TPTP Problem File: ITP162^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP162^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Refine_Basic problem prob_1909__3604894_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_1909__3604894_1 [Des21]
% Status : Theorem
% Rating : 0.50 v8.2.0, 0.38 v8.1.0, 0.45 v7.5.0
% Syntax : Number of formulae : 388 ( 158 unt; 31 typ; 0 def)
% Number of atoms : 1039 ( 293 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 2890 ( 43 ~; 2 |; 36 &;2348 @)
% ( 0 <=>; 461 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 7 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 430 ( 430 >; 0 *; 0 +; 0 <<)
% Number of symbols : 31 ( 28 usr; 7 con; 0-4 aty)
% Number of variables : 1098 ( 126 ^; 970 !; 2 ?;1098 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:36:21.134
%------------------------------------------------------------------------------
% Could-be-implicit typings (3)
thf(ty_n_t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J,type,
refine424419629nres_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (28)
thf(sy_c_If_001_062_Itf__a_M_Eo_J,type,
if_a_o: $o > ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_If_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J,type,
if_Ref1724547303nres_a: $o > refine424419629nres_a > refine424419629nres_a > refine424419629nres_a ).
thf(sy_c_If_001t__Set__Oset_Itf__a_J,type,
if_set_a: $o > set_a > set_a > set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_Itf__a_M_Eo_J,type,
inf_inf_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_Eo,type,
inf_inf_o: $o > $o > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J,type,
inf_in262696383nres_a: refine424419629nres_a > refine424419629nres_a > refine424419629nres_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J,type,
bot_bo529555393nres_a: refine424419629nres_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
ord_less_eq_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J,type,
ord_le519537037nres_a: refine424419629nres_a > refine424419629nres_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J,type,
order_1714329108nres_a: ( refine424419629nres_a > $o ) > refine424419629nres_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J,type,
top_to231829469nres_a: refine424419629nres_a ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_ORETURN_001tf__a,type,
refine2063221604TURN_a: a > refine424419629nres_a ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Obind_001tf__a_001tf__a,type,
refine436832838nd_a_a: refine424419629nres_a > ( a > refine424419629nres_a ) > refine424419629nres_a ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Oinres_001tf__a,type,
refine1001002027nres_a: refine424419629nres_a > a > $o ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Onf__inres_001tf__a,type,
refine1312857699nres_a: refine424419629nres_a > a > $o ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Onofail_001tf__a,type,
refine412683989fail_a: refine424419629nres_a > $o ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Onres_OFAILi_001tf__a,type,
refine464223677AILi_a: refine424419629nres_a ).
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_Othe__RES_001tf__a,type,
refine1822134885_RES_a: refine424419629nres_a > set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_A,type,
a2: refine424419629nres_a ).
thf(sy_v_P,type,
p: a > $o ).
thf(sy_v_Q,type,
q: a > $o ).
% Relevant facts (349)
thf(fact_0_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_1_nres_Oinject,axiom,
! [X2: set_a,Y2: set_a] :
( ( ( refine1198353288_RES_a @ X2 )
= ( refine1198353288_RES_a @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nres.inject
thf(fact_2_assms_I2_J,axiom,
ord_le519537037nres_a @ a2 @ ( refine1198353288_RES_a @ ( collect_a @ q ) ) ).
% assms(2)
thf(fact_3_assms_I1_J,axiom,
ord_le519537037nres_a @ a2 @ ( refine1198353288_RES_a @ ( collect_a @ p ) ) ).
% assms(1)
thf(fact_4__092_060open_062A_A_092_060le_062_Ainf_A_ISPEC_AP_J_A_ISPEC_AQ_J_092_060close_062,axiom,
ord_le519537037nres_a @ a2 @ ( inf_in262696383nres_a @ ( refine1198353288_RES_a @ ( collect_a @ p ) ) @ ( refine1198353288_RES_a @ ( collect_a @ q ) ) ) ).
% \<open>A \<le> inf (SPEC P) (SPEC Q)\<close>
thf(fact_5_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_6_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_7_SPEC__iff,axiom,
! [P: refine424419629nres_a,Q: a > $o,R: a > $o] :
( ( ord_le519537037nres_a @ P
@ ( refine1198353288_RES_a
@ ( collect_a
@ ^ [S3: a] :
( ( Q @ S3 )
=> ( R @ S3 ) ) ) ) )
=> ( ( ord_le519537037nres_a @ P
@ ( refine1198353288_RES_a
@ ( collect_a
@ ^ [S3: a] :
( ~ ( Q @ S3 )
=> ~ ( R @ S3 ) ) ) ) )
=> ( ord_le519537037nres_a @ P
@ ( refine1198353288_RES_a
@ ( collect_a
@ ^ [S3: a] :
( ( Q @ S3 )
= ( R @ S3 ) ) ) ) ) ) ) ).
% SPEC_iff
thf(fact_8_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_9_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_10_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_11_lhs__step__If,axiom,
! [B: $o,T: a > $o,M: a > $o,E: a > $o] :
( ( B
=> ( ord_less_eq_a_o @ T @ M ) )
=> ( ( ~ B
=> ( ord_less_eq_a_o @ E @ M ) )
=> ( ord_less_eq_a_o @ ( if_a_o @ B @ T @ E ) @ M ) ) ) ).
% lhs_step_If
thf(fact_12_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_13_use__spec__rule,axiom,
! [M: refine424419629nres_a,Psi: a > $o,Phi: a > $o] :
( ( ord_le519537037nres_a @ M @ ( refine1198353288_RES_a @ ( collect_a @ Psi ) ) )
=> ( ( ord_le519537037nres_a @ M
@ ( refine1198353288_RES_a
@ ( collect_a
@ ^ [S3: a] :
( ( Psi @ S3 )
=> ( Phi @ S3 ) ) ) ) )
=> ( ord_le519537037nres_a @ M @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) ) ) ) ).
% use_spec_rule
thf(fact_14_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_15_order__mono__setup_Orefl,axiom,
! [X4: set_a] : ( ord_less_eq_set_a @ X4 @ X4 ) ).
% order_mono_setup.refl
thf(fact_16_order__mono__setup_Orefl,axiom,
! [X4: a > $o] : ( ord_less_eq_a_o @ X4 @ X4 ) ).
% order_mono_setup.refl
thf(fact_17_order__mono__setup_Orefl,axiom,
! [X4: refine424419629nres_a] : ( ord_le519537037nres_a @ X4 @ X4 ) ).
% order_mono_setup.refl
thf(fact_18_the__RES_Osimps,axiom,
! [X: set_a] :
( ( refine1822134885_RES_a @ ( refine1198353288_RES_a @ X ) )
= X ) ).
% the_RES.simps
thf(fact_19_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_20_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_21_le__funD,axiom,
! [F: a > $o,G: a > $o,X4: a] :
( ( ord_less_eq_a_o @ F @ G )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ X4 ) ) ) ).
% le_funD
thf(fact_22_le__funE,axiom,
! [F: a > $o,G: a > $o,X4: a] :
( ( ord_less_eq_a_o @ F @ G )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ X4 ) ) ) ).
% le_funE
thf(fact_23_le__funI,axiom,
! [F: a > $o,G: a > $o] :
( ! [X3: a] : ( ord_less_eq_o @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq_a_o @ F @ G ) ) ).
% le_funI
thf(fact_24_Refine__Misc_Oif__mono,axiom,
! [B: $o,M1: set_a,M12: set_a,M2: set_a,M22: set_a] :
( ( B
=> ( ord_less_eq_set_a @ M1 @ M12 ) )
=> ( ( ~ B
=> ( ord_less_eq_set_a @ M2 @ M22 ) )
=> ( ord_less_eq_set_a @ ( if_set_a @ B @ M1 @ M2 ) @ ( if_set_a @ B @ M12 @ M22 ) ) ) ) ).
% Refine_Misc.if_mono
thf(fact_25_Refine__Misc_Oif__mono,axiom,
! [B: $o,M1: a > $o,M12: a > $o,M2: a > $o,M22: a > $o] :
( ( B
=> ( ord_less_eq_a_o @ M1 @ M12 ) )
=> ( ( ~ B
=> ( ord_less_eq_a_o @ M2 @ M22 ) )
=> ( ord_less_eq_a_o @ ( if_a_o @ B @ M1 @ M2 ) @ ( if_a_o @ B @ M12 @ M22 ) ) ) ) ).
% Refine_Misc.if_mono
thf(fact_26_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_27_le__fun__def,axiom,
( ord_less_eq_a_o
= ( ^ [F2: a > $o,G2: a > $o] :
! [X5: a] : ( ord_less_eq_o @ ( F2 @ X5 ) @ ( G2 @ X5 ) ) ) ) ).
% le_fun_def
thf(fact_28_order__subst1,axiom,
! [A: refine424419629nres_a,F: set_a > refine424419629nres_a,B: set_a,C: set_a] :
( ( ord_le519537037nres_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le519537037nres_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_29_order__subst1,axiom,
! [A: refine424419629nres_a,F: ( a > $o ) > refine424419629nres_a,B: a > $o,C: a > $o] :
( ( ord_le519537037nres_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a_o @ B @ C )
=> ( ! [X3: a > $o,Y4: a > $o] :
( ( ord_less_eq_a_o @ X3 @ Y4 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le519537037nres_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_30_order__subst1,axiom,
! [A: set_a,F: refine424419629nres_a > set_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_le519537037nres_a @ B @ C )
=> ( ! [X3: refine424419629nres_a,Y4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_31_order__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_32_order__subst1,axiom,
! [A: set_a,F: ( a > $o ) > set_a,B: a > $o,C: a > $o] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a_o @ B @ C )
=> ( ! [X3: a > $o,Y4: a > $o] :
( ( ord_less_eq_a_o @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_33_order__subst1,axiom,
! [A: a > $o,F: refine424419629nres_a > a > $o,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_less_eq_a_o @ A @ ( F @ B ) )
=> ( ( ord_le519537037nres_a @ B @ C )
=> ( ! [X3: refine424419629nres_a,Y4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y4 )
=> ( ord_less_eq_a_o @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a_o @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_34_order__subst1,axiom,
! [A: a > $o,F: set_a > a > $o,B: set_a,C: set_a] :
( ( ord_less_eq_a_o @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_less_eq_a_o @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a_o @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_35_order__subst1,axiom,
! [A: a > $o,F: ( a > $o ) > a > $o,B: a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a_o @ B @ C )
=> ( ! [X3: a > $o,Y4: a > $o] :
( ( ord_less_eq_a_o @ X3 @ Y4 )
=> ( ord_less_eq_a_o @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a_o @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_36_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_37_le__infD2,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
=> ( ord_less_eq_set_a @ A @ C ) ) ).
% le_infD2
thf(fact_38_le__infD2,axiom,
! [A: a > $o,B: a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ A @ ( inf_inf_a_o @ B @ C ) )
=> ( ord_less_eq_a_o @ A @ C ) ) ).
% le_infD2
thf(fact_39_le__infD2,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ ( inf_in262696383nres_a @ B @ C ) )
=> ( ord_le519537037nres_a @ A @ C ) ) ).
% le_infD2
thf(fact_40_le__infD1,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% le_infD1
thf(fact_41_le__infD1,axiom,
! [A: a > $o,B: a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ A @ ( inf_inf_a_o @ B @ C ) )
=> ( ord_less_eq_a_o @ A @ B ) ) ).
% le_infD1
thf(fact_42_le__infD1,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ ( inf_in262696383nres_a @ B @ C ) )
=> ( ord_le519537037nres_a @ A @ B ) ) ).
% le_infD1
thf(fact_43_inf__leI,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ! [X3: set_a] :
( ( ord_less_eq_set_a @ X3 @ A )
=> ( ( ord_less_eq_set_a @ X3 @ B )
=> ( ord_less_eq_set_a @ X3 @ C ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ C ) ) ).
% inf_leI
thf(fact_44_inf__leI,axiom,
! [A: a > $o,B: a > $o,C: a > $o] :
( ! [X3: a > $o] :
( ( ord_less_eq_a_o @ X3 @ A )
=> ( ( ord_less_eq_a_o @ X3 @ B )
=> ( ord_less_eq_a_o @ X3 @ C ) ) )
=> ( ord_less_eq_a_o @ ( inf_inf_a_o @ A @ B ) @ C ) ) ).
% inf_leI
thf(fact_45_inf__leI,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ! [X3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ A )
=> ( ( ord_le519537037nres_a @ X3 @ B )
=> ( ord_le519537037nres_a @ X3 @ C ) ) )
=> ( ord_le519537037nres_a @ ( inf_in262696383nres_a @ A @ B ) @ C ) ) ).
% inf_leI
thf(fact_46_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_47_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_48_order__mono__setup_Omono__if,axiom,
! [T: set_a,T2: set_a,E: set_a,E2: set_a,B: $o] :
( ( ord_less_eq_set_a @ T @ T2 )
=> ( ( ord_less_eq_set_a @ E @ E2 )
=> ( ord_less_eq_set_a @ ( if_set_a @ B @ T @ E ) @ ( if_set_a @ B @ T2 @ E2 ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_49_order__mono__setup_Omono__if,axiom,
! [T: a > $o,T2: a > $o,E: a > $o,E2: a > $o,B: $o] :
( ( ord_less_eq_a_o @ T @ T2 )
=> ( ( ord_less_eq_a_o @ E @ E2 )
=> ( ord_less_eq_a_o @ ( if_a_o @ B @ T @ E ) @ ( if_a_o @ B @ T2 @ E2 ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_50_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_51_dual__order_Oantisym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_52_dual__order_Oantisym,axiom,
! [B: a > $o,A: a > $o] :
( ( ord_less_eq_a_o @ B @ A )
=> ( ( ord_less_eq_a_o @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_53_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_54_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_a,Z: set_a] : Y5 = Z )
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
& ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_55_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: a > $o,Z: a > $o] : Y5 = Z )
= ( ^ [A2: a > $o,B2: a > $o] :
( ( ord_less_eq_a_o @ B2 @ A2 )
& ( ord_less_eq_a_o @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_56_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_57_dual__order_Otrans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_58_dual__order_Otrans,axiom,
! [B: a > $o,A: a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ B @ A )
=> ( ( ord_less_eq_a_o @ C @ B )
=> ( ord_less_eq_a_o @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_59_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_60_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_61_dual__order_Orefl,axiom,
! [A: a > $o] : ( ord_less_eq_a_o @ A @ A ) ).
% dual_order.refl
thf(fact_62_dual__order_Orefl,axiom,
! [A: refine424419629nres_a] : ( ord_le519537037nres_a @ A @ A ) ).
% dual_order.refl
thf(fact_63_order__trans,axiom,
! [X4: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ Z2 )
=> ( ord_less_eq_set_a @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_64_order__trans,axiom,
! [X4: a > $o,Y3: a > $o,Z2: a > $o] :
( ( ord_less_eq_a_o @ X4 @ Y3 )
=> ( ( ord_less_eq_a_o @ Y3 @ Z2 )
=> ( ord_less_eq_a_o @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_65_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_66_order__class_Oorder_Oantisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_67_order__class_Oorder_Oantisym,axiom,
! [A: a > $o,B: a > $o] :
( ( ord_less_eq_a_o @ A @ B )
=> ( ( ord_less_eq_a_o @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_68_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_69_ord__le__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_70_ord__le__eq__trans,axiom,
! [A: a > $o,B: a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_a_o @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_71_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_72_ord__eq__le__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_73_ord__eq__le__trans,axiom,
! [A: a > $o,B: a > $o,C: a > $o] :
( ( A = B )
=> ( ( ord_less_eq_a_o @ B @ C )
=> ( ord_less_eq_a_o @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_74_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_75_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y5: set_a,Z: set_a] : Y5 = Z )
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
& ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_76_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y5: a > $o,Z: a > $o] : Y5 = Z )
= ( ^ [A2: a > $o,B2: a > $o] :
( ( ord_less_eq_a_o @ A2 @ B2 )
& ( ord_less_eq_a_o @ B2 @ A2 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_77_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_78_antisym__conv,axiom,
! [Y3: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X4 )
=> ( ( ord_less_eq_set_a @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% antisym_conv
thf(fact_79_antisym__conv,axiom,
! [Y3: a > $o,X4: a > $o] :
( ( ord_less_eq_a_o @ Y3 @ X4 )
=> ( ( ord_less_eq_a_o @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% antisym_conv
thf(fact_80_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_81_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_82_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X5: a] : ( member_a @ X5 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_83_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_84_order_Otrans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_85_order_Otrans,axiom,
! [A: a > $o,B: a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ A @ B )
=> ( ( ord_less_eq_a_o @ B @ C )
=> ( ord_less_eq_a_o @ A @ C ) ) ) ).
% order.trans
thf(fact_86_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_87_eq__refl,axiom,
! [X4: set_a,Y3: set_a] :
( ( X4 = Y3 )
=> ( ord_less_eq_set_a @ X4 @ Y3 ) ) ).
% eq_refl
thf(fact_88_eq__refl,axiom,
! [X4: a > $o,Y3: a > $o] :
( ( X4 = Y3 )
=> ( ord_less_eq_a_o @ X4 @ Y3 ) ) ).
% eq_refl
thf(fact_89_eq__refl,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( X4 = Y3 )
=> ( ord_le519537037nres_a @ X4 @ Y3 ) ) ).
% eq_refl
thf(fact_90_antisym,axiom,
! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% antisym
thf(fact_91_antisym,axiom,
! [X4: a > $o,Y3: a > $o] :
( ( ord_less_eq_a_o @ X4 @ Y3 )
=> ( ( ord_less_eq_a_o @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% antisym
thf(fact_92_antisym,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X4 @ Y3 )
=> ( ( ord_le519537037nres_a @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% antisym
thf(fact_93_eq__iff,axiom,
( ( ^ [Y5: set_a,Z: set_a] : Y5 = Z )
= ( ^ [X5: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y6 )
& ( ord_less_eq_set_a @ Y6 @ X5 ) ) ) ) ).
% eq_iff
thf(fact_94_eq__iff,axiom,
( ( ^ [Y5: a > $o,Z: a > $o] : Y5 = Z )
= ( ^ [X5: a > $o,Y6: a > $o] :
( ( ord_less_eq_a_o @ X5 @ Y6 )
& ( ord_less_eq_a_o @ Y6 @ X5 ) ) ) ) ).
% eq_iff
thf(fact_95_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_96_ord__le__eq__subst,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,F: refine424419629nres_a > set_a,C: set_a] :
( ( ord_le519537037nres_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: refine424419629nres_a,Y4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_97_ord__le__eq__subst,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,F: refine424419629nres_a > a > $o,C: a > $o] :
( ( ord_le519537037nres_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: refine424419629nres_a,Y4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y4 )
=> ( ord_less_eq_a_o @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a_o @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_98_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le519537037nres_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_99_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_100_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > a > $o,C: a > $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_less_eq_a_o @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a_o @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_101_ord__le__eq__subst,axiom,
! [A: a > $o,B: a > $o,F: ( a > $o ) > refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_less_eq_a_o @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a > $o,Y4: a > $o] :
( ( ord_less_eq_a_o @ X3 @ Y4 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le519537037nres_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_102_ord__le__eq__subst,axiom,
! [A: a > $o,B: a > $o,F: ( a > $o ) > set_a,C: set_a] :
( ( ord_less_eq_a_o @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a > $o,Y4: a > $o] :
( ( ord_less_eq_a_o @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_103_ord__le__eq__subst,axiom,
! [A: a > $o,B: a > $o,F: ( a > $o ) > a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a > $o,Y4: a > $o] :
( ( ord_less_eq_a_o @ X3 @ Y4 )
=> ( ord_less_eq_a_o @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a_o @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_104_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_105_ord__eq__le__subst,axiom,
! [A: set_a,F: refine424419629nres_a > set_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_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_106_ord__eq__le__subst,axiom,
! [A: a > $o,F: refine424419629nres_a > a > $o,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_less_eq_a_o @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a_o @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_107_ord__eq__le__subst,axiom,
! [A: refine424419629nres_a,F: set_a > refine424419629nres_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le519537037nres_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_108_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_109_ord__eq__le__subst,axiom,
! [A: a > $o,F: set_a > a > $o,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_less_eq_a_o @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a_o @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_110_ord__eq__le__subst,axiom,
! [A: refine424419629nres_a,F: ( a > $o ) > refine424419629nres_a,B: a > $o,C: a > $o] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a_o @ B @ C )
=> ( ! [X3: a > $o,Y4: a > $o] :
( ( ord_less_eq_a_o @ X3 @ Y4 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le519537037nres_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_111_ord__eq__le__subst,axiom,
! [A: set_a,F: ( a > $o ) > set_a,B: a > $o,C: a > $o] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a_o @ B @ C )
=> ( ! [X3: a > $o,Y4: a > $o] :
( ( ord_less_eq_a_o @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_112_ord__eq__le__subst,axiom,
! [A: a > $o,F: ( a > $o ) > a > $o,B: a > $o,C: a > $o] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a_o @ B @ C )
=> ( ! [X3: a > $o,Y4: a > $o] :
( ( ord_less_eq_a_o @ X3 @ Y4 )
=> ( ord_less_eq_a_o @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a_o @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_113_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_114_order__subst2,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,F: refine424419629nres_a > set_a,C: set_a] :
( ( ord_le519537037nres_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: refine424419629nres_a,Y4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_115_order__subst2,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,F: refine424419629nres_a > a > $o,C: a > $o] :
( ( ord_le519537037nres_a @ A @ B )
=> ( ( ord_less_eq_a_o @ ( F @ B ) @ C )
=> ( ! [X3: refine424419629nres_a,Y4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y4 )
=> ( ord_less_eq_a_o @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a_o @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_116_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_le519537037nres_a @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le519537037nres_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_117_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_118_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > a > $o,C: a > $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_a_o @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_less_eq_a_o @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a_o @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_119_order__subst2,axiom,
! [A: a > $o,B: a > $o,F: ( a > $o ) > refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_less_eq_a_o @ A @ B )
=> ( ( ord_le519537037nres_a @ ( F @ B ) @ C )
=> ( ! [X3: a > $o,Y4: a > $o] :
( ( ord_less_eq_a_o @ X3 @ Y4 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le519537037nres_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_120_order__subst2,axiom,
! [A: a > $o,B: a > $o,F: ( a > $o ) > set_a,C: set_a] :
( ( ord_less_eq_a_o @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: a > $o,Y4: a > $o] :
( ( ord_less_eq_a_o @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_121_order__subst2,axiom,
! [A: a > $o,B: a > $o,F: ( a > $o ) > a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ A @ B )
=> ( ( ord_less_eq_a_o @ ( F @ B ) @ C )
=> ( ! [X3: a > $o,Y4: a > $o] :
( ( ord_less_eq_a_o @ X3 @ Y4 )
=> ( ord_less_eq_a_o @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a_o @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_122_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_123_inf_Obounded__iff,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
= ( ( ord_less_eq_set_a @ A @ B )
& ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_124_inf_Obounded__iff,axiom,
! [A: a > $o,B: a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ A @ ( inf_inf_a_o @ B @ C ) )
= ( ( ord_less_eq_a_o @ A @ B )
& ( ord_less_eq_a_o @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_125_inf_Obounded__iff,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ ( inf_in262696383nres_a @ B @ C ) )
= ( ( ord_le519537037nres_a @ A @ B )
& ( ord_le519537037nres_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_126_le__inf__iff,axiom,
! [X4: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ Y3 @ Z2 ) )
= ( ( ord_less_eq_set_a @ X4 @ Y3 )
& ( ord_less_eq_set_a @ X4 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_127_le__inf__iff,axiom,
! [X4: a > $o,Y3: a > $o,Z2: a > $o] :
( ( ord_less_eq_a_o @ X4 @ ( inf_inf_a_o @ Y3 @ Z2 ) )
= ( ( ord_less_eq_a_o @ X4 @ Y3 )
& ( ord_less_eq_a_o @ X4 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_128_le__inf__iff,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a,Z2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X4 @ ( inf_in262696383nres_a @ Y3 @ Z2 ) )
= ( ( ord_le519537037nres_a @ X4 @ Y3 )
& ( ord_le519537037nres_a @ X4 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_129_inf__apply,axiom,
( inf_inf_a_o
= ( ^ [F2: a > $o,G2: a > $o,X5: a] : ( inf_inf_o @ ( F2 @ X5 ) @ ( G2 @ X5 ) ) ) ) ).
% inf_apply
thf(fact_130_inf_Oidem,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_131_inf_Oidem,axiom,
! [A: a > $o] :
( ( inf_inf_a_o @ A @ A )
= A ) ).
% inf.idem
thf(fact_132_inf_Oidem,axiom,
! [A: refine424419629nres_a] :
( ( inf_in262696383nres_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_133_inf__idem,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ X4 @ X4 )
= X4 ) ).
% inf_idem
thf(fact_134_inf__idem,axiom,
! [X4: a > $o] :
( ( inf_inf_a_o @ X4 @ X4 )
= X4 ) ).
% inf_idem
thf(fact_135_inf__idem,axiom,
! [X4: refine424419629nres_a] :
( ( inf_in262696383nres_a @ X4 @ X4 )
= X4 ) ).
% inf_idem
thf(fact_136_inf_Oleft__idem,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% inf.left_idem
thf(fact_137_inf_Oleft__idem,axiom,
! [A: a > $o,B: a > $o] :
( ( inf_inf_a_o @ A @ ( inf_inf_a_o @ A @ B ) )
= ( inf_inf_a_o @ A @ B ) ) ).
% inf.left_idem
thf(fact_138_inf_Oleft__idem,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a] :
( ( inf_in262696383nres_a @ A @ ( inf_in262696383nres_a @ A @ B ) )
= ( inf_in262696383nres_a @ A @ B ) ) ).
% inf.left_idem
thf(fact_139_inf__left__idem,axiom,
! [X4: set_a,Y3: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ X4 @ Y3 ) )
= ( inf_inf_set_a @ X4 @ Y3 ) ) ).
% inf_left_idem
thf(fact_140_inf__left__idem,axiom,
! [X4: a > $o,Y3: a > $o] :
( ( inf_inf_a_o @ X4 @ ( inf_inf_a_o @ X4 @ Y3 ) )
= ( inf_inf_a_o @ X4 @ Y3 ) ) ).
% inf_left_idem
thf(fact_141_inf__left__idem,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( inf_in262696383nres_a @ X4 @ ( inf_in262696383nres_a @ X4 @ Y3 ) )
= ( inf_in262696383nres_a @ X4 @ Y3 ) ) ).
% inf_left_idem
thf(fact_142_inf_Oright__idem,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ B )
= ( inf_inf_set_a @ A @ B ) ) ).
% inf.right_idem
thf(fact_143_inf_Oright__idem,axiom,
! [A: a > $o,B: a > $o] :
( ( inf_inf_a_o @ ( inf_inf_a_o @ A @ B ) @ B )
= ( inf_inf_a_o @ A @ B ) ) ).
% inf.right_idem
thf(fact_144_inf_Oright__idem,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a] :
( ( inf_in262696383nres_a @ ( inf_in262696383nres_a @ A @ B ) @ B )
= ( inf_in262696383nres_a @ A @ B ) ) ).
% inf.right_idem
thf(fact_145_inf__right__idem,axiom,
! [X4: set_a,Y3: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X4 @ Y3 ) @ Y3 )
= ( inf_inf_set_a @ X4 @ Y3 ) ) ).
% inf_right_idem
thf(fact_146_inf__right__idem,axiom,
! [X4: a > $o,Y3: a > $o] :
( ( inf_inf_a_o @ ( inf_inf_a_o @ X4 @ Y3 ) @ Y3 )
= ( inf_inf_a_o @ X4 @ Y3 ) ) ).
% inf_right_idem
thf(fact_147_inf__right__idem,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( inf_in262696383nres_a @ ( inf_in262696383nres_a @ X4 @ Y3 ) @ Y3 )
= ( inf_in262696383nres_a @ X4 @ Y3 ) ) ).
% inf_right_idem
thf(fact_148_inf__nres_Osimps_I3_J,axiom,
! [A: set_a,B: set_a] :
( ( inf_in262696383nres_a @ ( refine1198353288_RES_a @ A ) @ ( refine1198353288_RES_a @ B ) )
= ( refine1198353288_RES_a @ ( inf_inf_set_a @ A @ B ) ) ) ).
% inf_nres.simps(3)
thf(fact_149_inf__left__commute,axiom,
! [X4: set_a,Y3: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y3 @ Z2 ) )
= ( inf_inf_set_a @ Y3 @ ( inf_inf_set_a @ X4 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_150_inf__left__commute,axiom,
! [X4: a > $o,Y3: a > $o,Z2: a > $o] :
( ( inf_inf_a_o @ X4 @ ( inf_inf_a_o @ Y3 @ Z2 ) )
= ( inf_inf_a_o @ Y3 @ ( inf_inf_a_o @ X4 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_151_inf__left__commute,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a,Z2: refine424419629nres_a] :
( ( inf_in262696383nres_a @ X4 @ ( inf_in262696383nres_a @ Y3 @ Z2 ) )
= ( inf_in262696383nres_a @ Y3 @ ( inf_in262696383nres_a @ X4 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_152_inf_Oleft__commute,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A @ C ) )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_153_inf_Oleft__commute,axiom,
! [B: a > $o,A: a > $o,C: a > $o] :
( ( inf_inf_a_o @ B @ ( inf_inf_a_o @ A @ C ) )
= ( inf_inf_a_o @ A @ ( inf_inf_a_o @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_154_inf_Oleft__commute,axiom,
! [B: refine424419629nres_a,A: refine424419629nres_a,C: refine424419629nres_a] :
( ( inf_in262696383nres_a @ B @ ( inf_in262696383nres_a @ A @ C ) )
= ( inf_in262696383nres_a @ A @ ( inf_in262696383nres_a @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_155_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X5: set_a,Y6: set_a] : ( inf_inf_set_a @ Y6 @ X5 ) ) ) ).
% inf_commute
thf(fact_156_inf__commute,axiom,
( inf_inf_a_o
= ( ^ [X5: a > $o,Y6: a > $o] : ( inf_inf_a_o @ Y6 @ X5 ) ) ) ).
% inf_commute
thf(fact_157_inf__commute,axiom,
( inf_in262696383nres_a
= ( ^ [X5: refine424419629nres_a,Y6: refine424419629nres_a] : ( inf_in262696383nres_a @ Y6 @ X5 ) ) ) ).
% inf_commute
thf(fact_158_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A2: set_a,B2: set_a] : ( inf_inf_set_a @ B2 @ A2 ) ) ) ).
% inf.commute
thf(fact_159_inf_Ocommute,axiom,
( inf_inf_a_o
= ( ^ [A2: a > $o,B2: a > $o] : ( inf_inf_a_o @ B2 @ A2 ) ) ) ).
% inf.commute
thf(fact_160_inf_Ocommute,axiom,
( inf_in262696383nres_a
= ( ^ [A2: refine424419629nres_a,B2: refine424419629nres_a] : ( inf_in262696383nres_a @ B2 @ A2 ) ) ) ).
% inf.commute
thf(fact_161_inf__assoc,axiom,
! [X4: set_a,Y3: set_a,Z2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X4 @ Y3 ) @ Z2 )
= ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y3 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_162_inf__assoc,axiom,
! [X4: a > $o,Y3: a > $o,Z2: a > $o] :
( ( inf_inf_a_o @ ( inf_inf_a_o @ X4 @ Y3 ) @ Z2 )
= ( inf_inf_a_o @ X4 @ ( inf_inf_a_o @ Y3 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_163_inf__assoc,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a,Z2: refine424419629nres_a] :
( ( inf_in262696383nres_a @ ( inf_in262696383nres_a @ X4 @ Y3 ) @ Z2 )
= ( inf_in262696383nres_a @ X4 @ ( inf_in262696383nres_a @ Y3 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_164_inf_Oassoc,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ C )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ).
% inf.assoc
thf(fact_165_inf_Oassoc,axiom,
! [A: a > $o,B: a > $o,C: a > $o] :
( ( inf_inf_a_o @ ( inf_inf_a_o @ A @ B ) @ C )
= ( inf_inf_a_o @ A @ ( inf_inf_a_o @ B @ C ) ) ) ).
% inf.assoc
thf(fact_166_inf_Oassoc,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( inf_in262696383nres_a @ ( inf_in262696383nres_a @ A @ B ) @ C )
= ( inf_in262696383nres_a @ A @ ( inf_in262696383nres_a @ B @ C ) ) ) ).
% inf.assoc
thf(fact_167_boolean__algebra__cancel_Oinf2,axiom,
! [B3: set_a,K: set_a,B: set_a,A: set_a] :
( ( B3
= ( inf_inf_set_a @ K @ B ) )
=> ( ( inf_inf_set_a @ A @ B3 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_168_boolean__algebra__cancel_Oinf2,axiom,
! [B3: a > $o,K: a > $o,B: a > $o,A: a > $o] :
( ( B3
= ( inf_inf_a_o @ K @ B ) )
=> ( ( inf_inf_a_o @ A @ B3 )
= ( inf_inf_a_o @ K @ ( inf_inf_a_o @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_169_boolean__algebra__cancel_Oinf2,axiom,
! [B3: refine424419629nres_a,K: refine424419629nres_a,B: refine424419629nres_a,A: refine424419629nres_a] :
( ( B3
= ( inf_in262696383nres_a @ K @ B ) )
=> ( ( inf_in262696383nres_a @ A @ B3 )
= ( inf_in262696383nres_a @ K @ ( inf_in262696383nres_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_170_boolean__algebra__cancel_Oinf1,axiom,
! [A3: set_a,K: set_a,A: set_a,B: set_a] :
( ( A3
= ( inf_inf_set_a @ K @ A ) )
=> ( ( inf_inf_set_a @ A3 @ B )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_171_boolean__algebra__cancel_Oinf1,axiom,
! [A3: a > $o,K: a > $o,A: a > $o,B: a > $o] :
( ( A3
= ( inf_inf_a_o @ K @ A ) )
=> ( ( inf_inf_a_o @ A3 @ B )
= ( inf_inf_a_o @ K @ ( inf_inf_a_o @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_172_boolean__algebra__cancel_Oinf1,axiom,
! [A3: refine424419629nres_a,K: refine424419629nres_a,A: refine424419629nres_a,B: refine424419629nres_a] :
( ( A3
= ( inf_in262696383nres_a @ K @ A ) )
=> ( ( inf_in262696383nres_a @ A3 @ B )
= ( inf_in262696383nres_a @ K @ ( inf_in262696383nres_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_173_inf__fun__def,axiom,
( inf_inf_a_o
= ( ^ [F2: a > $o,G2: a > $o,X5: a] : ( inf_inf_o @ ( F2 @ X5 ) @ ( G2 @ X5 ) ) ) ) ).
% inf_fun_def
thf(fact_174_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X5: set_a,Y6: set_a] : ( inf_inf_set_a @ Y6 @ X5 ) ) ) ).
% inf_sup_aci(1)
thf(fact_175_inf__sup__aci_I1_J,axiom,
( inf_inf_a_o
= ( ^ [X5: a > $o,Y6: a > $o] : ( inf_inf_a_o @ Y6 @ X5 ) ) ) ).
% inf_sup_aci(1)
thf(fact_176_inf__sup__aci_I1_J,axiom,
( inf_in262696383nres_a
= ( ^ [X5: refine424419629nres_a,Y6: refine424419629nres_a] : ( inf_in262696383nres_a @ Y6 @ X5 ) ) ) ).
% inf_sup_aci(1)
thf(fact_177_inf__sup__aci_I2_J,axiom,
! [X4: set_a,Y3: set_a,Z2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X4 @ Y3 ) @ Z2 )
= ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y3 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_178_inf__sup__aci_I2_J,axiom,
! [X4: a > $o,Y3: a > $o,Z2: a > $o] :
( ( inf_inf_a_o @ ( inf_inf_a_o @ X4 @ Y3 ) @ Z2 )
= ( inf_inf_a_o @ X4 @ ( inf_inf_a_o @ Y3 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_179_inf__sup__aci_I2_J,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a,Z2: refine424419629nres_a] :
( ( inf_in262696383nres_a @ ( inf_in262696383nres_a @ X4 @ Y3 ) @ Z2 )
= ( inf_in262696383nres_a @ X4 @ ( inf_in262696383nres_a @ Y3 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_180_inf__sup__aci_I3_J,axiom,
! [X4: set_a,Y3: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y3 @ Z2 ) )
= ( inf_inf_set_a @ Y3 @ ( inf_inf_set_a @ X4 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_181_inf__sup__aci_I3_J,axiom,
! [X4: a > $o,Y3: a > $o,Z2: a > $o] :
( ( inf_inf_a_o @ X4 @ ( inf_inf_a_o @ Y3 @ Z2 ) )
= ( inf_inf_a_o @ Y3 @ ( inf_inf_a_o @ X4 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_182_inf__sup__aci_I3_J,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a,Z2: refine424419629nres_a] :
( ( inf_in262696383nres_a @ X4 @ ( inf_in262696383nres_a @ Y3 @ Z2 ) )
= ( inf_in262696383nres_a @ Y3 @ ( inf_in262696383nres_a @ X4 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_183_inf__sup__aci_I4_J,axiom,
! [X4: set_a,Y3: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ X4 @ Y3 ) )
= ( inf_inf_set_a @ X4 @ Y3 ) ) ).
% inf_sup_aci(4)
thf(fact_184_inf__sup__aci_I4_J,axiom,
! [X4: a > $o,Y3: a > $o] :
( ( inf_inf_a_o @ X4 @ ( inf_inf_a_o @ X4 @ Y3 ) )
= ( inf_inf_a_o @ X4 @ Y3 ) ) ).
% inf_sup_aci(4)
thf(fact_185_inf__sup__aci_I4_J,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( inf_in262696383nres_a @ X4 @ ( inf_in262696383nres_a @ X4 @ Y3 ) )
= ( inf_in262696383nres_a @ X4 @ Y3 ) ) ).
% inf_sup_aci(4)
thf(fact_186_inf__sup__ord_I2_J,axiom,
! [X4: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_187_inf__sup__ord_I2_J,axiom,
! [X4: a > $o,Y3: a > $o] : ( ord_less_eq_a_o @ ( inf_inf_a_o @ X4 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_188_inf__sup__ord_I2_J,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a] : ( ord_le519537037nres_a @ ( inf_in262696383nres_a @ X4 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_189_inf__sup__ord_I1_J,axiom,
! [X4: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y3 ) @ X4 ) ).
% inf_sup_ord(1)
thf(fact_190_inf__sup__ord_I1_J,axiom,
! [X4: a > $o,Y3: a > $o] : ( ord_less_eq_a_o @ ( inf_inf_a_o @ X4 @ Y3 ) @ X4 ) ).
% inf_sup_ord(1)
thf(fact_191_inf__sup__ord_I1_J,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a] : ( ord_le519537037nres_a @ ( inf_in262696383nres_a @ X4 @ Y3 ) @ X4 ) ).
% inf_sup_ord(1)
thf(fact_192_inf__le1,axiom,
! [X4: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y3 ) @ X4 ) ).
% inf_le1
thf(fact_193_inf__le1,axiom,
! [X4: a > $o,Y3: a > $o] : ( ord_less_eq_a_o @ ( inf_inf_a_o @ X4 @ Y3 ) @ X4 ) ).
% inf_le1
thf(fact_194_inf__le1,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a] : ( ord_le519537037nres_a @ ( inf_in262696383nres_a @ X4 @ Y3 ) @ X4 ) ).
% inf_le1
thf(fact_195_inf__le2,axiom,
! [X4: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_196_inf__le2,axiom,
! [X4: a > $o,Y3: a > $o] : ( ord_less_eq_a_o @ ( inf_inf_a_o @ X4 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_197_inf__le2,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a] : ( ord_le519537037nres_a @ ( inf_in262696383nres_a @ X4 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_198_le__infE,axiom,
! [X4: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( ord_less_eq_set_a @ X4 @ A )
=> ~ ( ord_less_eq_set_a @ X4 @ B ) ) ) ).
% le_infE
thf(fact_199_le__infE,axiom,
! [X4: a > $o,A: a > $o,B: a > $o] :
( ( ord_less_eq_a_o @ X4 @ ( inf_inf_a_o @ A @ B ) )
=> ~ ( ( ord_less_eq_a_o @ X4 @ A )
=> ~ ( ord_less_eq_a_o @ X4 @ B ) ) ) ).
% le_infE
thf(fact_200_le__infE,axiom,
! [X4: refine424419629nres_a,A: refine424419629nres_a,B: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X4 @ ( inf_in262696383nres_a @ A @ B ) )
=> ~ ( ( ord_le519537037nres_a @ X4 @ A )
=> ~ ( ord_le519537037nres_a @ X4 @ B ) ) ) ).
% le_infE
thf(fact_201_le__infI,axiom,
! [X4: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X4 @ A )
=> ( ( ord_less_eq_set_a @ X4 @ B )
=> ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% le_infI
thf(fact_202_le__infI,axiom,
! [X4: a > $o,A: a > $o,B: a > $o] :
( ( ord_less_eq_a_o @ X4 @ A )
=> ( ( ord_less_eq_a_o @ X4 @ B )
=> ( ord_less_eq_a_o @ X4 @ ( inf_inf_a_o @ A @ B ) ) ) ) ).
% le_infI
thf(fact_203_le__infI,axiom,
! [X4: refine424419629nres_a,A: refine424419629nres_a,B: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X4 @ A )
=> ( ( ord_le519537037nres_a @ X4 @ B )
=> ( ord_le519537037nres_a @ X4 @ ( inf_in262696383nres_a @ A @ B ) ) ) ) ).
% le_infI
thf(fact_204_inf__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_205_inf__mono,axiom,
! [A: a > $o,C: a > $o,B: a > $o,D: a > $o] :
( ( ord_less_eq_a_o @ A @ C )
=> ( ( ord_less_eq_a_o @ B @ D )
=> ( ord_less_eq_a_o @ ( inf_inf_a_o @ A @ B ) @ ( inf_inf_a_o @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_206_inf__mono,axiom,
! [A: refine424419629nres_a,C: refine424419629nres_a,B: refine424419629nres_a,D: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ C )
=> ( ( ord_le519537037nres_a @ B @ D )
=> ( ord_le519537037nres_a @ ( inf_in262696383nres_a @ A @ B ) @ ( inf_in262696383nres_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_207_le__infI1,axiom,
! [A: set_a,X4: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ X4 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ X4 ) ) ).
% le_infI1
thf(fact_208_le__infI1,axiom,
! [A: a > $o,X4: a > $o,B: a > $o] :
( ( ord_less_eq_a_o @ A @ X4 )
=> ( ord_less_eq_a_o @ ( inf_inf_a_o @ A @ B ) @ X4 ) ) ).
% le_infI1
thf(fact_209_le__infI1,axiom,
! [A: refine424419629nres_a,X4: refine424419629nres_a,B: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ X4 )
=> ( ord_le519537037nres_a @ ( inf_in262696383nres_a @ A @ B ) @ X4 ) ) ).
% le_infI1
thf(fact_210_le__infI2,axiom,
! [B: set_a,X4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ X4 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ X4 ) ) ).
% le_infI2
thf(fact_211_le__infI2,axiom,
! [B: a > $o,X4: a > $o,A: a > $o] :
( ( ord_less_eq_a_o @ B @ X4 )
=> ( ord_less_eq_a_o @ ( inf_inf_a_o @ A @ B ) @ X4 ) ) ).
% le_infI2
thf(fact_212_le__infI2,axiom,
! [B: refine424419629nres_a,X4: refine424419629nres_a,A: refine424419629nres_a] :
( ( ord_le519537037nres_a @ B @ X4 )
=> ( ord_le519537037nres_a @ ( inf_in262696383nres_a @ A @ B ) @ X4 ) ) ).
% le_infI2
thf(fact_213_inf_OorderE,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( A
= ( inf_inf_set_a @ A @ B ) ) ) ).
% inf.orderE
thf(fact_214_inf_OorderE,axiom,
! [A: a > $o,B: a > $o] :
( ( ord_less_eq_a_o @ A @ B )
=> ( A
= ( inf_inf_a_o @ A @ B ) ) ) ).
% inf.orderE
thf(fact_215_inf_OorderE,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ B )
=> ( A
= ( inf_in262696383nres_a @ A @ B ) ) ) ).
% inf.orderE
thf(fact_216_inf_OorderI,axiom,
! [A: set_a,B: set_a] :
( ( A
= ( inf_inf_set_a @ A @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% inf.orderI
thf(fact_217_inf_OorderI,axiom,
! [A: a > $o,B: a > $o] :
( ( A
= ( inf_inf_a_o @ A @ B ) )
=> ( ord_less_eq_a_o @ A @ B ) ) ).
% inf.orderI
thf(fact_218_inf_OorderI,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a] :
( ( A
= ( inf_in262696383nres_a @ A @ B ) )
=> ( ord_le519537037nres_a @ A @ B ) ) ).
% inf.orderI
thf(fact_219_inf__unique,axiom,
! [F: set_a > set_a > set_a,X4: set_a,Y3: set_a] :
( ! [X3: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( F @ X3 @ Y4 ) @ X3 )
=> ( ! [X3: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( F @ X3 @ Y4 ) @ Y4 )
=> ( ! [X3: set_a,Y4: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ( ord_less_eq_set_a @ X3 @ Z3 )
=> ( ord_less_eq_set_a @ X3 @ ( F @ Y4 @ Z3 ) ) ) )
=> ( ( inf_inf_set_a @ X4 @ Y3 )
= ( F @ X4 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_220_inf__unique,axiom,
! [F: ( a > $o ) > ( a > $o ) > a > $o,X4: a > $o,Y3: a > $o] :
( ! [X3: a > $o,Y4: a > $o] : ( ord_less_eq_a_o @ ( F @ X3 @ Y4 ) @ X3 )
=> ( ! [X3: a > $o,Y4: a > $o] : ( ord_less_eq_a_o @ ( F @ X3 @ Y4 ) @ Y4 )
=> ( ! [X3: a > $o,Y4: a > $o,Z3: a > $o] :
( ( ord_less_eq_a_o @ X3 @ Y4 )
=> ( ( ord_less_eq_a_o @ X3 @ Z3 )
=> ( ord_less_eq_a_o @ X3 @ ( F @ Y4 @ Z3 ) ) ) )
=> ( ( inf_inf_a_o @ X4 @ Y3 )
= ( F @ X4 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_221_inf__unique,axiom,
! [F: refine424419629nres_a > refine424419629nres_a > refine424419629nres_a,X4: refine424419629nres_a,Y3: refine424419629nres_a] :
( ! [X3: refine424419629nres_a,Y4: refine424419629nres_a] : ( ord_le519537037nres_a @ ( F @ X3 @ Y4 ) @ X3 )
=> ( ! [X3: refine424419629nres_a,Y4: refine424419629nres_a] : ( ord_le519537037nres_a @ ( F @ X3 @ Y4 ) @ Y4 )
=> ( ! [X3: refine424419629nres_a,Y4: refine424419629nres_a,Z3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y4 )
=> ( ( ord_le519537037nres_a @ X3 @ Z3 )
=> ( ord_le519537037nres_a @ X3 @ ( F @ Y4 @ Z3 ) ) ) )
=> ( ( inf_in262696383nres_a @ X4 @ Y3 )
= ( F @ X4 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_222_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X5: set_a,Y6: set_a] :
( ( inf_inf_set_a @ X5 @ Y6 )
= X5 ) ) ) ).
% le_iff_inf
thf(fact_223_le__iff__inf,axiom,
( ord_less_eq_a_o
= ( ^ [X5: a > $o,Y6: a > $o] :
( ( inf_inf_a_o @ X5 @ Y6 )
= X5 ) ) ) ).
% le_iff_inf
thf(fact_224_le__iff__inf,axiom,
( ord_le519537037nres_a
= ( ^ [X5: refine424419629nres_a,Y6: refine424419629nres_a] :
( ( inf_in262696383nres_a @ X5 @ Y6 )
= X5 ) ) ) ).
% le_iff_inf
thf(fact_225_inf_Oabsorb1,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_226_inf_Oabsorb1,axiom,
! [A: a > $o,B: a > $o] :
( ( ord_less_eq_a_o @ A @ B )
=> ( ( inf_inf_a_o @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_227_inf_Oabsorb1,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ B )
=> ( ( inf_in262696383nres_a @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_228_inf_Oabsorb2,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_229_inf_Oabsorb2,axiom,
! [B: a > $o,A: a > $o] :
( ( ord_less_eq_a_o @ B @ A )
=> ( ( inf_inf_a_o @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_230_inf_Oabsorb2,axiom,
! [B: refine424419629nres_a,A: refine424419629nres_a] :
( ( ord_le519537037nres_a @ B @ A )
=> ( ( inf_in262696383nres_a @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_231_inf__absorb1,axiom,
! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ( inf_inf_set_a @ X4 @ Y3 )
= X4 ) ) ).
% inf_absorb1
thf(fact_232_inf__absorb1,axiom,
! [X4: a > $o,Y3: a > $o] :
( ( ord_less_eq_a_o @ X4 @ Y3 )
=> ( ( inf_inf_a_o @ X4 @ Y3 )
= X4 ) ) ).
% inf_absorb1
thf(fact_233_inf__absorb1,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X4 @ Y3 )
=> ( ( inf_in262696383nres_a @ X4 @ Y3 )
= X4 ) ) ).
% inf_absorb1
thf(fact_234_inf__absorb2,axiom,
! [Y3: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X4 )
=> ( ( inf_inf_set_a @ X4 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_235_inf__absorb2,axiom,
! [Y3: a > $o,X4: a > $o] :
( ( ord_less_eq_a_o @ Y3 @ X4 )
=> ( ( inf_inf_a_o @ X4 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_236_inf__absorb2,axiom,
! [Y3: refine424419629nres_a,X4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ Y3 @ X4 )
=> ( ( inf_in262696383nres_a @ X4 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_237_inf_OboundedE,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_238_inf_OboundedE,axiom,
! [A: a > $o,B: a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ A @ ( inf_inf_a_o @ B @ C ) )
=> ~ ( ( ord_less_eq_a_o @ A @ B )
=> ~ ( ord_less_eq_a_o @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_239_inf_OboundedE,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ ( inf_in262696383nres_a @ B @ C ) )
=> ~ ( ( ord_le519537037nres_a @ A @ B )
=> ~ ( ord_le519537037nres_a @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_240_inf_OboundedI,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_241_inf_OboundedI,axiom,
! [A: a > $o,B: a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ A @ B )
=> ( ( ord_less_eq_a_o @ A @ C )
=> ( ord_less_eq_a_o @ A @ ( inf_inf_a_o @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_242_inf_OboundedI,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ B )
=> ( ( ord_le519537037nres_a @ A @ C )
=> ( ord_le519537037nres_a @ A @ ( inf_in262696383nres_a @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_243_inf__greatest,axiom,
! [X4: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ( ord_less_eq_set_a @ X4 @ Z2 )
=> ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ Y3 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_244_inf__greatest,axiom,
! [X4: a > $o,Y3: a > $o,Z2: a > $o] :
( ( ord_less_eq_a_o @ X4 @ Y3 )
=> ( ( ord_less_eq_a_o @ X4 @ Z2 )
=> ( ord_less_eq_a_o @ X4 @ ( inf_inf_a_o @ Y3 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_245_inf__greatest,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a,Z2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X4 @ Y3 )
=> ( ( ord_le519537037nres_a @ X4 @ Z2 )
=> ( ord_le519537037nres_a @ X4 @ ( inf_in262696383nres_a @ Y3 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_246_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B2: set_a] :
( A2
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_247_inf_Oorder__iff,axiom,
( ord_less_eq_a_o
= ( ^ [A2: a > $o,B2: a > $o] :
( A2
= ( inf_inf_a_o @ A2 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_248_inf_Oorder__iff,axiom,
( ord_le519537037nres_a
= ( ^ [A2: refine424419629nres_a,B2: refine424419629nres_a] :
( A2
= ( inf_in262696383nres_a @ A2 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_249_inf_Ocobounded1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_250_inf_Ocobounded1,axiom,
! [A: a > $o,B: a > $o] : ( ord_less_eq_a_o @ ( inf_inf_a_o @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_251_inf_Ocobounded1,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a] : ( ord_le519537037nres_a @ ( inf_in262696383nres_a @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_252_inf_Ocobounded2,axiom,
! [A: a > $o,B: a > $o] : ( ord_less_eq_a_o @ ( inf_inf_a_o @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_253_inf_Ocobounded2,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a] : ( ord_le519537037nres_a @ ( inf_in262696383nres_a @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_254_inf_Oabsorb__iff1,axiom,
( ord_le519537037nres_a
= ( ^ [A2: refine424419629nres_a,B2: refine424419629nres_a] :
( ( inf_in262696383nres_a @ A2 @ B2 )
= A2 ) ) ) ).
% inf.absorb_iff1
thf(fact_255_inf_Oabsorb__iff2,axiom,
( ord_le519537037nres_a
= ( ^ [B2: refine424419629nres_a,A2: refine424419629nres_a] :
( ( inf_in262696383nres_a @ A2 @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_256_inf_OcoboundedI1,axiom,
! [A: refine424419629nres_a,C: refine424419629nres_a,B: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ C )
=> ( ord_le519537037nres_a @ ( inf_in262696383nres_a @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_257_inf_OcoboundedI2,axiom,
! [B: refine424419629nres_a,C: refine424419629nres_a,A: refine424419629nres_a] :
( ( ord_le519537037nres_a @ B @ C )
=> ( ord_le519537037nres_a @ ( inf_in262696383nres_a @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_258_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_259_Collect__subset,axiom,
! [A3: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X5: a] :
( ( member_a @ X5 @ A3 )
& ( P @ X5 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_260_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ord_less_eq_a_o
@ ^ [X5: a] : ( member_a @ X5 @ A4 )
@ ^ [X5: a] : ( member_a @ X5 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_261_pred__subset__eq,axiom,
! [R: set_a,S: set_a] :
( ( ord_less_eq_a_o
@ ^ [X5: a] : ( member_a @ X5 @ R )
@ ^ [X5: a] : ( member_a @ X5 @ S ) )
= ( ord_less_eq_set_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_262_conj__subset__def,axiom,
! [A3: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A3
@ ( collect_a
@ ^ [X5: a] :
( ( P @ X5 )
& ( Q @ X5 ) ) ) )
= ( ( ord_less_eq_set_a @ A3 @ ( collect_a @ P ) )
& ( ord_less_eq_set_a @ A3 @ ( collect_a @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_263_IntI,axiom,
! [C: a,A3: set_a,B3: set_a] :
( ( member_a @ C @ A3 )
=> ( ( member_a @ C @ B3 )
=> ( member_a @ C @ ( inf_inf_set_a @ A3 @ B3 ) ) ) ) ).
% IntI
thf(fact_264_Int__iff,axiom,
! [C: a,A3: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A3 @ B3 ) )
= ( ( member_a @ C @ A3 )
& ( member_a @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_265_IntE,axiom,
! [C: a,A3: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A3 @ B3 ) )
=> ~ ( ( member_a @ C @ A3 )
=> ~ ( member_a @ C @ B3 ) ) ) ).
% IntE
thf(fact_266_IntD1,axiom,
! [C: a,A3: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A3 @ B3 ) )
=> ( member_a @ C @ A3 ) ) ).
% IntD1
thf(fact_267_IntD2,axiom,
! [C: a,A3: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A3 @ B3 ) )
=> ( member_a @ C @ B3 ) ) ).
% IntD2
thf(fact_268_Int__def,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B4: set_a] :
( collect_a
@ ^ [X5: a] :
( ( member_a @ X5 @ A4 )
& ( member_a @ X5 @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_269_Int__Collect,axiom,
! [X4: a,A3: set_a,P: a > $o] :
( ( member_a @ X4 @ ( inf_inf_set_a @ A3 @ ( collect_a @ P ) ) )
= ( ( member_a @ X4 @ A3 )
& ( P @ X4 ) ) ) ).
% Int_Collect
thf(fact_270_inf__set__def,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B4: set_a] :
( collect_a
@ ( inf_inf_a_o
@ ^ [X5: a] : ( member_a @ X5 @ A4 )
@ ^ [X5: a] : ( member_a @ X5 @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_271_inf__Int__eq,axiom,
! [R: set_a,S: set_a] :
( ( inf_inf_a_o
@ ^ [X5: a] : ( member_a @ X5 @ R )
@ ^ [X5: a] : ( member_a @ X5 @ S ) )
= ( ^ [X5: a] : ( member_a @ X5 @ ( inf_inf_set_a @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_272_Collect__conj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X5: a] :
( ( P @ X5 )
& ( Q @ X5 ) ) )
= ( inf_inf_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_273_Int__Collect__mono,axiom,
! [A3: set_a,B3: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B3 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_274_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_275_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_276_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_277_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_278_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_279_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_280_subset__Collect__iff,axiom,
! [B3: set_a,A3: set_a,P: a > $o] :
( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( ( ord_less_eq_set_a @ B3
@ ( collect_a
@ ^ [X5: a] :
( ( member_a @ X5 @ A3 )
& ( P @ X5 ) ) ) )
= ( ! [X5: a] :
( ( member_a @ X5 @ B3 )
=> ( P @ X5 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_281_subset__CollectI,axiom,
! [B3: set_a,A3: set_a,Q: a > $o,P: a > $o] :
( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B3 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq_set_a
@ ( collect_a
@ ^ [X5: a] :
( ( member_a @ X5 @ B3 )
& ( Q @ X5 ) ) )
@ ( collect_a
@ ^ [X5: a] :
( ( member_a @ X5 @ A3 )
& ( P @ X5 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_282_Collect__restrict,axiom,
! [X: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X5: a] :
( ( member_a @ X5 @ X )
& ( P @ X5 ) ) )
@ X ) ).
% Collect_restrict
thf(fact_283_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_284_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_285_prop__restrict,axiom,
! [X4: a,Z4: set_a,X: set_a,P: a > $o] :
( ( member_a @ X4 @ Z4 )
=> ( ( ord_less_eq_set_a @ Z4
@ ( collect_a
@ ^ [X5: a] :
( ( member_a @ X5 @ X )
& ( P @ X5 ) ) ) )
=> ( P @ X4 ) ) ) ).
% prop_restrict
thf(fact_286_inf__nres_Oelims,axiom,
! [X4: refine424419629nres_a,Xa: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( ( inf_in262696383nres_a @ X4 @ Xa )
= Y3 )
=> ( ( ( Xa = refine464223677AILi_a )
=> ( Y3 != X4 ) )
=> ( ( ( X4 = refine464223677AILi_a )
=> ! [V: set_a] :
( ( Xa
= ( refine1198353288_RES_a @ V ) )
=> ( Y3
!= ( refine1198353288_RES_a @ V ) ) ) )
=> ~ ! [A5: set_a] :
( ( X4
= ( refine1198353288_RES_a @ A5 ) )
=> ! [B5: set_a] :
( ( Xa
= ( refine1198353288_RES_a @ B5 ) )
=> ( Y3
!= ( refine1198353288_RES_a @ ( inf_inf_set_a @ A5 @ B5 ) ) ) ) ) ) ) ) ).
% inf_nres.elims
thf(fact_287_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_288_inf__nres_Osimps_I1_J,axiom,
! [X4: refine424419629nres_a] :
( ( inf_in262696383nres_a @ X4 @ refine464223677AILi_a )
= X4 ) ).
% inf_nres.simps(1)
thf(fact_289_nres_Odistinct_I1_J,axiom,
! [X2: set_a] :
( refine464223677AILi_a
!= ( refine1198353288_RES_a @ X2 ) ) ).
% nres.distinct(1)
thf(fact_290_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_291_nres_Oexhaust,axiom,
! [Y3: refine424419629nres_a] :
( ( Y3 != refine464223677AILi_a )
=> ~ ! [X22: set_a] :
( Y3
!= ( refine1198353288_RES_a @ X22 ) ) ) ).
% nres.exhaust
thf(fact_292_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_293_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_294_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_295_less__eq__nres_Osimps_I1_J,axiom,
! [Uu2: refine424419629nres_a] : ( ord_le519537037nres_a @ Uu2 @ refine464223677AILi_a ) ).
% less_eq_nres.simps(1)
thf(fact_296_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_297_inf__nres_Osimps_I2_J,axiom,
! [V2: set_a] :
( ( inf_in262696383nres_a @ refine464223677AILi_a @ ( refine1198353288_RES_a @ V2 ) )
= ( refine1198353288_RES_a @ V2 ) ) ).
% inf_nres.simps(2)
thf(fact_298_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_299_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_300_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_301_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_302_the__RES__inv,axiom,
! [M: refine424419629nres_a] :
( ( refine412683989fail_a @ M )
=> ( ( refine1198353288_RES_a @ ( refine1822134885_RES_a @ M ) )
= M ) ) ).
% the_RES_inv
thf(fact_303_inf__RETURN__SPEC_I1_J,axiom,
! [X4: a,Phi: a > $o] :
( ( inf_in262696383nres_a @ ( refine2063221604TURN_a @ X4 ) @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) )
= ( refine1198353288_RES_a
@ ( collect_a
@ ^ [Y6: a] :
( ( Y6 = X4 )
& ( Phi @ X4 ) ) ) ) ) ).
% inf_RETURN_SPEC(1)
thf(fact_304_nofail__simps_I2_J,axiom,
! [X: set_a] : ( refine412683989fail_a @ ( refine1198353288_RES_a @ X ) ) ).
% nofail_simps(2)
thf(fact_305_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_306_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_307_inf__RETURN__SPEC_I2_J,axiom,
! [Phi: a > $o,X4: a] :
( ( inf_in262696383nres_a @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) @ ( refine2063221604TURN_a @ X4 ) )
= ( refine1198353288_RES_a
@ ( collect_a
@ ^ [Y6: a] :
( ( Y6 = X4 )
& ( Phi @ X4 ) ) ) ) ) ).
% inf_RETURN_SPEC(2)
thf(fact_308_pwD1,axiom,
! [S: refine424419629nres_a,S4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ S @ S4 )
=> ( ( refine412683989fail_a @ S4 )
=> ( refine412683989fail_a @ S ) ) ) ).
% pwD1
thf(fact_309_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_310_nofail__RES__conv,axiom,
( refine412683989fail_a
= ( ^ [M5: refine424419629nres_a] :
? [M6: set_a] :
( M5
= ( refine1198353288_RES_a @ M6 ) ) ) ) ).
% nofail_RES_conv
thf(fact_311_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_312_RETURN__SPEC__conv,axiom,
( refine2063221604TURN_a
= ( ^ [R2: a] :
( refine1198353288_RES_a
@ ( collect_a
@ ^ [X5: a] : X5 = R2 ) ) ) ) ).
% RETURN_SPEC_conv
thf(fact_313_SPEC__eq__is__RETURN_I2_J,axiom,
! [Y3: a] :
( ( refine1198353288_RES_a
@ ( collect_a
@ ^ [X5: a] : X5 = Y3 ) )
= ( refine2063221604TURN_a @ Y3 ) ) ).
% SPEC_eq_is_RETURN(2)
thf(fact_314_pw__inf__nofail,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a] :
( ( refine412683989fail_a @ ( inf_in262696383nres_a @ A @ B ) )
= ( ( refine412683989fail_a @ A )
| ( refine412683989fail_a @ B ) ) ) ).
% pw_inf_nofail
thf(fact_315_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_316_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_317_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_318_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_319_lhs__step__SPEC,axiom,
! [Phi: a > $o,M: refine424419629nres_a] :
( ! [X3: a] :
( ( Phi @ X3 )
=> ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X3 ) @ M ) )
=> ( ord_le519537037nres_a @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) @ M ) ) ).
% lhs_step_SPEC
thf(fact_320_inf__RETURN__RES_I2_J,axiom,
! [X4: a,X: set_a] :
( ( ( member_a @ X4 @ X )
=> ( ( inf_in262696383nres_a @ ( refine1198353288_RES_a @ X ) @ ( refine2063221604TURN_a @ X4 ) )
= ( refine2063221604TURN_a @ X4 ) ) )
& ( ~ ( member_a @ X4 @ X )
=> ( ( inf_in262696383nres_a @ ( refine1198353288_RES_a @ X ) @ ( refine2063221604TURN_a @ X4 ) )
= bot_bo529555393nres_a ) ) ) ).
% inf_RETURN_RES(2)
thf(fact_321_inf__RETURN__RES_I1_J,axiom,
! [X4: a,X: set_a] :
( ( ( member_a @ X4 @ X )
=> ( ( inf_in262696383nres_a @ ( refine2063221604TURN_a @ X4 ) @ ( refine1198353288_RES_a @ X ) )
= ( refine2063221604TURN_a @ X4 ) ) )
& ( ~ ( member_a @ X4 @ X )
=> ( ( inf_in262696383nres_a @ ( refine2063221604TURN_a @ X4 ) @ ( refine1198353288_RES_a @ X ) )
= bot_bo529555393nres_a ) ) ) ).
% inf_RETURN_RES(1)
thf(fact_322_inf__bot__right,axiom,
! [X4: refine424419629nres_a] :
( ( inf_in262696383nres_a @ X4 @ bot_bo529555393nres_a )
= bot_bo529555393nres_a ) ).
% inf_bot_right
thf(fact_323_inf__bot__left,axiom,
! [X4: refine424419629nres_a] :
( ( inf_in262696383nres_a @ bot_bo529555393nres_a @ X4 )
= bot_bo529555393nres_a ) ).
% inf_bot_left
thf(fact_324_nres__order__simps_I2_J,axiom,
! [M4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ M4 @ bot_bo529555393nres_a )
= ( M4 = bot_bo529555393nres_a ) ) ).
% nres_order_simps(2)
thf(fact_325_Refine__Basic__Mirabelle__kwjuvthmas_Obind__mono_I1_J,axiom,
! [M4: refine424419629nres_a,M3: refine424419629nres_a,F: a > refine424419629nres_a,F3: a > refine424419629nres_a] :
( ( ord_le519537037nres_a @ M4 @ M3 )
=> ( ! [X3: a] :
( ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X3 ) @ M4 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F3 @ X3 ) ) )
=> ( ord_le519537037nres_a @ ( refine436832838nd_a_a @ M4 @ F ) @ ( refine436832838nd_a_a @ M3 @ F3 ) ) ) ) ).
% Refine_Basic_Mirabelle_kwjuvthmas.bind_mono(1)
thf(fact_326_le__SPEC__bindI,axiom,
! [Phi: a > $o,X4: a,M: refine424419629nres_a,F: a > refine424419629nres_a] :
( ( Phi @ X4 )
=> ( ( ord_le519537037nres_a @ M @ ( F @ X4 ) )
=> ( ord_le519537037nres_a @ M @ ( refine436832838nd_a_a @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) @ F ) ) ) ) ).
% le_SPEC_bindI
thf(fact_327_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_328_SUCCEED__rule,axiom,
! [Phi: a > $o] : ( ord_le519537037nres_a @ bot_bo529555393nres_a @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) ) ).
% SUCCEED_rule
thf(fact_329_bot_Oextremum__uniqueI,axiom,
! [A: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ bot_bo529555393nres_a )
=> ( A = bot_bo529555393nres_a ) ) ).
% bot.extremum_uniqueI
thf(fact_330_bot_Oextremum__unique,axiom,
! [A: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ bot_bo529555393nres_a )
= ( A = bot_bo529555393nres_a ) ) ).
% bot.extremum_unique
thf(fact_331_bot_Oextremum,axiom,
! [A: refine424419629nres_a] : ( ord_le519537037nres_a @ bot_bo529555393nres_a @ A ) ).
% bot.extremum
thf(fact_332_nres__order__simps_I1_J,axiom,
! [M4: refine424419629nres_a] : ( ord_le519537037nres_a @ bot_bo529555393nres_a @ M4 ) ).
% nres_order_simps(1)
thf(fact_333_bind__rule__complete,axiom,
! [M4: refine424419629nres_a,F: a > refine424419629nres_a,Phi: a > $o] :
( ( ord_le519537037nres_a @ ( refine436832838nd_a_a @ M4 @ F ) @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) )
= ( ord_le519537037nres_a @ M4
@ ( refine1198353288_RES_a
@ ( collect_a
@ ^ [X5: a] : ( ord_le519537037nres_a @ ( F @ X5 ) @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) ) ) ) ) ) ).
% bind_rule_complete
thf(fact_334_specify__left,axiom,
! [M: refine424419629nres_a,Phi: a > $o,F: a > refine424419629nres_a,M4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ M @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) )
=> ( ! [X3: a] :
( ( Phi @ X3 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ M4 ) )
=> ( ord_le519537037nres_a @ ( refine436832838nd_a_a @ M @ F ) @ M4 ) ) ) ).
% specify_left
thf(fact_335_bind__rule,axiom,
! [M4: refine424419629nres_a,F: a > refine424419629nres_a,Phi: a > $o] :
( ( ord_le519537037nres_a @ M4
@ ( refine1198353288_RES_a
@ ( collect_a
@ ^ [X5: a] : ( ord_le519537037nres_a @ ( F @ X5 ) @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) ) ) ) )
=> ( ord_le519537037nres_a @ ( refine436832838nd_a_a @ M4 @ F ) @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) ) ) ).
% bind_rule
thf(fact_336_bind__le__nofailI,axiom,
! [M: refine424419629nres_a,F: a > refine424419629nres_a,M7: refine424419629nres_a] :
( ( refine412683989fail_a @ M )
=> ( ! [X3: a] :
( ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X3 ) @ M )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ M7 ) )
=> ( ord_le519537037nres_a @ ( refine436832838nd_a_a @ M @ F ) @ M7 ) ) ) ).
% bind_le_nofailI
thf(fact_337_bind__le__shift,axiom,
! [M: refine424419629nres_a,F: a > refine424419629nres_a,M7: refine424419629nres_a] :
( ( ord_le519537037nres_a @ ( refine436832838nd_a_a @ M @ F ) @ M7 )
= ( ord_le519537037nres_a @ M
@ ( if_Ref1724547303nres_a @ ( refine412683989fail_a @ M7 )
@ ( refine1198353288_RES_a
@ ( collect_a
@ ^ [X5: a] : ( ord_le519537037nres_a @ ( F @ X5 ) @ M7 ) ) )
@ top_to231829469nres_a ) ) ) ).
% bind_le_shift
thf(fact_338_inf__top__left,axiom,
! [X4: refine424419629nres_a] :
( ( inf_in262696383nres_a @ top_to231829469nres_a @ X4 )
= X4 ) ).
% inf_top_left
thf(fact_339_inf__top__right,axiom,
! [X4: refine424419629nres_a] :
( ( inf_in262696383nres_a @ X4 @ top_to231829469nres_a )
= X4 ) ).
% inf_top_right
thf(fact_340_inf__eq__top__iff,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( ( inf_in262696383nres_a @ X4 @ Y3 )
= top_to231829469nres_a )
= ( ( X4 = top_to231829469nres_a )
& ( Y3 = top_to231829469nres_a ) ) ) ).
% inf_eq_top_iff
thf(fact_341_top__eq__inf__iff,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( top_to231829469nres_a
= ( inf_in262696383nres_a @ X4 @ Y3 ) )
= ( ( X4 = top_to231829469nres_a )
& ( Y3 = top_to231829469nres_a ) ) ) ).
% top_eq_inf_iff
thf(fact_342_inf__top_Oeq__neutr__iff,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a] :
( ( ( inf_in262696383nres_a @ A @ B )
= top_to231829469nres_a )
= ( ( A = top_to231829469nres_a )
& ( B = top_to231829469nres_a ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_343_inf__top_Oleft__neutral,axiom,
! [A: refine424419629nres_a] :
( ( inf_in262696383nres_a @ top_to231829469nres_a @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_344_inf__top_Oneutr__eq__iff,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a] :
( ( top_to231829469nres_a
= ( inf_in262696383nres_a @ A @ B ) )
= ( ( A = top_to231829469nres_a )
& ( B = top_to231829469nres_a ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_345_inf__top_Oright__neutral,axiom,
! [A: refine424419629nres_a] :
( ( inf_in262696383nres_a @ A @ top_to231829469nres_a )
= A ) ).
% inf_top.right_neutral
thf(fact_346_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_347_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_348_nres__more__simps_I1_J,axiom,
! [X: set_a] :
( ( bot_bo529555393nres_a
= ( refine1198353288_RES_a @ X ) )
= ( X = bot_bot_set_a ) ) ).
% nres_more_simps(1)
% Helper facts (7)
thf(help_If_2_1_If_001_062_Itf__a_M_Eo_J_T,axiom,
! [X4: a > $o,Y3: a > $o] :
( ( if_a_o @ $false @ X4 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001_062_Itf__a_M_Eo_J_T,axiom,
! [X4: a > $o,Y3: a > $o] :
( ( if_a_o @ $true @ X4 @ Y3 )
= X4 ) ).
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_3_1_If_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
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 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ord_le519537037nres_a @ a2
@ ( refine1198353288_RES_a
@ ( collect_a
@ ^ [V3: a] :
( ( p @ V3 )
& ( q @ V3 ) ) ) ) ) ).
%------------------------------------------------------------------------------