1_J,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ A )] :
( ( refine81118332bs_fun @ B @ A @ R @ ( top_top @ ( refine1665802226e_nres @ B ) ) )
= ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% abs_fun_simps(1)
thf(fact_23_nofail__simps_I3_J,axiom,
! [C: $tType,X2: C] : ( refine1102455758nofail @ C @ ( refine1687780735RETURN @ C @ X2 ) ) ).
% nofail_simps(3)
thf(fact_24_inres__simps_I3_J,axiom,
! [C: $tType,X2: C] :
( ( refine1315500908_inres @ C @ ( refine1687780735RETURN @ C @ X2 ) )
= ( ^ [Y2: C,Z: C] : Y2 = Z
@ X2 ) ) ).
% inres_simps(3)
thf(fact_25_intro__nofail_I2_J,axiom,
! [A: $tType,S: refine1665802226e_nres @ A] :
( ( ( top_top @ ( refine1665802226e_nres @ A ) )
!= S )
= ( refine1102455758nofail @ A @ S ) ) ).
% intro_nofail(2)
thf(fact_26_nres__order__simps_I3_J,axiom,
! [C: $tType,M: refine1665802226e_nres @ C] : ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ M @ ( top_top @ ( refine1665802226e_nres @ C ) ) ) ).
% nres_order_simps(3)
thf(fact_27_nres__inequalities_I3_J,axiom,
! [C: $tType,X2: C] :
( ( top_top @ ( refine1665802226e_nres @ C ) )
!= ( refine1687780735RETURN @ C @ X2 ) ) ).
% nres_inequalities(3)
thf(fact_28_abs__trans__additional_I1_J,axiom,
! [A: $tType,B: $tType,A2: refine1665802226e_nres @ A,B2: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),C2: refine1665802226e_nres @ B] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ B2 ) @ C2 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ A2 ) @ C2 ) ) ) ).
% abs_trans_additional(1)
thf(fact_29_conc__trans__additional_I1_J,axiom,
! [A: $tType,B: $tType,A2: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),B2: refine1665802226e_nres @ B,C2: refine1665802226e_nres @ B] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ ( refine1073749519nc_fun @ A @ B @ R @ B2 ) )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ B2 @ C2 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ ( refine1073749519nc_fun @ A @ B @ R @ C2 ) ) ) ) ).
% conc_trans_additional(1)
thf(fact_30_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funD
thf(fact_31_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funE
thf(fact_32_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_33_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% le_fun_def
thf(fact_34_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B3: B,C3: B] :
( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C3 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% order_subst1
thf(fact_35_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B3: A,F: A > C,C3: C] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ C @ ( F @ B3 ) @ C3 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% order_subst2
thf(fact_36_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F: B > A,B3: B,C3: B] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C3 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_37_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B3: A,F: A > B,C3: B] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C3 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_38_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y2: A,Z: A] : Y2 = Z )
= ( ^ [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
& ( ord_less_eq @ A @ Y4 @ X ) ) ) ) ) ).
% eq_iff
thf(fact_39_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ X2 )
=> ( X2 = Y ) ) ) ) ).
% antisym
thf(fact_40_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
| ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% linear
thf(fact_41_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( X2 = Y )
=> ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% eq_refl
thf(fact_42_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ~ ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% le_cases
thf(fact_43_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% order.trans
thf(fact_44_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( ( ord_less_eq @ A @ X2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X2 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y )
= ( X2 = Y ) ) ) ) ).
% antisym_conv
thf(fact_50_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y2: A,Z: A] : Y2 = Z )
= ( ^ [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
& ( ord_less_eq @ A @ B4 @ A4 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_51_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( A3 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_52_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( B3 = C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_53_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_54_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less_eq @ A @ X2 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_55_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_56_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B3: A] :
( ! [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: A,B5: A] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_57_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ B3 )
=> ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_58_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_59_pwD1,axiom,
! [A: $tType,S: refine1665802226e_nres @ A,S2: refine1665802226e_nres @ A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S2 )
=> ( ( refine1102455758nofail @ A @ S2 )
=> ( refine1102455758nofail @ A @ S ) ) ) ).
% pwD1
thf(fact_60_pwD2,axiom,
! [A: $tType,S: refine1665802226e_nres @ A,S2: refine1665802226e_nres @ A,X2: A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S2 )
=> ( ( refine1315500908_inres @ A @ S @ X2 )
=> ( refine1315500908_inres @ A @ S2 @ X2 ) ) ) ).
% pwD2
thf(fact_61_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y2: A,Z: A] : Y2 = Z )
= ( ^ [A4: A,B4: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
& ( ord_less_eq @ A @ A4 @ B4 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_62_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_63_pw__eqI,axiom,
! [A: $tType,S: refine1665802226e_nres @ A,S2: refine1665802226e_nres @ A] :
( ( ( refine1102455758nofail @ A @ S )
= ( refine1102455758nofail @ A @ S2 ) )
=> ( ! [X3: A] :
( ( refine1315500908_inres @ A @ S @ X3 )
= ( refine1315500908_inres @ A @ S2 @ X3 ) )
=> ( S = S2 ) ) ) ).
% pw_eqI
thf(fact_64_pw__leI,axiom,
! [A: $tType,S2: refine1665802226e_nres @ A,S: refine1665802226e_nres @ A] :
( ( ( refine1102455758nofail @ A @ S2 )
=> ( ( refine1102455758nofail @ A @ S )
& ! [X3: A] :
( ( refine1315500908_inres @ A @ S @ X3 )
=> ( refine1315500908_inres @ A @ S2 @ X3 ) ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S2 ) ) ).
% pw_leI
thf(fact_65_pw__leI_H,axiom,
! [A: $tType,S2: refine1665802226e_nres @ A,S: refine1665802226e_nres @ A] :
( ( ( refine1102455758nofail @ A @ S2 )
=> ( refine1102455758nofail @ A @ S ) )
=> ( ! [X3: A] :
( ( refine1102455758nofail @ A @ S2 )
=> ( ( refine1315500908_inres @ A @ S @ X3 )
=> ( refine1315500908_inres @ A @ S2 @ X3 ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S2 ) ) ) ).
% pw_leI'
thf(fact_66_abs__trans,axiom,
! [B: $tType,A: $tType,C: $tType,R: set @ ( product_prod @ B @ A ),C2: refine1665802226e_nres @ B,B2: refine1665802226e_nres @ A,R2: set @ ( product_prod @ A @ C ),A2: refine1665802226e_nres @ C] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine81118332bs_fun @ B @ A @ R @ C2 ) @ B2 )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( refine81118332bs_fun @ A @ C @ R2 @ B2 ) @ A2 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( refine81118332bs_fun @ A @ C @ R2 @ ( refine81118332bs_fun @ B @ A @ R @ C2 ) ) @ A2 ) ) ) ).
% abs_trans
thf(fact_67_inres__def,axiom,
! [A: $tType] :
( ( refine1315500908_inres @ A )
= ( ^ [S3: refine1665802226e_nres @ A,X: A] : ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X ) @ S3 ) ) ) ).
% inres_def
thf(fact_68_pw__eq__iff,axiom,
! [A: $tType] :
( ( ^ [Y2: refine1665802226e_nres @ A,Z: refine1665802226e_nres @ A] : Y2 = Z )
= ( ^ [S3: refine1665802226e_nres @ A,S4: refine1665802226e_nres @ A] :
( ( ( refine1102455758nofail @ A @ S3 )
= ( refine1102455758nofail @ A @ S4 ) )
& ! [X: A] :
( ( refine1315500908_inres @ A @ S3 @ X )
= ( refine1315500908_inres @ A @ S4 @ X ) ) ) ) ) ).
% pw_eq_iff
thf(fact_69_pw__le__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) )
= ( ^ [S3: refine1665802226e_nres @ A,S4: refine1665802226e_nres @ A] :
( ( refine1102455758nofail @ A @ S4 )
=> ( ( refine1102455758nofail @ A @ S3 )
& ! [X: A] :
( ( refine1315500908_inres @ A @ S3 @ X )
=> ( refine1315500908_inres @ A @ S4 @ X ) ) ) ) ) ) ).
% pw_le_iff
thf(fact_70_conc__trans,axiom,
! [A: $tType,B: $tType,C: $tType,C2: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),B2: refine1665802226e_nres @ B,R2: set @ ( product_prod @ B @ C ),A2: refine1665802226e_nres @ C] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ C2 @ ( refine1073749519nc_fun @ A @ B @ R @ B2 ) )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ B2 @ ( refine1073749519nc_fun @ B @ C @ R2 @ A2 ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ C2 @ ( refine1073749519nc_fun @ A @ B @ R @ ( refine1073749519nc_fun @ B @ C @ R2 @ A2 ) ) ) ) ) ).
% conc_trans
thf(fact_71_le__nofailI,axiom,
! [A: $tType,M2: refine1665802226e_nres @ A,M: refine1665802226e_nres @ A] :
( ( ( refine1102455758nofail @ A @ M2 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ M2 ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ M2 ) ) ).
% le_nofailI
thf(fact_72_nofail__def,axiom,
! [A: $tType] :
( ( refine1102455758nofail @ A )
= ( ^ [S3: refine1665802226e_nres @ A] :
( S3
!= ( top_top @ ( refine1665802226e_nres @ A ) ) ) ) ) ).
% nofail_def
thf(fact_73_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
= ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_74_pw__bind__leI,axiom,
! [B: $tType,A: $tType,S: refine1665802226e_nres @ A,M: refine1665802226e_nres @ B,F: B > ( refine1665802226e_nres @ A )] :
( ( ( refine1102455758nofail @ A @ S )
=> ( refine1102455758nofail @ B @ M ) )
=> ( ! [X3: B] :
( ( refine1102455758nofail @ B @ M )
=> ( ( refine1315500908_inres @ B @ M @ X3 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( F @ X3 ) @ S ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine463715084e_bind @ B @ A @ M @ F ) @ S ) ) ) ).
% pw_bind_leI
thf(fact_75_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
=> ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_76_not__nofail__iff,axiom,
! [A: $tType,S: refine1665802226e_nres @ A] :
( ( ~ ( refine1102455758nofail @ A @ S ) )
= ( S
= ( top_top @ ( refine1665802226e_nres @ A ) ) ) ) ).
% not_nofail_iff
thf(fact_77_pw__bind__le__iff,axiom,
! [A: $tType,B: $tType,M: refine1665802226e_nres @ B,F: B > ( refine1665802226e_nres @ A ),S: refine1665802226e_nres @ A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine463715084e_bind @ B @ A @ M @ F ) @ S )
= ( ( ( refine1102455758nofail @ A @ S )
=> ( refine1102455758nofail @ B @ M ) )
& ! [X: B] :
( ( ( refine1102455758nofail @ B @ M )
& ( refine1315500908_inres @ B @ M @ X ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( F @ X ) @ S ) ) ) ) ).
% pw_bind_le_iff
thf(fact_78_pw__conc__nofail,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S: refine1665802226e_nres @ B] :
( ( refine1102455758nofail @ A @ ( refine1073749519nc_fun @ A @ B @ R @ S ) )
= ( refine1102455758nofail @ B @ S ) ) ).
% pw_conc_nofail
thf(fact_79_not__nofail__inres,axiom,
! [A: $tType,S: refine1665802226e_nres @ A,X2: A] :
( ~ ( refine1102455758nofail @ A @ S )
=> ( refine1315500908_inres @ A @ S @ X2 ) ) ).
% not_nofail_inres
thf(fact_80_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
=> ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_81_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
= ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_82_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% bot.extremum
thf(fact_83_pw__bind__nofail,axiom,
! [A: $tType,B: $tType,M: refine1665802226e_nres @ B,F: B > ( refine1665802226e_nres @ A )] :
( ( refine1102455758nofail @ A @ ( refine463715084e_bind @ B @ A @ M @ F ) )
= ( ( refine1102455758nofail @ B @ M )
& ! [X: B] :
( ( refine1315500908_inres @ B @ M @ X )
=> ( refine1102455758nofail @ A @ ( F @ X ) ) ) ) ) ).
% pw_bind_nofail
thf(fact_84_bind__cong,axiom,
! [B: $tType,A: $tType,M3: refine1665802226e_nres @ A,M4: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F3: A > ( refine1665802226e_nres @ B )] :
( ( M3 = M4 )
=> ( ! [X3: A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X3 ) @ M4 )
=> ( ( F @ X3 )
= ( F3 @ X3 ) ) )
=> ( ( refine463715084e_bind @ A @ B @ M3 @ F )
= ( refine463715084e_bind @ A @ B @ M4 @ F3 ) ) ) ) ).
% bind_cong
thf(fact_85_Refine__Basic__Mirabelle__tqojlsrkwy_Obind__mono_I1_J,axiom,
! [B: $tType,A: $tType,M: refine1665802226e_nres @ A,M2: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F3: A > ( refine1665802226e_nres @ B )] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ M2 )
=> ( ! [X3: A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X3 ) @ M )
=> ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( F @ X3 ) @ ( F3 @ X3 ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine463715084e_bind @ A @ B @ M @ F ) @ ( refine463715084e_bind @ A @ B @ M2 @ F3 ) ) ) ) ).
% Refine_Basic_Mirabelle_tqojlsrkwy.bind_mono(1)
thf(fact_86_nres__inequalities_I2_J,axiom,
! [B: $tType] :
( ( top_top @ ( refine1665802226e_nres @ B ) )
!= ( bot_bot @ ( refine1665802226e_nres @ B ) ) ) ).
% nres_inequalities(2)
thf(fact_87_nres__inequalities_I4_J,axiom,
! [D: $tType] :
( ( bot_bot @ ( refine1665802226e_nres @ D ) )
!= ( top_top @ ( refine1665802226e_nres @ D ) ) ) ).
% nres_inequalities(4)
thf(fact_88_order__mono__setup_Orefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% order_mono_setup.refl
thf(fact_89_nf__inres__def,axiom,
! [A: $tType] :
( ( refine406925620_inres @ A )
= ( ^ [M5: refine1665802226e_nres @ A,X: A] :
( ( refine1102455758nofail @ A @ M5 )
& ( refine1315500908_inres @ A @ M5 @ X ) ) ) ) ).
% nf_inres_def
thf(fact_90_meta__le__everything__if__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [M3: A,X2: A] :
( ( M3
= ( top_top @ A ) )
=> ( ord_less_eq @ A @ X2 @ M3 ) ) ) ).
% meta_le_everything_if_top
thf(fact_91_pw__flat__le__iff,axiom,
! [A: $tType,S: refine1665802226e_nres @ A,S2: refine1665802226e_nres @ A] :
( ( partial_flat_ord @ ( refine1665802226e_nres @ A ) @ ( bot_bot @ ( refine1665802226e_nres @ A ) ) @ S @ S2 )
= ( ? [X4: A] : ( refine1315500908_inres @ A @ S @ X4 )
=> ( ( ( refine1102455758nofail @ A @ S )
= ( refine1102455758nofail @ A @ S2 ) )
& ! [X: A] :
( ( refine1315500908_inres @ A @ S @ X )
= ( refine1315500908_inres @ A @ S2 @ X ) ) ) ) ) ).
% pw_flat_le_iff
thf(fact_92_pw__abs__inres,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),M: refine1665802226e_nres @ B,A3: A] :
( ( refine1315500908_inres @ A @ ( refine81118332bs_fun @ B @ A @ R @ M ) @ A3 )
= ( ( refine1102455758nofail @ A @ ( refine81118332bs_fun @ B @ A @ R @ M ) )
=> ? [C4: B] :
( ( refine1315500908_inres @ B @ M @ C4 )
& ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ C4 @ A3 ) @ R ) ) ) ) ).
% pw_abs_inres
thf(fact_93_pw__abs__nofail,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),M: refine1665802226e_nres @ B] :
( ( refine1102455758nofail @ A @ ( refine81118332bs_fun @ B @ A @ R @ M ) )
= ( ( refine1102455758nofail @ B @ M )
& ! [X: B] :
( ( refine1315500908_inres @ B @ M @ X )
=> ( member @ B @ X @ ( domain @ B @ A @ R ) ) ) ) ) ).
% pw_abs_nofail
thf(fact_94_conc__abs__swap,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),M4: refine1665802226e_nres @ A,M3: refine1665802226e_nres @ B] :
( ( single_valued @ A @ B @ R )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M4 @ ( refine1073749519nc_fun @ A @ B @ R @ M3 ) )
= ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ M4 ) @ M3 ) ) ) ).
% conc_abs_swap
thf(fact_95_pw__flat__ge__iff,axiom,
! [A: $tType,S: refine1665802226e_nres @ A,S2: refine1665802226e_nres @ A] :
( ( partial_flat_ord @ ( refine1665802226e_nres @ A ) @ ( top_top @ ( refine1665802226e_nres @ A ) ) @ S @ S2 )
= ( ( refine1102455758nofail @ A @ S )
=> ( ( refine1102455758nofail @ A @ S2 )
& ! [X: A] :
( ( refine1315500908_inres @ A @ S @ X )
= ( refine1315500908_inres @ A @ S2 @ X ) ) ) ) ) ).
% pw_flat_ge_iff
thf(fact_96_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X2: A,Q: A > $o] :
( ( P @ X2 )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X2 ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ A @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_97_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X2: A] :
( ( P @ X2 )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X2 ) )
=> ( ( order_Greatest @ A @ P )
= X2 ) ) ) ) ).
% Greatest_equality
thf(fact_98_Refine__Misc_Oif__mono,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [B3: $o,M1: A,M12: A,M22: A,M23: A] :
( ( B3
=> ( ord_less_eq @ A @ M1 @ M12 ) )
=> ( ( ~ B3
=> ( ord_less_eq @ A @ M22 @ M23 ) )
=> ( ord_less_eq @ A @ ( if @ A @ B3 @ M1 @ M22 ) @ ( if @ A @ B3 @ M12 @ M23 ) ) ) ) ) ).
% Refine_Misc.if_mono
thf(fact_99_flat__ord_Odual__order_Orefl,axiom,
! [A: $tType,B3: A,A3: A] : ( partial_flat_ord @ A @ B3 @ A3 @ A3 ) ).
% flat_ord.dual_order.refl
thf(fact_100_flat__le__mono__setup_Orefl,axiom,
! [A: $tType,B3: A,X2: A] : ( partial_flat_ord @ A @ B3 @ X2 @ X2 ) ).
% flat_le_mono_setup.refl
thf(fact_101_flat__ord_Odual__order_Oantisym,axiom,
! [A: $tType,Ba: A,B3: A,A3: A] :
( ( partial_flat_ord @ A @ Ba @ B3 @ A3 )
=> ( ( partial_flat_ord @ A @ Ba @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% flat_ord.dual_order.antisym
thf(fact_102_flat__le__mono__setup_Omono__let,axiom,
! [A: $tType,B: $tType,B3: A,F: B > A,F3: B > A,X2: B] :
( ! [X3: B] : ( partial_flat_ord @ A @ B3 @ ( F @ X3 ) @ ( F3 @ X3 ) )
=> ( partial_flat_ord @ A @ B3 @ ( F @ X2 ) @ ( F3 @ X2 ) ) ) ).
% flat_le_mono_setup.mono_let
thf(fact_103_flat__ord_Odual__order_Oeq__iff,axiom,
! [A: $tType,Ba: A] :
( ( ^ [Y2: A,Z: A] : Y2 = Z )
= ( ^ [A4: A,B4: A] :
( ( partial_flat_ord @ A @ Ba @ B4 @ A4 )
& ( partial_flat_ord @ A @ Ba @ A4 @ B4 ) ) ) ) ).
% flat_ord.dual_order.eq_iff
thf(fact_104_flat__le__mono__setup_Omono__if,axiom,
! [A: $tType,Ba: A,T: A,T2: A,E: A,E2: A,B3: $o] :
( ( partial_flat_ord @ A @ Ba @ T @ T2 )
=> ( ( partial_flat_ord @ A @ Ba @ E @ E2 )
=> ( partial_flat_ord @ A @ Ba @ ( if @ A @ B3 @ T @ E ) @ ( if @ A @ B3 @ T2 @ E2 ) ) ) ) ).
% flat_le_mono_setup.mono_if
thf(fact_105_flat__ord_Odual__order_Otrans,axiom,
! [A: $tType,Ba: A,B3: A,A3: A,C3: A] :
( ( partial_flat_ord @ A @ Ba @ B3 @ A3 )
=> ( ( partial_flat_ord @ A @ Ba @ C3 @ B3 )
=> ( partial_flat_ord @ A @ Ba @ C3 @ A3 ) ) ) ).
% flat_ord.dual_order.trans
thf(fact_106_flat__ord_Oord__le__eq__trans,axiom,
! [A: $tType,Ba: A,A3: A,B3: A,C3: A] :
( ( partial_flat_ord @ A @ Ba @ A3 @ B3 )
=> ( ( B3 = C3 )
=> ( partial_flat_ord @ A @ Ba @ A3 @ C3 ) ) ) ).
% flat_ord.ord_le_eq_trans
thf(fact_107_flat__ord_Oord__eq__le__trans,axiom,
! [A: $tType,A3: A,B3: A,Ba: A,C3: A] :
( ( A3 = B3 )
=> ( ( partial_flat_ord @ A @ Ba @ B3 @ C3 )
=> ( partial_flat_ord @ A @ Ba @ A3 @ C3 ) ) ) ).
% flat_ord.ord_eq_le_trans
thf(fact_108_flat__ord_Oorder_Oantisym,axiom,
! [A: $tType,Ba: A,A3: A,B3: A] :
( ( partial_flat_ord @ A @ Ba @ A3 @ B3 )
=> ( ( partial_flat_ord @ A @ Ba @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% flat_ord.order.antisym
thf(fact_109_flat__ord_Oorder_Oeq__iff,axiom,
! [A: $tType,Ba: A] :
( ( ^ [Y2: A,Z: A] : Y2 = Z )
= ( ^ [A4: A,B4: A] :
( ( partial_flat_ord @ A @ Ba @ A4 @ B4 )
& ( partial_flat_ord @ A @ Ba @ B4 @ A4 ) ) ) ) ).
% flat_ord.order.eq_iff
thf(fact_110_flat__ord_Oantisym__conv,axiom,
! [A: $tType,B3: A,Y: A,X2: A] :
( ( partial_flat_ord @ A @ B3 @ Y @ X2 )
=> ( ( partial_flat_ord @ A @ B3 @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% flat_ord.antisym_conv
thf(fact_111_flat__ord_Oorder__trans,axiom,
! [A: $tType,B3: A,X2: A,Y: A,Z2: A] :
( ( partial_flat_ord @ A @ B3 @ X2 @ Y )
=> ( ( partial_flat_ord @ A @ B3 @ Y @ Z2 )
=> ( partial_flat_ord @ A @ B3 @ X2 @ Z2 ) ) ) ).
% flat_ord.order_trans
thf(fact_112_flat__ord_Oorder_Otrans,axiom,
! [A: $tType,Ba: A,A3: A,B3: A,C3: A] :
( ( partial_flat_ord @ A @ Ba @ A3 @ B3 )
=> ( ( partial_flat_ord @ A @ Ba @ B3 @ C3 )
=> ( partial_flat_ord @ A @ Ba @ A3 @ C3 ) ) ) ).
% flat_ord.order.trans
thf(fact_113_flat__ord_Oeq__refl,axiom,
! [A: $tType,X2: A,Y: A,B3: A] :
( ( X2 = Y )
=> ( partial_flat_ord @ A @ B3 @ X2 @ Y ) ) ).
% flat_ord.eq_refl
thf(fact_114_flat__ord_Oantisym,axiom,
! [A: $tType,B3: A,X2: A,Y: A] :
( ( partial_flat_ord @ A @ B3 @ X2 @ Y )
=> ( ( partial_flat_ord @ A @ B3 @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% flat_ord.antisym
thf(fact_115_flat__ord_Oeq__iff,axiom,
! [A: $tType,B3: A] :
( ( ^ [Y2: A,Z: A] : Y2 = Z )
= ( ^ [X: A,Y4: A] :
( ( partial_flat_ord @ A @ B3 @ X @ Y4 )
& ( partial_flat_ord @ A @ B3 @ Y4 @ X ) ) ) ) ).
% flat_ord.eq_iff
thf(fact_116_nofail__antimono__fun,axiom,
! [B: $tType,A: $tType,F: A > ( refine1665802226e_nres @ B ),G: A > ( refine1665802226e_nres @ B ),X2: A] :
( ( ord_less_eq @ ( A > ( refine1665802226e_nres @ B ) ) @ F @ G )
=> ( ( refine1102455758nofail @ B @ ( G @ X2 ) )
=> ( refine1102455758nofail @ B @ ( F @ X2 ) ) ) ) ).
% nofail_antimono_fun
thf(fact_117_flat__ord__compat_I1_J,axiom,
! [A: $tType] :
( ( comple187826305attice @ A )
=> ! [X2: A,Y: A] :
( ( partial_flat_ord @ A @ ( bot_bot @ A ) @ X2 @ Y )
=> ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% flat_ord_compat(1)
thf(fact_118_flat__ord__compat_I2_J,axiom,
! [A: $tType] :
( ( comple187826305attice @ A )
=> ! [X2: A,Y: A] :
( ( partial_flat_ord @ A @ ( top_top @ A ) @ X2 @ Y )
=> ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% flat_ord_compat(2)
thf(fact_119_Refine__Basic__Mirabelle__tqojlsrkwy_Obind__mono_I2_J,axiom,
! [B: $tType,A: $tType,M: refine1665802226e_nres @ A,M2: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F3: A > ( refine1665802226e_nres @ B )] :
( ( partial_flat_ord @ ( refine1665802226e_nres @ A ) @ ( top_top @ ( refine1665802226e_nres @ A ) ) @ M @ M2 )
=> ( ! [X3: A] : ( partial_flat_ord @ ( refine1665802226e_nres @ B ) @ ( top_top @ ( refine1665802226e_nres @ B ) ) @ ( F @ X3 ) @ ( F3 @ X3 ) )
=> ( partial_flat_ord @ ( refine1665802226e_nres @ B ) @ ( top_top @ ( refine1665802226e_nres @ B ) ) @ ( refine463715084e_bind @ A @ B @ M @ F ) @ ( refine463715084e_bind @ A @ B @ M2 @ F3 ) ) ) ) ).
% Refine_Basic_Mirabelle_tqojlsrkwy.bind_mono(2)
thf(fact_120_conc__fun__R__mono,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),R2: set @ ( product_prod @ A @ B ),M: refine1665802226e_nres @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ R2 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1073749519nc_fun @ A @ B @ R @ M ) @ ( refine1073749519nc_fun @ A @ B @ R2 @ M ) ) ) ).
% conc_fun_R_mono
thf(fact_121_order__mono__setup_Omono__let,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [F: B > A,F3: B > A,X2: B] :
( ! [X3: B] : ( ord_less_eq @ A @ ( F @ X3 ) @ ( F3 @ X3 ) )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F3 @ X2 ) ) ) ) ).
% order_mono_setup.mono_let
thf(fact_122_order__mono__setup_Omono__if,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [T: A,T2: A,E: A,E2: A,B3: $o] :
( ( ord_less_eq @ A @ T @ T2 )
=> ( ( ord_less_eq @ A @ E @ E2 )
=> ( ord_less_eq @ A @ ( if @ A @ B3 @ T @ E ) @ ( if @ A @ B3 @ T2 @ E2 ) ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_123_Domain__empty,axiom,
! [B: $tType,A: $tType] :
( ( domain @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Domain_empty
thf(fact_124_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A6: A,B6: B] :
( ( ( product_Pair @ A @ B @ A3 @ B3 )
= ( product_Pair @ A @ B @ A6 @ B6 ) )
= ( ( A3 = A6 )
& ( B3 = B6 ) ) ) ).
% old.prod.inject
thf(fact_125_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
( ( ( product_Pair @ A @ B @ X1 @ X22 )
= ( product_Pair @ A @ B @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_126_Domain_Oinducts,axiom,
! [B: $tType,A: $tType,X2: A,R3: set @ ( product_prod @ A @ B ),P: A > $o] :
( ( member @ A @ X2 @ ( domain @ A @ B @ R3 ) )
=> ( ! [A5: A,B5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ R3 )
=> ( P @ A5 ) )
=> ( P @ X2 ) ) ) ).
% Domain.inducts
thf(fact_127_Domain_ODomainI,axiom,
! [B: $tType,A: $tType,A3: A,B3: B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R3 )
=> ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) ) ) ).
% Domain.DomainI
thf(fact_128_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X: A] : ( member @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_129_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
? [X3: A,Y3: B] :
( P2
= ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_130_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_131_Pair__inject,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A6: A,B6: B] :
( ( ( product_Pair @ A @ B @ A3 @ B3 )
= ( product_Pair @ A @ B @ A6 @ B6 ) )
=> ~ ( ( A3 = A6 )
=> ( B3 != B6 ) ) ) ).
% Pair_inject
thf(fact_132_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A5: A,B5: B,C5: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C5 ) ) ) ).
% prod_cases3
thf(fact_133_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A5: A,B5: B,C5: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C5 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_134_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) )] :
~ ! [A5: A,B5: B,C5: C,D2: D,E4: E3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E3 ) @ C5 @ ( product_Pair @ D @ E3 @ D2 @ E4 ) ) ) ) ) ).
% prod_cases5
thf(fact_135_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E3: $tType,F4: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F4 ) ) ) )] :
~ ! [A5: A,B5: B,C5: C,D2: D,E4: E3,F5: F4] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F4 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F4 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F4 ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E3 @ F4 ) @ D2 @ ( product_Pair @ E3 @ F4 @ E4 @ F5 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_136_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E3: $tType,F4: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) ) ) )] :
~ ! [A5: A,B5: B,C5: C,D2: D,E4: E3,F5: F4,G4: G3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) @ D2 @ ( product_Pair @ E3 @ ( product_prod @ F4 @ G3 ) @ E4 @ ( product_Pair @ F4 @ G3 @ F5 @ G4 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_137_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A5: A,B5: B,C5: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C5 ) ) )
=> ( P @ X2 ) ) ).
% prod_induct3
thf(fact_138_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A5: A,B5: B,C5: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C5 @ D2 ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct4
thf(fact_139_prod__induct5,axiom,
! [E3: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) )] :
( ! [A5: A,B5: B,C5: C,D2: D,E4: E3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E3 ) @ C5 @ ( product_Pair @ D @ E3 @ D2 @ E4 ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct5
thf(fact_140_prod__induct6,axiom,
! [F4: $tType,E3: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F4 ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F4 ) ) ) )] :
( ! [A5: A,B5: B,C5: C,D2: D,E4: E3,F5: F4] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F4 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F4 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F4 ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E3 @ F4 ) @ D2 @ ( product_Pair @ E3 @ F4 @ E4 @ F5 ) ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct6
thf(fact_141_prod__induct7,axiom,
! [G3: $tType,F4: $tType,E3: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) ) ) )] :
( ! [A5: A,B5: B,C5: C,D2: D,E4: E3,F5: F4,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) @ D2 @ ( product_Pair @ E3 @ ( product_prod @ F4 @ G3 ) @ E4 @ ( product_Pair @ F4 @ G3 @ F5 @ G4 ) ) ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct7
thf(fact_142_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A5: A,B5: B] :
( Y
!= ( product_Pair @ A @ B @ A5 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_143_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_144_subrelI,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S5: set @ ( product_prod @ A @ B )] :
( ! [X3: A,Y3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S5 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S5 ) ) ).
% subrelI
thf(fact_145_single__valued__subset,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S5: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S5 )
=> ( ( single_valued @ A @ B @ S5 )
=> ( single_valued @ A @ B @ R3 ) ) ) ).
% single_valued_subset
thf(fact_146_single__valued__empty,axiom,
! [B: $tType,A: $tType] : ( single_valued @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% single_valued_empty
thf(fact_147_Domain__empty__iff,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] :
( ( ( domain @ A @ B @ R3 )
= ( bot_bot @ ( set @ A ) ) )
= ( R3
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ).
% Domain_empty_iff
thf(fact_148_Domain__mono,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S5: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S5 )
=> ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ B @ R3 ) @ ( domain @ A @ B @ S5 ) ) ) ).
% Domain_mono
thf(fact_149_single__valuedD,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ B ),X2: A,Y: B,Z2: B] :
( ( single_valued @ A @ B @ R3 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R3 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Z2 ) @ R3 )
=> ( Y = Z2 ) ) ) ) ).
% single_valuedD
thf(fact_150_single__valuedI,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] :
( ! [X3: A,Y3: B,Z3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Z3 ) @ R3 )
=> ( Y3 = Z3 ) ) )
=> ( single_valued @ A @ B @ R3 ) ) ).
% single_valuedI
thf(fact_151_single__valued__def,axiom,
! [B: $tType,A: $tType] :
( ( single_valued @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ B )] :
! [X: A,Y4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ R4 )
=> ! [Z4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Z4 ) @ R4 )
=> ( Y4 = Z4 ) ) ) ) ) ).
% single_valued_def
thf(fact_152_DomainE,axiom,
! [B: $tType,A: $tType,A3: A,R3: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) )
=> ~ ! [B5: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B5 ) @ R3 ) ) ).
% DomainE
thf(fact_153_Domain__iff,axiom,
! [A: $tType,B: $tType,A3: A,R3: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) )
= ( ? [Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ Y4 ) @ R3 ) ) ) ).
% Domain_iff
thf(fact_154_Domain_Ocases,axiom,
! [B: $tType,A: $tType,A3: A,R3: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) )
=> ~ ! [B5: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B5 ) @ R3 ) ) ).
% Domain.cases
thf(fact_155_Domain_Osimps,axiom,
! [B: $tType,A: $tType,A3: A,R3: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) )
= ( ? [A4: A,B4: B] :
( ( A3 = A4 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ R3 ) ) ) ) ).
% Domain.simps
thf(fact_156_subset__empty,axiom,
! [A: $tType,A2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_157_empty__subsetI,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 ) ).
% empty_subsetI
thf(fact_158_subsetI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( member @ A @ X3 @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% subsetI
thf(fact_159_subset__antisym,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_160_empty__iff,axiom,
! [A: $tType,C3: A] :
~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_161_all__not__in__conv,axiom,
! [A: $tType,A2: set @ A] :
( ( ! [X: A] :
~ ( member @ A @ X @ A2 ) )
= ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_162_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X: A] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_163_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X: A] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_164_emptyE,axiom,
! [A: $tType,A3: A] :
~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_165_equals0D,axiom,
! [A: $tType,A2: set @ A,A3: A] :
( ( A2
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A3 @ A2 ) ) ).
% equals0D
thf(fact_166_equals0I,axiom,
! [A: $tType,A2: set @ A] :
( ! [Y3: A] :
~ ( member @ A @ Y3 @ A2 )
=> ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_167_ex__in__conv,axiom,
! [A: $tType,A2: set @ A] :
( ( ? [X: A] : ( member @ A @ X @ A2 ) )
= ( A2
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_168_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_169_empty__not__UNIV,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
!= ( top_top @ ( set @ A ) ) ) ).
% empty_not_UNIV
thf(fact_170_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X: A] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_171_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y2: set @ A,Z: set @ A] : Y2 = Z )
= ( ^ [A7: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_172_subset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_173_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_174_subset__refl,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_refl
thf(fact_175_subset__UNIV,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_176_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A7 )
=> ( member @ A @ T3 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_177_equalityD2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_178_equalityD1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_179_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
! [X: A] :
( ( member @ A @ X @ A7 )
=> ( member @ A @ X @ B7 ) ) ) ) ).
% subset_eq
thf(fact_180_equalityE,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_181_subsetD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ C3 @ A2 )
=> ( member @ A @ C3 @ B2 ) ) ) ).
% subsetD
thf(fact_182_in__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ X2 @ A2 )
=> ( member @ A @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_183_Collect__empty__eq__bot,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( P
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_184_less__by__empty,axiom,
! [A: $tType,A2: set @ ( product_prod @ A @ A ),B2: set @ ( product_prod @ A @ A )] :
( ( A2
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A2 @ B2 ) ) ).
% less_by_empty
thf(fact_185_subset__emptyI,axiom,
! [A: $tType,A2: set @ A] :
( ! [X3: A] :
~ ( member @ A @ X3 @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_186_UNIV__I,axiom,
! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_187_UNIV__witness,axiom,
! [A: $tType] :
? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_188_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_189_UNIV__eq__I,axiom,
! [A: $tType,A2: set @ A] :
( ! [X3: A] : ( member @ A @ X3 @ A2 )
=> ( ( top_top @ ( set @ A ) )
= A2 ) ) ).
% UNIV_eq_I
thf(fact_190_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_191_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R3: A,S5: B,R: set @ ( product_prod @ A @ B ),S6: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S5 ) @ R )
=> ( ( S6 = S5 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S6 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_192_iso__tuple__UNIV__I,axiom,
! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_193_ac__galois,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
( ( single_valued @ A @ B @ R )
=> ( refine1150083786ection @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R ) @ ( refine1073749519nc_fun @ A @ B @ R ) ) ) ).
% ac_galois
thf(fact_194_abs__fun__simps_I3_J,axiom,
! [B: $tType,A: $tType,X5: set @ B,R: set @ ( product_prod @ B @ A )] :
( ~ ( ord_less_eq @ ( set @ B ) @ X5 @ ( domain @ B @ A @ R ) )
=> ( ( refine81118332bs_fun @ B @ A @ R @ ( refine605929679le_RES @ B @ X5 ) )
= ( top_top @ ( refine1665802226e_nres @ A ) ) ) ) ).
% abs_fun_simps(3)
thf(fact_195_nres_Oinject,axiom,
! [A: $tType,X22: set @ A,Y22: set @ A] :
( ( ( refine605929679le_RES @ A @ X22 )
= ( refine605929679le_RES @ A @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nres.inject
thf(fact_196_nres__more__simps_I4_J,axiom,
! [A: $tType,X5: set @ A,Y6: set @ A] :
( ( ( refine605929679le_RES @ A @ X5 )
= ( refine605929679le_RES @ A @ Y6 ) )
= ( X5 = Y6 ) ) ).
% nres_more_simps(4)
thf(fact_197_top1I,axiom,
! [A: $tType,X2: A] : ( top_top @ ( A > $o ) @ X2 ) ).
% top1I
thf(fact_198_nofail__simps_I2_J,axiom,
! [B: $tType,X5: set @ B] : ( refine1102455758nofail @ B @ ( refine605929679le_RES @ B @ X5 ) ) ).
% nofail_simps(2)
thf(fact_199_inres__simps_I2_J,axiom,
! [B: $tType,X5: set @ B] :
( ( refine1315500908_inres @ B @ ( refine605929679le_RES @ B @ X5 ) )
= ( ^ [X: B] : ( member @ B @ X @ X5 ) ) ) ).
% inres_simps(2)
thf(fact_200_nf__inres__RES,axiom,
! [A: $tType,X5: set @ A,X2: A] :
( ( refine406925620_inres @ A @ ( refine605929679le_RES @ A @ X5 ) @ X2 )
= ( member @ A @ X2 @ X5 ) ) ).
% nf_inres_RES
thf(fact_201_nf__inres__SPEC,axiom,
! [A: $tType,Phi: A > $o,X2: A] :
( ( refine406925620_inres @ A @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) @ X2 )
= ( Phi @ X2 ) ) ).
% nf_inres_SPEC
thf(fact_202_nres__more__simps_I2_J,axiom,
! [A: $tType,X5: set @ A] :
( ( ( refine605929679le_RES @ A @ X5 )
= ( bot_bot @ ( refine1665802226e_nres @ A ) ) )
= ( X5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% nres_more_simps(2)
thf(fact_203_nres__more__simps_I1_J,axiom,
! [A: $tType,X5: set @ A] :
( ( ( bot_bot @ ( refine1665802226e_nres @ A ) )
= ( refine605929679le_RES @ A @ X5 ) )
= ( X5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% nres_more_simps(1)
thf(fact_204_nres__order__simps_I21_J,axiom,
! [X6: $tType,X2: X6,Y6: set @ X6] :
( ( ord_less_eq @ ( refine1665802226e_nres @ X6 ) @ ( refine1687780735RETURN @ X6 @ X2 ) @ ( refine605929679le_RES @ X6 @ Y6 ) )
= ( member @ X6 @ X2 @ Y6 ) ) ).
% nres_order_simps(21)
thf(fact_205_nres__cases,axiom,
! [A: $tType,M: refine1665802226e_nres @ A] :
( ( M
!= ( top_top @ ( refine1665802226e_nres @ A ) ) )
=> ~ ! [X7: set @ A] :
( M
!= ( refine605929679le_RES @ A @ X7 ) ) ) ).
% nres_cases
thf(fact_206_nres__inequalities_I1_J,axiom,
! [A: $tType,X5: set @ A] :
( ( top_top @ ( refine1665802226e_nres @ A ) )
!= ( refine605929679le_RES @ A @ X5 ) ) ).
% nres_inequalities(1)
thf(fact_207_galois__connection__def,axiom,
! [B: $tType,A: $tType] :
( ( ( comple187826305attice @ A )
& ( comple187826305attice @ B ) )
=> ( ( refine1150083786ection @ A @ B )
= ( ^ [Alpha: A > B,Gamma: B > A] :
! [C4: A,A4: B] :
( ( ord_less_eq @ A @ C4 @ ( Gamma @ A4 ) )
= ( ord_less_eq @ B @ ( Alpha @ C4 ) @ A4 ) ) ) ) ) ).
% galois_connection_def
thf(fact_208_galois__connection_Ointro,axiom,
! [A: $tType,B: $tType] :
( ( ( comple187826305attice @ B )
& ( comple187826305attice @ A ) )
=> ! [Gamma2: B > A,Alpha2: A > B] :
( ! [C5: A,A5: B] :
( ( ord_less_eq @ A @ C5 @ ( Gamma2 @ A5 ) )
= ( ord_less_eq @ B @ ( Alpha2 @ C5 ) @ A5 ) )
=> ( refine1150083786ection @ A @ B @ Alpha2 @ Gamma2 ) ) ) ).
% galois_connection.intro
thf(fact_209_galois__connection_Ogalois,axiom,
! [A: $tType,B: $tType] :
( ( ( comple187826305attice @ B )
& ( comple187826305attice @ A ) )
=> ! [Alpha2: A > B,Gamma2: B > A,C3: A,A3: B] :
( ( refine1150083786ection @ A @ B @ Alpha2 @ Gamma2 )
=> ( ( ord_less_eq @ A @ C3 @ ( Gamma2 @ A3 ) )
= ( ord_less_eq @ B @ ( Alpha2 @ C3 ) @ A3 ) ) ) ) ).
% galois_connection.galois
thf(fact_210_galois__connection_O_092_060alpha_062_092_060gamma_062__defl,axiom,
! [A: $tType,B: $tType] :
( ( ( comple187826305attice @ B )
& ( comple187826305attice @ A ) )
=> ! [Alpha2: A > B,Gamma2: B > A,X2: B] :
( ( refine1150083786ection @ A @ B @ Alpha2 @ Gamma2 )
=> ( ord_less_eq @ B @ ( Alpha2 @ ( Gamma2 @ X2 ) ) @ X2 ) ) ) ).
% galois_connection.\<alpha>\<gamma>_defl
thf(fact_211_galois__connection_O_092_060gamma_062_092_060alpha_062__infl,axiom,
! [B: $tType,A: $tType] :
( ( ( comple187826305attice @ A )
& ( comple187826305attice @ B ) )
=> ! [Alpha2: A > B,Gamma2: B > A,X2: A] :
( ( refine1150083786ection @ A @ B @ Alpha2 @ Gamma2 )
=> ( ord_less_eq @ A @ X2 @ ( Gamma2 @ ( Alpha2 @ X2 ) ) ) ) ) ).
% galois_connection.\<gamma>\<alpha>_infl
thf(fact_212_nres__order__simps_I5_J,axiom,
! [E3: $tType,X5: set @ E3,Y6: set @ E3] :
( ( ord_less_eq @ ( refine1665802226e_nres @ E3 ) @ ( refine605929679le_RES @ E3 @ X5 ) @ ( refine605929679le_RES @ E3 @ Y6 ) )
= ( ord_less_eq @ ( set @ E3 ) @ X5 @ Y6 ) ) ).
% nres_order_simps(5)
thf(fact_213_less__eq__nres_Osimps_I2_J,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A3 ) @ ( refine605929679le_RES @ A @ B3 ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% less_eq_nres.simps(2)
thf(fact_214_bot__nres__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( refine1665802226e_nres @ A ) )
= ( refine605929679le_RES @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bot_nres_def
thf(fact_215_nofail__RES__conv,axiom,
! [A: $tType] :
( ( refine1102455758nofail @ A )
= ( ^ [M5: refine1665802226e_nres @ A] :
? [M6: set @ A] :
( M5
= ( refine605929679le_RES @ A @ M6 ) ) ) ) ).
% nofail_RES_conv
thf(fact_216_SPEC__cons__rule,axiom,
! [A: $tType,M3: refine1665802226e_nres @ A,Phi: A > $o,Psi: A > $o] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M3 @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) )
=> ( ! [X3: A] :
( ( Phi @ X3 )
=> ( Psi @ X3 ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M3 @ ( refine605929679le_RES @ A @ ( collect @ A @ Psi ) ) ) ) ) ).
% SPEC_cons_rule
thf(fact_217_SPEC__trans,axiom,
! [A: $tType,X2: refine1665802226e_nres @ A,Y: refine1665802226e_nres @ A,Postcond: A > $o] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X2 @ Y )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ Y @ ( refine605929679le_RES @ A @ ( collect @ A @ Postcond ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X2 @ ( refine605929679le_RES @ A @ ( collect @ A @ Postcond ) ) ) ) ) ).
% SPEC_trans
thf(fact_218_the__RES__inv,axiom,
! [A: $tType,M3: refine1665802226e_nres @ A] :
( ( refine1102455758nofail @ A @ M3 )
=> ( ( refine605929679le_RES @ A @ ( refine1672542526he_RES @ A @ M3 ) )
= M3 ) ) ).
% the_RES_inv
thf(fact_219_nres__order__simps_I22_J,axiom,
! [Y7: $tType,X5: set @ Y7,Y: Y7] :
( ( ord_less_eq @ ( refine1665802226e_nres @ Y7 ) @ ( refine605929679le_RES @ Y7 @ X5 ) @ ( refine1687780735RETURN @ Y7 @ Y ) )
= ( ord_less_eq @ ( set @ Y7 ) @ X5 @ ( insert @ Y7 @ Y @ ( bot_bot @ ( set @ Y7 ) ) ) ) ) ).
% nres_order_simps(22)
thf(fact_220_insert__absorb2,axiom,
! [A: $tType,X2: A,A2: set @ A] :
( ( insert @ A @ X2 @ ( insert @ A @ X2 @ A2 ) )
= ( insert @ A @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_221_insert__iff,axiom,
! [A: $tType,A3: A,B3: A,A2: set @ A] :
( ( member @ A @ A3 @ ( insert @ A @ B3 @ A2 ) )
= ( ( A3 = B3 )
| ( member @ A @ A3 @ A2 ) ) ) ).
% insert_iff
thf(fact_222_insertCI,axiom,
! [A: $tType,A3: A,B2: set @ A,B3: A] :
( ( ~ ( member @ A @ A3 @ B2 )
=> ( A3 = B3 ) )
=> ( member @ A @ A3 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% insertCI
thf(fact_223_singletonI,axiom,
! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_224_insert__subset,axiom,
! [A: $tType,X2: A,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X2 @ A2 ) @ B2 )
= ( ( member @ A @ X2 @ B2 )
& ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_225_singleton__insert__inj__eq,axiom,
! [A: $tType,B3: A,A3: A,A2: set @ A] :
( ( ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A3 @ A2 ) )
= ( ( A3 = B3 )
& ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_226_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A3: A,A2: set @ A,B3: A] :
( ( ( insert @ A @ A3 @ A2 )
= ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A3 = B3 )
& ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_227_Domain__insert,axiom,
! [B: $tType,A: $tType,A3: A,B3: B,R3: set @ ( product_prod @ A @ B )] :
( ( domain @ A @ B @ ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R3 ) )
= ( insert @ A @ A3 @ ( domain @ A @ B @ R3 ) ) ) ).
% Domain_insert
thf(fact_228_nres__more__simps_I3_J,axiom,
! [A: $tType,X5: set @ A,X2: A] :
( ( ( refine605929679le_RES @ A @ X5 )
= ( refine1687780735RETURN @ A @ X2 ) )
= ( X5
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% nres_more_simps(3)
thf(fact_229_nres__more__simps_I5_J,axiom,
! [A: $tType,X2: A,X5: set @ A] :
( ( ( refine1687780735RETURN @ A @ X2 )
= ( refine605929679le_RES @ A @ X5 ) )
= ( ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) )
= X5 ) ) ).
% nres_more_simps(5)
thf(fact_230_insert__UNIV,axiom,
! [A: $tType,X2: A] :
( ( insert @ A @ X2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% insert_UNIV
thf(fact_231_insert__mono,axiom,
! [A: $tType,C2: set @ A,D3: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ C2 @ D3 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ C2 ) @ ( insert @ A @ A3 @ D3 ) ) ) ).
% insert_mono
thf(fact_232_subset__insert,axiom,
! [A: $tType,X2: A,A2: set @ A,B2: set @ A] :
( ~ ( member @ A @ X2 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_233_subset__insertI,axiom,
! [A: $tType,B2: set @ A,A3: A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( insert @ A @ A3 @ B2 ) ) ).
% subset_insertI
thf(fact_234_subset__insertI2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% subset_insertI2
thf(fact_235_singleton__inject,axiom,
! [A: $tType,A3: A,B3: A] :
( ( ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A3 = B3 ) ) ).
% singleton_inject
thf(fact_236_insert__not__empty,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( insert @ A @ A3 @ A2 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_237_doubleton__eq__iff,axiom,
! [A: $tType,A3: A,B3: A,C3: A,D4: A] :
( ( ( insert @ A @ A3 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C3 @ ( insert @ A @ D4 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A3 = C3 )
& ( B3 = D4 ) )
| ( ( A3 = D4 )
& ( B3 = C3 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_238_singleton__iff,axiom,
! [A: $tType,B3: A,A3: A] :
( ( member @ A @ B3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B3 = A3 ) ) ).
% singleton_iff
thf(fact_239_singletonD,axiom,
! [A: $tType,B3: A,A3: A] :
( ( member @ A @ B3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B3 = A3 ) ) ).
% singletonD
thf(fact_240_mk__disjoint__insert,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( member @ A @ A3 @ A2 )
=> ? [B8: set @ A] :
( ( A2
= ( insert @ A @ A3 @ B8 ) )
& ~ ( member @ A @ A3 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_241_insert__commute,axiom,
! [A: $tType,X2: A,Y: A,A2: set @ A] :
( ( insert @ A @ X2 @ ( insert @ A @ Y @ A2 ) )
= ( insert @ A @ Y @ ( insert @ A @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_242_insert__eq__iff,axiom,
! [A: $tType,A3: A,A2: set @ A,B3: A,B2: set @ A] :
( ~ ( member @ A @ A3 @ A2 )
=> ( ~ ( member @ A @ B3 @ B2 )
=> ( ( ( insert @ A @ A3 @ A2 )
= ( insert @ A @ B3 @ B2 ) )
= ( ( ( A3 = B3 )
=> ( A2 = B2 ) )
& ( ( A3 != B3 )
=> ? [C6: set @ A] :
( ( A2
= ( insert @ A @ B3 @ C6 ) )
& ~ ( member @ A @ B3 @ C6 )
& ( B2
= ( insert @ A @ A3 @ C6 ) )
& ~ ( member @ A @ A3 @ C6 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_243_insert__absorb,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( member @ A @ A3 @ A2 )
=> ( ( insert @ A @ A3 @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_244_insert__ident,axiom,
! [A: $tType,X2: A,A2: set @ A,B2: set @ A] :
( ~ ( member @ A @ X2 @ A2 )
=> ( ~ ( member @ A @ X2 @ B2 )
=> ( ( ( insert @ A @ X2 @ A2 )
= ( insert @ A @ X2 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_245_Set_Oset__insert,axiom,
! [A: $tType,X2: A,A2: set @ A] :
( ( member @ A @ X2 @ A2 )
=> ~ ! [B8: set @ A] :
( ( A2
= ( insert @ A @ X2 @ B8 ) )
=> ( member @ A @ X2 @ B8 ) ) ) ).
% Set.set_insert
thf(fact_246_insertI2,axiom,
! [A: $tType,A3: A,B2: set @ A,B3: A] :
( ( member @ A @ A3 @ B2 )
=> ( member @ A @ A3 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% insertI2
thf(fact_247_insertI1,axiom,
! [A: $tType,A3: A,B2: set @ A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ B2 ) ) ).
% insertI1
thf(fact_248_insertE,axiom,
! [A: $tType,A3: A,B3: A,A2: set @ A] :
( ( member @ A @ A3 @ ( insert @ A @ B3 @ A2 ) )
=> ( ( A3 != B3 )
=> ( member @ A @ A3 @ A2 ) ) ) ).
% insertE
thf(fact_249_subset__singletonD,axiom,
! [A: $tType,A2: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A2
= ( bot_bot @ ( set @ A ) ) )
| ( A2
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_250_subset__singleton__iff,axiom,
! [A: $tType,X5: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ X5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X5
= ( bot_bot @ ( set @ A ) ) )
| ( X5
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_251_insert__subsetI,axiom,
! [A: $tType,X2: A,A2: set @ A,X5: set @ A] :
( ( member @ A @ X2 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ X5 @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X2 @ X5 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_252_RETURN__def,axiom,
! [A: $tType] :
( ( refine1687780735RETURN @ A )
= ( ^ [X: A] : ( refine605929679le_RES @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% RETURN_def
thf(fact_253_the__RES_Osimps,axiom,
! [A: $tType,X5: set @ A] :
( ( refine1672542526he_RES @ A @ ( refine605929679le_RES @ A @ X5 ) )
= X5 ) ).
% the_RES.simps
thf(fact_254_the__elem__eq,axiom,
! [A: $tType,X2: A] :
( ( the_elem @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
= X2 ) ).
% the_elem_eq
% Type constructors (42)
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A8: $tType,A9: $tType] :
( ( comple187826305attice @ A9 )
=> ( comple187826305attice @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A8: $tType,A9: $tType] :
( ( order_top @ A9 )
=> ( order_top @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A8: $tType,A9: $tType] :
( ( order_bot @ A9 )
=> ( order_bot @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 )
=> ( preorder @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 )
=> ( order @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A8: $tType,A9: $tType] :
( ( top @ A9 )
=> ( top @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 )
=> ( ord @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A8: $tType,A9: $tType] :
( ( bot @ A9 )
=> ( bot @ ( A8 > A9 ) ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_1,axiom,
! [A8: $tType] : ( comple187826305attice @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_2,axiom,
! [A8: $tType] : ( order_top @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_3,axiom,
! [A8: $tType] : ( order_bot @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_6,axiom,
! [A8: $tType] : ( top @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_8,axiom,
! [A8: $tType] : ( bot @ ( set @ A8 ) ) ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_9,axiom,
comple187826305attice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_10,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_11,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_12,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_13,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_14,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_15,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_16,axiom,
bot @ $o ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_17,axiom,
comple187826305attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_18,axiom,
order_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_19,axiom,
order_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_20,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_21,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_22,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Otop_23,axiom,
top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_24,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Obot_25,axiom,
bot @ product_unit ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Complete__Lattices_Ocomplete__lattice_26,axiom,
! [A8: $tType] : ( comple187826305attice @ ( refine1665802226e_nres @ A8 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder__top_27,axiom,
! [A8: $tType] : ( order_top @ ( refine1665802226e_nres @ A8 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder__bot_28,axiom,
! [A8: $tType] : ( order_bot @ ( refine1665802226e_nres @ A8 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Opreorder_29,axiom,
! [A8: $tType] : ( preorder @ ( refine1665802226e_nres @ A8 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder_30,axiom,
! [A8: $tType] : ( order @ ( refine1665802226e_nres @ A8 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Otop_31,axiom,
! [A8: $tType] : ( top @ ( refine1665802226e_nres @ A8 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oord_32,axiom,
! [A8: $tType] : ( ord @ ( refine1665802226e_nres @ A8 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Obot_33,axiom,
! [A8: $tType] : ( bot @ ( refine1665802226e_nres @ A8 ) ) ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X2: A,Y: A] :
( ( if @ A @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X2: A,Y: A] :
( ( if @ A @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( refine463715084e_bind @ product_unit @ a @ ( bot_bot @ ( refine1665802226e_nres @ product_unit ) ) @ f )
= ( bot_bot @ ( refine1665802226e_nres @ a ) ) ) ).
%------------------------------------------------------------------------------