TPTP Problem File: ITP080^2.p
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%------------------------------------------------------------------------------
% File : ITP080^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Irreducible problem prob_414__6626236_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_414__6626236_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 317 ( 89 unt; 53 typ; 0 def)
% Number of atoms : 861 ( 354 equ; 0 cnn)
% Maximal formula atoms : 10 ( 3 avg)
% Number of connectives : 5561 ( 156 ~; 13 |; 84 &;4871 @)
% ( 0 <=>; 437 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 10 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 303 ( 303 >; 0 *; 0 +; 0 <<)
% Number of symbols : 50 ( 49 usr; 11 con; 0-13 aty)
% Number of variables : 1217 ( 16 ^;1069 !; 73 ?;1217 :)
% ( 59 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:24:38.328
%------------------------------------------------------------------------------
% Could-be-implicit typings (8)
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_val,type,
val: $tType ).
thf(ty_tf_g,type,
g: $tType ).
% Explicit typings (45)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Graph__path_Ograph__Entry_OEntryPath,type,
graph_2081878492ryPath:
!>[G: $tType,Node: $tType,EdgeD: $tType] : ( ( G > ( list @ Node ) ) > ( G > $o ) > ( G > Node > ( list @ ( product_prod @ Node @ EdgeD ) ) ) > ( G > Node ) > G > ( list @ Node ) > $o ) ).
thf(sy_c_Graph__path_Ograph__Entry_OisIdom,type,
graph_graph_isIdom:
!>[G: $tType,Node: $tType,EdgeD: $tType] : ( ( G > ( list @ Node ) ) > ( G > $o ) > ( G > Node > ( list @ ( product_prod @ Node @ EdgeD ) ) ) > ( G > Node ) > G > Node > Node > $o ) ).
thf(sy_c_Graph__path_Ograph__Entry__base_Odominates,type,
graph_709200220inates:
!>[G: $tType,Node: $tType,EdgeD: $tType] : ( ( G > ( list @ Node ) ) > ( G > $o ) > ( G > Node > ( list @ ( product_prod @ Node @ EdgeD ) ) ) > ( G > Node ) > G > Node > Node > $o ) ).
thf(sy_c_Graph__path_Ograph__path__base_OinEdges,type,
graph_1822314308nEdges:
!>[G: $tType,Node: $tType,EdgeD: $tType] : ( ( G > Node > ( list @ ( product_prod @ Node @ EdgeD ) ) ) > G > Node > ( list @ ( product_prod @ Node @ ( product_prod @ EdgeD @ Node ) ) ) ) ).
thf(sy_c_Graph__path_Ograph__path__base_Opath2,type,
graph_1661282752_path2:
!>[G: $tType,Node: $tType,EdgeD: $tType] : ( ( G > ( list @ Node ) ) > ( G > $o ) > ( G > Node > ( list @ ( product_prod @ Node @ EdgeD ) ) ) > G > Node > ( list @ Node ) > Node > $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_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Odistinct,type,
distinct:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Olist_Ohd,type,
hd:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Olist_Otl,type,
tl:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Minimality_Ograph__Entry_Oreducible,type,
graph_reducible:
!>[G: $tType,Node: $tType,EdgeD: $tType] : ( ( G > ( list @ Node ) ) > ( G > $o ) > ( G > Node > ( list @ ( product_prod @ Node @ EdgeD ) ) ) > ( G > Node ) > G > $o ) ).
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_OdefAss,type,
sSA_CFG_SSA_defAss:
!>[G: $tType,Node: $tType,EdgeD: $tType,Val: $tType] : ( ( G > ( list @ Node ) ) > ( G > $o ) > ( G > Node > ( list @ ( product_prod @ Node @ EdgeD ) ) ) > ( G > Node ) > ( G > Node > ( set @ Val ) ) > ( G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) ) ) > G > Node > Val > $o ) ).
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_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__base_OdefAss_H,type,
sSA_CFG_defAss:
!>[G: $tType,Node: $tType,EdgeD: $tType,Var: $tType] : ( ( G > ( list @ Node ) ) > ( G > $o ) > ( G > Node > ( list @ ( product_prod @ Node @ EdgeD ) ) ) > ( G > Node ) > ( G > Node > ( set @ Var ) ) > G > Node > Var > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Sublist_Osuffix,type,
suffix:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_Entry,type,
entry: g > node ).
thf(sy_v__092_060alpha_062n,type,
alpha_n: g > ( list @ node ) ).
thf(sy_v__092_060phi_062_092_060_094sub_062r,type,
phi_r: 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_invar,type,
invar: g > $o ).
thf(sy_v_m,type,
m: node ).
thf(sy_v_ms,type,
ms: list @ node ).
thf(sy_v_n,type,
n: node ).
thf(sy_v_ns,type,
ns: list @ node ).
thf(sy_v_phis,type,
phis: g > ( product_prod @ node @ val ) > ( option @ ( list @ val ) ) ).
thf(sy_v_pred_092_060_094sub_062_092_060phi_062_092_060_094sub_062r____,type,
pred_phi_r: node ).
thf(sy_v_r,type,
r: val ).
thf(sy_v_rs_H____,type,
rs: list @ node ).
thf(sy_v_rs____,type,
rs2: list @ node ).
thf(sy_v_s,type,
s: val ).
% Relevant facts (255)
thf(fact_0_old_Opath2__hd,axiom,
! [G2: g,N: node,Ns: list @ node,M: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( N
= ( hd @ node @ Ns ) ) ) ).
% old.path2_hd
thf(fact_1_False,axiom,
r != phi_r ).
% False
thf(fact_2_old_OEntry__in__graph,axiom,
! [G2: g] : ( member @ node @ ( entry @ G2 ) @ ( set2 @ node @ ( alpha_n @ G2 ) ) ) ).
% old.Entry_in_graph
thf(fact_3_rs_H__props_I2_J,axiom,
graph_2081878492ryPath @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ g2 @ rs ).
% rs'_props(2)
thf(fact_4_rs_H__props_I1_J,axiom,
graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ g2 @ ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ g2 @ r ) @ rs @ pred_phi_r ).
% rs'_props(1)
thf(fact_5_old_OEntryPath__distinct,axiom,
! [G2: g,Ns: list @ node] :
( ( graph_2081878492ryPath @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ Ns )
=> ( distinct @ node @ Ns ) ) ).
% old.EntryPath_distinct
thf(fact_6_old_O_092_060alpha_062n__distinct,axiom,
! [G2: g] :
( ( invar @ G2 )
=> ( distinct @ node @ ( alpha_n @ G2 ) ) ) ).
% old.\<alpha>n_distinct
thf(fact_7_rs_H__props_I3_J,axiom,
member @ val @ r @ ( sSA_CFG_SSA_phiUses @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ phis @ g2 @ pred_phi_r ) ).
% rs'_props(3)
thf(fact_8_assms_I10_J,axiom,
sSA_CF1165125185phiArg @ g @ node @ val @ alpha_n @ defs @ phis @ g2 @ phi_r @ r ).
% assms(10)
thf(fact_9_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_10_assms_I7_J,axiom,
graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ g2 @ n @ ns @ ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ g2 @ r ) ).
% assms(7)
thf(fact_11_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_12_rs_H__props_I4_J,axiom,
member @ node @ pred_phi_r @ ( set2 @ node @ ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ g2 @ ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ g2 @ phi_r ) ) ) ).
% rs'_props(4)
thf(fact_13_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_14_FormalSSA__Misc_Oin__set__tlD,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ Xs ) ) )
=> ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% FormalSSA_Misc.in_set_tlD
thf(fact_15_old_Opath2__hd__in___092_060alpha_062n,axiom,
! [G2: g,N: node,Ns: list @ node,M: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) ) ) ).
% old.path2_hd_in_\<alpha>n
thf(fact_16_old_Opath2__hd__in__ns,axiom,
! [G2: g,N: node,Ns: list @ node,M: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( member @ node @ N @ ( set2 @ node @ Ns ) ) ) ).
% old.path2_hd_in_ns
thf(fact_17_old_Opath2__in___092_060alpha_062n,axiom,
! [G2: g,N: node,Ns: list @ node,M: node,L: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( ( member @ node @ L @ ( set2 @ node @ Ns ) )
=> ( member @ node @ L @ ( set2 @ node @ ( alpha_n @ G2 ) ) ) ) ) ).
% old.path2_in_\<alpha>n
thf(fact_18_old_Oinvar,axiom,
! [G2: g] : ( invar @ G2 ) ).
% old.invar
thf(fact_19_old_Opath2__tl__in___092_060alpha_062n,axiom,
! [G2: g,N: node,Ns: list @ node,M: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( member @ node @ M @ ( set2 @ node @ ( alpha_n @ G2 ) ) ) ) ).
% old.path2_tl_in_\<alpha>n
thf(fact_20_old_Opath2__last__in__ns,axiom,
! [G2: g,N: node,Ns: list @ node,M: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( member @ node @ M @ ( set2 @ node @ Ns ) ) ) ).
% old.path2_last_in_ns
thf(fact_21_old_Osuccessor__is__node,axiom,
! [N: node,G2: g,N2: node] :
( ( member @ node @ N @ ( set2 @ node @ ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ G2 @ N2 ) ) )
=> ( ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( ( invar @ G2 )
=> ( member @ node @ N2 @ ( set2 @ node @ ( alpha_n @ G2 ) ) ) ) ) ) ).
% old.successor_is_node
thf(fact_22_old_Opredecessor__is__node,axiom,
! [N: node,G2: g,N2: node] :
( ( member @ node @ N @ ( set2 @ node @ ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ G2 @ N2 ) ) )
=> ( ( invar @ G2 )
=> ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) ) ) ) ).
% old.predecessor_is_node
thf(fact_23_old_Opath2__forget__hd,axiom,
! [G2: g,N: node,Ns: list @ node,M: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ ( hd @ node @ Ns ) @ Ns @ M ) ) ).
% old.path2_forget_hd
thf(fact_24_old_OEntry__reachesE,axiom,
! [N: node,G2: g] :
( ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( ( invar @ G2 )
=> ~ ! [Ns2: list @ node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ ( entry @ G2 ) @ Ns2 @ N )
=> ~ ( graph_2081878492ryPath @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ Ns2 ) ) ) ) ).
% old.Entry_reachesE
thf(fact_25__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062rs_H_Apred_092_060_094sub_062_092_060phi_062_092_060_094sub_062r_O_A_092_060lbrakk_062g_A_092_060turnstile_062_AdefNode_Ag_Ar_Nrs_H_092_060rightarrow_062pred_092_060_094sub_062_092_060phi_062_092_060_094sub_062r_059_Aold_OEntryPath_Ag_Ars_H_059_Ar_A_092_060in_062_AphiUses_Ag_Apred_092_060_094sub_062_092_060phi_062_092_060_094sub_062r_059_Apred_092_060_094sub_062_092_060phi_062_092_060_094sub_062r_A_092_060in_062_Aset_A_Iold_Opredecessors_Ag_A_IdefNode_Ag_A_092_060phi_062_092_060_094sub_062r_J_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Rs: list @ node,Pred_phi_r: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ g2 @ ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ g2 @ r ) @ Rs @ Pred_phi_r )
=> ( ( graph_2081878492ryPath @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ g2 @ Rs )
=> ( ( member @ val @ r @ ( sSA_CFG_SSA_phiUses @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ phis @ g2 @ Pred_phi_r ) )
=> ~ ( member @ node @ Pred_phi_r @ ( set2 @ node @ ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ g2 @ ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ g2 @ phi_r ) ) ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>rs' pred\<^sub>\<phi>\<^sub>r. \<lbrakk>g \<turnstile> defNode g r-rs'\<rightarrow>pred\<^sub>\<phi>\<^sub>r; old.EntryPath g rs'; r \<in> phiUses g pred\<^sub>\<phi>\<^sub>r; pred\<^sub>\<phi>\<^sub>r \<in> set (old.predecessors g (defNode g \<phi>\<^sub>r))\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_26_old_OEntry__reaches,axiom,
! [N: node,G2: g] :
( ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( ( invar @ G2 )
=> ? [Ns2: list @ node] : ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ ( entry @ G2 ) @ Ns2 @ N ) ) ) ).
% old.Entry_reaches
thf(fact_27_old_Oidom__ex,axiom,
! [G2: g,N: node] :
( ( invar @ G2 )
=> ( ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( ( N
!= ( entry @ G2 ) )
=> ? [X2: node] :
( ( graph_graph_isIdom @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N @ X2 )
& ! [Y: node] :
( ( graph_graph_isIdom @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N @ Y )
=> ( Y = X2 ) ) ) ) ) ) ).
% old.idom_ex
thf(fact_28_FormalSSA__Misc_Odistinct__hd__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ~ ( member @ A @ ( hd @ A @ Xs ) @ ( set2 @ A @ ( tl @ A @ Xs ) ) ) ) ).
% FormalSSA_Misc.distinct_hd_tl
thf(fact_29_defAss_H__extend,axiom,
! [G2: g,M: node,V: val,N: node,Ns: list @ node] :
( ( sSA_CFG_defAss @ g @ node @ edgeD @ val @ alpha_n @ invar @ inEdges @ entry @ defs @ G2 @ M @ V )
=> ( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( ! [X2: node] :
( ( member @ node @ X2 @ ( set2 @ node @ ( tl @ node @ Ns ) ) )
=> ~ ( member @ val @ V @ ( defs @ G2 @ X2 ) ) )
=> ( sSA_CFG_defAss @ g @ node @ edgeD @ val @ alpha_n @ invar @ inEdges @ entry @ defs @ G2 @ N @ V ) ) ) ) ).
% defAss'_extend
thf(fact_30_defAss_H__def,axiom,
! [G2: g,M: node,V: val] :
( ( sSA_CFG_defAss @ g @ node @ edgeD @ val @ alpha_n @ invar @ inEdges @ entry @ defs @ G2 @ M @ V )
= ( ! [Ns3: list @ node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ ( entry @ G2 ) @ Ns3 @ M )
=> ? [X3: node] :
( ( member @ node @ X3 @ ( set2 @ node @ Ns3 ) )
& ( member @ val @ V @ ( defs @ G2 @ X3 ) ) ) ) ) ) ).
% defAss'_def
thf(fact_31_defAss_HI,axiom,
! [G2: g,M: node,V: val] :
( ! [Ns2: list @ node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ ( entry @ G2 ) @ Ns2 @ M )
=> ? [X4: node] :
( ( member @ node @ X4 @ ( set2 @ node @ Ns2 ) )
& ( member @ val @ V @ ( defs @ G2 @ X4 ) ) ) )
=> ( sSA_CFG_defAss @ g @ node @ edgeD @ val @ alpha_n @ invar @ inEdges @ entry @ defs @ G2 @ M @ V ) ) ).
% defAss'I
thf(fact_32_defAss_HE,axiom,
! [G2: g,M: node,V: val,Ns: list @ node] :
( ( sSA_CFG_defAss @ g @ node @ edgeD @ val @ alpha_n @ invar @ inEdges @ entry @ defs @ G2 @ M @ V )
=> ( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ ( entry @ G2 ) @ Ns @ M )
=> ~ ! [N3: node] :
( ( member @ node @ N3 @ ( set2 @ node @ Ns ) )
=> ~ ( member @ val @ V @ ( defs @ G2 @ N3 ) ) ) ) ) ).
% defAss'E
thf(fact_33_defAss__extend,axiom,
! [G2: g,M: node,V: val,N: node,Ns: list @ node] :
( ( sSA_CFG_SSA_defAss @ g @ node @ edgeD @ val @ alpha_n @ invar @ inEdges @ entry @ defs @ phis @ G2 @ M @ V )
=> ( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( ! [X2: node] :
( ( member @ node @ X2 @ ( set2 @ node @ ( tl @ node @ Ns ) ) )
=> ~ ( member @ val @ V @ ( sSA_CFG_SSA_allDefs @ g @ node @ val @ defs @ phis @ G2 @ X2 ) ) )
=> ( sSA_CFG_SSA_defAss @ g @ node @ edgeD @ val @ alpha_n @ invar @ inEdges @ entry @ defs @ phis @ G2 @ N @ V ) ) ) ) ).
% defAss_extend
thf(fact_34_assms_I8_J,axiom,
graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ g2 @ m @ ms @ ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ g2 @ s ) ).
% assms(8)
thf(fact_35_defAss__def,axiom,
! [G2: g,M: node,V: val] :
( ( sSA_CFG_SSA_defAss @ g @ node @ edgeD @ val @ alpha_n @ invar @ inEdges @ entry @ defs @ phis @ G2 @ M @ V )
= ( ! [Ns3: list @ node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ ( entry @ G2 ) @ Ns3 @ M )
=> ? [X3: node] :
( ( member @ node @ X3 @ ( set2 @ node @ Ns3 ) )
& ( member @ val @ V @ ( sSA_CFG_SSA_allDefs @ g @ node @ val @ defs @ phis @ G2 @ X3 ) ) ) ) ) ) ).
% defAss_def
thf(fact_36_defAssI,axiom,
! [G2: g,M: node,V: val] :
( ! [Ns2: list @ node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ ( entry @ G2 ) @ Ns2 @ M )
=> ? [X4: node] :
( ( member @ node @ X4 @ ( set2 @ node @ Ns2 ) )
& ( member @ val @ V @ ( sSA_CFG_SSA_allDefs @ g @ node @ val @ defs @ phis @ G2 @ X4 ) ) ) )
=> ( sSA_CFG_SSA_defAss @ g @ node @ edgeD @ val @ alpha_n @ invar @ inEdges @ entry @ defs @ phis @ G2 @ M @ V ) ) ).
% defAssI
thf(fact_37_defAssD,axiom,
! [G2: g,M: node,V: val,Ns: list @ node] :
( ( sSA_CFG_SSA_defAss @ g @ node @ edgeD @ val @ alpha_n @ invar @ inEdges @ entry @ defs @ phis @ G2 @ M @ V )
=> ( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ ( entry @ G2 ) @ Ns @ M )
=> ? [X2: node] :
( ( member @ node @ X2 @ ( set2 @ node @ Ns ) )
& ( member @ val @ V @ ( sSA_CFG_SSA_allDefs @ g @ node @ val @ defs @ phis @ G2 @ X2 ) ) ) ) ) ).
% defAssD
thf(fact_38_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_39_defAss__dominating,axiom,
! [N: node,G2: g,V: val] :
( ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( ( sSA_CFG_SSA_defAss @ g @ node @ edgeD @ val @ alpha_n @ invar @ inEdges @ entry @ defs @ phis @ G2 @ N @ V )
= ( ? [X3: node] :
( ( member @ node @ X3 @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
& ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ X3 @ N )
& ( member @ val @ V @ ( sSA_CFG_SSA_allDefs @ g @ node @ val @ defs @ phis @ G2 @ X3 ) ) ) ) ) ) ).
% defAss_dominating
thf(fact_40_old_Osuccessor__in___092_060alpha_062n,axiom,
! [G2: g,N: node] :
( ( ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ G2 @ N )
!= ( nil @ node ) )
=> ( ( invar @ G2 )
=> ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) ) ) ) ).
% old.successor_in_\<alpha>n
thf(fact_41_phiUses__finite,axiom,
! [N: node,G2: g] :
( ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( finite_finite @ val @ ( sSA_CFG_SSA_phiUses @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ phis @ G2 @ N ) ) ) ).
% phiUses_finite
thf(fact_42_old_Opath2__split__last__prop,axiom,
! [G2: g,N: node,Ns: list @ node,M: node,P: node > $o] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( ? [X4: node] :
( ( member @ node @ X4 @ ( set2 @ node @ Ns ) )
& ( P @ X4 ) )
=> ~ ! [N4: node,Ns4: list @ node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N4 @ Ns4 @ M )
=> ( ( P @ N4 )
=> ( ! [X4: node] :
( ( member @ node @ X4 @ ( set2 @ node @ ( tl @ node @ Ns4 ) ) )
=> ~ ( P @ X4 ) )
=> ~ ( suffix @ node @ Ns4 @ Ns ) ) ) ) ) ) ).
% old.path2_split_last_prop
thf(fact_43_allDefs__finite,axiom,
! [N: node,G2: g] :
( ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( finite_finite @ val @ ( sSA_CFG_SSA_allDefs @ g @ node @ val @ defs @ phis @ G2 @ N ) ) ) ).
% allDefs_finite
thf(fact_44_old_Opath2__not__Nil,axiom,
! [G2: g,N: node,Ns: list @ node,M: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( Ns
!= ( nil @ node ) ) ) ).
% old.path2_not_Nil
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G2: A > B] :
( ! [X2: A] :
( ( F @ X2 )
= ( G2 @ X2 ) )
=> ( F = G2 ) ) ).
% ext
thf(fact_49_old_Opath2__not__Nil2,axiom,
! [G2: g,N: node,M: node] :
~ ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ ( nil @ node ) @ M ) ).
% old.path2_not_Nil2
thf(fact_50_old_Opath2__app,axiom,
! [G2: g,N: node,Ns: list @ node,M: node,Ms: list @ node,L: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ M @ Ms @ L )
=> ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ ( append @ node @ Ns @ ( tl @ node @ Ms ) ) @ L ) ) ) ).
% old.path2_app
thf(fact_51_defs__finite,axiom,
! [G2: g,N: node] : ( finite_finite @ val @ ( defs @ G2 @ N ) ) ).
% defs_finite
thf(fact_52_phiDefs__finite,axiom,
! [G2: g,N: node] : ( finite_finite @ val @ ( sSA_CFG_SSA_phiDefs @ g @ node @ val @ phis @ G2 @ N ) ) ).
% phiDefs_finite
thf(fact_53_strict__dom__trans_H,axiom,
! [N: node,M: node,G2: g,M2: node] :
( ( ( N != M )
& ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N @ M ) )
=> ( ( ( M != M2 )
& ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ M @ M2 ) )
=> ( ( N != M2 )
& ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N @ M2 ) ) ) ) ).
% strict_dom_trans'
thf(fact_54_old_Ostrict__dom__trans,axiom,
! [G2: g,N: node,M: node,M2: node] :
( ( invar @ G2 )
=> ( ( ( N != M )
& ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N @ M ) )
=> ( ( ( M != M2 )
& ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ M @ M2 ) )
=> ( ( N != M2 )
& ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N @ M2 ) ) ) ) ) ).
% old.strict_dom_trans
thf(fact_55_old_Odominates__trans,axiom,
! [G2: g,N: node,N2: node,N5: node] :
( ( invar @ G2 )
=> ( ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N @ N2 )
=> ( ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N2 @ N5 )
=> ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N @ N5 ) ) ) ) ).
% old.dominates_trans
thf(fact_56_old_Odominates__antitrans,axiom,
! [G2: g,N_1: node,M: node,N_2: node] :
( ( invar @ G2 )
=> ( ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N_1 @ M )
=> ( ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N_2 @ M )
=> ( ~ ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N_1 @ N_2 )
=> ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N_2 @ N_1 ) ) ) ) ) ).
% old.dominates_antitrans
thf(fact_57_old_Odominates__antisymm,axiom,
! [G2: g,N: node,N2: node] :
( ( invar @ G2 )
=> ( ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N @ N2 )
=> ( ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N2 @ N )
=> ( N = N2 ) ) ) ) ).
% old.dominates_antisymm
thf(fact_58_dominates__trans_H,axiom,
! [G2: g,N: node,N2: node,N5: node] :
( ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N @ N2 )
=> ( ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N2 @ N5 )
=> ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N @ N5 ) ) ) ).
% dominates_trans'
thf(fact_59_dominates__antisymm_H,axiom,
! [G2: g,N: node,N2: node] :
( ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N @ N2 )
=> ( ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N2 @ N )
=> ( N = N2 ) ) ) ).
% dominates_antisymm'
thf(fact_60_dominates__refl_H,axiom,
! [N: node,G2: g] :
( ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N @ N ) ) ).
% dominates_refl'
thf(fact_61_old_Odominates__path,axiom,
! [G2: g,N: node,M: node] :
( ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N @ M )
=> ( ( invar @ G2 )
=> ~ ! [Ns2: list @ node] :
~ ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns2 @ M ) ) ) ).
% old.dominates_path
thf(fact_62_old_Odominates__mid,axiom,
! [G2: g,N: node,X: node,M: node,Ns: list @ node] :
( ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N @ X )
=> ( ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ X @ M )
=> ( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( ( invar @ G2 )
=> ( member @ node @ X @ ( set2 @ node @ Ns ) ) ) ) ) ) ).
% old.dominates_mid
thf(fact_63_old_Odominates__def,axiom,
! [G2: g,N: node,M: node] :
( ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N @ M )
= ( ( member @ node @ M @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
& ! [Ns3: list @ node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ ( entry @ G2 ) @ Ns3 @ M )
=> ( member @ node @ N @ ( set2 @ node @ Ns3 ) ) ) ) ) ).
% old.dominates_def
thf(fact_64_old_OdominatesE,axiom,
! [G2: g,N: node,M: node] :
( ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N @ M )
=> ~ ( ( member @ node @ M @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ~ ! [Ns5: list @ node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ ( entry @ G2 ) @ Ns5 @ M )
=> ( member @ node @ N @ ( set2 @ node @ Ns5 ) ) ) ) ) ).
% old.dominatesE
thf(fact_65_old_Odominates__unsnoc,axiom,
! [G2: g,N: node,M: node,M2: node] :
( ( invar @ G2 )
=> ( ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N @ M )
=> ( ( member @ node @ M2 @ ( set2 @ node @ ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ G2 @ M ) ) )
=> ( ( N != M )
=> ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N @ M2 ) ) ) ) ) ).
% old.dominates_unsnoc
thf(fact_66_non__dominated__predecessor,axiom,
! [N: node,G2: g] :
( ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( ( N
!= ( entry @ G2 ) )
=> ~ ! [M3: node] :
( ( member @ node @ M3 @ ( set2 @ node @ ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ G2 @ N ) ) )
=> ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N @ M3 ) ) ) ) ).
% non_dominated_predecessor
thf(fact_67_old_OisIdom__def,axiom,
! [G2: g,N: node,M: node] :
( ( graph_graph_isIdom @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N @ M )
= ( ( M != N )
& ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ M @ N )
& ! [X3: node] :
( ( member @ node @ X3 @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( ( ( X3 != N )
& ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ X3 @ N ) )
=> ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ X3 @ M ) ) ) ) ) ).
% old.isIdom_def
thf(fact_68_old_OEntryPath__suffix,axiom,
! [G2: g,Ns: list @ node,Ns6: list @ node] :
( ( graph_2081878492ryPath @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ Ns )
=> ( ( suffix @ node @ Ns6 @ Ns )
=> ( ( Ns6
!= ( nil @ node ) )
=> ( graph_2081878492ryPath @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ Ns6 ) ) ) ) ).
% old.EntryPath_suffix
thf(fact_69_old_Odominates__unsnoc_H,axiom,
! [G2: g,N: node,M: node,M2: node,Ms: list @ node] :
( ( invar @ G2 )
=> ( ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N @ M )
=> ( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ M2 @ Ms @ M )
=> ( ! [X2: node] :
( ( member @ node @ X2 @ ( set2 @ node @ ( tl @ node @ Ms ) ) )
=> ( X2 != N ) )
=> ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N @ M2 ) ) ) ) ) ).
% old.dominates_unsnoc'
thf(fact_70_old_Odominates__extend,axiom,
! [G2: g,N: node,M: node,M2: node,Ms: list @ node] :
( ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N @ M )
=> ( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ M2 @ Ms @ M )
=> ( ~ ( member @ node @ N @ ( set2 @ node @ ( tl @ node @ Ms ) ) )
=> ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N @ M2 ) ) ) ) ).
% old.dominates_extend
thf(fact_71_old_OEntry__no__predecessor,axiom,
! [G2: g] :
( ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ G2 @ ( entry @ G2 ) )
= ( nil @ node ) ) ).
% old.Entry_no_predecessor
thf(fact_72_old_Odominates__refl,axiom,
! [G2: g,N: node] :
( ( invar @ G2 )
=> ( ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N @ N ) ) ) ).
% old.dominates_refl
thf(fact_73_old_OEntry__dominates,axiom,
! [G2: g,N: node] :
( ( invar @ G2 )
=> ( ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ ( entry @ G2 ) @ N ) ) ) ).
% old.Entry_dominates
thf(fact_74_old_OdominatesI,axiom,
! [M: node,G2: g,N: node] :
( ( member @ node @ M @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( ! [Ns2: list @ node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ ( entry @ G2 ) @ Ns2 @ M )
=> ( member @ node @ N @ ( set2 @ node @ Ns2 ) ) )
=> ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ N @ M ) ) ) ).
% old.dominatesI
thf(fact_75_old_OEntry__iff__unreachable,axiom,
! [G2: g,N: node] :
( ( invar @ G2 )
=> ( ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( ( ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ G2 @ N )
= ( nil @ node ) )
= ( N
= ( entry @ G2 ) ) ) ) ) ).
% old.Entry_iff_unreachable
thf(fact_76_old_Oreducible__def,axiom,
! [G2: g] :
( ( graph_reducible @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 )
= ( ! [N6: node,Ns3: list @ node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N6 @ Ns3 @ N6 )
=> ? [X3: node] :
( ( member @ node @ X3 @ ( set2 @ node @ Ns3 ) )
& ! [Y2: node] :
( ( member @ node @ Y2 @ ( set2 @ node @ Ns3 ) )
=> ( graph_709200220inates @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ X3 @ Y2 ) ) ) ) ) ) ).
% old.reducible_def
thf(fact_77_in__hd__or__tl__conv,axiom,
! [A: $tType,L: list @ A,X: A] :
( ( L
!= ( nil @ A ) )
=> ( ( ( X
= ( hd @ A @ L ) )
| ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ L ) ) ) )
= ( member @ A @ X @ ( set2 @ A @ L ) ) ) ) ).
% in_hd_or_tl_conv
thf(fact_78_old_Opath2__cases,axiom,
! [G2: g,N: node,Ns: list @ node,M: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( ( ( Ns
= ( cons @ node @ N @ ( nil @ node ) ) )
=> ( M != N ) )
=> ~ ( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ ( hd @ node @ ( tl @ node @ Ns ) ) @ ( tl @ node @ Ns ) @ M )
=> ~ ( member @ node @ N @ ( set2 @ node @ ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ G2 @ ( hd @ node @ ( tl @ node @ Ns ) ) ) ) ) ) ) ) ).
% old.path2_cases
thf(fact_79_old_Opath2__snoc,axiom,
! [G2: g,N: node,Ns: list @ node,M: node,M2: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( ( member @ node @ M @ ( set2 @ node @ ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ G2 @ M2 ) ) )
=> ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ ( append @ node @ Ns @ ( cons @ node @ M2 @ ( nil @ node ) ) ) @ M2 ) ) ) ).
% old.path2_snoc
thf(fact_80_old_Opath2__rev__induct,axiom,
! [G2: g,N: node,Ns: list @ node,M: node,P: node > ( list @ node ) > node > $o] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( ( ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( P @ N @ ( cons @ node @ N @ ( nil @ node ) ) @ N ) )
=> ( ! [Ns2: list @ node,M4: node,M3: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns2 @ M4 )
=> ( ( P @ N @ Ns2 @ M4 )
=> ( ( member @ node @ M4 @ ( set2 @ node @ ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ G2 @ M3 ) ) )
=> ( P @ N @ ( append @ node @ Ns2 @ ( cons @ node @ M3 @ ( nil @ node ) ) ) @ M3 ) ) ) )
=> ( P @ N @ Ns @ M ) ) ) ) ).
% old.path2_rev_induct
thf(fact_81_tl__append2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( tl @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( tl @ A @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_82_hd__append2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys ) )
= ( hd @ A @ Xs ) ) ) ).
% hd_append2
thf(fact_83_same__suffix__nil,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( suffix @ A @ ( append @ A @ Ys @ Xs ) @ Xs )
= ( Ys
= ( nil @ A ) ) ) ).
% same_suffix_nil
thf(fact_84_old_Opath2__induct,axiom,
! [G2: g,N: node,Ns: list @ node,M: node,P: node > ( list @ node ) > node > $o] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( ( ( invar @ G2 )
=> ( P @ M @ ( cons @ node @ M @ ( nil @ node ) ) @ M ) )
=> ( ! [Ns2: list @ node,N4: node,N3: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N3 @ Ns2 @ M )
=> ( ( P @ N3 @ Ns2 @ M )
=> ( ( member @ node @ N4 @ ( set2 @ node @ ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ G2 @ N3 ) ) )
=> ( P @ N4 @ ( cons @ node @ N4 @ Ns2 ) @ M ) ) ) )
=> ( P @ N @ Ns @ M ) ) ) ) ).
% old.path2_induct
thf(fact_85_old_Oelem__set__implies__elem__tl__app__cons,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,Y3: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ ( append @ A @ Ys @ ( cons @ A @ Y3 @ Xs ) ) ) ) ) ) ).
% old.elem_set_implies_elem_tl_app_cons
thf(fact_86_list_Oinject,axiom,
! [A: $tType,X21: A,X22: list @ A,Y21: A,Y22: list @ A] :
( ( ( cons @ A @ X21 @ X22 )
= ( cons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_87_same__append__eq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_88_append__same__eq,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( append @ A @ Ys @ Xs )
= ( append @ A @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_89_append__assoc,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( append @ A @ ( append @ A @ Xs @ Ys ) @ Zs )
= ( append @ A @ Xs @ ( append @ A @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_90_append_Oassoc,axiom,
! [A: $tType,A2: list @ A,B2: list @ A,C: list @ A] :
( ( append @ A @ ( append @ A @ A2 @ B2 ) @ C )
= ( append @ A @ A2 @ ( append @ A @ B2 @ C ) ) ) ).
% append.assoc
thf(fact_91_suffix__order_Odual__order_Orefl,axiom,
! [A: $tType,A2: list @ A] : ( suffix @ A @ A2 @ A2 ) ).
% suffix_order.dual_order.refl
thf(fact_92_suffix__order_Oorder__refl,axiom,
! [A: $tType,X: list @ A] : ( suffix @ A @ X @ X ) ).
% suffix_order.order_refl
thf(fact_93_old_Opath2__split_I2_J,axiom,
! [G2: g,N: node,Ns: list @ node,N2: node,Ns6: list @ node,M: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ ( append @ node @ Ns @ ( cons @ node @ N2 @ Ns6 ) ) @ M )
=> ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N2 @ ( cons @ node @ N2 @ Ns6 ) @ M ) ) ).
% old.path2_split(2)
thf(fact_94_List_Ofinite__set,axiom,
! [A: $tType,Xs: list @ A] : ( finite_finite @ A @ ( set2 @ A @ Xs ) ) ).
% List.finite_set
thf(fact_95_append_Oright__neutral,axiom,
! [A: $tType,A2: list @ A] :
( ( append @ A @ A2 @ ( nil @ A ) )
= A2 ) ).
% append.right_neutral
thf(fact_96_empty__append__eq__id,axiom,
! [A: $tType] :
( ( append @ A @ ( nil @ A ) )
= ( ^ [X3: list @ A] : X3 ) ) ).
% empty_append_eq_id
thf(fact_97_append__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% append_is_Nil_conv
thf(fact_98_Nil__is__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( nil @ A )
= ( append @ A @ Xs @ Ys ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% Nil_is_append_conv
thf(fact_99_self__append__conv2,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( Ys
= ( append @ A @ Xs @ Ys ) )
= ( Xs
= ( nil @ A ) ) ) ).
% self_append_conv2
thf(fact_100_append__self__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Ys )
= ( Xs
= ( nil @ A ) ) ) ).
% append_self_conv2
thf(fact_101_self__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
= ( append @ A @ Xs @ Ys ) )
= ( Ys
= ( nil @ A ) ) ) ).
% self_append_conv
thf(fact_102_append__self__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Xs )
= ( Ys
= ( nil @ A ) ) ) ).
% append_self_conv
thf(fact_103_append__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( append @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% append_Nil2
thf(fact_104_suffix__bot_Obot_Oextremum__unique,axiom,
! [A: $tType,A2: list @ A] :
( ( suffix @ A @ A2 @ ( nil @ A ) )
= ( A2
= ( nil @ A ) ) ) ).
% suffix_bot.bot.extremum_unique
thf(fact_105_suffix__Nil,axiom,
! [A: $tType,Xs: list @ A] :
( ( suffix @ A @ Xs @ ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% suffix_Nil
thf(fact_106_same__suffix__suffix,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( suffix @ A @ ( append @ A @ Ys @ Xs ) @ ( append @ A @ Zs @ Xs ) )
= ( suffix @ A @ Ys @ Zs ) ) ).
% same_suffix_suffix
thf(fact_107_old_Opath2__split_I1_J,axiom,
! [G2: g,N: node,Ns: list @ node,N2: node,Ns6: list @ node,M: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ ( append @ node @ Ns @ ( cons @ node @ N2 @ Ns6 ) ) @ M )
=> ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ ( append @ node @ Ns @ ( cons @ node @ N2 @ ( nil @ node ) ) ) @ N2 ) ) ).
% old.path2_split(1)
thf(fact_108_old_OEntry__loop,axiom,
! [G2: g,Ns: list @ node] :
( ( invar @ G2 )
=> ( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ ( entry @ G2 ) @ Ns @ ( entry @ G2 ) )
=> ( Ns
= ( cons @ node @ ( entry @ G2 ) @ ( nil @ node ) ) ) ) ) ).
% old.Entry_loop
thf(fact_109_old_OEntryPath__triv,axiom,
! [G2: g,N: node] : ( graph_2081878492ryPath @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ entry @ G2 @ ( cons @ node @ N @ ( nil @ node ) ) ) ).
% old.EntryPath_triv
thf(fact_110_append1__eq__conv,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A,Y3: A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( append @ A @ Ys @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) )
= ( ( Xs = Ys )
& ( X = Y3 ) ) ) ).
% append1_eq_conv
thf(fact_111_list__ee__eq__leel_I1_J,axiom,
! [A: $tType,E1: A,E2: A,L1: list @ A,E12: A,E22: A,L2: list @ A] :
( ( ( cons @ A @ E1 @ ( cons @ A @ E2 @ ( nil @ A ) ) )
= ( append @ A @ L1 @ ( cons @ A @ E12 @ ( cons @ A @ E22 @ L2 ) ) ) )
= ( ( L1
= ( nil @ A ) )
& ( E1 = E12 )
& ( E2 = E22 )
& ( L2
= ( nil @ A ) ) ) ) ).
% list_ee_eq_leel(1)
thf(fact_112_list__ee__eq__leel_I2_J,axiom,
! [A: $tType,L1: list @ A,E12: A,E22: A,L2: list @ A,E1: A,E2: A] :
( ( ( append @ A @ L1 @ ( cons @ A @ E12 @ ( cons @ A @ E22 @ L2 ) ) )
= ( cons @ A @ E1 @ ( cons @ A @ E2 @ ( nil @ A ) ) ) )
= ( ( L1
= ( nil @ A ) )
& ( E1 = E12 )
& ( E2 = E22 )
& ( L2
= ( nil @ A ) ) ) ) ).
% list_ee_eq_leel(2)
thf(fact_113_list__se__match_I1_J,axiom,
! [A: $tType,L1: list @ A,L2: list @ A,A2: A] :
( ( L1
!= ( nil @ A ) )
=> ( ( ( append @ A @ L1 @ L2 )
= ( cons @ A @ A2 @ ( nil @ A ) ) )
= ( ( L1
= ( cons @ A @ A2 @ ( nil @ A ) ) )
& ( L2
= ( nil @ A ) ) ) ) ) ).
% list_se_match(1)
thf(fact_114_list__se__match_I2_J,axiom,
! [A: $tType,L2: list @ A,L1: list @ A,A2: A] :
( ( L2
!= ( nil @ A ) )
=> ( ( ( append @ A @ L1 @ L2 )
= ( cons @ A @ A2 @ ( nil @ A ) ) )
= ( ( L1
= ( nil @ A ) )
& ( L2
= ( cons @ A @ A2 @ ( nil @ A ) ) ) ) ) ) ).
% list_se_match(2)
thf(fact_115_list__se__match_I3_J,axiom,
! [A: $tType,L1: list @ A,A2: A,L2: list @ A] :
( ( L1
!= ( nil @ A ) )
=> ( ( ( cons @ A @ A2 @ ( nil @ A ) )
= ( append @ A @ L1 @ L2 ) )
= ( ( L1
= ( cons @ A @ A2 @ ( nil @ A ) ) )
& ( L2
= ( nil @ A ) ) ) ) ) ).
% list_se_match(3)
thf(fact_116_list__se__match_I4_J,axiom,
! [A: $tType,L2: list @ A,A2: A,L1: list @ A] :
( ( L2
!= ( nil @ A ) )
=> ( ( ( cons @ A @ A2 @ ( nil @ A ) )
= ( append @ A @ L1 @ L2 ) )
= ( ( L1
= ( nil @ A ) )
& ( L2
= ( cons @ A @ A2 @ ( nil @ A ) ) ) ) ) ) ).
% list_se_match(4)
thf(fact_117_list__e__eq__lel_I1_J,axiom,
! [A: $tType,E: A,L1: list @ A,E3: A,L2: list @ A] :
( ( ( cons @ A @ E @ ( nil @ A ) )
= ( append @ A @ L1 @ ( cons @ A @ E3 @ L2 ) ) )
= ( ( L1
= ( nil @ A ) )
& ( E3 = E )
& ( L2
= ( nil @ A ) ) ) ) ).
% list_e_eq_lel(1)
thf(fact_118_list__e__eq__lel_I2_J,axiom,
! [A: $tType,L1: list @ A,E3: A,L2: list @ A,E: A] :
( ( ( append @ A @ L1 @ ( cons @ A @ E3 @ L2 ) )
= ( cons @ A @ E @ ( nil @ A ) ) )
= ( ( L1
= ( nil @ A ) )
& ( E3 = E )
& ( L2
= ( nil @ A ) ) ) ) ).
% list_e_eq_lel(2)
thf(fact_119_snoc__suffix__snoc,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A,Y3: A] :
( ( suffix @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) @ ( append @ A @ Ys @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) )
= ( ( X = Y3 )
& ( suffix @ A @ Xs @ Ys ) ) ) ).
% snoc_suffix_snoc
thf(fact_120_suffix__snoc,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Y3: A] :
( ( suffix @ A @ Xs @ ( append @ A @ Ys @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) )
= ( ( Xs
= ( nil @ A ) )
| ? [Zs2: list @ A] :
( ( Xs
= ( append @ A @ Zs2 @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) )
& ( suffix @ A @ Zs2 @ Ys ) ) ) ) ).
% suffix_snoc
thf(fact_121_list_Ocollapse,axiom,
! [A: $tType,List: list @ A] :
( ( List
!= ( nil @ A ) )
=> ( ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) )
= List ) ) ).
% list.collapse
thf(fact_122_hd__Cons__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( cons @ A @ ( hd @ A @ Xs ) @ ( tl @ A @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_123_rs__def,axiom,
( rs2
= ( append @ node @ rs @ ( cons @ node @ ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ g2 @ phi_r ) @ ( nil @ node ) ) ) ) ).
% rs_def
thf(fact_124_old_Oempty__path2,axiom,
! [N: node,G2: g] :
( ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( ( invar @ G2 )
=> ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ ( cons @ node @ N @ ( nil @ node ) ) @ N ) ) ) ).
% old.empty_path2
thf(fact_125_old_OCons__path2,axiom,
! [G2: g,N: node,Ns: list @ node,M: node,N2: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( ( member @ node @ N2 @ ( set2 @ node @ ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ G2 @ N ) ) )
=> ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N2 @ ( cons @ node @ N2 @ Ns ) @ M ) ) ) ).
% old.Cons_path2
thf(fact_126_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_127_neq__NilE,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
=> ~ ! [X2: A,Xs2: list @ A] :
( L
!= ( cons @ A @ X2 @ Xs2 ) ) ) ).
% neq_NilE
thf(fact_128_list_OdiscI,axiom,
! [A: $tType,List: list @ A,X21: A,X22: list @ A] :
( ( List
= ( cons @ A @ X21 @ X22 ) )
=> ( List
!= ( nil @ A ) ) ) ).
% list.discI
thf(fact_129_revg_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
=> ( ! [A4: A,As: list @ A,B3: list @ A] :
( ( P @ As @ ( cons @ A @ A4 @ B3 ) )
=> ( P @ ( cons @ A @ A4 @ As ) @ B3 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% revg.induct
thf(fact_130_zipf_Oinduct,axiom,
! [A: $tType,C2: $tType,B: $tType,P: ( A > B > C2 ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B > C2,A1: list @ A,A22: list @ B] :
( ! [F2: A > B > C2] : ( P @ F2 @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [F2: A > B > C2,A4: A,As: list @ A,B3: B,Bs: list @ B] :
( ( P @ F2 @ As @ Bs )
=> ( P @ F2 @ ( cons @ A @ A4 @ As ) @ ( cons @ B @ B3 @ Bs ) ) )
=> ( ! [A4: A > B > C2,V2: A,Va: list @ A] : ( P @ A4 @ ( cons @ A @ V2 @ Va ) @ ( nil @ B ) )
=> ( ! [A4: A > B > C2,V2: B,Va: list @ B] : ( P @ A4 @ ( nil @ A ) @ ( cons @ B @ V2 @ Va ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).
% zipf.induct
thf(fact_131_list_Oexhaust,axiom,
! [A: $tType,Y3: list @ A] :
( ( Y3
!= ( nil @ A ) )
=> ~ ! [X212: A,X222: list @ A] :
( Y3
!= ( cons @ A @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_132_list_Oinducts,axiom,
! [A: $tType,P: ( list @ A ) > $o,List: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X1: A,X23: list @ A] :
( ( P @ X23 )
=> ( P @ ( cons @ A @ X1 @ X23 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_133_neq__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
= ( ? [Y2: A,Ys2: list @ A] :
( Xs
= ( cons @ A @ Y2 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_134_list__induct2_H,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X2: A,Xs2: list @ A] : ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( nil @ B ) )
=> ( ! [Y4: B,Ys3: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y4 @ Ys3 ) )
=> ( ! [X2: A,Xs2: list @ A,Y4: B,Ys3: list @ B] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ B @ Y4 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_135_splice_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
=> ( ! [X2: A,Xs2: list @ A,Ys3: list @ A] :
( ( P @ Ys3 @ Xs2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ Ys3 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_136_induct__list012,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Y4: A,Zs3: list @ A] :
( ( P @ Zs3 )
=> ( ( P @ ( cons @ A @ Y4 @ Zs3 ) )
=> ( P @ ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Zs3 ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% induct_list012
thf(fact_137_min__list_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: list @ A] :
( ! [X2: A,Xs2: list @ A] :
( X
!= ( cons @ A @ X2 @ Xs2 ) )
=> ( X
= ( nil @ A ) ) ) ) ).
% min_list.cases
thf(fact_138_suffix__Cons,axiom,
! [A: $tType,Xs: list @ A,Y3: A,Ys: list @ A] :
( ( suffix @ A @ Xs @ ( cons @ A @ Y3 @ Ys ) )
= ( ( Xs
= ( cons @ A @ Y3 @ Ys ) )
| ( suffix @ A @ Xs @ Ys ) ) ) ).
% suffix_Cons
thf(fact_139_Cons__eq__appendI,axiom,
! [A: $tType,X: A,Xs1: list @ A,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append @ A @ Xs1 @ Zs ) )
=> ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_140_min__list_Oinduct,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [P: ( list @ A ) > $o,A0: list @ A] :
( ! [X2: A,Xs2: list @ A] :
( ! [X213: A,X223: list @ A] :
( ( Xs2
= ( cons @ A @ X213 @ X223 ) )
=> ( P @ Xs2 ) )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) ) )
=> ( ( P @ ( nil @ A ) )
=> ( P @ A0 ) ) ) ) ).
% min_list.induct
thf(fact_141_shuffles_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
=> ( ! [Xs2: list @ A] : ( P @ Xs2 @ ( nil @ A ) )
=> ( ! [X2: A,Xs2: list @ A,Y4: A,Ys3: list @ A] :
( ( P @ Xs2 @ ( cons @ A @ Y4 @ Ys3 ) )
=> ( ( P @ ( cons @ A @ X2 @ Xs2 ) @ Ys3 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys3 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_142_transpose_Ocases,axiom,
! [A: $tType,X: list @ ( list @ A )] :
( ( X
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ~ ! [X2: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_143_suffix__ConsD,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( suffix @ A @ ( cons @ A @ X @ Xs ) @ Ys )
=> ( suffix @ A @ Xs @ Ys ) ) ).
% suffix_ConsD
thf(fact_144_suffix__ConsI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Y3: A] :
( ( suffix @ A @ Xs @ Ys )
=> ( suffix @ A @ Xs @ ( cons @ A @ Y3 @ Ys ) ) ) ).
% suffix_ConsI
thf(fact_145_list__2pre__induct,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,W1: list @ A,W2: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [E4: A,W12: list @ A,W22: list @ B] :
( ( P @ W12 @ W22 )
=> ( P @ ( cons @ A @ E4 @ W12 ) @ W22 ) )
=> ( ! [E4: B,W13: list @ A,W23: list @ B] :
( ( P @ W13 @ W23 )
=> ( P @ W13 @ ( cons @ B @ E4 @ W23 ) ) )
=> ( P @ W1 @ W2 ) ) ) ) ).
% list_2pre_induct
thf(fact_146_suffix__ConsD2,axiom,
! [A: $tType,X: A,Xs: list @ A,Y3: A,Ys: list @ A] :
( ( suffix @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y3 @ Ys ) )
=> ( suffix @ A @ Xs @ Ys ) ) ).
% suffix_ConsD2
thf(fact_147_remdups__adj_Ocases,axiom,
! [A: $tType,X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ( ! [X2: A] :
( X
!= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ~ ! [X2: A,Y4: A,Xs2: list @ A] :
( X
!= ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_148_sorted__wrt_Oinduct,axiom,
! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
( ! [P2: A > A > $o] : ( P @ P2 @ ( nil @ A ) )
=> ( ! [P2: A > A > $o,X2: A,Ys3: list @ A] :
( ( P @ P2 @ Ys3 )
=> ( P @ P2 @ ( cons @ A @ X2 @ Ys3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_149_append__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( append @ A @ ( cons @ A @ X @ Xs ) @ Ys )
= ( cons @ A @ X @ ( append @ A @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_150_remdups__adj_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Y4: A,Xs2: list @ A] :
( ( ( X2 = Y4 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) ) )
=> ( ( ( X2 != Y4 )
=> ( P @ ( cons @ A @ Y4 @ Xs2 ) ) )
=> ( P @ ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_151_list__induct__first2,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X1: A,X23: A,Xs2: list @ A] :
( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X1 @ ( cons @ A @ X23 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_induct_first2
thf(fact_152_list__match__lel__lel,axiom,
! [A: $tType,C1: list @ A,Qs: A,C22: list @ A,C12: list @ A,Qs2: A,C23: list @ A] :
( ( ( append @ A @ C1 @ ( cons @ A @ Qs @ C22 ) )
= ( append @ A @ C12 @ ( cons @ A @ Qs2 @ C23 ) ) )
=> ( ! [C21: list @ A] :
( ( C1
= ( append @ A @ C12 @ ( cons @ A @ Qs2 @ C21 ) ) )
=> ( C23
!= ( append @ A @ C21 @ ( cons @ A @ Qs @ C22 ) ) ) )
=> ( ( ( C12 = C1 )
=> ( ( Qs2 = Qs )
=> ( C23 != C22 ) ) )
=> ~ ! [C212: list @ A] :
( ( C12
= ( append @ A @ C1 @ ( cons @ A @ Qs @ C212 ) ) )
=> ( C22
!= ( append @ A @ C212 @ ( cons @ A @ Qs2 @ C23 ) ) ) ) ) ) ) ).
% list_match_lel_lel
thf(fact_153_arg__min__list_Oinduct,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [P: ( A > B ) > ( list @ A ) > $o,A0: A > B,A1: list @ A] :
( ! [F2: A > B,X2: A] : ( P @ F2 @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [F2: A > B,X2: A,Y4: A,Zs3: list @ A] :
( ( P @ F2 @ ( cons @ A @ Y4 @ Zs3 ) )
=> ( P @ F2 @ ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Zs3 ) ) ) )
=> ( ! [A4: A > B] : ( P @ A4 @ ( nil @ A ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ).
% arg_min_list.induct
thf(fact_154_successively_Oinduct,axiom,
! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
( ! [P2: A > A > $o] : ( P @ P2 @ ( nil @ A ) )
=> ( ! [P2: A > A > $o,X2: A] : ( P @ P2 @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [P2: A > A > $o,X2: A,Y4: A,Xs2: list @ A] :
( ( P @ P2 @ ( cons @ A @ Y4 @ Xs2 ) )
=> ( P @ P2 @ ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_155_list__all__zip_Oinduct,axiom,
! [A: $tType,B: $tType,P: ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B > $o,A1: list @ A,A22: list @ B] :
( ! [P2: A > B > $o] : ( P @ P2 @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [P2: A > B > $o,A4: A,As: list @ A,B3: B,Bs: list @ B] :
( ( P @ P2 @ As @ Bs )
=> ( P @ P2 @ ( cons @ A @ A4 @ As ) @ ( cons @ B @ B3 @ Bs ) ) )
=> ( ! [P2: A > B > $o,V2: A,Va: list @ A] : ( P @ P2 @ ( cons @ A @ V2 @ Va ) @ ( nil @ B ) )
=> ( ! [P2: A > B > $o,V2: B,Va: list @ B] : ( P @ P2 @ ( nil @ A ) @ ( cons @ B @ V2 @ Va ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).
% list_all_zip.induct
thf(fact_156_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_157_map__tailrec__rev_Oinduct,axiom,
! [A: $tType,B: $tType,P: ( A > B ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B,A1: list @ A,A22: list @ B] :
( ! [F2: A > B,X_1: list @ B] : ( P @ F2 @ ( nil @ A ) @ X_1 )
=> ( ! [F2: A > B,A4: A,As: list @ A,Bs: list @ B] :
( ( P @ F2 @ As @ ( cons @ B @ ( F2 @ A4 ) @ Bs ) )
=> ( P @ F2 @ ( cons @ A @ A4 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_158_mergesort__by__rel__merge_Oinduct,axiom,
! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A,A22: list @ A] :
( ! [R: A > A > $o,X2: A,Xs2: list @ A,Y4: A,Ys3: list @ A] :
( ( ( R @ X2 @ Y4 )
=> ( P @ R @ Xs2 @ ( cons @ A @ Y4 @ Ys3 ) ) )
=> ( ( ~ ( R @ X2 @ Y4 )
=> ( P @ R @ ( cons @ A @ X2 @ Xs2 ) @ Ys3 ) )
=> ( P @ R @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys3 ) ) ) )
=> ( ! [R: A > A > $o,Xs2: list @ A] : ( P @ R @ Xs2 @ ( nil @ A ) )
=> ( ! [R: A > A > $o,V2: A,Va: list @ A] : ( P @ R @ ( nil @ A ) @ ( cons @ A @ V2 @ Va ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ) ).
% mergesort_by_rel_merge.induct
thf(fact_159_mergesort__by__rel__merge__induct,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,R2: A > B > $o,Xs: list @ A,Ys: list @ B] :
( ! [Xs2: list @ A] : ( P @ Xs2 @ ( nil @ B ) )
=> ( ! [X_1: list @ B] : ( P @ ( nil @ A ) @ X_1 )
=> ( ! [X2: A,Xs2: list @ A,Y4: B,Ys3: list @ B] :
( ( R2 @ X2 @ Y4 )
=> ( ( P @ Xs2 @ ( cons @ B @ Y4 @ Ys3 ) )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ B @ Y4 @ Ys3 ) ) ) )
=> ( ! [X2: A,Xs2: list @ A,Y4: B,Ys3: list @ B] :
( ~ ( R2 @ X2 @ Y4 )
=> ( ( P @ ( cons @ A @ X2 @ Xs2 ) @ Ys3 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ B @ Y4 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% mergesort_by_rel_merge_induct
thf(fact_160_longest__common__prefix_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X2: A,Xs2: list @ A,Y4: A,Ys3: list @ A] :
( ( ( X2 = Y4 )
=> ( P @ Xs2 @ Ys3 ) )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys3 ) ) )
=> ( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
=> ( ! [Uu: list @ A] : ( P @ Uu @ ( nil @ A ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% longest_common_prefix.induct
thf(fact_161_strict__sorted_Oinduct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X2: A,Ys3: list @ A] :
( ( P @ Ys3 )
=> ( P @ ( cons @ A @ X2 @ Ys3 ) ) )
=> ( P @ A0 ) ) ) ) ).
% strict_sorted.induct
thf(fact_162_list_Osel_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( hd @ A @ ( cons @ A @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_163_list_Osel_I3_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( tl @ A @ ( cons @ A @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_164_list__tail__coinc,axiom,
! [A: $tType,N1: A,R1: list @ A,N22: A,R22: list @ A] :
( ( ( cons @ A @ N1 @ R1 )
= ( cons @ A @ N22 @ R22 ) )
=> ( ( N1 = N22 )
& ( R1 = R22 ) ) ) ).
% list_tail_coinc
thf(fact_165_not__Cons__self2,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( cons @ A @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_166_distinct__length__2__or__more,axiom,
! [A: $tType,A2: A,B2: A,Xs: list @ A] :
( ( distinct @ A @ ( cons @ A @ A2 @ ( cons @ A @ B2 @ Xs ) ) )
= ( ( A2 != B2 )
& ( distinct @ A @ ( cons @ A @ A2 @ Xs ) )
& ( distinct @ A @ ( cons @ A @ B2 @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_167_list_Oset__cases,axiom,
! [A: $tType,E: A,A2: list @ A] :
( ( member @ A @ E @ ( set2 @ A @ A2 ) )
=> ( ! [Z2: list @ A] :
( A2
!= ( cons @ A @ E @ Z2 ) )
=> ~ ! [Z1: A,Z2: list @ A] :
( ( A2
= ( cons @ A @ Z1 @ Z2 ) )
=> ~ ( member @ A @ E @ ( set2 @ A @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_168_set__ConsD,axiom,
! [A: $tType,Y3: A,X: A,Xs: list @ A] :
( ( member @ A @ Y3 @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
=> ( ( Y3 = X )
| ( member @ A @ Y3 @ ( set2 @ A @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_169_list_Oset__intros_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] : ( member @ A @ X21 @ ( set2 @ A @ ( cons @ A @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_170_list_Oset__intros_I2_J,axiom,
! [A: $tType,Y3: A,X22: list @ A,X21: A] :
( ( member @ A @ Y3 @ ( set2 @ A @ X22 ) )
=> ( member @ A @ Y3 @ ( set2 @ A @ ( cons @ A @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_171_list__append__eq__Cons__cases,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,X: A,Xs: list @ A] :
( ( ( append @ A @ Ys @ Zs )
= ( cons @ A @ X @ Xs ) )
=> ( ( ( Ys
= ( nil @ A ) )
=> ( Zs
!= ( cons @ A @ X @ Xs ) ) )
=> ~ ! [Ys4: list @ A] :
( ( Ys
= ( cons @ A @ X @ Ys4 ) )
=> ( ( append @ A @ Ys4 @ Zs )
!= Xs ) ) ) ) ).
% list_append_eq_Cons_cases
thf(fact_172_list__Cons__eq__append__cases,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs ) )
=> ( ( ( Ys
= ( nil @ A ) )
=> ( Zs
!= ( cons @ A @ X @ Xs ) ) )
=> ~ ! [Ys4: list @ A] :
( ( Ys
= ( cons @ A @ X @ Ys4 ) )
=> ( ( append @ A @ Ys4 @ Zs )
!= Xs ) ) ) ) ).
% list_Cons_eq_append_cases
thf(fact_173_rev__nonempty__induct2_H,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys
!= ( nil @ B ) )
=> ( ! [X2: A,Y4: B] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) @ ( cons @ B @ Y4 @ ( nil @ B ) ) )
=> ( ! [X2: A,Xs2: list @ A,Y4: B] :
( ( Xs2
!= ( nil @ A ) )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) )
=> ( ! [X2: A,Y4: B,Ys3: list @ B] :
( ( Ys3
!= ( nil @ B ) )
=> ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) @ ( append @ B @ Ys3 @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) ) )
=> ( ! [X2: A,Xs2: list @ A,Y4: B,Ys3: list @ B] :
( ( P @ Xs2 @ Ys3 )
=> ( ( Xs2
!= ( nil @ A ) )
=> ( ( Ys3
!= ( nil @ B ) )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ ( append @ B @ Ys3 @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ) ) ).
% rev_nonempty_induct2'
thf(fact_174_rev__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_175_append__eq__Cons__conv,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,X: A,Xs: list @ A] :
( ( ( append @ A @ Ys @ Zs )
= ( cons @ A @ X @ Xs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( Zs
= ( cons @ A @ X @ Xs ) ) )
| ? [Ys5: list @ A] :
( ( Ys
= ( cons @ A @ X @ Ys5 ) )
& ( ( append @ A @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_176_Cons__eq__append__conv,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( ( cons @ A @ X @ Xs )
= Zs ) )
| ? [Ys5: list @ A] :
( ( ( cons @ A @ X @ Ys5 )
= Ys )
& ( Xs
= ( append @ A @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_177_neq__Nil__rev__conv,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
= ( ? [Xs3: list @ A,X3: A] :
( L
= ( append @ A @ Xs3 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) ) ).
% neq_Nil_rev_conv
thf(fact_178_rev__induct2_H,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X2: A,Xs2: list @ A] : ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ ( nil @ B ) )
=> ( ! [Y4: B,Ys3: list @ B] : ( P @ ( nil @ A ) @ ( append @ B @ Ys3 @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) )
=> ( ! [X2: A,Xs2: list @ A,Y4: B,Ys3: list @ B] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ ( append @ B @ Ys3 @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% rev_induct2'
thf(fact_179_neq__Nil__revE,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
=> ~ ! [Ll: list @ A,E4: A] :
( L
!= ( append @ A @ Ll @ ( cons @ A @ E4 @ ( nil @ A ) ) ) ) ) ).
% neq_Nil_revE
thf(fact_180_rev__exhaust,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ~ ! [Ys3: list @ A,Y4: A] :
( Xs
!= ( append @ A @ Ys3 @ ( cons @ A @ Y4 @ ( nil @ A ) ) ) ) ) ).
% rev_exhaust
thf(fact_181_rev__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X2: A,Xs2: list @ A] :
( ( P @ Xs2 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_182_split__list,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [Ys3: list @ A,Zs3: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_183_split__list__last,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [Ys3: list @ A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs3 ) ) )
& ~ ( member @ A @ X @ ( set2 @ A @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_184_split__list__prop,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list @ A,X2: A] :
( ? [Zs3: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs3 ) ) )
& ( P @ X2 ) ) ) ).
% split_list_prop
thf(fact_185_xy__in__set__cases,axiom,
! [A: $tType,X: A,L: list @ A,Y3: A] :
( ( member @ A @ X @ ( set2 @ A @ L ) )
=> ( ( member @ A @ Y3 @ ( set2 @ A @ L ) )
=> ( ( ( X = Y3 )
=> ! [L12: list @ A,L22: list @ A] :
( L
!= ( append @ A @ L12 @ ( cons @ A @ Y3 @ L22 ) ) ) )
=> ( ( ( X != Y3 )
=> ! [L12: list @ A,L22: list @ A,L3: list @ A] :
( L
!= ( append @ A @ L12 @ ( cons @ A @ X @ ( append @ A @ L22 @ ( cons @ A @ Y3 @ L3 ) ) ) ) ) )
=> ~ ( ( X != Y3 )
=> ! [L12: list @ A,L22: list @ A,L3: list @ A] :
( L
!= ( append @ A @ L12 @ ( cons @ A @ Y3 @ ( append @ A @ L22 @ ( cons @ A @ X @ L3 ) ) ) ) ) ) ) ) ) ) ).
% xy_in_set_cases
thf(fact_186_split__list__first,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [Ys3: list @ A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs3 ) ) )
& ~ ( member @ A @ X @ ( set2 @ A @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_187_split__list__propE,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list @ A,X2: A] :
( ? [Zs3: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs3 ) ) )
=> ~ ( P @ X2 ) ) ) ).
% split_list_propE
thf(fact_188_append__Cons__eq__iff,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,Xs4: list @ A,Ys6: list @ A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ~ ( member @ A @ X @ ( set2 @ A @ Ys ) )
=> ( ( ( append @ A @ Xs @ ( cons @ A @ X @ Ys ) )
= ( append @ A @ Xs4 @ ( cons @ A @ X @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_189_in__set__conv__decomp,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [Ys2: list @ A,Zs2: list @ A] :
( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_190_in__set__list__format,axiom,
! [A: $tType,E: A,L: list @ A] :
( ( member @ A @ E @ ( set2 @ A @ L ) )
=> ~ ! [L12: list @ A,L22: list @ A] :
( L
!= ( append @ A @ L12 @ ( cons @ A @ E @ L22 ) ) ) ) ).
% in_set_list_format
thf(fact_191_split__list__last__prop,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list @ A,X2: A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_192_split__list__first__prop,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list @ A,X2: A] :
( ? [Zs3: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Ys3 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_193_split__list__last__propE,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list @ A,X2: A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs3 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_194_split__list__first__propE,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list @ A,X2: A] :
( ? [Zs3: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs3 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Ys3 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_195_in__set__conv__decomp__last,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [Ys2: list @ A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X @ Zs2 ) ) )
& ~ ( member @ A @ X @ ( set2 @ A @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_196_in__set__conv__decomp__first,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [Ys2: list @ A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X @ Zs2 ) ) )
& ~ ( member @ A @ X @ ( set2 @ A @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_197_split__list__last__prop__iff,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ? [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys2: list @ A,X3: A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y2: A] :
( ( member @ A @ Y2 @ ( set2 @ A @ Zs2 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_198_split__list__first__prop__iff,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ? [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys2: list @ A,X3: A] :
( ? [Zs2: list @ A] :
( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y2: A] :
( ( member @ A @ Y2 @ ( set2 @ A @ Ys2 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_199_distinct__singleton,axiom,
! [A: $tType,X: A] : ( distinct @ A @ ( cons @ A @ X @ ( nil @ A ) ) ) ).
% distinct_singleton
thf(fact_200_distinct_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( distinct @ A @ ( cons @ A @ X @ Xs ) )
= ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
& ( distinct @ A @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_201_distinct__match,axiom,
! [A: $tType,Al: list @ A,E: A,Bl: list @ A,Al2: list @ A,Bl2: list @ A] :
( ( distinct @ A @ ( append @ A @ Al @ ( cons @ A @ E @ Bl ) ) )
=> ( ( ( append @ A @ Al @ ( cons @ A @ E @ Bl ) )
= ( append @ A @ Al2 @ ( cons @ A @ E @ Bl2 ) ) )
= ( ( Al = Al2 )
& ( Bl = Bl2 ) ) ) ) ).
% distinct_match
thf(fact_202_Nil__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( nil @ A )
= ( tl @ A @ Xs ) )
= ( ( Xs
= ( nil @ A ) )
| ? [X3: A] :
( Xs
= ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) ).
% Nil_tl
thf(fact_203_tl__Nil,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( tl @ A @ Xs )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
| ? [X3: A] :
( Xs
= ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) ).
% tl_Nil
thf(fact_204_tl__obtain__elem,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ( tl @ A @ Xs )
= ( nil @ A ) )
=> ~ ! [E4: A] :
( Xs
!= ( cons @ A @ E4 @ ( nil @ A ) ) ) ) ) ).
% tl_obtain_elem
thf(fact_205_not__distinct__decomp,axiom,
! [A: $tType,Ws: list @ A] :
( ~ ( distinct @ A @ Ws )
=> ? [Xs2: list @ A,Ys3: list @ A,Zs3: list @ A,Y4: A] :
( Ws
= ( append @ A @ Xs2 @ ( append @ A @ ( cons @ A @ Y4 @ ( nil @ A ) ) @ ( append @ A @ Ys3 @ ( append @ A @ ( cons @ A @ Y4 @ ( nil @ A ) ) @ Zs3 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_206_not__distinct__conv__prefix,axiom,
! [A: $tType,As2: list @ A] :
( ( ~ ( distinct @ A @ As2 ) )
= ( ? [Xs3: list @ A,Y2: A,Ys2: list @ A] :
( ( member @ A @ Y2 @ ( set2 @ A @ Xs3 ) )
& ( distinct @ A @ Xs3 )
& ( As2
= ( append @ A @ Xs3 @ ( cons @ A @ Y2 @ Ys2 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_207_not__suffix__induct,axiom,
! [A: $tType,Ps: list @ A,Ls: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ~ ( suffix @ A @ Ps @ Ls )
=> ( ! [X2: A,Xs2: list @ A] : ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ ( nil @ A ) )
=> ( ! [X2: A,Xs2: list @ A,Y4: A,Ys3: list @ A] :
( ( X2 != Y4 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ ( append @ A @ Ys3 @ ( cons @ A @ Y4 @ ( nil @ A ) ) ) ) )
=> ( ! [X2: A,Xs2: list @ A,Y4: A,Ys3: list @ A] :
( ( X2 = Y4 )
=> ( ~ ( suffix @ A @ Xs2 @ Ys3 )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ ( append @ A @ Ys3 @ ( cons @ A @ Y4 @ ( nil @ A ) ) ) ) ) ) )
=> ( P @ Ps @ Ls ) ) ) ) ) ).
% not_suffix_induct
thf(fact_208_not__suffix__cases,axiom,
! [A: $tType,Ps: list @ A,Ls: list @ A] :
( ~ ( suffix @ A @ Ps @ Ls )
=> ( ( ( Ps
!= ( nil @ A ) )
=> ( Ls
!= ( nil @ A ) ) )
=> ( ! [A4: A,As: list @ A] :
( ( Ps
= ( append @ A @ As @ ( cons @ A @ A4 @ ( nil @ A ) ) ) )
=> ! [X2: A,Xs2: list @ A] :
( ( Ls
= ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
=> ( ( X2 = A4 )
=> ( suffix @ A @ As @ Xs2 ) ) ) )
=> ~ ! [A4: A] :
( ? [As: list @ A] :
( Ps
= ( append @ A @ As @ ( cons @ A @ A4 @ ( nil @ A ) ) ) )
=> ! [X2: A] :
( ? [Xs2: list @ A] :
( Ls
= ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
=> ( X2 = A4 ) ) ) ) ) ) ).
% not_suffix_cases
thf(fact_209_list_Oexhaust__sel,axiom,
! [A: $tType,List: list @ A] :
( ( List
!= ( nil @ A ) )
=> ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_210_not__distinct__split__distinct,axiom,
! [A: $tType,Xs: list @ A] :
( ~ ( distinct @ A @ Xs )
=> ~ ! [Y4: A,Ys3: list @ A] :
( ( distinct @ A @ Ys3 )
=> ( ( member @ A @ Y4 @ ( set2 @ A @ Ys3 ) )
=> ! [Zs3: list @ A] :
( Xs
!= ( append @ A @ Ys3 @ ( append @ A @ ( cons @ A @ Y4 @ ( nil @ A ) ) @ Zs3 ) ) ) ) ) ) ).
% not_distinct_split_distinct
thf(fact_211_append__eq__append__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A,Ts: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Ts ) )
= ( ? [Us: list @ A] :
( ( ( Xs
= ( append @ A @ Zs @ Us ) )
& ( ( append @ A @ Us @ Ys )
= Ts ) )
| ( ( ( append @ A @ Xs @ Us )
= Zs )
& ( Ys
= ( append @ A @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_212_append__eq__appendI,axiom,
! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs: list @ A,Ys: list @ A,Us2: list @ A] :
( ( ( append @ A @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append @ A @ Xs1 @ Us2 ) )
=> ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_213_suffix__order_Odual__order_Oantisym,axiom,
! [A: $tType,B2: list @ A,A2: list @ A] :
( ( suffix @ A @ B2 @ A2 )
=> ( ( suffix @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% suffix_order.dual_order.antisym
thf(fact_214_suffix__order_Odual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( ^ [Y5: list @ A,Z: list @ A] : Y5 = Z )
= ( ^ [A5: list @ A,B4: list @ A] :
( ( suffix @ A @ B4 @ A5 )
& ( suffix @ A @ A5 @ B4 ) ) ) ) ).
% suffix_order.dual_order.eq_iff
thf(fact_215_suffix__order_Odual__order_Otrans,axiom,
! [A: $tType,B2: list @ A,A2: list @ A,C: list @ A] :
( ( suffix @ A @ B2 @ A2 )
=> ( ( suffix @ A @ C @ B2 )
=> ( suffix @ A @ C @ A2 ) ) ) ).
% suffix_order.dual_order.trans
thf(fact_216_suffix__order_Oord__le__eq__trans,axiom,
! [A: $tType,A2: list @ A,B2: list @ A,C: list @ A] :
( ( suffix @ A @ A2 @ B2 )
=> ( ( B2 = C )
=> ( suffix @ A @ A2 @ C ) ) ) ).
% suffix_order.ord_le_eq_trans
thf(fact_217_suffix__order_Oord__eq__le__trans,axiom,
! [A: $tType,A2: list @ A,B2: list @ A,C: list @ A] :
( ( A2 = B2 )
=> ( ( suffix @ A @ B2 @ C )
=> ( suffix @ A @ A2 @ C ) ) ) ).
% suffix_order.ord_eq_le_trans
thf(fact_218_suffix__order_Oorder_Oantisym,axiom,
! [A: $tType,A2: list @ A,B2: list @ A] :
( ( suffix @ A @ A2 @ B2 )
=> ( ( suffix @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% suffix_order.order.antisym
thf(fact_219_suffix__order_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( ^ [Y5: list @ A,Z: list @ A] : Y5 = Z )
= ( ^ [A5: list @ A,B4: list @ A] :
( ( suffix @ A @ A5 @ B4 )
& ( suffix @ A @ B4 @ A5 ) ) ) ) ).
% suffix_order.order.eq_iff
thf(fact_220_suffix__order_Oantisym__conv,axiom,
! [A: $tType,Y3: list @ A,X: list @ A] :
( ( suffix @ A @ Y3 @ X )
=> ( ( suffix @ A @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% suffix_order.antisym_conv
thf(fact_221_suffix__order_Oorder__trans,axiom,
! [A: $tType,X: list @ A,Y3: list @ A,Z3: list @ A] :
( ( suffix @ A @ X @ Y3 )
=> ( ( suffix @ A @ Y3 @ Z3 )
=> ( suffix @ A @ X @ Z3 ) ) ) ).
% suffix_order.order_trans
thf(fact_222_suffix__order_Oorder_Otrans,axiom,
! [A: $tType,A2: list @ A,B2: list @ A,C: list @ A] :
( ( suffix @ A @ A2 @ B2 )
=> ( ( suffix @ A @ B2 @ C )
=> ( suffix @ A @ A2 @ C ) ) ) ).
% suffix_order.order.trans
thf(fact_223_suffix__order_Oeq__refl,axiom,
! [A: $tType,X: list @ A,Y3: list @ A] :
( ( X = Y3 )
=> ( suffix @ A @ X @ Y3 ) ) ).
% suffix_order.eq_refl
thf(fact_224_suffix__order_Oantisym,axiom,
! [A: $tType,X: list @ A,Y3: list @ A] :
( ( suffix @ A @ X @ Y3 )
=> ( ( suffix @ A @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% suffix_order.antisym
thf(fact_225_suffix__order_Oeq__iff,axiom,
! [A: $tType] :
( ( ^ [Y5: list @ A,Z: list @ A] : Y5 = Z )
= ( ^ [X3: list @ A,Y2: list @ A] :
( ( suffix @ A @ X3 @ Y2 )
& ( suffix @ A @ Y2 @ X3 ) ) ) ) ).
% suffix_order.eq_iff
thf(fact_226_suffix__same__cases,axiom,
! [A: $tType,Xs_1: list @ A,Ys: list @ A,Xs_2: list @ A] :
( ( suffix @ A @ Xs_1 @ Ys )
=> ( ( suffix @ A @ Xs_2 @ Ys )
=> ( ( suffix @ A @ Xs_1 @ Xs_2 )
| ( suffix @ A @ Xs_2 @ Xs_1 ) ) ) ) ).
% suffix_same_cases
thf(fact_227_finite__list,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite @ A @ A3 )
=> ? [Xs2: list @ A] :
( ( set2 @ A @ Xs2 )
= A3 ) ) ).
% finite_list
thf(fact_228_append_Oleft__neutral,axiom,
! [A: $tType,A2: list @ A] :
( ( append @ A @ ( nil @ A ) @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_229_append__Nil,axiom,
! [A: $tType,Ys: list @ A] :
( ( append @ A @ ( nil @ A ) @ Ys )
= Ys ) ).
% append_Nil
thf(fact_230_eq__Nil__appendI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs = Ys )
=> ( Xs
= ( append @ A @ ( nil @ A ) @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_231_distinct_Osimps_I1_J,axiom,
! [A: $tType] : ( distinct @ A @ ( nil @ A ) ) ).
% distinct.simps(1)
thf(fact_232_suffix__bot_Obot_Oextremum__uniqueI,axiom,
! [A: $tType,A2: list @ A] :
( ( suffix @ A @ A2 @ ( nil @ A ) )
=> ( A2
= ( nil @ A ) ) ) ).
% suffix_bot.bot.extremum_uniqueI
thf(fact_233_suffix__bot_Obot_Oextremum,axiom,
! [A: $tType,A2: list @ A] : ( suffix @ A @ ( nil @ A ) @ A2 ) ).
% suffix_bot.bot.extremum
thf(fact_234_Nil__suffix,axiom,
! [A: $tType,Xs: list @ A] : ( suffix @ A @ ( nil @ A ) @ Xs ) ).
% Nil_suffix
thf(fact_235_list_Osel_I2_J,axiom,
! [A: $tType] :
( ( tl @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% list.sel(2)
thf(fact_236_suffix__appendI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( suffix @ A @ Xs @ Ys )
=> ( suffix @ A @ Xs @ ( append @ A @ Zs @ Ys ) ) ) ).
% suffix_appendI
thf(fact_237_suffix__appendD,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A] :
( ( suffix @ A @ ( append @ A @ Zs @ Xs ) @ Ys )
=> ( suffix @ A @ Xs @ Ys ) ) ).
% suffix_appendD
thf(fact_238_suffix__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( suffix @ A @ Xs @ ( append @ A @ Ys @ Zs ) )
= ( ( suffix @ A @ Xs @ Zs )
| ? [Xs5: list @ A] :
( ( Xs
= ( append @ A @ Xs5 @ Zs ) )
& ( suffix @ A @ Xs5 @ Ys ) ) ) ) ).
% suffix_append
thf(fact_239_Sublist_Osuffix__def,axiom,
! [A: $tType] :
( ( suffix @ A )
= ( ^ [Xs3: list @ A,Ys2: list @ A] :
? [Zs2: list @ A] :
( Ys2
= ( append @ A @ Zs2 @ Xs3 ) ) ) ) ).
% Sublist.suffix_def
thf(fact_240_suffixI,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,Xs: list @ A] :
( ( Ys
= ( append @ A @ Zs @ Xs ) )
=> ( suffix @ A @ Xs @ Ys ) ) ).
% suffixI
thf(fact_241_suffixE,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( suffix @ A @ Xs @ Ys )
=> ~ ! [Zs3: list @ A] :
( Ys
!= ( append @ A @ Zs3 @ Xs ) ) ) ).
% suffixE
thf(fact_242_distinct__suffix,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( distinct @ A @ Ys )
=> ( ( suffix @ A @ Xs @ Ys )
=> ( distinct @ A @ Xs ) ) ) ).
% distinct_suffix
thf(fact_243_distinct__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( tl @ A @ Xs ) ) ) ).
% distinct_tl
thf(fact_244_suffix__tl,axiom,
! [A: $tType,Xs: list @ A] : ( suffix @ A @ ( tl @ A @ Xs ) @ Xs ) ).
% suffix_tl
thf(fact_245_finite__distinct__list,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite @ A @ A3 )
=> ? [Xs2: list @ A] :
( ( ( set2 @ A @ Xs2 )
= A3 )
& ( distinct @ A @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_246_list_Oset__sel_I1_J,axiom,
! [A: $tType,A2: list @ A] :
( ( A2
!= ( nil @ A ) )
=> ( member @ A @ ( hd @ A @ A2 ) @ ( set2 @ A @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_247_hd__in__set,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( member @ A @ ( hd @ A @ Xs ) @ ( set2 @ A @ Xs ) ) ) ).
% hd_in_set
thf(fact_248_list_Oset__sel_I2_J,axiom,
! [A: $tType,A2: list @ A,X: A] :
( ( A2
!= ( nil @ A ) )
=> ( ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ A2 ) ) )
=> ( member @ A @ X @ ( set2 @ A @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_249_longest__common__prefix,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
? [Ps2: list @ A,Xs6: list @ A,Ys4: list @ A] :
( ( Xs
= ( append @ A @ Ps2 @ Xs6 ) )
& ( Ys
= ( append @ A @ Ps2 @ Ys4 ) )
& ( ( Xs6
= ( nil @ A ) )
| ( Ys4
= ( nil @ A ) )
| ( ( hd @ A @ Xs6 )
!= ( hd @ A @ Ys4 ) ) ) ) ).
% longest_common_prefix
thf(fact_250_hd__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys ) )
= ( hd @ A @ Ys ) ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys ) )
= ( hd @ A @ Xs ) ) ) ) ).
% hd_append
thf(fact_251_list_Oexpand,axiom,
! [A: $tType,List: list @ A,List2: list @ A] :
( ( ( List
= ( nil @ A ) )
= ( List2
= ( nil @ A ) ) )
=> ( ( ( List
!= ( nil @ A ) )
=> ( ( List2
!= ( nil @ A ) )
=> ( ( ( hd @ A @ List )
= ( hd @ A @ List2 ) )
& ( ( tl @ A @ List )
= ( tl @ A @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_252_not__hd__in__tl,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( X
!= ( hd @ A @ Xs ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ Xs ) ) ) ) ) ).
% not_hd_in_tl
thf(fact_253_Misc_Odistinct__hd__tl,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( X
= ( hd @ A @ Xs ) )
=> ~ ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ Xs ) ) ) ) ) ).
% Misc.distinct_hd_tl
thf(fact_254_old_OEntry__unreachable,axiom,
! [G2: g] :
( ( invar @ G2 )
=> ( ( graph_1822314308nEdges @ g @ node @ edgeD @ inEdges @ G2 @ ( entry @ G2 ) )
= ( nil @ ( product_prod @ node @ ( product_prod @ edgeD @ node ) ) ) ) ) ).
% old.Entry_unreachable
% Subclasses (2)
thf(subcl_Orderings_Olinorder___HOL_Otype,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( type @ A ) ) ).
thf(subcl_Orderings_Olinorder___Orderings_Oord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ord @ A ) ) ).
% Type constructors (4)
thf(tcon_fun___Orderings_Oord,axiom,
! [A6: $tType,A7: $tType] :
( ( ord @ A7 )
=> ( ord @ ( A6 > A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_1,axiom,
! [A6: $tType] : ( ord @ ( set @ A6 ) ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_2,axiom,
ord @ $o ).
% Free types (2)
thf(tfree_0,hypothesis,
linorder @ node ).
thf(tfree_1,hypothesis,
linorder @ val ).
% Conjectures (1)
thf(conj_0,conjecture,
~ ( member @ node @ ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ g2 @ r ) @ ( set2 @ node @ ( tl @ node @ rs ) ) ) ).
%------------------------------------------------------------------------------