TPTP Problem File: ITP083^2.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP083^2 : 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 : 0.33 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 367 ( 90 unt; 66 typ; 0 def)
% Number of atoms : 834 ( 225 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 4837 ( 69 ~; 6 |; 78 &;4315 @)
% ( 0 <=>; 369 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 8 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 383 ( 383 >; 0 *; 0 +; 0 <<)
% Number of symbols : 63 ( 61 usr; 6 con; 0-12 aty)
% Number of variables : 1063 ( 84 ^; 851 !; 32 ?;1063 :)
% ( 96 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:28:47.811
%------------------------------------------------------------------------------
% Could-be-implicit typings (9)
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Option_Ooption,type,
option: $tType > $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_tf_edgeD,type,
edgeD: $tType ).
thf(ty_tf_node,type,
node: $tType ).
thf(ty_tf_var,type,
var: $tType ).
thf(ty_tf_val,type,
val: $tType ).
thf(ty_tf_g,type,
g: $tType ).
% Explicit typings (57)
thf(sy_cl_Lattices_Obounded__lattice__top,type,
bounded_lattice_top:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osup,type,
sup:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
bounde1808546759up_bot:
!>[A: $tType] : $o ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Graph__path_Ograph__path__base_Opredecessors,type,
graph_1201503639essors:
!>[G: $tType,Node: $tType,EdgeD: $tType] : ( ( G > Node > ( list @ ( product_prod @ Node @ EdgeD ) ) ) > G > Node > ( list @ Node ) ) ).
thf(sy_c_Graph__path_Ograph__path__base_Osuccessors,type,
graph_449533722essors:
!>[G: $tType,Node: $tType,EdgeD: $tType] : ( ( G > ( list @ Node ) ) > ( G > Node > ( list @ ( product_prod @ Node @ EdgeD ) ) ) > G > Node > ( list @ Node ) ) ).
thf(sy_c_Irreducible__Mirabelle__vdklcywumt_OCFG__SSA__Transformed_Ocondensation__edges,type,
irredu280805948_edges:
!>[G: $tType,Node: $tType,Val: $tType] : ( ( G > ( list @ Node ) ) > ( G > Node > ( set @ Val ) ) > ( G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) ) ) > G > ( set @ Val ) > ( set @ ( product_prod @ ( set @ Val ) @ ( set @ Val ) ) ) ) ).
thf(sy_c_Irreducible__Mirabelle__vdklcywumt_OCFG__SSA__Transformed_Ocondensation__nodes,type,
irredu1918690039_nodes:
!>[G: $tType,Node: $tType,Val: $tType] : ( ( G > ( list @ Node ) ) > ( G > Node > ( set @ Val ) ) > ( G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) ) ) > G > ( set @ Val ) > ( set @ ( set @ Val ) ) ) ).
thf(sy_c_Irreducible__Mirabelle__vdklcywumt_OCFG__SSA__Transformed_Oredundant__scc,type,
irredu2110774547nt_scc:
!>[G: $tType,Node: $tType,EdgeD: $tType,Val: $tType] : ( ( G > ( list @ Node ) ) > ( G > Node > ( list @ ( product_prod @ Node @ EdgeD ) ) ) > ( G > Node > ( set @ Val ) ) > ( G > Node > ( set @ Val ) ) > ( G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) ) ) > G > ( set @ Val ) > ( set @ Val ) > $o ) ).
thf(sy_c_Irreducible__Mirabelle__vdklcywumt_OCFG__SSA__Transformed_Oredundant__set,type,
irredu2110905762nt_set:
!>[G: $tType,Node: $tType,EdgeD: $tType,Val: $tType] : ( ( G > ( list @ Node ) ) > ( G > Node > ( list @ ( product_prod @ Node @ EdgeD ) ) ) > ( G > Node > ( set @ Val ) ) > ( G > Node > ( set @ Val ) ) > ( G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) ) ) > G > ( set @ Val ) > $o ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Ozip,type,
zip:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Map_Odom,type,
dom:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) ) ).
thf(sy_c_Option_Ooption_ONone,type,
none:
!>[A: $tType] : ( option @ A ) ).
thf(sy_c_Option_Ooption_OSome,type,
some:
!>[A: $tType] : ( A > ( option @ A ) ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_SSA__CFG_OCFG__SSA__base_OallDefs,type,
sSA_CFG_SSA_allDefs:
!>[G: $tType,Node: $tType,Val: $tType] : ( ( G > Node > ( set @ Val ) ) > ( G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) ) ) > G > Node > ( set @ Val ) ) ).
thf(sy_c_SSA__CFG_OCFG__SSA__base_OallUses,type,
sSA_CFG_SSA_allUses:
!>[G: $tType,Node: $tType,EdgeD: $tType,Val: $tType] : ( ( G > ( list @ Node ) ) > ( G > Node > ( list @ ( product_prod @ Node @ EdgeD ) ) ) > ( G > Node > ( set @ Val ) ) > ( G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) ) ) > G > Node > ( set @ Val ) ) ).
thf(sy_c_SSA__CFG_OCFG__SSA__base_OallVars,type,
sSA_CFG_SSA_allVars:
!>[G: $tType,Node: $tType,EdgeD: $tType,Val: $tType] : ( ( G > ( list @ Node ) ) > ( G > Node > ( list @ ( product_prod @ Node @ EdgeD ) ) ) > ( G > Node > ( set @ Val ) ) > ( G > Node > ( set @ Val ) ) > ( G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) ) ) > G > ( set @ Val ) ) ).
thf(sy_c_SSA__CFG_OCFG__SSA__base_OphiDefs,type,
sSA_CFG_SSA_phiDefs:
!>[G: $tType,Node: $tType,Val: $tType] : ( ( G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) ) ) > G > Node > ( set @ Val ) ) ).
thf(sy_c_SSA__CFG_OCFG__SSA__base_OphiUses,type,
sSA_CFG_SSA_phiUses:
!>[G: $tType,Node: $tType,EdgeD: $tType,Val: $tType] : ( ( G > ( list @ Node ) ) > ( G > Node > ( list @ ( product_prod @ Node @ EdgeD ) ) ) > ( G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) ) ) > G > Node > ( set @ Val ) ) ).
thf(sy_c_SSA__CFG_OCFG__SSA__wf__base_OdefNode,type,
sSA_CF1081484811efNode:
!>[G: $tType,Node: $tType,Val: $tType] : ( ( G > ( list @ Node ) ) > ( G > Node > ( set @ Val ) ) > ( G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) ) ) > G > Val > Node ) ).
thf(sy_c_SSA__CFG_OCFG__SSA__wf__base_OisTrivialPhi,type,
sSA_CF1297404942ialPhi:
!>[G: $tType,Node: $tType,Val: $tType] : ( ( G > ( list @ Node ) ) > ( G > Node > ( set @ Val ) ) > ( G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) ) ) > G > Val > Val > $o ) ).
thf(sy_c_SSA__CFG_OCFG__SSA__wf__base_OliveVal,type,
sSA_CF722422777iveVal:
!>[G: $tType,Node: $tType,Val: $tType] : ( ( G > ( list @ Node ) ) > ( G > Node > ( set @ Val ) ) > ( G > Node > ( set @ Val ) ) > ( G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) ) ) > G > Val > $o ) ).
thf(sy_c_SSA__CFG_OCFG__SSA__wf__base_Ophi,type,
sSA_CFG_SSA_wf_phi:
!>[G: $tType,Node: $tType,Val: $tType] : ( ( G > ( list @ Node ) ) > ( G > Node > ( set @ Val ) ) > ( G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) ) ) > G > Val > ( option @ ( list @ Val ) ) ) ).
thf(sy_c_SSA__CFG_OCFG__SSA__wf__base_OphiArg,type,
sSA_CF1165125185phiArg:
!>[G: $tType,Node: $tType,Val: $tType] : ( ( G > ( list @ Node ) ) > ( G > Node > ( set @ Val ) ) > ( G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) ) ) > G > Val > Val > $o ) ).
thf(sy_c_SSA__CFG_OCFG__SSA__wf__base_Opruned,type,
sSA_CF823892918pruned:
!>[G: $tType,Node: $tType,Val: $tType] : ( ( G > ( list @ Node ) ) > ( G > Node > ( set @ Val ) ) > ( G > Node > ( set @ Val ) ) > ( G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) ) ) > G > $o ) ).
thf(sy_c_SSA__CFG_OCFG__SSA__wf__base_Oredundant,type,
sSA_CF462202545undant:
!>[G: $tType,Node: $tType,EdgeD: $tType,Val: $tType] : ( ( G > ( list @ Node ) ) > ( G > Node > ( list @ ( product_prod @ Node @ EdgeD ) ) ) > ( G > Node > ( set @ Val ) ) > ( G > Node > ( set @ Val ) ) > ( G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) ) ) > G > $o ) ).
thf(sy_c_SSA__CFG_OCFG__SSA__wf__base_Otrivial,type,
sSA_CF1363434349rivial:
!>[G: $tType,Node: $tType,EdgeD: $tType,Val: $tType] : ( ( G > ( list @ Node ) ) > ( G > Node > ( list @ ( product_prod @ Node @ EdgeD ) ) ) > ( G > Node > ( set @ Val ) ) > ( G > Node > ( set @ Val ) ) > ( G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) ) ) > G > Val > $o ) ).
thf(sy_c_SSA__CFG_OCFG__base_Ovars,type,
sSA_CFG_vars:
!>[G: $tType,Node: $tType,Var: $tType] : ( ( G > ( list @ Node ) ) > ( G > Node > ( set @ Var ) ) > G > ( set @ Var ) ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Transitive__Closure_Oacyclic,type,
transitive_acyclic:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Transitive__Closure_Otranclp,type,
transitive_tranclp:
!>[A: $tType] : ( ( A > A > $o ) > A > A > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $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 @ ( product_prod @ node @ edgeD ) ) ).
thf(sy_v_phis,type,
phis: g > ( product_prod @ node @ 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 (255)
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_CF1165125185phiArg @ g @ node @ 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_CFG_SSA_allVars @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ defs @ uses @ phis @ g2 ) ).
% assms(3)
thf(fact_4_assms_I2_J,axiom,
member @ val @ phi @ ( sSA_CFG_SSA_allVars @ g @ node @ edgeD @ 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_CF1165125185phiArg @ g @ node @ 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,
! [G2: g,V: val,V2: val] :
( ( sSA_CF1165125185phiArg @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ V @ V2 )
=> ( member @ val @ V2 @ ( sSA_CFG_SSA_allVars @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ defs @ uses @ phis @ G2 ) ) ) ).
% phiArg_in_allVars
thf(fact_7_uses__in__vars,axiom,
! [V: val,G2: g,N: node] :
( ( member @ val @ V @ ( uses @ G2 @ N ) )
=> ( member @ val @ V @ ( sSA_CFG_vars @ g @ node @ val @ alpha_n @ uses @ G2 ) ) ) ).
% uses_in_vars
thf(fact_8_trivial__in__allVars,axiom,
! [G2: g,V: val] :
( ( sSA_CF1363434349rivial @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ defs @ uses @ phis @ G2 @ V )
=> ( member @ val @ V @ ( sSA_CFG_SSA_allVars @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ defs @ uses @ phis @ G2 ) ) ) ).
% trivial_in_allVars
thf(fact_9_liveVal__in__allVars,axiom,
! [G2: g,V: val] :
( ( sSA_CF722422777iveVal @ g @ node @ val @ alpha_n @ defs @ uses @ phis @ G2 @ V )
=> ( member @ val @ V @ ( sSA_CFG_SSA_allVars @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ defs @ uses @ phis @ G2 ) ) ) ).
% liveVal_in_allVars
thf(fact_10_allVars__finite,axiom,
! [G2: g] : ( finite_finite2 @ val @ ( sSA_CFG_SSA_allVars @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ defs @ uses @ phis @ G2 ) ) ).
% allVars_finite
thf(fact_11_uses__in__allUses,axiom,
! [V: val,G2: g,N: node] :
( ( member @ val @ V @ ( uses @ G2 @ N ) )
=> ( member @ val @ V @ ( sSA_CFG_SSA_allUses @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ uses @ phis @ G2 @ N ) ) ) ).
% uses_in_allUses
thf(fact_12_redundant__def,axiom,
! [G2: g] :
( ( sSA_CF462202545undant @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ defs @ uses @ phis @ G2 )
= ( ? [X: val] :
( ( member @ val @ X @ ( sSA_CFG_SSA_allVars @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ defs @ uses @ phis @ G2 ) )
& ( sSA_CF1363434349rivial @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ defs @ uses @ phis @ G2 @ X ) ) ) ) ).
% redundant_def
thf(fact_13_trivial__def,axiom,
! [G2: g,V: val] :
( ( sSA_CF1363434349rivial @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ defs @ uses @ phis @ G2 @ V )
= ( ? [X: val] :
( ( member @ val @ X @ ( sSA_CFG_SSA_allVars @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ defs @ uses @ phis @ G2 ) )
& ( sSA_CF1297404942ialPhi @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ 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_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ g2 @ phi_s )
= ( sSA_CF1081484811efNode @ g @ node @ 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,G2: g,V: val] :
( ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( ( member @ val @ V @ ( defs @ G2 @ N ) )
=> ~ ( member @ val @ V @ ( uses @ G2 @ N ) ) ) ) ).
% defs_uses_disjoint'
thf(fact_16_livePhi,axiom,
! [G2: g,V: val,V2: val] :
( ( sSA_CF722422777iveVal @ g @ node @ val @ alpha_n @ defs @ uses @ phis @ G2 @ V )
=> ( ( sSA_CF1165125185phiArg @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ V @ V2 )
=> ( sSA_CF722422777iveVal @ g @ node @ val @ alpha_n @ defs @ uses @ phis @ G2 @ V2 ) ) ) ).
% livePhi
thf(fact_17_phiArg__distinct__nodes,axiom,
! [G2: g,P: val,Q: val] :
( ( sSA_CF1165125185phiArg @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ P @ Q )
=> ( ( P != Q )
=> ( ( member @ val @ P @ ( sSA_CFG_SSA_allVars @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ defs @ uses @ phis @ G2 ) )
=> ( ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ P )
!= ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ Q ) ) ) ) ) ).
% phiArg_distinct_nodes
thf(fact_18_phiArgs__def__distinct,axiom,
! [G2: g,P: val,Q: val,R: val] :
( ( sSA_CF1165125185phiArg @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ P @ Q )
=> ( ( sSA_CF1165125185phiArg @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ P @ R )
=> ( ( Q != R )
=> ( ( member @ val @ P @ ( sSA_CFG_SSA_allVars @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ defs @ uses @ phis @ G2 ) )
=> ( ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ Q )
!= ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ R ) ) ) ) ) ) ).
% phiArgs_def_distinct
thf(fact_19_uses__in___092_060alpha_062n,axiom,
! [V: val,G2: g,N: node] :
( ( member @ val @ V @ ( uses @ G2 @ N ) )
=> ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) ) ) ).
% uses_in_\<alpha>n
thf(fact_20_defs__finite,axiom,
! [G2: g,N: node] : ( finite_finite2 @ val @ ( defs @ G2 @ N ) ) ).
% defs_finite
thf(fact_21_varsE,axiom,
! [V: val,G2: g] :
( ( member @ val @ V @ ( sSA_CFG_vars @ g @ node @ val @ alpha_n @ uses @ G2 ) )
=> ~ ! [N2: node] :
( ( member @ node @ N2 @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ~ ( member @ val @ V @ ( uses @ G2 @ N2 ) ) ) ) ).
% varsE
thf(fact_22_vars__finite,axiom,
! [G2: g] : ( finite_finite2 @ val @ ( sSA_CFG_vars @ g @ node @ val @ alpha_n @ uses @ G2 ) ) ).
% vars_finite
thf(fact_23_liveSimple,axiom,
! [N: node,G2: g,Val2: val] :
( ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( ( member @ val @ Val2 @ ( uses @ G2 @ N ) )
=> ( sSA_CF722422777iveVal @ g @ node @ val @ alpha_n @ defs @ uses @ phis @ G2 @ Val2 ) ) ) ).
% liveSimple
thf(fact_24_liveVal_Osimps,axiom,
! [G2: g,A2: val] :
( ( sSA_CF722422777iveVal @ g @ node @ val @ alpha_n @ defs @ uses @ phis @ G2 @ A2 )
= ( ? [N3: node,Val3: val] :
( ( A2 = Val3 )
& ( member @ node @ N3 @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
& ( member @ val @ Val3 @ ( uses @ G2 @ N3 ) ) )
| ? [V3: val,V4: val] :
( ( A2 = V4 )
& ( sSA_CF722422777iveVal @ g @ node @ val @ alpha_n @ defs @ uses @ phis @ G2 @ V3 )
& ( sSA_CF1165125185phiArg @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ V3 @ V4 ) ) ) ) ).
% liveVal.simps
thf(fact_25_liveVal_Oinducts,axiom,
! [G2: g,X2: val,P2: val > $o] :
( ( sSA_CF722422777iveVal @ g @ node @ val @ alpha_n @ defs @ uses @ phis @ G2 @ X2 )
=> ( ! [N2: node,Val4: val] :
( ( member @ node @ N2 @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( ( member @ val @ Val4 @ ( uses @ G2 @ N2 ) )
=> ( P2 @ Val4 ) ) )
=> ( ! [V5: val,V6: val] :
( ( sSA_CF722422777iveVal @ g @ node @ val @ alpha_n @ defs @ uses @ phis @ G2 @ V5 )
=> ( ( P2 @ V5 )
=> ( ( sSA_CF1165125185phiArg @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ V5 @ V6 )
=> ( P2 @ V6 ) ) ) )
=> ( P2 @ X2 ) ) ) ) ).
% liveVal.inducts
thf(fact_26_liveVal_Ocases,axiom,
! [G2: g,A2: val] :
( ( sSA_CF722422777iveVal @ g @ node @ val @ alpha_n @ defs @ uses @ phis @ G2 @ A2 )
=> ( ! [N2: node] :
( ( member @ node @ N2 @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ~ ( member @ val @ A2 @ ( uses @ G2 @ N2 ) ) )
=> ~ ! [V5: val] :
( ( sSA_CF722422777iveVal @ g @ node @ val @ alpha_n @ defs @ uses @ phis @ G2 @ V5 )
=> ~ ( sSA_CF1165125185phiArg @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ V5 @ A2 ) ) ) ) ).
% liveVal.cases
thf(fact_27_allUses__finite,axiom,
! [N: node,G2: g] :
( ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( finite_finite2 @ val @ ( sSA_CFG_SSA_allUses @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ uses @ phis @ G2 @ N ) ) ) ).
% allUses_finite
thf(fact_28_defNode_I1_J,axiom,
! [V: val,G2: g] :
( ( member @ val @ V @ ( sSA_CFG_SSA_allVars @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ defs @ uses @ phis @ G2 ) )
=> ( member @ node @ ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ V ) @ ( set2 @ node @ ( alpha_n @ G2 ) ) ) ) ).
% defNode(1)
thf(fact_29_uses__finite,axiom,
! [G2: g,N: node] : ( finite_finite2 @ val @ ( uses @ G2 @ N ) ) ).
% uses_finite
thf(fact_30_eq,axiom,
( ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ g2 @ phi_s )
= ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ g2 @ s ) ) ).
% eq
thf(fact_31_allUses__in__allVars,axiom,
! [V: val,G2: g,N: node] :
( ( member @ val @ V @ ( sSA_CFG_SSA_allUses @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ uses @ phis @ G2 @ N ) )
=> ( ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( member @ val @ V @ ( sSA_CFG_SSA_allVars @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ defs @ uses @ phis @ G2 ) ) ) ) ).
% allUses_in_allVars
thf(fact_32_phiUses__in__allUses,axiom,
! [V: val,G2: g,N: node] :
( ( member @ val @ V @ ( sSA_CFG_SSA_phiUses @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ phis @ G2 @ N ) )
=> ( member @ val @ V @ ( sSA_CFG_SSA_allUses @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ uses @ phis @ G2 @ N ) ) ) ).
% phiUses_in_allUses
thf(fact_33_phiUses__finite,axiom,
! [N: node,G2: g] :
( ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( finite_finite2 @ val @ ( sSA_CFG_SSA_phiUses @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ phis @ G2 @ N ) ) ) ).
% phiUses_finite
thf(fact_34_defNode_I2_J,axiom,
! [V: val,G2: g] :
( ( member @ val @ V @ ( sSA_CFG_SSA_allVars @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ defs @ uses @ phis @ G2 ) )
=> ( member @ val @ V @ ( sSA_CFG_SSA_allDefs @ g @ node @ val @ defs @ phis @ G2 @ ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ V ) ) ) ) ).
% defNode(2)
thf(fact_35_defNode__ex1,axiom,
! [V: val,G2: g] :
( ( member @ val @ V @ ( sSA_CFG_SSA_allVars @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ defs @ uses @ phis @ G2 ) )
=> ? [X3: node] :
( ( member @ node @ X3 @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
& ( member @ val @ V @ ( sSA_CFG_SSA_allDefs @ g @ node @ val @ defs @ phis @ G2 @ X3 ) )
& ! [Y: node] :
( ( ( member @ node @ Y @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
& ( member @ val @ V @ ( sSA_CFG_SSA_allDefs @ g @ node @ val @ defs @ phis @ G2 @ Y ) ) )
=> ( Y = X3 ) ) ) ) ).
% defNode_ex1
thf(fact_36_allVars__in__allDefs,axiom,
! [V: val,G2: g] :
( ( member @ val @ V @ ( sSA_CFG_SSA_allVars @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ defs @ uses @ phis @ G2 ) )
=> ? [X3: node] :
( ( member @ node @ X3 @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
& ( member @ val @ V @ ( sSA_CFG_SSA_allDefs @ g @ node @ val @ defs @ phis @ G2 @ X3 ) ) ) ) ).
% allVars_in_allDefs
thf(fact_37_List_Ofinite__set,axiom,
! [A: $tType,Xs: list @ A] : ( finite_finite2 @ A @ ( set2 @ A @ Xs ) ) ).
% List.finite_set
thf(fact_38_phiDefs__finite,axiom,
! [G2: g,N: node] : ( finite_finite2 @ val @ ( sSA_CFG_SSA_phiDefs @ g @ node @ val @ phis @ G2 @ N ) ) ).
% phiDefs_finite
thf(fact_39_allDefs__finite,axiom,
! [N: node,G2: g] :
( ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( finite_finite2 @ val @ ( sSA_CFG_SSA_allDefs @ g @ node @ val @ defs @ phis @ G2 @ N ) ) ) ).
% allDefs_finite
thf(fact_40_assms_I5_J,axiom,
( ( var2 @ g2 @ phi )
= ( var2 @ g2 @ s ) ) ).
% assms(5)
thf(fact_41_CFG__SSA__wf__base_Oredundant__def,axiom,
! [EdgeD: $tType,Val: $tType,Node: $tType,G: $tType] :
( ( ( linorder @ Node )
& ( linorder @ Val ) )
=> ( ( sSA_CF462202545undant @ G @ Node @ EdgeD @ Val )
= ( ^ [Alpha_n: G > ( list @ Node ),InEdges: G > Node > ( list @ ( product_prod @ Node @ EdgeD ) ),Defs: G > Node > ( set @ Val ),Uses: G > Node > ( set @ Val ),Phis: G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) ),G3: G] :
? [X: Val] :
( ( member @ Val @ X @ ( sSA_CFG_SSA_allVars @ G @ Node @ EdgeD @ Val @ Alpha_n @ InEdges @ Defs @ Uses @ Phis @ G3 ) )
& ( sSA_CF1363434349rivial @ G @ Node @ EdgeD @ Val @ Alpha_n @ InEdges @ Defs @ Uses @ Phis @ G3 @ X ) ) ) ) ) ).
% CFG_SSA_wf_base.redundant_def
thf(fact_42_defs__in__allDefs,axiom,
! [V: val,G2: g,N: node] :
( ( member @ val @ V @ ( defs @ G2 @ N ) )
=> ( member @ val @ V @ ( sSA_CFG_SSA_allDefs @ g @ node @ val @ defs @ phis @ G2 @ N ) ) ) ).
% defs_in_allDefs
thf(fact_43_phiDefs__in__allDefs,axiom,
! [V: val,G2: g,N: node] :
( ( member @ val @ V @ ( sSA_CFG_SSA_phiDefs @ g @ node @ val @ phis @ G2 @ N ) )
=> ( member @ val @ V @ ( sSA_CFG_SSA_allDefs @ g @ node @ val @ defs @ phis @ G2 @ N ) ) ) ).
% phiDefs_in_allDefs
thf(fact_44_allDefs__disjoint_H,axiom,
! [N: node,G2: g,M: node,V: val] :
( ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( ( member @ node @ M @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( ( member @ val @ V @ ( sSA_CFG_SSA_allDefs @ g @ node @ val @ defs @ phis @ G2 @ N ) )
=> ( ( member @ val @ V @ ( sSA_CFG_SSA_allDefs @ g @ node @ val @ defs @ phis @ G2 @ M ) )
=> ( N = M ) ) ) ) ) ).
% allDefs_disjoint'
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P2: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q2: A > $o] :
( ! [X3: A] :
( ( P2 @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collect @ A @ P2 )
= ( collect @ A @ Q2 ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G2: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G2 @ X3 ) )
=> ( F = G2 ) ) ).
% ext
thf(fact_49_phiArg__same__var,axiom,
! [G2: g,P: val,Q: val] :
( ( sSA_CF1165125185phiArg @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ P @ Q )
=> ( ( var2 @ G2 @ Q )
= ( var2 @ G2 @ P ) ) ) ).
% phiArg_same_var
thf(fact_50_vars__eq,axiom,
( ( var2 @ g2 @ phi )
= ( var2 @ g2 @ phi_s ) ) ).
% vars_eq
thf(fact_51_allDefs__var__disjoint,axiom,
! [N: node,G2: g,V: val,V2: val] :
( ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( ( member @ val @ V @ ( sSA_CFG_SSA_allDefs @ g @ node @ val @ defs @ phis @ G2 @ N ) )
=> ( ( member @ val @ V2 @ ( sSA_CFG_SSA_allDefs @ g @ node @ val @ defs @ phis @ G2 @ N ) )
=> ( ( V != V2 )
=> ( ( var2 @ G2 @ V2 )
!= ( var2 @ G2 @ V ) ) ) ) ) ) ).
% allDefs_var_disjoint
thf(fact_52_defNode__var__disjoint,axiom,
! [P: val,G2: g,Q: val] :
( ( member @ val @ P @ ( sSA_CFG_SSA_allVars @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ defs @ uses @ phis @ G2 ) )
=> ( ( member @ val @ Q @ ( sSA_CFG_SSA_allVars @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ defs @ uses @ phis @ G2 ) )
=> ( ( P != Q )
=> ( ( ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ P )
= ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ Q ) )
=> ( ( var2 @ G2 @ P )
!= ( var2 @ G2 @ Q ) ) ) ) ) ) ).
% defNode_var_disjoint
thf(fact_53_defNode__eq,axiom,
! [N: node,G2: g,V: val] :
( ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( ( member @ val @ V @ ( sSA_CFG_SSA_allDefs @ g @ node @ val @ defs @ phis @ G2 @ N ) )
=> ( ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ V )
= N ) ) ) ).
% defNode_eq
thf(fact_54_allDefs__in__allVars,axiom,
! [V: val,G2: g,N: node] :
( ( member @ val @ V @ ( sSA_CFG_SSA_allDefs @ g @ node @ val @ defs @ phis @ G2 @ N ) )
=> ( ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( member @ val @ V @ ( sSA_CFG_SSA_allVars @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ defs @ uses @ phis @ G2 ) ) ) ) ).
% allDefs_in_allVars
thf(fact_55_pruned__def,axiom,
! [G2: g] :
( ( sSA_CF823892918pruned @ g @ node @ val @ alpha_n @ defs @ uses @ phis @ G2 )
= ( ! [X: node] :
( ( member @ node @ X @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ! [Val3: val] :
( ( member @ val @ Val3 @ ( sSA_CFG_SSA_phiDefs @ g @ node @ val @ phis @ G2 @ X ) )
=> ( sSA_CF722422777iveVal @ g @ node @ val @ alpha_n @ defs @ uses @ phis @ G2 @ Val3 ) ) ) ) ) ).
% pruned_def
thf(fact_56_CFG__SSA__base_OphiUses_Ocong,axiom,
! [Val: $tType,EdgeD: $tType,Node: $tType,G: $tType] :
( ( ( linorder @ Node )
& ( linorder @ Val ) )
=> ( ( sSA_CFG_SSA_phiUses @ G @ Node @ EdgeD @ Val )
= ( sSA_CFG_SSA_phiUses @ G @ Node @ EdgeD @ Val ) ) ) ).
% CFG_SSA_base.phiUses.cong
thf(fact_57_CFG__SSA__base_OphiDefs_Ocong,axiom,
! [Val: $tType,Node: $tType,G: $tType] :
( ( ( linorder @ Node )
& ( linorder @ Val ) )
=> ( ( sSA_CFG_SSA_phiDefs @ G @ Node @ Val )
= ( sSA_CFG_SSA_phiDefs @ G @ Node @ Val ) ) ) ).
% CFG_SSA_base.phiDefs.cong
thf(fact_58_CFG__SSA__base_OallDefs_Ocong,axiom,
! [Val: $tType,Node: $tType,G: $tType] :
( ( ( linorder @ Node )
& ( linorder @ Val ) )
=> ( ( sSA_CFG_SSA_allDefs @ G @ Node @ Val )
= ( sSA_CFG_SSA_allDefs @ G @ Node @ Val ) ) ) ).
% CFG_SSA_base.allDefs.cong
thf(fact_59_CFG__SSA__wf__base_OdefNode_Ocong,axiom,
! [Val: $tType,Node: $tType,G: $tType] :
( ( ( linorder @ Node )
& ( linorder @ Val ) )
=> ( ( sSA_CF1081484811efNode @ G @ Node @ Val )
= ( sSA_CF1081484811efNode @ G @ Node @ Val ) ) ) ).
% CFG_SSA_wf_base.defNode.cong
thf(fact_60_CFG__SSA__base_OallVars_Ocong,axiom,
! [Val: $tType,EdgeD: $tType,Node: $tType,G: $tType] :
( ( ( linorder @ Node )
& ( linorder @ Val ) )
=> ( ( sSA_CFG_SSA_allVars @ G @ Node @ EdgeD @ Val )
= ( sSA_CFG_SSA_allVars @ G @ Node @ EdgeD @ Val ) ) ) ).
% CFG_SSA_base.allVars.cong
thf(fact_61_CFG__SSA__wf__base_OphiArg_Ocong,axiom,
! [Val: $tType,Node: $tType,G: $tType] :
( ( ( linorder @ Node )
& ( linorder @ Val ) )
=> ( ( sSA_CF1165125185phiArg @ G @ Node @ Val )
= ( sSA_CF1165125185phiArg @ G @ Node @ Val ) ) ) ).
% CFG_SSA_wf_base.phiArg.cong
thf(fact_62_CFG__SSA__base_OallUses_Ocong,axiom,
! [Val: $tType,EdgeD: $tType,Node: $tType,G: $tType] :
( ( ( linorder @ Node )
& ( linorder @ Val ) )
=> ( ( sSA_CFG_SSA_allUses @ G @ Node @ EdgeD @ Val )
= ( sSA_CFG_SSA_allUses @ G @ Node @ EdgeD @ Val ) ) ) ).
% CFG_SSA_base.allUses.cong
thf(fact_63_CFG__SSA__wf__base_OliveVal_Ocong,axiom,
! [Val: $tType,Node: $tType,G: $tType] :
( ( ( linorder @ Node )
& ( linorder @ Val ) )
=> ( ( sSA_CF722422777iveVal @ G @ Node @ Val )
= ( sSA_CF722422777iveVal @ G @ Node @ Val ) ) ) ).
% CFG_SSA_wf_base.liveVal.cong
thf(fact_64_CFG__SSA__wf__base_Otrivial_Ocong,axiom,
! [Val: $tType,EdgeD: $tType,Node: $tType,G: $tType] :
( ( ( linorder @ Node )
& ( linorder @ Val ) )
=> ( ( sSA_CF1363434349rivial @ G @ Node @ EdgeD @ Val )
= ( sSA_CF1363434349rivial @ G @ Node @ EdgeD @ Val ) ) ) ).
% CFG_SSA_wf_base.trivial.cong
thf(fact_65_CFG__base_Ovars_Ocong,axiom,
! [Var: $tType,Node: $tType,G: $tType] :
( ( ( linorder @ Node )
& ( linorder @ Var ) )
=> ( ( sSA_CFG_vars @ G @ Node @ Var )
= ( sSA_CFG_vars @ G @ Node @ Var ) ) ) ).
% CFG_base.vars.cong
thf(fact_66_CFG__SSA__wf__base_OisTrivialPhi_Ocong,axiom,
! [Val: $tType,Node: $tType,G: $tType] :
( ( ( linorder @ Node )
& ( linorder @ Val ) )
=> ( ( sSA_CF1297404942ialPhi @ G @ Node @ Val )
= ( sSA_CF1297404942ialPhi @ G @ Node @ Val ) ) ) ).
% CFG_SSA_wf_base.isTrivialPhi.cong
thf(fact_67_CFG__SSA__wf__base_Oredundant_Ocong,axiom,
! [Val: $tType,EdgeD: $tType,Node: $tType,G: $tType] :
( ( ( linorder @ Node )
& ( linorder @ Val ) )
=> ( ( sSA_CF462202545undant @ G @ Node @ EdgeD @ Val )
= ( sSA_CF462202545undant @ G @ Node @ EdgeD @ Val ) ) ) ).
% CFG_SSA_wf_base.redundant.cong
thf(fact_68_finite__list,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ? [Xs2: list @ A] :
( ( set2 @ A @ Xs2 )
= A3 ) ) ).
% finite_list
thf(fact_69_CFG__SSA__wf__base_OliveSimple,axiom,
! [G: $tType,Node: $tType,Val: $tType] :
( ( ( linorder @ Val )
& ( linorder @ Node ) )
=> ! [N: Node,Alpha_n2: G > ( list @ Node ),G2: G,Val2: Val,Uses2: G > Node > ( set @ Val ),Defs2: G > Node > ( set @ Val ),Phis2: G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) )] :
( ( member @ Node @ N @ ( set2 @ Node @ ( Alpha_n2 @ G2 ) ) )
=> ( ( member @ Val @ Val2 @ ( Uses2 @ G2 @ N ) )
=> ( sSA_CF722422777iveVal @ G @ Node @ Val @ Alpha_n2 @ Defs2 @ Uses2 @ Phis2 @ G2 @ Val2 ) ) ) ) ).
% CFG_SSA_wf_base.liveSimple
thf(fact_70_CFG__SSA__wf__base_OlivePhi,axiom,
! [G: $tType,Node: $tType,Val: $tType] :
( ( ( linorder @ Val )
& ( linorder @ Node ) )
=> ! [Alpha_n2: G > ( list @ Node ),Defs2: G > Node > ( set @ Val ),Uses2: G > Node > ( set @ Val ),Phis2: G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) ),G2: G,V: Val,V2: Val] :
( ( sSA_CF722422777iveVal @ G @ Node @ Val @ Alpha_n2 @ Defs2 @ Uses2 @ Phis2 @ G2 @ V )
=> ( ( sSA_CF1165125185phiArg @ G @ Node @ Val @ Alpha_n2 @ Defs2 @ Phis2 @ G2 @ V @ V2 )
=> ( sSA_CF722422777iveVal @ G @ Node @ Val @ Alpha_n2 @ Defs2 @ Uses2 @ Phis2 @ G2 @ V2 ) ) ) ) ).
% CFG_SSA_wf_base.livePhi
thf(fact_71_CFG__SSA__wf__base_OliveVal_Oinducts,axiom,
! [Node: $tType,G: $tType,Val: $tType] :
( ( ( linorder @ Val )
& ( linorder @ Node ) )
=> ! [Alpha_n2: G > ( list @ Node ),Defs2: G > Node > ( set @ Val ),Uses2: G > Node > ( set @ Val ),Phis2: G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) ),G2: G,X2: Val,P2: Val > $o] :
( ( sSA_CF722422777iveVal @ G @ Node @ Val @ Alpha_n2 @ Defs2 @ Uses2 @ Phis2 @ G2 @ X2 )
=> ( ! [N2: Node,Val4: Val] :
( ( member @ Node @ N2 @ ( set2 @ Node @ ( Alpha_n2 @ G2 ) ) )
=> ( ( member @ Val @ Val4 @ ( Uses2 @ G2 @ N2 ) )
=> ( P2 @ Val4 ) ) )
=> ( ! [V5: Val,V6: Val] :
( ( sSA_CF722422777iveVal @ G @ Node @ Val @ Alpha_n2 @ Defs2 @ Uses2 @ Phis2 @ G2 @ V5 )
=> ( ( P2 @ V5 )
=> ( ( sSA_CF1165125185phiArg @ G @ Node @ Val @ Alpha_n2 @ Defs2 @ Phis2 @ G2 @ V5 @ V6 )
=> ( P2 @ V6 ) ) ) )
=> ( P2 @ X2 ) ) ) ) ) ).
% CFG_SSA_wf_base.liveVal.inducts
thf(fact_72_CFG__SSA__wf__base_OliveVal_Osimps,axiom,
! [Val: $tType,Node: $tType,G: $tType] :
( ( ( linorder @ Node )
& ( linorder @ Val ) )
=> ( ( sSA_CF722422777iveVal @ G @ Node @ Val )
= ( ^ [Alpha_n: G > ( list @ Node ),Defs: G > Node > ( set @ Val ),Uses: G > Node > ( set @ Val ),Phis: G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) ),G3: G,A4: Val] :
( ? [N3: Node,Val3: Val] :
( ( A4 = Val3 )
& ( member @ Node @ N3 @ ( set2 @ Node @ ( Alpha_n @ G3 ) ) )
& ( member @ Val @ Val3 @ ( Uses @ G3 @ N3 ) ) )
| ? [V3: Val,V4: Val] :
( ( A4 = V4 )
& ( sSA_CF722422777iveVal @ G @ Node @ Val @ Alpha_n @ Defs @ Uses @ Phis @ G3 @ V3 )
& ( sSA_CF1165125185phiArg @ G @ Node @ Val @ Alpha_n @ Defs @ Phis @ G3 @ V3 @ V4 ) ) ) ) ) ) ).
% CFG_SSA_wf_base.liveVal.simps
thf(fact_73_CFG__SSA__wf__base_OliveVal_Ocases,axiom,
! [Node: $tType,G: $tType,Val: $tType] :
( ( ( linorder @ Val )
& ( linorder @ Node ) )
=> ! [Alpha_n2: G > ( list @ Node ),Defs2: G > Node > ( set @ Val ),Uses2: G > Node > ( set @ Val ),Phis2: G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) ),G2: G,A2: Val] :
( ( sSA_CF722422777iveVal @ G @ Node @ Val @ Alpha_n2 @ Defs2 @ Uses2 @ Phis2 @ G2 @ A2 )
=> ( ! [N2: Node] :
( ( member @ Node @ N2 @ ( set2 @ Node @ ( Alpha_n2 @ G2 ) ) )
=> ~ ( member @ Val @ A2 @ ( Uses2 @ G2 @ N2 ) ) )
=> ~ ! [V5: Val] :
( ( sSA_CF722422777iveVal @ G @ Node @ Val @ Alpha_n2 @ Defs2 @ Uses2 @ Phis2 @ G2 @ V5 )
=> ~ ( sSA_CF1165125185phiArg @ G @ Node @ Val @ Alpha_n2 @ Defs2 @ Phis2 @ G2 @ V5 @ A2 ) ) ) ) ) ).
% CFG_SSA_wf_base.liveVal.cases
thf(fact_74_CFG__SSA__wf__base_Otrivial__def,axiom,
! [EdgeD: $tType,Val: $tType,Node: $tType,G: $tType] :
( ( ( linorder @ Node )
& ( linorder @ Val ) )
=> ( ( sSA_CF1363434349rivial @ G @ Node @ EdgeD @ Val )
= ( ^ [Alpha_n: G > ( list @ Node ),InEdges: G > Node > ( list @ ( product_prod @ Node @ EdgeD ) ),Defs: G > Node > ( set @ Val ),Uses: G > Node > ( set @ Val ),Phis: G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) ),G3: G,V3: Val] :
? [X: Val] :
( ( member @ Val @ X @ ( sSA_CFG_SSA_allVars @ G @ Node @ EdgeD @ Val @ Alpha_n @ InEdges @ Defs @ Uses @ Phis @ G3 ) )
& ( sSA_CF1297404942ialPhi @ G @ Node @ Val @ Alpha_n @ Defs @ Phis @ G3 @ V3 @ X ) ) ) ) ) ).
% CFG_SSA_wf_base.trivial_def
thf(fact_75_condensation__finite,axiom,
! [G2: g,P2: set @ val] : ( finite_finite2 @ ( product_prod @ ( set @ val ) @ ( set @ val ) ) @ ( irredu280805948_edges @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ P2 ) ) ).
% condensation_finite
thf(fact_76_phiUses__exI_H,axiom,
! [G2: g,P: val,Q: val] :
( ( sSA_CF1165125185phiArg @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ P @ Q )
=> ( ( member @ val @ P @ ( sSA_CFG_SSA_allVars @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ defs @ uses @ phis @ G2 ) )
=> ~ ! [M2: node] :
( ( member @ val @ Q @ ( sSA_CFG_SSA_phiUses @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ phis @ G2 @ M2 ) )
=> ~ ( member @ node @ M2 @ ( set2 @ node @ ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ G2 @ ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ P ) ) ) ) ) ) ) ).
% phiUses_exI'
thf(fact_77_allUses__def,axiom,
! [G2: g,N: node] :
( ( sSA_CFG_SSA_allUses @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ uses @ phis @ G2 @ N )
= ( sup_sup @ ( set @ val ) @ ( uses @ G2 @ N ) @ ( sSA_CFG_SSA_phiUses @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ phis @ G2 @ N ) ) ) ).
% allUses_def
thf(fact_78_phiArg__trancl__same__var,axiom,
! [G2: g,Phi2: val,N: val] :
( ( transitive_tranclp @ val @ ( sSA_CF1165125185phiArg @ g @ node @ val @ alpha_n @ defs @ phis @ G2 ) @ Phi2 @ N )
=> ( ( var2 @ G2 @ Phi2 )
= ( var2 @ G2 @ N ) ) ) ).
% phiArg_trancl_same_var
thf(fact_79_allDefs__def,axiom,
! [G2: g,N: node] :
( ( sSA_CFG_SSA_allDefs @ g @ node @ val @ defs @ phis @ G2 @ N )
= ( sup_sup @ ( set @ val ) @ ( defs @ G2 @ N ) @ ( sSA_CFG_SSA_phiDefs @ g @ node @ val @ phis @ G2 @ N ) ) ) ).
% allDefs_def
thf(fact_80_CFG__SSA__wf__base_Opruned__def,axiom,
! [Val: $tType,Node: $tType,G: $tType] :
( ( ( linorder @ Node )
& ( linorder @ Val ) )
=> ( ( sSA_CF823892918pruned @ G @ Node @ Val )
= ( ^ [Alpha_n: G > ( list @ Node ),Defs: G > Node > ( set @ Val ),Uses: G > Node > ( set @ Val ),Phis: G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) ),G3: G] :
! [X: Node] :
( ( member @ Node @ X @ ( set2 @ Node @ ( Alpha_n @ G3 ) ) )
=> ! [Val3: Val] :
( ( member @ Val @ Val3 @ ( sSA_CFG_SSA_phiDefs @ G @ Node @ Val @ Phis @ G3 @ X ) )
=> ( sSA_CF722422777iveVal @ G @ Node @ Val @ Alpha_n @ Defs @ Uses @ Phis @ G3 @ Val3 ) ) ) ) ) ) ).
% CFG_SSA_wf_base.pruned_def
thf(fact_81_finite__code,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( finite_finite2 @ A )
= ( ^ [A5: set @ A] : $true ) ) ) ).
% finite_code
thf(fact_82_defNode__cases,axiom,
! [V: val,G2: g] :
( ( member @ val @ V @ ( sSA_CFG_SSA_allVars @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ defs @ uses @ phis @ G2 ) )
=> ( ~ ( member @ val @ V @ ( defs @ G2 @ ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ V ) ) )
=> ( ( sSA_CFG_SSA_wf_phi @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ V )
!= ( none @ ( list @ val ) ) ) ) ) ).
% defNode_cases
thf(fact_83_simpleDef__not__phi,axiom,
! [N: node,G2: g,V: val] :
( ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( ( member @ val @ V @ ( defs @ G2 @ N ) )
=> ( ( sSA_CFG_SSA_wf_phi @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ V )
= ( none @ ( list @ val ) ) ) ) ) ).
% simpleDef_not_phi
thf(fact_84_finite__Un,axiom,
! [A: $tType,F2: set @ A,G4: set @ A] :
( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F2 @ G4 ) )
= ( ( finite_finite2 @ A @ F2 )
& ( finite_finite2 @ A @ G4 ) ) ) ).
% finite_Un
thf(fact_85_trivial__phi,axiom,
! [G2: g,V: val] :
( ( sSA_CF1363434349rivial @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ defs @ uses @ phis @ G2 @ V )
=> ( ( sSA_CFG_SSA_wf_phi @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ V )
!= ( none @ ( list @ val ) ) ) ) ).
% trivial_phi
thf(fact_86_phiArg__exI,axiom,
! [M: node,G2: g,V: val] :
( ( member @ node @ M @ ( set2 @ node @ ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ G2 @ ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ V ) ) ) )
=> ( ( ( sSA_CFG_SSA_wf_phi @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ V )
!= ( none @ ( list @ val ) ) )
=> ( ( member @ val @ V @ ( sSA_CFG_SSA_allVars @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ defs @ uses @ phis @ G2 ) )
=> ~ ! [V6: val] :
( ( member @ val @ V6 @ ( sSA_CFG_SSA_phiUses @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ phis @ G2 @ M ) )
=> ~ ( sSA_CF1165125185phiArg @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ V @ V6 ) ) ) ) ) ).
% phiArg_exI
thf(fact_87_CFG__SSA__wf__base_Ophi_Ocong,axiom,
! [Val: $tType,Node: $tType,G: $tType] :
( ( ( linorder @ Node )
& ( linorder @ Val ) )
=> ( ( sSA_CFG_SSA_wf_phi @ G @ Node @ Val )
= ( sSA_CFG_SSA_wf_phi @ G @ Node @ Val ) ) ) ).
% CFG_SSA_wf_base.phi.cong
thf(fact_88_infinite__Un,axiom,
! [A: $tType,S: set @ A,T: set @ A] :
( ( ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S @ T ) ) )
= ( ~ ( finite_finite2 @ A @ S )
| ~ ( finite_finite2 @ A @ T ) ) ) ).
% infinite_Un
thf(fact_89_Un__infinite,axiom,
! [A: $tType,S: set @ A,T: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S @ T ) ) ) ).
% Un_infinite
thf(fact_90_finite__UnI,axiom,
! [A: $tType,F2: set @ A,G4: set @ A] :
( ( finite_finite2 @ A @ F2 )
=> ( ( finite_finite2 @ A @ G4 )
=> ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F2 @ G4 ) ) ) ) ).
% finite_UnI
thf(fact_91_CFG__SSA__Transformed_Ocondensation__edges_Ocong,axiom,
! [Val: $tType,Node: $tType,G: $tType] :
( ( ( linorder @ Node )
& ( linorder @ Val ) )
=> ( ( irredu280805948_edges @ G @ Node @ Val )
= ( irredu280805948_edges @ G @ Node @ Val ) ) ) ).
% CFG_SSA_Transformed.condensation_edges.cong
thf(fact_92_CFG__SSA__wf__base_Opruned_Ocong,axiom,
! [Val: $tType,Node: $tType,G: $tType] :
( ( ( linorder @ Node )
& ( linorder @ Val ) )
=> ( ( sSA_CF823892918pruned @ G @ Node @ Val )
= ( sSA_CF823892918pruned @ G @ Node @ Val ) ) ) ).
% CFG_SSA_wf_base.pruned.cong
thf(fact_93_CFG__SSA__base_OallDefs__def,axiom,
! [Val: $tType,Node: $tType,G: $tType] :
( ( ( linorder @ Node )
& ( linorder @ Val ) )
=> ( ( sSA_CFG_SSA_allDefs @ G @ Node @ Val )
= ( ^ [Defs: G > Node > ( set @ Val ),Phis: G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) ),G3: G,N3: Node] : ( sup_sup @ ( set @ Val ) @ ( Defs @ G3 @ N3 ) @ ( sSA_CFG_SSA_phiDefs @ G @ Node @ Val @ Phis @ G3 @ N3 ) ) ) ) ) ).
% CFG_SSA_base.allDefs_def
thf(fact_94_CFG__SSA__base_OallUses__def,axiom,
! [EdgeD: $tType,Val: $tType,Node: $tType,G: $tType] :
( ( ( linorder @ Node )
& ( linorder @ Val ) )
=> ( ( sSA_CFG_SSA_allUses @ G @ Node @ EdgeD @ Val )
= ( ^ [Alpha_n: G > ( list @ Node ),InEdges: G > Node > ( list @ ( product_prod @ Node @ EdgeD ) ),Uses: G > Node > ( set @ Val ),Phis: G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) ),G3: G,N3: Node] : ( sup_sup @ ( set @ Val ) @ ( Uses @ G3 @ N3 ) @ ( sSA_CFG_SSA_phiUses @ G @ Node @ EdgeD @ Val @ Alpha_n @ InEdges @ Phis @ G3 @ N3 ) ) ) ) ) ).
% CFG_SSA_base.allUses_def
thf(fact_95_finite__set__choice,axiom,
! [B: $tType,A: $tType,A3: set @ A,P2: A > B > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ? [X_1: B] : ( P2 @ X3 @ X_1 ) )
=> ? [F3: A > B] :
! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( P2 @ X4 @ ( F3 @ X4 ) ) ) ) ) ).
% finite_set_choice
thf(fact_96_finite,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [A3: set @ A] : ( finite_finite2 @ A @ A3 ) ) ).
% finite
thf(fact_97_phi__phiDefs,axiom,
! [G2: g,V: val,Vs: list @ val] :
( ( ( sSA_CFG_SSA_wf_phi @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ V )
= ( some @ ( list @ val ) @ Vs ) )
=> ( member @ val @ V @ ( sSA_CFG_SSA_phiDefs @ g @ node @ val @ phis @ G2 @ ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ V ) ) ) ) ).
% phi_phiDefs
thf(fact_98_condensation__acyclic,axiom,
! [G2: g,P2: set @ val] : ( transitive_acyclic @ ( set @ val ) @ ( irredu280805948_edges @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ P2 ) ) ).
% condensation_acyclic
thf(fact_99_phi__def,axiom,
! [G2: g,V: val] :
( ( sSA_CFG_SSA_wf_phi @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ V )
= ( phis @ G2 @ ( product_Pair @ node @ val @ ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ V ) @ V ) ) ) ).
% phi_def
thf(fact_100_phiArg__def,axiom,
! [G2: g,V: val,V2: val] :
( ( sSA_CF1165125185phiArg @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ V @ V2 )
= ( ? [Vs2: list @ val] :
( ( ( sSA_CFG_SSA_wf_phi @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ V )
= ( some @ ( list @ val ) @ Vs2 ) )
& ( member @ val @ V2 @ ( set2 @ val @ Vs2 ) ) ) ) ) ).
% phiArg_def
thf(fact_101_phi__finite,axiom,
! [G2: g] : ( finite_finite2 @ val @ ( dom @ val @ ( list @ val ) @ ( sSA_CFG_SSA_wf_phi @ g @ node @ val @ alpha_n @ defs @ phis @ G2 ) ) ) ).
% phi_finite
thf(fact_102_redundant__scc__phis,axiom,
! [G2: g,P2: set @ val,Scc: set @ val,X2: val] :
( ( irredu2110905762nt_set @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ defs @ uses @ phis @ G2 @ P2 )
=> ( ( member @ ( set @ val ) @ Scc @ ( irredu1918690039_nodes @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ P2 ) )
=> ( ( member @ val @ X2 @ Scc )
=> ( ( sSA_CFG_SSA_wf_phi @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ X2 )
!= ( none @ ( list @ val ) ) ) ) ) ) ).
% redundant_scc_phis
thf(fact_103_phis__disj_I2_J,axiom,
! [G2: g,N: node,V: val,Vs: list @ val,N4: node,Vs3: list @ val] :
( ( ( phis @ G2 @ ( product_Pair @ node @ val @ N @ V ) )
= ( some @ ( list @ val ) @ Vs ) )
=> ( ( ( phis @ G2 @ ( product_Pair @ node @ val @ N4 @ V ) )
= ( some @ ( list @ val ) @ Vs3 ) )
=> ( Vs = Vs3 ) ) ) ).
% phis_disj(2)
thf(fact_104_phis__disj_I1_J,axiom,
! [G2: g,N: node,V: val,Vs: list @ val,N4: node,Vs3: list @ val] :
( ( ( phis @ G2 @ ( product_Pair @ node @ val @ N @ V ) )
= ( some @ ( list @ val ) @ Vs ) )
=> ( ( ( phis @ G2 @ ( product_Pair @ node @ val @ N4 @ V ) )
= ( some @ ( list @ val ) @ Vs3 ) )
=> ( N = N4 ) ) ) ).
% phis_disj(1)
thf(fact_105_phis__in___092_060alpha_062n,axiom,
! [G2: g,N: node,V: val,Vs: list @ val] :
( ( ( phis @ G2 @ ( product_Pair @ node @ val @ N @ V ) )
= ( some @ ( list @ val ) @ Vs ) )
=> ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) ) ) ).
% phis_in_\<alpha>n
thf(fact_106_phis__same__var,axiom,
! [G2: g,N: node,V: val,Vs: list @ val,V2: val] :
( ( ( phis @ G2 @ ( product_Pair @ node @ val @ N @ V ) )
= ( some @ ( list @ val ) @ Vs ) )
=> ( ( member @ val @ V2 @ ( set2 @ val @ Vs ) )
=> ( ( var2 @ G2 @ V2 )
= ( var2 @ G2 @ V ) ) ) ) ).
% phis_same_var
thf(fact_107_phis__phi,axiom,
! [G2: g,N: node,V: val,Vs: list @ val] :
( ( ( phis @ G2 @ ( product_Pair @ node @ val @ N @ V ) )
= ( some @ ( list @ val ) @ Vs ) )
=> ( ( sSA_CFG_SSA_wf_phi @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ V )
= ( some @ ( list @ val ) @ Vs ) ) ) ).
% phis_phi
thf(fact_108_phiUses__exI,axiom,
! [M: node,G2: g,N: node,V: val,Vs: list @ val] :
( ( member @ node @ M @ ( set2 @ node @ ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ G2 @ N ) ) )
=> ( ( ( phis @ G2 @ ( product_Pair @ node @ val @ N @ V ) )
= ( some @ ( list @ val ) @ Vs ) )
=> ( ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ~ ! [V6: val] :
( ( member @ val @ V6 @ ( sSA_CFG_SSA_phiUses @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ phis @ G2 @ M ) )
=> ~ ( member @ val @ V6 @ ( set2 @ val @ Vs ) ) ) ) ) ) ).
% phiUses_exI
thf(fact_109_CFG__SSA__Transformed_Ocondensation__nodes_Ocong,axiom,
! [Val: $tType,Node: $tType,G: $tType] :
( ( ( linorder @ Node )
& ( linorder @ Val ) )
=> ( ( irredu1918690039_nodes @ G @ Node @ Val )
= ( irredu1918690039_nodes @ G @ Node @ Val ) ) ) ).
% CFG_SSA_Transformed.condensation_nodes.cong
thf(fact_110_CFG__SSA__Transformed_Oredundant__set_Ocong,axiom,
! [Val: $tType,EdgeD: $tType,Node: $tType,G: $tType] :
( ( ( linorder @ Node )
& ( linorder @ Val ) )
=> ( ( irredu2110905762nt_set @ G @ Node @ EdgeD @ Val )
= ( irredu2110905762nt_set @ G @ Node @ EdgeD @ Val ) ) ) ).
% CFG_SSA_Transformed.redundant_set.cong
thf(fact_111_CFG__SSA__wf__base_Ophi__def,axiom,
! [Val: $tType,Node: $tType,G: $tType] :
( ( ( linorder @ Node )
& ( linorder @ Val ) )
=> ( ( sSA_CFG_SSA_wf_phi @ G @ Node @ Val )
= ( ^ [Alpha_n: G > ( list @ Node ),Defs: G > Node > ( set @ Val ),Phis: G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) ),G3: G,V3: Val] : ( Phis @ G3 @ ( product_Pair @ Node @ Val @ ( sSA_CF1081484811efNode @ G @ Node @ Val @ Alpha_n @ Defs @ Phis @ G3 @ V3 ) @ V3 ) ) ) ) ) ).
% CFG_SSA_wf_base.phi_def
thf(fact_112_CFG__SSA__wf__base_OphiArg__def,axiom,
! [Val: $tType,Node: $tType,G: $tType] :
( ( ( linorder @ Node )
& ( linorder @ Val ) )
=> ( ( sSA_CF1165125185phiArg @ G @ Node @ Val )
= ( ^ [Alpha_n: G > ( list @ Node ),Defs: G > Node > ( set @ Val ),Phis: G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) ),G3: G,V3: Val,V4: Val] :
? [Vs2: list @ Val] :
( ( ( sSA_CFG_SSA_wf_phi @ G @ Node @ Val @ Alpha_n @ Defs @ Phis @ G3 @ V3 )
= ( some @ ( list @ Val ) @ Vs2 ) )
& ( member @ Val @ V4 @ ( set2 @ Val @ Vs2 ) ) ) ) ) ) ).
% CFG_SSA_wf_base.phiArg_def
thf(fact_113_phiUsesI,axiom,
! [N4: node,G2: g,V2: val,Vs: list @ val,N: node,V: val] :
( ( member @ node @ N4 @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( ( ( phis @ G2 @ ( product_Pair @ node @ val @ N4 @ V2 ) )
= ( some @ ( list @ val ) @ Vs ) )
=> ( ( member @ ( product_prod @ node @ val ) @ ( product_Pair @ node @ val @ N @ V ) @ ( set2 @ ( product_prod @ node @ val ) @ ( zip @ node @ val @ ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ G2 @ N4 ) @ Vs ) ) )
=> ( member @ val @ V @ ( sSA_CFG_SSA_phiUses @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ phis @ G2 @ N ) ) ) ) ) ).
% phiUsesI
thf(fact_114_scc__in__P,axiom,
! [Scc: set @ val,G2: g,P2: set @ val] :
( ( member @ ( set @ val ) @ Scc @ ( irredu1918690039_nodes @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ P2 ) )
=> ( ord_less_eq @ ( set @ val ) @ Scc @ P2 ) ) ).
% scc_in_P
thf(fact_115_not__None__eq,axiom,
! [A: $tType,X2: option @ A] :
( ( X2
!= ( none @ A ) )
= ( ? [Y2: A] :
( X2
= ( some @ A @ Y2 ) ) ) ) ).
% not_None_eq
thf(fact_116_not__Some__eq,axiom,
! [A: $tType,X2: option @ A] :
( ( ! [Y2: A] :
( X2
!= ( some @ A @ Y2 ) ) )
= ( X2
= ( none @ A ) ) ) ).
% not_Some_eq
thf(fact_117_sup__Some,axiom,
! [A: $tType] :
( ( sup @ A )
=> ! [X2: A,Y3: A] :
( ( sup_sup @ ( option @ A ) @ ( some @ A @ X2 ) @ ( some @ A @ Y3 ) )
= ( some @ A @ ( sup_sup @ A @ X2 @ Y3 ) ) ) ) ).
% sup_Some
thf(fact_118_option_Oinject,axiom,
! [A: $tType,X22: A,Y22: A] :
( ( ( some @ A @ X22 )
= ( some @ A @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_119_less__eq__option__Some,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ ( option @ A ) @ ( some @ A @ X2 ) @ ( some @ A @ Y3 ) )
= ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ).
% less_eq_option_Some
thf(fact_120_sup__None__1,axiom,
! [A: $tType] :
( ( sup @ A )
=> ! [Y3: option @ A] :
( ( sup_sup @ ( option @ A ) @ ( none @ A ) @ Y3 )
= Y3 ) ) ).
% sup_None_1
thf(fact_121_sup__None__2,axiom,
! [A: $tType] :
( ( sup @ A )
=> ! [X2: option @ A] :
( ( sup_sup @ ( option @ A ) @ X2 @ ( none @ A ) )
= X2 ) ) ).
% sup_None_2
thf(fact_122_zip__same,axiom,
! [A: $tType,A2: A,B2: A,Xs: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( set2 @ ( product_prod @ A @ A ) @ ( zip @ A @ A @ Xs @ Xs ) ) )
= ( ( member @ A @ A2 @ ( set2 @ A @ Xs ) )
& ( A2 = B2 ) ) ) ).
% zip_same
thf(fact_123__C1_C,axiom,
! [G2: g,P2: set @ val] :
( ( irredu2110905762nt_set @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ defs @ uses @ phis @ G2 @ P2 )
=> ? [Scc2: set @ val] :
( ( ord_less_eq @ ( set @ val ) @ Scc2 @ P2 )
& ( irredu2110774547nt_scc @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ defs @ uses @ phis @ G2 @ P2 @ Scc2 ) ) ) ).
% "1"
thf(fact_124_phiUsesE,axiom,
! [V: val,G2: g,N: node] :
( ( member @ val @ V @ ( sSA_CFG_SSA_phiUses @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ phis @ G2 @ N ) )
=> ~ ! [N5: node] :
( ( member @ node @ N5 @ ( set2 @ node @ ( graph_449533722essors @ g @ node @ edgeD @ alpha_n @ inEdges @ G2 @ N ) ) )
=> ! [V6: val,Vs4: list @ val] :
( ( member @ ( product_prod @ node @ val ) @ ( product_Pair @ node @ val @ N @ V ) @ ( set2 @ ( product_prod @ node @ val ) @ ( zip @ node @ val @ ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ G2 @ N5 ) @ Vs4 ) ) )
=> ( ( phis @ G2 @ ( product_Pair @ node @ val @ N5 @ V6 ) )
!= ( some @ ( list @ val ) @ Vs4 ) ) ) ) ) ).
% phiUsesE
thf(fact_125_finite__has__minimal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A2 @ A3 )
=> ? [X3: A] :
( ( member @ A @ X3 @ A3 )
& ( ord_less_eq @ A @ X3 @ A2 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A3 )
=> ( ( ord_less_eq @ A @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_126_finite__has__maximal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A2 @ A3 )
=> ? [X3: A] :
( ( member @ A @ X3 @ A3 )
& ( ord_less_eq @ A @ A2 @ X3 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A3 )
=> ( ( ord_less_eq @ A @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_127_subset__code_I1_J,axiom,
! [A: $tType,Xs: list @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B3 )
= ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_128_rev__finite__subset,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( finite_finite2 @ A @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_129_infinite__super,axiom,
! [A: $tType,S: set @ A,T: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S @ T )
=> ( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ T ) ) ) ).
% infinite_super
thf(fact_130_finite__subset,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( finite_finite2 @ A @ B3 )
=> ( finite_finite2 @ A @ A3 ) ) ) ).
% finite_subset
thf(fact_131_in__set__zipE,axiom,
! [A: $tType,B: $tType,X2: A,Y3: B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ~ ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ~ ( member @ B @ Y3 @ ( set2 @ B @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_132_set__zip__leftD,axiom,
! [B: $tType,A: $tType,X2: A,Y3: B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ( member @ A @ X2 @ ( set2 @ A @ Xs ) ) ) ).
% set_zip_leftD
thf(fact_133_set__zip__rightD,axiom,
! [A: $tType,B: $tType,X2: A,Y3: B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ( member @ B @ Y3 @ ( set2 @ B @ Ys ) ) ) ).
% set_zip_rightD
thf(fact_134_combine__options__cases,axiom,
! [A: $tType,B: $tType,X2: option @ A,P2: ( option @ A ) > ( option @ B ) > $o,Y3: option @ B] :
( ( ( X2
= ( none @ A ) )
=> ( P2 @ X2 @ Y3 ) )
=> ( ( ( Y3
= ( none @ B ) )
=> ( P2 @ X2 @ Y3 ) )
=> ( ! [A6: A,B4: B] :
( ( X2
= ( some @ A @ A6 ) )
=> ( ( Y3
= ( some @ B @ B4 ) )
=> ( P2 @ X2 @ Y3 ) ) )
=> ( P2 @ X2 @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_135_split__option__all,axiom,
! [A: $tType] :
( ( ^ [P3: ( option @ A ) > $o] :
! [X5: option @ A] : ( P3 @ X5 ) )
= ( ^ [P4: ( option @ A ) > $o] :
( ( P4 @ ( none @ A ) )
& ! [X: A] : ( P4 @ ( some @ A @ X ) ) ) ) ) ).
% split_option_all
thf(fact_136_split__option__ex,axiom,
! [A: $tType] :
( ( ^ [P3: ( option @ A ) > $o] :
? [X5: option @ A] : ( P3 @ X5 ) )
= ( ^ [P4: ( option @ A ) > $o] :
( ( P4 @ ( none @ A ) )
| ? [X: A] : ( P4 @ ( some @ A @ X ) ) ) ) ) ).
% split_option_ex
thf(fact_137_option_Oinducts,axiom,
! [A: $tType,P2: ( option @ A ) > $o,Option: option @ A] :
( ( P2 @ ( none @ A ) )
=> ( ! [X3: A] : ( P2 @ ( some @ A @ X3 ) )
=> ( P2 @ Option ) ) ) ).
% option.inducts
thf(fact_138_option_Oexhaust,axiom,
! [A: $tType,Y3: option @ A] :
( ( Y3
!= ( none @ A ) )
=> ~ ! [X23: A] :
( Y3
!= ( some @ A @ X23 ) ) ) ).
% option.exhaust
thf(fact_139_option_OdiscI,axiom,
! [A: $tType,Option: option @ A,X22: A] :
( ( Option
= ( some @ A @ X22 ) )
=> ( Option
!= ( none @ A ) ) ) ).
% option.discI
thf(fact_140_option_Odistinct_I1_J,axiom,
! [A: $tType,X22: A] :
( ( none @ A )
!= ( some @ A @ X22 ) ) ).
% option.distinct(1)
thf(fact_141_Ex__condensation__leaf,axiom,
! [P2: set @ val,G2: g] :
( ( P2
!= ( bot_bot @ ( set @ val ) ) )
=> ? [Leaf: set @ val] :
( ( member @ ( set @ val ) @ Leaf @ ( irredu1918690039_nodes @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ P2 ) )
& ! [Scc3: set @ val] :
~ ( member @ ( product_prod @ ( set @ val ) @ ( set @ val ) ) @ ( product_Pair @ ( set @ val ) @ ( set @ val ) @ Leaf @ Scc3 ) @ ( irredu280805948_edges @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ P2 ) ) ) ) ).
% Ex_condensation_leaf
thf(fact_142_Un__subset__iff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ C )
= ( ( ord_less_eq @ ( set @ A ) @ A3 @ C )
& ( ord_less_eq @ ( set @ A ) @ B3 @ C ) ) ) ).
% Un_subset_iff
thf(fact_143_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,Y3: A,Z: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X2 @ Y3 ) @ Z )
= ( ( ord_less_eq @ A @ X2 @ Z )
& ( ord_less_eq @ A @ Y3 @ Z ) ) ) ) ).
% le_sup_iff
thf(fact_144_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 )
= ( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% sup.bounded_iff
thf(fact_145_subset__antisym,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% subset_antisym
thf(fact_146_subsetI,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( member @ A @ X3 @ B3 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% subsetI
thf(fact_147_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ B2 )
= ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.right_idem
thf(fact_148_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,Y3: A] :
( ( sup_sup @ A @ X2 @ ( sup_sup @ A @ X2 @ Y3 ) )
= ( sup_sup @ A @ X2 @ Y3 ) ) ) ).
% sup_left_idem
thf(fact_149_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] :
( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) )
= ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.left_idem
thf(fact_150_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A] :
( ( sup_sup @ A @ X2 @ X2 )
= X2 ) ) ).
% sup_idem
thf(fact_151_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A] :
( ( sup_sup @ A @ A2 @ A2 )
= A2 ) ) ).
% sup.idem
thf(fact_152_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F4: A > B,G3: A > B,X: A] : ( sup_sup @ B @ ( F4 @ X ) @ ( G3 @ X ) ) ) ) ) ).
% sup_apply
thf(fact_153_Un__iff,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
= ( ( member @ A @ C2 @ A3 )
| ( member @ A @ C2 @ B3 ) ) ) ).
% Un_iff
thf(fact_154_UnCI,axiom,
! [A: $tType,C2: A,B3: set @ A,A3: set @ A] :
( ( ~ ( member @ A @ C2 @ B3 )
=> ( member @ A @ C2 @ A3 ) )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% UnCI
thf(fact_155_subset__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_156_empty__subsetI,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% empty_subsetI
thf(fact_157_sup__bot__left,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [X2: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ X2 )
= X2 ) ) ).
% sup_bot_left
thf(fact_158_sup__bot__right,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [X2: A] :
( ( sup_sup @ A @ X2 @ ( bot_bot @ A ) )
= X2 ) ) ).
% sup_bot_right
thf(fact_159_bot__eq__sup__iff,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [X2: A,Y3: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ X2 @ Y3 ) )
= ( ( X2
= ( bot_bot @ A ) )
& ( Y3
= ( bot_bot @ A ) ) ) ) ) ).
% bot_eq_sup_iff
thf(fact_160_sup__eq__bot__iff,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [X2: A,Y3: A] :
( ( ( sup_sup @ A @ X2 @ Y3 )
= ( bot_bot @ A ) )
= ( ( X2
= ( bot_bot @ A ) )
& ( Y3
= ( bot_bot @ A ) ) ) ) ) ).
% sup_eq_bot_iff
thf(fact_161_sup__bot_Oeq__neutr__iff,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [A2: A,B2: A] :
( ( ( sup_sup @ A @ A2 @ B2 )
= ( bot_bot @ A ) )
= ( ( A2
= ( bot_bot @ A ) )
& ( B2
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_162_sup__bot_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [A2: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ A2 )
= A2 ) ) ).
% sup_bot.left_neutral
thf(fact_163_sup__bot_Oneutr__eq__iff,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [A2: A,B2: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ A2 @ B2 ) )
= ( ( A2
= ( bot_bot @ A ) )
& ( B2
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_164_sup__bot_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [A2: A] :
( ( sup_sup @ A @ A2 @ ( bot_bot @ A ) )
= A2 ) ) ).
% sup_bot.right_neutral
thf(fact_165_Un__empty,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
& ( B3
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Un_empty
thf(fact_166_less__eq__option__None__code,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: option @ A] : ( ord_less_eq @ ( option @ A ) @ ( none @ A ) @ X2 ) ) ).
% less_eq_option_None_code
thf(fact_167_less__eq__option__Some__None,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] :
~ ( ord_less_eq @ ( option @ A ) @ ( some @ A @ X2 ) @ ( none @ A ) ) ) ).
% less_eq_option_Some_None
thf(fact_168_old_Osuccessors__predecessors,axiom,
! [N: node,G2: g,M: node] :
( ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( ( member @ node @ N @ ( set2 @ node @ ( graph_449533722essors @ g @ node @ edgeD @ alpha_n @ inEdges @ G2 @ M ) ) )
= ( member @ node @ M @ ( set2 @ node @ ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ G2 @ N ) ) ) ) ) ).
% old.successors_predecessors
thf(fact_169_Un__empty__left,axiom,
! [A: $tType,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B3 )
= B3 ) ).
% Un_empty_left
thf(fact_170_Un__empty__right,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= A3 ) ).
% Un_empty_right
thf(fact_171_less__eq__option__None__is__None,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: option @ A] :
( ( ord_less_eq @ ( option @ A ) @ X2 @ ( none @ A ) )
=> ( X2
= ( none @ A ) ) ) ) ).
% less_eq_option_None_is_None
thf(fact_172_less__eq__option__None,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: option @ A] : ( ord_less_eq @ ( option @ A ) @ ( none @ A ) @ X2 ) ) ).
% less_eq_option_None
thf(fact_173_infinite__imp__nonempty,axiom,
! [A: $tType,S: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ( S
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_imp_nonempty
thf(fact_174_finite_OemptyI,axiom,
! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% finite.emptyI
thf(fact_175_CFG__SSA__Transformed_Oredundant__scc_Ocong,axiom,
! [Val: $tType,EdgeD: $tType,Node: $tType,G: $tType] :
( ( ( linorder @ Node )
& ( linorder @ Val ) )
=> ( ( irredu2110774547nt_scc @ G @ Node @ EdgeD @ Val )
= ( irredu2110774547nt_scc @ G @ Node @ EdgeD @ Val ) ) ) ).
% CFG_SSA_Transformed.redundant_scc.cong
thf(fact_176_finite__has__maximal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A3 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A3 )
=> ( ( ord_less_eq @ A @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_177_finite__has__minimal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A3 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A3 )
=> ( ( ord_less_eq @ A @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_178_Collect__mono__iff,axiom,
! [A: $tType,P2: A > $o,Q2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q2 ) )
= ( ! [X: A] :
( ( P2 @ X )
=> ( Q2 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_179_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y4: set @ A,Z2: set @ A] : Y4 = Z2 )
= ( ^ [A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_180_subset__trans,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ C ) ) ) ).
% subset_trans
thf(fact_181_Collect__mono,axiom,
! [A: $tType,P2: A > $o,Q2: A > $o] :
( ! [X3: A] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q2 ) ) ) ).
% Collect_mono
thf(fact_182_subset__refl,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% subset_refl
thf(fact_183_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
! [T2: A] :
( ( member @ A @ T2 @ A5 )
=> ( member @ A @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_184_equalityD2,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( A3 = B3 )
=> ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ).
% equalityD2
thf(fact_185_equalityD1,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( A3 = B3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% equalityD1
thf(fact_186_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
! [X: A] :
( ( member @ A @ X @ A5 )
=> ( member @ A @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_187_equalityE,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( A3 = B3 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ).
% equalityE
thf(fact_188_subsetD,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% subsetD
thf(fact_189_in__mono,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( member @ A @ X2 @ A3 )
=> ( member @ A @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_190_sup__left__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,Y3: A,Z: A] :
( ( sup_sup @ A @ X2 @ ( sup_sup @ A @ Y3 @ Z ) )
= ( sup_sup @ A @ Y3 @ ( sup_sup @ A @ X2 @ Z ) ) ) ) ).
% sup_left_commute
thf(fact_191_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( sup_sup @ A @ B2 @ ( sup_sup @ A @ A2 @ C2 ) )
= ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C2 ) ) ) ) ).
% sup.left_commute
thf(fact_192_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [X: A,Y2: A] : ( sup_sup @ A @ Y2 @ X ) ) ) ) ).
% sup_commute
thf(fact_193_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [A4: A,B6: A] : ( sup_sup @ A @ B6 @ A4 ) ) ) ) ).
% sup.commute
thf(fact_194_sup__assoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,Y3: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X2 @ Y3 ) @ Z )
= ( sup_sup @ A @ X2 @ ( sup_sup @ A @ Y3 @ Z ) ) ) ) ).
% sup_assoc
thf(fact_195_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ C2 )
= ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C2 ) ) ) ) ).
% sup.assoc
thf(fact_196_boolean__algebra__cancel_Osup2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,K: A,B2: A,A2: A] :
( ( B3
= ( sup_sup @ A @ K @ B2 ) )
=> ( ( sup_sup @ A @ A2 @ B3 )
= ( sup_sup @ A @ K @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_197_boolean__algebra__cancel_Osup1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,K: A,A2: A,B2: A] :
( ( A3
= ( sup_sup @ A @ K @ A2 ) )
=> ( ( sup_sup @ A @ A3 @ B2 )
= ( sup_sup @ A @ K @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_198_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F4: A > B,G3: A > B,X: A] : ( sup_sup @ B @ ( F4 @ X ) @ ( G3 @ X ) ) ) ) ) ).
% sup_fun_def
thf(fact_199_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( sup_sup @ A )
= ( ^ [X: A,Y2: A] : ( sup_sup @ A @ Y2 @ X ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_200_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X2: A,Y3: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X2 @ Y3 ) @ Z )
= ( sup_sup @ A @ X2 @ ( sup_sup @ A @ Y3 @ Z ) ) ) ) ).
% inf_sup_aci(6)
thf(fact_201_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X2: A,Y3: A,Z: A] :
( ( sup_sup @ A @ X2 @ ( sup_sup @ A @ Y3 @ Z ) )
= ( sup_sup @ A @ Y3 @ ( sup_sup @ A @ X2 @ Z ) ) ) ) ).
% inf_sup_aci(7)
thf(fact_202_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X2: A,Y3: A] :
( ( sup_sup @ A @ X2 @ ( sup_sup @ A @ X2 @ Y3 ) )
= ( sup_sup @ A @ X2 @ Y3 ) ) ) ).
% inf_sup_aci(8)
thf(fact_203_Un__left__commute,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C ) )
= ( sup_sup @ ( set @ A ) @ B3 @ ( sup_sup @ ( set @ A ) @ A3 @ C ) ) ) ).
% Un_left_commute
thf(fact_204_Un__left__absorb,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ).
% Un_left_absorb
thf(fact_205_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] : ( sup_sup @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% Un_commute
thf(fact_206_Un__absorb,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_207_Un__assoc,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ C )
= ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C ) ) ) ).
% Un_assoc
thf(fact_208_ball__Un,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,P2: A > $o] :
( ( ! [X: A] :
( ( member @ A @ X @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
=> ( P2 @ X ) ) )
= ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( P2 @ X ) )
& ! [X: A] :
( ( member @ A @ X @ B3 )
=> ( P2 @ X ) ) ) ) ).
% ball_Un
thf(fact_209_bex__Un,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,P2: A > $o] :
( ( ? [X: A] :
( ( member @ A @ X @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
& ( P2 @ X ) ) )
= ( ? [X: A] :
( ( member @ A @ X @ A3 )
& ( P2 @ X ) )
| ? [X: A] :
( ( member @ A @ X @ B3 )
& ( P2 @ X ) ) ) ) ).
% bex_Un
thf(fact_210_UnI2,axiom,
! [A: $tType,C2: A,B3: set @ A,A3: set @ A] :
( ( member @ A @ C2 @ B3 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% UnI2
thf(fact_211_UnI1,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% UnI1
thf(fact_212_UnE,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
=> ( ~ ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% UnE
thf(fact_213_sup_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.coboundedI2
thf(fact_214_sup_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.coboundedI1
thf(fact_215_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B6: A] :
( ( sup_sup @ A @ A4 @ B6 )
= B6 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_216_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B6: A,A4: A] :
( ( sup_sup @ A @ A4 @ B6 )
= A4 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_217_sup_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A] : ( ord_less_eq @ A @ B2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.cobounded2
thf(fact_218_sup_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.cobounded1
thf(fact_219_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B6: A,A4: A] :
( A4
= ( sup_sup @ A @ A4 @ B6 ) ) ) ) ) ).
% sup.order_iff
thf(fact_220_sup_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% sup.boundedI
thf(fact_221_sup_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A2 )
=> ~ ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% sup.boundedE
thf(fact_222_sup__absorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ( sup_sup @ A @ X2 @ Y3 )
= Y3 ) ) ) ).
% sup_absorb2
thf(fact_223_sup__absorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y3: A,X2: A] :
( ( ord_less_eq @ A @ Y3 @ X2 )
=> ( ( sup_sup @ A @ X2 @ Y3 )
= X2 ) ) ) ).
% sup_absorb1
thf(fact_224_sup_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( sup_sup @ A @ A2 @ B2 )
= B2 ) ) ) ).
% sup.absorb2
thf(fact_225_sup_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( sup_sup @ A @ A2 @ B2 )
= A2 ) ) ) ).
% sup.absorb1
thf(fact_226_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [F: A > A > A,X2: A,Y3: A] :
( ! [X3: A,Y5: A] : ( ord_less_eq @ A @ X3 @ ( F @ X3 @ Y5 ) )
=> ( ! [X3: A,Y5: A] : ( ord_less_eq @ A @ Y5 @ ( F @ X3 @ Y5 ) )
=> ( ! [X3: A,Y5: A,Z3: A] :
( ( ord_less_eq @ A @ Y5 @ X3 )
=> ( ( ord_less_eq @ A @ Z3 @ X3 )
=> ( ord_less_eq @ A @ ( F @ Y5 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup @ A @ X2 @ Y3 )
= ( F @ X2 @ Y3 ) ) ) ) ) ) ).
% sup_unique
thf(fact_227_sup_OorderI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( sup_sup @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% sup.orderI
thf(fact_228_sup_OorderE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2
= ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.orderE
thf(fact_229_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X: A,Y2: A] :
( ( sup_sup @ A @ X @ Y2 )
= Y2 ) ) ) ) ).
% le_iff_sup
thf(fact_230_sup__least,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y3: A,X2: A,Z: A] :
( ( ord_less_eq @ A @ Y3 @ X2 )
=> ( ( ord_less_eq @ A @ Z @ X2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ Y3 @ Z ) @ X2 ) ) ) ) ).
% sup_least
thf(fact_231_sup__mono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,C2: A,B2: A,D: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ D )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ ( sup_sup @ A @ C2 @ D ) ) ) ) ) ).
% sup_mono
thf(fact_232_sup_Omono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A2: A,D: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ( ord_less_eq @ A @ D @ B2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ C2 @ D ) @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% sup.mono
thf(fact_233_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ X2 @ B2 )
=> ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% le_supI2
thf(fact_234_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ X2 @ A2 )
=> ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% le_supI1
thf(fact_235_sup__ge2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y3: A,X2: A] : ( ord_less_eq @ A @ Y3 @ ( sup_sup @ A @ X2 @ Y3 ) ) ) ).
% sup_ge2
thf(fact_236_sup__ge1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,Y3: A] : ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ X2 @ Y3 ) ) ) ).
% sup_ge1
thf(fact_237_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,X2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ X2 )
=> ( ( ord_less_eq @ A @ B2 @ X2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X2 ) ) ) ) ).
% le_supI
thf(fact_238_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A,X2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_less_eq @ A @ A2 @ X2 )
=> ~ ( ord_less_eq @ A @ B2 @ X2 ) ) ) ) ).
% le_supE
thf(fact_239_inf__sup__ord_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X2: A,Y3: A] : ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ X2 @ Y3 ) ) ) ).
% inf_sup_ord(3)
thf(fact_240_inf__sup__ord_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [Y3: A,X2: A] : ( ord_less_eq @ A @ Y3 @ ( sup_sup @ A @ X2 @ Y3 ) ) ) ).
% inf_sup_ord(4)
thf(fact_241_subset__Un__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_242_subset__UnE,axiom,
! [A: $tType,C: set @ A,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
=> ~ ! [A7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ A3 )
=> ! [B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B7 @ B3 )
=> ( C
!= ( sup_sup @ ( set @ A ) @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_243_Un__absorb2,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( ( sup_sup @ ( set @ A ) @ A3 @ B3 )
= A3 ) ) ).
% Un_absorb2
thf(fact_244_Un__absorb1,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( sup_sup @ ( set @ A ) @ A3 @ B3 )
= B3 ) ) ).
% Un_absorb1
thf(fact_245_Un__upper2,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ B3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ).
% Un_upper2
thf(fact_246_Un__upper1,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ).
% Un_upper1
thf(fact_247_Un__least,axiom,
! [A: $tType,A3: set @ A,C: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ C ) ) ) ).
% Un_least
thf(fact_248_Un__mono,axiom,
! [A: $tType,A3: set @ A,C: set @ A,B3: set @ A,D2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ D2 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ ( sup_sup @ ( set @ A ) @ C @ D2 ) ) ) ) ).
% Un_mono
thf(fact_249_redundant__set__def,axiom,
! [G2: g,P2: set @ val] :
( ( irredu2110905762nt_set @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ defs @ uses @ phis @ G2 @ P2 )
= ( ( P2
!= ( bot_bot @ ( set @ val ) ) )
& ( ord_less_eq @ ( set @ val ) @ P2 @ ( dom @ val @ ( list @ val ) @ ( sSA_CFG_SSA_wf_phi @ g @ node @ val @ alpha_n @ defs @ phis @ G2 ) ) )
& ? [X: val] :
( ( member @ val @ X @ ( sSA_CFG_SSA_allVars @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ defs @ uses @ phis @ G2 ) )
& ! [Y2: val] :
( ( member @ val @ Y2 @ P2 )
=> ! [Phi3: val] :
( ( sSA_CF1165125185phiArg @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ Y2 @ Phi3 )
=> ( member @ val @ Phi3 @ ( sup_sup @ ( set @ val ) @ P2 @ ( insert @ val @ X @ ( bot_bot @ ( set @ val ) ) ) ) ) ) ) ) ) ) ).
% redundant_set_def
thf(fact_250_phi__no__closed__loop,axiom,
! [P: val,G2: g,Vs: list @ val] :
( ( member @ val @ P @ ( sSA_CFG_SSA_allVars @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ defs @ uses @ phis @ G2 ) )
=> ( ( ( sSA_CFG_SSA_wf_phi @ g @ node @ val @ alpha_n @ defs @ phis @ G2 @ P )
= ( some @ ( list @ val ) @ Vs ) )
=> ( ( set2 @ val @ Vs )
!= ( insert @ val @ P @ ( bot_bot @ ( set @ val ) ) ) ) ) ) ).
% phi_no_closed_loop
thf(fact_251_dom__eq__empty__conv,axiom,
! [B: $tType,A: $tType,F: A > ( option @ B )] :
( ( ( dom @ A @ B @ F )
= ( bot_bot @ ( set @ A ) ) )
= ( F
= ( ^ [X: A] : ( none @ B ) ) ) ) ).
% dom_eq_empty_conv
thf(fact_252_phis__finite,axiom,
! [G2: g] : ( finite_finite2 @ ( product_prod @ node @ val ) @ ( dom @ ( product_prod @ node @ val ) @ ( list @ val ) @ ( phis @ G2 ) ) ) ).
% phis_finite
thf(fact_253_insert__subset,axiom,
! [A: $tType,X2: A,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X2 @ A3 ) @ B3 )
= ( ( member @ A @ X2 @ B3 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% insert_subset
thf(fact_254_finite__insert,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( finite_finite2 @ A @ ( insert @ A @ A2 @ A3 ) )
= ( finite_finite2 @ A @ A3 ) ) ).
% finite_insert
% Subclasses (3)
thf(subcl_Orderings_Olinorder___HOL_Otype,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( type @ A ) ) ).
thf(subcl_Orderings_Olinorder___Orderings_Oorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( order @ A ) ) ).
thf(subcl_Orderings_Olinorder___Orderings_Opreorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( preorder @ A ) ) ).
% Type constructors (39)
thf(tcon_Option_Ooption___Lattices_Obounded__lattice,axiom,
! [A8: $tType] :
( ( bounded_lattice_top @ A8 )
=> ( bounded_lattice @ ( option @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice_1,axiom,
bounded_lattice @ $o ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_2,axiom,
! [A8: $tType] : ( bounded_lattice @ ( set @ A8 ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_3,axiom,
! [A8: $tType,A9: $tType] :
( ( bounded_lattice @ A9 )
=> ( bounded_lattice @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
! [A8: $tType,A9: $tType] :
( ( bounded_lattice @ A9 )
=> ( bounded_lattice_top @ ( A8 > A9 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_4,axiom,
! [A8: $tType] : ( bounded_lattice_top @ ( set @ A8 ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_5,axiom,
bounded_lattice_top @ $o ).
thf(tcon_Option_Ooption___Lattices_Obounded__lattice__top_6,axiom,
! [A8: $tType] :
( ( bounded_lattice_top @ A8 )
=> ( bounded_lattice_top @ ( option @ A8 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A8: $tType,A9: $tType] :
( ( bounded_lattice @ A9 )
=> ( bounde1808546759up_bot @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A8: $tType,A9: $tType] :
( ( semilattice_sup @ A9 )
=> ( semilattice_sup @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 )
=> ( preorder @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A8: $tType,A9: $tType] :
( ( ( finite_finite @ A8 )
& ( finite_finite @ A9 ) )
=> ( finite_finite @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A8: $tType,A9: $tType] :
( ( lattice @ A9 )
=> ( lattice @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 )
=> ( order @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Lattices_Osup,axiom,
! [A8: $tType,A9: $tType] :
( ( semilattice_sup @ A9 )
=> ( sup @ ( A8 > A9 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_7,axiom,
! [A8: $tType] : ( bounde1808546759up_bot @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_8,axiom,
! [A8: $tType] : ( semilattice_sup @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_9,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_10,axiom,
! [A8: $tType] :
( ( finite_finite @ A8 )
=> ( finite_finite @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_11,axiom,
! [A8: $tType] : ( lattice @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_12,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Lattices_Osup_13,axiom,
! [A8: $tType] : ( sup @ ( set @ A8 ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_14,axiom,
bounde1808546759up_bot @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_15,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_16,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_17,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_18,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_19,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Lattices_Osup_20,axiom,
sup @ $o ).
thf(tcon_Option_Ooption___Lattices_Obounded__semilattice__sup__bot_21,axiom,
! [A8: $tType] :
( ( lattice @ A8 )
=> ( bounde1808546759up_bot @ ( option @ A8 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Osemilattice__sup_22,axiom,
! [A8: $tType] :
( ( semilattice_sup @ A8 )
=> ( semilattice_sup @ ( option @ A8 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Opreorder_23,axiom,
! [A8: $tType] :
( ( preorder @ A8 )
=> ( preorder @ ( option @ A8 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Olinorder_24,axiom,
! [A8: $tType] :
( ( linorder @ A8 )
=> ( linorder @ ( option @ A8 ) ) ) ).
thf(tcon_Option_Ooption___Finite__Set_Ofinite_25,axiom,
! [A8: $tType] :
( ( finite_finite @ A8 )
=> ( finite_finite @ ( option @ A8 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Olattice_26,axiom,
! [A8: $tType] :
( ( lattice @ A8 )
=> ( lattice @ ( option @ A8 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder_27,axiom,
! [A8: $tType] :
( ( order @ A8 )
=> ( order @ ( option @ A8 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Osup_28,axiom,
! [A8: $tType] :
( ( sup @ A8 )
=> ( sup @ ( option @ A8 ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_29,axiom,
! [A8: $tType,A9: $tType] :
( ( ( finite_finite @ A8 )
& ( finite_finite @ A9 ) )
=> ( finite_finite @ ( product_prod @ A8 @ A9 ) ) ) ).
% Free types (3)
thf(tfree_0,hypothesis,
linorder @ val ).
thf(tfree_1,hypothesis,
linorder @ node ).
thf(tfree_2,hypothesis,
linorder @ var ).
% Conjectures (1)
thf(conj_0,conjecture,
member @ val @ phi_s @ ( sSA_CFG_SSA_allVars @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ defs @ uses @ phis @ g2 ) ).
%------------------------------------------------------------------------------