TPTP Problem File: ITP083^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP083^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Irreducible problem prob_641__6629180_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 : Irreducible/prob_641__6629180_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.23 v8.1.0, 0.18 v7.5.0
% Syntax : Number of formulae : 541 ( 183 unt; 187 typ; 0 def)
% Number of atoms : 948 ( 275 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 3285 ( 114 ~; 12 |; 66 &;2729 @)
% ( 0 <=>; 364 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 6 avg)
% Number of types : 53 ( 52 usr)
% Number of type conns : 511 ( 511 >; 0 *; 0 +; 0 <<)
% Number of symbols : 136 ( 135 usr; 10 con; 0-8 aty)
% Number of variables : 937 ( 98 ^; 800 !; 39 ?; 937 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:43:41.217
%------------------------------------------------------------------------------
% Could-be-implicit typings (52)
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_Itf__node_Mtf__val_J_J,type,
bot_bo404898488de_val: set_Pr699757092de_val ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__val_J_J,type,
bot_bot_set_set_val: set_set_val ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__val_J,type,
bot_bot_set_val: set_val ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Set__Oset_Itf__val_J_J,type,
ord_le596993316et_val: option_set_val > option_set_val > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_Itf__val_J_Mt__Set__Oset_Itf__val_J_J_J,type,
ord_le299366439et_val: set_Pr1311924359et_val > set_Pr1311924359et_val > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__node_Mtf__val_J_J,type,
ord_le1643692676de_val: set_Pr699757092de_val > set_Pr699757092de_val > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__val_J_J,type,
ord_le1742111550et_val: set_set_val > set_set_val > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__node_J,type,
ord_less_eq_set_node: set_node > set_node > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__val_J,type,
ord_less_eq_set_val: set_val > set_val > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__node,type,
ord_less_eq_node: node > node > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__val,type,
ord_less_eq_val: val > val > $o ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Set__Oset_Itf__val_J_Mt__Set__Oset_Itf__val_J_J_001t__Product____Type__Oprod_It__Set__Oset_Itf__val_J_Mt__Set__Oset_Itf__val_J_J,type,
produc1377626967et_val: produc1324971431et_val > produc1324971431et_val > produc1904584935et_val ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_Itf__node_Mtf__val_J_001t__Product____Type__Oprod_Itf__node_Mtf__val_J,type,
produc453318901de_val: produc1432036078de_val > produc1432036078de_val > produc452075965de_val ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_Itf__val_J_001t__Set__Oset_Itf__val_J,type,
produc1041633943et_val: set_val > set_val > produc1324971431et_val ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_Itf__val_J_001tf__node,type,
produc2052093836l_node: set_val > node > produc1013974930l_node ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_Itf__val_J_001tf__val,type,
produc1337903031al_val: set_val > val > produc782216903al_val ).
thf(sy_c_Product__Type_OPair_001tf__node_001t__Product____Type__Oprod_Itf__node_Mtf__val_J,type,
produc1162531244de_val: node > produc1432036078de_val > produc1615737844de_val ).
thf(sy_c_Product__Type_OPair_001tf__node_001t__Set__Oset_Itf__val_J,type,
produc2042994048et_val: node > set_val > produc1062784462et_val ).
thf(sy_c_Product__Type_OPair_001tf__node_001tf__node,type,
produc457016035e_node: node > node > produc951622955e_node ).
thf(sy_c_Product__Type_OPair_001tf__node_001tf__val,type,
produc1470527136de_val: node > val > produc1432036078de_val ).
thf(sy_c_Product__Type_OPair_001tf__val_001t__Product____Type__Oprod_Itf__node_Mtf__val_J,type,
produc1571934709de_val: val > produc1432036078de_val > produc1664334075de_val ).
thf(sy_c_Product__Type_OPair_001tf__val_001t__Set__Oset_Itf__val_J,type,
produc1446164855et_val: val > set_val > produc1858904199et_val ).
thf(sy_c_Product__Type_OPair_001tf__val_001tf__node,type,
produc1074923692l_node: val > node > produc1946948786l_node ).
thf(sy_c_Product__Type_OPair_001tf__val_001tf__val,type,
product_Pair_val_val: val > val > product_prod_val_val ).
thf(sy_c_SSA__CFG_OCFG__SSA__base_OallDefs_001tf__g_001tf__node_001tf__val,type,
sSA_CF139593942de_val: ( g > node > set_val ) > ( g > produc1432036078de_val > option_list_val ) > g > node > set_val ).
thf(sy_c_SSA__CFG_OCFG__SSA__base_OallUses_001tf__g_001tf__node_001tf__edgeD_001tf__val,type,
sSA_CF178745443eD_val: ( g > list_node ) > ( g > node > list_P561207620_edgeD ) > ( g > node > set_val ) > ( g > produc1432036078de_val > option_list_val ) > g > node > set_val ).
thf(sy_c_SSA__CFG_OCFG__SSA__base_OallVars_001tf__g_001tf__node_001tf__edgeD_001tf__val,type,
sSA_CF1517915011eD_val: ( g > list_node ) > ( g > node > list_P561207620_edgeD ) > ( g > node > set_val ) > ( g > node > set_val ) > ( g > produc1432036078de_val > option_list_val ) > g > set_val ).
thf(sy_c_SSA__CFG_OCFG__SSA__base_OphiDefs_001tf__g_001tf__node_001tf__val,type,
sSA_CF370335846de_val: ( g > produc1432036078de_val > option_list_val ) > g > node > set_val ).
thf(sy_c_SSA__CFG_OCFG__SSA__base_OphiUses_001tf__g_001tf__node_001tf__edgeD_001tf__val,type,
sSA_CF848637139eD_val: ( g > list_node ) > ( g > node > list_P561207620_edgeD ) > ( g > produc1432036078de_val > option_list_val ) > g > node > set_val ).
thf(sy_c_SSA__CFG_OCFG__SSA__wf__base_OdefNode_001tf__g_001tf__node_001tf__val,type,
sSA_CF551432799de_val: ( g > list_node ) > ( g > node > set_val ) > ( g > produc1432036078de_val > option_list_val ) > g > val > node ).
thf(sy_c_SSA__CFG_OCFG__SSA__wf__base_OisTrivialPhi_001tf__g_001tf__node_001tf__val,type,
sSA_CF1909049442de_val: ( g > list_node ) > ( g > node > set_val ) > ( g > produc1432036078de_val > option_list_val ) > g > val > val > $o ).
thf(sy_c_SSA__CFG_OCFG__SSA__wf__base_OliveVal_001tf__g_001tf__node_001tf__val,type,
sSA_CF794421325de_val: ( g > list_node ) > ( g > node > set_val ) > ( g > node > set_val ) > ( g > produc1432036078de_val > option_list_val ) > g > val > $o ).
thf(sy_c_SSA__CFG_OCFG__SSA__wf__base_OphiArg_001tf__g_001tf__node_001tf__val,type,
sSA_CF1252180629de_val: ( g > list_node ) > ( g > node > set_val ) > ( g > produc1432036078de_val > option_list_val ) > g > val > val > $o ).
thf(sy_c_SSA__CFG_OCFG__SSA__wf__base_Ophi_001tf__g_001tf__node_001tf__val,type,
sSA_CF262257161de_val: ( g > list_node ) > ( g > node > set_val ) > ( g > produc1432036078de_val > option_list_val ) > g > val > option_list_val ).
thf(sy_c_SSA__CFG_OCFG__SSA__wf__base_Opruned_001tf__g_001tf__node_001tf__val,type,
sSA_CF2074824714de_val: ( g > list_node ) > ( g > node > set_val ) > ( g > node > set_val ) > ( g > produc1432036078de_val > option_list_val ) > g > $o ).
thf(sy_c_SSA__CFG_OCFG__SSA__wf__base_Oredundant_001tf__g_001tf__node_001tf__edgeD_001tf__val,type,
sSA_CF1660885746eD_val: ( g > list_node ) > ( g > node > list_P561207620_edgeD ) > ( g > node > set_val ) > ( g > node > set_val ) > ( g > produc1432036078de_val > option_list_val ) > g > $o ).
thf(sy_c_SSA__CFG_OCFG__SSA__wf__base_Otrivial_001tf__g_001tf__node_001tf__edgeD_001tf__val,type,
sSA_CF1899243830eD_val: ( g > list_node ) > ( g > node > list_P561207620_edgeD ) > ( g > node > set_val ) > ( g > node > set_val ) > ( g > produc1432036078de_val > option_list_val ) > g > val > $o ).
thf(sy_c_SSA__CFG_OCFG__base_Ovars_001tf__g_001tf__node_001tf__val,type,
sSA_CF655860150de_val: ( g > list_node ) > ( g > node > set_val ) > g > set_val ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Set__Oset_Itf__val_J_Mt__Set__Oset_Itf__val_J_J,type,
collec1117167378et_val: ( produc1324971431et_val > $o ) > set_Pr1311924359et_val ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__node_Mtf__val_J,type,
collec370342979de_val: ( produc1432036078de_val > $o ) > set_Pr699757092de_val ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__val_Mtf__node_J,type,
collec885255687l_node: ( produc1946948786l_node > $o ) > set_Pr2105655784l_node ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__val_Mtf__val_J,type,
collec52550418al_val: ( product_prod_val_val > $o ) > set_Pr700443783al_val ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__val_J,type,
collect_set_val: ( set_val > $o ) > set_set_val ).
thf(sy_c_Set_OCollect_001tf__node,type,
collect_node: ( node > $o ) > set_node ).
thf(sy_c_Set_OCollect_001tf__val,type,
collect_val: ( val > $o ) > set_val ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Set__Oset_Itf__val_J_Mt__Set__Oset_Itf__val_J_J,type,
insert1846469495et_val: produc1324971431et_val > set_Pr1311924359et_val > set_Pr1311924359et_val ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_Itf__node_Mtf__val_J,type,
insert869443870de_val: produc1432036078de_val > set_Pr699757092de_val > set_Pr699757092de_val ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__val_J,type,
insert_set_val: set_val > set_set_val > set_set_val ).
thf(sy_c_Set_Oinsert_001tf__node,type,
insert_node: node > set_node > set_node ).
thf(sy_c_Set_Oinsert_001tf__val,type,
insert_val: val > set_val > set_val ).
thf(sy_c_Transitive__Closure_Oacyclic_001t__Set__Oset_Itf__val_J,type,
transi1953622797et_val: set_Pr1311924359et_val > $o ).
thf(sy_c_Transitive__Closure_Otranclp_001tf__val,type,
transi1991289355lp_val: ( val > val > $o ) > val > val > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_Itf__val_J_Mt__Set__Oset_Itf__val_J_J_Mt__Product____Type__Oprod_It__Set__Oset_Itf__val_J_Mt__Set__Oset_Itf__val_J_J_J,type,
member1254968080et_val: produc1904584935et_val > set_Pr1587248327et_val > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__node_Mtf__val_J_Mt__Product____Type__Oprod_Itf__node_Mtf__val_J_J,type,
member698732390de_val: produc452075965de_val > set_Pr819703837de_val > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_Itf__val_J_Mt__Set__Oset_Itf__val_J_J,type,
member1711426256et_val: produc1324971431et_val > set_Pr1311924359et_val > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_Itf__val_J_Mtf__node_J,type,
member76372137l_node: produc1013974930l_node > set_Pr574789832l_node > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_Itf__val_J_Mtf__val_J,type,
member1110026224al_val: produc782216903al_val > set_Pr1157324199al_val > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__node_Mt__Product____Type__Oprod_Itf__node_Mtf__val_J_J,type,
member562853661de_val: produc1615737844de_val > set_Pr1245322836de_val > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__node_Mt__Set__Oset_Itf__val_J_J,type,
member125181669et_val: produc1062784462et_val > set_Pr1320419588et_val > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__node_Mtf__node_J,type,
member2110109140e_node: produc951622955e_node > set_Pr9085835e_node > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__node_Mtf__val_J,type,
member313544709de_val: produc1432036078de_val > set_Pr699757092de_val > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__val_Mt__Product____Type__Oprod_Itf__node_Mtf__val_J_J,type,
member1894062994de_val: produc1664334075de_val > set_Pr701036849de_val > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__val_Mt__Set__Oset_Itf__val_J_J,type,
member39229872et_val: produc1858904199et_val > set_Pr978022247et_val > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__val_Mtf__node_J,type,
member828457417l_node: produc1946948786l_node > set_Pr2105655784l_node > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__val_Mtf__val_J,type,
member1680438992al_val: product_prod_val_val > set_Pr700443783al_val > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__val_J,type,
member_set_val: set_val > set_set_val > $o ).
thf(sy_c_member_001tf__node,type,
member_node: node > set_node > $o ).
thf(sy_c_member_001tf__val,type,
member_val: val > set_val > $o ).
thf(sy_v__092_060alpha_062n,type,
alpha_n: g > list_node ).
thf(sy_v__092_060phi_062,type,
phi: val ).
thf(sy_v__092_060phi_062_092_060_094sub_062s,type,
phi_s: val ).
thf(sy_v__092_060phi_062_H____,type,
phi2: val ).
thf(sy_v_defs,type,
defs: g > node > set_val ).
thf(sy_v_g,type,
g2: g ).
thf(sy_v_inEdges_H,type,
inEdges: g > node > list_P561207620_edgeD ).
thf(sy_v_phis,type,
phis: g > produc1432036078de_val > option_list_val ).
thf(sy_v_s,type,
s: val ).
thf(sy_v_uses,type,
uses: g > node > set_val ).
thf(sy_v_var,type,
var2: g > val > var ).
% Relevant facts (353)
thf(fact_0_assms_I4_J,axiom,
phi_s != s ).
% assms(4)
thf(fact_1_False,axiom,
phi != phi_s ).
% False
thf(fact_2__092_060open_062phiArg_Ag_A_092_060phi_062_H_A_092_060phi_062_092_060_094sub_062s_092_060close_062,axiom,
sSA_CF1252180629de_val @ alpha_n @ defs @ phis @ g2 @ phi2 @ phi_s ).
% \<open>phiArg g \<phi>' \<phi>\<^sub>s\<close>
thf(fact_3_assms_I3_J,axiom,
member_val @ s @ ( sSA_CF1517915011eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ g2 ) ).
% assms(3)
thf(fact_4_assms_I2_J,axiom,
member_val @ phi @ ( sSA_CF1517915011eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ g2 ) ).
% assms(2)
thf(fact_5__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062_092_060phi_062_H_O_AphiArg_Ag_A_092_060phi_062_H_A_092_060phi_062_092_060_094sub_062s_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Phi: val] :
~ ( sSA_CF1252180629de_val @ alpha_n @ defs @ phis @ g2 @ Phi @ phi_s ) ).
% \<open>\<And>thesis. (\<And>\<phi>'. phiArg g \<phi>' \<phi>\<^sub>s \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_6_phiArg__in__allVars,axiom,
! [G: g,V: val,V2: val] :
( ( sSA_CF1252180629de_val @ alpha_n @ defs @ phis @ G @ V @ V2 )
=> ( member_val @ V2 @ ( sSA_CF1517915011eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ G ) ) ) ).
% phiArg_in_allVars
thf(fact_7_uses__in__vars,axiom,
! [V: val,G: g,N: node] :
( ( member_val @ V @ ( uses @ G @ N ) )
=> ( member_val @ V @ ( sSA_CF655860150de_val @ alpha_n @ uses @ G ) ) ) ).
% uses_in_vars
thf(fact_8_trivial__in__allVars,axiom,
! [G: g,V: val] :
( ( sSA_CF1899243830eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ G @ V )
=> ( member_val @ V @ ( sSA_CF1517915011eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ G ) ) ) ).
% trivial_in_allVars
thf(fact_9_liveVal__in__allVars,axiom,
! [G: g,V: val] :
( ( sSA_CF794421325de_val @ alpha_n @ defs @ uses @ phis @ G @ V )
=> ( member_val @ V @ ( sSA_CF1517915011eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ G ) ) ) ).
% liveVal_in_allVars
thf(fact_10_allVars__finite,axiom,
! [G: g] : ( finite_finite_val @ ( sSA_CF1517915011eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ G ) ) ).
% allVars_finite
thf(fact_11_uses__in__allUses,axiom,
! [V: val,G: g,N: node] :
( ( member_val @ V @ ( uses @ G @ N ) )
=> ( member_val @ V @ ( sSA_CF178745443eD_val @ alpha_n @ inEdges @ uses @ phis @ G @ N ) ) ) ).
% uses_in_allUses
thf(fact_12_redundant__def,axiom,
! [G: g] :
( ( sSA_CF1660885746eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ G )
= ( ? [X: val] :
( ( member_val @ X @ ( sSA_CF1517915011eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ G ) )
& ( sSA_CF1899243830eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ G @ X ) ) ) ) ).
% redundant_def
thf(fact_13_trivial__def,axiom,
! [G: g,V: val] :
( ( sSA_CF1899243830eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ G @ V )
= ( ? [X: val] :
( ( member_val @ X @ ( sSA_CF1517915011eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ G ) )
& ( sSA_CF1909049442de_val @ alpha_n @ defs @ phis @ G @ V @ X ) ) ) ) ).
% trivial_def
thf(fact_14__092_060open_062_092_060not_062_AdefNode_Ag_A_092_060phi_062_092_060_094sub_062s_A_092_060noteq_062_AdefNode_Ag_As_092_060close_062,axiom,
( ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ g2 @ phi_s )
= ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ g2 @ s ) ) ).
% \<open>\<not> defNode g \<phi>\<^sub>s \<noteq> defNode g s\<close>
thf(fact_15_defs__uses__disjoint_H,axiom,
! [N: node,G: g,V: val] :
( ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( ( member_val @ V @ ( defs @ G @ N ) )
=> ~ ( member_val @ V @ ( uses @ G @ N ) ) ) ) ).
% defs_uses_disjoint'
thf(fact_16_livePhi,axiom,
! [G: g,V: val,V2: val] :
( ( sSA_CF794421325de_val @ alpha_n @ defs @ uses @ phis @ G @ V )
=> ( ( sSA_CF1252180629de_val @ alpha_n @ defs @ phis @ G @ V @ V2 )
=> ( sSA_CF794421325de_val @ alpha_n @ defs @ uses @ phis @ G @ V2 ) ) ) ).
% livePhi
thf(fact_17_phiArg__distinct__nodes,axiom,
! [G: g,P: val,Q: val] :
( ( sSA_CF1252180629de_val @ alpha_n @ defs @ phis @ G @ P @ Q )
=> ( ( P != Q )
=> ( ( member_val @ P @ ( sSA_CF1517915011eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ G ) )
=> ( ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ G @ P )
!= ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ G @ Q ) ) ) ) ) ).
% phiArg_distinct_nodes
thf(fact_18_phiArgs__def__distinct,axiom,
! [G: g,P: val,Q: val,R: val] :
( ( sSA_CF1252180629de_val @ alpha_n @ defs @ phis @ G @ P @ Q )
=> ( ( sSA_CF1252180629de_val @ alpha_n @ defs @ phis @ G @ P @ R )
=> ( ( Q != R )
=> ( ( member_val @ P @ ( sSA_CF1517915011eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ G ) )
=> ( ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ G @ Q )
!= ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ G @ R ) ) ) ) ) ) ).
% phiArgs_def_distinct
thf(fact_19_uses__in___092_060alpha_062n,axiom,
! [V: val,G: g,N: node] :
( ( member_val @ V @ ( uses @ G @ N ) )
=> ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) ) ) ).
% uses_in_\<alpha>n
thf(fact_20_defs__finite,axiom,
! [G: g,N: node] : ( finite_finite_val @ ( defs @ G @ N ) ) ).
% defs_finite
thf(fact_21_varsE,axiom,
! [V: val,G: g] :
( ( member_val @ V @ ( sSA_CF655860150de_val @ alpha_n @ uses @ G ) )
=> ~ ! [N2: node] :
( ( member_node @ N2 @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ~ ( member_val @ V @ ( uses @ G @ N2 ) ) ) ) ).
% varsE
thf(fact_22_vars__finite,axiom,
! [G: g] : ( finite_finite_val @ ( sSA_CF655860150de_val @ alpha_n @ uses @ G ) ) ).
% vars_finite
thf(fact_23_liveSimple,axiom,
! [N: node,G: g,Val: val] :
( ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( ( member_val @ Val @ ( uses @ G @ N ) )
=> ( sSA_CF794421325de_val @ alpha_n @ defs @ uses @ phis @ G @ Val ) ) ) ).
% liveSimple
thf(fact_24_liveVal_Osimps,axiom,
! [G: g,A: val] :
( ( sSA_CF794421325de_val @ alpha_n @ defs @ uses @ phis @ G @ A )
= ( ? [N3: node,Val2: val] :
( ( A = Val2 )
& ( member_node @ N3 @ ( set_node2 @ ( alpha_n @ G ) ) )
& ( member_val @ Val2 @ ( uses @ G @ N3 ) ) )
| ? [V3: val,V4: val] :
( ( A = V4 )
& ( sSA_CF794421325de_val @ alpha_n @ defs @ uses @ phis @ G @ V3 )
& ( sSA_CF1252180629de_val @ alpha_n @ defs @ phis @ G @ V3 @ V4 ) ) ) ) ).
% liveVal.simps
thf(fact_25_liveVal_Oinducts,axiom,
! [G: g,X2: val,P2: val > $o] :
( ( sSA_CF794421325de_val @ alpha_n @ defs @ uses @ phis @ G @ X2 )
=> ( ! [N2: node,Val3: val] :
( ( member_node @ N2 @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( ( member_val @ Val3 @ ( uses @ G @ N2 ) )
=> ( P2 @ Val3 ) ) )
=> ( ! [V5: val,V6: val] :
( ( sSA_CF794421325de_val @ alpha_n @ defs @ uses @ phis @ G @ V5 )
=> ( ( P2 @ V5 )
=> ( ( sSA_CF1252180629de_val @ alpha_n @ defs @ phis @ G @ V5 @ V6 )
=> ( P2 @ V6 ) ) ) )
=> ( P2 @ X2 ) ) ) ) ).
% liveVal.inducts
thf(fact_26_liveVal_Ocases,axiom,
! [G: g,A: val] :
( ( sSA_CF794421325de_val @ alpha_n @ defs @ uses @ phis @ G @ A )
=> ( ! [N2: node] :
( ( member_node @ N2 @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ~ ( member_val @ A @ ( uses @ G @ N2 ) ) )
=> ~ ! [V5: val] :
( ( sSA_CF794421325de_val @ alpha_n @ defs @ uses @ phis @ G @ V5 )
=> ~ ( sSA_CF1252180629de_val @ alpha_n @ defs @ phis @ G @ V5 @ A ) ) ) ) ).
% liveVal.cases
thf(fact_27_allUses__finite,axiom,
! [N: node,G: g] :
( ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( finite_finite_val @ ( sSA_CF178745443eD_val @ alpha_n @ inEdges @ uses @ phis @ G @ N ) ) ) ).
% allUses_finite
thf(fact_28_defNode_I1_J,axiom,
! [V: val,G: g] :
( ( member_val @ V @ ( sSA_CF1517915011eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ G ) )
=> ( member_node @ ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ G @ V ) @ ( set_node2 @ ( alpha_n @ G ) ) ) ) ).
% defNode(1)
thf(fact_29_uses__finite,axiom,
! [G: g,N: node] : ( finite_finite_val @ ( uses @ G @ N ) ) ).
% uses_finite
thf(fact_30_eq,axiom,
( ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ g2 @ phi_s )
= ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ g2 @ s ) ) ).
% eq
thf(fact_31_allUses__in__allVars,axiom,
! [V: val,G: g,N: node] :
( ( member_val @ V @ ( sSA_CF178745443eD_val @ alpha_n @ inEdges @ uses @ phis @ G @ N ) )
=> ( ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( member_val @ V @ ( sSA_CF1517915011eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ G ) ) ) ) ).
% allUses_in_allVars
thf(fact_32_phiUses__in__allUses,axiom,
! [V: val,G: g,N: node] :
( ( member_val @ V @ ( sSA_CF848637139eD_val @ alpha_n @ inEdges @ phis @ G @ N ) )
=> ( member_val @ V @ ( sSA_CF178745443eD_val @ alpha_n @ inEdges @ uses @ phis @ G @ N ) ) ) ).
% phiUses_in_allUses
thf(fact_33_phiUses__finite,axiom,
! [N: node,G: g] :
( ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( finite_finite_val @ ( sSA_CF848637139eD_val @ alpha_n @ inEdges @ phis @ G @ N ) ) ) ).
% phiUses_finite
thf(fact_34_defNode_I2_J,axiom,
! [V: val,G: g] :
( ( member_val @ V @ ( sSA_CF1517915011eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ G ) )
=> ( member_val @ V @ ( sSA_CF139593942de_val @ defs @ phis @ G @ ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ G @ V ) ) ) ) ).
% defNode(2)
thf(fact_35_defNode__ex1,axiom,
! [V: val,G: g] :
( ( member_val @ V @ ( sSA_CF1517915011eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ G ) )
=> ? [X3: node] :
( ( member_node @ X3 @ ( set_node2 @ ( alpha_n @ G ) ) )
& ( member_val @ V @ ( sSA_CF139593942de_val @ defs @ phis @ G @ X3 ) )
& ! [Y: node] :
( ( ( member_node @ Y @ ( set_node2 @ ( alpha_n @ G ) ) )
& ( member_val @ V @ ( sSA_CF139593942de_val @ defs @ phis @ G @ Y ) ) )
=> ( Y = X3 ) ) ) ) ).
% defNode_ex1
thf(fact_36_allVars__in__allDefs,axiom,
! [V: val,G: g] :
( ( member_val @ V @ ( sSA_CF1517915011eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ G ) )
=> ? [X3: node] :
( ( member_node @ X3 @ ( set_node2 @ ( alpha_n @ G ) ) )
& ( member_val @ V @ ( sSA_CF139593942de_val @ defs @ phis @ G @ X3 ) ) ) ) ).
% allVars_in_allDefs
thf(fact_37_List_Ofinite__set,axiom,
! [Xs: list_P844977291de_val] : ( finite781217938de_val @ ( set_Pr434939430de_val @ Xs ) ) ).
% List.finite_set
thf(fact_38_List_Ofinite__set,axiom,
! [Xs: list_P761482893et_val] : ( finite1581417648et_val @ ( set_Pr2104310812et_val @ Xs ) ) ).
% List.finite_set
thf(fact_39_List_Ofinite__set,axiom,
! [Xs: list_P1078858818l_node] : ( finite423892681l_node @ ( set_Pr1600883293l_node @ Xs ) ) ).
% List.finite_set
thf(fact_40_List_Ofinite__set,axiom,
! [Xs: list_P1811375021al_val] : ( finite759259600al_val @ ( set_Pr102746428al_val @ Xs ) ) ).
% List.finite_set
thf(fact_41_List_Ofinite__set,axiom,
! [Xs: list_set_val] : ( finite1265617511et_val @ ( set_set_val2 @ Xs ) ) ).
% List.finite_set
thf(fact_42_List_Ofinite__set,axiom,
! [Xs: list_node] : ( finite_finite_node @ ( set_node2 @ Xs ) ) ).
% List.finite_set
thf(fact_43_List_Ofinite__set,axiom,
! [Xs: list_val] : ( finite_finite_val @ ( set_val2 @ Xs ) ) ).
% List.finite_set
thf(fact_44_List_Ofinite__set,axiom,
! [Xs: list_P2089461677et_val] : ( finite79836624et_val @ ( set_Pr284769596et_val @ Xs ) ) ).
% List.finite_set
thf(fact_45_List_Ofinite__set,axiom,
! [Xs: list_P1820443774de_val] : ( finite2056463621de_val @ ( set_Pr1085970585de_val @ Xs ) ) ).
% List.finite_set
thf(fact_46_phiDefs__finite,axiom,
! [G: g,N: node] : ( finite_finite_val @ ( sSA_CF370335846de_val @ phis @ G @ N ) ) ).
% phiDefs_finite
thf(fact_47_allDefs__finite,axiom,
! [N: node,G: g] :
( ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( finite_finite_val @ ( sSA_CF139593942de_val @ defs @ phis @ G @ N ) ) ) ).
% allDefs_finite
thf(fact_48_assms_I5_J,axiom,
( ( var2 @ g2 @ phi )
= ( var2 @ g2 @ s ) ) ).
% assms(5)
thf(fact_49_CFG__SSA__wf__base_Oredundant__def,axiom,
( sSA_CF1660885746eD_val
= ( ^ [Alpha_n: g > list_node,InEdges: g > node > list_P561207620_edgeD,Defs: g > node > set_val,Uses: g > node > set_val,Phis: g > produc1432036078de_val > option_list_val,G2: g] :
? [X: val] :
( ( member_val @ X @ ( sSA_CF1517915011eD_val @ Alpha_n @ InEdges @ Defs @ Uses @ Phis @ G2 ) )
& ( sSA_CF1899243830eD_val @ Alpha_n @ InEdges @ Defs @ Uses @ Phis @ G2 @ X ) ) ) ) ).
% CFG_SSA_wf_base.redundant_def
thf(fact_50_defs__in__allDefs,axiom,
! [V: val,G: g,N: node] :
( ( member_val @ V @ ( defs @ G @ N ) )
=> ( member_val @ V @ ( sSA_CF139593942de_val @ defs @ phis @ G @ N ) ) ) ).
% defs_in_allDefs
thf(fact_51_phiDefs__in__allDefs,axiom,
! [V: val,G: g,N: node] :
( ( member_val @ V @ ( sSA_CF370335846de_val @ phis @ G @ N ) )
=> ( member_val @ V @ ( sSA_CF139593942de_val @ defs @ phis @ G @ N ) ) ) ).
% phiDefs_in_allDefs
thf(fact_52_allDefs__disjoint_H,axiom,
! [N: node,G: g,M: node,V: val] :
( ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( ( member_node @ M @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( ( member_val @ V @ ( sSA_CF139593942de_val @ defs @ phis @ G @ N ) )
=> ( ( member_val @ V @ ( sSA_CF139593942de_val @ defs @ phis @ G @ M ) )
=> ( N = M ) ) ) ) ) ).
% allDefs_disjoint'
thf(fact_53_mem__Collect__eq,axiom,
! [A: produc1946948786l_node,P2: produc1946948786l_node > $o] :
( ( member828457417l_node @ A @ ( collec885255687l_node @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_54_mem__Collect__eq,axiom,
! [A: product_prod_val_val,P2: product_prod_val_val > $o] :
( ( member1680438992al_val @ A @ ( collec52550418al_val @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_55_mem__Collect__eq,axiom,
! [A: val,P2: val > $o] :
( ( member_val @ A @ ( collect_val @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_56_mem__Collect__eq,axiom,
! [A: node,P2: node > $o] :
( ( member_node @ A @ ( collect_node @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_57_mem__Collect__eq,axiom,
! [A: set_val,P2: set_val > $o] :
( ( member_set_val @ A @ ( collect_set_val @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_58_mem__Collect__eq,axiom,
! [A: produc1432036078de_val,P2: produc1432036078de_val > $o] :
( ( member313544709de_val @ A @ ( collec370342979de_val @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_59_mem__Collect__eq,axiom,
! [A: produc1324971431et_val,P2: produc1324971431et_val > $o] :
( ( member1711426256et_val @ A @ ( collec1117167378et_val @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_60_Collect__mem__eq,axiom,
! [A2: set_val] :
( ( collect_val
@ ^ [X: val] : ( member_val @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_61_Collect__mem__eq,axiom,
! [A2: set_node] :
( ( collect_node
@ ^ [X: node] : ( member_node @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_62_Collect__mem__eq,axiom,
! [A2: set_set_val] :
( ( collect_set_val
@ ^ [X: set_val] : ( member_set_val @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_63_Collect__mem__eq,axiom,
! [A2: set_Pr699757092de_val] :
( ( collec370342979de_val
@ ^ [X: produc1432036078de_val] : ( member313544709de_val @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_64_Collect__mem__eq,axiom,
! [A2: set_Pr1311924359et_val] :
( ( collec1117167378et_val
@ ^ [X: produc1324971431et_val] : ( member1711426256et_val @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_65_phiArg__same__var,axiom,
! [G: g,P: val,Q: val] :
( ( sSA_CF1252180629de_val @ alpha_n @ defs @ phis @ G @ P @ Q )
=> ( ( var2 @ G @ Q )
= ( var2 @ G @ P ) ) ) ).
% phiArg_same_var
thf(fact_66_vars__eq,axiom,
( ( var2 @ g2 @ phi )
= ( var2 @ g2 @ phi_s ) ) ).
% vars_eq
thf(fact_67_allDefs__var__disjoint,axiom,
! [N: node,G: g,V: val,V2: val] :
( ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( ( member_val @ V @ ( sSA_CF139593942de_val @ defs @ phis @ G @ N ) )
=> ( ( member_val @ V2 @ ( sSA_CF139593942de_val @ defs @ phis @ G @ N ) )
=> ( ( V != V2 )
=> ( ( var2 @ G @ V2 )
!= ( var2 @ G @ V ) ) ) ) ) ) ).
% allDefs_var_disjoint
thf(fact_68_defNode__var__disjoint,axiom,
! [P: val,G: g,Q: val] :
( ( member_val @ P @ ( sSA_CF1517915011eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ G ) )
=> ( ( member_val @ Q @ ( sSA_CF1517915011eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ G ) )
=> ( ( P != Q )
=> ( ( ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ G @ P )
= ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ G @ Q ) )
=> ( ( var2 @ G @ P )
!= ( var2 @ G @ Q ) ) ) ) ) ) ).
% defNode_var_disjoint
thf(fact_69_defNode__eq,axiom,
! [N: node,G: g,V: val] :
( ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( ( member_val @ V @ ( sSA_CF139593942de_val @ defs @ phis @ G @ N ) )
=> ( ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ G @ V )
= N ) ) ) ).
% defNode_eq
thf(fact_70_allDefs__in__allVars,axiom,
! [V: val,G: g,N: node] :
( ( member_val @ V @ ( sSA_CF139593942de_val @ defs @ phis @ G @ N ) )
=> ( ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( member_val @ V @ ( sSA_CF1517915011eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ G ) ) ) ) ).
% allDefs_in_allVars
thf(fact_71_pruned__def,axiom,
! [G: g] :
( ( sSA_CF2074824714de_val @ alpha_n @ defs @ uses @ phis @ G )
= ( ! [X: node] :
( ( member_node @ X @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ! [Val2: val] :
( ( member_val @ Val2 @ ( sSA_CF370335846de_val @ phis @ G @ X ) )
=> ( sSA_CF794421325de_val @ alpha_n @ defs @ uses @ phis @ G @ Val2 ) ) ) ) ) ).
% pruned_def
thf(fact_72_CFG__SSA__base_OphiUses_Ocong,axiom,
sSA_CF848637139eD_val = sSA_CF848637139eD_val ).
% CFG_SSA_base.phiUses.cong
thf(fact_73_CFG__SSA__base_OphiDefs_Ocong,axiom,
sSA_CF370335846de_val = sSA_CF370335846de_val ).
% CFG_SSA_base.phiDefs.cong
thf(fact_74_CFG__SSA__base_OallDefs_Ocong,axiom,
sSA_CF139593942de_val = sSA_CF139593942de_val ).
% CFG_SSA_base.allDefs.cong
thf(fact_75_CFG__SSA__wf__base_OdefNode_Ocong,axiom,
sSA_CF551432799de_val = sSA_CF551432799de_val ).
% CFG_SSA_wf_base.defNode.cong
thf(fact_76_CFG__SSA__base_OallVars_Ocong,axiom,
sSA_CF1517915011eD_val = sSA_CF1517915011eD_val ).
% CFG_SSA_base.allVars.cong
thf(fact_77_CFG__SSA__wf__base_OphiArg_Ocong,axiom,
sSA_CF1252180629de_val = sSA_CF1252180629de_val ).
% CFG_SSA_wf_base.phiArg.cong
thf(fact_78_CFG__SSA__base_OallUses_Ocong,axiom,
sSA_CF178745443eD_val = sSA_CF178745443eD_val ).
% CFG_SSA_base.allUses.cong
thf(fact_79_CFG__SSA__wf__base_OliveVal_Ocong,axiom,
sSA_CF794421325de_val = sSA_CF794421325de_val ).
% CFG_SSA_wf_base.liveVal.cong
thf(fact_80_CFG__SSA__wf__base_Otrivial_Ocong,axiom,
sSA_CF1899243830eD_val = sSA_CF1899243830eD_val ).
% CFG_SSA_wf_base.trivial.cong
thf(fact_81_CFG__base_Ovars_Ocong,axiom,
sSA_CF655860150de_val = sSA_CF655860150de_val ).
% CFG_base.vars.cong
thf(fact_82_CFG__SSA__wf__base_OisTrivialPhi_Ocong,axiom,
sSA_CF1909049442de_val = sSA_CF1909049442de_val ).
% CFG_SSA_wf_base.isTrivialPhi.cong
thf(fact_83_CFG__SSA__wf__base_Oredundant_Ocong,axiom,
sSA_CF1660885746eD_val = sSA_CF1660885746eD_val ).
% CFG_SSA_wf_base.redundant.cong
thf(fact_84_finite__list,axiom,
! [A2: set_node] :
( ( finite_finite_node @ A2 )
=> ? [Xs2: list_node] :
( ( set_node2 @ Xs2 )
= A2 ) ) ).
% finite_list
thf(fact_85_finite__list,axiom,
! [A2: set_val] :
( ( finite_finite_val @ A2 )
=> ? [Xs2: list_val] :
( ( set_val2 @ Xs2 )
= A2 ) ) ).
% finite_list
thf(fact_86_finite__list,axiom,
! [A2: set_Pr1311924359et_val] :
( ( finite79836624et_val @ A2 )
=> ? [Xs2: list_P2089461677et_val] :
( ( set_Pr284769596et_val @ Xs2 )
= A2 ) ) ).
% finite_list
thf(fact_87_finite__list,axiom,
! [A2: set_Pr699757092de_val] :
( ( finite2056463621de_val @ A2 )
=> ? [Xs2: list_P1820443774de_val] :
( ( set_Pr1085970585de_val @ Xs2 )
= A2 ) ) ).
% finite_list
thf(fact_88_CFG__SSA__wf__base_OliveSimple,axiom,
! [N: node,Alpha_n2: g > list_node,G: g,Val: val,Uses2: g > node > set_val,Defs2: g > node > set_val,Phis2: g > produc1432036078de_val > option_list_val] :
( ( member_node @ N @ ( set_node2 @ ( Alpha_n2 @ G ) ) )
=> ( ( member_val @ Val @ ( Uses2 @ G @ N ) )
=> ( sSA_CF794421325de_val @ Alpha_n2 @ Defs2 @ Uses2 @ Phis2 @ G @ Val ) ) ) ).
% CFG_SSA_wf_base.liveSimple
thf(fact_89_CFG__SSA__wf__base_OlivePhi,axiom,
! [Alpha_n2: g > list_node,Defs2: g > node > set_val,Uses2: g > node > set_val,Phis2: g > produc1432036078de_val > option_list_val,G: g,V: val,V2: val] :
( ( sSA_CF794421325de_val @ Alpha_n2 @ Defs2 @ Uses2 @ Phis2 @ G @ V )
=> ( ( sSA_CF1252180629de_val @ Alpha_n2 @ Defs2 @ Phis2 @ G @ V @ V2 )
=> ( sSA_CF794421325de_val @ Alpha_n2 @ Defs2 @ Uses2 @ Phis2 @ G @ V2 ) ) ) ).
% CFG_SSA_wf_base.livePhi
thf(fact_90_CFG__SSA__wf__base_OliveVal_Oinducts,axiom,
! [Alpha_n2: g > list_node,Defs2: g > node > set_val,Uses2: g > node > set_val,Phis2: g > produc1432036078de_val > option_list_val,G: g,X2: val,P2: val > $o] :
( ( sSA_CF794421325de_val @ Alpha_n2 @ Defs2 @ Uses2 @ Phis2 @ G @ X2 )
=> ( ! [N2: node,Val3: val] :
( ( member_node @ N2 @ ( set_node2 @ ( Alpha_n2 @ G ) ) )
=> ( ( member_val @ Val3 @ ( Uses2 @ G @ N2 ) )
=> ( P2 @ Val3 ) ) )
=> ( ! [V5: val,V6: val] :
( ( sSA_CF794421325de_val @ Alpha_n2 @ Defs2 @ Uses2 @ Phis2 @ G @ V5 )
=> ( ( P2 @ V5 )
=> ( ( sSA_CF1252180629de_val @ Alpha_n2 @ Defs2 @ Phis2 @ G @ V5 @ V6 )
=> ( P2 @ V6 ) ) ) )
=> ( P2 @ X2 ) ) ) ) ).
% CFG_SSA_wf_base.liveVal.inducts
thf(fact_91_CFG__SSA__wf__base_OliveVal_Osimps,axiom,
( sSA_CF794421325de_val
= ( ^ [Alpha_n: g > list_node,Defs: g > node > set_val,Uses: g > node > set_val,Phis: g > produc1432036078de_val > option_list_val,G2: g,A3: val] :
( ? [N3: node,Val2: val] :
( ( A3 = Val2 )
& ( member_node @ N3 @ ( set_node2 @ ( Alpha_n @ G2 ) ) )
& ( member_val @ Val2 @ ( Uses @ G2 @ N3 ) ) )
| ? [V3: val,V4: val] :
( ( A3 = V4 )
& ( sSA_CF794421325de_val @ Alpha_n @ Defs @ Uses @ Phis @ G2 @ V3 )
& ( sSA_CF1252180629de_val @ Alpha_n @ Defs @ Phis @ G2 @ V3 @ V4 ) ) ) ) ) ).
% CFG_SSA_wf_base.liveVal.simps
thf(fact_92_CFG__SSA__wf__base_OliveVal_Ocases,axiom,
! [Alpha_n2: g > list_node,Defs2: g > node > set_val,Uses2: g > node > set_val,Phis2: g > produc1432036078de_val > option_list_val,G: g,A: val] :
( ( sSA_CF794421325de_val @ Alpha_n2 @ Defs2 @ Uses2 @ Phis2 @ G @ A )
=> ( ! [N2: node] :
( ( member_node @ N2 @ ( set_node2 @ ( Alpha_n2 @ G ) ) )
=> ~ ( member_val @ A @ ( Uses2 @ G @ N2 ) ) )
=> ~ ! [V5: val] :
( ( sSA_CF794421325de_val @ Alpha_n2 @ Defs2 @ Uses2 @ Phis2 @ G @ V5 )
=> ~ ( sSA_CF1252180629de_val @ Alpha_n2 @ Defs2 @ Phis2 @ G @ V5 @ A ) ) ) ) ).
% CFG_SSA_wf_base.liveVal.cases
thf(fact_93_CFG__SSA__wf__base_Otrivial__def,axiom,
( sSA_CF1899243830eD_val
= ( ^ [Alpha_n: g > list_node,InEdges: g > node > list_P561207620_edgeD,Defs: g > node > set_val,Uses: g > node > set_val,Phis: g > produc1432036078de_val > option_list_val,G2: g,V3: val] :
? [X: val] :
( ( member_val @ X @ ( sSA_CF1517915011eD_val @ Alpha_n @ InEdges @ Defs @ Uses @ Phis @ G2 ) )
& ( sSA_CF1909049442de_val @ Alpha_n @ Defs @ Phis @ G2 @ V3 @ X ) ) ) ) ).
% CFG_SSA_wf_base.trivial_def
thf(fact_94_condensation__finite,axiom,
! [G: g,P2: set_val] : ( finite79836624et_val @ ( irredu893086079de_val @ alpha_n @ defs @ phis @ G @ P2 ) ) ).
% condensation_finite
thf(fact_95_phiUses__exI_H,axiom,
! [G: g,P: val,Q: val] :
( ( sSA_CF1252180629de_val @ alpha_n @ defs @ phis @ G @ P @ Q )
=> ( ( member_val @ P @ ( sSA_CF1517915011eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ G ) )
=> ~ ! [M2: node] :
( ( member_val @ Q @ ( sSA_CF848637139eD_val @ alpha_n @ inEdges @ phis @ G @ M2 ) )
=> ~ ( member_node @ M2 @ ( set_node2 @ ( graph_272749361_edgeD @ inEdges @ G @ ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ G @ P ) ) ) ) ) ) ) ).
% phiUses_exI'
thf(fact_96_allUses__def,axiom,
! [G: g,N: node] :
( ( sSA_CF178745443eD_val @ alpha_n @ inEdges @ uses @ phis @ G @ N )
= ( sup_sup_set_val @ ( uses @ G @ N ) @ ( sSA_CF848637139eD_val @ alpha_n @ inEdges @ phis @ G @ N ) ) ) ).
% allUses_def
thf(fact_97_phiArg__trancl__same__var,axiom,
! [G: g,Phi2: val,N: val] :
( ( transi1991289355lp_val @ ( sSA_CF1252180629de_val @ alpha_n @ defs @ phis @ G ) @ Phi2 @ N )
=> ( ( var2 @ G @ Phi2 )
= ( var2 @ G @ N ) ) ) ).
% phiArg_trancl_same_var
thf(fact_98_allDefs__def,axiom,
! [G: g,N: node] :
( ( sSA_CF139593942de_val @ defs @ phis @ G @ N )
= ( sup_sup_set_val @ ( defs @ G @ N ) @ ( sSA_CF370335846de_val @ phis @ G @ N ) ) ) ).
% allDefs_def
thf(fact_99_CFG__SSA__wf__base_Opruned__def,axiom,
( sSA_CF2074824714de_val
= ( ^ [Alpha_n: g > list_node,Defs: g > node > set_val,Uses: g > node > set_val,Phis: g > produc1432036078de_val > option_list_val,G2: g] :
! [X: node] :
( ( member_node @ X @ ( set_node2 @ ( Alpha_n @ G2 ) ) )
=> ! [Val2: val] :
( ( member_val @ Val2 @ ( sSA_CF370335846de_val @ Phis @ G2 @ X ) )
=> ( sSA_CF794421325de_val @ Alpha_n @ Defs @ Uses @ Phis @ G2 @ Val2 ) ) ) ) ) ).
% CFG_SSA_wf_base.pruned_def
thf(fact_100_defNode__cases,axiom,
! [V: val,G: g] :
( ( member_val @ V @ ( sSA_CF1517915011eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ G ) )
=> ( ~ ( member_val @ V @ ( defs @ G @ ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ G @ V ) ) )
=> ( ( sSA_CF262257161de_val @ alpha_n @ defs @ phis @ G @ V )
!= none_list_val ) ) ) ).
% defNode_cases
thf(fact_101_simpleDef__not__phi,axiom,
! [N: node,G: g,V: val] :
( ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( ( member_val @ V @ ( defs @ G @ N ) )
=> ( ( sSA_CF262257161de_val @ alpha_n @ defs @ phis @ G @ V )
= none_list_val ) ) ) ).
% simpleDef_not_phi
thf(fact_102_finite__Un,axiom,
! [F: set_Pr1311924359et_val,G3: set_Pr1311924359et_val] :
( ( finite79836624et_val @ ( sup_su2005730907et_val @ F @ G3 ) )
= ( ( finite79836624et_val @ F )
& ( finite79836624et_val @ G3 ) ) ) ).
% finite_Un
thf(fact_103_finite__Un,axiom,
! [F: set_Pr699757092de_val,G3: set_Pr699757092de_val] :
( ( finite2056463621de_val @ ( sup_su2138939728de_val @ F @ G3 ) )
= ( ( finite2056463621de_val @ F )
& ( finite2056463621de_val @ G3 ) ) ) ).
% finite_Un
thf(fact_104_finite__Un,axiom,
! [F: set_val,G3: set_val] :
( ( finite_finite_val @ ( sup_sup_set_val @ F @ G3 ) )
= ( ( finite_finite_val @ F )
& ( finite_finite_val @ G3 ) ) ) ).
% finite_Un
thf(fact_105_trivial__phi,axiom,
! [G: g,V: val] :
( ( sSA_CF1899243830eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ G @ V )
=> ( ( sSA_CF262257161de_val @ alpha_n @ defs @ phis @ G @ V )
!= none_list_val ) ) ).
% trivial_phi
thf(fact_106_phiArg__exI,axiom,
! [M: node,G: g,V: val] :
( ( member_node @ M @ ( set_node2 @ ( graph_272749361_edgeD @ inEdges @ G @ ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ G @ V ) ) ) )
=> ( ( ( sSA_CF262257161de_val @ alpha_n @ defs @ phis @ G @ V )
!= none_list_val )
=> ( ( member_val @ V @ ( sSA_CF1517915011eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ G ) )
=> ~ ! [V6: val] :
( ( member_val @ V6 @ ( sSA_CF848637139eD_val @ alpha_n @ inEdges @ phis @ G @ M ) )
=> ~ ( sSA_CF1252180629de_val @ alpha_n @ defs @ phis @ G @ V @ V6 ) ) ) ) ) ).
% phiArg_exI
thf(fact_107_CFG__SSA__wf__base_Ophi_Ocong,axiom,
sSA_CF262257161de_val = sSA_CF262257161de_val ).
% CFG_SSA_wf_base.phi.cong
thf(fact_108_infinite__Un,axiom,
! [S: set_Pr1311924359et_val,T: set_Pr1311924359et_val] :
( ( ~ ( finite79836624et_val @ ( sup_su2005730907et_val @ S @ T ) ) )
= ( ~ ( finite79836624et_val @ S )
| ~ ( finite79836624et_val @ T ) ) ) ).
% infinite_Un
thf(fact_109_infinite__Un,axiom,
! [S: set_Pr699757092de_val,T: set_Pr699757092de_val] :
( ( ~ ( finite2056463621de_val @ ( sup_su2138939728de_val @ S @ T ) ) )
= ( ~ ( finite2056463621de_val @ S )
| ~ ( finite2056463621de_val @ T ) ) ) ).
% infinite_Un
thf(fact_110_infinite__Un,axiom,
! [S: set_val,T: set_val] :
( ( ~ ( finite_finite_val @ ( sup_sup_set_val @ S @ T ) ) )
= ( ~ ( finite_finite_val @ S )
| ~ ( finite_finite_val @ T ) ) ) ).
% infinite_Un
thf(fact_111_Un__infinite,axiom,
! [S: set_Pr1311924359et_val,T: set_Pr1311924359et_val] :
( ~ ( finite79836624et_val @ S )
=> ~ ( finite79836624et_val @ ( sup_su2005730907et_val @ S @ T ) ) ) ).
% Un_infinite
thf(fact_112_Un__infinite,axiom,
! [S: set_Pr699757092de_val,T: set_Pr699757092de_val] :
( ~ ( finite2056463621de_val @ S )
=> ~ ( finite2056463621de_val @ ( sup_su2138939728de_val @ S @ T ) ) ) ).
% Un_infinite
thf(fact_113_Un__infinite,axiom,
! [S: set_val,T: set_val] :
( ~ ( finite_finite_val @ S )
=> ~ ( finite_finite_val @ ( sup_sup_set_val @ S @ T ) ) ) ).
% Un_infinite
thf(fact_114_finite__UnI,axiom,
! [F: set_Pr1311924359et_val,G3: set_Pr1311924359et_val] :
( ( finite79836624et_val @ F )
=> ( ( finite79836624et_val @ G3 )
=> ( finite79836624et_val @ ( sup_su2005730907et_val @ F @ G3 ) ) ) ) ).
% finite_UnI
thf(fact_115_finite__UnI,axiom,
! [F: set_Pr699757092de_val,G3: set_Pr699757092de_val] :
( ( finite2056463621de_val @ F )
=> ( ( finite2056463621de_val @ G3 )
=> ( finite2056463621de_val @ ( sup_su2138939728de_val @ F @ G3 ) ) ) ) ).
% finite_UnI
thf(fact_116_finite__UnI,axiom,
! [F: set_val,G3: set_val] :
( ( finite_finite_val @ F )
=> ( ( finite_finite_val @ G3 )
=> ( finite_finite_val @ ( sup_sup_set_val @ F @ G3 ) ) ) ) ).
% finite_UnI
thf(fact_117_CFG__SSA__Transformed_Ocondensation__edges_Ocong,axiom,
irredu893086079de_val = irredu893086079de_val ).
% CFG_SSA_Transformed.condensation_edges.cong
thf(fact_118_CFG__SSA__wf__base_Opruned_Ocong,axiom,
sSA_CF2074824714de_val = sSA_CF2074824714de_val ).
% CFG_SSA_wf_base.pruned.cong
thf(fact_119_CFG__SSA__base_OallDefs__def,axiom,
( sSA_CF139593942de_val
= ( ^ [Defs: g > node > set_val,Phis: g > produc1432036078de_val > option_list_val,G2: g,N3: node] : ( sup_sup_set_val @ ( Defs @ G2 @ N3 ) @ ( sSA_CF370335846de_val @ Phis @ G2 @ N3 ) ) ) ) ).
% CFG_SSA_base.allDefs_def
thf(fact_120_CFG__SSA__base_OallUses__def,axiom,
( sSA_CF178745443eD_val
= ( ^ [Alpha_n: g > list_node,InEdges: g > node > list_P561207620_edgeD,Uses: g > node > set_val,Phis: g > produc1432036078de_val > option_list_val,G2: g,N3: node] : ( sup_sup_set_val @ ( Uses @ G2 @ N3 ) @ ( sSA_CF848637139eD_val @ Alpha_n @ InEdges @ Phis @ G2 @ N3 ) ) ) ) ).
% CFG_SSA_base.allUses_def
thf(fact_121_phi__phiDefs,axiom,
! [G: g,V: val,Vs: list_val] :
( ( ( sSA_CF262257161de_val @ alpha_n @ defs @ phis @ G @ V )
= ( some_list_val @ Vs ) )
=> ( member_val @ V @ ( sSA_CF370335846de_val @ phis @ G @ ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ G @ V ) ) ) ) ).
% phi_phiDefs
thf(fact_122_condensation__acyclic,axiom,
! [G: g,P2: set_val] : ( transi1953622797et_val @ ( irredu893086079de_val @ alpha_n @ defs @ phis @ G @ P2 ) ) ).
% condensation_acyclic
thf(fact_123_phi__def,axiom,
! [G: g,V: val] :
( ( sSA_CF262257161de_val @ alpha_n @ defs @ phis @ G @ V )
= ( phis @ G @ ( produc1470527136de_val @ ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ G @ V ) @ V ) ) ) ).
% phi_def
thf(fact_124_phiArg__def,axiom,
! [G: g,V: val,V2: val] :
( ( sSA_CF1252180629de_val @ alpha_n @ defs @ phis @ G @ V @ V2 )
= ( ? [Vs2: list_val] :
( ( ( sSA_CF262257161de_val @ alpha_n @ defs @ phis @ G @ V )
= ( some_list_val @ Vs2 ) )
& ( member_val @ V2 @ ( set_val2 @ Vs2 ) ) ) ) ) ).
% phiArg_def
thf(fact_125_phi__finite,axiom,
! [G: g] : ( finite_finite_val @ ( dom_val_list_val @ ( sSA_CF262257161de_val @ alpha_n @ defs @ phis @ G ) ) ) ).
% phi_finite
thf(fact_126_redundant__scc__phis,axiom,
! [G: g,P2: set_val,Scc: set_val,X2: val] :
( ( irredu224829488eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ G @ P2 )
=> ( ( member_set_val @ Scc @ ( irredu681226810de_val @ alpha_n @ defs @ phis @ G @ P2 ) )
=> ( ( member_val @ X2 @ Scc )
=> ( ( sSA_CF262257161de_val @ alpha_n @ defs @ phis @ G @ X2 )
!= none_list_val ) ) ) ) ).
% redundant_scc_phis
thf(fact_127_phis__disj_I2_J,axiom,
! [G: g,N: node,V: val,Vs: list_val,N4: node,Vs3: list_val] :
( ( ( phis @ G @ ( produc1470527136de_val @ N @ V ) )
= ( some_list_val @ Vs ) )
=> ( ( ( phis @ G @ ( produc1470527136de_val @ N4 @ V ) )
= ( some_list_val @ Vs3 ) )
=> ( Vs = Vs3 ) ) ) ).
% phis_disj(2)
thf(fact_128_phis__disj_I1_J,axiom,
! [G: g,N: node,V: val,Vs: list_val,N4: node,Vs3: list_val] :
( ( ( phis @ G @ ( produc1470527136de_val @ N @ V ) )
= ( some_list_val @ Vs ) )
=> ( ( ( phis @ G @ ( produc1470527136de_val @ N4 @ V ) )
= ( some_list_val @ Vs3 ) )
=> ( N = N4 ) ) ) ).
% phis_disj(1)
thf(fact_129_phis__in___092_060alpha_062n,axiom,
! [G: g,N: node,V: val,Vs: list_val] :
( ( ( phis @ G @ ( produc1470527136de_val @ N @ V ) )
= ( some_list_val @ Vs ) )
=> ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) ) ) ).
% phis_in_\<alpha>n
thf(fact_130_phis__same__var,axiom,
! [G: g,N: node,V: val,Vs: list_val,V2: val] :
( ( ( phis @ G @ ( produc1470527136de_val @ N @ V ) )
= ( some_list_val @ Vs ) )
=> ( ( member_val @ V2 @ ( set_val2 @ Vs ) )
=> ( ( var2 @ G @ V2 )
= ( var2 @ G @ V ) ) ) ) ).
% phis_same_var
thf(fact_131_phis__phi,axiom,
! [G: g,N: node,V: val,Vs: list_val] :
( ( ( phis @ G @ ( produc1470527136de_val @ N @ V ) )
= ( some_list_val @ Vs ) )
=> ( ( sSA_CF262257161de_val @ alpha_n @ defs @ phis @ G @ V )
= ( some_list_val @ Vs ) ) ) ).
% phis_phi
thf(fact_132_phiUses__exI,axiom,
! [M: node,G: g,N: node,V: val,Vs: list_val] :
( ( member_node @ M @ ( set_node2 @ ( graph_272749361_edgeD @ inEdges @ G @ N ) ) )
=> ( ( ( phis @ G @ ( produc1470527136de_val @ N @ V ) )
= ( some_list_val @ Vs ) )
=> ( ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ~ ! [V6: val] :
( ( member_val @ V6 @ ( sSA_CF848637139eD_val @ alpha_n @ inEdges @ phis @ G @ M ) )
=> ~ ( member_val @ V6 @ ( set_val2 @ Vs ) ) ) ) ) ) ).
% phiUses_exI
thf(fact_133_CFG__SSA__Transformed_Ocondensation__nodes_Ocong,axiom,
irredu681226810de_val = irredu681226810de_val ).
% CFG_SSA_Transformed.condensation_nodes.cong
thf(fact_134_CFG__SSA__Transformed_Oredundant__set_Ocong,axiom,
irredu224829488eD_val = irredu224829488eD_val ).
% CFG_SSA_Transformed.redundant_set.cong
thf(fact_135_CFG__SSA__wf__base_Ophi__def,axiom,
( sSA_CF262257161de_val
= ( ^ [Alpha_n: g > list_node,Defs: g > node > set_val,Phis: g > produc1432036078de_val > option_list_val,G2: g,V3: val] : ( Phis @ G2 @ ( produc1470527136de_val @ ( sSA_CF551432799de_val @ Alpha_n @ Defs @ Phis @ G2 @ V3 ) @ V3 ) ) ) ) ).
% CFG_SSA_wf_base.phi_def
thf(fact_136_CFG__SSA__wf__base_OphiArg__def,axiom,
( sSA_CF1252180629de_val
= ( ^ [Alpha_n: g > list_node,Defs: g > node > set_val,Phis: g > produc1432036078de_val > option_list_val,G2: g,V3: val,V4: val] :
? [Vs2: list_val] :
( ( ( sSA_CF262257161de_val @ Alpha_n @ Defs @ Phis @ G2 @ V3 )
= ( some_list_val @ Vs2 ) )
& ( member_val @ V4 @ ( set_val2 @ Vs2 ) ) ) ) ) ).
% CFG_SSA_wf_base.phiArg_def
thf(fact_137_phiUsesI,axiom,
! [N4: node,G: g,V2: val,Vs: list_val,N: node,V: val] :
( ( member_node @ N4 @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( ( ( phis @ G @ ( produc1470527136de_val @ N4 @ V2 ) )
= ( some_list_val @ Vs ) )
=> ( ( member313544709de_val @ ( produc1470527136de_val @ N @ V ) @ ( set_Pr1085970585de_val @ ( zip_node_val @ ( graph_272749361_edgeD @ inEdges @ G @ N4 ) @ Vs ) ) )
=> ( member_val @ V @ ( sSA_CF848637139eD_val @ alpha_n @ inEdges @ phis @ G @ N ) ) ) ) ) ).
% phiUsesI
thf(fact_138_scc__in__P,axiom,
! [Scc: set_val,G: g,P2: set_val] :
( ( member_set_val @ Scc @ ( irredu681226810de_val @ alpha_n @ defs @ phis @ G @ P2 ) )
=> ( ord_less_eq_set_val @ Scc @ P2 ) ) ).
% scc_in_P
thf(fact_139_not__None__eq,axiom,
! [X2: option_list_val] :
( ( X2 != none_list_val )
= ( ? [Y2: list_val] :
( X2
= ( some_list_val @ Y2 ) ) ) ) ).
% not_None_eq
thf(fact_140_not__Some__eq,axiom,
! [X2: option_list_val] :
( ( ! [Y2: list_val] :
( X2
!= ( some_list_val @ Y2 ) ) )
= ( X2 = none_list_val ) ) ).
% not_Some_eq
thf(fact_141_sup__Some,axiom,
! [X2: set_val,Y3: set_val] :
( ( sup_su1487070040et_val @ ( some_set_val @ X2 ) @ ( some_set_val @ Y3 ) )
= ( some_set_val @ ( sup_sup_set_val @ X2 @ Y3 ) ) ) ).
% sup_Some
thf(fact_142_option_Oinject,axiom,
! [X22: list_val,Y22: list_val] :
( ( ( some_list_val @ X22 )
= ( some_list_val @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_143_less__eq__option__Some,axiom,
! [X2: set_val,Y3: set_val] :
( ( ord_le596993316et_val @ ( some_set_val @ X2 ) @ ( some_set_val @ Y3 ) )
= ( ord_less_eq_set_val @ X2 @ Y3 ) ) ).
% less_eq_option_Some
thf(fact_144_zip__same,axiom,
! [A: produc1324971431et_val,B: produc1324971431et_val,Xs: list_P2089461677et_val] :
( ( member1254968080et_val @ ( produc1377626967et_val @ A @ B ) @ ( set_Pr1006393724et_val @ ( zip_Pr21865079et_val @ Xs @ Xs ) ) )
= ( ( member1711426256et_val @ A @ ( set_Pr284769596et_val @ Xs ) )
& ( A = B ) ) ) ).
% zip_same
thf(fact_145_zip__same,axiom,
! [A: node,B: node,Xs: list_node] :
( ( member2110109140e_node @ ( produc457016035e_node @ A @ B ) @ ( set_Pr450990656e_node @ ( zip_node_node @ Xs @ Xs ) ) )
= ( ( member_node @ A @ ( set_node2 @ Xs ) )
& ( A = B ) ) ) ).
% zip_same
thf(fact_146_zip__same,axiom,
! [A: val,B: val,Xs: list_val] :
( ( member1680438992al_val @ ( product_Pair_val_val @ A @ B ) @ ( set_Pr102746428al_val @ ( zip_val_val @ Xs @ Xs ) ) )
= ( ( member_val @ A @ ( set_val2 @ Xs ) )
& ( A = B ) ) ) ).
% zip_same
thf(fact_147_zip__same,axiom,
! [A: produc1432036078de_val,B: produc1432036078de_val,Xs: list_P1820443774de_val] :
( ( member698732390de_val @ ( produc453318901de_val @ A @ B ) @ ( set_Pr482027474de_val @ ( zip_Pr1472086037de_val @ Xs @ Xs ) ) )
= ( ( member313544709de_val @ A @ ( set_Pr1085970585de_val @ Xs ) )
& ( A = B ) ) ) ).
% zip_same
thf(fact_148_zip__same,axiom,
! [A: set_val,B: set_val,Xs: list_set_val] :
( ( member1711426256et_val @ ( produc1041633943et_val @ A @ B ) @ ( set_Pr284769596et_val @ ( zip_set_val_set_val @ Xs @ Xs ) ) )
= ( ( member_set_val @ A @ ( set_set_val2 @ Xs ) )
& ( A = B ) ) ) ).
% zip_same
thf(fact_149__C1_C,axiom,
! [G: g,P2: set_val] :
( ( irredu224829488eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ G @ P2 )
=> ? [Scc2: set_val] :
( ( ord_less_eq_set_val @ Scc2 @ P2 )
& ( irredu1955035135eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ G @ P2 @ Scc2 ) ) ) ).
% "1"
thf(fact_150_phiUsesE,axiom,
! [V: val,G: g,N: node] :
( ( member_val @ V @ ( sSA_CF848637139eD_val @ alpha_n @ inEdges @ phis @ G @ N ) )
=> ~ ! [N5: node] :
( ( member_node @ N5 @ ( set_node2 @ ( graph_1037173556_edgeD @ alpha_n @ inEdges @ G @ N ) ) )
=> ! [V6: val,Vs4: list_val] :
( ( member313544709de_val @ ( produc1470527136de_val @ N @ V ) @ ( set_Pr1085970585de_val @ ( zip_node_val @ ( graph_272749361_edgeD @ inEdges @ G @ N5 ) @ Vs4 ) ) )
=> ( ( phis @ G @ ( produc1470527136de_val @ N5 @ V6 ) )
!= ( some_list_val @ Vs4 ) ) ) ) ) ).
% phiUsesE
thf(fact_151_finite__has__minimal2,axiom,
! [A2: set_node,A: node] :
( ( finite_finite_node @ A2 )
=> ( ( member_node @ A @ A2 )
=> ? [X3: node] :
( ( member_node @ X3 @ A2 )
& ( ord_less_eq_node @ X3 @ A )
& ! [Xa: node] :
( ( member_node @ Xa @ A2 )
=> ( ( ord_less_eq_node @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_152_finite__has__minimal2,axiom,
! [A2: set_val,A: val] :
( ( finite_finite_val @ A2 )
=> ( ( member_val @ A @ A2 )
=> ? [X3: val] :
( ( member_val @ X3 @ A2 )
& ( ord_less_eq_val @ X3 @ A )
& ! [Xa: val] :
( ( member_val @ Xa @ A2 )
=> ( ( ord_less_eq_val @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_153_finite__has__minimal2,axiom,
! [A2: set_set_val,A: set_val] :
( ( finite1265617511et_val @ A2 )
=> ( ( member_set_val @ A @ A2 )
=> ? [X3: set_val] :
( ( member_set_val @ X3 @ A2 )
& ( ord_less_eq_set_val @ X3 @ A )
& ! [Xa: set_val] :
( ( member_set_val @ Xa @ A2 )
=> ( ( ord_less_eq_set_val @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_154_finite__has__maximal2,axiom,
! [A2: set_node,A: node] :
( ( finite_finite_node @ A2 )
=> ( ( member_node @ A @ A2 )
=> ? [X3: node] :
( ( member_node @ X3 @ A2 )
& ( ord_less_eq_node @ A @ X3 )
& ! [Xa: node] :
( ( member_node @ Xa @ A2 )
=> ( ( ord_less_eq_node @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_155_finite__has__maximal2,axiom,
! [A2: set_val,A: val] :
( ( finite_finite_val @ A2 )
=> ( ( member_val @ A @ A2 )
=> ? [X3: val] :
( ( member_val @ X3 @ A2 )
& ( ord_less_eq_val @ A @ X3 )
& ! [Xa: val] :
( ( member_val @ Xa @ A2 )
=> ( ( ord_less_eq_val @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_156_finite__has__maximal2,axiom,
! [A2: set_set_val,A: set_val] :
( ( finite1265617511et_val @ A2 )
=> ( ( member_set_val @ A @ A2 )
=> ? [X3: set_val] :
( ( member_set_val @ X3 @ A2 )
& ( ord_less_eq_set_val @ A @ X3 )
& ! [Xa: set_val] :
( ( member_set_val @ Xa @ A2 )
=> ( ( ord_less_eq_set_val @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_157_subset__code_I1_J,axiom,
! [Xs: list_set_val,B2: set_set_val] :
( ( ord_le1742111550et_val @ ( set_set_val2 @ Xs ) @ B2 )
= ( ! [X: set_val] :
( ( member_set_val @ X @ ( set_set_val2 @ Xs ) )
=> ( member_set_val @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_158_subset__code_I1_J,axiom,
! [Xs: list_P2089461677et_val,B2: set_Pr1311924359et_val] :
( ( ord_le299366439et_val @ ( set_Pr284769596et_val @ Xs ) @ B2 )
= ( ! [X: produc1324971431et_val] :
( ( member1711426256et_val @ X @ ( set_Pr284769596et_val @ Xs ) )
=> ( member1711426256et_val @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_159_subset__code_I1_J,axiom,
! [Xs: list_node,B2: set_node] :
( ( ord_less_eq_set_node @ ( set_node2 @ Xs ) @ B2 )
= ( ! [X: node] :
( ( member_node @ X @ ( set_node2 @ Xs ) )
=> ( member_node @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_160_subset__code_I1_J,axiom,
! [Xs: list_P1820443774de_val,B2: set_Pr699757092de_val] :
( ( ord_le1643692676de_val @ ( set_Pr1085970585de_val @ Xs ) @ B2 )
= ( ! [X: produc1432036078de_val] :
( ( member313544709de_val @ X @ ( set_Pr1085970585de_val @ Xs ) )
=> ( member313544709de_val @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_161_subset__code_I1_J,axiom,
! [Xs: list_val,B2: set_val] :
( ( ord_less_eq_set_val @ ( set_val2 @ Xs ) @ B2 )
= ( ! [X: val] :
( ( member_val @ X @ ( set_val2 @ Xs ) )
=> ( member_val @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_162_rev__finite__subset,axiom,
! [B2: set_Pr1311924359et_val,A2: set_Pr1311924359et_val] :
( ( finite79836624et_val @ B2 )
=> ( ( ord_le299366439et_val @ A2 @ B2 )
=> ( finite79836624et_val @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_163_rev__finite__subset,axiom,
! [B2: set_Pr699757092de_val,A2: set_Pr699757092de_val] :
( ( finite2056463621de_val @ B2 )
=> ( ( ord_le1643692676de_val @ A2 @ B2 )
=> ( finite2056463621de_val @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_164_rev__finite__subset,axiom,
! [B2: set_val,A2: set_val] :
( ( finite_finite_val @ B2 )
=> ( ( ord_less_eq_set_val @ A2 @ B2 )
=> ( finite_finite_val @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_165_infinite__super,axiom,
! [S: set_Pr1311924359et_val,T: set_Pr1311924359et_val] :
( ( ord_le299366439et_val @ S @ T )
=> ( ~ ( finite79836624et_val @ S )
=> ~ ( finite79836624et_val @ T ) ) ) ).
% infinite_super
thf(fact_166_infinite__super,axiom,
! [S: set_Pr699757092de_val,T: set_Pr699757092de_val] :
( ( ord_le1643692676de_val @ S @ T )
=> ( ~ ( finite2056463621de_val @ S )
=> ~ ( finite2056463621de_val @ T ) ) ) ).
% infinite_super
thf(fact_167_infinite__super,axiom,
! [S: set_val,T: set_val] :
( ( ord_less_eq_set_val @ S @ T )
=> ( ~ ( finite_finite_val @ S )
=> ~ ( finite_finite_val @ T ) ) ) ).
% infinite_super
thf(fact_168_finite__subset,axiom,
! [A2: set_Pr1311924359et_val,B2: set_Pr1311924359et_val] :
( ( ord_le299366439et_val @ A2 @ B2 )
=> ( ( finite79836624et_val @ B2 )
=> ( finite79836624et_val @ A2 ) ) ) ).
% finite_subset
thf(fact_169_finite__subset,axiom,
! [A2: set_Pr699757092de_val,B2: set_Pr699757092de_val] :
( ( ord_le1643692676de_val @ A2 @ B2 )
=> ( ( finite2056463621de_val @ B2 )
=> ( finite2056463621de_val @ A2 ) ) ) ).
% finite_subset
thf(fact_170_finite__subset,axiom,
! [A2: set_val,B2: set_val] :
( ( ord_less_eq_set_val @ A2 @ B2 )
=> ( ( finite_finite_val @ B2 )
=> ( finite_finite_val @ A2 ) ) ) ).
% finite_subset
thf(fact_171_in__set__zipE,axiom,
! [X2: node,Y3: node,Xs: list_node,Ys: list_node] :
( ( member2110109140e_node @ ( produc457016035e_node @ X2 @ Y3 ) @ ( set_Pr450990656e_node @ ( zip_node_node @ Xs @ Ys ) ) )
=> ~ ( ( member_node @ X2 @ ( set_node2 @ Xs ) )
=> ~ ( member_node @ Y3 @ ( set_node2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_172_in__set__zipE,axiom,
! [X2: val,Y3: node,Xs: list_val,Ys: list_node] :
( ( member828457417l_node @ ( produc1074923692l_node @ X2 @ Y3 ) @ ( set_Pr1600883293l_node @ ( zip_val_node @ Xs @ Ys ) ) )
=> ~ ( ( member_val @ X2 @ ( set_val2 @ Xs ) )
=> ~ ( member_node @ Y3 @ ( set_node2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_173_in__set__zipE,axiom,
! [X2: val,Y3: val,Xs: list_val,Ys: list_val] :
( ( member1680438992al_val @ ( product_Pair_val_val @ X2 @ Y3 ) @ ( set_Pr102746428al_val @ ( zip_val_val @ Xs @ Ys ) ) )
=> ~ ( ( member_val @ X2 @ ( set_val2 @ Xs ) )
=> ~ ( member_val @ Y3 @ ( set_val2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_174_in__set__zipE,axiom,
! [X2: node,Y3: val,Xs: list_node,Ys: list_val] :
( ( member313544709de_val @ ( produc1470527136de_val @ X2 @ Y3 ) @ ( set_Pr1085970585de_val @ ( zip_node_val @ Xs @ Ys ) ) )
=> ~ ( ( member_node @ X2 @ ( set_node2 @ Xs ) )
=> ~ ( member_val @ Y3 @ ( set_val2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_175_in__set__zipE,axiom,
! [X2: set_val,Y3: node,Xs: list_set_val,Ys: list_node] :
( ( member76372137l_node @ ( produc2052093836l_node @ X2 @ Y3 ) @ ( set_Pr1904955709l_node @ ( zip_set_val_node @ Xs @ Ys ) ) )
=> ~ ( ( member_set_val @ X2 @ ( set_set_val2 @ Xs ) )
=> ~ ( member_node @ Y3 @ ( set_node2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_176_in__set__zipE,axiom,
! [X2: set_val,Y3: val,Xs: list_set_val,Ys: list_val] :
( ( member1110026224al_val @ ( produc1337903031al_val @ X2 @ Y3 ) @ ( set_Pr1027623516al_val @ ( zip_set_val_val @ Xs @ Ys ) ) )
=> ~ ( ( member_set_val @ X2 @ ( set_set_val2 @ Xs ) )
=> ~ ( member_val @ Y3 @ ( set_val2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_177_in__set__zipE,axiom,
! [X2: node,Y3: set_val,Xs: list_node,Ys: list_set_val] :
( ( member125181669et_val @ ( produc2042994048et_val @ X2 @ Y3 ) @ ( set_Pr1953765241et_val @ ( zip_node_set_val @ Xs @ Ys ) ) )
=> ~ ( ( member_node @ X2 @ ( set_node2 @ Xs ) )
=> ~ ( member_set_val @ Y3 @ ( set_set_val2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_178_in__set__zipE,axiom,
! [X2: val,Y3: set_val,Xs: list_val,Ys: list_set_val] :
( ( member39229872et_val @ ( produc1446164855et_val @ X2 @ Y3 ) @ ( set_Pr2104310812et_val @ ( zip_val_set_val @ Xs @ Ys ) ) )
=> ~ ( ( member_val @ X2 @ ( set_val2 @ Xs ) )
=> ~ ( member_set_val @ Y3 @ ( set_set_val2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_179_in__set__zipE,axiom,
! [X2: node,Y3: produc1432036078de_val,Xs: list_node,Ys: list_P1820443774de_val] :
( ( member562853661de_val @ ( produc1162531244de_val @ X2 @ Y3 ) @ ( set_Pr1157337481de_val @ ( zip_no263301324de_val @ Xs @ Ys ) ) )
=> ~ ( ( member_node @ X2 @ ( set_node2 @ Xs ) )
=> ~ ( member313544709de_val @ Y3 @ ( set_Pr1085970585de_val @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_180_in__set__zipE,axiom,
! [X2: val,Y3: produc1432036078de_val,Xs: list_val,Ys: list_P1820443774de_val] :
( ( member1894062994de_val @ ( produc1571934709de_val @ X2 @ Y3 ) @ ( set_Pr434939430de_val @ ( zip_va679686869de_val @ Xs @ Ys ) ) )
=> ~ ( ( member_val @ X2 @ ( set_val2 @ Xs ) )
=> ~ ( member313544709de_val @ Y3 @ ( set_Pr1085970585de_val @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_181_set__zip__leftD,axiom,
! [X2: node,Y3: val,Xs: list_node,Ys: list_val] :
( ( member313544709de_val @ ( produc1470527136de_val @ X2 @ Y3 ) @ ( set_Pr1085970585de_val @ ( zip_node_val @ Xs @ Ys ) ) )
=> ( member_node @ X2 @ ( set_node2 @ Xs ) ) ) ).
% set_zip_leftD
thf(fact_182_set__zip__leftD,axiom,
! [X2: set_val,Y3: set_val,Xs: list_set_val,Ys: list_set_val] :
( ( member1711426256et_val @ ( produc1041633943et_val @ X2 @ Y3 ) @ ( set_Pr284769596et_val @ ( zip_set_val_set_val @ Xs @ Ys ) ) )
=> ( member_set_val @ X2 @ ( set_set_val2 @ Xs ) ) ) ).
% set_zip_leftD
thf(fact_183_set__zip__rightD,axiom,
! [X2: node,Y3: val,Xs: list_node,Ys: list_val] :
( ( member313544709de_val @ ( produc1470527136de_val @ X2 @ Y3 ) @ ( set_Pr1085970585de_val @ ( zip_node_val @ Xs @ Ys ) ) )
=> ( member_val @ Y3 @ ( set_val2 @ Ys ) ) ) ).
% set_zip_rightD
thf(fact_184_set__zip__rightD,axiom,
! [X2: set_val,Y3: set_val,Xs: list_set_val,Ys: list_set_val] :
( ( member1711426256et_val @ ( produc1041633943et_val @ X2 @ Y3 ) @ ( set_Pr284769596et_val @ ( zip_set_val_set_val @ Xs @ Ys ) ) )
=> ( member_set_val @ Y3 @ ( set_set_val2 @ Ys ) ) ) ).
% set_zip_rightD
thf(fact_185_combine__options__cases,axiom,
! [X2: option_list_val,P2: option_list_val > option_list_val > $o,Y3: option_list_val] :
( ( ( X2 = none_list_val )
=> ( P2 @ X2 @ Y3 ) )
=> ( ( ( Y3 = none_list_val )
=> ( P2 @ X2 @ Y3 ) )
=> ( ! [A4: list_val,B3: list_val] :
( ( X2
= ( some_list_val @ A4 ) )
=> ( ( Y3
= ( some_list_val @ B3 ) )
=> ( P2 @ X2 @ Y3 ) ) )
=> ( P2 @ X2 @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_186_split__option__all,axiom,
( ( ^ [P3: option_list_val > $o] :
! [X4: option_list_val] : ( P3 @ X4 ) )
= ( ^ [P4: option_list_val > $o] :
( ( P4 @ none_list_val )
& ! [X: list_val] : ( P4 @ ( some_list_val @ X ) ) ) ) ) ).
% split_option_all
thf(fact_187_split__option__ex,axiom,
( ( ^ [P3: option_list_val > $o] :
? [X4: option_list_val] : ( P3 @ X4 ) )
= ( ^ [P4: option_list_val > $o] :
( ( P4 @ none_list_val )
| ? [X: list_val] : ( P4 @ ( some_list_val @ X ) ) ) ) ) ).
% split_option_ex
thf(fact_188_option_Oinducts,axiom,
! [P2: option_list_val > $o,Option: option_list_val] :
( ( P2 @ none_list_val )
=> ( ! [X3: list_val] : ( P2 @ ( some_list_val @ X3 ) )
=> ( P2 @ Option ) ) ) ).
% option.inducts
thf(fact_189_option_Oexhaust,axiom,
! [Y3: option_list_val] :
( ( Y3 != none_list_val )
=> ~ ! [X23: list_val] :
( Y3
!= ( some_list_val @ X23 ) ) ) ).
% option.exhaust
thf(fact_190_option_OdiscI,axiom,
! [Option: option_list_val,X22: list_val] :
( ( Option
= ( some_list_val @ X22 ) )
=> ( Option != none_list_val ) ) ).
% option.discI
thf(fact_191_option_Odistinct_I1_J,axiom,
! [X22: list_val] :
( none_list_val
!= ( some_list_val @ X22 ) ) ).
% option.distinct(1)
thf(fact_192_Ex__condensation__leaf,axiom,
! [P2: set_val,G: g] :
( ( P2 != bot_bot_set_val )
=> ? [Leaf: set_val] :
( ( member_set_val @ Leaf @ ( irredu681226810de_val @ alpha_n @ defs @ phis @ G @ P2 ) )
& ! [Scc3: set_val] :
~ ( member1711426256et_val @ ( produc1041633943et_val @ Leaf @ Scc3 ) @ ( irredu893086079de_val @ alpha_n @ defs @ phis @ G @ P2 ) ) ) ) ).
% Ex_condensation_leaf
thf(fact_193_Un__subset__iff,axiom,
! [A2: set_val,B2: set_val,C: set_val] :
( ( ord_less_eq_set_val @ ( sup_sup_set_val @ A2 @ B2 ) @ C )
= ( ( ord_less_eq_set_val @ A2 @ C )
& ( ord_less_eq_set_val @ B2 @ C ) ) ) ).
% Un_subset_iff
thf(fact_194_le__sup__iff,axiom,
! [X2: set_val,Y3: set_val,Z: set_val] :
( ( ord_less_eq_set_val @ ( sup_sup_set_val @ X2 @ Y3 ) @ Z )
= ( ( ord_less_eq_set_val @ X2 @ Z )
& ( ord_less_eq_set_val @ Y3 @ Z ) ) ) ).
% le_sup_iff
thf(fact_195_sup_Obounded__iff,axiom,
! [B: set_val,C2: set_val,A: set_val] :
( ( ord_less_eq_set_val @ ( sup_sup_set_val @ B @ C2 ) @ A )
= ( ( ord_less_eq_set_val @ B @ A )
& ( ord_less_eq_set_val @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_196_subset__antisym,axiom,
! [A2: set_val,B2: set_val] :
( ( ord_less_eq_set_val @ A2 @ B2 )
=> ( ( ord_less_eq_set_val @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_197_subsetI,axiom,
! [A2: set_node,B2: set_node] :
( ! [X3: node] :
( ( member_node @ X3 @ A2 )
=> ( member_node @ X3 @ B2 ) )
=> ( ord_less_eq_set_node @ A2 @ B2 ) ) ).
% subsetI
thf(fact_198_subsetI,axiom,
! [A2: set_set_val,B2: set_set_val] :
( ! [X3: set_val] :
( ( member_set_val @ X3 @ A2 )
=> ( member_set_val @ X3 @ B2 ) )
=> ( ord_le1742111550et_val @ A2 @ B2 ) ) ).
% subsetI
thf(fact_199_subsetI,axiom,
! [A2: set_Pr699757092de_val,B2: set_Pr699757092de_val] :
( ! [X3: produc1432036078de_val] :
( ( member313544709de_val @ X3 @ A2 )
=> ( member313544709de_val @ X3 @ B2 ) )
=> ( ord_le1643692676de_val @ A2 @ B2 ) ) ).
% subsetI
thf(fact_200_subsetI,axiom,
! [A2: set_Pr1311924359et_val,B2: set_Pr1311924359et_val] :
( ! [X3: produc1324971431et_val] :
( ( member1711426256et_val @ X3 @ A2 )
=> ( member1711426256et_val @ X3 @ B2 ) )
=> ( ord_le299366439et_val @ A2 @ B2 ) ) ).
% subsetI
thf(fact_201_subsetI,axiom,
! [A2: set_val,B2: set_val] :
( ! [X3: val] :
( ( member_val @ X3 @ A2 )
=> ( member_val @ X3 @ B2 ) )
=> ( ord_less_eq_set_val @ A2 @ B2 ) ) ).
% subsetI
thf(fact_202_sup_Oright__idem,axiom,
! [A: set_val,B: set_val] :
( ( sup_sup_set_val @ ( sup_sup_set_val @ A @ B ) @ B )
= ( sup_sup_set_val @ A @ B ) ) ).
% sup.right_idem
thf(fact_203_sup__left__idem,axiom,
! [X2: set_val,Y3: set_val] :
( ( sup_sup_set_val @ X2 @ ( sup_sup_set_val @ X2 @ Y3 ) )
= ( sup_sup_set_val @ X2 @ Y3 ) ) ).
% sup_left_idem
thf(fact_204_sup_Oleft__idem,axiom,
! [A: set_val,B: set_val] :
( ( sup_sup_set_val @ A @ ( sup_sup_set_val @ A @ B ) )
= ( sup_sup_set_val @ A @ B ) ) ).
% sup.left_idem
thf(fact_205_sup__idem,axiom,
! [X2: set_val] :
( ( sup_sup_set_val @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_206_sup_Oidem,axiom,
! [A: set_val] :
( ( sup_sup_set_val @ A @ A )
= A ) ).
% sup.idem
thf(fact_207_Un__iff,axiom,
! [C2: node,A2: set_node,B2: set_node] :
( ( member_node @ C2 @ ( sup_sup_set_node @ A2 @ B2 ) )
= ( ( member_node @ C2 @ A2 )
| ( member_node @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_208_Un__iff,axiom,
! [C2: set_val,A2: set_set_val,B2: set_set_val] :
( ( member_set_val @ C2 @ ( sup_sup_set_set_val @ A2 @ B2 ) )
= ( ( member_set_val @ C2 @ A2 )
| ( member_set_val @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_209_Un__iff,axiom,
! [C2: produc1432036078de_val,A2: set_Pr699757092de_val,B2: set_Pr699757092de_val] :
( ( member313544709de_val @ C2 @ ( sup_su2138939728de_val @ A2 @ B2 ) )
= ( ( member313544709de_val @ C2 @ A2 )
| ( member313544709de_val @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_210_Un__iff,axiom,
! [C2: produc1324971431et_val,A2: set_Pr1311924359et_val,B2: set_Pr1311924359et_val] :
( ( member1711426256et_val @ C2 @ ( sup_su2005730907et_val @ A2 @ B2 ) )
= ( ( member1711426256et_val @ C2 @ A2 )
| ( member1711426256et_val @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_211_Un__iff,axiom,
! [C2: val,A2: set_val,B2: set_val] :
( ( member_val @ C2 @ ( sup_sup_set_val @ A2 @ B2 ) )
= ( ( member_val @ C2 @ A2 )
| ( member_val @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_212_UnCI,axiom,
! [C2: node,B2: set_node,A2: set_node] :
( ( ~ ( member_node @ C2 @ B2 )
=> ( member_node @ C2 @ A2 ) )
=> ( member_node @ C2 @ ( sup_sup_set_node @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_213_UnCI,axiom,
! [C2: set_val,B2: set_set_val,A2: set_set_val] :
( ( ~ ( member_set_val @ C2 @ B2 )
=> ( member_set_val @ C2 @ A2 ) )
=> ( member_set_val @ C2 @ ( sup_sup_set_set_val @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_214_UnCI,axiom,
! [C2: produc1432036078de_val,B2: set_Pr699757092de_val,A2: set_Pr699757092de_val] :
( ( ~ ( member313544709de_val @ C2 @ B2 )
=> ( member313544709de_val @ C2 @ A2 ) )
=> ( member313544709de_val @ C2 @ ( sup_su2138939728de_val @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_215_UnCI,axiom,
! [C2: produc1324971431et_val,B2: set_Pr1311924359et_val,A2: set_Pr1311924359et_val] :
( ( ~ ( member1711426256et_val @ C2 @ B2 )
=> ( member1711426256et_val @ C2 @ A2 ) )
=> ( member1711426256et_val @ C2 @ ( sup_su2005730907et_val @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_216_UnCI,axiom,
! [C2: val,B2: set_val,A2: set_val] :
( ( ~ ( member_val @ C2 @ B2 )
=> ( member_val @ C2 @ A2 ) )
=> ( member_val @ C2 @ ( sup_sup_set_val @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_217_subset__empty,axiom,
! [A2: set_val] :
( ( ord_less_eq_set_val @ A2 @ bot_bot_set_val )
= ( A2 = bot_bot_set_val ) ) ).
% subset_empty
thf(fact_218_empty__subsetI,axiom,
! [A2: set_val] : ( ord_less_eq_set_val @ bot_bot_set_val @ A2 ) ).
% empty_subsetI
thf(fact_219_sup__bot__left,axiom,
! [X2: set_val] :
( ( sup_sup_set_val @ bot_bot_set_val @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_220_sup__bot__right,axiom,
! [X2: set_val] :
( ( sup_sup_set_val @ X2 @ bot_bot_set_val )
= X2 ) ).
% sup_bot_right
thf(fact_221_bot__eq__sup__iff,axiom,
! [X2: set_val,Y3: set_val] :
( ( bot_bot_set_val
= ( sup_sup_set_val @ X2 @ Y3 ) )
= ( ( X2 = bot_bot_set_val )
& ( Y3 = bot_bot_set_val ) ) ) ).
% bot_eq_sup_iff
thf(fact_222_sup__eq__bot__iff,axiom,
! [X2: set_val,Y3: set_val] :
( ( ( sup_sup_set_val @ X2 @ Y3 )
= bot_bot_set_val )
= ( ( X2 = bot_bot_set_val )
& ( Y3 = bot_bot_set_val ) ) ) ).
% sup_eq_bot_iff
thf(fact_223_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_val,B: set_val] :
( ( ( sup_sup_set_val @ A @ B )
= bot_bot_set_val )
= ( ( A = bot_bot_set_val )
& ( B = bot_bot_set_val ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_224_sup__bot_Oleft__neutral,axiom,
! [A: set_val] :
( ( sup_sup_set_val @ bot_bot_set_val @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_225_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_val,B: set_val] :
( ( bot_bot_set_val
= ( sup_sup_set_val @ A @ B ) )
= ( ( A = bot_bot_set_val )
& ( B = bot_bot_set_val ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_226_sup__bot_Oright__neutral,axiom,
! [A: set_val] :
( ( sup_sup_set_val @ A @ bot_bot_set_val )
= A ) ).
% sup_bot.right_neutral
thf(fact_227_Un__empty,axiom,
! [A2: set_val,B2: set_val] :
( ( ( sup_sup_set_val @ A2 @ B2 )
= bot_bot_set_val )
= ( ( A2 = bot_bot_set_val )
& ( B2 = bot_bot_set_val ) ) ) ).
% Un_empty
thf(fact_228_old_Osuccessors__predecessors,axiom,
! [N: node,G: g,M: node] :
( ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( ( member_node @ N @ ( set_node2 @ ( graph_1037173556_edgeD @ alpha_n @ inEdges @ G @ M ) ) )
= ( member_node @ M @ ( set_node2 @ ( graph_272749361_edgeD @ inEdges @ G @ N ) ) ) ) ) ).
% old.successors_predecessors
thf(fact_229_Un__empty__left,axiom,
! [B2: set_val] :
( ( sup_sup_set_val @ bot_bot_set_val @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_230_Un__empty__right,axiom,
! [A2: set_val] :
( ( sup_sup_set_val @ A2 @ bot_bot_set_val )
= A2 ) ).
% Un_empty_right
thf(fact_231_infinite__imp__nonempty,axiom,
! [S: set_Pr1311924359et_val] :
( ~ ( finite79836624et_val @ S )
=> ( S != bot_bo2078591731et_val ) ) ).
% infinite_imp_nonempty
thf(fact_232_infinite__imp__nonempty,axiom,
! [S: set_Pr699757092de_val] :
( ~ ( finite2056463621de_val @ S )
=> ( S != bot_bo404898488de_val ) ) ).
% infinite_imp_nonempty
thf(fact_233_infinite__imp__nonempty,axiom,
! [S: set_val] :
( ~ ( finite_finite_val @ S )
=> ( S != bot_bot_set_val ) ) ).
% infinite_imp_nonempty
thf(fact_234_finite_OemptyI,axiom,
finite79836624et_val @ bot_bo2078591731et_val ).
% finite.emptyI
thf(fact_235_finite_OemptyI,axiom,
finite2056463621de_val @ bot_bo404898488de_val ).
% finite.emptyI
thf(fact_236_finite_OemptyI,axiom,
finite_finite_val @ bot_bot_set_val ).
% finite.emptyI
thf(fact_237_CFG__SSA__Transformed_Oredundant__scc_Ocong,axiom,
irredu1955035135eD_val = irredu1955035135eD_val ).
% CFG_SSA_Transformed.redundant_scc.cong
thf(fact_238_finite__has__maximal,axiom,
! [A2: set_val] :
( ( finite_finite_val @ A2 )
=> ( ( A2 != bot_bot_set_val )
=> ? [X3: val] :
( ( member_val @ X3 @ A2 )
& ! [Xa: val] :
( ( member_val @ Xa @ A2 )
=> ( ( ord_less_eq_val @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_239_finite__has__maximal,axiom,
! [A2: set_set_val] :
( ( finite1265617511et_val @ A2 )
=> ( ( A2 != bot_bot_set_set_val )
=> ? [X3: set_val] :
( ( member_set_val @ X3 @ A2 )
& ! [Xa: set_val] :
( ( member_set_val @ Xa @ A2 )
=> ( ( ord_less_eq_set_val @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_240_finite__has__minimal,axiom,
! [A2: set_val] :
( ( finite_finite_val @ A2 )
=> ( ( A2 != bot_bot_set_val )
=> ? [X3: val] :
( ( member_val @ X3 @ A2 )
& ! [Xa: val] :
( ( member_val @ Xa @ A2 )
=> ( ( ord_less_eq_val @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_241_finite__has__minimal,axiom,
! [A2: set_set_val] :
( ( finite1265617511et_val @ A2 )
=> ( ( A2 != bot_bot_set_set_val )
=> ? [X3: set_val] :
( ( member_set_val @ X3 @ A2 )
& ! [Xa: set_val] :
( ( member_set_val @ Xa @ A2 )
=> ( ( ord_less_eq_set_val @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_242_Collect__mono__iff,axiom,
! [P2: val > $o,Q2: val > $o] :
( ( ord_less_eq_set_val @ ( collect_val @ P2 ) @ ( collect_val @ Q2 ) )
= ( ! [X: val] :
( ( P2 @ X )
=> ( Q2 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_243_set__eq__subset,axiom,
( ( ^ [Y4: set_val,Z2: set_val] : Y4 = Z2 )
= ( ^ [A5: set_val,B4: set_val] :
( ( ord_less_eq_set_val @ A5 @ B4 )
& ( ord_less_eq_set_val @ B4 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_244_subset__trans,axiom,
! [A2: set_val,B2: set_val,C: set_val] :
( ( ord_less_eq_set_val @ A2 @ B2 )
=> ( ( ord_less_eq_set_val @ B2 @ C )
=> ( ord_less_eq_set_val @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_245_Collect__mono,axiom,
! [P2: val > $o,Q2: val > $o] :
( ! [X3: val] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_less_eq_set_val @ ( collect_val @ P2 ) @ ( collect_val @ Q2 ) ) ) ).
% Collect_mono
thf(fact_246_subset__refl,axiom,
! [A2: set_val] : ( ord_less_eq_set_val @ A2 @ A2 ) ).
% subset_refl
thf(fact_247_subset__iff,axiom,
( ord_less_eq_set_node
= ( ^ [A5: set_node,B4: set_node] :
! [T2: node] :
( ( member_node @ T2 @ A5 )
=> ( member_node @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_248_subset__iff,axiom,
( ord_le1742111550et_val
= ( ^ [A5: set_set_val,B4: set_set_val] :
! [T2: set_val] :
( ( member_set_val @ T2 @ A5 )
=> ( member_set_val @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_249_subset__iff,axiom,
( ord_le1643692676de_val
= ( ^ [A5: set_Pr699757092de_val,B4: set_Pr699757092de_val] :
! [T2: produc1432036078de_val] :
( ( member313544709de_val @ T2 @ A5 )
=> ( member313544709de_val @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_250_subset__iff,axiom,
( ord_le299366439et_val
= ( ^ [A5: set_Pr1311924359et_val,B4: set_Pr1311924359et_val] :
! [T2: produc1324971431et_val] :
( ( member1711426256et_val @ T2 @ A5 )
=> ( member1711426256et_val @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_251_subset__iff,axiom,
( ord_less_eq_set_val
= ( ^ [A5: set_val,B4: set_val] :
! [T2: val] :
( ( member_val @ T2 @ A5 )
=> ( member_val @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_252_equalityD2,axiom,
! [A2: set_val,B2: set_val] :
( ( A2 = B2 )
=> ( ord_less_eq_set_val @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_253_equalityD1,axiom,
! [A2: set_val,B2: set_val] :
( ( A2 = B2 )
=> ( ord_less_eq_set_val @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_254_subset__eq,axiom,
( ord_less_eq_set_node
= ( ^ [A5: set_node,B4: set_node] :
! [X: node] :
( ( member_node @ X @ A5 )
=> ( member_node @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_255_subset__eq,axiom,
( ord_le1742111550et_val
= ( ^ [A5: set_set_val,B4: set_set_val] :
! [X: set_val] :
( ( member_set_val @ X @ A5 )
=> ( member_set_val @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_256_subset__eq,axiom,
( ord_le1643692676de_val
= ( ^ [A5: set_Pr699757092de_val,B4: set_Pr699757092de_val] :
! [X: produc1432036078de_val] :
( ( member313544709de_val @ X @ A5 )
=> ( member313544709de_val @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_257_subset__eq,axiom,
( ord_le299366439et_val
= ( ^ [A5: set_Pr1311924359et_val,B4: set_Pr1311924359et_val] :
! [X: produc1324971431et_val] :
( ( member1711426256et_val @ X @ A5 )
=> ( member1711426256et_val @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_258_subset__eq,axiom,
( ord_less_eq_set_val
= ( ^ [A5: set_val,B4: set_val] :
! [X: val] :
( ( member_val @ X @ A5 )
=> ( member_val @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_259_equalityE,axiom,
! [A2: set_val,B2: set_val] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_val @ A2 @ B2 )
=> ~ ( ord_less_eq_set_val @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_260_subsetD,axiom,
! [A2: set_node,B2: set_node,C2: node] :
( ( ord_less_eq_set_node @ A2 @ B2 )
=> ( ( member_node @ C2 @ A2 )
=> ( member_node @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_261_subsetD,axiom,
! [A2: set_set_val,B2: set_set_val,C2: set_val] :
( ( ord_le1742111550et_val @ A2 @ B2 )
=> ( ( member_set_val @ C2 @ A2 )
=> ( member_set_val @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_262_subsetD,axiom,
! [A2: set_Pr699757092de_val,B2: set_Pr699757092de_val,C2: produc1432036078de_val] :
( ( ord_le1643692676de_val @ A2 @ B2 )
=> ( ( member313544709de_val @ C2 @ A2 )
=> ( member313544709de_val @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_263_subsetD,axiom,
! [A2: set_Pr1311924359et_val,B2: set_Pr1311924359et_val,C2: produc1324971431et_val] :
( ( ord_le299366439et_val @ A2 @ B2 )
=> ( ( member1711426256et_val @ C2 @ A2 )
=> ( member1711426256et_val @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_264_subsetD,axiom,
! [A2: set_val,B2: set_val,C2: val] :
( ( ord_less_eq_set_val @ A2 @ B2 )
=> ( ( member_val @ C2 @ A2 )
=> ( member_val @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_265_in__mono,axiom,
! [A2: set_node,B2: set_node,X2: node] :
( ( ord_less_eq_set_node @ A2 @ B2 )
=> ( ( member_node @ X2 @ A2 )
=> ( member_node @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_266_in__mono,axiom,
! [A2: set_set_val,B2: set_set_val,X2: set_val] :
( ( ord_le1742111550et_val @ A2 @ B2 )
=> ( ( member_set_val @ X2 @ A2 )
=> ( member_set_val @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_267_in__mono,axiom,
! [A2: set_Pr699757092de_val,B2: set_Pr699757092de_val,X2: produc1432036078de_val] :
( ( ord_le1643692676de_val @ A2 @ B2 )
=> ( ( member313544709de_val @ X2 @ A2 )
=> ( member313544709de_val @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_268_in__mono,axiom,
! [A2: set_Pr1311924359et_val,B2: set_Pr1311924359et_val,X2: produc1324971431et_val] :
( ( ord_le299366439et_val @ A2 @ B2 )
=> ( ( member1711426256et_val @ X2 @ A2 )
=> ( member1711426256et_val @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_269_in__mono,axiom,
! [A2: set_val,B2: set_val,X2: val] :
( ( ord_less_eq_set_val @ A2 @ B2 )
=> ( ( member_val @ X2 @ A2 )
=> ( member_val @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_270_sup__left__commute,axiom,
! [X2: set_val,Y3: set_val,Z: set_val] :
( ( sup_sup_set_val @ X2 @ ( sup_sup_set_val @ Y3 @ Z ) )
= ( sup_sup_set_val @ Y3 @ ( sup_sup_set_val @ X2 @ Z ) ) ) ).
% sup_left_commute
thf(fact_271_sup_Oleft__commute,axiom,
! [B: set_val,A: set_val,C2: set_val] :
( ( sup_sup_set_val @ B @ ( sup_sup_set_val @ A @ C2 ) )
= ( sup_sup_set_val @ A @ ( sup_sup_set_val @ B @ C2 ) ) ) ).
% sup.left_commute
thf(fact_272_sup__commute,axiom,
( sup_sup_set_val
= ( ^ [X: set_val,Y2: set_val] : ( sup_sup_set_val @ Y2 @ X ) ) ) ).
% sup_commute
thf(fact_273_sup_Ocommute,axiom,
( sup_sup_set_val
= ( ^ [A3: set_val,B5: set_val] : ( sup_sup_set_val @ B5 @ A3 ) ) ) ).
% sup.commute
thf(fact_274_sup__assoc,axiom,
! [X2: set_val,Y3: set_val,Z: set_val] :
( ( sup_sup_set_val @ ( sup_sup_set_val @ X2 @ Y3 ) @ Z )
= ( sup_sup_set_val @ X2 @ ( sup_sup_set_val @ Y3 @ Z ) ) ) ).
% sup_assoc
thf(fact_275_sup_Oassoc,axiom,
! [A: set_val,B: set_val,C2: set_val] :
( ( sup_sup_set_val @ ( sup_sup_set_val @ A @ B ) @ C2 )
= ( sup_sup_set_val @ A @ ( sup_sup_set_val @ B @ C2 ) ) ) ).
% sup.assoc
thf(fact_276_boolean__algebra__cancel_Osup2,axiom,
! [B2: set_val,K: set_val,B: set_val,A: set_val] :
( ( B2
= ( sup_sup_set_val @ K @ B ) )
=> ( ( sup_sup_set_val @ A @ B2 )
= ( sup_sup_set_val @ K @ ( sup_sup_set_val @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_277_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_val,K: set_val,A: set_val,B: set_val] :
( ( A2
= ( sup_sup_set_val @ K @ A ) )
=> ( ( sup_sup_set_val @ A2 @ B )
= ( sup_sup_set_val @ K @ ( sup_sup_set_val @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_278_inf__sup__aci_I5_J,axiom,
( sup_sup_set_val
= ( ^ [X: set_val,Y2: set_val] : ( sup_sup_set_val @ Y2 @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_279_inf__sup__aci_I6_J,axiom,
! [X2: set_val,Y3: set_val,Z: set_val] :
( ( sup_sup_set_val @ ( sup_sup_set_val @ X2 @ Y3 ) @ Z )
= ( sup_sup_set_val @ X2 @ ( sup_sup_set_val @ Y3 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_280_inf__sup__aci_I7_J,axiom,
! [X2: set_val,Y3: set_val,Z: set_val] :
( ( sup_sup_set_val @ X2 @ ( sup_sup_set_val @ Y3 @ Z ) )
= ( sup_sup_set_val @ Y3 @ ( sup_sup_set_val @ X2 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_281_inf__sup__aci_I8_J,axiom,
! [X2: set_val,Y3: set_val] :
( ( sup_sup_set_val @ X2 @ ( sup_sup_set_val @ X2 @ Y3 ) )
= ( sup_sup_set_val @ X2 @ Y3 ) ) ).
% inf_sup_aci(8)
thf(fact_282_Un__left__commute,axiom,
! [A2: set_val,B2: set_val,C: set_val] :
( ( sup_sup_set_val @ A2 @ ( sup_sup_set_val @ B2 @ C ) )
= ( sup_sup_set_val @ B2 @ ( sup_sup_set_val @ A2 @ C ) ) ) ).
% Un_left_commute
thf(fact_283_Un__left__absorb,axiom,
! [A2: set_val,B2: set_val] :
( ( sup_sup_set_val @ A2 @ ( sup_sup_set_val @ A2 @ B2 ) )
= ( sup_sup_set_val @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_284_Un__commute,axiom,
( sup_sup_set_val
= ( ^ [A5: set_val,B4: set_val] : ( sup_sup_set_val @ B4 @ A5 ) ) ) ).
% Un_commute
thf(fact_285_Un__absorb,axiom,
! [A2: set_val] :
( ( sup_sup_set_val @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_286_Un__assoc,axiom,
! [A2: set_val,B2: set_val,C: set_val] :
( ( sup_sup_set_val @ ( sup_sup_set_val @ A2 @ B2 ) @ C )
= ( sup_sup_set_val @ A2 @ ( sup_sup_set_val @ B2 @ C ) ) ) ).
% Un_assoc
thf(fact_287_ball__Un,axiom,
! [A2: set_val,B2: set_val,P2: val > $o] :
( ( ! [X: val] :
( ( member_val @ X @ ( sup_sup_set_val @ A2 @ B2 ) )
=> ( P2 @ X ) ) )
= ( ! [X: val] :
( ( member_val @ X @ A2 )
=> ( P2 @ X ) )
& ! [X: val] :
( ( member_val @ X @ B2 )
=> ( P2 @ X ) ) ) ) ).
% ball_Un
thf(fact_288_bex__Un,axiom,
! [A2: set_val,B2: set_val,P2: val > $o] :
( ( ? [X: val] :
( ( member_val @ X @ ( sup_sup_set_val @ A2 @ B2 ) )
& ( P2 @ X ) ) )
= ( ? [X: val] :
( ( member_val @ X @ A2 )
& ( P2 @ X ) )
| ? [X: val] :
( ( member_val @ X @ B2 )
& ( P2 @ X ) ) ) ) ).
% bex_Un
thf(fact_289_UnI2,axiom,
! [C2: node,B2: set_node,A2: set_node] :
( ( member_node @ C2 @ B2 )
=> ( member_node @ C2 @ ( sup_sup_set_node @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_290_UnI2,axiom,
! [C2: set_val,B2: set_set_val,A2: set_set_val] :
( ( member_set_val @ C2 @ B2 )
=> ( member_set_val @ C2 @ ( sup_sup_set_set_val @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_291_UnI2,axiom,
! [C2: produc1432036078de_val,B2: set_Pr699757092de_val,A2: set_Pr699757092de_val] :
( ( member313544709de_val @ C2 @ B2 )
=> ( member313544709de_val @ C2 @ ( sup_su2138939728de_val @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_292_UnI2,axiom,
! [C2: produc1324971431et_val,B2: set_Pr1311924359et_val,A2: set_Pr1311924359et_val] :
( ( member1711426256et_val @ C2 @ B2 )
=> ( member1711426256et_val @ C2 @ ( sup_su2005730907et_val @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_293_UnI2,axiom,
! [C2: val,B2: set_val,A2: set_val] :
( ( member_val @ C2 @ B2 )
=> ( member_val @ C2 @ ( sup_sup_set_val @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_294_UnI1,axiom,
! [C2: node,A2: set_node,B2: set_node] :
( ( member_node @ C2 @ A2 )
=> ( member_node @ C2 @ ( sup_sup_set_node @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_295_UnI1,axiom,
! [C2: set_val,A2: set_set_val,B2: set_set_val] :
( ( member_set_val @ C2 @ A2 )
=> ( member_set_val @ C2 @ ( sup_sup_set_set_val @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_296_UnI1,axiom,
! [C2: produc1432036078de_val,A2: set_Pr699757092de_val,B2: set_Pr699757092de_val] :
( ( member313544709de_val @ C2 @ A2 )
=> ( member313544709de_val @ C2 @ ( sup_su2138939728de_val @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_297_UnI1,axiom,
! [C2: produc1324971431et_val,A2: set_Pr1311924359et_val,B2: set_Pr1311924359et_val] :
( ( member1711426256et_val @ C2 @ A2 )
=> ( member1711426256et_val @ C2 @ ( sup_su2005730907et_val @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_298_UnI1,axiom,
! [C2: val,A2: set_val,B2: set_val] :
( ( member_val @ C2 @ A2 )
=> ( member_val @ C2 @ ( sup_sup_set_val @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_299_UnE,axiom,
! [C2: node,A2: set_node,B2: set_node] :
( ( member_node @ C2 @ ( sup_sup_set_node @ A2 @ B2 ) )
=> ( ~ ( member_node @ C2 @ A2 )
=> ( member_node @ C2 @ B2 ) ) ) ).
% UnE
thf(fact_300_UnE,axiom,
! [C2: set_val,A2: set_set_val,B2: set_set_val] :
( ( member_set_val @ C2 @ ( sup_sup_set_set_val @ A2 @ B2 ) )
=> ( ~ ( member_set_val @ C2 @ A2 )
=> ( member_set_val @ C2 @ B2 ) ) ) ).
% UnE
thf(fact_301_UnE,axiom,
! [C2: produc1432036078de_val,A2: set_Pr699757092de_val,B2: set_Pr699757092de_val] :
( ( member313544709de_val @ C2 @ ( sup_su2138939728de_val @ A2 @ B2 ) )
=> ( ~ ( member313544709de_val @ C2 @ A2 )
=> ( member313544709de_val @ C2 @ B2 ) ) ) ).
% UnE
thf(fact_302_UnE,axiom,
! [C2: produc1324971431et_val,A2: set_Pr1311924359et_val,B2: set_Pr1311924359et_val] :
( ( member1711426256et_val @ C2 @ ( sup_su2005730907et_val @ A2 @ B2 ) )
=> ( ~ ( member1711426256et_val @ C2 @ A2 )
=> ( member1711426256et_val @ C2 @ B2 ) ) ) ).
% UnE
thf(fact_303_UnE,axiom,
! [C2: val,A2: set_val,B2: set_val] :
( ( member_val @ C2 @ ( sup_sup_set_val @ A2 @ B2 ) )
=> ( ~ ( member_val @ C2 @ A2 )
=> ( member_val @ C2 @ B2 ) ) ) ).
% UnE
thf(fact_304_sup_OcoboundedI2,axiom,
! [C2: set_val,B: set_val,A: set_val] :
( ( ord_less_eq_set_val @ C2 @ B )
=> ( ord_less_eq_set_val @ C2 @ ( sup_sup_set_val @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_305_sup_OcoboundedI1,axiom,
! [C2: set_val,A: set_val,B: set_val] :
( ( ord_less_eq_set_val @ C2 @ A )
=> ( ord_less_eq_set_val @ C2 @ ( sup_sup_set_val @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_306_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_val
= ( ^ [A3: set_val,B5: set_val] :
( ( sup_sup_set_val @ A3 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_307_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_val
= ( ^ [B5: set_val,A3: set_val] :
( ( sup_sup_set_val @ A3 @ B5 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_308_sup_Ocobounded2,axiom,
! [B: set_val,A: set_val] : ( ord_less_eq_set_val @ B @ ( sup_sup_set_val @ A @ B ) ) ).
% sup.cobounded2
thf(fact_309_sup_Ocobounded1,axiom,
! [A: set_val,B: set_val] : ( ord_less_eq_set_val @ A @ ( sup_sup_set_val @ A @ B ) ) ).
% sup.cobounded1
thf(fact_310_sup_Oorder__iff,axiom,
( ord_less_eq_set_val
= ( ^ [B5: set_val,A3: set_val] :
( A3
= ( sup_sup_set_val @ A3 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_311_sup_OboundedI,axiom,
! [B: set_val,A: set_val,C2: set_val] :
( ( ord_less_eq_set_val @ B @ A )
=> ( ( ord_less_eq_set_val @ C2 @ A )
=> ( ord_less_eq_set_val @ ( sup_sup_set_val @ B @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_312_sup_OboundedE,axiom,
! [B: set_val,C2: set_val,A: set_val] :
( ( ord_less_eq_set_val @ ( sup_sup_set_val @ B @ C2 ) @ A )
=> ~ ( ( ord_less_eq_set_val @ B @ A )
=> ~ ( ord_less_eq_set_val @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_313_sup__absorb2,axiom,
! [X2: set_val,Y3: set_val] :
( ( ord_less_eq_set_val @ X2 @ Y3 )
=> ( ( sup_sup_set_val @ X2 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_314_sup__absorb1,axiom,
! [Y3: set_val,X2: set_val] :
( ( ord_less_eq_set_val @ Y3 @ X2 )
=> ( ( sup_sup_set_val @ X2 @ Y3 )
= X2 ) ) ).
% sup_absorb1
thf(fact_315_sup_Oabsorb2,axiom,
! [A: set_val,B: set_val] :
( ( ord_less_eq_set_val @ A @ B )
=> ( ( sup_sup_set_val @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_316_sup_Oabsorb1,axiom,
! [B: set_val,A: set_val] :
( ( ord_less_eq_set_val @ B @ A )
=> ( ( sup_sup_set_val @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_317_sup__unique,axiom,
! [F2: set_val > set_val > set_val,X2: set_val,Y3: set_val] :
( ! [X3: set_val,Y5: set_val] : ( ord_less_eq_set_val @ X3 @ ( F2 @ X3 @ Y5 ) )
=> ( ! [X3: set_val,Y5: set_val] : ( ord_less_eq_set_val @ Y5 @ ( F2 @ X3 @ Y5 ) )
=> ( ! [X3: set_val,Y5: set_val,Z3: set_val] :
( ( ord_less_eq_set_val @ Y5 @ X3 )
=> ( ( ord_less_eq_set_val @ Z3 @ X3 )
=> ( ord_less_eq_set_val @ ( F2 @ Y5 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_set_val @ X2 @ Y3 )
= ( F2 @ X2 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_318_sup_OorderI,axiom,
! [A: set_val,B: set_val] :
( ( A
= ( sup_sup_set_val @ A @ B ) )
=> ( ord_less_eq_set_val @ B @ A ) ) ).
% sup.orderI
thf(fact_319_sup_OorderE,axiom,
! [B: set_val,A: set_val] :
( ( ord_less_eq_set_val @ B @ A )
=> ( A
= ( sup_sup_set_val @ A @ B ) ) ) ).
% sup.orderE
thf(fact_320_le__iff__sup,axiom,
( ord_less_eq_set_val
= ( ^ [X: set_val,Y2: set_val] :
( ( sup_sup_set_val @ X @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_321_sup__least,axiom,
! [Y3: set_val,X2: set_val,Z: set_val] :
( ( ord_less_eq_set_val @ Y3 @ X2 )
=> ( ( ord_less_eq_set_val @ Z @ X2 )
=> ( ord_less_eq_set_val @ ( sup_sup_set_val @ Y3 @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_322_sup__mono,axiom,
! [A: set_val,C2: set_val,B: set_val,D: set_val] :
( ( ord_less_eq_set_val @ A @ C2 )
=> ( ( ord_less_eq_set_val @ B @ D )
=> ( ord_less_eq_set_val @ ( sup_sup_set_val @ A @ B ) @ ( sup_sup_set_val @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_323_sup_Omono,axiom,
! [C2: set_val,A: set_val,D: set_val,B: set_val] :
( ( ord_less_eq_set_val @ C2 @ A )
=> ( ( ord_less_eq_set_val @ D @ B )
=> ( ord_less_eq_set_val @ ( sup_sup_set_val @ C2 @ D ) @ ( sup_sup_set_val @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_324_le__supI2,axiom,
! [X2: set_val,B: set_val,A: set_val] :
( ( ord_less_eq_set_val @ X2 @ B )
=> ( ord_less_eq_set_val @ X2 @ ( sup_sup_set_val @ A @ B ) ) ) ).
% le_supI2
thf(fact_325_le__supI1,axiom,
! [X2: set_val,A: set_val,B: set_val] :
( ( ord_less_eq_set_val @ X2 @ A )
=> ( ord_less_eq_set_val @ X2 @ ( sup_sup_set_val @ A @ B ) ) ) ).
% le_supI1
thf(fact_326_sup__ge2,axiom,
! [Y3: set_val,X2: set_val] : ( ord_less_eq_set_val @ Y3 @ ( sup_sup_set_val @ X2 @ Y3 ) ) ).
% sup_ge2
thf(fact_327_sup__ge1,axiom,
! [X2: set_val,Y3: set_val] : ( ord_less_eq_set_val @ X2 @ ( sup_sup_set_val @ X2 @ Y3 ) ) ).
% sup_ge1
thf(fact_328_le__supI,axiom,
! [A: set_val,X2: set_val,B: set_val] :
( ( ord_less_eq_set_val @ A @ X2 )
=> ( ( ord_less_eq_set_val @ B @ X2 )
=> ( ord_less_eq_set_val @ ( sup_sup_set_val @ A @ B ) @ X2 ) ) ) ).
% le_supI
thf(fact_329_le__supE,axiom,
! [A: set_val,B: set_val,X2: set_val] :
( ( ord_less_eq_set_val @ ( sup_sup_set_val @ A @ B ) @ X2 )
=> ~ ( ( ord_less_eq_set_val @ A @ X2 )
=> ~ ( ord_less_eq_set_val @ B @ X2 ) ) ) ).
% le_supE
thf(fact_330_inf__sup__ord_I3_J,axiom,
! [X2: set_val,Y3: set_val] : ( ord_less_eq_set_val @ X2 @ ( sup_sup_set_val @ X2 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_331_inf__sup__ord_I4_J,axiom,
! [Y3: set_val,X2: set_val] : ( ord_less_eq_set_val @ Y3 @ ( sup_sup_set_val @ X2 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_332_subset__Un__eq,axiom,
( ord_less_eq_set_val
= ( ^ [A5: set_val,B4: set_val] :
( ( sup_sup_set_val @ A5 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_333_subset__UnE,axiom,
! [C: set_val,A2: set_val,B2: set_val] :
( ( ord_less_eq_set_val @ C @ ( sup_sup_set_val @ A2 @ B2 ) )
=> ~ ! [A6: set_val] :
( ( ord_less_eq_set_val @ A6 @ A2 )
=> ! [B6: set_val] :
( ( ord_less_eq_set_val @ B6 @ B2 )
=> ( C
!= ( sup_sup_set_val @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_334_Un__absorb2,axiom,
! [B2: set_val,A2: set_val] :
( ( ord_less_eq_set_val @ B2 @ A2 )
=> ( ( sup_sup_set_val @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_335_Un__absorb1,axiom,
! [A2: set_val,B2: set_val] :
( ( ord_less_eq_set_val @ A2 @ B2 )
=> ( ( sup_sup_set_val @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_336_Un__upper2,axiom,
! [B2: set_val,A2: set_val] : ( ord_less_eq_set_val @ B2 @ ( sup_sup_set_val @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_337_Un__upper1,axiom,
! [A2: set_val,B2: set_val] : ( ord_less_eq_set_val @ A2 @ ( sup_sup_set_val @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_338_Un__least,axiom,
! [A2: set_val,C: set_val,B2: set_val] :
( ( ord_less_eq_set_val @ A2 @ C )
=> ( ( ord_less_eq_set_val @ B2 @ C )
=> ( ord_less_eq_set_val @ ( sup_sup_set_val @ A2 @ B2 ) @ C ) ) ) ).
% Un_least
thf(fact_339_Un__mono,axiom,
! [A2: set_val,C: set_val,B2: set_val,D2: set_val] :
( ( ord_less_eq_set_val @ A2 @ C )
=> ( ( ord_less_eq_set_val @ B2 @ D2 )
=> ( ord_less_eq_set_val @ ( sup_sup_set_val @ A2 @ B2 ) @ ( sup_sup_set_val @ C @ D2 ) ) ) ) ).
% Un_mono
thf(fact_340_redundant__set__def,axiom,
! [G: g,P2: set_val] :
( ( irredu224829488eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ G @ P2 )
= ( ( P2 != bot_bot_set_val )
& ( ord_less_eq_set_val @ P2 @ ( dom_val_list_val @ ( sSA_CF262257161de_val @ alpha_n @ defs @ phis @ G ) ) )
& ? [X: val] :
( ( member_val @ X @ ( sSA_CF1517915011eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ G ) )
& ! [Y2: val] :
( ( member_val @ Y2 @ P2 )
=> ! [Phi3: val] :
( ( sSA_CF1252180629de_val @ alpha_n @ defs @ phis @ G @ Y2 @ Phi3 )
=> ( member_val @ Phi3 @ ( sup_sup_set_val @ P2 @ ( insert_val @ X @ bot_bot_set_val ) ) ) ) ) ) ) ) ).
% redundant_set_def
thf(fact_341_phi__no__closed__loop,axiom,
! [P: val,G: g,Vs: list_val] :
( ( member_val @ P @ ( sSA_CF1517915011eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ G ) )
=> ( ( ( sSA_CF262257161de_val @ alpha_n @ defs @ phis @ G @ P )
= ( some_list_val @ Vs ) )
=> ( ( set_val2 @ Vs )
!= ( insert_val @ P @ bot_bot_set_val ) ) ) ) ).
% phi_no_closed_loop
thf(fact_342_dom__eq__empty__conv,axiom,
! [F2: produc1432036078de_val > option_list_val] :
( ( ( dom_Pr729677986st_val @ F2 )
= bot_bo404898488de_val )
= ( F2
= ( ^ [X: produc1432036078de_val] : none_list_val ) ) ) ).
% dom_eq_empty_conv
thf(fact_343_dom__eq__empty__conv,axiom,
! [F2: val > option_list_val] :
( ( ( dom_val_list_val @ F2 )
= bot_bot_set_val )
= ( F2
= ( ^ [X: val] : none_list_val ) ) ) ).
% dom_eq_empty_conv
thf(fact_344_phis__finite,axiom,
! [G: g] : ( finite2056463621de_val @ ( dom_Pr729677986st_val @ ( phis @ G ) ) ) ).
% phis_finite
thf(fact_345_insert__subset,axiom,
! [X2: node,A2: set_node,B2: set_node] :
( ( ord_less_eq_set_node @ ( insert_node @ X2 @ A2 ) @ B2 )
= ( ( member_node @ X2 @ B2 )
& ( ord_less_eq_set_node @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_346_insert__subset,axiom,
! [X2: set_val,A2: set_set_val,B2: set_set_val] :
( ( ord_le1742111550et_val @ ( insert_set_val @ X2 @ A2 ) @ B2 )
= ( ( member_set_val @ X2 @ B2 )
& ( ord_le1742111550et_val @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_347_insert__subset,axiom,
! [X2: produc1432036078de_val,A2: set_Pr699757092de_val,B2: set_Pr699757092de_val] :
( ( ord_le1643692676de_val @ ( insert869443870de_val @ X2 @ A2 ) @ B2 )
= ( ( member313544709de_val @ X2 @ B2 )
& ( ord_le1643692676de_val @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_348_insert__subset,axiom,
! [X2: produc1324971431et_val,A2: set_Pr1311924359et_val,B2: set_Pr1311924359et_val] :
( ( ord_le299366439et_val @ ( insert1846469495et_val @ X2 @ A2 ) @ B2 )
= ( ( member1711426256et_val @ X2 @ B2 )
& ( ord_le299366439et_val @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_349_insert__subset,axiom,
! [X2: val,A2: set_val,B2: set_val] :
( ( ord_less_eq_set_val @ ( insert_val @ X2 @ A2 ) @ B2 )
= ( ( member_val @ X2 @ B2 )
& ( ord_less_eq_set_val @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_350_finite__insert,axiom,
! [A: val,A2: set_val] :
( ( finite_finite_val @ ( insert_val @ A @ A2 ) )
= ( finite_finite_val @ A2 ) ) ).
% finite_insert
thf(fact_351_finite__insert,axiom,
! [A: produc1324971431et_val,A2: set_Pr1311924359et_val] :
( ( finite79836624et_val @ ( insert1846469495et_val @ A @ A2 ) )
= ( finite79836624et_val @ A2 ) ) ).
% finite_insert
thf(fact_352_finite__insert,axiom,
! [A: produc1432036078de_val,A2: set_Pr699757092de_val] :
( ( finite2056463621de_val @ ( insert869443870de_val @ A @ A2 ) )
= ( finite2056463621de_val @ A2 ) ) ).
% finite_insert
% Conjectures (1)
thf(conj_0,conjecture,
member_val @ phi_s @ ( sSA_CF1517915011eD_val @ alpha_n @ inEdges @ defs @ uses @ phis @ g2 ) ).
%------------------------------------------------------------------------------