TPTP Problem File: ITP161^1.p
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%------------------------------------------------------------------------------
% File : ITP161^1 : TPTP v9.0.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Refine_Basic problem prob_1708__3602512_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_1708__3602512_1 [Des21]
% Status : Theorem
% Rating : 0.12 v9.0.0, 0.30 v8.2.0, 0.31 v8.1.0, 0.36 v7.5.0
% Syntax : Number of formulae : 503 ( 146 unt; 144 typ; 0 def)
% Number of atoms : 1005 ( 266 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 3105 ( 107 ~; 1 |; 29 &;2426 @)
% ( 0 <=>; 542 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 7 avg)
% Number of types : 27 ( 26 usr)
% Number of type conns : 394 ( 394 >; 0 *; 0 +; 0 <<)
% Number of symbols : 121 ( 118 usr; 12 con; 0-4 aty)
% Number of variables : 1083 ( 62 ^;1002 !; 19 ?;1083 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:36:00.723
%------------------------------------------------------------------------------
% Could-be-implicit typings (26)
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% Explicit typings (118)
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thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Onofail_001tf__a,type,
refine412683989fail_a: refine424419629nres_a > $o ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Onofail_001tf__b,type,
refine412683990fail_b: refine432681134nres_b > $o ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Onres_OFAILi_001t__Product____Type__Ounit,type,
refine1607584204t_unit: refine787176636t_unit ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Onres_OFAILi_001tf__a,type,
refine464223677AILi_a: refine424419629nres_a ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Onres_OFAILi_001tf__b,type,
refine464223678AILi_b: refine432681134nres_b ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Onres_ORES_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
refine622451633od_a_a: set_Product_prod_a_a > refine671810262od_a_a ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Onres_ORES_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
refine630713138od_a_b: set_Product_prod_a_b > refine1811374551od_a_b ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Onres_ORES_001t__Product____Type__Oprod_Itf__b_Mtf__a_J,type,
refine1450221040od_b_a: set_Product_prod_b_a > refine891721109od_b_a ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Onres_ORES_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
refine1458482545od_b_b: set_Product_prod_b_b > refine2031285398od_b_b ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Onres_ORES_001t__Product____Type__Ounit,type,
refine1777164439t_unit: set_Product_unit > refine787176636t_unit ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Onres_ORES_001tf__a,type,
refine1198353288_RES_a: set_a > refine424419629nres_a ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Onres_ORES_001tf__b,type,
refine1198353289_RES_b: set_b > refine432681134nres_b ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Othe__RES_001t__Product____Type__Ounit,type,
refine1208325748t_unit: refine787176636t_unit > set_Product_unit ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Othe__RES_001tf__a,type,
refine1822134885_RES_a: refine424419629nres_a > set_a ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Othe__RES_001tf__b,type,
refine1822134886_RES_b: refine432681134nres_b > set_b ).
thf(sy_c_Relation_ODomain_001tf__a_001t__Product____Type__Ounit,type,
domain799550107t_unit: set_Pr1628433942t_unit > set_a ).
thf(sy_c_Relation_ODomain_001tf__a_001tf__a,type,
domain_a_a: set_Product_prod_a_a > set_a ).
thf(sy_c_Relation_ODomain_001tf__a_001tf__b,type,
domain_a_b: set_Product_prod_a_b > set_a ).
thf(sy_c_Relation_ODomain_001tf__b_001t__Product____Type__Ounit,type,
domain918802906t_unit: set_Pr1707992533t_unit > set_b ).
thf(sy_c_Relation_ODomain_001tf__b_001tf__a,type,
domain_b_a: set_Product_prod_b_a > set_b ).
thf(sy_c_Relation_ODomain_001tf__b_001tf__b,type,
domain_b_b: set_Product_prod_b_b > set_b ).
thf(sy_c_Relation_OImage_001tf__a_001tf__a,type,
image_a_a: set_Product_prod_a_a > set_a > set_a ).
thf(sy_c_Relation_OImage_001tf__a_001tf__b,type,
image_a_b: set_Product_prod_a_b > set_a > set_b ).
thf(sy_c_Relation_OImage_001tf__b_001tf__a,type,
image_b_a: set_Product_prod_b_a > set_b > set_a ).
thf(sy_c_Relation_OImage_001tf__b_001tf__b,type,
image_b_b: set_Product_prod_b_b > set_b > set_b ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
collec654117139od_a_b: ( product_prod_a_b > $o ) > set_Product_prod_a_b ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__b_Mtf__a_J,type,
collec1473625041od_b_a: ( product_prod_b_a > $o ) > set_Product_prod_b_a ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
collec1481886546od_b_b: ( product_prod_b_b > $o ) > set_Product_prod_b_b ).
thf(sy_c_Set_OCollect_001t__Product____Type__Ounit,type,
collect_Product_unit: ( product_unit > $o ) > set_Product_unit ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OCollect_001tf__b,type,
collect_b: ( b > $o ) > set_b ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
member458171089od_a_b: product_prod_a_b > set_Product_prod_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mtf__a_J,type,
member1277678991od_b_a: product_prod_b_a > set_Product_prod_b_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
member1285940496od_b_b: product_prod_b_b > set_Product_prod_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Ounit,type,
member_Product_unit: product_unit > set_Product_unit > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v__092_060Phi_062,type,
phi: a > $o ).
thf(sy_v_f,type,
f: a > refine432681134nres_b ).
thf(sy_v_m,type,
m: refine432681134nres_b ).
thf(sy_v_x,type,
x: a ).
% Relevant facts (343)
thf(fact_0_assms_I1_J,axiom,
phi @ x ).
% assms(1)
thf(fact_1_assms_I2_J,axiom,
ord_le527798542nres_b @ m @ ( f @ x ) ).
% assms(2)
thf(fact_2_nres__more__simps_I4_J,axiom,
! [X: set_Product_unit,Y: set_Product_unit] :
( ( ( refine1777164439t_unit @ X )
= ( refine1777164439t_unit @ Y ) )
= ( X = Y ) ) ).
% nres_more_simps(4)
thf(fact_3_nres__more__simps_I4_J,axiom,
! [X: set_b,Y: set_b] :
( ( ( refine1198353289_RES_b @ X )
= ( refine1198353289_RES_b @ Y ) )
= ( X = Y ) ) ).
% nres_more_simps(4)
thf(fact_4_nres__more__simps_I4_J,axiom,
! [X: set_a,Y: set_a] :
( ( ( refine1198353288_RES_a @ X )
= ( refine1198353288_RES_a @ Y ) )
= ( X = Y ) ) ).
% nres_more_simps(4)
thf(fact_5_nres_Oinject,axiom,
! [X2: set_Product_unit,Y2: set_Product_unit] :
( ( ( refine1777164439t_unit @ X2 )
= ( refine1777164439t_unit @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nres.inject
thf(fact_6_nres_Oinject,axiom,
! [X2: set_b,Y2: set_b] :
( ( ( refine1198353289_RES_b @ X2 )
= ( refine1198353289_RES_b @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nres.inject
thf(fact_7_nres_Oinject,axiom,
! [X2: set_a,Y2: set_a] :
( ( ( refine1198353288_RES_a @ X2 )
= ( refine1198353288_RES_a @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nres.inject
thf(fact_8_if__rule,axiom,
! [B: $o,S1: refine787176636t_unit,Phi: product_unit > $o,S2: refine787176636t_unit] :
( ( B
=> ( ord_le1051254044t_unit @ S1 @ ( refine1777164439t_unit @ ( collect_Product_unit @ Phi ) ) ) )
=> ( ( ~ B
=> ( ord_le1051254044t_unit @ S2 @ ( refine1777164439t_unit @ ( collect_Product_unit @ Phi ) ) ) )
=> ( ord_le1051254044t_unit @ ( if_Ref1369692790t_unit @ B @ S1 @ S2 ) @ ( refine1777164439t_unit @ ( collect_Product_unit @ Phi ) ) ) ) ) ).
% if_rule
thf(fact_9_if__rule,axiom,
! [B: $o,S1: refine432681134nres_b,Phi: b > $o,S2: refine432681134nres_b] :
( ( B
=> ( ord_le527798542nres_b @ S1 @ ( refine1198353289_RES_b @ ( collect_b @ Phi ) ) ) )
=> ( ( ~ B
=> ( ord_le527798542nres_b @ S2 @ ( refine1198353289_RES_b @ ( collect_b @ Phi ) ) ) )
=> ( ord_le527798542nres_b @ ( if_Ref1732808808nres_b @ B @ S1 @ S2 ) @ ( refine1198353289_RES_b @ ( collect_b @ Phi ) ) ) ) ) ).
% if_rule
thf(fact_10_if__rule,axiom,
! [B: $o,S1: refine424419629nres_a,Phi: a > $o,S2: refine424419629nres_a] :
( ( B
=> ( ord_le519537037nres_a @ S1 @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) ) )
=> ( ( ~ B
=> ( ord_le519537037nres_a @ S2 @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) ) )
=> ( ord_le519537037nres_a @ ( if_Ref1724547303nres_a @ B @ S1 @ S2 ) @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) ) ) ) ).
% if_rule
thf(fact_11_RES__rule,axiom,
! [S: set_Product_prod_a_a,Phi: product_prod_a_a > $o] :
( ! [X3: product_prod_a_a] :
( ( member449909584od_a_a @ X3 @ S )
=> ( Phi @ X3 ) )
=> ( ord_le915866422od_a_a @ ( refine622451633od_a_a @ S ) @ ( refine622451633od_a_a @ ( collec645855634od_a_a @ Phi ) ) ) ) ).
% RES_rule
thf(fact_12_RES__rule,axiom,
! [S: set_Product_prod_b_a,Phi: product_prod_b_a > $o] :
( ! [X3: product_prod_b_a] :
( ( member1277678991od_b_a @ X3 @ S )
=> ( Phi @ X3 ) )
=> ( ord_le1135777269od_b_a @ ( refine1450221040od_b_a @ S ) @ ( refine1450221040od_b_a @ ( collec1473625041od_b_a @ Phi ) ) ) ) ).
% RES_rule
thf(fact_13_RES__rule,axiom,
! [S: set_Product_prod_b_b,Phi: product_prod_b_b > $o] :
( ! [X3: product_prod_b_b] :
( ( member1285940496od_b_b @ X3 @ S )
=> ( Phi @ X3 ) )
=> ( ord_le127857910od_b_b @ ( refine1458482545od_b_b @ S ) @ ( refine1458482545od_b_b @ ( collec1481886546od_b_b @ Phi ) ) ) ) ).
% RES_rule
thf(fact_14_RES__rule,axiom,
! [S: set_Product_prod_a_b,Phi: product_prod_a_b > $o] :
( ! [X3: product_prod_a_b] :
( ( member458171089od_a_b @ X3 @ S )
=> ( Phi @ X3 ) )
=> ( ord_le2055430711od_a_b @ ( refine630713138od_a_b @ S ) @ ( refine630713138od_a_b @ ( collec654117139od_a_b @ Phi ) ) ) ) ).
% RES_rule
thf(fact_15_RES__rule,axiom,
! [S: set_Product_unit,Phi: product_unit > $o] :
( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ S )
=> ( Phi @ X3 ) )
=> ( ord_le1051254044t_unit @ ( refine1777164439t_unit @ S ) @ ( refine1777164439t_unit @ ( collect_Product_unit @ Phi ) ) ) ) ).
% RES_rule
thf(fact_16_RES__rule,axiom,
! [S: set_b,Phi: b > $o] :
( ! [X3: b] :
( ( member_b @ X3 @ S )
=> ( Phi @ X3 ) )
=> ( ord_le527798542nres_b @ ( refine1198353289_RES_b @ S ) @ ( refine1198353289_RES_b @ ( collect_b @ Phi ) ) ) ) ).
% RES_rule
thf(fact_17_RES__rule,axiom,
! [S: set_a,Phi: a > $o] :
( ! [X3: a] :
( ( member_a @ X3 @ S )
=> ( Phi @ X3 ) )
=> ( ord_le519537037nres_a @ ( refine1198353288_RES_a @ S ) @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) ) ) ).
% RES_rule
thf(fact_18_SPEC__rule,axiom,
! [Phi: product_unit > $o,Phi2: product_unit > $o] :
( ! [X3: product_unit] :
( ( Phi @ X3 )
=> ( Phi2 @ X3 ) )
=> ( ord_le1051254044t_unit @ ( refine1777164439t_unit @ ( collect_Product_unit @ Phi ) ) @ ( refine1777164439t_unit @ ( collect_Product_unit @ Phi2 ) ) ) ) ).
% SPEC_rule
thf(fact_19_SPEC__rule,axiom,
! [Phi: b > $o,Phi2: b > $o] :
( ! [X3: b] :
( ( Phi @ X3 )
=> ( Phi2 @ X3 ) )
=> ( ord_le527798542nres_b @ ( refine1198353289_RES_b @ ( collect_b @ Phi ) ) @ ( refine1198353289_RES_b @ ( collect_b @ Phi2 ) ) ) ) ).
% SPEC_rule
thf(fact_20_SPEC__rule,axiom,
! [Phi: a > $o,Phi2: a > $o] :
( ! [X3: a] :
( ( Phi @ X3 )
=> ( Phi2 @ X3 ) )
=> ( ord_le519537037nres_a @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) @ ( refine1198353288_RES_a @ ( collect_a @ Phi2 ) ) ) ) ).
% SPEC_rule
thf(fact_21_SPEC__trans,axiom,
! [X4: refine787176636t_unit,Y3: refine787176636t_unit,Postcond: product_unit > $o] :
( ( ord_le1051254044t_unit @ X4 @ Y3 )
=> ( ( ord_le1051254044t_unit @ Y3 @ ( refine1777164439t_unit @ ( collect_Product_unit @ Postcond ) ) )
=> ( ord_le1051254044t_unit @ X4 @ ( refine1777164439t_unit @ ( collect_Product_unit @ Postcond ) ) ) ) ) ).
% SPEC_trans
thf(fact_22_SPEC__trans,axiom,
! [X4: refine432681134nres_b,Y3: refine432681134nres_b,Postcond: b > $o] :
( ( ord_le527798542nres_b @ X4 @ Y3 )
=> ( ( ord_le527798542nres_b @ Y3 @ ( refine1198353289_RES_b @ ( collect_b @ Postcond ) ) )
=> ( ord_le527798542nres_b @ X4 @ ( refine1198353289_RES_b @ ( collect_b @ Postcond ) ) ) ) ) ).
% SPEC_trans
thf(fact_23_SPEC__trans,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a,Postcond: a > $o] :
( ( ord_le519537037nres_a @ X4 @ Y3 )
=> ( ( ord_le519537037nres_a @ Y3 @ ( refine1198353288_RES_a @ ( collect_a @ Postcond ) ) )
=> ( ord_le519537037nres_a @ X4 @ ( refine1198353288_RES_a @ ( collect_a @ Postcond ) ) ) ) ) ).
% SPEC_trans
thf(fact_24_lhs__step__If,axiom,
! [B: $o,T: set_b,M: set_b,E: set_b] :
( ( B
=> ( ord_less_eq_set_b @ T @ M ) )
=> ( ( ~ B
=> ( ord_less_eq_set_b @ E @ M ) )
=> ( ord_less_eq_set_b @ ( if_set_b @ B @ T @ E ) @ M ) ) ) ).
% lhs_step_If
thf(fact_25_lhs__step__If,axiom,
! [B: $o,T: set_a,M: set_a,E: set_a] :
( ( B
=> ( ord_less_eq_set_a @ T @ M ) )
=> ( ( ~ B
=> ( ord_less_eq_set_a @ E @ M ) )
=> ( ord_less_eq_set_a @ ( if_set_a @ B @ T @ E ) @ M ) ) ) ).
% lhs_step_If
thf(fact_26_lhs__step__If,axiom,
! [B: $o,T: $o > refine424419629nres_a,M: $o > refine424419629nres_a,E: $o > refine424419629nres_a] :
( ( B
=> ( ord_le516598614nres_a @ T @ M ) )
=> ( ( ~ B
=> ( ord_le516598614nres_a @ E @ M ) )
=> ( ord_le516598614nres_a @ ( if_o_R1705037948nres_a @ B @ T @ E ) @ M ) ) ) ).
% lhs_step_If
thf(fact_27_lhs__step__If,axiom,
! [B: $o,T: $o > refine432681134nres_b,M: $o > refine432681134nres_b,E: $o > refine432681134nres_b] :
( ( B
=> ( ord_le1656162903nres_b @ T @ M ) )
=> ( ( ~ B
=> ( ord_le1656162903nres_b @ E @ M ) )
=> ( ord_le1656162903nres_b @ ( if_o_R697118589nres_b @ B @ T @ E ) @ M ) ) ) ).
% lhs_step_If
thf(fact_28_lhs__step__If,axiom,
! [B: $o,T: refine787176636t_unit,M: refine787176636t_unit,E: refine787176636t_unit] :
( ( B
=> ( ord_le1051254044t_unit @ T @ M ) )
=> ( ( ~ B
=> ( ord_le1051254044t_unit @ E @ M ) )
=> ( ord_le1051254044t_unit @ ( if_Ref1369692790t_unit @ B @ T @ E ) @ M ) ) ) ).
% lhs_step_If
thf(fact_29_lhs__step__If,axiom,
! [B: $o,T: refine424419629nres_a,M: refine424419629nres_a,E: refine424419629nres_a] :
( ( B
=> ( ord_le519537037nres_a @ T @ M ) )
=> ( ( ~ B
=> ( ord_le519537037nres_a @ E @ M ) )
=> ( ord_le519537037nres_a @ ( if_Ref1724547303nres_a @ B @ T @ E ) @ M ) ) ) ).
% lhs_step_If
thf(fact_30_lhs__step__If,axiom,
! [B: $o,T: refine432681134nres_b,M: refine432681134nres_b,E: refine432681134nres_b] :
( ( B
=> ( ord_le527798542nres_b @ T @ M ) )
=> ( ( ~ B
=> ( ord_le527798542nres_b @ E @ M ) )
=> ( ord_le527798542nres_b @ ( if_Ref1732808808nres_b @ B @ T @ E ) @ M ) ) ) ).
% lhs_step_If
thf(fact_31_SPEC__cons__rule,axiom,
! [M: refine787176636t_unit,Phi: product_unit > $o,Psi: product_unit > $o] :
( ( ord_le1051254044t_unit @ M @ ( refine1777164439t_unit @ ( collect_Product_unit @ Phi ) ) )
=> ( ! [X3: product_unit] :
( ( Phi @ X3 )
=> ( Psi @ X3 ) )
=> ( ord_le1051254044t_unit @ M @ ( refine1777164439t_unit @ ( collect_Product_unit @ Psi ) ) ) ) ) ).
% SPEC_cons_rule
thf(fact_32_SPEC__cons__rule,axiom,
! [M: refine424419629nres_a,Phi: a > $o,Psi: a > $o] :
( ( ord_le519537037nres_a @ M @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) )
=> ( ! [X3: a] :
( ( Phi @ X3 )
=> ( Psi @ X3 ) )
=> ( ord_le519537037nres_a @ M @ ( refine1198353288_RES_a @ ( collect_a @ Psi ) ) ) ) ) ).
% SPEC_cons_rule
thf(fact_33_SPEC__cons__rule,axiom,
! [M: refine432681134nres_b,Phi: b > $o,Psi: b > $o] :
( ( ord_le527798542nres_b @ M @ ( refine1198353289_RES_b @ ( collect_b @ Phi ) ) )
=> ( ! [X3: b] :
( ( Phi @ X3 )
=> ( Psi @ X3 ) )
=> ( ord_le527798542nres_b @ M @ ( refine1198353289_RES_b @ ( collect_b @ Psi ) ) ) ) ) ).
% SPEC_cons_rule
thf(fact_34_RES__bind__choose,axiom,
! [X4: a,X: set_a,M: refine424419629nres_a,F: a > refine424419629nres_a] :
( ( member_a @ X4 @ X )
=> ( ( ord_le519537037nres_a @ M @ ( F @ X4 ) )
=> ( ord_le519537037nres_a @ M @ ( refine436832838nd_a_a @ ( refine1198353288_RES_a @ X ) @ F ) ) ) ) ).
% RES_bind_choose
thf(fact_35_RES__bind__choose,axiom,
! [X4: a,X: set_a,M: refine432681134nres_b,F: a > refine432681134nres_b] :
( ( member_a @ X4 @ X )
=> ( ( ord_le527798542nres_b @ M @ ( F @ X4 ) )
=> ( ord_le527798542nres_b @ M @ ( refine436832839nd_a_b @ ( refine1198353288_RES_a @ X ) @ F ) ) ) ) ).
% RES_bind_choose
thf(fact_36_RES__bind__choose,axiom,
! [X4: b,X: set_b,M: refine432681134nres_b,F: b > refine432681134nres_b] :
( ( member_b @ X4 @ X )
=> ( ( ord_le527798542nres_b @ M @ ( F @ X4 ) )
=> ( ord_le527798542nres_b @ M @ ( refine1264602246nd_b_b @ ( refine1198353289_RES_b @ X ) @ F ) ) ) ) ).
% RES_bind_choose
thf(fact_37_RES__bind__choose,axiom,
! [X4: b,X: set_b,M: refine424419629nres_a,F: b > refine424419629nres_a] :
( ( member_b @ X4 @ X )
=> ( ( ord_le519537037nres_a @ M @ ( F @ X4 ) )
=> ( ord_le519537037nres_a @ M @ ( refine1264602245nd_b_a @ ( refine1198353289_RES_b @ X ) @ F ) ) ) ) ).
% RES_bind_choose
thf(fact_38_RES__bind__choose,axiom,
! [X4: a,X: set_a,M: refine787176636t_unit,F: a > refine787176636t_unit] :
( ( member_a @ X4 @ X )
=> ( ( ord_le1051254044t_unit @ M @ ( F @ X4 ) )
=> ( ord_le1051254044t_unit @ M @ ( refine96995669t_unit @ ( refine1198353288_RES_a @ X ) @ F ) ) ) ) ).
% RES_bind_choose
thf(fact_39_RES__bind__choose,axiom,
! [X4: b,X: set_b,M: refine787176636t_unit,F: b > refine787176636t_unit] :
( ( member_b @ X4 @ X )
=> ( ( ord_le1051254044t_unit @ M @ ( F @ X4 ) )
=> ( ord_le1051254044t_unit @ M @ ( refine216248468t_unit @ ( refine1198353289_RES_b @ X ) @ F ) ) ) ) ).
% RES_bind_choose
thf(fact_40_RES__bind__choose,axiom,
! [X4: product_unit,X: set_Product_unit,M: refine432681134nres_b,F: product_unit > refine432681134nres_b] :
( ( member_Product_unit @ X4 @ X )
=> ( ( ord_le527798542nres_b @ M @ ( F @ X4 ) )
=> ( ord_le527798542nres_b @ M @ ( refine119808504unit_b @ ( refine1777164439t_unit @ X ) @ F ) ) ) ) ).
% RES_bind_choose
thf(fact_41_RES__bind__choose,axiom,
! [X4: product_unit,X: set_Product_unit,M: refine424419629nres_a,F: product_unit > refine424419629nres_a] :
( ( member_Product_unit @ X4 @ X )
=> ( ( ord_le519537037nres_a @ M @ ( F @ X4 ) )
=> ( ord_le519537037nres_a @ M @ ( refine119808503unit_a @ ( refine1777164439t_unit @ X ) @ F ) ) ) ) ).
% RES_bind_choose
thf(fact_42_RES__bind__choose,axiom,
! [X4: product_unit,X: set_Product_unit,M: refine787176636t_unit,F: product_unit > refine787176636t_unit] :
( ( member_Product_unit @ X4 @ X )
=> ( ( ord_le1051254044t_unit @ M @ ( F @ X4 ) )
=> ( ord_le1051254044t_unit @ M @ ( refine681446406t_unit @ ( refine1777164439t_unit @ X ) @ F ) ) ) ) ).
% RES_bind_choose
thf(fact_43_RES__bind__choose,axiom,
! [X4: product_prod_a_a,X: set_Product_prod_a_a,M: refine432681134nres_b,F: product_prod_a_a > refine432681134nres_b] :
( ( member449909584od_a_a @ X4 @ X )
=> ( ( ord_le527798542nres_b @ M @ ( F @ X4 ) )
=> ( ord_le527798542nres_b @ M @ ( refine988994910_a_a_b @ ( refine622451633od_a_a @ X ) @ F ) ) ) ) ).
% RES_bind_choose
thf(fact_44_order__mono__setup_Orefl,axiom,
! [X4: set_b] : ( ord_less_eq_set_b @ X4 @ X4 ) ).
% order_mono_setup.refl
thf(fact_45_order__mono__setup_Orefl,axiom,
! [X4: set_a] : ( ord_less_eq_set_a @ X4 @ X4 ) ).
% order_mono_setup.refl
thf(fact_46_order__mono__setup_Orefl,axiom,
! [X4: $o > refine424419629nres_a] : ( ord_le516598614nres_a @ X4 @ X4 ) ).
% order_mono_setup.refl
thf(fact_47_order__mono__setup_Orefl,axiom,
! [X4: $o > refine432681134nres_b] : ( ord_le1656162903nres_b @ X4 @ X4 ) ).
% order_mono_setup.refl
thf(fact_48_order__mono__setup_Orefl,axiom,
! [X4: refine787176636t_unit] : ( ord_le1051254044t_unit @ X4 @ X4 ) ).
% order_mono_setup.refl
thf(fact_49_order__mono__setup_Orefl,axiom,
! [X4: refine432681134nres_b] : ( ord_le527798542nres_b @ X4 @ X4 ) ).
% order_mono_setup.refl
thf(fact_50_order__mono__setup_Orefl,axiom,
! [X4: refine424419629nres_a] : ( ord_le519537037nres_a @ X4 @ X4 ) ).
% order_mono_setup.refl
thf(fact_51_the__RES_Osimps,axiom,
! [X: set_Product_unit] :
( ( refine1208325748t_unit @ ( refine1777164439t_unit @ X ) )
= X ) ).
% the_RES.simps
thf(fact_52_the__RES_Osimps,axiom,
! [X: set_a] :
( ( refine1822134885_RES_a @ ( refine1198353288_RES_a @ X ) )
= X ) ).
% the_RES.simps
thf(fact_53_the__RES_Osimps,axiom,
! [X: set_b] :
( ( refine1822134886_RES_b @ ( refine1198353289_RES_b @ X ) )
= X ) ).
% the_RES.simps
thf(fact_54_nf__inres__RES,axiom,
! [X: set_Product_prod_a_a,X4: product_prod_a_a] :
( ( refine378960780od_a_a @ ( refine622451633od_a_a @ X ) @ X4 )
= ( member449909584od_a_a @ X4 @ X ) ) ).
% nf_inres_RES
thf(fact_55_nf__inres__RES,axiom,
! [X: set_Product_prod_b_a,X4: product_prod_b_a] :
( ( refine1206730187od_b_a @ ( refine1450221040od_b_a @ X ) @ X4 )
= ( member1277678991od_b_a @ X4 @ X ) ) ).
% nf_inres_RES
thf(fact_56_nf__inres__RES,axiom,
! [X: set_Product_prod_b_b,X4: product_prod_b_b] :
( ( refine1214991692od_b_b @ ( refine1458482545od_b_b @ X ) @ X4 )
= ( member1285940496od_b_b @ X4 @ X ) ) ).
% nf_inres_RES
thf(fact_57_nf__inres__RES,axiom,
! [X: set_Product_prod_a_b,X4: product_prod_a_b] :
( ( refine387222285od_a_b @ ( refine630713138od_a_b @ X ) @ X4 )
= ( member458171089od_a_b @ X4 @ X ) ) ).
% nf_inres_RES
thf(fact_58_nf__inres__RES,axiom,
! [X: set_Product_unit,X4: product_unit] :
( ( refine983493746t_unit @ ( refine1777164439t_unit @ X ) @ X4 )
= ( member_Product_unit @ X4 @ X ) ) ).
% nf_inres_RES
thf(fact_59_nf__inres__RES,axiom,
! [X: set_a,X4: a] :
( ( refine1312857699nres_a @ ( refine1198353288_RES_a @ X ) @ X4 )
= ( member_a @ X4 @ X ) ) ).
% nf_inres_RES
thf(fact_60_nf__inres__RES,axiom,
! [X: set_b,X4: b] :
( ( refine1312857700nres_b @ ( refine1198353289_RES_b @ X ) @ X4 )
= ( member_b @ X4 @ X ) ) ).
% nf_inres_RES
thf(fact_61_nf__inres__SPEC,axiom,
! [Phi: product_unit > $o,X4: product_unit] :
( ( refine983493746t_unit @ ( refine1777164439t_unit @ ( collect_Product_unit @ Phi ) ) @ X4 )
= ( Phi @ X4 ) ) ).
% nf_inres_SPEC
thf(fact_62_nf__inres__SPEC,axiom,
! [Phi: a > $o,X4: a] :
( ( refine1312857699nres_a @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) @ X4 )
= ( Phi @ X4 ) ) ).
% nf_inres_SPEC
thf(fact_63_nf__inres__SPEC,axiom,
! [Phi: b > $o,X4: b] :
( ( refine1312857700nres_b @ ( refine1198353289_RES_b @ ( collect_b @ Phi ) ) @ X4 )
= ( Phi @ X4 ) ) ).
% nf_inres_SPEC
thf(fact_64_le__funD,axiom,
! [F: $o > refine424419629nres_a,G: $o > refine424419629nres_a,X4: $o] :
( ( ord_le516598614nres_a @ F @ G )
=> ( ord_le519537037nres_a @ ( F @ X4 ) @ ( G @ X4 ) ) ) ).
% le_funD
thf(fact_65_le__funD,axiom,
! [F: $o > refine432681134nres_b,G: $o > refine432681134nres_b,X4: $o] :
( ( ord_le1656162903nres_b @ F @ G )
=> ( ord_le527798542nres_b @ ( F @ X4 ) @ ( G @ X4 ) ) ) ).
% le_funD
thf(fact_66_le__funE,axiom,
! [F: $o > refine424419629nres_a,G: $o > refine424419629nres_a,X4: $o] :
( ( ord_le516598614nres_a @ F @ G )
=> ( ord_le519537037nres_a @ ( F @ X4 ) @ ( G @ X4 ) ) ) ).
% le_funE
thf(fact_67_le__funE,axiom,
! [F: $o > refine432681134nres_b,G: $o > refine432681134nres_b,X4: $o] :
( ( ord_le1656162903nres_b @ F @ G )
=> ( ord_le527798542nres_b @ ( F @ X4 ) @ ( G @ X4 ) ) ) ).
% le_funE
thf(fact_68_le__funI,axiom,
! [F: $o > refine424419629nres_a,G: $o > refine424419629nres_a] :
( ! [X3: $o] : ( ord_le519537037nres_a @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_le516598614nres_a @ F @ G ) ) ).
% le_funI
thf(fact_69_Refine__Misc_Oif__mono,axiom,
! [B: $o,M1: refine432681134nres_b,M12: refine432681134nres_b,M2: refine432681134nres_b,M22: refine432681134nres_b] :
( ( B
=> ( ord_le527798542nres_b @ M1 @ M12 ) )
=> ( ( ~ B
=> ( ord_le527798542nres_b @ M2 @ M22 ) )
=> ( ord_le527798542nres_b @ ( if_Ref1732808808nres_b @ B @ M1 @ M2 ) @ ( if_Ref1732808808nres_b @ B @ M12 @ M22 ) ) ) ) ).
% Refine_Misc.if_mono
thf(fact_70_Refine__Misc_Oif__mono,axiom,
! [B: $o,M1: refine424419629nres_a,M12: refine424419629nres_a,M2: refine424419629nres_a,M22: refine424419629nres_a] :
( ( B
=> ( ord_le519537037nres_a @ M1 @ M12 ) )
=> ( ( ~ B
=> ( ord_le519537037nres_a @ M2 @ M22 ) )
=> ( ord_le519537037nres_a @ ( if_Ref1724547303nres_a @ B @ M1 @ M2 ) @ ( if_Ref1724547303nres_a @ B @ M12 @ M22 ) ) ) ) ).
% Refine_Misc.if_mono
thf(fact_71_order__subst1,axiom,
! [A: refine432681134nres_b,F: refine432681134nres_b > refine432681134nres_b,B: refine432681134nres_b,C: refine432681134nres_b] :
( ( ord_le527798542nres_b @ A @ ( F @ B ) )
=> ( ( ord_le527798542nres_b @ B @ C )
=> ( ! [X3: refine432681134nres_b,Y4: refine432681134nres_b] :
( ( ord_le527798542nres_b @ X3 @ Y4 )
=> ( ord_le527798542nres_b @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le527798542nres_b @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_72_order__subst1,axiom,
! [A: refine432681134nres_b,F: refine424419629nres_a > refine432681134nres_b,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le527798542nres_b @ A @ ( F @ B ) )
=> ( ( ord_le519537037nres_a @ B @ C )
=> ( ! [X3: refine424419629nres_a,Y4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y4 )
=> ( ord_le527798542nres_b @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le527798542nres_b @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_73_order__subst1,axiom,
! [A: refine424419629nres_a,F: refine432681134nres_b > refine424419629nres_a,B: refine432681134nres_b,C: refine432681134nres_b] :
( ( ord_le519537037nres_a @ A @ ( F @ B ) )
=> ( ( ord_le527798542nres_b @ B @ C )
=> ( ! [X3: refine432681134nres_b,Y4: refine432681134nres_b] :
( ( ord_le527798542nres_b @ X3 @ Y4 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le519537037nres_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_74_order__subst1,axiom,
! [A: refine424419629nres_a,F: refine424419629nres_a > refine424419629nres_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ ( F @ B ) )
=> ( ( ord_le519537037nres_a @ B @ C )
=> ( ! [X3: refine424419629nres_a,Y4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y4 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le519537037nres_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_75_less__eq__nres_Osimps_I2_J,axiom,
! [A: set_a,B: set_a] :
( ( ord_le519537037nres_a @ ( refine1198353288_RES_a @ A ) @ ( refine1198353288_RES_a @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ).
% less_eq_nres.simps(2)
thf(fact_76_less__eq__nres_Osimps_I2_J,axiom,
! [A: set_b,B: set_b] :
( ( ord_le527798542nres_b @ ( refine1198353289_RES_b @ A ) @ ( refine1198353289_RES_b @ B ) )
= ( ord_less_eq_set_b @ A @ B ) ) ).
% less_eq_nres.simps(2)
thf(fact_77_nres__order__simps_I5_J,axiom,
! [X: set_a,Y: set_a] :
( ( ord_le519537037nres_a @ ( refine1198353288_RES_a @ X ) @ ( refine1198353288_RES_a @ Y ) )
= ( ord_less_eq_set_a @ X @ Y ) ) ).
% nres_order_simps(5)
thf(fact_78_nres__order__simps_I5_J,axiom,
! [X: set_b,Y: set_b] :
( ( ord_le527798542nres_b @ ( refine1198353289_RES_b @ X ) @ ( refine1198353289_RES_b @ Y ) )
= ( ord_less_eq_set_b @ X @ Y ) ) ).
% nres_order_simps(5)
thf(fact_79_order__mono__setup_Omono__if,axiom,
! [T: refine432681134nres_b,T2: refine432681134nres_b,E: refine432681134nres_b,E2: refine432681134nres_b,B: $o] :
( ( ord_le527798542nres_b @ T @ T2 )
=> ( ( ord_le527798542nres_b @ E @ E2 )
=> ( ord_le527798542nres_b @ ( if_Ref1732808808nres_b @ B @ T @ E ) @ ( if_Ref1732808808nres_b @ B @ T2 @ E2 ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_80_order__mono__setup_Omono__if,axiom,
! [T: refine424419629nres_a,T2: refine424419629nres_a,E: refine424419629nres_a,E2: refine424419629nres_a,B: $o] :
( ( ord_le519537037nres_a @ T @ T2 )
=> ( ( ord_le519537037nres_a @ E @ E2 )
=> ( ord_le519537037nres_a @ ( if_Ref1724547303nres_a @ B @ T @ E ) @ ( if_Ref1724547303nres_a @ B @ T2 @ E2 ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_81_dual__order_Oantisym,axiom,
! [B: refine432681134nres_b,A: refine432681134nres_b] :
( ( ord_le527798542nres_b @ B @ A )
=> ( ( ord_le527798542nres_b @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_82_dual__order_Oantisym,axiom,
! [B: refine424419629nres_a,A: refine424419629nres_a] :
( ( ord_le519537037nres_a @ B @ A )
=> ( ( ord_le519537037nres_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_83_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: refine432681134nres_b,Z: refine432681134nres_b] : ( Y5 = Z ) )
= ( ^ [A2: refine432681134nres_b,B2: refine432681134nres_b] :
( ( ord_le527798542nres_b @ B2 @ A2 )
& ( ord_le527798542nres_b @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_84_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: refine424419629nres_a,Z: refine424419629nres_a] : ( Y5 = Z ) )
= ( ^ [A2: refine424419629nres_a,B2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ B2 @ A2 )
& ( ord_le519537037nres_a @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_85_dual__order_Otrans,axiom,
! [B: refine432681134nres_b,A: refine432681134nres_b,C: refine432681134nres_b] :
( ( ord_le527798542nres_b @ B @ A )
=> ( ( ord_le527798542nres_b @ C @ B )
=> ( ord_le527798542nres_b @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_86_dual__order_Otrans,axiom,
! [B: refine424419629nres_a,A: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ B @ A )
=> ( ( ord_le519537037nres_a @ C @ B )
=> ( ord_le519537037nres_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_87_dual__order_Orefl,axiom,
! [A: refine432681134nres_b] : ( ord_le527798542nres_b @ A @ A ) ).
% dual_order.refl
thf(fact_88_dual__order_Orefl,axiom,
! [A: refine424419629nres_a] : ( ord_le519537037nres_a @ A @ A ) ).
% dual_order.refl
thf(fact_89_order__trans,axiom,
! [X4: refine432681134nres_b,Y3: refine432681134nres_b,Z2: refine432681134nres_b] :
( ( ord_le527798542nres_b @ X4 @ Y3 )
=> ( ( ord_le527798542nres_b @ Y3 @ Z2 )
=> ( ord_le527798542nres_b @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_90_order__trans,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a,Z2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X4 @ Y3 )
=> ( ( ord_le519537037nres_a @ Y3 @ Z2 )
=> ( ord_le519537037nres_a @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_91_order__class_Oorder_Oantisym,axiom,
! [A: refine432681134nres_b,B: refine432681134nres_b] :
( ( ord_le527798542nres_b @ A @ B )
=> ( ( ord_le527798542nres_b @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_92_order__class_Oorder_Oantisym,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ B )
=> ( ( ord_le519537037nres_a @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_93_ord__le__eq__trans,axiom,
! [A: refine432681134nres_b,B: refine432681134nres_b,C: refine432681134nres_b] :
( ( ord_le527798542nres_b @ A @ B )
=> ( ( B = C )
=> ( ord_le527798542nres_b @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_94_ord__le__eq__trans,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ B )
=> ( ( B = C )
=> ( ord_le519537037nres_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_95_ord__eq__le__trans,axiom,
! [A: refine432681134nres_b,B: refine432681134nres_b,C: refine432681134nres_b] :
( ( A = B )
=> ( ( ord_le527798542nres_b @ B @ C )
=> ( ord_le527798542nres_b @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_96_ord__eq__le__trans,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( A = B )
=> ( ( ord_le519537037nres_a @ B @ C )
=> ( ord_le519537037nres_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_97_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y5: refine432681134nres_b,Z: refine432681134nres_b] : ( Y5 = Z ) )
= ( ^ [A2: refine432681134nres_b,B2: refine432681134nres_b] :
( ( ord_le527798542nres_b @ A2 @ B2 )
& ( ord_le527798542nres_b @ B2 @ A2 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_98_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y5: refine424419629nres_a,Z: refine424419629nres_a] : ( Y5 = Z ) )
= ( ^ [A2: refine424419629nres_a,B2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A2 @ B2 )
& ( ord_le519537037nres_a @ B2 @ A2 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_99_antisym__conv,axiom,
! [Y3: refine432681134nres_b,X4: refine432681134nres_b] :
( ( ord_le527798542nres_b @ Y3 @ X4 )
=> ( ( ord_le527798542nres_b @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% antisym_conv
thf(fact_100_antisym__conv,axiom,
! [Y3: refine424419629nres_a,X4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ Y3 @ X4 )
=> ( ( ord_le519537037nres_a @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% antisym_conv
thf(fact_101_order_Otrans,axiom,
! [A: refine432681134nres_b,B: refine432681134nres_b,C: refine432681134nres_b] :
( ( ord_le527798542nres_b @ A @ B )
=> ( ( ord_le527798542nres_b @ B @ C )
=> ( ord_le527798542nres_b @ A @ C ) ) ) ).
% order.trans
thf(fact_102_order_Otrans,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ B )
=> ( ( ord_le519537037nres_a @ B @ C )
=> ( ord_le519537037nres_a @ A @ C ) ) ) ).
% order.trans
thf(fact_103_eq__refl,axiom,
! [X4: refine432681134nres_b,Y3: refine432681134nres_b] :
( ( X4 = Y3 )
=> ( ord_le527798542nres_b @ X4 @ Y3 ) ) ).
% eq_refl
thf(fact_104_eq__refl,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( X4 = Y3 )
=> ( ord_le519537037nres_a @ X4 @ Y3 ) ) ).
% eq_refl
thf(fact_105_antisym,axiom,
! [X4: refine432681134nres_b,Y3: refine432681134nres_b] :
( ( ord_le527798542nres_b @ X4 @ Y3 )
=> ( ( ord_le527798542nres_b @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% antisym
thf(fact_106_antisym,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X4 @ Y3 )
=> ( ( ord_le519537037nres_a @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% antisym
thf(fact_107_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_108_mem__Collect__eq,axiom,
! [A: b,P: b > $o] :
( ( member_b @ A @ ( collect_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_109_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X5: a] : ( member_a @ X5 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_110_Collect__mem__eq,axiom,
! [A3: set_b] :
( ( collect_b
@ ^ [X5: b] : ( member_b @ X5 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_111_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_112_Collect__cong,axiom,
! [P: b > $o,Q: b > $o] :
( ! [X3: b] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_b @ P )
= ( collect_b @ Q ) ) ) ).
% Collect_cong
thf(fact_113_eq__iff,axiom,
( ( ^ [Y5: refine432681134nres_b,Z: refine432681134nres_b] : ( Y5 = Z ) )
= ( ^ [X5: refine432681134nres_b,Y6: refine432681134nres_b] :
( ( ord_le527798542nres_b @ X5 @ Y6 )
& ( ord_le527798542nres_b @ Y6 @ X5 ) ) ) ) ).
% eq_iff
thf(fact_114_eq__iff,axiom,
( ( ^ [Y5: refine424419629nres_a,Z: refine424419629nres_a] : ( Y5 = Z ) )
= ( ^ [X5: refine424419629nres_a,Y6: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X5 @ Y6 )
& ( ord_le519537037nres_a @ Y6 @ X5 ) ) ) ) ).
% eq_iff
thf(fact_115_ord__le__eq__subst,axiom,
! [A: refine432681134nres_b,B: refine432681134nres_b,F: refine432681134nres_b > refine432681134nres_b,C: refine432681134nres_b] :
( ( ord_le527798542nres_b @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: refine432681134nres_b,Y4: refine432681134nres_b] :
( ( ord_le527798542nres_b @ X3 @ Y4 )
=> ( ord_le527798542nres_b @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le527798542nres_b @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_116_ord__le__eq__subst,axiom,
! [A: refine432681134nres_b,B: refine432681134nres_b,F: refine432681134nres_b > refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le527798542nres_b @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: refine432681134nres_b,Y4: refine432681134nres_b] :
( ( ord_le527798542nres_b @ X3 @ Y4 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le519537037nres_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_117_ord__le__eq__subst,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,F: refine424419629nres_a > refine432681134nres_b,C: refine432681134nres_b] :
( ( ord_le519537037nres_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: refine424419629nres_a,Y4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y4 )
=> ( ord_le527798542nres_b @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le527798542nres_b @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_118_ord__le__eq__subst,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,F: refine424419629nres_a > refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: refine424419629nres_a,Y4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y4 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le519537037nres_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_119_ord__eq__le__subst,axiom,
! [A: refine432681134nres_b,F: refine432681134nres_b > refine432681134nres_b,B: refine432681134nres_b,C: refine432681134nres_b] :
( ( A
= ( F @ B ) )
=> ( ( ord_le527798542nres_b @ B @ C )
=> ( ! [X3: refine432681134nres_b,Y4: refine432681134nres_b] :
( ( ord_le527798542nres_b @ X3 @ Y4 )
=> ( ord_le527798542nres_b @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le527798542nres_b @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_120_ord__eq__le__subst,axiom,
! [A: refine424419629nres_a,F: refine432681134nres_b > refine424419629nres_a,B: refine432681134nres_b,C: refine432681134nres_b] :
( ( A
= ( F @ B ) )
=> ( ( ord_le527798542nres_b @ B @ C )
=> ( ! [X3: refine432681134nres_b,Y4: refine432681134nres_b] :
( ( ord_le527798542nres_b @ X3 @ Y4 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le519537037nres_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_121_ord__eq__le__subst,axiom,
! [A: refine432681134nres_b,F: refine424419629nres_a > refine432681134nres_b,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le519537037nres_a @ B @ C )
=> ( ! [X3: refine424419629nres_a,Y4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y4 )
=> ( ord_le527798542nres_b @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le527798542nres_b @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_122_ord__eq__le__subst,axiom,
! [A: refine424419629nres_a,F: refine424419629nres_a > refine424419629nres_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le519537037nres_a @ B @ C )
=> ( ! [X3: refine424419629nres_a,Y4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y4 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le519537037nres_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_123_order__subst2,axiom,
! [A: refine432681134nres_b,B: refine432681134nres_b,F: refine432681134nres_b > refine432681134nres_b,C: refine432681134nres_b] :
( ( ord_le527798542nres_b @ A @ B )
=> ( ( ord_le527798542nres_b @ ( F @ B ) @ C )
=> ( ! [X3: refine432681134nres_b,Y4: refine432681134nres_b] :
( ( ord_le527798542nres_b @ X3 @ Y4 )
=> ( ord_le527798542nres_b @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le527798542nres_b @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_124_order__subst2,axiom,
! [A: refine432681134nres_b,B: refine432681134nres_b,F: refine432681134nres_b > refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le527798542nres_b @ A @ B )
=> ( ( ord_le519537037nres_a @ ( F @ B ) @ C )
=> ( ! [X3: refine432681134nres_b,Y4: refine432681134nres_b] :
( ( ord_le527798542nres_b @ X3 @ Y4 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le519537037nres_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_125_order__subst2,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,F: refine424419629nres_a > refine432681134nres_b,C: refine432681134nres_b] :
( ( ord_le519537037nres_a @ A @ B )
=> ( ( ord_le527798542nres_b @ ( F @ B ) @ C )
=> ( ! [X3: refine424419629nres_a,Y4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y4 )
=> ( ord_le527798542nres_b @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le527798542nres_b @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_126_order__subst2,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,F: refine424419629nres_a > refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ B )
=> ( ( ord_le519537037nres_a @ ( F @ B ) @ C )
=> ( ! [X3: refine424419629nres_a,Y4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y4 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le519537037nres_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_127_the__RES__inv,axiom,
! [M: refine424419629nres_a] :
( ( refine412683989fail_a @ M )
=> ( ( refine1198353288_RES_a @ ( refine1822134885_RES_a @ M ) )
= M ) ) ).
% the_RES_inv
thf(fact_128_the__RES__inv,axiom,
! [M: refine432681134nres_b] :
( ( refine412683990fail_b @ M )
=> ( ( refine1198353289_RES_b @ ( refine1822134886_RES_b @ M ) )
= M ) ) ).
% the_RES_inv
thf(fact_129_Greatest__equality,axiom,
! [P: refine432681134nres_b > $o,X4: refine432681134nres_b] :
( ( P @ X4 )
=> ( ! [Y4: refine432681134nres_b] :
( ( P @ Y4 )
=> ( ord_le527798542nres_b @ Y4 @ X4 ) )
=> ( ( order_1722590613nres_b @ P )
= X4 ) ) ) ).
% Greatest_equality
thf(fact_130_Greatest__equality,axiom,
! [P: refine424419629nres_a > $o,X4: refine424419629nres_a] :
( ( P @ X4 )
=> ( ! [Y4: refine424419629nres_a] :
( ( P @ Y4 )
=> ( ord_le519537037nres_a @ Y4 @ X4 ) )
=> ( ( order_1714329108nres_a @ P )
= X4 ) ) ) ).
% Greatest_equality
thf(fact_131_GreatestI2__order,axiom,
! [P: refine432681134nres_b > $o,X4: refine432681134nres_b,Q: refine432681134nres_b > $o] :
( ( P @ X4 )
=> ( ! [Y4: refine432681134nres_b] :
( ( P @ Y4 )
=> ( ord_le527798542nres_b @ Y4 @ X4 ) )
=> ( ! [X3: refine432681134nres_b] :
( ( P @ X3 )
=> ( ! [Y7: refine432681134nres_b] :
( ( P @ Y7 )
=> ( ord_le527798542nres_b @ Y7 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_1722590613nres_b @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_132_GreatestI2__order,axiom,
! [P: refine424419629nres_a > $o,X4: refine424419629nres_a,Q: refine424419629nres_a > $o] :
( ( P @ X4 )
=> ( ! [Y4: refine424419629nres_a] :
( ( P @ Y4 )
=> ( ord_le519537037nres_a @ Y4 @ X4 ) )
=> ( ! [X3: refine424419629nres_a] :
( ( P @ X3 )
=> ( ! [Y7: refine424419629nres_a] :
( ( P @ Y7 )
=> ( ord_le519537037nres_a @ Y7 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_1714329108nres_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_133_le__rel__bool__arg__iff,axiom,
( ord_le1656162903nres_b
= ( ^ [X6: $o > refine432681134nres_b,Y8: $o > refine432681134nres_b] :
( ( ord_le527798542nres_b @ ( X6 @ $false ) @ ( Y8 @ $false ) )
& ( ord_le527798542nres_b @ ( X6 @ $true ) @ ( Y8 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_134_le__rel__bool__arg__iff,axiom,
( ord_le516598614nres_a
= ( ^ [X6: $o > refine424419629nres_a,Y8: $o > refine424419629nres_a] :
( ( ord_le519537037nres_a @ ( X6 @ $false ) @ ( Y8 @ $false ) )
& ( ord_le519537037nres_a @ ( X6 @ $true ) @ ( Y8 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_135_ord__eq__le__eq__trans,axiom,
! [A: refine432681134nres_b,B: refine432681134nres_b,C: refine432681134nres_b,D: refine432681134nres_b] :
( ( A = B )
=> ( ( ord_le527798542nres_b @ B @ C )
=> ( ( C = D )
=> ( ord_le527798542nres_b @ A @ D ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_136_ord__eq__le__eq__trans,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,C: refine424419629nres_a,D: refine424419629nres_a] :
( ( A = B )
=> ( ( ord_le519537037nres_a @ B @ C )
=> ( ( C = D )
=> ( ord_le519537037nres_a @ A @ D ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_137_nofail__simps_I2_J,axiom,
! [X: set_a] : ( refine412683989fail_a @ ( refine1198353288_RES_a @ X ) ) ).
% nofail_simps(2)
thf(fact_138_nofail__simps_I2_J,axiom,
! [X: set_b] : ( refine412683990fail_b @ ( refine1198353289_RES_b @ X ) ) ).
% nofail_simps(2)
thf(fact_139_subset__Collect__conv,axiom,
! [S: set_a,P: a > $o] :
( ( ord_less_eq_set_a @ S @ ( collect_a @ P ) )
= ( ! [X5: a] :
( ( member_a @ X5 @ S )
=> ( P @ X5 ) ) ) ) ).
% subset_Collect_conv
thf(fact_140_subset__Collect__conv,axiom,
! [S: set_b,P: b > $o] :
( ( ord_less_eq_set_b @ S @ ( collect_b @ P ) )
= ( ! [X5: b] :
( ( member_b @ X5 @ S )
=> ( P @ X5 ) ) ) ) ).
% subset_Collect_conv
thf(fact_141_pwD1,axiom,
! [S: refine432681134nres_b,S3: refine432681134nres_b] :
( ( ord_le527798542nres_b @ S @ S3 )
=> ( ( refine412683990fail_b @ S3 )
=> ( refine412683990fail_b @ S ) ) ) ).
% pwD1
thf(fact_142_pwD1,axiom,
! [S: refine424419629nres_a,S3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ S @ S3 )
=> ( ( refine412683989fail_a @ S3 )
=> ( refine412683989fail_a @ S ) ) ) ).
% pwD1
thf(fact_143_le__nofailI,axiom,
! [M3: refine432681134nres_b,M4: refine432681134nres_b] :
( ( ( refine412683990fail_b @ M3 )
=> ( ord_le527798542nres_b @ M4 @ M3 ) )
=> ( ord_le527798542nres_b @ M4 @ M3 ) ) ).
% le_nofailI
thf(fact_144_le__nofailI,axiom,
! [M3: refine424419629nres_a,M4: refine424419629nres_a] :
( ( ( refine412683989fail_a @ M3 )
=> ( ord_le519537037nres_a @ M4 @ M3 ) )
=> ( ord_le519537037nres_a @ M4 @ M3 ) ) ).
% le_nofailI
thf(fact_145_nofail__RES__conv,axiom,
( refine412683989fail_a
= ( ^ [M5: refine424419629nres_a] :
? [M6: set_a] :
( M5
= ( refine1198353288_RES_a @ M6 ) ) ) ) ).
% nofail_RES_conv
thf(fact_146_nofail__RES__conv,axiom,
( refine412683990fail_b
= ( ^ [M5: refine432681134nres_b] :
? [M6: set_b] :
( M5
= ( refine1198353289_RES_b @ M6 ) ) ) ) ).
% nofail_RES_conv
thf(fact_147_ref__two__step,axiom,
! [A3: refine432681134nres_b,R: set_Product_prod_b_b,B3: refine432681134nres_b,C2: refine432681134nres_b] :
( ( ord_le527798542nres_b @ A3 @ ( refine122110613un_b_b @ R @ B3 ) )
=> ( ( ord_le527798542nres_b @ B3 @ C2 )
=> ( ord_le527798542nres_b @ A3 @ ( refine122110613un_b_b @ R @ C2 ) ) ) ) ).
% ref_two_step
thf(fact_148_ref__two__step,axiom,
! [A3: refine432681134nres_b,R: set_Product_prod_b_a,B3: refine424419629nres_a,C2: refine424419629nres_a] :
( ( ord_le527798542nres_b @ A3 @ ( refine122110612un_b_a @ R @ B3 ) )
=> ( ( ord_le519537037nres_a @ B3 @ C2 )
=> ( ord_le527798542nres_b @ A3 @ ( refine122110612un_b_a @ R @ C2 ) ) ) ) ).
% ref_two_step
thf(fact_149_ref__two__step,axiom,
! [A3: refine424419629nres_a,R: set_Product_prod_a_b,B3: refine432681134nres_b,C2: refine432681134nres_b] :
( ( ord_le519537037nres_a @ A3 @ ( refine1441824854un_a_b @ R @ B3 ) )
=> ( ( ord_le527798542nres_b @ B3 @ C2 )
=> ( ord_le519537037nres_a @ A3 @ ( refine1441824854un_a_b @ R @ C2 ) ) ) ) ).
% ref_two_step
thf(fact_150_ref__two__step,axiom,
! [A3: refine424419629nres_a,R: set_Product_prod_a_a,B3: refine424419629nres_a,C2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A3 @ ( refine1441824853un_a_a @ R @ B3 ) )
=> ( ( ord_le519537037nres_a @ B3 @ C2 )
=> ( ord_le519537037nres_a @ A3 @ ( refine1441824853un_a_a @ R @ C2 ) ) ) ) ).
% ref_two_step
thf(fact_151_le__RES__nofailI,axiom,
! [A: refine424419629nres_a,X4: set_a] :
( ( ord_le519537037nres_a @ A @ ( refine1198353288_RES_a @ X4 ) )
=> ( refine412683989fail_a @ A ) ) ).
% le_RES_nofailI
thf(fact_152_le__RES__nofailI,axiom,
! [A: refine432681134nres_b,X4: set_b] :
( ( ord_le527798542nres_b @ A @ ( refine1198353289_RES_b @ X4 ) )
=> ( refine412683990fail_b @ A ) ) ).
% le_RES_nofailI
thf(fact_153_pw__RES__bind__choose_I1_J,axiom,
! [X: set_a,F: a > refine432681134nres_b] :
( ( refine412683990fail_b @ ( refine436832839nd_a_b @ ( refine1198353288_RES_a @ X ) @ F ) )
= ( ! [X5: a] :
( ( member_a @ X5 @ X )
=> ( refine412683990fail_b @ ( F @ X5 ) ) ) ) ) ).
% pw_RES_bind_choose(1)
thf(fact_154_rhs__step__bind__RES,axiom,
! [X7: a,X8: set_a,M: refine432681134nres_b,R: set_Product_prod_b_b,F2: a > refine432681134nres_b] :
( ( member_a @ X7 @ X8 )
=> ( ( ord_le527798542nres_b @ M @ ( refine122110613un_b_b @ R @ ( F2 @ X7 ) ) )
=> ( ord_le527798542nres_b @ M @ ( refine122110613un_b_b @ R @ ( refine436832839nd_a_b @ ( refine1198353288_RES_a @ X8 ) @ F2 ) ) ) ) ) ).
% rhs_step_bind_RES
thf(fact_155_rhs__step__bind__RES,axiom,
! [X7: a,X8: set_a,M: refine424419629nres_a,R: set_Product_prod_a_b,F2: a > refine432681134nres_b] :
( ( member_a @ X7 @ X8 )
=> ( ( ord_le519537037nres_a @ M @ ( refine1441824854un_a_b @ R @ ( F2 @ X7 ) ) )
=> ( ord_le519537037nres_a @ M @ ( refine1441824854un_a_b @ R @ ( refine436832839nd_a_b @ ( refine1198353288_RES_a @ X8 ) @ F2 ) ) ) ) ) ).
% rhs_step_bind_RES
thf(fact_156_rhs__step__bind__SPEC,axiom,
! [Phi: a > $o,X7: a,M: refine432681134nres_b,R: set_Product_prod_b_b,F2: a > refine432681134nres_b] :
( ( Phi @ X7 )
=> ( ( ord_le527798542nres_b @ M @ ( refine122110613un_b_b @ R @ ( F2 @ X7 ) ) )
=> ( ord_le527798542nres_b @ M @ ( refine122110613un_b_b @ R @ ( refine436832839nd_a_b @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) @ F2 ) ) ) ) ) ).
% rhs_step_bind_SPEC
thf(fact_157_rhs__step__bind__SPEC,axiom,
! [Phi: a > $o,X7: a,M: refine424419629nres_a,R: set_Product_prod_a_b,F2: a > refine432681134nres_b] :
( ( Phi @ X7 )
=> ( ( ord_le519537037nres_a @ M @ ( refine1441824854un_a_b @ R @ ( F2 @ X7 ) ) )
=> ( ord_le519537037nres_a @ M @ ( refine1441824854un_a_b @ R @ ( refine436832839nd_a_b @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) @ F2 ) ) ) ) ) ).
% rhs_step_bind_SPEC
thf(fact_158_subsetI,axiom,
! [A3: set_a,B3: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( member_a @ X3 @ B3 ) )
=> ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% subsetI
thf(fact_159_subsetI,axiom,
! [A3: set_b,B3: set_b] :
( ! [X3: b] :
( ( member_b @ X3 @ A3 )
=> ( member_b @ X3 @ B3 ) )
=> ( ord_less_eq_set_b @ A3 @ B3 ) ) ).
% subsetI
thf(fact_160_bind__sim__select__rule,axiom,
! [M: refine424419629nres_a,F2: a > refine424419629nres_a,Psi: a > $o,F: a > refine432681134nres_b,Phi: b > $o] :
( ( ord_le519537037nres_a @ ( refine436832838nd_a_a @ M @ F2 ) @ ( refine1198353288_RES_a @ ( collect_a @ Psi ) ) )
=> ( ! [X3: a] :
( ( refine412683989fail_a @ M )
=> ( ( refine1001002027nres_a @ M @ X3 )
=> ( ( ord_le519537037nres_a @ ( F2 @ X3 ) @ ( refine1198353288_RES_a @ ( collect_a @ Psi ) ) )
=> ( ord_le527798542nres_b @ ( F @ X3 ) @ ( refine1198353289_RES_b @ ( collect_b @ Phi ) ) ) ) ) )
=> ( ord_le527798542nres_b @ ( refine436832839nd_a_b @ M @ F ) @ ( refine1198353289_RES_b @ ( collect_b @ Phi ) ) ) ) ) ).
% bind_sim_select_rule
thf(fact_161_bind__sim__select__rule,axiom,
! [M: refine424419629nres_a,F2: a > refine432681134nres_b,Psi: b > $o,F: a > refine424419629nres_a,Phi: a > $o] :
( ( ord_le527798542nres_b @ ( refine436832839nd_a_b @ M @ F2 ) @ ( refine1198353289_RES_b @ ( collect_b @ Psi ) ) )
=> ( ! [X3: a] :
( ( refine412683989fail_a @ M )
=> ( ( refine1001002027nres_a @ M @ X3 )
=> ( ( ord_le527798542nres_b @ ( F2 @ X3 ) @ ( refine1198353289_RES_b @ ( collect_b @ Psi ) ) )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) ) ) ) )
=> ( ord_le519537037nres_a @ ( refine436832838nd_a_a @ M @ F ) @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) ) ) ) ).
% bind_sim_select_rule
thf(fact_162_bind__sim__select__rule,axiom,
! [M: refine424419629nres_a,F2: a > refine432681134nres_b,Psi: b > $o,F: a > refine432681134nres_b,Phi: b > $o] :
( ( ord_le527798542nres_b @ ( refine436832839nd_a_b @ M @ F2 ) @ ( refine1198353289_RES_b @ ( collect_b @ Psi ) ) )
=> ( ! [X3: a] :
( ( refine412683989fail_a @ M )
=> ( ( refine1001002027nres_a @ M @ X3 )
=> ( ( ord_le527798542nres_b @ ( F2 @ X3 ) @ ( refine1198353289_RES_b @ ( collect_b @ Psi ) ) )
=> ( ord_le527798542nres_b @ ( F @ X3 ) @ ( refine1198353289_RES_b @ ( collect_b @ Phi ) ) ) ) ) )
=> ( ord_le527798542nres_b @ ( refine436832839nd_a_b @ M @ F ) @ ( refine1198353289_RES_b @ ( collect_b @ Phi ) ) ) ) ) ).
% bind_sim_select_rule
thf(fact_163_in__mono,axiom,
! [A3: set_a,B3: set_a,X4: a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( member_a @ X4 @ A3 )
=> ( member_a @ X4 @ B3 ) ) ) ).
% in_mono
thf(fact_164_in__mono,axiom,
! [A3: set_b,B3: set_b,X4: b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ( member_b @ X4 @ A3 )
=> ( member_b @ X4 @ B3 ) ) ) ).
% in_mono
thf(fact_165_subsetD,axiom,
! [A3: set_a,B3: set_a,C: a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_166_subsetD,axiom,
! [A3: set_b,B3: set_b,C: b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
=> ( ( member_b @ C @ A3 )
=> ( member_b @ C @ B3 ) ) ) ).
% subsetD
thf(fact_167_inres__simps_I2_J,axiom,
! [X: set_a] :
( ( refine1001002027nres_a @ ( refine1198353288_RES_a @ X ) )
= ( ^ [X5: a] : ( member_a @ X5 @ X ) ) ) ).
% inres_simps(2)
thf(fact_168_inres__simps_I2_J,axiom,
! [X: set_b] :
( ( refine1001002028nres_b @ ( refine1198353289_RES_b @ X ) )
= ( ^ [X5: b] : ( member_b @ X5 @ X ) ) ) ).
% inres_simps(2)
thf(fact_169_pwD2,axiom,
! [S: refine432681134nres_b,S3: refine432681134nres_b,X4: b] :
( ( ord_le527798542nres_b @ S @ S3 )
=> ( ( refine1001002028nres_b @ S @ X4 )
=> ( refine1001002028nres_b @ S3 @ X4 ) ) ) ).
% pwD2
thf(fact_170_pwD2,axiom,
! [S: refine424419629nres_a,S3: refine424419629nres_a,X4: a] :
( ( ord_le519537037nres_a @ S @ S3 )
=> ( ( refine1001002027nres_a @ S @ X4 )
=> ( refine1001002027nres_a @ S3 @ X4 ) ) ) ).
% pwD2
thf(fact_171_pw__RES__bind__choose_I2_J,axiom,
! [X: set_a,F: a > refine432681134nres_b,Y3: b] :
( ( refine1001002028nres_b @ ( refine436832839nd_a_b @ ( refine1198353288_RES_a @ X ) @ F ) @ Y3 )
= ( ? [X5: a] :
( ( member_a @ X5 @ X )
& ( refine1001002028nres_b @ ( F @ X5 ) @ Y3 ) ) ) ) ).
% pw_RES_bind_choose(2)
thf(fact_172_pw__leI,axiom,
! [S3: refine432681134nres_b,S: refine432681134nres_b] :
( ( ( refine412683990fail_b @ S3 )
=> ( ( refine412683990fail_b @ S )
& ! [X3: b] :
( ( refine1001002028nres_b @ S @ X3 )
=> ( refine1001002028nres_b @ S3 @ X3 ) ) ) )
=> ( ord_le527798542nres_b @ S @ S3 ) ) ).
% pw_leI
thf(fact_173_pw__leI,axiom,
! [S3: refine424419629nres_a,S: refine424419629nres_a] :
( ( ( refine412683989fail_a @ S3 )
=> ( ( refine412683989fail_a @ S )
& ! [X3: a] :
( ( refine1001002027nres_a @ S @ X3 )
=> ( refine1001002027nres_a @ S3 @ X3 ) ) ) )
=> ( ord_le519537037nres_a @ S @ S3 ) ) ).
% pw_leI
thf(fact_174_pw__leI_H,axiom,
! [S3: refine432681134nres_b,S: refine432681134nres_b] :
( ( ( refine412683990fail_b @ S3 )
=> ( refine412683990fail_b @ S ) )
=> ( ! [X3: b] :
( ( refine412683990fail_b @ S3 )
=> ( ( refine1001002028nres_b @ S @ X3 )
=> ( refine1001002028nres_b @ S3 @ X3 ) ) )
=> ( ord_le527798542nres_b @ S @ S3 ) ) ) ).
% pw_leI'
thf(fact_175_pw__leI_H,axiom,
! [S3: refine424419629nres_a,S: refine424419629nres_a] :
( ( ( refine412683989fail_a @ S3 )
=> ( refine412683989fail_a @ S ) )
=> ( ! [X3: a] :
( ( refine412683989fail_a @ S3 )
=> ( ( refine1001002027nres_a @ S @ X3 )
=> ( refine1001002027nres_a @ S3 @ X3 ) ) )
=> ( ord_le519537037nres_a @ S @ S3 ) ) ) ).
% pw_leI'
thf(fact_176_pw__le__iff,axiom,
( ord_le527798542nres_b
= ( ^ [S4: refine432681134nres_b,S5: refine432681134nres_b] :
( ( refine412683990fail_b @ S5 )
=> ( ( refine412683990fail_b @ S4 )
& ! [X5: b] :
( ( refine1001002028nres_b @ S4 @ X5 )
=> ( refine1001002028nres_b @ S5 @ X5 ) ) ) ) ) ) ).
% pw_le_iff
thf(fact_177_pw__le__iff,axiom,
( ord_le519537037nres_a
= ( ^ [S4: refine424419629nres_a,S5: refine424419629nres_a] :
( ( refine412683989fail_a @ S5 )
=> ( ( refine412683989fail_a @ S4 )
& ! [X5: a] :
( ( refine1001002027nres_a @ S4 @ X5 )
=> ( refine1001002027nres_a @ S5 @ X5 ) ) ) ) ) ) ).
% pw_le_iff
thf(fact_178_pw__bind__nofail,axiom,
! [M4: refine424419629nres_a,F: a > refine432681134nres_b] :
( ( refine412683990fail_b @ ( refine436832839nd_a_b @ M4 @ F ) )
= ( ( refine412683989fail_a @ M4 )
& ! [X5: a] :
( ( refine1001002027nres_a @ M4 @ X5 )
=> ( refine412683990fail_b @ ( F @ X5 ) ) ) ) ) ).
% pw_bind_nofail
thf(fact_179_pw__bind__le__iff,axiom,
! [M4: refine424419629nres_a,F: a > refine432681134nres_b,S: refine432681134nres_b] :
( ( ord_le527798542nres_b @ ( refine436832839nd_a_b @ M4 @ F ) @ S )
= ( ( ( refine412683990fail_b @ S )
=> ( refine412683989fail_a @ M4 ) )
& ! [X5: a] :
( ( ( refine412683989fail_a @ M4 )
& ( refine1001002027nres_a @ M4 @ X5 ) )
=> ( ord_le527798542nres_b @ ( F @ X5 ) @ S ) ) ) ) ).
% pw_bind_le_iff
thf(fact_180_pw__bind__leI,axiom,
! [S: refine432681134nres_b,M4: refine424419629nres_a,F: a > refine432681134nres_b] :
( ( ( refine412683990fail_b @ S )
=> ( refine412683989fail_a @ M4 ) )
=> ( ! [X3: a] :
( ( refine412683989fail_a @ M4 )
=> ( ( refine1001002027nres_a @ M4 @ X3 )
=> ( ord_le527798542nres_b @ ( F @ X3 ) @ S ) ) )
=> ( ord_le527798542nres_b @ ( refine436832839nd_a_b @ M4 @ F ) @ S ) ) ) ).
% pw_bind_leI
thf(fact_181_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X5: a] :
( ( P @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_182_Collect__mono__iff,axiom,
! [P: b > $o,Q: b > $o] :
( ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) )
= ( ! [X5: b] :
( ( P @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_183_Collect__mono,axiom,
! [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_184_Collect__mono,axiom,
! [P: b > $o,Q: b > $o] :
( ! [X3: b] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) ) ) ).
% Collect_mono
thf(fact_185_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [T3: a] :
( ( member_a @ T3 @ A4 )
=> ( member_a @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_186_subset__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A4: set_b,B4: set_b] :
! [T3: b] :
( ( member_b @ T3 @ A4 )
=> ( member_b @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_187_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [X5: a] :
( ( member_a @ X5 @ A4 )
=> ( member_a @ X5 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_188_subset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A4: set_b,B4: set_b] :
! [X5: b] :
( ( member_b @ X5 @ A4 )
=> ( member_b @ X5 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_189_rhs__step__bind,axiom,
! [M: refine432681134nres_b,R: set_Product_prod_b_a,M7: refine424419629nres_a,X4: b,Lhs: refine432681134nres_b,S: set_Product_prod_b_b,F2: a > refine432681134nres_b] :
( ( ord_le527798542nres_b @ M @ ( refine122110612un_b_a @ R @ M7 ) )
=> ( ( refine1001002028nres_b @ M @ X4 )
=> ( ! [X9: a] :
( ( member1277678991od_b_a @ ( product_Pair_b_a @ X4 @ X9 ) @ R )
=> ( ord_le527798542nres_b @ Lhs @ ( refine122110613un_b_b @ S @ ( F2 @ X9 ) ) ) )
=> ( ord_le527798542nres_b @ Lhs @ ( refine122110613un_b_b @ S @ ( refine436832839nd_a_b @ M7 @ F2 ) ) ) ) ) ) ).
% rhs_step_bind
thf(fact_190_rhs__step__bind,axiom,
! [M: refine432681134nres_b,R: set_Product_prod_b_a,M7: refine424419629nres_a,X4: b,Lhs: refine424419629nres_a,S: set_Product_prod_a_b,F2: a > refine432681134nres_b] :
( ( ord_le527798542nres_b @ M @ ( refine122110612un_b_a @ R @ M7 ) )
=> ( ( refine1001002028nres_b @ M @ X4 )
=> ( ! [X9: a] :
( ( member1277678991od_b_a @ ( product_Pair_b_a @ X4 @ X9 ) @ R )
=> ( ord_le519537037nres_a @ Lhs @ ( refine1441824854un_a_b @ S @ ( F2 @ X9 ) ) ) )
=> ( ord_le519537037nres_a @ Lhs @ ( refine1441824854un_a_b @ S @ ( refine436832839nd_a_b @ M7 @ F2 ) ) ) ) ) ) ).
% rhs_step_bind
thf(fact_191_rhs__step__bind,axiom,
! [M: refine424419629nres_a,R: set_Product_prod_a_a,M7: refine424419629nres_a,X4: a,Lhs: refine432681134nres_b,S: set_Product_prod_b_b,F2: a > refine432681134nres_b] :
( ( ord_le519537037nres_a @ M @ ( refine1441824853un_a_a @ R @ M7 ) )
=> ( ( refine1001002027nres_a @ M @ X4 )
=> ( ! [X9: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ X9 ) @ R )
=> ( ord_le527798542nres_b @ Lhs @ ( refine122110613un_b_b @ S @ ( F2 @ X9 ) ) ) )
=> ( ord_le527798542nres_b @ Lhs @ ( refine122110613un_b_b @ S @ ( refine436832839nd_a_b @ M7 @ F2 ) ) ) ) ) ) ).
% rhs_step_bind
thf(fact_192_rhs__step__bind,axiom,
! [M: refine424419629nres_a,R: set_Product_prod_a_a,M7: refine424419629nres_a,X4: a,Lhs: refine424419629nres_a,S: set_Product_prod_a_b,F2: a > refine432681134nres_b] :
( ( ord_le519537037nres_a @ M @ ( refine1441824853un_a_a @ R @ M7 ) )
=> ( ( refine1001002027nres_a @ M @ X4 )
=> ( ! [X9: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ X9 ) @ R )
=> ( ord_le519537037nres_a @ Lhs @ ( refine1441824854un_a_b @ S @ ( F2 @ X9 ) ) ) )
=> ( ord_le519537037nres_a @ Lhs @ ( refine1441824854un_a_b @ S @ ( refine436832839nd_a_b @ M7 @ F2 ) ) ) ) ) ) ).
% rhs_step_bind
thf(fact_193_RES__refine,axiom,
! [S: set_a,S3: set_a,R: set_Product_prod_a_a] :
( ! [S6: a] :
( ( member_a @ S6 @ S )
=> ? [X10: a] :
( ( member_a @ X10 @ S3 )
& ( member449909584od_a_a @ ( product_Pair_a_a @ S6 @ X10 ) @ R ) ) )
=> ( ord_le519537037nres_a @ ( refine1198353288_RES_a @ S ) @ ( refine1441824853un_a_a @ R @ ( refine1198353288_RES_a @ S3 ) ) ) ) ).
% RES_refine
thf(fact_194_RES__refine,axiom,
! [S: set_a,S3: set_b,R: set_Product_prod_a_b] :
( ! [S6: a] :
( ( member_a @ S6 @ S )
=> ? [X10: b] :
( ( member_b @ X10 @ S3 )
& ( member458171089od_a_b @ ( product_Pair_a_b @ S6 @ X10 ) @ R ) ) )
=> ( ord_le519537037nres_a @ ( refine1198353288_RES_a @ S ) @ ( refine1441824854un_a_b @ R @ ( refine1198353289_RES_b @ S3 ) ) ) ) ).
% RES_refine
thf(fact_195_RES__refine,axiom,
! [S: set_b,S3: set_a,R: set_Product_prod_b_a] :
( ! [S6: b] :
( ( member_b @ S6 @ S )
=> ? [X10: a] :
( ( member_a @ X10 @ S3 )
& ( member1277678991od_b_a @ ( product_Pair_b_a @ S6 @ X10 ) @ R ) ) )
=> ( ord_le527798542nres_b @ ( refine1198353289_RES_b @ S ) @ ( refine122110612un_b_a @ R @ ( refine1198353288_RES_a @ S3 ) ) ) ) ).
% RES_refine
thf(fact_196_RES__refine,axiom,
! [S: set_b,S3: set_b,R: set_Product_prod_b_b] :
( ! [S6: b] :
( ( member_b @ S6 @ S )
=> ? [X10: b] :
( ( member_b @ X10 @ S3 )
& ( member1285940496od_b_b @ ( product_Pair_b_b @ S6 @ X10 ) @ R ) ) )
=> ( ord_le527798542nres_b @ ( refine1198353289_RES_b @ S ) @ ( refine122110613un_b_b @ R @ ( refine1198353289_RES_b @ S3 ) ) ) ) ).
% RES_refine
thf(fact_197_less__eq__nres_Oelims_I1_J,axiom,
! [X4: refine424419629nres_a,Xa: refine424419629nres_a,Y3: $o] :
( ( ( ord_le519537037nres_a @ X4 @ Xa )
= Y3 )
=> ( ( ( Xa = refine464223677AILi_a )
=> ~ Y3 )
=> ( ! [A5: set_a] :
( ( X4
= ( refine1198353288_RES_a @ A5 ) )
=> ! [B5: set_a] :
( ( Xa
= ( refine1198353288_RES_a @ B5 ) )
=> ( Y3
= ( ~ ( ord_less_eq_set_a @ A5 @ B5 ) ) ) ) )
=> ~ ( ( X4 = refine464223677AILi_a )
=> ( ? [Uv: set_a] :
( Xa
= ( refine1198353288_RES_a @ Uv ) )
=> Y3 ) ) ) ) ) ).
% less_eq_nres.elims(1)
thf(fact_198_less__eq__nres_Oelims_I1_J,axiom,
! [X4: refine432681134nres_b,Xa: refine432681134nres_b,Y3: $o] :
( ( ( ord_le527798542nres_b @ X4 @ Xa )
= Y3 )
=> ( ( ( Xa = refine464223678AILi_b )
=> ~ Y3 )
=> ( ! [A5: set_b] :
( ( X4
= ( refine1198353289_RES_b @ A5 ) )
=> ! [B5: set_b] :
( ( Xa
= ( refine1198353289_RES_b @ B5 ) )
=> ( Y3
= ( ~ ( ord_less_eq_set_b @ A5 @ B5 ) ) ) ) )
=> ~ ( ( X4 = refine464223678AILi_b )
=> ( ? [Uv: set_b] :
( Xa
= ( refine1198353289_RES_b @ Uv ) )
=> Y3 ) ) ) ) ) ).
% less_eq_nres.elims(1)
thf(fact_199_less__eq__nres_Oelims_I2_J,axiom,
! [X4: refine424419629nres_a,Xa: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X4 @ Xa )
=> ( ( Xa != refine464223677AILi_a )
=> ~ ! [A5: set_a] :
( ( X4
= ( refine1198353288_RES_a @ A5 ) )
=> ! [B5: set_a] :
( ( Xa
= ( refine1198353288_RES_a @ B5 ) )
=> ~ ( ord_less_eq_set_a @ A5 @ B5 ) ) ) ) ) ).
% less_eq_nres.elims(2)
thf(fact_200_less__eq__nres_Oelims_I2_J,axiom,
! [X4: refine432681134nres_b,Xa: refine432681134nres_b] :
( ( ord_le527798542nres_b @ X4 @ Xa )
=> ( ( Xa != refine464223678AILi_b )
=> ~ ! [A5: set_b] :
( ( X4
= ( refine1198353289_RES_b @ A5 ) )
=> ! [B5: set_b] :
( ( Xa
= ( refine1198353289_RES_b @ B5 ) )
=> ~ ( ord_less_eq_set_b @ A5 @ B5 ) ) ) ) ) ).
% less_eq_nres.elims(2)
thf(fact_201_less__eq__nres_Ocases,axiom,
! [X4: produc1609974940nres_a] :
( ! [Uu: refine424419629nres_a] :
( X4
!= ( produc1136923598nres_a @ Uu @ refine464223677AILi_a ) )
=> ( ! [A5: set_a,B5: set_a] :
( X4
!= ( produc1136923598nres_a @ ( refine1198353288_RES_a @ A5 ) @ ( refine1198353288_RES_a @ B5 ) ) )
=> ~ ! [Uv: set_a] :
( X4
!= ( produc1136923598nres_a @ refine464223677AILi_a @ ( refine1198353288_RES_a @ Uv ) ) ) ) ) ).
% less_eq_nres.cases
thf(fact_202_less__eq__nres_Ocases,axiom,
! [X4: produc1587512412nres_b] :
( ! [Uu: refine432681134nres_b] :
( X4
!= ( produc2130641934nres_b @ Uu @ refine464223678AILi_b ) )
=> ( ! [A5: set_b,B5: set_b] :
( X4
!= ( produc2130641934nres_b @ ( refine1198353289_RES_b @ A5 ) @ ( refine1198353289_RES_b @ B5 ) ) )
=> ~ ! [Uv: set_b] :
( X4
!= ( produc2130641934nres_b @ refine464223678AILi_b @ ( refine1198353289_RES_b @ Uv ) ) ) ) ) ).
% less_eq_nres.cases
thf(fact_203_less__nres_Ocases,axiom,
! [X4: produc1609974940nres_a] :
( ! [Uu: refine424419629nres_a] :
( X4
!= ( produc1136923598nres_a @ refine464223677AILi_a @ Uu ) )
=> ( ! [Uv: set_a] :
( X4
!= ( produc1136923598nres_a @ ( refine1198353288_RES_a @ Uv ) @ refine464223677AILi_a ) )
=> ~ ! [A5: set_a,B5: set_a] :
( X4
!= ( produc1136923598nres_a @ ( refine1198353288_RES_a @ A5 ) @ ( refine1198353288_RES_a @ B5 ) ) ) ) ) ).
% less_nres.cases
thf(fact_204_less__nres_Ocases,axiom,
! [X4: produc1587512412nres_b] :
( ! [Uu: refine432681134nres_b] :
( X4
!= ( produc2130641934nres_b @ refine464223678AILi_b @ Uu ) )
=> ( ! [Uv: set_b] :
( X4
!= ( produc2130641934nres_b @ ( refine1198353289_RES_b @ Uv ) @ refine464223678AILi_b ) )
=> ~ ! [A5: set_b,B5: set_b] :
( X4
!= ( produc2130641934nres_b @ ( refine1198353289_RES_b @ A5 ) @ ( refine1198353289_RES_b @ B5 ) ) ) ) ) ).
% less_nres.cases
thf(fact_205_sup__nres_Ocases,axiom,
! [X4: produc1609974940nres_a] :
( ! [Uu: refine424419629nres_a] :
( X4
!= ( produc1136923598nres_a @ Uu @ refine464223677AILi_a ) )
=> ( ! [V: set_a] :
( X4
!= ( produc1136923598nres_a @ refine464223677AILi_a @ ( refine1198353288_RES_a @ V ) ) )
=> ~ ! [A5: set_a,B5: set_a] :
( X4
!= ( produc1136923598nres_a @ ( refine1198353288_RES_a @ A5 ) @ ( refine1198353288_RES_a @ B5 ) ) ) ) ) ).
% sup_nres.cases
thf(fact_206_sup__nres_Ocases,axiom,
! [X4: produc1587512412nres_b] :
( ! [Uu: refine432681134nres_b] :
( X4
!= ( produc2130641934nres_b @ Uu @ refine464223678AILi_b ) )
=> ( ! [V: set_b] :
( X4
!= ( produc2130641934nres_b @ refine464223678AILi_b @ ( refine1198353289_RES_b @ V ) ) )
=> ~ ! [A5: set_b,B5: set_b] :
( X4
!= ( produc2130641934nres_b @ ( refine1198353289_RES_b @ A5 ) @ ( refine1198353289_RES_b @ B5 ) ) ) ) ) ).
% sup_nres.cases
thf(fact_207_less__eq__nres_Osimps_I1_J,axiom,
! [Uu2: refine432681134nres_b] : ( ord_le527798542nres_b @ Uu2 @ refine464223678AILi_b ) ).
% less_eq_nres.simps(1)
thf(fact_208_less__eq__nres_Osimps_I1_J,axiom,
! [Uu2: refine424419629nres_a] : ( ord_le519537037nres_a @ Uu2 @ refine464223677AILi_a ) ).
% less_eq_nres.simps(1)
thf(fact_209_nres_Odistinct_I1_J,axiom,
! [X2: set_a] :
( refine464223677AILi_a
!= ( refine1198353288_RES_a @ X2 ) ) ).
% nres.distinct(1)
thf(fact_210_nres_Odistinct_I1_J,axiom,
! [X2: set_b] :
( refine464223678AILi_b
!= ( refine1198353289_RES_b @ X2 ) ) ).
% nres.distinct(1)
thf(fact_211_nres_Oinduct,axiom,
! [P: refine424419629nres_a > $o,Nres: refine424419629nres_a] :
( ( P @ refine464223677AILi_a )
=> ( ! [X3: set_a] : ( P @ ( refine1198353288_RES_a @ X3 ) )
=> ( P @ Nres ) ) ) ).
% nres.induct
thf(fact_212_nres_Oinduct,axiom,
! [P: refine432681134nres_b > $o,Nres: refine432681134nres_b] :
( ( P @ refine464223678AILi_b )
=> ( ! [X3: set_b] : ( P @ ( refine1198353289_RES_b @ X3 ) )
=> ( P @ Nres ) ) ) ).
% nres.induct
thf(fact_213_nres_Oexhaust,axiom,
! [Y3: refine424419629nres_a] :
( ( Y3 != refine464223677AILi_a )
=> ~ ! [X22: set_a] :
( Y3
!= ( refine1198353288_RES_a @ X22 ) ) ) ).
% nres.exhaust
thf(fact_214_nres_Oexhaust,axiom,
! [Y3: refine432681134nres_b] :
( ( Y3 != refine464223678AILi_b )
=> ~ ! [X22: set_b] :
( Y3
!= ( refine1198353289_RES_b @ X22 ) ) ) ).
% nres.exhaust
thf(fact_215_sup__nres_Oinduct,axiom,
! [P: refine424419629nres_a > refine424419629nres_a > $o,A0: refine424419629nres_a,A1: refine424419629nres_a] :
( ! [Uu: refine424419629nres_a] : ( P @ Uu @ refine464223677AILi_a )
=> ( ! [V: set_a] : ( P @ refine464223677AILi_a @ ( refine1198353288_RES_a @ V ) )
=> ( ! [A5: set_a,B5: set_a] : ( P @ ( refine1198353288_RES_a @ A5 ) @ ( refine1198353288_RES_a @ B5 ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% sup_nres.induct
thf(fact_216_sup__nres_Oinduct,axiom,
! [P: refine432681134nres_b > refine432681134nres_b > $o,A0: refine432681134nres_b,A1: refine432681134nres_b] :
( ! [Uu: refine432681134nres_b] : ( P @ Uu @ refine464223678AILi_b )
=> ( ! [V: set_b] : ( P @ refine464223678AILi_b @ ( refine1198353289_RES_b @ V ) )
=> ( ! [A5: set_b,B5: set_b] : ( P @ ( refine1198353289_RES_b @ A5 ) @ ( refine1198353289_RES_b @ B5 ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% sup_nres.induct
thf(fact_217_less__nres_Oinduct,axiom,
! [P: refine424419629nres_a > refine424419629nres_a > $o,A0: refine424419629nres_a,A1: refine424419629nres_a] :
( ! [X_1: refine424419629nres_a] : ( P @ refine464223677AILi_a @ X_1 )
=> ( ! [Uv: set_a] : ( P @ ( refine1198353288_RES_a @ Uv ) @ refine464223677AILi_a )
=> ( ! [A5: set_a,B5: set_a] : ( P @ ( refine1198353288_RES_a @ A5 ) @ ( refine1198353288_RES_a @ B5 ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% less_nres.induct
thf(fact_218_less__nres_Oinduct,axiom,
! [P: refine432681134nres_b > refine432681134nres_b > $o,A0: refine432681134nres_b,A1: refine432681134nres_b] :
( ! [X_1: refine432681134nres_b] : ( P @ refine464223678AILi_b @ X_1 )
=> ( ! [Uv: set_b] : ( P @ ( refine1198353289_RES_b @ Uv ) @ refine464223678AILi_b )
=> ( ! [A5: set_b,B5: set_b] : ( P @ ( refine1198353289_RES_b @ A5 ) @ ( refine1198353289_RES_b @ B5 ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% less_nres.induct
thf(fact_219_less__eq__nres_Oinduct,axiom,
! [P: refine424419629nres_a > refine424419629nres_a > $o,A0: refine424419629nres_a,A1: refine424419629nres_a] :
( ! [Uu: refine424419629nres_a] : ( P @ Uu @ refine464223677AILi_a )
=> ( ! [A5: set_a,B5: set_a] : ( P @ ( refine1198353288_RES_a @ A5 ) @ ( refine1198353288_RES_a @ B5 ) )
=> ( ! [Uv: set_a] : ( P @ refine464223677AILi_a @ ( refine1198353288_RES_a @ Uv ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% less_eq_nres.induct
thf(fact_220_less__eq__nres_Oinduct,axiom,
! [P: refine432681134nres_b > refine432681134nres_b > $o,A0: refine432681134nres_b,A1: refine432681134nres_b] :
( ! [Uu: refine432681134nres_b] : ( P @ Uu @ refine464223678AILi_b )
=> ( ! [A5: set_b,B5: set_b] : ( P @ ( refine1198353289_RES_b @ A5 ) @ ( refine1198353289_RES_b @ B5 ) )
=> ( ! [Uv: set_b] : ( P @ refine464223678AILi_b @ ( refine1198353289_RES_b @ Uv ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% less_eq_nres.induct
thf(fact_221_less__eq__nres_Osimps_I3_J,axiom,
! [Uv2: set_a] :
~ ( ord_le519537037nres_a @ refine464223677AILi_a @ ( refine1198353288_RES_a @ Uv2 ) ) ).
% less_eq_nres.simps(3)
thf(fact_222_less__eq__nres_Osimps_I3_J,axiom,
! [Uv2: set_b] :
~ ( ord_le527798542nres_b @ refine464223678AILi_b @ ( refine1198353289_RES_b @ Uv2 ) ) ).
% less_eq_nres.simps(3)
thf(fact_223_less__eq__nres_Oelims_I3_J,axiom,
! [X4: refine424419629nres_a,Xa: refine424419629nres_a] :
( ~ ( ord_le519537037nres_a @ X4 @ Xa )
=> ( ! [A5: set_a] :
( ( X4
= ( refine1198353288_RES_a @ A5 ) )
=> ! [B5: set_a] :
( ( Xa
= ( refine1198353288_RES_a @ B5 ) )
=> ( ord_less_eq_set_a @ A5 @ B5 ) ) )
=> ~ ( ( X4 = refine464223677AILi_a )
=> ! [Uv: set_a] :
( Xa
!= ( refine1198353288_RES_a @ Uv ) ) ) ) ) ).
% less_eq_nres.elims(3)
thf(fact_224_less__eq__nres_Oelims_I3_J,axiom,
! [X4: refine432681134nres_b,Xa: refine432681134nres_b] :
( ~ ( ord_le527798542nres_b @ X4 @ Xa )
=> ( ! [A5: set_b] :
( ( X4
= ( refine1198353289_RES_b @ A5 ) )
=> ! [B5: set_b] :
( ( Xa
= ( refine1198353289_RES_b @ B5 ) )
=> ( ord_less_eq_set_b @ A5 @ B5 ) ) )
=> ~ ( ( X4 = refine464223678AILi_b )
=> ! [Uv: set_b] :
( Xa
!= ( refine1198353289_RES_b @ Uv ) ) ) ) ) ).
% less_eq_nres.elims(3)
thf(fact_225_return__refine__prop__return,axiom,
! [M: refine432681134nres_b,X4: b,R: set_Product_prod_b_b] :
( ( refine412683990fail_b @ M )
=> ( ( ord_le527798542nres_b @ ( refine2063221605TURN_b @ X4 ) @ ( refine122110613un_b_b @ R @ M ) )
=> ~ ! [X9: b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ X9 ) @ R )
=> ~ ( ord_le527798542nres_b @ ( refine2063221605TURN_b @ X9 ) @ M ) ) ) ) ).
% return_refine_prop_return
thf(fact_226_return__refine__prop__return,axiom,
! [M: refine424419629nres_a,X4: b,R: set_Product_prod_b_a] :
( ( refine412683989fail_a @ M )
=> ( ( ord_le527798542nres_b @ ( refine2063221605TURN_b @ X4 ) @ ( refine122110612un_b_a @ R @ M ) )
=> ~ ! [X9: a] :
( ( member1277678991od_b_a @ ( product_Pair_b_a @ X4 @ X9 ) @ R )
=> ~ ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X9 ) @ M ) ) ) ) ).
% return_refine_prop_return
thf(fact_227_return__refine__prop__return,axiom,
! [M: refine432681134nres_b,X4: a,R: set_Product_prod_a_b] :
( ( refine412683990fail_b @ M )
=> ( ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X4 ) @ ( refine1441824854un_a_b @ R @ M ) )
=> ~ ! [X9: b] :
( ( member458171089od_a_b @ ( product_Pair_a_b @ X4 @ X9 ) @ R )
=> ~ ( ord_le527798542nres_b @ ( refine2063221605TURN_b @ X9 ) @ M ) ) ) ) ).
% return_refine_prop_return
thf(fact_228_return__refine__prop__return,axiom,
! [M: refine424419629nres_a,X4: a,R: set_Product_prod_a_a] :
( ( refine412683989fail_a @ M )
=> ( ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X4 ) @ ( refine1441824853un_a_a @ R @ M ) )
=> ~ ! [X9: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ X9 ) @ R )
=> ~ ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X9 ) @ M ) ) ) ) ).
% return_refine_prop_return
thf(fact_229_ret__le__down__conv,axiom,
! [M: refine432681134nres_b,C: b,R: set_Product_prod_b_b] :
( ( refine412683990fail_b @ M )
=> ( ( ord_le527798542nres_b @ ( refine2063221605TURN_b @ C ) @ ( refine122110613un_b_b @ R @ M ) )
= ( ? [A2: b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ C @ A2 ) @ R )
& ( ord_le527798542nres_b @ ( refine2063221605TURN_b @ A2 ) @ M ) ) ) ) ) ).
% ret_le_down_conv
thf(fact_230_ret__le__down__conv,axiom,
! [M: refine424419629nres_a,C: b,R: set_Product_prod_b_a] :
( ( refine412683989fail_a @ M )
=> ( ( ord_le527798542nres_b @ ( refine2063221605TURN_b @ C ) @ ( refine122110612un_b_a @ R @ M ) )
= ( ? [A2: a] :
( ( member1277678991od_b_a @ ( product_Pair_b_a @ C @ A2 ) @ R )
& ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ A2 ) @ M ) ) ) ) ) ).
% ret_le_down_conv
thf(fact_231_ret__le__down__conv,axiom,
! [M: refine432681134nres_b,C: a,R: set_Product_prod_a_b] :
( ( refine412683990fail_b @ M )
=> ( ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ C ) @ ( refine1441824854un_a_b @ R @ M ) )
= ( ? [A2: b] :
( ( member458171089od_a_b @ ( product_Pair_a_b @ C @ A2 ) @ R )
& ( ord_le527798542nres_b @ ( refine2063221605TURN_b @ A2 ) @ M ) ) ) ) ) ).
% ret_le_down_conv
thf(fact_232_ret__le__down__conv,axiom,
! [M: refine424419629nres_a,C: a,R: set_Product_prod_a_a] :
( ( refine412683989fail_a @ M )
=> ( ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ C ) @ ( refine1441824853un_a_a @ R @ M ) )
= ( ? [A2: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ C @ A2 ) @ R )
& ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ A2 ) @ M ) ) ) ) ) ).
% ret_le_down_conv
thf(fact_233_nres__order__simps_I20_J,axiom,
! [X4: b,Y3: b] :
( ( ord_le527798542nres_b @ ( refine2063221605TURN_b @ X4 ) @ ( refine2063221605TURN_b @ Y3 ) )
= ( X4 = Y3 ) ) ).
% nres_order_simps(20)
thf(fact_234_nres__order__simps_I20_J,axiom,
! [X4: a,Y3: a] :
( ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X4 ) @ ( refine2063221604TURN_a @ Y3 ) )
= ( X4 = Y3 ) ) ).
% nres_order_simps(20)
thf(fact_235_nres__monad1,axiom,
! [X4: a,F: a > refine432681134nres_b] :
( ( refine436832839nd_a_b @ ( refine2063221604TURN_a @ X4 ) @ F )
= ( F @ X4 ) ) ).
% nres_monad1
thf(fact_236_nres__order__simps_I21_J,axiom,
! [X4: a,Y: set_a] :
( ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X4 ) @ ( refine1198353288_RES_a @ Y ) )
= ( member_a @ X4 @ Y ) ) ).
% nres_order_simps(21)
thf(fact_237_nres__order__simps_I21_J,axiom,
! [X4: b,Y: set_b] :
( ( ord_le527798542nres_b @ ( refine2063221605TURN_b @ X4 ) @ ( refine1198353289_RES_b @ Y ) )
= ( member_b @ X4 @ Y ) ) ).
% nres_order_simps(21)
thf(fact_238_SPEC__eq__is__RETURN_I1_J,axiom,
! [X4: a] :
( ( refine1198353288_RES_a
@ ( collect_a
@ ( ^ [Y5: a,Z: a] : ( Y5 = Z )
@ X4 ) ) )
= ( refine2063221604TURN_a @ X4 ) ) ).
% SPEC_eq_is_RETURN(1)
thf(fact_239_SPEC__eq__is__RETURN_I1_J,axiom,
! [X4: b] :
( ( refine1198353289_RES_b
@ ( collect_b
@ ( ^ [Y5: b,Z: b] : ( Y5 = Z )
@ X4 ) ) )
= ( refine2063221605TURN_b @ X4 ) ) ).
% SPEC_eq_is_RETURN(1)
thf(fact_240_RETURN__rule,axiom,
! [Phi: a > $o,X4: a] :
( ( Phi @ X4 )
=> ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X4 ) @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) ) ) ).
% RETURN_rule
thf(fact_241_RETURN__rule,axiom,
! [Phi: b > $o,X4: b] :
( ( Phi @ X4 )
=> ( ord_le527798542nres_b @ ( refine2063221605TURN_b @ X4 ) @ ( refine1198353289_RES_b @ ( collect_b @ Phi ) ) ) ) ).
% RETURN_rule
thf(fact_242_lhs__step__RES,axiom,
! [X: set_a,M: refine424419629nres_a] :
( ! [X3: a] :
( ( member_a @ X3 @ X )
=> ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X3 ) @ M ) )
=> ( ord_le519537037nres_a @ ( refine1198353288_RES_a @ X ) @ M ) ) ).
% lhs_step_RES
thf(fact_243_lhs__step__RES,axiom,
! [X: set_b,M: refine432681134nres_b] :
( ! [X3: b] :
( ( member_b @ X3 @ X )
=> ( ord_le527798542nres_b @ ( refine2063221605TURN_b @ X3 ) @ M ) )
=> ( ord_le527798542nres_b @ ( refine1198353289_RES_b @ X ) @ M ) ) ).
% lhs_step_RES
thf(fact_244_RETURN__to__SPEC__rule,axiom,
! [M: refine424419629nres_a,V2: a] :
( ( ord_le519537037nres_a @ M
@ ( refine1198353288_RES_a
@ ( collect_a
@ ( ^ [Y5: a,Z: a] : ( Y5 = Z )
@ V2 ) ) ) )
=> ( ord_le519537037nres_a @ M @ ( refine2063221604TURN_a @ V2 ) ) ) ).
% RETURN_to_SPEC_rule
thf(fact_245_RETURN__to__SPEC__rule,axiom,
! [M: refine432681134nres_b,V2: b] :
( ( ord_le527798542nres_b @ M
@ ( refine1198353289_RES_b
@ ( collect_b
@ ( ^ [Y5: b,Z: b] : ( Y5 = Z )
@ V2 ) ) ) )
=> ( ord_le527798542nres_b @ M @ ( refine2063221605TURN_b @ V2 ) ) ) ).
% RETURN_to_SPEC_rule
thf(fact_246_Refine__Basic__Mirabelle__kwjuvthmas_Obind__mono_I1_J,axiom,
! [M4: refine432681134nres_b,M3: refine432681134nres_b,F: b > refine432681134nres_b,F2: b > refine432681134nres_b] :
( ( ord_le527798542nres_b @ M4 @ M3 )
=> ( ! [X3: b] :
( ( ord_le527798542nres_b @ ( refine2063221605TURN_b @ X3 ) @ M4 )
=> ( ord_le527798542nres_b @ ( F @ X3 ) @ ( F2 @ X3 ) ) )
=> ( ord_le527798542nres_b @ ( refine1264602246nd_b_b @ M4 @ F ) @ ( refine1264602246nd_b_b @ M3 @ F2 ) ) ) ) ).
% Refine_Basic_Mirabelle_kwjuvthmas.bind_mono(1)
thf(fact_247_Refine__Basic__Mirabelle__kwjuvthmas_Obind__mono_I1_J,axiom,
! [M4: refine432681134nres_b,M3: refine432681134nres_b,F: b > refine424419629nres_a,F2: b > refine424419629nres_a] :
( ( ord_le527798542nres_b @ M4 @ M3 )
=> ( ! [X3: b] :
( ( ord_le527798542nres_b @ ( refine2063221605TURN_b @ X3 ) @ M4 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F2 @ X3 ) ) )
=> ( ord_le519537037nres_a @ ( refine1264602245nd_b_a @ M4 @ F ) @ ( refine1264602245nd_b_a @ M3 @ F2 ) ) ) ) ).
% Refine_Basic_Mirabelle_kwjuvthmas.bind_mono(1)
thf(fact_248_Refine__Basic__Mirabelle__kwjuvthmas_Obind__mono_I1_J,axiom,
! [M4: refine424419629nres_a,M3: refine424419629nres_a,F: a > refine424419629nres_a,F2: a > refine424419629nres_a] :
( ( ord_le519537037nres_a @ M4 @ M3 )
=> ( ! [X3: a] :
( ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X3 ) @ M4 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F2 @ X3 ) ) )
=> ( ord_le519537037nres_a @ ( refine436832838nd_a_a @ M4 @ F ) @ ( refine436832838nd_a_a @ M3 @ F2 ) ) ) ) ).
% Refine_Basic_Mirabelle_kwjuvthmas.bind_mono(1)
thf(fact_249_Refine__Basic__Mirabelle__kwjuvthmas_Obind__mono_I1_J,axiom,
! [M4: refine424419629nres_a,M3: refine424419629nres_a,F: a > refine432681134nres_b,F2: a > refine432681134nres_b] :
( ( ord_le519537037nres_a @ M4 @ M3 )
=> ( ! [X3: a] :
( ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X3 ) @ M4 )
=> ( ord_le527798542nres_b @ ( F @ X3 ) @ ( F2 @ X3 ) ) )
=> ( ord_le527798542nres_b @ ( refine436832839nd_a_b @ M4 @ F ) @ ( refine436832839nd_a_b @ M3 @ F2 ) ) ) ) ).
% Refine_Basic_Mirabelle_kwjuvthmas.bind_mono(1)
thf(fact_250_bind__cong,axiom,
! [M: refine424419629nres_a,M7: refine424419629nres_a,F: a > refine432681134nres_b,F2: a > refine432681134nres_b] :
( ( M = M7 )
=> ( ! [X3: a] :
( ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X3 ) @ M7 )
=> ( ( F @ X3 )
= ( F2 @ X3 ) ) )
=> ( ( refine436832839nd_a_b @ M @ F )
= ( refine436832839nd_a_b @ M7 @ F2 ) ) ) ) ).
% bind_cong
thf(fact_251_inres__def,axiom,
( refine1001002028nres_b
= ( ^ [S4: refine432681134nres_b,X5: b] : ( ord_le527798542nres_b @ ( refine2063221605TURN_b @ X5 ) @ S4 ) ) ) ).
% inres_def
thf(fact_252_inres__def,axiom,
( refine1001002027nres_a
= ( ^ [S4: refine424419629nres_a,X5: a] : ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X5 ) @ S4 ) ) ) ).
% inres_def
thf(fact_253_RETURN__ref__SPECD,axiom,
! [C: b,R: set_Product_prod_b_a,Phi: a > $o] :
( ( ord_le527798542nres_b @ ( refine2063221605TURN_b @ C ) @ ( refine122110612un_b_a @ R @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) ) )
=> ~ ! [A5: a] :
( ( member1277678991od_b_a @ ( product_Pair_b_a @ C @ A5 ) @ R )
=> ~ ( Phi @ A5 ) ) ) ).
% RETURN_ref_SPECD
thf(fact_254_RETURN__ref__SPECD,axiom,
! [C: a,R: set_Product_prod_a_a,Phi: a > $o] :
( ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ C ) @ ( refine1441824853un_a_a @ R @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) ) )
=> ~ ! [A5: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ C @ A5 ) @ R )
=> ~ ( Phi @ A5 ) ) ) ).
% RETURN_ref_SPECD
thf(fact_255_RETURN__ref__SPECD,axiom,
! [C: b,R: set_Product_prod_b_b,Phi: b > $o] :
( ( ord_le527798542nres_b @ ( refine2063221605TURN_b @ C ) @ ( refine122110613un_b_b @ R @ ( refine1198353289_RES_b @ ( collect_b @ Phi ) ) ) )
=> ~ ! [A5: b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ C @ A5 ) @ R )
=> ~ ( Phi @ A5 ) ) ) ).
% RETURN_ref_SPECD
thf(fact_256_RETURN__ref__SPECD,axiom,
! [C: a,R: set_Product_prod_a_b,Phi: b > $o] :
( ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ C ) @ ( refine1441824854un_a_b @ R @ ( refine1198353289_RES_b @ ( collect_b @ Phi ) ) ) )
=> ~ ! [A5: b] :
( ( member458171089od_a_b @ ( product_Pair_a_b @ C @ A5 ) @ R )
=> ~ ( Phi @ A5 ) ) ) ).
% RETURN_ref_SPECD
thf(fact_257_RETURN__SPEC__refine,axiom,
! [X4: b,R: set_Product_prod_b_a,Phi: a > $o] :
( ? [X11: a] :
( ( member1277678991od_b_a @ ( product_Pair_b_a @ X4 @ X11 ) @ R )
& ( Phi @ X11 ) )
=> ( ord_le527798542nres_b @ ( refine2063221605TURN_b @ X4 ) @ ( refine122110612un_b_a @ R @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) ) ) ) ).
% RETURN_SPEC_refine
thf(fact_258_RETURN__SPEC__refine,axiom,
! [X4: a,R: set_Product_prod_a_a,Phi: a > $o] :
( ? [X11: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ X11 ) @ R )
& ( Phi @ X11 ) )
=> ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X4 ) @ ( refine1441824853un_a_a @ R @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) ) ) ) ).
% RETURN_SPEC_refine
thf(fact_259_RETURN__SPEC__refine,axiom,
! [X4: b,R: set_Product_prod_b_b,Phi: b > $o] :
( ? [X11: b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ X11 ) @ R )
& ( Phi @ X11 ) )
=> ( ord_le527798542nres_b @ ( refine2063221605TURN_b @ X4 ) @ ( refine122110613un_b_b @ R @ ( refine1198353289_RES_b @ ( collect_b @ Phi ) ) ) ) ) ).
% RETURN_SPEC_refine
thf(fact_260_RETURN__SPEC__refine,axiom,
! [X4: a,R: set_Product_prod_a_b,Phi: b > $o] :
( ? [X11: b] :
( ( member458171089od_a_b @ ( product_Pair_a_b @ X4 @ X11 ) @ R )
& ( Phi @ X11 ) )
=> ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X4 ) @ ( refine1441824854un_a_b @ R @ ( refine1198353289_RES_b @ ( collect_b @ Phi ) ) ) ) ) ).
% RETURN_SPEC_refine
thf(fact_261_abs__trans__additional_I1_J,axiom,
! [A3: refine432681134nres_b,B3: refine432681134nres_b,R: set_Product_prod_b_b,C2: refine432681134nres_b] :
( ( ord_le527798542nres_b @ A3 @ B3 )
=> ( ( ord_le527798542nres_b @ ( refine1964549110un_b_b @ R @ B3 ) @ C2 )
=> ( ord_le527798542nres_b @ ( refine1964549110un_b_b @ R @ A3 ) @ C2 ) ) ) ).
% abs_trans_additional(1)
thf(fact_262_abs__trans__additional_I1_J,axiom,
! [A3: refine432681134nres_b,B3: refine432681134nres_b,R: set_Product_prod_b_a,C2: refine424419629nres_a] :
( ( ord_le527798542nres_b @ A3 @ B3 )
=> ( ( ord_le519537037nres_a @ ( refine1964549109un_b_a @ R @ B3 ) @ C2 )
=> ( ord_le519537037nres_a @ ( refine1964549109un_b_a @ R @ A3 ) @ C2 ) ) ) ).
% abs_trans_additional(1)
thf(fact_263_abs__trans__additional_I1_J,axiom,
! [A3: refine424419629nres_a,B3: refine424419629nres_a,R: set_Product_prod_a_b,C2: refine432681134nres_b] :
( ( ord_le519537037nres_a @ A3 @ B3 )
=> ( ( ord_le527798542nres_b @ ( refine1136779703un_a_b @ R @ B3 ) @ C2 )
=> ( ord_le527798542nres_b @ ( refine1136779703un_a_b @ R @ A3 ) @ C2 ) ) ) ).
% abs_trans_additional(1)
thf(fact_264_abs__trans__additional_I1_J,axiom,
! [A3: refine424419629nres_a,B3: refine424419629nres_a,R: set_Product_prod_a_a,C2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A3 @ B3 )
=> ( ( ord_le519537037nres_a @ ( refine1136779702un_a_a @ R @ B3 ) @ C2 )
=> ( ord_le519537037nres_a @ ( refine1136779702un_a_a @ R @ A3 ) @ C2 ) ) ) ).
% abs_trans_additional(1)
thf(fact_265_abs__fun__simps_I3_J,axiom,
! [X: set_a,R: set_Pr1628433942t_unit] :
( ~ ( ord_less_eq_set_a @ X @ ( domain799550107t_unit @ R ) )
=> ( ( refine341651653t_unit @ R @ ( refine1198353288_RES_a @ X ) )
= top_to177290092t_unit ) ) ).
% abs_fun_simps(3)
thf(fact_266_abs__fun__simps_I3_J,axiom,
! [X: set_b,R: set_Pr1707992533t_unit] :
( ~ ( ord_less_eq_set_b @ X @ ( domain918802906t_unit @ R ) )
=> ( ( refine460904452t_unit @ R @ ( refine1198353289_RES_b @ X ) )
= top_to177290092t_unit ) ) ).
% abs_fun_simps(3)
thf(fact_267_abs__fun__simps_I2_J,axiom,
! [X: set_a,R: set_Product_prod_a_a] :
( ( ord_less_eq_set_a @ X @ ( domain_a_a @ R ) )
=> ( ( refine1136779702un_a_a @ R @ ( refine1198353288_RES_a @ X ) )
= ( refine1198353288_RES_a @ ( image_a_a @ R @ X ) ) ) ) ).
% abs_fun_simps(2)
thf(fact_268_abs__fun__simps_I2_J,axiom,
! [X: set_a,R: set_Product_prod_a_b] :
( ( ord_less_eq_set_a @ X @ ( domain_a_b @ R ) )
=> ( ( refine1136779703un_a_b @ R @ ( refine1198353288_RES_a @ X ) )
= ( refine1198353289_RES_b @ ( image_a_b @ R @ X ) ) ) ) ).
% abs_fun_simps(2)
thf(fact_269_abs__fun__simps_I2_J,axiom,
! [X: set_b,R: set_Product_prod_b_a] :
( ( ord_less_eq_set_b @ X @ ( domain_b_a @ R ) )
=> ( ( refine1964549109un_b_a @ R @ ( refine1198353289_RES_b @ X ) )
= ( refine1198353288_RES_a @ ( image_b_a @ R @ X ) ) ) ) ).
% abs_fun_simps(2)
thf(fact_270_abs__fun__simps_I2_J,axiom,
! [X: set_b,R: set_Product_prod_b_b] :
( ( ord_less_eq_set_b @ X @ ( domain_b_b @ R ) )
=> ( ( refine1964549110un_b_b @ R @ ( refine1198353289_RES_b @ X ) )
= ( refine1198353289_RES_b @ ( image_b_b @ R @ X ) ) ) ) ).
% abs_fun_simps(2)
thf(fact_271_ImageI,axiom,
! [A: a,B: a,R2: set_Product_prod_a_a,A3: set_a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
=> ( ( member_a @ A @ A3 )
=> ( member_a @ B @ ( image_a_a @ R2 @ A3 ) ) ) ) ).
% ImageI
thf(fact_272_ImageI,axiom,
! [A: a,B: b,R2: set_Product_prod_a_b,A3: set_a] :
( ( member458171089od_a_b @ ( product_Pair_a_b @ A @ B ) @ R2 )
=> ( ( member_a @ A @ A3 )
=> ( member_b @ B @ ( image_a_b @ R2 @ A3 ) ) ) ) ).
% ImageI
thf(fact_273_ImageI,axiom,
! [A: b,B: a,R2: set_Product_prod_b_a,A3: set_b] :
( ( member1277678991od_b_a @ ( product_Pair_b_a @ A @ B ) @ R2 )
=> ( ( member_b @ A @ A3 )
=> ( member_a @ B @ ( image_b_a @ R2 @ A3 ) ) ) ) ).
% ImageI
thf(fact_274_ImageI,axiom,
! [A: b,B: b,R2: set_Product_prod_b_b,A3: set_b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 )
=> ( ( member_b @ A @ A3 )
=> ( member_b @ B @ ( image_b_b @ R2 @ A3 ) ) ) ) ).
% ImageI
thf(fact_275_nres__order__simps_I4_J,axiom,
! [M4: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ top_to177290092t_unit @ M4 )
= ( M4 = top_to177290092t_unit ) ) ).
% nres_order_simps(4)
thf(fact_276_nres__order__simps_I4_J,axiom,
! [M4: refine432681134nres_b] :
( ( ord_le527798542nres_b @ top_to240090974nres_b @ M4 )
= ( M4 = top_to240090974nres_b ) ) ).
% nres_order_simps(4)
thf(fact_277_nres__order__simps_I4_J,axiom,
! [M4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ top_to231829469nres_a @ M4 )
= ( M4 = top_to231829469nres_a ) ) ).
% nres_order_simps(4)
thf(fact_278_conc__fun__FAIL,axiom,
! [R: set_Pr451126599t_unit] :
( ( refine944483349t_unit @ R @ top_to177290092t_unit )
= top_to177290092t_unit ) ).
% conc_fun_FAIL
thf(fact_279_conc__fun__fail__iff_I1_J,axiom,
! [R: set_Pr451126599t_unit,S: refine787176636t_unit] :
( ( ( refine944483349t_unit @ R @ S )
= top_to177290092t_unit )
= ( S = top_to177290092t_unit ) ) ).
% conc_fun_fail_iff(1)
thf(fact_280_conc__fun__fail__iff_I2_J,axiom,
! [R: set_Pr451126599t_unit,S: refine787176636t_unit] :
( ( top_to177290092t_unit
= ( refine944483349t_unit @ R @ S ) )
= ( S = top_to177290092t_unit ) ) ).
% conc_fun_fail_iff(2)
thf(fact_281_bind__FAIL,axiom,
! [F: product_unit > refine787176636t_unit] :
( ( refine681446406t_unit @ top_to177290092t_unit @ F )
= top_to177290092t_unit ) ).
% bind_FAIL
thf(fact_282_bind__FAIL,axiom,
! [F: a > refine432681134nres_b] :
( ( refine436832839nd_a_b @ top_to231829469nres_a @ F )
= top_to240090974nres_b ) ).
% bind_FAIL
thf(fact_283_nofail__simps_I1_J,axiom,
~ ( refine579265252t_unit @ top_to177290092t_unit ) ).
% nofail_simps(1)
thf(fact_284_inres__simps_I1_J,axiom,
( ( refine558004794t_unit @ top_to177290092t_unit )
= ( ^ [Uu3: product_unit] : $true ) ) ).
% inres_simps(1)
thf(fact_285_nres__simp__internals_I2_J,axiom,
refine1607584204t_unit = top_to177290092t_unit ).
% nres_simp_internals(2)
thf(fact_286_abs__fun__simps_I1_J,axiom,
! [R: set_Pr451126599t_unit] :
( ( refine838861686t_unit @ R @ top_to177290092t_unit )
= top_to177290092t_unit ) ).
% abs_fun_simps(1)
thf(fact_287_FAIL__refine,axiom,
! [X: refine432681134nres_b,R: set_Pr1707992533t_unit] : ( ord_le527798542nres_b @ X @ ( refine2140306339t_unit @ R @ top_to177290092t_unit ) ) ).
% FAIL_refine
thf(fact_288_FAIL__refine,axiom,
! [X: refine424419629nres_a,R: set_Pr1628433942t_unit] : ( ord_le519537037nres_a @ X @ ( refine2021053540t_unit @ R @ top_to177290092t_unit ) ) ).
% FAIL_refine
thf(fact_289_rev__ImageI,axiom,
! [A: a,A3: set_a,B: a,R2: set_Product_prod_a_a] :
( ( member_a @ A @ A3 )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
=> ( member_a @ B @ ( image_a_a @ R2 @ A3 ) ) ) ) ).
% rev_ImageI
thf(fact_290_rev__ImageI,axiom,
! [A: a,A3: set_a,B: b,R2: set_Product_prod_a_b] :
( ( member_a @ A @ A3 )
=> ( ( member458171089od_a_b @ ( product_Pair_a_b @ A @ B ) @ R2 )
=> ( member_b @ B @ ( image_a_b @ R2 @ A3 ) ) ) ) ).
% rev_ImageI
thf(fact_291_rev__ImageI,axiom,
! [A: b,A3: set_b,B: a,R2: set_Product_prod_b_a] :
( ( member_b @ A @ A3 )
=> ( ( member1277678991od_b_a @ ( product_Pair_b_a @ A @ B ) @ R2 )
=> ( member_a @ B @ ( image_b_a @ R2 @ A3 ) ) ) ) ).
% rev_ImageI
thf(fact_292_rev__ImageI,axiom,
! [A: b,A3: set_b,B: b,R2: set_Product_prod_b_b] :
( ( member_b @ A @ A3 )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 )
=> ( member_b @ B @ ( image_b_b @ R2 @ A3 ) ) ) ) ).
% rev_ImageI
thf(fact_293_ImageE,axiom,
! [B: a,R2: set_Product_prod_a_a,A3: set_a] :
( ( member_a @ B @ ( image_a_a @ R2 @ A3 ) )
=> ~ ! [X3: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ B ) @ R2 )
=> ~ ( member_a @ X3 @ A3 ) ) ) ).
% ImageE
thf(fact_294_ImageE,axiom,
! [B: a,R2: set_Product_prod_b_a,A3: set_b] :
( ( member_a @ B @ ( image_b_a @ R2 @ A3 ) )
=> ~ ! [X3: b] :
( ( member1277678991od_b_a @ ( product_Pair_b_a @ X3 @ B ) @ R2 )
=> ~ ( member_b @ X3 @ A3 ) ) ) ).
% ImageE
thf(fact_295_ImageE,axiom,
! [B: b,R2: set_Product_prod_a_b,A3: set_a] :
( ( member_b @ B @ ( image_a_b @ R2 @ A3 ) )
=> ~ ! [X3: a] :
( ( member458171089od_a_b @ ( product_Pair_a_b @ X3 @ B ) @ R2 )
=> ~ ( member_a @ X3 @ A3 ) ) ) ).
% ImageE
thf(fact_296_ImageE,axiom,
! [B: b,R2: set_Product_prod_b_b,A3: set_b] :
( ( member_b @ B @ ( image_b_b @ R2 @ A3 ) )
=> ~ ! [X3: b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ X3 @ B ) @ R2 )
=> ~ ( member_b @ X3 @ A3 ) ) ) ).
% ImageE
thf(fact_297_nres__inequalities_I3_J,axiom,
! [X4: product_unit] :
( top_to177290092t_unit
!= ( refine1420258419t_unit @ X4 ) ) ).
% nres_inequalities(3)
thf(fact_298_top__nres__def,axiom,
top_to177290092t_unit = refine1607584204t_unit ).
% top_nres_def
thf(fact_299_top__greatest,axiom,
! [A: refine787176636t_unit] : ( ord_le1051254044t_unit @ A @ top_to177290092t_unit ) ).
% top_greatest
thf(fact_300_top__greatest,axiom,
! [A: refine432681134nres_b] : ( ord_le527798542nres_b @ A @ top_to240090974nres_b ) ).
% top_greatest
thf(fact_301_top__greatest,axiom,
! [A: refine424419629nres_a] : ( ord_le519537037nres_a @ A @ top_to231829469nres_a ) ).
% top_greatest
thf(fact_302_top_Oextremum__unique,axiom,
! [A: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ top_to177290092t_unit @ A )
= ( A = top_to177290092t_unit ) ) ).
% top.extremum_unique
thf(fact_303_top_Oextremum__unique,axiom,
! [A: refine432681134nres_b] :
( ( ord_le527798542nres_b @ top_to240090974nres_b @ A )
= ( A = top_to240090974nres_b ) ) ).
% top.extremum_unique
thf(fact_304_top_Oextremum__unique,axiom,
! [A: refine424419629nres_a] :
( ( ord_le519537037nres_a @ top_to231829469nres_a @ A )
= ( A = top_to231829469nres_a ) ) ).
% top.extremum_unique
thf(fact_305_top_Oextremum__uniqueI,axiom,
! [A: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ top_to177290092t_unit @ A )
=> ( A = top_to177290092t_unit ) ) ).
% top.extremum_uniqueI
thf(fact_306_top_Oextremum__uniqueI,axiom,
! [A: refine432681134nres_b] :
( ( ord_le527798542nres_b @ top_to240090974nres_b @ A )
=> ( A = top_to240090974nres_b ) ) ).
% top.extremum_uniqueI
thf(fact_307_top_Oextremum__uniqueI,axiom,
! [A: refine424419629nres_a] :
( ( ord_le519537037nres_a @ top_to231829469nres_a @ A )
=> ( A = top_to231829469nres_a ) ) ).
% top.extremum_uniqueI
thf(fact_308_nres__order__simps_I3_J,axiom,
! [M4: refine787176636t_unit] : ( ord_le1051254044t_unit @ M4 @ top_to177290092t_unit ) ).
% nres_order_simps(3)
thf(fact_309_nres__order__simps_I3_J,axiom,
! [M4: refine432681134nres_b] : ( ord_le527798542nres_b @ M4 @ top_to240090974nres_b ) ).
% nres_order_simps(3)
thf(fact_310_nres__order__simps_I3_J,axiom,
! [M4: refine424419629nres_a] : ( ord_le519537037nres_a @ M4 @ top_to231829469nres_a ) ).
% nres_order_simps(3)
thf(fact_311_nres__inequalities_I1_J,axiom,
! [X: set_Product_unit] :
( top_to177290092t_unit
!= ( refine1777164439t_unit @ X ) ) ).
% nres_inequalities(1)
thf(fact_312_nres__inequalities_I1_J,axiom,
! [X: set_a] :
( top_to231829469nres_a
!= ( refine1198353288_RES_a @ X ) ) ).
% nres_inequalities(1)
thf(fact_313_nres__inequalities_I1_J,axiom,
! [X: set_b] :
( top_to240090974nres_b
!= ( refine1198353289_RES_b @ X ) ) ).
% nres_inequalities(1)
thf(fact_314_nres__cases,axiom,
! [M4: refine787176636t_unit] :
( ( M4 != top_to177290092t_unit )
=> ~ ! [X12: set_Product_unit] :
( M4
!= ( refine1777164439t_unit @ X12 ) ) ) ).
% nres_cases
thf(fact_315_nres__cases,axiom,
! [M4: refine424419629nres_a] :
( ( M4 != top_to231829469nres_a )
=> ~ ! [X12: set_a] :
( M4
!= ( refine1198353288_RES_a @ X12 ) ) ) ).
% nres_cases
thf(fact_316_nres__cases,axiom,
! [M4: refine432681134nres_b] :
( ( M4 != top_to240090974nres_b )
=> ~ ! [X12: set_b] :
( M4
!= ( refine1198353289_RES_b @ X12 ) ) ) ).
% nres_cases
thf(fact_317_not__nofail__iff,axiom,
! [S: refine787176636t_unit] :
( ( ~ ( refine579265252t_unit @ S ) )
= ( S = top_to177290092t_unit ) ) ).
% not_nofail_iff
thf(fact_318_nofail__def,axiom,
( refine579265252t_unit
= ( ^ [S4: refine787176636t_unit] : ( S4 != top_to177290092t_unit ) ) ) ).
% nofail_def
thf(fact_319_intro__nofail_I2_J,axiom,
! [S: refine787176636t_unit] :
( ( top_to177290092t_unit != S )
= ( refine579265252t_unit @ S ) ) ).
% intro_nofail(2)
thf(fact_320_ibind__strict_I2_J,axiom,
! [F: product_unit > refine787176636t_unit] :
( ( refine681446406t_unit @ top_to177290092t_unit @ F )
= top_to177290092t_unit ) ).
% ibind_strict(2)
thf(fact_321_meta__le__everything__if__top,axiom,
! [M: refine787176636t_unit,X4: refine787176636t_unit] :
( ( M = top_to177290092t_unit )
=> ( ord_le1051254044t_unit @ X4 @ M ) ) ).
% meta_le_everything_if_top
thf(fact_322_meta__le__everything__if__top,axiom,
! [M: refine432681134nres_b,X4: refine432681134nres_b] :
( ( M = top_to240090974nres_b )
=> ( ord_le527798542nres_b @ X4 @ M ) ) ).
% meta_le_everything_if_top
thf(fact_323_meta__le__everything__if__top,axiom,
! [M: refine424419629nres_a,X4: refine424419629nres_a] :
( ( M = top_to231829469nres_a )
=> ( ord_le519537037nres_a @ X4 @ M ) ) ).
% meta_le_everything_if_top
thf(fact_324_pw__flat__ge__iff,axiom,
! [S: refine787176636t_unit,S3: refine787176636t_unit] :
( ( partia1658438072t_unit @ top_to177290092t_unit @ S @ S3 )
= ( ( refine579265252t_unit @ S )
=> ( ( refine579265252t_unit @ S3 )
& ! [X5: product_unit] :
( ( refine558004794t_unit @ S @ X5 )
= ( refine558004794t_unit @ S3 @ X5 ) ) ) ) ) ).
% pw_flat_ge_iff
thf(fact_325_Refine__Basic__Mirabelle__kwjuvthmas_Obind__mono_I2_J,axiom,
! [M4: refine787176636t_unit,M3: refine787176636t_unit,F: product_unit > refine787176636t_unit,F2: product_unit > refine787176636t_unit] :
( ( partia1658438072t_unit @ top_to177290092t_unit @ M4 @ M3 )
=> ( ! [X3: product_unit] : ( partia1658438072t_unit @ top_to177290092t_unit @ ( F @ X3 ) @ ( F2 @ X3 ) )
=> ( partia1658438072t_unit @ top_to177290092t_unit @ ( refine681446406t_unit @ M4 @ F ) @ ( refine681446406t_unit @ M3 @ F2 ) ) ) ) ).
% Refine_Basic_Mirabelle_kwjuvthmas.bind_mono(2)
thf(fact_326_Refine__Basic__Mirabelle__kwjuvthmas_Obind__mono_I2_J,axiom,
! [M4: refine424419629nres_a,M3: refine424419629nres_a,F: a > refine432681134nres_b,F2: a > refine432681134nres_b] :
( ( partia906949161nres_a @ top_to231829469nres_a @ M4 @ M3 )
=> ( ! [X3: a] : ( partia915210666nres_b @ top_to240090974nres_b @ ( F @ X3 ) @ ( F2 @ X3 ) )
=> ( partia915210666nres_b @ top_to240090974nres_b @ ( refine436832839nd_a_b @ M4 @ F ) @ ( refine436832839nd_a_b @ M3 @ F2 ) ) ) ) ).
% Refine_Basic_Mirabelle_kwjuvthmas.bind_mono(2)
thf(fact_327_flat__ord__compat_I2_J,axiom,
! [X4: refine787176636t_unit,Y3: refine787176636t_unit] :
( ( partia1658438072t_unit @ top_to177290092t_unit @ X4 @ Y3 )
=> ( ord_le1051254044t_unit @ Y3 @ X4 ) ) ).
% flat_ord_compat(2)
thf(fact_328_flat__ord__compat_I2_J,axiom,
! [X4: refine432681134nres_b,Y3: refine432681134nres_b] :
( ( partia915210666nres_b @ top_to240090974nres_b @ X4 @ Y3 )
=> ( ord_le527798542nres_b @ Y3 @ X4 ) ) ).
% flat_ord_compat(2)
thf(fact_329_flat__ord__compat_I2_J,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( partia906949161nres_a @ top_to231829469nres_a @ X4 @ Y3 )
=> ( ord_le519537037nres_a @ Y3 @ X4 ) ) ).
% flat_ord_compat(2)
thf(fact_330_UNIV__I,axiom,
! [X4: a] : ( member_a @ X4 @ top_top_set_a ) ).
% UNIV_I
thf(fact_331_UNIV__I,axiom,
! [X4: b] : ( member_b @ X4 @ top_top_set_b ) ).
% UNIV_I
thf(fact_332_eq__UNIV__iff,axiom,
! [S: set_a] :
( ( S = top_top_set_a )
= ( ! [X5: a] : ( member_a @ X5 @ S ) ) ) ).
% eq_UNIV_iff
thf(fact_333_eq__UNIV__iff,axiom,
! [S: set_b] :
( ( S = top_top_set_b )
= ( ! [X5: b] : ( member_b @ X5 @ S ) ) ) ).
% eq_UNIV_iff
thf(fact_334_UNIV__witness,axiom,
? [X3: a] : ( member_a @ X3 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_335_UNIV__witness,axiom,
? [X3: b] : ( member_b @ X3 @ top_top_set_b ) ).
% UNIV_witness
thf(fact_336_top__set__def,axiom,
( top_top_set_a
= ( collect_a @ top_top_a_o ) ) ).
% top_set_def
thf(fact_337_top__set__def,axiom,
( top_top_set_b
= ( collect_b @ top_top_b_o ) ) ).
% top_set_def
thf(fact_338_UNIV__eq__I,axiom,
! [A3: set_a] :
( ! [X3: a] : ( member_a @ X3 @ A3 )
=> ( top_top_set_a = A3 ) ) ).
% UNIV_eq_I
thf(fact_339_UNIV__eq__I,axiom,
! [A3: set_b] :
( ! [X3: b] : ( member_b @ X3 @ A3 )
=> ( top_top_set_b = A3 ) ) ).
% UNIV_eq_I
thf(fact_340_top__empty__eq,axiom,
( top_top_a_o
= ( ^ [X5: a] : ( member_a @ X5 @ top_top_set_a ) ) ) ).
% top_empty_eq
thf(fact_341_top__empty__eq,axiom,
( top_top_b_o
= ( ^ [X5: b] : ( member_b @ X5 @ top_top_set_b ) ) ) ).
% top_empty_eq
thf(fact_342_ASSERT__simps_I2_J,axiom,
( ( refine2039610767ASSERT @ $false )
= top_to177290092t_unit ) ).
% ASSERT_simps(2)
% Helper facts (15)
thf(help_If_2_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X4: set_a,Y3: set_a] :
( ( if_set_a @ $false @ X4 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X4: set_a,Y3: set_a] :
( ( if_set_a @ $true @ X4 @ Y3 )
= X4 ) ).
thf(help_If_2_1_If_001t__Set__Oset_Itf__b_J_T,axiom,
! [X4: set_b,Y3: set_b] :
( ( if_set_b @ $false @ X4 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__b_J_T,axiom,
! [X4: set_b,Y3: set_b] :
( ( if_set_b @ $true @ X4 @ Y3 )
= X4 ) ).
thf(help_If_2_1_If_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_T,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( if_Ref1724547303nres_a @ $false @ X4 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_T,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( if_Ref1724547303nres_a @ $true @ X4 @ Y3 )
= X4 ) ).
thf(help_If_2_1_If_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__b_J_T,axiom,
! [X4: refine432681134nres_b,Y3: refine432681134nres_b] :
( ( if_Ref1732808808nres_b @ $false @ X4 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__b_J_T,axiom,
! [X4: refine432681134nres_b,Y3: refine432681134nres_b] :
( ( if_Ref1732808808nres_b @ $true @ X4 @ Y3 )
= X4 ) ).
thf(help_If_2_1_If_001_062_I_Eo_Mt__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_J_T,axiom,
! [X4: $o > refine424419629nres_a,Y3: $o > refine424419629nres_a] :
( ( if_o_R1705037948nres_a @ $false @ X4 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001_062_I_Eo_Mt__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_J_T,axiom,
! [X4: $o > refine424419629nres_a,Y3: $o > refine424419629nres_a] :
( ( if_o_R1705037948nres_a @ $true @ X4 @ Y3 )
= X4 ) ).
thf(help_If_2_1_If_001_062_I_Eo_Mt__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__b_J_J_T,axiom,
! [X4: $o > refine432681134nres_b,Y3: $o > refine432681134nres_b] :
( ( if_o_R697118589nres_b @ $false @ X4 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001_062_I_Eo_Mt__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__b_J_J_T,axiom,
! [X4: $o > refine432681134nres_b,Y3: $o > refine432681134nres_b] :
( ( if_o_R697118589nres_b @ $true @ X4 @ Y3 )
= X4 ) ).
thf(help_If_3_1_If_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_It__Product____Type__Ounit_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_It__Product____Type__Ounit_J_T,axiom,
! [X4: refine787176636t_unit,Y3: refine787176636t_unit] :
( ( if_Ref1369692790t_unit @ $false @ X4 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_It__Product____Type__Ounit_J_T,axiom,
! [X4: refine787176636t_unit,Y3: refine787176636t_unit] :
( ( if_Ref1369692790t_unit @ $true @ X4 @ Y3 )
= X4 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_le527798542nres_b @ m @ ( refine436832839nd_a_b @ ( refine1198353288_RES_a @ ( collect_a @ phi ) ) @ f ) ).
%------------------------------------------------------------------------------